A CUMULATIVE DAMAGE APPROACH TO MODELING ATMOSPHERIC CORROSION OF STEEL

Dissertation

Submitted to

The School of Engineering of the

UNIVERSITY OF DAYTON

In Partial Fulfillment of the Requirements for

The Degree of

Doctor of Philosophy in Materials Engineering

By

David Harry Rose

UNIVERSITY OF DAYTON

Dayton, Ohio

December, 2014

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A CUMULATIVE DAMAGE APPROACH TO MODELING ATMOSPHERIC CORROSION OF STEEL

Name: Rose, David Harry

APPROVED BY:

______Douglas C. Hansen, Ph.D. Charles E. Browning, Ph.D. Advisory Committee Chairman Advisory Committee Member Professor Torley Chair in Composite Materials Engineering Materials Chair, Department of Chemical and Materials Engineering

______P. Terrence Murray, Ph.D. Maher Qumsiyeh, Ph.D. Advisory Committee Member Advisory Committee Member Professor Professor Materials Engineering Mathematics

______Peter Sjöblom, Ph.D. Advisory Committee Member President S&K Technologies LLC

______John G. Weber, Ph.D. Eddy M. Rojas, Ph.D., M.A., P.E. Associate Dean Dean School of Engineering School of Engineering

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© Copyright by

David Harry Rose

All rights reserved

2014

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ABSTRACT

A CUMULATIVE DAMAGE APPROACH TO MODELING ATMOSPHERIC CORROSION OF STEEL

Name: Rose, David Harry University of Dayton

Advisor: Dr. Douglas C. Hansen

Past attempts to develop models that predict atmospheric corrosion rates used statistical regression, “power-law”, or other approaches that result in linear or simple nonlinear corrosion rate predictions. Such models were calibrated by statistically comparing corrosion test results to predictions based upon long-term (e.g., annual) deposition measurements of chloride aerosols and/or SO2. Relative humidity, if explicitly considered, was only used to define the amount of time during the year when conditions were thought to be favorable for corrosion.

Most models ignored temperature effects but those that do only consider annual averages.

A new approach was constructed to predict corrosion rates using the concept of cumulative damage. This new model is analogous to some types of fatigue models [1, 2] and is based upon the Eyring equation, which was originally developed to predict the dependence of chemical reaction rates on levels of the presumed acceleration factors. The model makes hourly weight loss predictions, which when added together makes longer-term “cumulative” predictions.

Principal advantage of using hourly predictions is that the effects of diurnal and seasonal temperature cycles and related changes to relative humidity are explicitly considered. The

iv stochastic nature of atmospheric contaminants is considered as well. An inverse modeling approach using Monte Carlo simulations was used to fit various candidate models to proxy environmental characterization data representing conditions at corrosion test sites. Proxy data

(measured elsewhere and for other purposes) was used to infer the stochastic environmental severity at sites where corrosion tests were conducted. Such data included hourly SO2 and ozone data obtained from the Environmental Protection Agency’s Air Quality System database, longer-term chloride deposition data from the National Atmospheric Deposition Program’s on- line database, and hourly weather data from the U.S. Air Force’s 14th Weather Squadron. Proxy data was used so that if this current research proved successful, follow-on work could lead to a practical methodology that design engineers could employ to make realistic predictions without having to explicitly characterize the environment at a location of interest. Cumulative predictions made using such data were statistically compared to quarterly corrosion test results that came from a DoD Strategic Environmental Research and Development funded effort and other related programs that used the same testing protocols.

Billions of simulations were conducted whereby coefficients employed by the candidate model were randomly varied and the individual predictions statistically compared to test measurements in order to identify the most accurate model. Each candidate model was calibrated by considering hourly data for an entire year at multiple locations in order to quantify interactions between acceleration factors. The degree of fit between the model results and test measurements at the calibration sites was very high (R2=~ 0.99). When the optimum model was applied to locations where corrosion tests were conducted but not used for calibration, the fit was not quite as good, but was still quite high (R2=~0.86). Analyses were conducted to identify ways to further improve accuracy, thus laying the framework for future efforts.

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Dedicated to my wife, Fran, our sons David and Wesley, and our daughter Stephanie

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ACKNOWLEDGEMENTS

I would like to offer special thanks to my advisor, Dr. Douglas Hansen, for agreeing to help me fulfill my long-time dream. If not for his technical advice, encouragement, support, and willingness to take on an older student, the research effort described in this dissertation would not have been possible. Special thanks are also offered to Dr. Daniel Eylon, Director of the

Graduate Materials Engineering Program, for facilitating my return to the University of Dayton. I would also like to thank Dr. James Snide (Professor Emeritus, Materials Engineering) for the helpful advice he offered throughout my research program. Similarly, I offer my thanks to the members of my advisory committee including Dr. Charles Browning (Chair, Chemical and

Materials Engineering), Dr. P. Terrence Murray (Materials Engineering), Dr. Maher Qumsiyeh

(Mathematics), and Dr. Peter Sjöblom (President, S&K Technologies LLC).

I would also like to express my deepest appreciation to my friend, Mr. Scott McCombie, from Booz Allen and Hamilton. Scott is a computer scientist and he realized the problem I was working on could be more effectively solved using massively parallel computing. He worked with me to convert my initial programming approach and algorithms into an application that ran on a massively parallel computer, which allowed me to conduct far more simulations than would have been otherwise possible. Going forward, the enhanced capabilities available using parallel computing provides the opportunity to evolve the proof-of-concept effort described here into the design analysis tool that has been my vision from the very start. I would also like to thank Mr. Paul Lein from Quanterion Solutions and Dr. Jorge Romeu, Research Professor of

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Industrial Statistics and Operations Research at Syracuse University, for the numerous mathematical discussions we had over the years. Their suggestions and advice at the early stages of my research proved quite helpful.

Last but certainly not least, I want to thank my wife, Fran. Little did she know the inspiration she provided me starting from the very first day we met would lead to my long-term quest for learning. She has been supportive from the beginning, through now, when as a middle aged man, I decided to return to school full-time to pursue a Ph.D. For her love, inspiration, and support over these past 30 years, I am eternally grateful.

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TABLE OF CONTENTS

ABSTRACT ...... iv

DEDICATION ...... vi

ACKNOWLEDGEMENTS ...... vii

LIST OF FIGURES ...... xvi

LIST OF TABLES ...... xxix

LIST OF ABBREVIATIONS AND NOTATIONS ...... xxxv

CHAPTER 1 INTRODUCTION ...... 1

1.1 Past Modeling Effort ...... 3

1.2 Literature Search ...... 4

1.2.1 Factors that Lead to Corrosion ...... 4

1.2.2 Statistical Assessment of Corrosion Models ...... 13

1.2.3 Atmospheric Severity and its Importance to Modeling Efforts ...... 16

1.2.4 Legacy Modeling Efforts ...... 17

1.2.5 Environmental Parameter Characterization Methods ...... 27

CHAPTER 2 DERIVATION OF THE CUMULATIVE DAMAGE MODEL ...... 30

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2.1 Construction of Notional Corrosion Model ...... 31

2.2 Derivation of Functions to Update the Notional Model ...... 36

2.2.1 Temperature-Relative Humidity Functions ...... 37

2.2.2 Temperature-Contaminant Functions ...... 38

2.3 Constructing a New Functional Form of the Notional Equation ...... 39

2.4 Development of Temperature-Relative Humidity Shape Functions ...... 42

2.4.1 Temperature Functions ...... 43

2.4.2 Relative Humidity Functions ...... 49

2.4.3 Illustration of Temperature-Relative Humidity Shape Functions ...... 57

2.5 Development of Temperature-Contaminant Shape Functions ...... 64

2.5.1 Linear Temperature-Linear Chloride Shape Function ...... 66

2.5.2 Linear Temperature-Linear SO2 Shape Function ...... 68

2.6 Linear Temperature-Linear Ozone Shape Function ...... 69

2.7 Identification of Optimal Shape Functions ...... 69

CHAPTER 3 COLLECTION, ESTIMATION, FILTERING, AND SYNTHESIS OF PROXY DATA ...... 71

3.1 Calibration and Validation Data ...... 72

3.1.1 Selection of Calibration and Validation Test Sites ...... 72

3.1.2 Corrosion Rate Measurements ...... 75

3.1.3 Environmental Characterization ...... 77

3.1.4 Composite Data File Used for Model Calibration and Validation ...... 100

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CHAPTER 4 RESULTS FROM NUMERICAL EXPERIMENTS ...... 102

4.1 Linear Temperature-Linear Relative Humidity Shape Function (Initial Baseline) ...... 104

4.2 Linear Temperature-Convex Relative Humidity Shape Function ...... 107

4.3 Convex Temperature-Convex Relative Humidity Shape Function ...... 109

4.4 Selection of the Most Accurate Base Model ...... 111

4.5 Revision of the Linear Temperature-Linear Chloride Shape Function ...... 112

4.6 Simulations to Determine the Optimum Chloride Deposition Interval ...... 116

4.7 Candidate Models Calibrated Using Data from Three Corrosion Test Sites ...... 117

4.8 Candidate Models Calibrated Using Data from Rock Island, IL ...... 119

4.9 Candidate Models Based Upon Three Activation Energies ...... 123

4.10 Evaluation of Data Used for Modeling and Simulations ...... 125

4.11 Refinement of the Monthly Chloride Model ...... 131

4.12 Simulations to Investigate the RH Threshold ...... 132

4.13 Simulations to Calibrate a New Basic Model with Improved Characterization Data ...... 137

4.13.1 Convex Temperature-Convex Relative Humidity Model ...... 138

4.13.2 Linear Temperature-Linear Relative Humidity Model ...... 140

4.13.3 Development of Final Proof-of-Concept Model ...... 142

CHAPTER 5 DISCUSSION ...... 152

5.1 Discussion of Preliminary Models ...... 152

5.1.1 Initial Modeling Efforts ...... 152

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5.1.2 Refinement of Initial Models ...... 159

5.1.3 Revised Mathematical Formulations of Initial Models ...... 162

5.2 Discussion of the Final Proof-of-Concept Model ...... 167

5.2.1 Accuracy of Legacy Models ...... 167

5.2.2 Metrics Used to Evaluate Final Models ...... 169

5.2.3 Evaluation of the Best Model Calibrated Using Data from Dobbins, GA ...... 170

5.2.4 Evaluation of the Final Proof-of-Concept Model ...... 172

5.2.5 Evidence Supporting the Cumulative Corrosion Damage Hypothesis ...... 175

5.3 Model Limitations...... 176

5.4 Application of Cumulative Damage Model ...... 177

CHAPTER 6 CONCLUSIONS ...... 179

CHAPTER 7 RECOMMENDATIONS FOR FUTURE WORK ...... 186

7.1 Conduct Simulations Using High-Performance Computing to Overcome Limitations of Current Hardware ...... 186

7.2 Apply Variable RH Threshold Functions to the Final Proof-of-concept Model ...... 187

7.3 Examine Models that Combine Variable RH Threshold Functions and Multiple Activation Energies ...... 187

7.4 Examine Other Temperature-Contaminant Shape Functions ...... 187

7.5 Investigate the Relative Humidity Threshold via Environmental Chamber Tests ...... 187

7.6 Conduct New Corrosion and Environmental Characterization Tests ...... 188

7.7 Develop Statistical Representations for Stochastic Variables ...... 188

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7.8 Expand Model to Consider Longer Periods of Time ...... 188

7.9 Expand Model to make Predictions at Locations with Very High Chloride Deposition Rates ...... 189

7.10 Consider Additional Atmospheric Contaminants ...... 189

7.11 Construct Models for Other Materials ...... 189

REFERENCES ...... 190

APPENDIX A DERIVATION OF NONLINEAR TEMPERATURE-RELATIVE HUMIDITY SHAPE FUNCTIONS AND AN IMPROVED MODEL FORMULATION ...... 203

A.1 Derivation of Nonlinear Temperature Functions ...... 203

A.1.1 Concave Temperature Function ...... 203

A.1.2 Convex Temperature Function...... 210

A.2 Derivation of Nonlinear Relative Humidity Functions ...... 219

A.2.1 Concave Relative Humidity Function ...... 219

A.2.2 Convex Relative Humidity Function ...... 226

A.3 Derivation of Shape Functions for Use with a New Modeling Approach Applied to a Revised Environmental Characterization Dataset ...... 232

A.3.1 Derivation of Revised Nonlinear Temperature-Nonlinear Relative Humidity Shape Functions ...... 232

A.3.2 Derivation of Revised Nonlinear Temperature-Nonlinear Contaminant Shape Functions ...... 239

APPENDIX B DATA COLLECTION AND ESTIMATION ...... 249

B.1 Corrosion Test Measurements and Environmental Characterization at Proxy Locations ...... 249

B.2 Descriptions of Corrosion Test and Environmental Characterization Sites ...... 251

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B.2.1 Estimation of Hourly Ozone Levels to Replace Missing Data ...... 280

APPENDIX C PROCESSES USED TO DEVELOP THE CUMULATIVE DAMAGE MODEL ...... 314

C.1 Inverse Modeling Using Monte Carlo Simulations ...... 314

C.2 Computational Approaches to Running Simulations ...... 317

C.2.1 Limitations on Random Numbers Selected from Distributions ...... 317

C.2.2 Advanced Computing ...... 324

C.3 Statistical Analysis of Results ...... 328

C.3.1 Residual Sum of Squares Method ...... 328

C.3.2 Coefficient of Determination Method ...... 329

C.4 Model Attributes ...... 330

APPENDIX D VARIABLE THRESHOLD FUNCTIONS ...... 333

D.1 Examination of the RH Threshold ...... 333

D.2 Variable RH Threshold Functions ...... 337

D.2.1 Linear RH Threshold Function ...... 338

D.2.2 Parabolic RH Threshold Function ...... 340

D.3 Evaluation of RH Threshold Functions ...... 344

APPENDIX E RESULT DETAILS ...... 345

E.1 Shape Functions ...... 345

E.1.1 Down-Selection of Temperature-Relative Humidity Shape Functions ...... 346

E.1.2 Temperature-Contaminant Shape Functions ...... 353

E.2 Screening of Initial Models to Identify the Most Accurate Shape Functions ...... 353

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E.3 Detailed Results from Simulations ...... 355

E.3.1 Simulations to Determine Optimum Chloride Deposition Interval – Candidate Models Calibrated Using Data from Four Corrosion Test Sites ...... 355

E.3.2 Candidate Models Calibrated Using Data from Three Corrosion Test Sites ...... 359

E.3.3 Candidate Models Calibrated Using Data from Rock Island, Illinois ...... 365

E.3.4 Candidate Models Based Upon Three Activation Energies ...... 373

E.3.5 Refinement of the Monthly Chloride Model ...... 376

E.3.6 Simulations to Investigate the Relative Humidity Threshold ...... 378

E.3.7 Simulations to Calibrate the Final Proof-of-Concept Model with Improved Characterization Data ...... 382

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LIST OF FIGURES

Figure 2-1 Combination of Factors Considered by the Incremental Corrosion Model ...... 34

Figure 2-2 Values of Exponential Function Based Upon the Range for f(T) ...... 37

Figure 2-3 Illustration of the Variety of Possible Solutions for f(T) ...... 44

Figure 2-4 Linear Function for f(T) ...... 45

Figure 2-5 Linear Relative Humidity Function...... 50

Figure 2-6 Linear Temperature-Linear Relative Humidity Shape Function (f(T)max=-2) ...... 52

Figure 2-7 Exponential Function Applied to the Linear Temperature-Linear RH Shape

Function (f(T)max=-2) ...... 53

Figure 2-8 Adjusted Exponential Function Applied to the Linear Temperature-Linear RH

Shape Function (f(T)max=-2) ...... 54

Figure 2-9 Linear Temperature-Concave RH Shape Function - f(T)max=-2 ...... 60

Figure 2-10 Linear Temperature-Convex RH Shape Function - f(T)max=-2 ...... 61

Figure 2-11 Concave Temperature-Linear RH Shape Function - f(T)max=-2 ...... 61

Figure 2-12 Concave Temperature-Concave RH Shape Function - f(T)max=-2 ...... 62

Figure 2-13 Concave Temperature-Convex RH Shape Function - f(T)max=-2 ...... 62

Figure 2-14 Convex Temperature-Linear RH Shape Function - f(T)max=-2 ...... 63

Figure 2-15 Convex Temperature-Concave RH Shape Function - f(T)max=-2 ...... 63

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Figure 2-16 Convex Temperature-Convex RH Shape Function - f(T)max=-2 ...... 64

Figure 2-17 Linear Chloride Function ...... 67

Figure 2-18 Linear Temperature-Linear Chloride Shape Function - f(T)max=-2 ...... 68

Figure 3-1 Model Calibration and Validation Test Sites in Comparison to Climate Zones ...... 74

Figure 3-2 Battelle Corrosion Test Fixture [Photo Courtesy of Battelle] ...... 75

Figure 3-3 Battelle Corrosion Test Specimen Card. (L-R): 99.99% silver, 99.99% copper, three different alloys of aluminum (2024, 6061, and 7075), and one specimen of steel (AISI 1010). [Photo Courtesy of Battelle] ...... 76

Figure 3-4 Illustration of SO2 Monitoring Sites across the United States ...... 78

Figure 3-5 Scatter Plot Indicating Problems with the Predictions for Fort Campbell (monthly chloride model) ...... 83

Figure 3-6 Proximity of Point Sources of SO2 to the Monitor at Clarksville, TN ...... 84

Figure 3-7 Illustration of Ozone Monitoring Sites across the United States ...... 86

Figure 3-8 Monthly Average Proxy Ozone Levels for Fort Drum and China Lake ...... 89

Figure 3-9 Cubic Function Fit to Monthly Ratios that Compare Fort Drum and China Lake Proxies ...... 92

Figure 3-10 Complete Monthly Average Proxy Ozone Levels for Fort Drum and China Lake ...... 93

Figure 3-11 Average Monthly Ozone Levels at Trona and Hopkinsville (includes adjustments) ...... 94

Figure 3-12 Map Illustrating the Location of NADP Monitor Locations ...... 95

Figure 3-13 Example of Monthly Chloride Deposition Measurements ...... 96

Figure 4-1 Error Evaluation to Identify the Most Accurate Model Calibrated from Data Pertaining to Four Test Sites (annual chloride deposition, optimum model: H=0.81 eV) ...... 104

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Figure 4-2 Comparison of Corrosion Test Data and Associated Predictions for the Annual Chloride Model Applied to Calibration Data from Four Test Sites (optimum H=0.81 eV) ...... 106

Figure 4-3 Error Evaluation to Identify the Most Accurate Model Calibrated from Data Pertaining to Four Test Sites (annual chloride deposition, optimum model: H=0.86 eV) ...... 107

Figure 4-4 Comparison of Corrosion Test Data and Associated Predictions for the Annual Chloride Model Applied to Calibration Data from Four Test Sites (optimum H=0.86 eV) ...... 108

Figure 4-5 Error Evaluation to Identify the Most Accurate Model Calibrated from Data Pertaining to Four Test Sites (annual chloride deposition, optimum model: H=0.66 eV) ...... 109

Figure 4-6 Comparison of Corrosion Test Data and Associated Predictions for the Annual Chloride Model Applied to Calibration Data from Four Test Sites (optimum: H=0.66 eV) ...... 110

Figure 4-7 Error Evaluation to Identify the Most Accurate Model Calibrated from Data Pertaining to Four Test Sites (annual chloride deposition, optimum model: H=0.66 eV) ...... 113

Figure 4-8 Comparison of Corrosion Test Data and Associated Predictions for the Annual Chloride Model Applied to Calibration Data from Four Test Sites (optimum H=0.66 eV) ...... 113

Figure 4-9 Comparison of Corrosion Test Data and Associated Predictions for the Annual Chloride Model Applied to Validation Data from Ten Independent Locations (H=0.66 eV) ...... 114

Figure 4-10 Comparison of Corrosion Test Data and Associated Predictions for the Weekly Chloride Model Applied to Validation Data from Nine Independent Locations (H=0.96 eV) ...... 121

Figure 4-11 Error Evaluation Used to Identify the Most Accurate Model Calibrated from Data Pertaining to Three Test Sites (monthly chloride deposition, optimum

HCl=0.97 eV) ...... 124

Figure 4-12 Illustration of the Impact of Applying Model Using Data Measured Near Point

Sources of SO2 Pollution and Locations Adjacent to Surf Zones (ten independent locations, optimum H=0.62 eV) ...... 125

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Figure 4-13 Improvement in R2 Value Results when Data Proxies are Revised and Model is not Applied to Locations Near Surf Zones (single activation energy formulation, validation data from eight independent locations, optimum H=0.62 eV) ...... 129

Figure 4-14 Results Obtained when the Monthly Chloride Model Using Three Activation Energies is Applied to Validation Data from Eight Independent Locations ...... 129

Figure 4-15 Illustration of the Constant Temperature-Relative Humidity Shape Function ...... 132

Figure 4-16 Temperature Dependence of Absolute Humidity at a Constant RH of 60% ...... 133

Figure 4-17 Illustration of the Temperature-Relative Humidity Shape Function Based Upon a Linear RH Threshold Function ...... 135

Figure 4-18 Illustration of the Temperature-Relative Humidity Shape Function Based

Upon a Parabolic RH Threshold Function (RHvar=80%RH) ...... 136

Figure 4-19 Parabolas of Different Widths ...... 139

Figure 4-20 Comparison of Corrosion Test Data and Associated Predictions for the Monthly Chloride Model Applied to Calibration Data (includes Rock Island Data, optimum H=0.81 eV)...... 141

Figure 4-21 Comparison of Corrosion Test Data and Associated Predictions for the Monthly Chloride Model Applied to Validation Data (includes Rock Island Data, optimum H=0.81 eV) ...... 142

Figure 4-22 Original Model Employs Exponential Functions to Calculate Hourly Interactions between the Acceleration Factors ...... 143

Figure 4-23 Notional Illustration of the Final Proof-of-Concept Model ...... 144

Figure 4-24 Comparison of 1010 Steel Test Points and Associated Predictions (Calibration Data) ...... 149

Figure 4-25 Comparison of 1010 Steel Test Points and Associated Predictions (Validation Data) ...... 150

Figure 5-1 Convex Temperature-Convex RH Shape Function Modified Using a Nonlinear RH Threshold Function ...... 165

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Figure 5-2 Exponential Function Applied to the Values Obtained from the Convex Temperature-Convex RH Shape Function Modified by a Nonlinear RH Threshold Function (adjusted values) ...... 166

Figure 5-3 Model Applied to Calibration Data for Kennedy Space Center, FL; Fort Drum, NY; and Dobbins, Air Reserve Base, GA (Most Accurate Initial Model) ...... 171

Figure 5-4 Model Applied to Validation Data for Kennedy Space Center, FL; Fort Drum, NY; and Dobbins, Air Reserve Base, GA (Most Accurate Initial Model) ...... 172

Figure 5-5 Comparison of 1010 Steel Test Points and Associated Predictions (Calibration Data) ...... 173

Figure 5-6 Comparison of 1010 Steel Test Points and Associated Predictions (Validation Data) ...... 174

Figure A-1 Illustration of the Concave Temperature Function ...... 204

Figure A-2 Concave Temperature Function ...... 210

Figure A-3 Illustration of the Convex Temperature Function ...... 212

Figure A-4 Convex Temperature Function ...... 218

Figure A-5 Illustration of the Concave Temperature-Relative Humidity Function ...... 219

Figure A-6 Illustration of the Convex Temperature-Relative Humidity Function ...... 227

Figure B-1 Illustration of Corrosion Test Site at China Lake and Proximity of Associated Environmental Characterization Sites ...... 253

Figure B-2 Illustration of Corrosion Test Site at Dobbins Air Reserve Base and Proximity of Associated Environmental Characterization Sites ...... 255

Figure B-3 Illustration of Corrosion Test Site at Fort Drum and Proximity of Associated Environmental Characterization Sites ...... 257

Figure B-4 Illustration of Corrosion Test Site at Kennedy Space Center and Proximity of Associated Environmental Characterization Sites ...... 259

Figure B-5 Illustration of Corrosion Test Site at Daytona Beach and Proximity of Associated Environmental Characterization Sites ...... 261

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Figure B-6 Illustration of Corrosion Test Site at Kirtland AFB and Proximity of Associated Environmental Characterization Sites ...... 263

Figure B-7 Illustration of Corrosion Test Site at Point Judith and Proximity of Associated Environmental Characterization Sites ...... 265

Figure B-8 Illustration of Corrosion Test Site at Tyndall AFB and Proximity of Associated Environmental Characterization Sites ...... 267

Figure B-9 Illustration of Corrosion Test Site at Wright Patterson AFB and Proximity of Associated Environmental Characterization Site ...... 269

Figure B-10 Illustration of Corrosion Test Site at Fort Campbell and Proximity of Associated Environmental Characterization Sites ...... 271

Figure B-11 Illustration of Corrosion Test Site at Fort Hood and Proximity of Associated Environmental Characterization Sites ...... 273

Figure B-12 Illustration of Corrosion Test Site at Fort Rucker and Proximity of Associated Environmental Characterization Sites ...... 275

Figure B-13 Illustration of Corrosion Test Site at Rock Island Arsenal and Proximity of Associated Environmental Characterization Sites ...... 277

Figure B-14 Illustration of Corrosion Test Site at West Jefferson and Proximity of Associated Environmental Characterization Sites ...... 279

Figure B-15 Monthly Average Proxy Ozone Levels for Fort Campbell and China Lake ...... 280

Figure B-16 Monthly Ozone Levels for Fort Campbell and China Lake (includes adjustments) ...... 282

Figure B-17 Cubic Function Fit to Monthly Ratios Comparing Fort Campbell and China Lake ...... 284

Figure B-18 Complete Monthly Average Proxy Ozone Levels for Fort Campbell and China Lake ...... 286

Figure B-19 Monthly Average Proxy Ozone Levels for Dobbins and China Lake ...... 286

Figure B-20 Monthly Average Ozone Levels for Dobbins and China Lake (includes adjustments) ...... 288

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Figure B-21 Cubic Function Fit to Monthly Ratios Comparing Dobbins and China Lake Proxies ...... 290

Figure B-22 Complete Monthly Average Proxy Ozone Levels for Dobbins and China Lake ...... 292

Figure B-23 Monthly Average Proxy Ozone Levels for Point Judith and China Lake ...... 292

Figure B-24 Cubic Function Fit to Monthly Ratios Comparing Point Judith and China Lake ..... 295

Figure B-25 Complete Monthly Average Proxy Ozone Levels for Point Judith and China Lake ...... 297

Figure B-26 Monthly Average Proxy Ozone Levels Fort Rucker and China Lake ...... 297

Figure B-27 Monthly Ozone Levels for Fort Rucker and China Lake (includes adjustments) .... 299

Figure B-28 Cubic Function Fit to Monthly Ratios Comparing Fort Rucker and China Lake ...... 301

Figure B-29 Complete Monthly Average Proxy Ozone Levels for Fort Rucker and China Lake ...... 303

Figure B-30 Monthly Average Proxy Ozone Levels for West Jefferson and China Lake ...... 303

Figure B-31 Cubic Function Fit to Monthly Ratios Comparing West Jefferson and China Lake ...... 306

Figure B-32 Complete Monthly Average Proxy Ozone Levels for West Jefferson and China Lake ...... 307

Figure B-33 Monthly Average Proxy Ozone Levels for Wright Patterson AFB and China Lake ...... 308

Figure B-34 Cubic Function Fit to Monthly Ratios that Compare Wright Patterson and China Lake Proxies ...... 311

Figure B-35 Complete Average Proxy Ozone Levels for Wright Patterson AFB and China Lake ...... 313

Figure C-1 Simplistic Illustration of Monte Carlo Simulations as Applied to a Forward Problem ...... 315

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Figure C-2 Simplistic Illustration of Monte Carlo Simulations as Applied to an Inverse Problem ...... 316

Figure C-3 Shifting of Uniform Distributions until an Acceptable Coefficient Value is Found ...... 321

Figure C-4 Illustration of the Distribution Adjustment and Refinement Process ...... 324

Figure C-5 Illustration of the GPU and CPU Processes ...... 327

Figure C-6 Illustration of a Trend Line and R2 Value in Relation to Plotted Data ...... 330

Figure C-7 Illustration of Attributes of a Model that Accurately Reflects Test Data ...... 331

Figure C-8 Illustration of Attributes of a Model that Poorly Reflects Test Data ...... 332

Figure D-1 Temperature Dependence of Absolute Humidity at a Constant RH of 60% ...... 336

Figure D-2 Illustration of the Constant Temperature-Relative Humidity Shape Function ...... 337

Figure D-3 Illustration of the Linear RH Temperature Function ...... 338

Figure D-4 Illustration of the Temperature-Relative Humidity Shape Function Based Upon a Functional Linear RH Threshold Boundary Condition ...... 340

Figure D-5 Illustration of the Parabolic RH Temperature Function (- root) ...... 342

Figure D-6 Illustration of the Parabolic RH Temperature Function (+root) ...... 342

Figure D-7 Illustration of the Temperature-Relative Humidity Shape Function Based Upon a Parabolic RH Threshold Boundary Condition ...... 344

Figure E-1 Temperature Functions Used to Create Shape Functions ...... 347

Figure E-2 Illustration of the Convex Temperature-Convex Relative Humidity Shape Function ...... 347

Figure E-3 Illustration of the Adjusted Exponential Function Applied to the Convex Temperature Convex Relative Humidity Shape Function ...... 349

Figure E-4 Comparison of Three Different Temperature-Relative Humidity Shape Functions at 320.15K ...... 349

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Figure E-5 Adjusted Exponential Function Applied to the Three Different Temperature- Relative Humidity Shape Functions at 320.15K ...... 350

Figure E-6 Comparison of Three Different Temperature-Relative Humidity Shape Functions at 285K ...... 351

Figure E-7 Adjusted Exponential Function Applied to the Three Different Temperature- Relative Humidity Shape Functions at 285K...... 352

Figure E-8 Error Evaluation to Identify the Most Accurate Model Calibrated from Data Pertaining to Four Test Sites (monthly chloride deposition, optimum model:H=0.64 eV) ...... 355

Figure E-9 Comparison of Corrosion Test Data and Associated Predictions for the Monthly Chloride Model Applied to Calibration Data from Four Test Sites (optimum H=0.64 eV) ...... 356

Figure E-10 Comparison of Corrosion Test Data and Associated Predictions for the Monthly Chloride Model Applied to Validation Data from Ten Independent Locations (H=0.64 eV) ...... 356

Figure E-11 Error Evaluation to Identify the Most Accurate Model Calibrated from Data Pertaining to Four Test Sites (weekly chloride deposition, optimum model: H=0.66 eV) ...... 357

Figure E-12 Comparison of Corrosion Test Data and Associated Predictions for the Weekly Chloride Model Applied to Calibration Data from Four Test Sites (optimum H=0.66 eV) ...... 358

Figure E-13 Comparison of Corrosion Test Data and Associated Predictions for the Weekly Chloride Model Applied to Validation Data from Ten Independent Locations (H=0.66 eV) ...... 358

Figure E-14 Error Evaluation to Identify the Most Accurate Model Calibrated from Data Pertaining to Three Test Sites (annual chloride deposition, optimum model:H=0.65 eV)...... 359

Figure E-15 Comparison of Corrosion Test Data and Associated Predictions for the Annual Chloride Model Applied to Calibration Data from Three Test Sites (optimum H=0.65 eV) ...... 360

Figure E-16 Comparison of Corrosion Test Data and Associated Predictions for the Annual Chloride Model Applied to Validation Data from Ten Independent Locations (H=0.65 eV) ...... 360

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Figure E-17 Error Evaluation to Identify the Most Accurate Model Calibrated from Data Pertaining to Three Test Sites (monthly chloride deposition, optimum model:H=0.62 eV)...... 361

Figure E-18 Comparison of Corrosion Test Data and Associated Predictions for the Monthly Chloride Model Applied to Calibration Data from Three Test Sites (optimum H=0.62 eV) ...... 362

Figure E-19 Comparison of Corrosion Test Data and Associated Predictions for the Monthly Chloride Model Applied to Validation Data from Ten Independent Locations (H=0.62 eV) ...... 362

Figure E-20 Error Evaluation to Identify the Most Accurate Model Calibrated from Data Pertaining to Three Test Sites (weekly chloride deposition, optimum model:H=0.63 eV)...... 363

Figure E-21 Comparison of Corrosion Test Data and Associated Predictions for the Weekly Chloride Model Applied to Calibration Data from Three Test Sites (optimum H=0.63 eV) ...... 364

Figure E-22 Comparison of Corrosion Test Data and Associated Predictions for the Weekly Chloride Model Applied to Validation Data from Ten Independent Locations (H=0.63 eV) ...... 364

Figure E-23 Error Evaluation to Identify the Most Accurate Model Calibrated from Data Pertaining to Rock Island (annual chloride deposition, optimum model: H=0.91 eV) ...... 365

Figure E-24 Comparison of Corrosion Test Data and Associated Predictions for the Annual Chloride Model Applied to Calibration Data from Rock Island (optimum H=0.91 eV) ...... 366

Figure E-25 Comparison of Corrosion Test Data and Associated Predictions for the Annual Chloride Model Applied to Validation Data from Nine Independent Locations (H=0.91 eV) ...... 366

Figure E-26 Comparison of Corrosion Test Data and Associated Predictions for the Annual Chloride Model Applied to Validation Data from Seven Independent Locations (no beach sites, H=0.91 eV) ...... 367

Figure E-27 Error Evaluation to Identify the Most Accurate Model Calibrated from Data Pertaining to Rock Island (monthly chloride deposition, optimum model: H=0.93 eV) ...... 368

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Figure E-28 Comparison of Corrosion Test Data and Associated Predictions for the Monthly Chloride Model Applied to Calibration Data from Rock Island (optimum H=0.93 eV) ...... 368

Figure E-29 Comparison of Corrosion Test Data and Associated Predictions for the Monthly Chloride Model Applied to Validation Data from Nine Independent Locations (H=0.93 eV) ...... 369

Figure E-30 Comparison of Corrosion Test Data and Associated Predictions for the Monthly Chloride Model Applied to Validation Data from Seven Independent Locations (no beach sites, H=0.93 eV) ...... 369

Figure E-31 Error Evaluation to Identify the Most Accurate Model Calibrated from Data Pertaining to Rock Island (weekly chloride deposition, optimum model: H=0.96 eV) ...... 370

Figure E-32 Comparison of Corrosion Test Data and Associated Predictions for the Weekly Chloride Model Applied to Calibration Data from Rock Island (optimum H=0.96 eV) ...... 371

Figure E-33 Comparison of Corrosion Test Data and Associated Predictions for the Weekly Chloride Model Applied to Validation Data from Nine Independent Locations (H=0.96 eV) ...... 371

Figure E-34 Comparison of Corrosion Test Data and Associated Predictions for the Weekly Chloride Model Applied to Validation Data from Seven Independent Locations (no beach sites, H=0.96 eV) ...... 372

Figure E-35 Error Evaluation to Identify the Most Accurate Three Activation Energy Model Calibrated using Data Pertaining to Three Test Sites (monthly chloride

deposition, optimum model:HCl=0.97 eV) ...... 373

Figure E-36 Error Evaluation to Identify the Most Accurate Three Activation Energy Model Calibrated using Data Pertaining to Three Test Sites (narrowed range,

monthly chloride deposition, optimum model:HCl=0.97 eV) ...... 373

Figure E-37 Comparison of Corrosion Test Data and Associated Predictions for the Three Activation Energy Monthly Chloride Model Applied to Calibration Data from

Three Test Sites (optimum HCl=0.97 eV) ...... 374

Figure E-38 Comparison of Corrosion Test Data and Associated Predictions for the Three Activation Energy Monthly Chloride Model Applied to Validation Data from

Ten Independent Locations (HCl =0.97 eV) ...... 374

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Figure E-39 Comparison of Corrosion Test Data and Associated Predictions for the Annual Chloride Model Applied to Validation Data from Eight Independent

Locations (updated SO2 proxies, no beach sites, HCl =0.97 eV) ...... 375

Figure E-40 Error Evaluation to Identify the Most Accurate Model Calibrated from Data Pertaining to Three Test Sites (refined model with monthly chloride deposition, optimum model:H=0.638 eV) ...... 376

Figure E-41 Comparison of Corrosion Test Data and Associated Predictions for the Monthly Chloride Model Applied to Calibration Data from Three Test Sites (refined model with optimum H=0.638 eV) ...... 376

Figure E-42 Comparison of Corrosion Test Data and Associated Predictions for the Monthly Chloride Model Applied to Validation Data from Eight Independent

Locations (refined model, updated SO2 proxies, no beach sites, H=0.638 eV) .... 377

Figure E-43 Error Evaluation to Identify the Most Accurate Model Calibrated from Data Pertaining to Three Test Sites (refined model with the linear RH threshold function and monthly chloride deposition, optimum model:H=0.352 eV) ...... 378

Figure E-44 Comparison of Corrosion Test Data and Associated Predictions for the Monthly Chloride Model Applied to Calibration Data from Three Test Sites (refined model, linear RH threshold function, H=0.352 eV) ...... 378

Figure E-45 Comparison of Corrosion Test Data and Associated Predictions for the Monthly Chloride Model Applied to Validation Data from Eight Independent

Locations (refined model, updated SO2 proxies, no beach sites, linear RH threshold function, monthly chloride deposition, H=0.352 eV) ...... 379

Figure E-46 Error Evaluation to Identify the Most Accurate Model Calibrated from Data Pertaining to Three Test Sites (refined model with the nonlinear RH threshold function and monthly chloride deposition, optimum model:H=0.449 eV) ...... 380

Figure E-47 Comparison of Corrosion Test Data and Associated Predictions for the Monthly Chloride Model Applied to Calibration Data from Three Test Sites (refined model, nonlinear RH threshold function, H=0.449 eV) ...... 380

Figure E-48 Comparison of Corrosion Test Data and Associated Predictions for the Monthly Chloride Model Applied to Validation Data from Eight Independent

Locations (refined model, updated SO2 proxies, no beach sites, nonlinear RH threshold functionH=0.449 eV) ...... 381

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Figure E-49 Error Evaluation to Identify the Most Accurate Model Calibrated from Data Pertaining to the Final Proof-of-Concept Model , optimum model: H=1.68 eV) ...... 382

Figure E-50 Comparison of Corrosion Test Data and Associated Predictions for the Final Proof-of-Concept Model Applied to Calibration Data (monthly chlorides, H=1.68 eV) ...... 382

Figure E-51 Comparison of Corrosion Test Data and Associated Predictions for the Final Proof-of-Concept Model Applied to Validation Data (monthly chlorides, H=1.68 eV) ...... 383

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LIST OF TABLES

Table 2-1 Definition of the Variables and Coefficients used in Equations 2.14 and 2.15 ...... 41

Table 2-2 Nine Combinations of Functions Used to Develop Candidate T-RH Shape Functions ...... 43

Table 3-1 Calibration Test Sites and Associated Climate Zones ...... 73

Table 3-2 Validation Test Sites and Associated Climate Zones ...... 74

Table 3-3 Proxy SO2 Monitor Locations ...... 79

Table 3-4 Proximity of SO2 Emission Sites to Pensacola Proxy Data Monitoring Location ...... 84

Table 3-5 Proxy Ozone Monitor Locations and Time Periods for Observations ...... 87

Table 3-6 Average Monthly Proxy Ozone Levels and Comparison Ratios (Fort Drum vs. China Lake) ...... 90

Table 3-7 Periodic Relationship of Comparison Ratios (Fort Drum vs. China Lake Proxies) ...... 91

Table 3-8 Comparison Ratios Calculated Using Cubic Equation from Figure 3-9 ...... 92

Table 3-9 Proxy Chloride Monitor Locations ...... 97

Table 3-10 Proxy Weather Office Locations ...... 99

Table 3-11 Illustration of Composite Proxy Data File ...... 100

Table 4-1 Fifteen Types of Initial Models that were Constructed and Tested under this Effort ...... 102

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Table 4-2 Coefficients Determined for the Model Based Upon Linear Temperature-Linear Relative Humidity Shape Functions ...... 105

Table 4-3 Coefficients Determined for the Model Based Upon Linear Temperature- Convex Relative Humidity Shape Functions ...... 108

Table 4-4 Coefficients Determined for the Model Based Upon Convex Temperature- Convex Relative Humidity Shape Functions ...... 110

Table 4-5 Accuracy of Models that were Calibrated during Initial Screening Simulations (annual chlorides) ...... 111

Table 4-6 Modeling Overview and Coefficients for the Optimum Convex Temperature- Convex RH Shape Function Model Calibrated using Data from Four Test Sites (annual chlorides) ...... 115

Table 4-7 Comparison of Metrics Pertaining to Models that were Calibrated Using Different Time Spans for Chloride Deposition ...... 116

Table 4-8 Evaluation of Parameters Relating to China Lake, CA ...... 117

Table 4-9 Comparison of Metrics Pertaining to Models that were Calibrated Using Data from Three Calibration Sites and Different Time Spans for Chloride Deposition .... 118

Table 4-10 Rock Island Arsenal Corrosion Test Site and Associated Environmental Characterization Sites ...... 120

Table 4-11 Comparison of Metrics Pertaining to Models that were Calibrated Using Data from Rock Island, Illinois and Different Time Spans for Chloride Deposition ...... 121

Table 4-12 Coefficients for the Optimum Convex Temperature-Convex RH Shape Function Model Calibrated using Data from Rock Island (weekly chlorides) ...... 123

Table 4-13 Comparison of Metrics Pertaining to Monthly Chloride Models that were Calibrated Using Data from Three Calibration Sites ...... 124

Table 4-14 Comparison of Metrics Pertaining to Monthly Chloride Models that were Calibrated Using Different Numbers of Activation Energies ...... 130

Table 4-15 Comparison of Metrics Pertaining to the Monthly Chloride Models that were Calibrated Using Different Starting and Ending Range Sizes ...... 132

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Table 4-16 Metrics Pertaining to the Monthly Chloride Model (three calibration sites) and Models Using Variable Thresholds (refined simulations as described above) ...... 136

Table 4-17 Rock Island Arsenal Corrosion Test Site and Associated Environmental Characterization Sites ...... 138

Table 4-18 Optimum Coefficients for New Model Calibrated Using Rock Island Data ...... 141

Table 4-19 Coefficients Determined via Simulations for the Final Cumulative Damage Model ...... 148

Table 5-1 Comparison of Metrics for Initial Models ...... 153

Table 5-2 Metrics for Best Initial Model Compared to Models with Revised Validation Data and Refined Simulation Parameters ...... 158

Table 5-3 Metrics for Best Initial Model Compared to Models with Revised Mathematical Forms or Revised Calibration Data ...... 163

Table 5-4 Legacy Atmospheric Corrosion Models Developed Over the Past Forty Years ...... 168

Table B-1 Weight Loss Measurement Data Used to Calibrate Candidate Models ...... 249

Table B-2 Weight Loss Measurement Data Used to Validate Candidate Models ...... 250

Table B-3 China Lake Corrosion Test Site and Associated Environmental Characterization Sites ...... 252

Table B-4 Dobbins Air Reserve Base Corrosion Test Site and Associated Environmental Characterization Sites ...... 254

Table B-5 Fort Drum Corrosion Test Site and Associated Environmental Characterization Sites ...... 256

Table B-6 Kennedy Space Center Corrosion Test Site and Associated Environmental Characterization Sites ...... 258

Table B-7 Daytona Beach Corrosion Test Site and Associated Environmental Characterization Sites ...... 260

Table B-8 Kirtland Air Force Base Corrosion Test Site and Associated Environmental Characterization Sites ...... 262

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Table B-9 Point Judith Corrosion Test Site and Associated Environmental Characterization Sites ...... 264

Table B-10 Tyndall Air Force Base Corrosion Test Site and Associated Environmental Characterization Sites ...... 266

Table B-11 Wright Patterson Air Force Base Corrosion Test Site and Associated Environmental Characterization Sites ...... 268

Table B-12 Fort Campbell Corrosion Test Site and Associated Environmental Characterization Sites ...... 270

Table B-13 Fort Hood Corrosion Test Site and Associated Environmental Characterization Sites ...... 272

Table B-14 Fort Rucker Corrosion Test Site and Associated Environmental Characterization Sites ...... 274

Table B-15 Rock Island Arsenal Corrosion Test Site and Associated Environmental Characterization Sites ...... 276

Table B-16 West Jefferson Corrosion Test Site and Associated Environmental Characterization Sites ...... 278

Table B-17 Average Monthly Ozone Levels/Comparison Ratios (Ft Campbell vs. China Lake) ...... 281

Table B-18 Periodic Relationship of Comparison Ratios (Fort Campbell vs. China Lake Proxies) ...... 283

Table B-19 Comparison Ratios Calculated Using Cubic Equation from Figure B-17 ...... 285

Table B-20 Average Monthly Proxy Ozone Levels/Comparison Ratios (Dobbins vs. China Lake) ...... 287

Table B-21 Periodic Relationship of Comparison Ratios (Dobbins vs. China Lake Proxies) ...... 289

Table B-22 Ratios Calculated Using Cubic Equation from Figure B-21 ...... 291

Table B-23 Average Monthly Ozone Levels and Comparison Ratios (Point Judith vs. China Lake) ...... 293

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Table B-24 Periodic Relationship of Comparison Ratios (Point Judith vs. China Lake Proxies) ...... 294

Table B-25 Comparison Ratios Calculated Using Cubic Equation from Figure B-24 ...... 296

Table B-26 Average Monthly Ozone Levels and Comparison Ratios (Ft Rucker vs. China Lake) ...... 298

Table B-27 Periodic Relationship of Comparison Ratios (Fort Rucker vs. China Lake Proxies) ...... 300

Table B-28 Comparison Ratios Calculated Using Cubic Equation from Figure B-28 ...... 302

Table B-29 Average Monthly Proxy Ozone Levels and Comparison Ratios (West Jefferson vs. China Lake) ...... 304

Table B-30 Periodic Relationship of Comparison Ratios (West Jefferson vs. China Lake Proxies) ...... 305

Table B-31 Comparison Ratios Calculated Using Cubic Equation from Figure B-31 ...... 307

Table B-32 Average Monthly Proxy Ozone Levels and Comparison Ratios (Wright Patterson vs. China Lake) ...... 309

Table B-33 Periodic Relationship of Comparison Ratios (Wright Patterson vs. China Lake) ..... 310

Table B-34 Comparison Ratios Calculated Using Cubic Equation from Figure B-34 ...... 312

Table D-1 Comparison of Steel Alloy Compositions (weight percent) ...... 334

Table E-1 Nine Candidate Temperature-Relative Humidity Shape Functions ...... 346

Table E-2 Modeling Overview and Coefficients for the Optimum Convex Temperature- Convex RH Shape Function Model Calibrated using Monthly Chloride Data from Four Test Sites ...... 357

Table E-3 Modeling Overview and Coefficients for the Optimum Convex Temperature- Convex RH Shape Function Model Calibrated using Weekly Chloride Data from Four Test Sites ...... 359

Table E-4 Modeling Overview and Coefficients for the Optimum Convex Temperature- Convex RH Shape Function Model Calibrated using Data from Three Test Sites .... 361

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Table E-5 Modeling Overview and Coefficients for the Optimum Convex Temperature- Convex RH Shape Function Model Calibrated using Data from Three Test Sites .... 363

Table E-6 Modeling Overview and Coefficients for the Optimum Convex Temperature- Convex RH Shape Function Model Calibrated using Data from Three Test Sites .... 365

Table E-7 Modeling Overview and Coefficients for the Optimum Convex Temperature- Convex RH Shape Function Model Calibrated using Data from Rock Island ...... 367

Table E-8 Modeling Overview and Coefficients for the Optimum Convex Temperature- Convex RH Shape Function Model Calibrated using Data from Rock Island ...... 370

Table E-9 Modeling Overview and Coefficients for the Optimum Convex Temperature- Convex RH Shape Function Model Calibrated using Data from Rock Island ...... 372

Table E-10 Modeling Overview and Coefficients for the Optimum Convex Temperature- Convex RH Shape Function Model Calibrated using Three Activation Energies and Data from Three Test Sites ...... 375

Table E-11 Modeling Overview and Coefficients for the Optimum Convex Temperature- Convex RH Shape Function Model Calibrated using Data from Three Locations (refined model) ...... 377

Table E-12 Modeling Overview and Coefficients for the Optimum Convex Temperature- Convex RH Shape Function Model Calibrated using Data from Three Locations (refined model using the linear RH threshold function) ...... 379

Table E-13 Modeling Overview and Coefficients for the Optimum Convex Temperature- Convex RH Shape Function Model Calibrated using Data from Three Locations (refined model using the nonlinear RH threshold function) ...... 381

Table E-14 Modeling Overview and Coefficients for Final Proof-of-Concept Model Calibrated using Data from Three Locations Including Rock Island, IL ...... 383

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LIST OF ABBREVIATIONS AND NOTATIONS

AAF U.S. Army Airfield

Absolute Humidity Mass of water vapor per unit volume

Acceleration Factor Variable environmental factors responsible for influencing the rate of the corrosive reactions

ACl Coefficient used to scale the corrosion model chloride prediction expression

AO3 Coefficient used to scale the corrosion model O3 prediction expression

ASO2 Coefficient used to scale the corrosion model SO2 prediction expression

Adsorption Adhesion of gaseous substances on surfaces

AFRL Air Force Research Laboratory

AgCl Silver chloride

AQS EPA Air Quality System database. Contains hourly measurements of air pollutants

ARB Air Reserve Base

CGC Corrosion Grand Challenges

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CORRAG ISO Atmospheric Corrosion Program

COS Carbonyl Sulfide

Cumulative Damage An approach whereby the total corrosion damage caused by exposure to variable environmental conditions is equal to the summation of damages caused over short time intervals

Desorption Process where a substance is released from a surface

EPA U.S. Environmental Protection Agency eV Electron volt, unit of energy used to describe the activation energy f(T) Temperature function used to construct shape functions

f(T)max Maximum value of Temperature-Relative Humidity shape function f(T, Cl) Temperature-Chloride Shape Function

f(T, O3) Temperature- O3 Shape Function f(T, RH) Temperature-Relative Humidity Shape Function

f(T, SO2) Temperature- SO2 Shape Function

GILDES Corrosion model that considers the “Gas, Interface, Liquid, Deposition, Electrodic, and Solid regimes”

GPU Graphics processing unit. High end computer video card that can also be used for massively parallel computing

H2S Hydrogen sulfide

ICAO International Civil Aviation Organization

ISO International Organization for Standardization k Boltzmann constant. Equal to 8.617 x 10-5 eV/K

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K Absolute temperature using the Kelvin scale

KJ/mole Unit of energy used to describe the activation energy

Monte Carlo simulation Simulation method that selects random numbers from probability distributions and applies them to models

MPP Massively parallel processing

NADP National Atmospheric Deposition Program. Joint U.S. federal, state, and local government program to measure atmospheric contaminant deposition rates

NCDC National Climate Data Center

NO Nitrogen Monoxide

NO2 Nitrogen Dioxide

NO3 Nitrate radical

NOx Nitrogen oxides

NRC National Research Council

O3 Ozone

PACER LIME Environmental Corrosion Severity Classification Method ppb Parts-per-billion ppm Parts-per-million

Power Law A type of model where the dependent variable is dependent upon the power of the independent variable

Proxy Data Data measured in one or more locations used to infer the environmental conditions at another location

QCM Quartz crystal microbalance

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R2 Coefficient of Determination, statistical measure of model fit to data

Range The difference between the minimum and maximum values of the uniform distributions from which random numbers are selected during Monte Carlo simulations

Reentrainment Desorption

Refined Model A model calibrated using smaller uniform distribution range sizes to develop more precise coefficient values

Resuspension Desorption

RH Relative Humidity

RHcal Relative humidity calibration point used during the development of RH threshold functions

RHTH RH threshold, minimum relative humidity level needed for corrosion to occur

RHvar Variable relative humidity coefficient used to construct RH threshold functions

RSS Residual sum of squares. Statistical method used to calculate model error

Run Refers to a group of simulations conducted using the same parameters. For example, many models were developed using 1.5M simulations per run

SERDP DoD Strategic Environmental Research and Development Program

Shape Function Functions used by the model to numerically assign a value that represents a relationship between temperature and a second acceleration factor

Simulation The systematic exercising of a model to examine the characteristics of a physical system

SO2 Sulfur dioxide

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Stream processors Multiple processing cores on GPUs that make calculations using streams of input data

TOW Time of wetness

UN ICP Materials United Nations International Cooperative Programme on Effects of Air Pollution on Materials, Including Historic and Cultural Monuments

WMO World Meteorological Organization

CL Coefficient used as the exponent used to provide a temperature adjustment for the chloride prediction expression

O3 Coefficient used as the exponent used to provide a temperature adjustment for the O3 prediction expression

SO2 Coefficient used as the exponent used to provide a temperature adjustment for the SO2 prediction expression

H Activation Energy. Minimum energy needed for a chemical reaction to occur. This factor pertains to reactions that occur as a result of exposure to a combination of different atmospheric contaminants that adsorbed/deposited onto a surface

HCL Activation Energy. Minimum energy needed for a chemical reaction to occur as a result of exposure to chloride aerosols that deposited upon a surface

HO3 Activation Energy. Minimum energy needed for a chemical reaction to occur as a result of exposure to ozone that adsorbed onto a surface

HSO2 Activation Energy. Minimum energy needed for a chemical reaction to occur as a result of exposure to SO2 that adsorbed onto a surface

14th Weather Squadron U.S. Air Force organization that collects and disseminates worldwide climatology data

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CHAPTER 1

INTRODUCTION

According to Fontana, “Corrosion is defined as the destruction or deterioration of a material because of reaction with its environment.” [3] This material degradation mechanism comes at a great cost to society. In 2001, a report containing the findings of a study commissioned by the

United States Federal Highway Administration was released. This study examined the direct cost of corrosion across 26 sectors of the U.S. economy and the researchers concluded that such costs amounted to $276B per year (in 1998 dollars) [4]. In response to these findings, the U.S.

Congress enacted a law requiring the Defense Department to establish an office at the Pentagon that would have the responsibility to implement new policies, strategic planning, and programs designed specifically to more effectively manage corrosion [5]. The Congress later directed the

National Research Council (NRC) to conduct two studies to investigate issues pertaining to corrosion. The first of these efforts assessed undergraduate engineering curricula to identify deficiencies and propose approaches to improve educational practices so that future engineers will be better prepared to design products that are more tolerant of corrosion [6]. A member of the National Academy of Engineering who participated in this study stated that in addition to improved educational practices, practical engineering processes that can be used by design engineers are also needed [7]. In the following NRC study, research needs were examined and prioritized to provide a roadmap for future development efforts to help further reduce

1 corrosion costs. This study led to the identification of four “Corrosion Grand Challenges” (CGC)

[8]. As will be discussed later, three of these challenges have complete or partial relevance to this current modeling effort.

 CGC I: “Development of cost-effective, environment-friendly corrosion-resistant

materials and coatings”

 CGC II: “High-fidelity modeling for the prediction of corrosion degradation in actual

service environments”

 CGC III: “Accelerated corrosion testing under controlled laboratory conditions that

quantitatively correlates to observed long-term behavior in service environments”

 CGC IV: “Accurate forecasting of remaining service time until major repair, replacement,

or overhaul becomes necessary—i.e., corrosion prognosis”

Material properties are often assumed to be independent of the service conditions under which they are subjected. For example, design engineers typically consider properties such as strength or stiffness as constant values, which they obtain from texts, databases, or literature from manufacturers. However, corrosion is a far more difficult phenomenon to consider since the electrochemical reactions that consume materials are highly dependent upon the severity of the service environment. Atmospheric corrosion rates are influenced by numerous acceleration factors, each of which is highly variable. Such factors include airborne contaminants from natural and anthropogenic sources as well as climatic factors that are dependent upon geographic location. A further complexity results from diurnal cycling, which typically results in large changes in temperature and humidity levels throughout the day. Models that do not explicitly account for variability of the principal acceleration factors are incapable of predicting accurate corrosion rates.

2

The principal hypothesis associated with the research described in this dissertation is that in comparison to past modeling efforts, a nonlinear modeling and simulation approach using time- dependent temperature, relative humidity, and atmospheric contaminant (chloride, SO2, and ozone) levels will result in better fit/correlation of predictions to actual AISI/SAE 1010 steel atmospheric corrosion test measurements. Correspondingly, the null hypothesis is that such a cumulative damage approach does not result in improved correlation. As will be discussed below, most past atmospheric modeling efforts employed the coefficient of determination (R2) method to evaluate model accuracy. This approach was used to provide evidence supporting the current research hypothesis.

1.1 Past Modeling Effort For more than forty years, there have been numerous attempts to develop models to predict atmospheric corrosion rates for steel and other materials [9-46]. None of these existing models, which are often referred to as “damage functions” or “dose-response” functions, explicitly consider how stochastic variables associated with weather and atmospheric contaminants combine to influence corrosion rates. Instead, they commonly were calibrated by considering long-term (e.g., yearly) average values for the environmental parameters of interest. As will be discussed in Section 1.2.4, most past models do not consider the effects of temperature, which is known to be a key factor in determining rates of chemical reactions. A variety of mathematical approaches have been used to develop these past models. As will be shown later, some of these approaches are based upon power-law or regression techniques, but other methodologies have also been employed.

Generally speaking, models described in the literature have limited accuracy when used to make predictions for multiple locations with diverse environmental conditions. Some of these models employ contaminant characterization measurements obtained through specialized testing

3 techniques that are not commonly used [25, 47-51]. Such data is rarely, if ever, published.

Thus, specialized environmental testing is often required to characterize the environment for locations where corrosion predictions are desired.

Successful development of a cumulative damage model based upon stochastic environmental severity could provide the mechanism needed to make accurate predictions for most locations, thus enabling a potential new paradigm for the design and sustainment of products constructed from steel alloys. If this current research effort proves successful, the results may well lead to a practical corrosion prediction methodology suitable for use by non-specialists. Such a methodology could be used to calculate accurate corrosion allowances so that products can be designed to predictably corrode without impacting structural safety margins over their intended life span. The methodology could also prove useful for helping optimize new accelerated corrosion test methods so they can more accurately simulate material behavior under natural exposure conditions. In addition, it could form the basis for prognostic approaches to assess the remaining life of degraded structures already exposed to field conditions, thus facilitating a more informed approach for planning of asset maintenance and replacement. As such, an accurate cumulative damage model could significantly address three of the four Corrosion

Grand Challenges (CGC II, CGC III, and CGC IV) identified in the NRC report [8]. The model developed through the work described here represents a proof-of-concept towards implementing the new methodology as discussed above.

1.2 Literature Search

1.2.1 Factors that Lead to Corrosion There are numerous acceleration factors that combine to initiate, sustain, and influence the rate of uniform corrosion reactions that occur upon steel surfaces. Moisture is a necessary condition that must be present for corrosion to occur. Under atmospheric conditions, such moisture can

4 be due to rainfall, snowmelt, or condensation, which provides a totally wet surface, or the wetness can come from adsorption, which leads to the build-up of monolayers of moisture on surfaces [52]. However, moisture alone cannot initiate and sustain electrochemical corrosive reactions. Oxygen as well as ionic contaminants coming from either natural or anthropogenic sources must also be present in order for the environmental attack to proceed. Gaseous pollutants can either directly adsorb onto surfaces [53] or dissociate when combined with atmospheric moisture and subsequently deposit as constituents within acid rain [54, 55].

Additional factors include chloride aerosols and particulates, which can deposit through either wet (i.e., rainfall) or dry processes [56]. When combined with moisture, all of these substances create an oxygen-rich electrolyte that enables electrochemical oxidation-reduction reactions to proceed [57]. Graedel and Frankenthal report that such reactions are not homogeneously spaced but instead occur in microscopic surface features such as pits, voids, and crevices that are dispersed across the metal surface [58].

Atmospheric contaminant levels pertaining to pollutants such as SO2 and ozone are known to exhibit stochastic behavior, which is indicated by rapidly changing measurements over time periods as short as an hour [59, 60]. In addition, physical processes such as rainfall can clean surfaces of contaminants [61, 62]. When combined, these factors can lead to rapidly changing corrosion rates. An additional factor concerns the actual thickness of the electrolyte with thinner layers of sufficient thickness having higher corrosion rates due to concentration effects

[63]. Temperature can affect corrosion rates but with two competing mechanisms. For instance, if the temperature increases while the absolute humidity remains somewhat constant, the electrolytic moisture layer on the surface will start to dry, which can reduce the corrosion rate if insufficient moisture is available. However, when sufficient moisture remains adsorbed

5 onto surfaces, temperature increases can accelerate the electrochemical processes responsible for corrosion [64, 65]

W.H.J. Vernon conducted a series of groundbreaking experiments to determine the critical relative humidity (RH) level required before iron and steel alloys can corrode when exposed to low levels of SO2 [66]. His work involved the precise weighing of specimens exposed to a range of relative humidity levels in increments of 10% RH and he determined that a minimum of 60% relative humidity was required for corrosion to occur. However, he did not report whether temperature affected the critical RH, which will be referred to hereafter as the RH threshold.

This 60% RH threshold is still reported in the modern literature [67, 68]. One thing Vernon noticed was that oxidation was not visually present on surfaces exposed to 60% RH, even though the specimens had gained mass. Oxidation wasn’t observed until the RH reached 70%.

Many efforts over the years have supported a different view of the RH threshold. A concept known as the “time of wetness”, TOW, has been frequently applied during past modeling efforts to quantify when atmospheric moisture levels are sufficient for corrosion to occur. The initial

(1992) version of ISO Standard 9223 defined TOW as the amount of time during a year when the relative humidity exceeds 80% [69]. However, this particular standard did not publish any dose- response functions that employ TOW but instead discussed it as a general modeling parameter.

The newest version of this standard (2012) does contain a series of material specific functions

[38]. It also defines TOW in the same manner as the 1992 version even though it is not used in any of the four published models (for four different materials). Instead, these models employ the annual average value of RH. Other threshold RH values have been used when constructing models. For example, Abbott uses a value of 70% RH [25] while other efforts have used a value of 90% RH [19, 21]. When used for regression modeling, TOW is essentially a curve fitting

6 parameter that results in the most accurate prediction despite whether or not corrosive reactions physically occur below the specified value.

In an attempt to determine the specific amount of water vapor that adsorbs onto surfaces in response to changes in relative humidity and temperature, Lee and Staehle precision weighed gold plated quartz crystal specimens using a quartz crystal microbalance (QCM) [52]. Gold was used during these experiments to preclude oxidation from occurring so the only observed mass gain would be from adsorption of water molecules onto the surface. They concluded that texture on metallic surfaces such as pores and capillaries play a role in adsorption and that these surface features must first fill with water before a continuous coverage of monolayers of water can build up on surfaces. This phenomenon could explain why Vernon’s specimens gained mass at 60% RH even though the surfaces did not visually appear to be corroded [66]. Lee and

Staehle also concluded that monolayer thickness builds up with increasing RH but that temperature plays a minimal role with respect to adsorption at low temperatures (e.g., 7oC and

25oC) [52]. Leygraf stated that the QCM technique is highly temperature sensitive by itself, which makes it hard to measure moisture adsorption as a function of both temperature and RH.

He further stated his belief that the amount of adsorbed moisture is dependent upon temperature. [70].

1.2.1.1 Atmospheric Contaminants Atmospheric contaminants in the form of gaseous pollutants or aerosol particle contaminants must either wet deposit or dry adsorb onto surfaces and combine with adsorbed moisture and oxygen in order to provide the electrolytic conditions necessary for corrosion to occur. Leygraf states that O3, H2O2, SO2, H2S, COS, NO2, HNO3, NH3, HCl, Cl2, HCHO, and HCOOH all have significant importance with respect to atmospheric corrosion [71]. The following discussions will focus on some of the more common contaminants.

7

1.2.1.2 Chloride Aerosols Chloride aerosol particles, sometimes referred to as airborne salinity, deposit onto surfaces through both wet and dry deposition processes [72]. On a global scale, the most significant source of these aerosols is the world’s oceans, where they are produced by the crests of foamy breaking waves (whitecaps). They also are formed when large waves build up and break in surf zones that are adjacent to beaches and reefs. Aerosols produced from these sources are then transported overland by winds [73].

Chloride aerosols produced on the open ocean are fine (≤6 m) to medium size particles while those generated in surf zones are much larger (up to 20 m) [73, 74]. The size of individual aerosols has a dramatic effect concerning their ability to remain in airborne suspension. The large aerosols generated in surf zones quickly fall from airborne suspension as the distance increases from the coast. In contrast, the smaller aerosols generated by whitecaps can travel far greater distances inland on the prevailing winds [75]. This is undoubtedly why Abbott reported such vast differences when comparing steel corrosion rates measured at a beach site at

Kennedy Space Center, Florida in comparison to rates measured ¼ mile (0.4 km) and five miles

(8.0 km) inland [25]. Summit and Fink describe how the maintenance algorithm developed under the “PACER LIME” program dictates actions that depend on whether or not the aircraft is within 4.5 km of a coastline. [76].

The role that chloride deposition has upon corrosion rates was examined by conducting a series of experiments across the island of Cuba [77]. One set of experiments involved setting up corrosion coupon exposure test sites from the north to the south side of the Island. The topography between the two coastlines was mostly flat with small hills and the prevailing winds responsible for chloride dispersion coming principally from the north. It was observed that chloride levels and the associated corrosion rates dropped as the distance from the northern

8 shore increased. This trend continued across the island until chloride deposition measurements and corrosion rates showed a small increase at a test site near the southern shore. Under another set of experiments, chloride aerosol deposition rates were measured at 29 sites in

Cuba. These included mountainous zones, urban/industrial areas, inland flat terrain, and coastal areas [78]. It was observed that deposition rates measured at each of the five test sites located in the mountains were lower than 23 of the remaining 24 sites. In addition, the researchers found that the mountain deposition measurements were three orders of magnitude lower than measurements made near the coast. Morcillo, Chico, Mariaca, and Otero conducted a series of experiments and observed the distance that aerosol particles travel is dependent on the direction and speed of wind, topographical features, and the presence of elevated terrain including mountains [79]. Cole, Patterson, and Ganther state that elevation plays a strong role with respect to the ability for chloride aerosols to remain in airborne suspension. They concluded that as winds flow uphill while passing over mountains, aerosol particles are more likely to fall to the ground with increasing elevation [73]. Thus, deposition of chloride aerosols is highest on the windward slopes of mountains facing the ocean while lower altitude areas inland from coastal mountain ranges have low levels. Other factors including wind speed, direction, and distance from the sea also influence the rate of chloride deposition [80].

As stated above, chlorides generally are produced by the oceans and flow inland as suspended particles flowing on the winds. However, there are other sources of chlorides that can be of concern. For instance, deicing salts are routinely dispersed over roadways in northern areas to help melt snow and ice resulting from snowstorms [65, 81]. Such particles can then become airborne by passing vehicle traffic and subsequently be blown into adjacent structures and surrounding areas by the wind [82, 83]. There are other natural sources of chlorides that must

9 also be considered. Salt flats are found around the world and they can also be a significant source of windblown chlorides [84].

1.2.1.3 Gaseous Pollutants Gaseous pollutants from anthropogenic or natural sources can provide the ionic contaminants that lead to atmospheric corrosion. There are a number of such substances including gases containing sulfur, nitrogen, and oxygen [71]. Sulfur dioxide (SO2) has long been known to initiate reactions that lead to corrosion. SO2 can adsorb directly onto surfaces and then oxidize when combined with adsorbed water molecules, thus providing a very thin electrolytic layer containing weak sulfuric acid [85]. However, gaseous SO2 can combine with water in clouds and rain drops, and directly deposit on surfaces as ionic constituents of acid rain [71, 86]. Hydrogen sulfide (H2S) and carbonyl sulfide (COS) are two other sulfurous gases known to be a corrosive under suitable conditions [53, 65, 71, 87].

Sulfur dioxide is produced by numerous sources including combustion processes [88]. For example, electrical power plants that burn coal or heavy fuel oil, diesel engines, and residential furnaces that use fuel oil or kerosene all produce SO2 [89]. Metal smelting facilities and paper mills are other anthropogenic sources [90, 91]. There are also natural sources of SO2 including decaying organic material [92] and volcanos, which can release tremendous amounts of gas over short time periods of time [93]. Another corrosive, hydrogen sulfide, which can convert into SO2 under atmospheric conditions, comes from both anthropogenic and natural sources including pulp and paper mills, catalytic converters used on gasoline powered motor vehicles, sewage plants, volcanoes, and swamps [71]. Carbonyl sulfide (COS) is a sulfur-based substance found in the atmospheric environment. COS is seasonally absorbed and emitted by the oceans [94-96] and under suitable atmospheric conditions, it too will convert into SO2 [97].

10

Nitrogen oxides (e.g., NO, NO2, and NO3, typically referred to as NOx) can both dry adsorb and wet deposit onto surfaces in the same manner as SO2, thus decreasing the pH of the thin electrolytic surface layer through the addition of weak nitric acid. NOx emissions can also lead to the formation of ozone, which is formed when NO2 reacts with hydrocarbon pollutants in the presence of ultraviolet light [98].

Like SO2, nitrogen oxides are formed through both anthropogenic and natural mechanisms. NOx is often produced as a byproduct of the high temperature combustion processes seen in hydrocarbon fueled power plants, internal combustion engines, and other high temperature energy conversion processes [71]. Such oxides are also produced by natural processes including lightning and biomass conversion [99, 100]. As described earlier, ozone (O3) is produced in the atmosphere when a mixture of NOx and hydrocarbon pollutants is exposed to UV radiation [71,

101, 102]. However, tropospheric ozone can also come via downward transport and mixing of ozone originally produced at high altitudes in the stratosphere [103].

1.2.1.4 Variability of Acceleration Factors Factors such as solar radiation, atmospheric contaminant levels, relative humidity, temperature, rain, wind, and particulate deposits all work synergistically to provide the conditions that lead to variable corrosion rates [104]. As discussed previously, atmospheric contaminant levels can vary significantly from hour-to-hour. This behavior is primarily due to the stochastic nature of pollutant levels but some variability likely comes from uncertainty associated with measurement techniques [105]. Brown and Stutz describe that in urban atmospheres, NO3 reaches its peak a few hours after sunset but that its levels, along with those of N2O5 decrease throughout the night as a result of reaction with sulfate aerosols [106]. Song et al. point out that NO3 will react under atmospheric conditions to form N2O5, which will further react with water molecules to form nitric acid [107]. Ozone also experiences significant variations to measurement levels due

11 to environmental conditions. Since the reactions that create tropospheric ozone are temperature dependent and require UV radiation, levels change throughout each day of the year due to diurnal and seasonal changes in temperature and sunlight [103], with the highest diurnal levels being midday and highest seasonal levels typically measured during the spring

[108].

Weather factors also play a significant role with respect to variable corrosion rates. For instance, the combination of diurnal and seasonal cycles along with storm fronts and other weather phenomenon provides enormous variability of temperature and (absolute) humidity throughout the year. Since electrochemical reactions depend upon temperature [65] and moisture [66], then variability in these factors will result in variable corrosion rates.

Further complexity with respect to conditions within electrolytic surface layers occurs as a result of washing effects associated with rainfall [61] or air currents [109]. The combination of stochastic adsorption, desorption, and deposition processes along with highly variable weather parameters and related cleaning effects results in conditions within aqueous surface electrolyte layers that are far from steady state.

As discussed above, many acceleration factors affect corrosion rates through their stochastic behavior. However, there are some long term effects that have led to reduced corrosion rates.

In 1970, the US Congress passed the Clean Air Act, which has been amended twice since then.

These laws regulated pollution emissions and combined with related state and local regulations, have led to dramatic declines in pollutant levels [92]. According to the Environmental

Protection Agency, the trend in the National average SO2 concentration in the United States has dropped 83% from 1980 through 2010 [110]. Smith reports that North American and European

SO2 emissions have steadily decreased since the 1970’s but that emissions from China have

12 shown a nearly exponential increase since 1950 [111]. Such a drastic decline in the U.S. and

Europe appears to negate the usage of previously developed maintenance approaches such as the Pacer Lime methodology, which is based upon the assumption that SO2 pollution becomes the predominate cause of corrosion at distances greater than 4.5 km from coastal shorelines

[112]. In response to the significant reductions in SO2 resulting from pollution control laws, researchers have begun looking into the significance of other corrosive substances such as those formed from nitrogen oxides [64]. New steel dose-response functions have been developed to account for the reductions in SO2 [113] and ISO Technical Committee 156 - working group 4

[114] was chartered with the mission to update ISO Standard 9223, which included publishing new dose response functions [38].

1.2.2 Statistical Assessment of Corrosion Models Models intended to predict the rate of uniform corrosion of steel due to exposure to atmospheric conditions typically present their predictions as either thickness loss (thinning) or mass loss per unit area. For those models that predict mass loss per increment of time, such rates can be used to calculate the time and environment-dependent thinning of the material.

Thinning is an important metric as it is a direct factor that designers can use to predict the life of structures exposed to corrosive conditions.

Models are typically calibrated using data measured during coupon exposure tests. During the calibration process, statistical analysis techniques are employed to ensure that predictions have the highest possible accuracy. There are several methods that can be used to perform the necessary statistical assessments.

1.2.2.1 Coefficient of Determination (R2) One popular way to assess the accuracy of corrosion models is by calculating the coefficient of determination, which is typically referred to as the “R2”value. This statistical measure pertains

13 to the proportion of the variance in the dependent variable that is explained by the independent variables [115]. If the independent variables perfectly fit the observations they model, the R2 value will be 1.0. Conversely, if the independent variables have no correlation to observations,

R2 will equal 0.0. Values between these two points provide an index of the “goodness of fit” pertaining to the model and the data from which it was constructed [116]. For example, if an R2 value of 0.6 is calculated for a particular model, then 60% of the variability associated with its predictions can be explained by the independent variables that it considers. However, this implies that 40% of the variability is due to other factors not recognized by the model. A review of the literature did not reveal any universally accepted criteria or “rules of thumb” to help ascertain what minimum value of R2 is needed for a model to have acceptable accuracy.

A potential problem with using the R2 statistic is that its value will always increase if additional factors are added to the model, even if such factors are insignificant [117, 118]. What this implies is that R2 values for simple models such as those based upon the power-law approach cannot be directly compared to values obtained from more complex models that consider many more factors. Montgomery states that the “adjusted” R2 statistic can be used as a better measure of the goodness of fit for more complex models [117]. It was not apparent from the descriptions of the individual legacy models identified during this literature review as to just how their various R2 values were calculated.

Another measure of model accuracy concerns the residuals themselves, which can be plotted using a variety of techniques [119]. A good model will result in residuals that are normally distributed on the plot while a poor model can exhibit some other behavior, including the appearance of a funnel where the residuals are larger at lengthier time periods. Such

14

“heteroscedastic“ behavior is also a prime indicator of model inaccuracy [120]. Appendix C discusses this issue more completely.

The papers describing the models examined under this this literature review did not explicitly state how their R2 values were calculated. They also didn’t discuss independent validation.

Thus, it seems reasonable to assume that the reported statistics represent the fit of the reported models to their calibration data. However, the R2 statistic can also be used to provide an index-of-fit of the model applied to independent validation data (data not used for calibration). This method was used for this current research.

1.2.2.2 Uncertainty Intervals Some past corrosion modeling efforts employed statistical intervals to convey accuracy. For instance, Kucera reported 95% confidence intervals for the individual coefficients used by his model [23]. ISO Standard 9223 (2012 version) reported an estimate for the level of total uncertainty associated with the actual predictions made by the published dose-response functions [38]. Of the existing corrosion models identified during this literature review and discussed later, these were the only efforts that considered statistical intervals to quantify uncertainty.

1.2.2.3 Residual Sum of Squares “Inverse” modeling is a methodology where values of poorly known and/or immeasurable coefficients are estimated during calibration using a process that fits the candidate model to data [121]. This type of modeling is used when an explicit model with coefficients that have closed-form solutions is unavailable. Regression is one such approach to constructing inverse models and the method of least squares is used to identify optimum coefficient values. This approach is based upon minimizing the residual sums of squares, which by definition will increase the goodness-of-fit between the model and the data it represents. Residual sum of

15 squares can also be used as a tool to help determine optimum coefficient values for empirical models developed using Monte Carlo simulations [120].

1.2.3 Atmospheric Severity and its Importance to Modeling Efforts The simplest way to classify atmospheric severity is to employ a qualitative description such as used by Tullmin and Roberge, who define four types of atmospheres: rural, urban, industrial, and marine. They point out that this qualitative description does not account for micro environments that result from point sources of contamination [65]

PACER LIME was an extensive project conducted for the U.S. Air Force in the 1970’s and 1980’s that resulted in aircraft maintenance approaches based upon consideration of environmental severity. Summit and Fink report that the PACER LIME environmental severity assessment was conducted by considering numerous factors including weather, geographical features, particulates, and a probabilistic analysis of atmospheric contaminants [76, 112]. The conclusions from PACER LIME are that there are two primary severity conditions corresponding to coastal areas and areas more than 4.5 km inland. This is likely due to the effect of chloride aerosols as illustrated by Abbott’s tests at Kennedy Space Center [25]. There are other factors that conflict with the qualitative classification of environmental severity. Shaw and Andersson reported that corrosion rates at Norman Wells, Northwest Territories, Canada are very low because of the cold temperatures, something not accounted for in Tullmin and Roberge’s qualitative definitions [104].

ISO Standard 9223 lists six different corrosivity categories ranging from very low to extreme.

The determination of the corrosivity category is made by either measuring the first year corrosion rate at a location of interest or by calculating them using the published dose-response functions. Listed in the standard are estimated levels of uncertainty associated with both the

16 test measurements and the associated predictions. These uncertainty bounds indicate that predictions can be considerably different from related test results [38].

Atmospheric severity is an important issue when it comes to predicting corrosion rates. As will be seen, there are numerous considerations made during the development of past models.

Some of these models consider chloride deposition, others focus upon SO2, and some consider both. Other contaminant and weather-related factors of interest are used in models as well.

The difference in approaches and factors considered when developing these past models were likely due to differences in environmental severity consistent with the previous discussions of acceleration factors including atmospheric contaminants.

1.2.4 Legacy Modeling Efforts For more than forty years, there have been numerous attempts to develop models, often referred to as “damage functions” or “dose-response” functions, to predict the rate of atmospheric corrosion [9-38]. Many of these efforts employed power-law approaches or statistical regression that considered actual atmospheric conditions at sites of interest. There have also been more sophisticated attempts that have led to a mechanistic model to predict corrosion rates based upon aqueous chemistry [39-44]. These efforts began in the mid-1990’s and have continued over the years with the most recent attempts focusing on the effects of corrosive acids [45, 46]. Recognizing that there is significant uncertainty associated with model predictions, other approaches have been developed to provide design recommendations based upon the environmental severity at locations where a product or system will be employed [38,

76, 112].

More sophisticated models are statistically calibrated using measurements of the assumed acceleration factors of interest (e.g. relative humidity or levels of airborne contaminants such as

17 atmospheric chloride aerosols or sulfur dioxide). In comparison, basic power-law models often consider only one independent variable, which is usually related to time of exposure. As will be discussed later, some of these models have been shown to be highly accurate (as indicated by their published “R2” values) but only under certain circumstances [28]. For example, if the corrosion test data used to calibrate a power-law model were measured at a single location, the predictions from that model will be highly accurate but only for that location or others with very similar environmental conditions [28]. Conversely, if that same model was applied to make predictions for other locations with significantly different environments, the results will be inaccurate. Many researchers have attempted to create general models using hybrid power-law, regression, and other types of approaches in order to calculate corrosion rates at any location

[10-14, 25-27, 34-37]. Such models generally have lower R2 values in comparison to the narrowly focused models discussed above, but they also have broader applicability.

Models that consider atmospheric contaminants are typically calibrated using accurate long- term measurements (e.g., annual averages) for atmospheric chlorides and/or SO2 but weather data is imprecisely considered. For example, relative humidity is often considered as the aforementioned time of wetness (TOW), which corresponds to the number of hours of exposure exceeding a threshold value (e.g. 80% RH) [24, 38]. Some newer functions have abandoned this approach and instead consider annual average RH values [38]. Most past models do not address temperature effects but those that do consider long-term average temperatures (e.g., annual averages [27, 36, 38] rather than addressing highly variable temperatures resulting from diurnal and seasonal cycles.

1.2.4.1 Types of Existing Models Power law models are quite simple and typically relate a dependent variable to a single independent variable. Equation 1.1 illustrates the basic form of this type of model [122], which

18 relates the corrosion rate “C” to the time of exposure, “t”, The coefficients “A” and “C” used in the basic form of this equation are typically constants.

퐶 = 퐴푡푛 (1.1)

Regression models are used to overcome the limitations of power-law models by allowing the dependent variable to be related to single and multiple independent variables, which typically pertain to the atmospheric contaminant(s) of interest. Equations 1.2 and 1.3 illustrate the general form of single and multiple regression models [123, 124]. Statistical curve fitting techniques are used to calculate the values for the coefficients seen in all three equations.

푦 = 푎 + 푏푥 (1.2)

푦 = 푎 + 푏1푥1 + 푏2푥2 + ⋯ 푏푛푥푛 (1.3)

Other forms of models used by corrosion researchers include exponential models, logarithmic models, physical models, cumulative damage models, and models based upon a combination of modeling types. Some past models were developed as part of international programs where calibration and environmental characterization tests were conducted at a large number of sites around the world while others were based upon data collected at one site. Some of these efforts gathered calibration data at numerous sites in a single country or geographical region. It should be noted that there are little or no details in most publications concerning how the specific models were constructed.

1.2.4.2 Power-Law Modeling Efforts Kobus developed three different models, based upon the exact form of Equation 1.1, from data measured at three sites in Poland [28]. The R2 values calculated for these three models were very high (0.96-0.99), indicating excellent fit to the data. Because they were constructed from

19 the same basic mathematical formulation, the specific values of the two coefficients “A” and “n” are what differentiates each models from the others. The fact that separate models were needed to accurately predict corrosion rates at the individual test sites indicates that they are unsuitable for use at locations with different environmental conditions.

Dean and Reiser attempted to create a general model using the exact same power-law modeling approach. In this case, the model was calibrated using data from the ISO CORRAG program, which conducted tests at 52 sites in 14 nations [29]. During calibration, over half of the data points were disregarded in order to improve the goodness-of-fit. Despite the tailoring of the calibration data to improve the fit, the subsequent model had an R2 value of 0.83. This indicates that power-law models based only on time have limitations in terms of their applicability.

1.2.4.3 Linear Regression-Type Modeling Efforts Kucera developed a linear regression model based upon the form seen in Equation 1.2. This model was calibrated using data measured at seven sites in Sweden and the former country of

Czechoslovakia [10]. Because of their geographical locations, the test sites were limited to temperate or cold climates. The R2 value for this model was 0.96, which is likely due to the limited range of climatic conditions at the seven sites. Haagenrud used the same approach but employed data from 32 sites in Scandinavia [11]. The diversity represented by the much larger number of sites had the effect of lowering the R2 value to 0.76. Haagenrud et al. also used the same approach but in this case employed data collected at 15 test sites in a 26 by 31 kilometer area in Norway [12]. The R2 for this new model was improved to 0.86, which was likely due to the limited environmental diversity at the test sites. In each of these models, the single independent variable was limited to average SO2 exposure measured over the test period.

20

Duncan and Balance created a model that appears to be in the form of Equation 1.2 except that the natural log of chloride is used as the independent variable [13]. This model was calibrated from data collected at 18 sites in New Zealand and it had a very high R2 of 0.95. Bashker et al. constructed a “cumulative damage function” but their terminology has a different meaning than that used here since the variable “time” is not employed by their methodology [37]. Instead, their use of the expression “cumulative damage” refers to corrosion rates that are dependent upon the accumulation of damage from individual acceleration factors (e.g., SO2 and chlorides).

As an example, they published an expression for mild steel, which focused solely on SO2 as the independent variable, which seems somewhat contradictory to their intent since no other acceleration factors such as chlorides were included in the formulation. No R2 value was reported for this model.

1.2.4.4 Multiple Regression-Type Modeling Efforts Several attempts at multiple regression modeling using Equation 1.3 were identified in the literature. The Ibero-American MICAT Program was another large program that conducted tests at 75 sites in 14 countries located in southern Europe, Central and South America, as well as

Mexico and Cuba [33]. The researchers developed two different models, one calibrated using all of their data and another that used a subset from 52 sites. Both models considered time of wetness and long-term average measurements for SO2 and chlorides. The 75 site model had an

R2 value of 0.46 while the 52 site model was somewhat better at 0.58.

Lien and San used the same approach to develop two models based upon testing at seven sites in Vietnam [36]. One of the models considered average annual temperature, annual average relative humidity, annual average temperature, time of wetness, and annual chloride deposition. The second model was similar with the exception that it did not contain a term for chloride deposition. This model was intended for use at locations away from coastal areas.

21

Neither model considered SO2 deposition. It is somewhat puzzling as to why these models would consider both annual relative humidity and time of wetness, which is a metric related to annual relative humidity levels. An R2 value of 0.88 was reported.

Roberge et al. reported a model very similar to the multiple regression model form shown by

Equation 1.3 but in this case, the dependent variable was reported as a logarithm [26]. This model was developed using the ISO CORRAG data (52 sites in 14 nations) and had an R2 of 0.65.

1.2.4.5 Polynomial Modeling Efforts Lipfert developed a polynomial model that used the variable f90, which pertains to the percentage of time during the year when the relative humidity exceeds 90% [21]. This model

+ considered other factors including average deposition rates for SO2, H ions, chloride aerosols, dust, and precipitation. The model did not include any power-law expressions or exponents and some of the coefficients were specified with uncertainty ranges. Guttman and Sereda created a different type of polynomial model that considered time of wetness, panel temperature, and

2 SO2 deposition [14]. No R values were reported for either of these polynomial models.

1.2.4.6 Simple Exponential Modeling Effort Haynie and Upham created a simple exponential model that was calibrated using data collected at 57 sites in the United States [20]. This model considered the relative humidity and the atmospheric SO2 concentration (rather than deposition rate as used by most other models).

The R2 value for this effort was reported as 0.77. Exponential functions were used in other more complex models including the candidates developed under this current research program.

However, the model developed by Haynie and Upham was the only one discovered during this literature review that did not use exponential functions as a part of a more complex mathematical formulation. Instead, their model employed a simple formulation using a single

22 exponential to calculate the contribution of the environmental parameters towards the corrosion rate.

1.2.4.7 Hybrid Power-Law Type Modeling Efforts There were many hybrid models identified that combined power-law expressions with other factors. Benarie and Lipfert developed a model that appears to be in the form of Equation 1.1 except that the independent variable, t, corresponds to time of wetness instead of time [15]. In addition, the coefficient “A” is not a constant but instead is a function of SO2 while the exponent, “n”, is a function of pH. This model was calibrated using data from 41 sites world- wide and it has an R2 of 0.45. Kucera et al. also developed a power-law type model but in their case the coefficient “A” was a function of SO2 and chloride deposition [16]. This model was calibrated using data collected at 32 sites in Scandinavia and it had an R2 of 0.89. Brillas et al. constructed a model using the same basic form as Kucera et al. and then calibrated it from data measured at 42 sites in Spain [17]. This model had an R2 that ranged from 0.8 to 0.99. A model constructed in the exact same fashion but with different results indicates that the specific coefficients obtained via the calibration process are not suitable for universal purposes over a wide variety of environmental conditions.

The United Nations International Cooperative Programme on Effects of Air Pollution on

Materials, Including Historic and Cultural Monuments (UN ICP Materials) was a large effort where materials were tested at 39 sites in 12 European countries as well as in the United States and Canada [23, 35]. The general form of a model developed from this effort includes two power-law relationships that are added together. One of the relationships pertains to dry

+ deposition of SO2 and ozone while the other pertains to wet deposition of H ions. In place of a constant in the formulation of the model is an unspecified function. This model was not tailored

23 to reflect corrosion of carbon steel but one model was developed for weathering steel. An R2 of

0.68 was reported.

Barton and Czerny created a model that employed two power-law expressions that were multiplied together [18]. Rather than being functions of time, one of the expressions was based upon time of wetness while the other was based upon average SO2 levels. Data used to calibrate this model was measured at seven sites in the former USSR, Sweden, Bulgaria, and the former nation of Czechoslovakia. The R2 value for this model was 0.76. Lipfert et al. created a model that actually multiplies three power-law type expressions together [19]. One of the expressions uses the aforementioned f90 variable and multiplies it with the length of the test

period. The second expression again takes f90 and multiplies it with the average SO2 measurement. The last expression uses the hydrogen ion (H+) deposition rate. The exponents for all three of the power-law expressions were constants. An R2 value was not reported.

Klinesmith et al. created a hybrid model containing four power-law expressions combined with an exponential term [27]. All five components of this model were multiplied together and the independent variables employed by the power-law expressions included time, time of wetness,

SO2 deposition, and chloride deposition. The exponential expression considered the average annual temperature. This model was calibrated using the same data (ISO CORRAG program, 52 sites in 14 nations) that was employed by Dean and Reiser [29] for their power-law model and by Roberge et al. for their multiple regression model [26]. An R2 of 0.67 was reported.

The most recent (2012) version of ISO Standard 9223 contains a hybrid Power-law/exponential model [38]. This function contains two basic parts, the first of which pertains to SO2 deposition while the second considers chloride deposition. These two parts are added together and each one contains a power-law type expression and an exponential term. The power-law component

24 of the first part pertains to annual SO2 deposition while the exponential component considers relative humidity and temperature. The two components are multiplied together. The second part of this model is similar to the first with the major difference being that the power-law

2 component pertains to annual chloride deposition. The R value for the steel “dose-response” function is 0.85. Also published in this standard are the estimated levels of uncertainty associated with predictions made by the individual models. The uncertainty estimate for

2 predictions of steel corrosion is -33% to +50%, which indicates that even when the calculated R value is high, the model can provide results that are quite erroneous.

1.2.4.8 Probabilistic Modeling Efforts Mikhailovsky created a model that used numerous factors such as temperature, time, SO2 concentration, and relative humidity [22]. This model used statistical probabilities concerning

SO2 levels rather than long-term average measurements. It was calibrated using data collected at eight sites in Russia. No R2 value was reported but instead an uncertainty range of 2.4-53% was listed.

1.2.4.9 Cumulative Damage Modeling Efforts For well over a decade, the United States Department of Defense has funded efforts by Abbott to measure corrosion rates at a large number of sites across the U.S. and in other countries. The results from this work were published in a book, recorded in an associated database of corrosion weight loss data, and used to construct a corrosion model [25].

Rather than conduct longer term tests (e.g., one-year interval), Abbott simultaneously exposed four precisely weighed coupons (for each material tested) to environmental conditions at each test site. After every three month interval (with some exceptions), a coupon of each material was removed from the test fixture, shipped to the lab, cleaned of corrosion products, and precision weighed. Because it was calibrated using quarterly test measurements, Abbott’s

25 model actually considers the variability in corrosion rates that result from seasonal changes to temperature and humidity levels. Thus, this model does represent a simple cumulative damage formulation. However, his approach does not account for the impact of diurnal temperature and humidity cycling combined with stochastic atmospheric contaminant levels.

The acceleration factors considered by Abbott’s model include chloride deposition, precipitation, and time of wetness, which in this case is defined as the percentage of time over

70% RH instead of 80% RH as specified by ISO Standard 9223 [38]. It does not consider SO2 deposition. No specific R2 value was reported for the steel model although there was a general statement on the range of R2 values being from 0.75-0.8.

1.2.4.10 Mechanistic Modeling Efforts In the mid 1990’s there were a series of development efforts focused towards evolving a complex model that considers the numerous electrochemical mechanisms that combine to influence corrosion rates. This model considers six physical regimes upon or adjacent to surfaces including Gas, Interface, Liquid, Deposition, Electrodic, and Solid (GILDES). The model is named after these regimes [39].

GILDES is used to perform simulations that investigate the chemistry of wetted surfaces as it relates to the aforementioned regimes. Numerous papers have been written that discuss how the methodology has been developed and expanded to consider a variety of corrosive electrolytes and base materials. For instance, Farrow et al. used GILDES to investigate the aqueous chemistry of zinc surfaces exposed to conditions in an environmental chamber, which at different times employed corrosive gases including SO2, NO2, and ozone [40]. In similar work,

Tidblad and Graedel expanded GILDES to examine the electrochemical corrosion of copper

26

- exposed to SO2 [41] and (NH4)2SO4 [42]. They followed this work with simulations to investigate corrosion of nickel surfaces exposed to SO2 [43].

After a seven-year hiatus, GILDES was further expanded to perform aqueous chemistry simulations pertaining to the corrosion of copper exposed to SO2/O3 and SO2/NO2 mixtures [44].

Recently, GILDES has been used to perform simulations concerning the effect of low concentrations of carboxylic acids on copper [45] and zinc [46].

GILDES provides an unprecedented way to consider the corrosion of select metallic substances due to the presence of electrolytic substances on surfaces. As discussed above, past GILDES studies have examined several corrosive substances found in the atmosphere but the researchers have yet to investigate the surface chemistry resulting from simultaneous stochastic exposure to all the contaminants they have studied. In addition, the effect of chloride aerosols has not yet been considered. The methodology has not been expanded to consider the corrosion of steel [70].

1.2.5 Environmental Parameter Characterization Methods Any methodology intended to predict environment-specific corrosion rates requires quantified inputs pertaining to the atmospheric contaminant and weather variables that combine to initiate and sustain electrochemical reactions. Of the steel corrosion models discussed above, those that considered contaminants focused on atmospheric chlorides and/or SO2. These development efforts typically quantified atmospheric contaminants by employing techniques that scavenge them from airborne suspension. For instance, chloride deposition rates can be measured using the “wet candle” method, which collects both wet aerosols as well as dry aerosols, which might not otherwise deposit [47, 48]. This technique uses a wick of known diameter and surface area, which is wet through partial immersion in a 40% solution of glycol

27 and water. At periodic intervals, the wick is removed from the environment and tested to quantify the total chloride deposition over the exposure period. Thus, this method quantifies the total aerosol content present at a location and not necessarily the amount that deposits over a period of time on metal surfaces.

Two ASTM approved methods for measuring SO2 exploit the fact that lead oxides react with SO2 to form lead sulfate. One method uses lead dioxide disks that are protected from direct rainfall and the other, referred to as a “candle”, employs lead peroxide paste which is carefully applied to a paper thimble and then inserted into a protective enclosure. Both methods react only with gaseous SO2 [49, 50]. ISO Standard 9225 describes the use of lead dioxide plates and cylinders as well as alkaline surfaces to determine sulfur dioxide deposition levels. It also describes wet candle and dry plate methods for measuring chloride deposition. The standard compares deposition rates determined by the various methods [51].

Abbott employed the use of silver coupons at corrosion test sites as a way to infer chloride levels [25]. The premise behind this method is that silver surfaces react with atmospheric chlorides to form silver chloride (AgCl) in a manner analogous to the formation of lead sulfate on lead dioxide. Thus, he believes this method can provide a quantitative measure of chloride deposition during the corrosion test period. Frankel believes the mechanisms responsible for the corrosion products found on silver surfaces are more complex than the formation of AgCl alone [125].

The environmental characterization tests described above are used to obtain longer-term measurements of atmospheric contaminants. In addition, these methods are not typically used for pollution monitoring purposes, hence, there is no widespread program to measure such data in a variety of urban and rural environments and subsequently make it freely available in a

28 database for use by others. Thus, the measurements obtained from these types of tests are not amenable for use in a cumulative damage methodology based upon very short time intervals.

There are widespread testing programs that routinely measure atmospheric contaminants and make the subsequent data available to the public. For example, in the United States there are large on-line databases that contain historical measurements of gaseous pollutants such as ozone and SO2 [126]. Using automated sampling methods [127-129], such data was, and continues to be measured at hundreds of sites across the country and the resultant hourly measurements for a wide variety of pollutants are freely accessible. Similarly, large databases of historical wet deposition measurements obtained through chemical analysis of rainwater are also available on-line [130]. Included in these databases are weekly, monthly, and annual average deposition measurements for chloride aerosols as well as numerous other substances.

These two sources of characterization data were used when developing the model developed under this research project.

29

CHAPTER 2

DERIVATION OF THE CUMULATIVE DAMAGE MODEL

Cumulative damage approaches have been applied to describe other material degradation processes such as fatigue. For example, Kujawski and Ellyin describe how the total amount of fatigue damage is based upon the summation of increments of damage during subintervals of time. They further state that damage during each increment is identical if the applied loading during each cycle remains constant, or the individual damage increments can vary in magnitude depending upon the variable amplitude of the cyclic loading [131]. Kujawski’s description can be represented by Equation 2.1, where “X” represents the cumulative damage, and “xi” represents the damage that occurs during the "ith" increment of time.

n  X x i 1 32 ... 1 xxxxx nn (2.1) i1

A cumulative corrosion damage model capable of calculating very small increments of damage over short intervals of time (e.g., 1 hour) provides the approach needed to account for the influence of diurnal and seasonal temperature cycling. Constructing such a model required a process far different than the methods used to construct legacy corrosion models. The optimal solution would have been to develop a model that exactly considers interactions between the numerous, highly variable, time-dependent factors known to contribute to corrosion. However, such rigorous consideration of the physical interactions responsible for corrosion is not currently possible. One approach to overcoming this problem was to develop an empirical model using

30 an inverse approach [132], whereby the initial form of the model was assumed and then refined through an iterative calibration and refinement process that considered hourly environmental characterization data measured at different locations with diverse environmental severities.

The model developed here was specifically constructed so that it considers the kinetics of the electrochemical reactions responsible for consuming the material.

The Arrhenius Equation, which is limited to considering temperature effects, has long been used to predict the rate of physical processes [133]. However, a simple temperature-based approach employing this equation is unsuitable for forming the basis of a cumulative damage model since it is incapable of explicitly considering other factors known to contribute to corrosion rates.

Such factors include the levels of relative humidity and atmospheric contaminants. Over the years, the reliability engineering community has developed a number of models to predict product life based upon accelerated testing results. Many of these models were based upon the Eyring Equation, which was originally developed to predict the rate of chemical reactions based upon variation of multiple acceleration factors [134]. While both the Arrhenius and

Eyring Equations consider chemical kinetics through the usage of a parameter referred to as the activation energy, the Eyring Equation has a major advantage in that it allows for consideration of other acceleration factors in addition to temperature.

2.1 Construction of Notional Corrosion Model Equation 2.2 shows a form of the Eyring Equation developed by reliability engineers to predict the time-to-failure, “tf”, of a product [135]. In this equation, “k” is Boltzmann’s constant (value

-5 equals 8.617 x 10 eV/K), “T” is the absolute temperature (primary acceleration factor), “S1” is the second acceleration stress factor, H” is the activation energy, and “A”, “”, “B”, and “C” are all modeling coefficients. This equation is calibrated using results from accelerated testing and the activation energy used in its formulation is analogous to the quantity used in the

31

Arrhenius Equation. The value of H is stated to fall in the range of 0.3 to 1.5 electron volts (eV)

[135].

H C t  AT  B  S })(exp{ (2.2) f kT T 1

As can be seen by Equation 2.2, this model predicts the time-to-failure of a product resulting from a reaction based upon temperature and one other acceleration factor. However, Eyring

Equation-based models are not limited to considering the effects of temperature and one other factor. As seen by Equation 2.3, a third acceleration factor can be considered by this new model, which adds two more modeling coefficients (“D” and “E”). In fact, the equation can be expanded further to account for as many factors as necessary. A problem that becomes immediately apparent is the large number of unknowns, none of which can be determined through testing or calculation using a closed form solution.

 H C E t  AT B DS  S })()(exp{ (2.3) f kT T 1 T 2

Because Equation 2.3 is used to predict the time for a reaction to occur and result in failure of an item such as an electrical component (as in Reliability Engineering), it cannot be used directly as a corrosion model. However, this equation does possess many of the attributes of a desired model. For example, from its dependence upon the activation energy, H, and Boltzmann’s constant, k, this equation considers the kinetics of chemical reactions. Considering such kinetics is of great importance since corrosion is known to be an electrochemical process. Another desirable attribute is that this equation also considers the interaction between temperature and other acceleration factors. A notional corrosion model (to be iteratively evolved using inverse techniques that will be described later) can thus be developed by assuming the expressions on

32 the right hand side of Equation 2.3 are sufficient to describe the processes that result in corrosion reactions. Therefore, Equation 2.4 appears to be of a form that can be used to calculate the hourly corrosion rate, Ki.

 H C E K  AT B DS  S })()(exp{ (2.4) i kT T 1 T 2

As discussed earlier, many past modeling efforts concentrated on either SO2 or chloride deposition (some did both) in combination with exposure to relative humidity. A few models also considered temperature effects. Each of these factors was considered in the model developed here. The current research effort was affiliated with an Air Force Research

Laboratory (AFRL) project sponsored by the Strategic Environmental Research and Development

Program (SERDP) [136]. This project investigated how corrosion rates of different materials varied when exposed to atmospheric contaminants including ozone. Thus, this gaseous pollutant was also be considered by the new model. It should be noted that other pollutants such as nitrogen oxides and perhaps even CO2 can affect corrosion rates [71]. However, including these additional substances would have made the calibration process for the proof-of- concept model developed here even more difficult. As a result, these additional substances were not considered in this new model. Such substances could be considered in follow-on modeling efforts.

 H C E G I K  AT B RH D Cl F SO H  O })()()()(exp{ (2.5) i kT T T T 2 T 3

Equation 2.5 presents the updated notional model, which includes each of the acceleration factors discussed above (i.e., temperature, relative humidity, chlorides, SO2, and ozone). This equation required further modifications before it became suitable for calibration using Monte

33

Carlo simulations. But first, it was important to consider just how this equation would be employed when making cumulative predictions. As discovered by Vernon [66], the electrochemical processes responsible for corrosion only occur when the amount of moisture adsorbed onto a material surface is sufficient to provide the necessary electrolytic conditions.

Under ambient laboratory conditions, Vernon found that the critical value of relative humidity necessary for corrosion of iron or steel to occur was found to be 60%. Thus, a conditional relationship such as shown in Equation 2.6 illustrates the foundation of the cumulative damage model. The critical value of relative humidity is referred to here as the RH threshold (RHTH).

H C E G I AT  B RH D Cl F SO H  O })()()()(exp{ , 푅퐻 > 푅퐻 퐾 = { 2 3 푇퐻 𝑖 kT T T T T (2.6) 0, 푅퐻 ≤ 푅퐻푇퐻

The nonzero portion of Equation 2.6 can be rewritten into the form seen by Equation 2.7. This new equation was used to facilitate the continuing development of the notional corrosion model.

H C  E  G I            Ki  AT [exp   B  RH  D  Cl  F  SO2   H  O3 ])(exp)(exp)(exp)(exp (2.7)  kT   T   T   T   T 

Material Environmental Reactivity Severity (kinetics)

  H   C   E   G   I  K i  AT [exp   B  RH   D  Cl   F  SO2   H  O3 ])(exp)(exp)(exp)(exp  kT   T   T   T   T 

Scaling Temperature- Effect of Effect of Effect of

Factors Humidity Chlorides SO2 Ozone Interaction

Figure 2-1 Combination of Factors Considered by the Incremental Corrosion Model

34

Figure 2-1 illustrates the various attributes of the cumulative damage model described by

Equation 2.7. As can be seen on this figure, the model was designed to account for the rapidly changing environmental severity at any location based upon its consideration of hourly levels of temperature, relative humidity, and several atmospheric contaminants such as chloride aerosols and gaseous pollutants including SO2 and ozone. This model also accounts for the material- specific kinetics relating to electrochemical reactions on exposed surfaces. Scaling factors including a temperature adjustment term are used to improve prediction accuracy.

Inspection of Equation 2.7 reveals a problem that had to be addressed. As configured in this equation, the model would predict a rate of zero corrosion if any one of the three contaminant functions had a (exponential) value of zero. Obviously, this is not feasible since sufficient moisture combined with ionic constituents from even a single contaminant will result in corrosion. Thus, the equation was reconstructed as seen in Equation 2.8. This new model introduces the use of three different temperature-relative humidity expressions, one for each of the three atmospheric contaminants.

 H  CL  C   E  Ki    Cl  BTA  RH   D  )(exp)(exp[exp Cl    kT   T   T   G   I  TA SO2  F  RH   H  )(exp)(exp SO   (2.8) SO2  T   T 2   K   M  O3  JTA  RH   L  O ])(exp)(exp O3  T   T 3 

The model shown by Equation 2.8 is predicated upon the assumption that a single activation energy is applicable to the individual reactions that result from the three different contaminants. However, it may be possible, if not likely, that each contaminant has its own reaction kinetics and thus its own activation energy. By incorporating three different activation energies, Equation 2.8 can be expanded into the form seen in Equation 2.9. Ultimately, the

35 decision on whether the single or multiple activation energy models provides the most accurate predictions will be determined using Monte Carlo simulations.

H C  E  CL  Cl      i  ClTAK [exp   B  RH  D  Cl])(exp)(exp   kT   T   T 

SO2  HSO 2   G   I  SO TA [exp   F  RH  H  SO ])(exp)(exp  (2.9) 2  kT   T   T 2 

O3  HO3   K   M  O TA [exp   J  RH  L  O ])(exp)(exp 3  kT   T   T 3 

Completing the notional equation that formed the basis for the actual corrosion model required an examination of the boundary conditions pertaining to the individual physical parameters. A comparison of each expression within Equation 2.8 (and Equation 2.9) with its corresponding boundary conditions helped determine the final forms of the notional model.

2.2 Derivation of Functions to Update the Notional Model With the exception of the function that represents the kinetics of the electrochemical surface reactions, each of the remaining exponential functions in Equations 2.8 and 2.9 must be capable of attaining a value of zero when environmental conditions are unfavorable for corrosion to occur. For example, if a particular pollutant such as SO2 attained a level of zero for a particular hourly increment of time, then the corrosion rate for that particular reaction must also be zero while the reactions for the other contaminants may be nonzero.

Figure 2-2 illustrates an exponential function where f(T) as graphed has a range from -4 to +1.

As can be seen from the figure and the adjoining table, an exponential value very close to zero results when f(T) equals -4. Thus, it was assumed that each of the non-kinetic functions (within the parentheses) in Equations 2.8 and 2.9 has an exact value of -4 when the corrosion rate is zero. During the simulation process, an adjustment was made to the model so that the values of

36 the exponential expressions evaluating these functions were identically equal to zero under situations specified by the boundary conditions.

3

2.5

2

1.5 exp[f(T)] 1

0.5

0 -5 -4 -3 -2 -1 0 1 2 f(T)

Figure 2-2 Values of Exponential Function Based Upon the Range for f(T)

2.2.1 Temperature-Relative Humidity Functions Equation 2.10 represents an expression from Equations 2.8 and 2.9 that is used to describe the interaction between temperature and relative humidity. Consistent with their usage in the accelerated testing model from which this notional corrosion model is based, the coefficients

“B” and “C” seen in this equation are constants.

C B  )( RH (2.10) T

There are two boundary conditions that must be addressed by Equation 2.10. The first of these conditions pertains to the assumption that the corrosion rate (determined by the exponential form of this expression) must equal zero at the freezing point (273.15K) where a phase transformation from liquid to solid occurs. This implies that the expression shown by Equation

37

2.10 must have a value of -4 at freezing. The second boundary condition happens at the threshold value of relative humidity, where sufficient moisture is not available for corrosive reactions to take place. Equations 2.11 and 2.12 represent these boundary conditions as applied to Equation 2.10. Note that the decimal form of 60% RH is used in Equation 2.12.

C (B  )RH  4 (2.11) .15273

C B .)(  460 (2.12) T

As seen from Equation 2.11, the value of this expression must equal -4 for a full range of relative humidity values, which is a condition that can never be met if the coefficients “B” and “C” are constants. Similarly, Equation 2.12 states that the expression must have a value of -4 for a full range of temperatures within the bounds set by the model. Like before, this condition cannot be met for anything other than a single temperature. Thus, the temperature-RH functions used in Equations 2.8 and 2.9 are unsuitable for the corrosion model being developed here.

2.2.2 Temperature-Contaminant Functions The boundary conditions for the three exponential contaminant functions used in Equations 2.8 and 2.9 are the same (i.e., the contaminants all have a minimum value of zero). As such, the corrosion rate for a particular contaminant must equal zero when its measured level falls to zero. Like the previous case of the temperature-RH expression, the contaminant-temperature expression must therefore equal -4 at this point so that its exponential value is approximately zero. This is illustrated by Equation 2.13 for the chloride aerosol contaminant. The SO2 and ozone expressions are similar.

38

E D )( Cl  4 (2.13) T

It is obvious by examining Equation 2.13 that this relationship cannot be met at the boundary condition. More specifically, it is not possible for this expression to have a value of -4 when the chloride level is zero. Thus, this expression as well as the related SO2 and ozone functions was invalid with respect to the notional corrosion model. As such, replacement functions were needed for the contaminant expressions as well as for the temperature-RH expressions.

2.3 Constructing a New Functional Form of the Notional Equation Inspection of each non-kinetic expression in Equations 2.8 and 2.9 does reveal a common attribute in that they are all a function of temperature. Thus, the explicit expressions used in these initial equations were replaced by functions as seen in Equation 2.14 (single activation energy) and Equation 2.15 (multiple activation energy). These equations are the updated notional corrosion models and thus form the basis for the final models. The challenge was to find appropriate functions that address the necessary boundary conditions while describing the interactions between acceleration factors.

 H  CL Ki   [exp CL Cl TfTA RH  Tf ,(exp,((exp Cl    kT  SO2      O3     SO2 SO2 TfTA RH Tf ,(exp,(exp SO O32 O3 TfTA RH OTf 3 ],(exp,(exp (2.14)

CL  HCl  i  ClTAK [exp  Cl Tf RH  Tf Cl ],(exp,(exp   kT 

SO2  HSO2  SO TA [exp  SO Tf RH  Tf SO ],(exp,(exp  2  kT  2 2

O3  HO3  O3TA [exp  O3 Tf RH  OTf 3 ],(exp,(exp  kT  (2.15)

Table 2-1, contains descriptions and related units for the variables, coefficients, and functions used in the notional cumulative corrosion damage models shown by Equations 2.14 and 2.15.

39

With the exception of the activation energies, the description and usage of the variables/coefficients/functions used by both equations are the same. As seen in these equations, three different Temperature-Relative Humidity shape functions are paired with three associated Temperature-Contaminant shape functions. This was done in order to calculate three nondimensional numerical indices proportional to the portion of the overall corrosion rate that results from the interaction between temperature, relative humidity, and each of the individual contaminants. Hourly corrosion rates result when these indices are appropriately combined with the kinetic terms (i.e., activation energies and the Boltzmann constant) as well as the requisite scaling and temperature adjustment factors.

As described earlier, there are other atmospheric contaminants (e.g., nitrogen compounds) that when combined with suitable moisture and temperature levels are known to initiate and sustain corrosive reactions. However, for this proof-of-concept effort, it was assumed that sufficient prediction accuracy results from limiting consideration to the three contaminants employed by the above models. Another assumption made during the construction of these models was the proportion of the total corrosion rate due to each individual contaminant could be quantified using shape functions calibrated to account for different types of data including gaseous measurements (ppm) for air pollutant (i.e., SO2 and ozone) adsorption and concentration measurements (mass per unit volume of rainwater) for chloride aerosol particle deposition.

Furthermore, it was assumed that gaseous SO2 levels were proportional to the total amount of

2- all sulfur-based contaminants in the atmospheric environment including H2S, SO4 (sulphate),

H2SO4 (acid rain), etc. This assumption enabled the construction of a single, properly calibrated shape function based upon SO2 measurements that was used to calculate the portion of the corrosion rate resulting from adsorption and deposition of all sulfur-based atmospheric contaminants. Similarly, it was assumed that wet chloride deposition measurements were

40 proportional to the total amount of chloride deposition (i.e., wet and dry deposition) so that a single shape function could account for both processes. Details concerning the construction of these shape functions as well as the collection, filtering, analysis, and estimation of environmental data they employ to make predictions are described in the following sections.

Table 2-1 Definition of the Variables and Coefficients used in Equations 2.14 and 2.15 Model Description Units Component 2 Ki Hourly corrosion rate g/cm 2 ACl Scaling factor for the chloride reaction g/cm 2 ASO2 Scaling factor for the SO2 reaction g/cm 2 AO3 Scaling factor for the ozone reaction g/cm CL Temperature adjustment exponent used for the chloride reaction Nondimensional

SO2 Temperature adjustment exponent used for the SO2 reaction Nondimensional O3 Temperature adjustment exponent used for the ozone reaction Nondimensional T Temperature Kelvin (K) Activation energy for the single activation energy formulation H eV/K (Equation 2.14) Activation energy for the chloride reaction in the multiple H eV/K Cl activation energy formulation (Equation 2.15) Activation energy for the SO reaction in the multiple activation H 2 eV/K SO2 energy formulation (Equation 2.15) Activation energy for the ozone reaction in the multiple activation H eV/K O3 energy formulation (Equation 2.15) K Boltzmann constant (=8.6173 x 10-5 eV/K) eV/K Temperature-Relative Humidity shape function for the chloride f (T,RH) Nondimensional Cl reaction. Temperature-Relative Humidity shape function for the SO f (T,RH) 2 Nondimensional SO2 reaction. Temperature-Relative Humidity shape function for the ozone f (T,RH) Nondimensional O3 reaction. Temperature-Contaminant shape function for the chloride fCl(T,CL) reaction. Calibrated using chloride deposition measurements Nondimensional (mass per unit volume of rainwater*)

Temperature-Contaminant shape function for the SO2 reaction. fSO2(T,SO2) Calibrated using hourly gaseous measurements (ppm) measured Nondimensional by automated air pollution monitoring systems. Temperature-Contaminant shape function for the ozone reaction. fO3(T,O3) Calibrated using hourly gaseous measurements (ppm) measured Nondimensional by automated air pollution monitoring systems. *Chloride data used was mass concentration measurements (mg/L of rainwater) made under the National Atmospheric Deposition Program (NADP) [137].

41

2.4 Development of Temperature-Relative Humidity Shape Functions As can be seen from Equations 2.14 and 2.15, there are two types of functions that needed to be developed to expand these notional equations into an actual model. One type pertains to the interaction between temperature and relative humidity while the other describes the interaction between temperature and contaminants. Each of the six total functions was formed through the combination of two equations that were developed. These include a variable temperature function, f (T), and another variable function pertaining to either relative humidity or contaminant levels. These two variable functions, when combined, represent the interaction between temperature and the second acceleration factor (e.g., relative humidity or contaminant). As will be shown later, the combined functions were illustrated by using them to create a 3-D surface that represents every combination of temperature and the second factor within the range established by the model. Such combined functions are referred to as shape functions, which is an expression used in finite element analysis to describe the piecewise distribution of displacements and strains or rotations [138]. It will become apparent when the first function is plotted that the use of this name is appropriate for the current effort.

The primary challenge of this research program was to find acceptable shape functions that could be input into the notional models shown by Equations 2.14 and 2.15. Candidate functions were constructed, calibrated, and statistically tested to determine suitability and accuracy. As will be seen later, the result was a model that provides a proof-of-concept for cumulative corrosion damage prediction based upon stochastic environmental conditions.

The derivation of the simplest (i.e., linear) corrosion shape functions being described in the following pages of this chapter illustrates how such functions were constructed. However, because of their more lengthy derivations, only the final nonlinear shape functions are reported here. Complete mathematical details pertaining to their derivations are presented in Appendix

42

A. All nonlinear shape functions were based upon parabolic equations. These are divided into two different types: “concave up”, which is referred to here as “concave”, and “concave down”, which is called “convex”. There are other nonlinear equations that could have been employed and it is possible, if not likely, that their use could lead to even better results. However, because of the large amount of time and simulations needed to calibrate the initial models, it was not feasible to examine other nonlinear possibilities during this research program.

Table 2-2 Nine Combinations of Functions Used to Develop Candidate T-RH Shape Functions RH Functions Temperature Functions Linear Concave Convex Linear Linear-Linear Concave -Linear Convex -Linear Concave Linear- Concave Concave - Concave Convex - Concave Convex Linear- Convex Concave - Convex Convex - Convex

Table 2-2 identifies three candidate temperature functions and three functions for relative humidity. The combination of these functions provides nine different temperature-relative humidity shape functions that were candidates for the corrosion model. As will be shown later, various analyses were conducted to find which one proved most suitable.

Adsorption of monolayers of water vapor on surfaces is a phenomenon that provides the moisture needed to initiate and sustain corrosive electrochemical processes. This phenomenon is dependent upon temperature (and other factors) so it seems reasonable to assume that one of the nine different Temperature-Relative Humidity shape functions defined by Table 2-2 and calibrated during the simulation process will provide an empirical relationship based, in part, upon moisture adsorption.

2.4.1 Temperature Functions As described above, shape functions used to quantify interactions between acceleration factors employ the use of temperature functions. During the initial stages of the model development

43 process, it was unknown which of the three assumed temperature functions (linear, convex, and concave) would perform best with respect to defining the temperature-relative humidity interaction. Simulations were conducted to identify the optimal function.

Temperature (K) 270 280 290 300 310 320 330 -1.5

-2 f(T)max-1

-2.5

f(T)max-2 f(T) -3 f(T)max-3

-3.5 f(T)max-4

-4

Figure 2-3 Illustration of the Variety of Possible Solutions for f(T)

Figure 2-3 illustrates how a shape function can be found to describe some unknown interaction.

As will be shown later, all shape functions are predicated upon the temperature function, f(T), on which it is based. Thus, the most accurate model will be the one that has the optimum temperature function. This figure demonstrates the breadth of possible interactions that result from applying the three candidate temperature functions (i.e., linear, concave, and convex) to just four different values of f(T)max, which is the quantity that defines their individual characteristics. When billions of potential values for f(T)max for each shape function used by the model are considered while running simulations, optimum functions can be identified that provide the most accurate model. The following derivation of the linear temperature function illustrates how such functions were constructed. The final equations for the nonlinear temperature functions are presented after the derivation of the linear function.

44

2.4.1.1 Linear Temperature Function Figure 2-4 illustrates the linear form of the temperature function, f(T). As discussed previously, the minimum value of -4 for this function occurs at a temperature of 273.15K, which is the freezing point in low chloride environments. This was done so the corrosion rate (determined in-part by taking the exponential of f(T)) is approximately equal to zero at freezing. Conversely, the maximum value considered by the function occurs at a temperature of 320.15K, which is slightly higher than the highest hourly temperature measurement found in the model calibration/validation dataset.

The general form of an equation for a straight line is shown in Equation 2.16, which for this case can be rewritten as shown by Equation 2.17.

y mx  b (2.16)

Tf )( mT  b (2.17)

f(T)

T=273.15K f(T)max

0 T

-4 T=320.15K

Figure 2-4 Linear Function for f(T)

Equation 2.18 is used to calculate the slope of the linear function. Despite the fact that the denominator in this function could be easily reduced to a value of 47, the symbolic

45 representation of 320.15-273.15 was retained throughout the subsequent derivations in case an adjustment to the lower temperature was needed at a later point. Such an adjustment was thought to be possible because there is evidence that corrosion can occur at temperatures below freezing when chloride levels are high [65].

Tf  4)()( Tf )(  4 m  max  max (2.18)  .. 1527315320  .. 1527315320

Please note that the vertical axis shown in Figure 2-4 does not contain the origin of the coordinate system. Instead, the coordinate system is limited to the portion of the x-axis that corresponds to a range of temperatures from 273.15K to 320.15K. Hence, f(T) equal to -4 is not the true y-axis intercept. The f(T) axis intercept, “b”, is calculated through extrapolation as shown in Equation 2.19 and 2.20.

Tf  4)()( )(  bTf max  max (2.19)  .. .153201527315320

Tf max  4 .))(( 15320 Tfb )( max  (2.20)  .. 1527315320

Substituting Equations 2.18 and 2.20 into Equation 2.17 results in the temperature function shown in Equations 2.21 and 2.22.

Tf )( max  4 Tf max  4 .))(( 15320 Tf )(  TfT )( max  (2.21)  .. 1527315320  .. 1527315320

Tf )( max  4 Tf )(  T 15320  Tf )().( max (2.22)  .. 1527315320

46

Equation 2.22 describes the linear relationship between temperature and f(T) that is illustrated on Figure 2-4. This equation was used to define the linear temperature-relative humidity shape functions.

2.4.1.2 Concave Temperature Function As previously mentioned, derivations of the nonlinear temperature functions are presented in

Appendix A. Equation A.30 (shown here as Equation 2.23) represents the concave temperature function, which applies to a temperature range of 273.15K to 320.15K. The coefficients “a”, “b”, and “c” used in this equation are determined using Equations 2.24-2.26.

Tf )( 2 bTaT  c (2.23)

Equation A.26 (shown here as Equation 2.24) is used to determine the value for the coefficient

“c”, which is one of the unknowns in Equation 2.23. Obviously this equation could be simplified by conducting some of the elementary mathematical operations used in its formulation. This was not done so that all computations are conducted using computer precision to minimize round-off errors.

2 2 2 (226.15) (226.15) (273.15 [)  Tf )(]1 max [  Tf )(]1 max (273.15 2 Tf )() (320.15) 2 (320.15) 2 4  max [ ] 273.15 (320.15) 2 (226.15) 2 (226.15) 2 [226.15  ](320.15) 226.15  c  320.15 320.15 (226.15) 2 (226.15) 2 (273.15 2 [)  ]1 [  ]1 (273.15) 2 (320.15) 2 (320.15) 2 [ [ ] 273.15  ]1 (320.15) 2 (226.15) 2 (226.15) 2 [226.15  ](320.15) 226.15  320.15 320.15 (2.24)

Explicit relationships for coefficients “a” and “b” were not developed because of the complexity of the equation used to define the coefficient “c”. Thus, values for coefficient “b” are obtained

47 by first numerically solving for “c” using Equation 2.24 and then using the precise result to obtain a value for “b” using Equation A.17 (seen here as Equation 2.25).

15226 2).( [ 1] 15320 ).( 2 b   Tfc max))(( (2.25) 15226 2).( .[ 15226  ] .15320 The results from calculations made using Equations 2.24 and 2.25 are then used by Equation A.8

(seen here as Equation 2.26) in order to solve for the unknown coefficient “a”. The combination of Equations 2.23-2.26 provides the desired concave temperature function.

)(  .15320 bcTf a  max (2.26) 15320 ).( 2

2.4.1.3 Convex Temperature Function As with the concave temperature function, the derivation of the convex function is also presented in Appendix A. Equation A.70 (shown here as Equation 2.27) represents this function, which as before applies to a temperature range of 273.15K to 320.15K. The coefficients “a”, “b”, and “c” used in this equation were determined using Equation A.67, Equation A.62, and

Equation A.68 (shown here as Equations 2.28-2.30).

2 4  Tcabb )( Tf )(  (2.27) 2a

 1527315320 )..(  152731532016   152731532064 )..(][f(T))..(   max a 2 3 2 (2.28) max)f(T) 16 2 max  24 max 96f(T))f(T)()f(T)([ max  128][ max)(f(T)  4)]

48

 152731532016   152731532064 )..(][f(T))..(  max b 3 2 (2.29) 2 max  24 max 96f(T))f(T)()f(T)( max  128

 1527315320 )..(  152731532016   152731532064 )..(][f(T))..( c  16[  max ] )f(T) 2 16 2 3  24 2 96f(T))f(T)()f(T)([  128][ )(f(T) 2  4)] max max max max max (2.30)  152731532016   152731532064 )..(][f(T))..(  Max  4[ 3 2 .] 15273 2 max  24 max 96f(T))f(T)()f(T)( max  128

2.4.2 Relative Humidity Functions As will be seen later, temperature functions such as those derived above combine with relative humidity functions to form temperature-relative humidity shape functions. Thus, the first function will define the temperature dependence of the yet unknown shape function while the second, which is based upon functional values obtained from the first, will define the associated relative humidity function.

There are two boundary conditions that must be considered when constructing relative humidity functions. The first concerns the threshold RH value of 60% as determined by Vernon

[66]. What this implies is that corrosion will only occur if the relative humidity exceeds this value. The second boundary condition relates to the temperature function, where the value of the relative humidity function at 100% RH is equal to the value of the temperature function at that temperature. These conditions are illustrated for the linear case in Figure 2-5.

2.4.2.1 Linear Relative Humidity Function Figure 2-5 displays a simple figure depicting the linear relative humidity function. As will be shown later, this function can be used with either of the linear, concave, or convex temperature functions in order to examine whether any of the resultant shape functions adequately describe the interactions between temperature and relative humidity. Similar to before, this function has a value of -4 when the lower relative humidity boundary condition (i.e., the threshold RH) is

49 met. Conversely, the value of this new function is equal to f(T) when the relative humidity is

100%. Equation 2.31 was used to derive the equation that describes Figure 2-5. The slope for this equation is subsequently calculated using Equation 2.32. The variable RHTH was used throughout this derivation rather than using a constant value of 60% RH as described by Vernon.

This was done to facilitate simulations described later to determine whether considering the threshold RH as a function of temperature would have any effect on model accuracy.

f(RH,T)

RH=RHTH f(T)

0 RH

-4 RH=100%

Figure 2-5 Linear Relative Humidity Function

Tf RH),(  bmRH (2.31)

Tf  4)()( Tf )(  4 m   (2.32) 1 RHTH 1 RHTH

Like the previous discussion for the temperature function, Figure 2-5 also does not show the origin of the coordinate system but instead is limited to the portion of the relative humidity axis that extends from the threshold value of RH to 100% RH. Equations 2.33 through 2.35 are used to derive the equation that describes the actual intercept value needed by Equation 2.31.

Tf  4 )()()(  bTf  (2.33) 1 RHTH  01

50

Tf )(  4 )( bTf  (2.34) 1 RHTH

Tf  4))(( Tfb )(  (2.35) 1 RHTH

Substituting Equations 2.32 and 2.35 into Equation 2.31 results into the relative humidity function shown by Equation 2.36, which can be simplified into Equation 2.37. When combined with a specific temperature function (e.g., the linear temperature function, Equation 2.22), the result is the Linear Temperature-Linear Relative Humidity shape function, which is illustrated by

Figure 2-6.

Tf )(  4 Tf  4))(( Tf RH),(  RH Tf )(  (2.36) 1 RHTH 1 RHTH

Tf )(  4 Tf RH),(  RH 1  Tf )()( (2.37) 1 RHTH

Figure 2-6 shows the results of applying the full range of applicable temperature and relative humidity values to the shape function defined by Equations 2.22 and 2.37 for the example of f(T)max=-2.0. The surface indicated by the 3-D shape shown on this figure contains numerical values for f(T, RH) pertaining to every possible combination of temperature and relative humidity within the bounds set during the formulation of the equations. As can be seen in this figure, the value of f(T, RH) is -4 when either the temperature is 273.15K or the relative humidity is 60%. Thus, each of the lower boundary conditions has been properly addressed. The inclined line on this figure indicates the temperature function (Equation 2.22), which terminates on the right end at the point indicated as f(T)max. The value of this point will be varied during the

51

Monte Carlo simulation process to find the optimum solution that provides the lowest model error.

0 f(T) f(T)max -0.5 -1 -1.5 -2 -2.5 f(T,RH) 100 -3 90 -3.5 80 -4 70

60

Temperature Temperature (K) Boundary Condition Relative Humidity Boundary Condition

Figure 2-6 Linear Temperature-Linear Relative Humidity Shape Function (f(T)max=-2)

As shown by Equations 2.14 and 2.15, the function f(T, RH) was not directly used by the model.

Instead, an exponential function was applied to values calculated by Equations 2.22 and 2.37.

Figure 2-7 illustrates the 3-D surface that results from applying the exponential function to all possible combinations of temperature and relative humidity used to construct Figure 2-6. Note that the minimum value shown on this figure is not precisely zero, which is a necessary condition in order for there to be no corrosive activity if the temperature falls to 273.15K or the relative humidity is at or below the threshold of 60%.

52

0.14 0.12 0.1 0.08

[f(T,RH)] 0.06

xp 0.04 e 0.02 90 0 75

60 Temperature (K)

Figure 2-7 Exponential Function Applied to the Linear Temperature-Linear RH Shape Function (f(T)max=-2)

Earlier during the development of the notional corrosion model, it was stated that an

“adjustment” would be necessary because the exponential function, when applied to a value of

-4, does not have a precise value of zero. When looking at Figure 2-7, the minimum value shown for the surface is precisely equal to the value of e-4 (see the table shown on Figure 2-2).

Thus, Equations 2.14 (single activation energy) and 2.15 (multiple activation energies) were adjusted to account for this discrepancy. Equations 2.38 and 2.39 show such adjustments.

 H     Ki  AT exp  Cl Tf RH  exp(,({[(exp  4 Tf Cl  exp(,())(exp 4))]   kT  (2.38) SO2 Tf RH  exp(,([(exp  4 Tf SO2  exp(],())(exp 4)) 

O3 Tf RH  exp(,([(exp  4 OTf 3   4))]}exp(,())(exp

53

CL  HCl  i  ClTAK exp  Cl Tf RH  exp(,([exp  4 Tf Cl  exp(,()][exp  4)]  kT 

SO2  HSO2  SO TA exp  SO Tf RH  exp(,([exp  4 Tf SO  exp(,()][exp 4)]  (2.39) 2  kT  2 2

O3  HO3  O TA exp  O Tf RH  exp(,([exp  4 OTf  exp(,()][exp  4)] 3  kT  3 3

Figure 2-8 illustrates such an adjustment to the shape function used to construct Figure 2-6. As can be seen, the adjusted value is now precisely zero when the relative humidity is at the 60% threshold value or the temperature is at 273.15K.

0.12 0.1 0.08

adjusted value adjusted 0.06 – 0.04 0.02 90

[f(T,RH)] [f(T,RH)] 0 75 xp e 60

Temperature (K)

Figure 2-8 Adjusted Exponential Function Applied to the Linear Temperature-Linear RH Shape Function (f(T)max=-2)

2.4.2.2 Concave Relative Humidity Function Equation A.74 (shown here as Equation 2.40) is used to calculate the concave relative humidity function. As before, this function was used with the aforementioned linear, concave, or convex temperature functions to examine whether any of the resultant shape functions adequately described the interactions between temperature and relative humidity.

54

2  TfcbRHaRH RH),( (2.40)

The model coefficient “c” used in this equation is calculated by Equation A.104 (seen here as

Equation 2.41). Obviously, a constant value of 60% RH could be input into this equation and then the numerous elementary mathematical operations performed to simplify the expression.

However, this form of the equation was retained to enable future flexibility in examining different approaches pertaining to the threshold RH. Instead, an algorithm based upon

Equation 2.41 used computer numerical precision to calculate an explicit value for “c” (for each simulation run), which was then used in Equation A.95 (seen here as Equation 2.42) to calculate a value for “a” and in Equation A.91 (seen here as Equation 2.43) to calculate “b”.

4 2 4 Tf )((  )( RHTH 12 ) Tf )((  )( RHTH 12 ) RH 2 4 RH 2 Tf )(  TH  RH 12 )( 2  TH 1 RH 2 TH 1 RH 1 )( TH 1 )( TH RH RH c  TH TH (2.41) 1 2 1 (1 2 )( RHTH 12 ) (1 2 )( RHTH 12 ) RHTH 1 2 RHTH [  2 RHTH 12 )(  1] 1 RHTH 1 RHTH 1 )( 1 )( RHTH RHTH

4 1 Tf )( c 1 )( RH 2 RH 2 4 c  TH TH  a ( ) 2 2 (2.42) 1 RHTH RHTH RHTH 1 )( RHTH

55

4 1 Tf )( c 1 )( RH 2 RH 2 b  TH TH (2.43) 1 1 )( RHTH

2.4.2.3 Convex Relative Humidity Function Equation A.148 (shown here as Equation 2.44) was used to calculate the convex relative humidity function. As before, this function can be used with either of the linear, concave, or convex temperature functions that were derived previously.

2 4 cabb  RH)( Tf RH),(  (2.44) 2a

Unlike the concave temperature-relative humidity function described earlier, explicit equations were derived for each of the three modeling coefficients used by Equation 2.44. Specifically,

Equation A.144 (seen here as Equation 2.45) was used to calculate the value for “a” while

Equation A.139 (Equation 2.46 here) was used to calculate “b” and “c” was determined using

Equation A.145 (Equation 2.47).

1RH )( 16f(T)[ 64]1 RH )( a  TH  TH (2.45) f(T) 2 16 [2f(T) 3 2 f(T)f(T) 1289624 ][  4]f(T)

16f(T)( 64)1 RH )( b  TH (2.46) 2f(T)3 2 f(T)f(T)  1289624

56

1RH )( 16f(T)[ 64]1 RH )( c  16[ TH  TH ] f(T) 2 16 [2f(T) 3 f(T) 2 f(T) 1289624 ][  4]f(T) (2.47) 16f(T)( 64)1 RHTH )( 4  RHTH 2f(T)3 f(T) 2 f(T)  1289624

2.4.3 Illustration of Temperature-Relative Humidity Shape Functions The 3-D surface shown on Figure 2-6 illustrates the full range of interactions for the linear temperature-linear relative humidity shape function. However, as seen in Table 2-2, there are eight other temperature-relative humidity shape functions that were also considered as candidates for the corrosion model. The following sections describe how each of these eight shape functions was constructed, which are then illustrated by Figures 2-9 through 2-16. Please note that each shape function was constructed with a maximum value of the temperature function, f(T)max, equal to -2.0 for illustration purposes only. During simulations to determine the optimum function, they were adjusted using the process described in Equations 2.38 and

2.39 and illustrated by Figures 2-7 and 2-8.

The 3-D graphs shown on Figures 2-7 through 2-16 were developed using Microsoft Excel.

Inspection of these figures reveals that the increment between 273.15K and 275K has the exact same scale as the increment from 275K and 280K (and every other five degree increment). This creates the illusion that the temperature functions used to create Figures 2-9, 2-10, and 2-16 are discontinuous. This is not the case but instead is an artifact resulting from the way the data was plotted.

57

2.4.3.1 Relative Humidity Shape Functions Based on Linear Temperatures As discussed previously, Equation 2.22 was used to calculate the linear temperature function.

This function was combined with three different relative humidity functions to obtain three temperature-relative humidity shape functions. The linear temperature-linear relative humidity shape function was defined previously as the combination of Equation 2.22 and Equation 2.37.

This scenario was illustrated by Figure 2-6. The remaining two shape functions based upon the linear temperature function are described below.

2.4.3.1.1 Linear Temperature-Concave Relative Humidity Shape Function Equations 2.40 through 2.43 were combined with Equation 2.22 to define the complete linear temperature-concave relative humidity shape function. Figure 2-9 illustrates the 3-D surface pertaining to the full range of interactions provided by this function.

2.4.3.1.2 Linear Temperature-Convex Relative Humidity Shape Function Equations 2.44 through 2.47 were combined with Equation 2.22 to define the complete linear temperature-convex relative humidity shape function. Figure 2-10 illustrates the 3-D surface pertaining to the full range of interactions provided by these functions.

2.4.3.2 Relative Humidity Shape Functions Based on Concave Temperatures Equations 2.23 through 2.26 were used to calculate the concave temperature function. As illustrated below, this function was combined with three different relative humidity functions to provide three shape functions as candidates for the corrosion model.

2.4.3.2.1 Concave Temperature-Linear Relative Humidity Shape Function The linear relative humidity function defined by Equation 2.37 in conjunction with the concave temperature function described above was used to construct the concave temperature-linear relative humidity shape function. Figure 2-11 illustrates the 3-D surface pertaining to the full range of interactions provided by these functions.

58

2.4.3.2.2 Concave Temperature-Concave Relative Humidity Shape Function The concave relative humidity function defined by Equations 2.40 through 2.43 in conjunction with the concave temperature function described above was used to construct the concave temperature-concave relative humidity shape function. Figure 2-12 illustrates the 3-D surface pertaining to the full range of interactions provided by these functions.

2.4.3.2.3 Concave Temperature-Convex Relative Humidity Shape Function The convex relative humidity function defined by Equations 2.44 through 2.47 in conjunction with the concave temperature function described above was used to construct the concave temperature-convex relative humidity function shape. Figure 2-13 illustrates the 3-D surface pertaining to the full range of interactions provided by these functions.

2.4.3.3 Relative Humidity Shape Functions Based on Convex Temperatures Equations 2.27 through 2.30 were used to calculate the convex temperature function. As illustrated below, this function was combined with three different relative humidity functions to form three additional shape functions as candidates for the corrosion model.

2.4.3.3.1 Convex Temperature-Linear Relative Humidity Shape Function The linear relative humidity function defined by Equation 2.37 in conjunction with the convex temperature function described above was used to construct the convex temperature-linear relative humidity shape function. Figure 2-14 illustrates the 3-D surface pertaining to the full range of interactions provided by these functions.

2.4.3.3.2 Convex Temperature-Concave Relative Humidity Shape Function The concave relative humidity function defined by Equations 2.40 through 2.43 in conjunction with the convex temperature function described above was used to construct the convex temperature-concave relative humidity shape function. Figure 2-15 illustrates the 3-D surface pertaining to the full range of interactions provided by these functions.

59

2.4.3.3.3 Convex Temperature-Convex Relative Humidity Shape Function The convex relative humidity function defined by Equations 2.44 through 2.47 in conjunction with the convex temperature function described above was used to construct the convex temperature-convex relative humidity shape function. Figure 2-16 illustrates the 3-D surface pertaining to the full range of interactions provided by these functions.

f(T)

0 f(T) -0.5 max -1 -1.5 -2 f(T,RH) -2.5 100 -3 90 -3.5 80 -4 70

60 Temperature Temperature (K) Boundary Condition Relative Humidity Boundary Condition

Figure 2-9 Linear Temperature-Concave RH Shape Function - f(T)max=-2

60

f(T)

0 f(T)max

-1

-2

f(T,RH) -3

-4 90

-5 75

60 Temperature Temperature (K) Boundary Condition Relative Humidity Boundary Condition

Figure 2-10 Linear Temperature-Convex RH Shape Function - f(T)max=-2

f(T) f(T)max

0

-1

-2

-3 f(T,RH) -4 90

-5 75

60

Temperature (K) Temperature Relative Humidity Boundary Condition Boundary Condition

Figure 2-11 Concave Temperature-Linear RH Shape Function - f(T)max=-2

61

f(T) f(T)max

0 -0.5 -1 -1.5 -2 -2.5 f(T,RH) -3 100 -3.5 90 -4 80 -4.5 70

60

Temperature Boundary Condition Temperature (K) Relative Humidity Boundary Condition

Figure 2-12 Concave Temperature-Concave RH Shape Function - f(T)max=-2

f(T) f(T)max

0 -0.5 -1 -1.5 -2 -2.5

f(T,RH) -3 100 -3.5 90 -4 80 -4.5 70

60

Temperature Temperature (K) Relative Humidity Boundary Condition Boundary Condition

Figure 2-13 Concave Temperature-Convex RH Shape Function - f(T)max=-2

62

f(T) f(T)max

0 -0.5 -1 -1.5 -2

f(T,RH) -2.5 -3 -3.5 90 -4 75

60

Temperature (K) Temperature Boundary Condition Relative Humidity Boundary Condition

Figure 2-14 Convex Temperature-Linear RH Shape Function - f(T)max=-2

f(T)

f(T)max

0 -0.5 -1 -1.5 -2

f(T,RH) -2.5 -3 100 90 -3.5 -4 80 70

60

Temperature Temperature (K) Boundary Condition Relative Humidity Boundary Condition

Figure 2-15 Convex Temperature-Concave RH Shape Function - f(T)max=-2

63

f(T) f(T)max

0 -0.5 -1 -1.5 -2

f(T,RH) -2.5 100 -3 90 -3.5 80 -4 70

60

Temperature (K) Temperature Relative Humidity Boundary Condition Boundary Condition

Figure 2-16 Convex Temperature-Convex RH Shape Function - f(T)max=-2

2.5 Development of Temperature-Contaminant Shape Functions As seen in Equations 2.14 and 2.15, three different shape functions were used to represent the interaction between temperature and SO2, chloride, and ozone. These functions were designed to account for the influence that contaminant deposition rates has upon the overall corrosion rate.

Deposition of gaseous substances onto surfaces is a complicated matter with many factors working together to dictate the rate at which the substances adsorb onto surfaces. Wu et al. report the rate of dry deposition of gases onto surfaces is dictated by three steps, the first of which is aerodynamic transport [139]. This mechanism pertains to the movement of contaminants via airflow. After contaminants are brought near a structure, they must then be transported across the boundary layer adjacent to its surface. The final step involves possible

64 physical and/or chemical interactions that occur when the contaminant comes in contact with the actual surface [139].

Wen and Kasper report that “reentrainment” (i.e., desorption) of a deposited substance is a process based upon a competition between dynamic lift forces such as that due to fluid flow and the adhesive forces that bond the substances to surfaces [140]. Such adhesion is reportedly due to intermolecular, Coulomb, or some other interactive forces between the particles and the surface. Braaten et al. identify particle “resuspension” as a “very complex mechanism even under very controlled conditions” [141]. They further report the fluid flow over the surface combined with surface roughness and physical characteristics of the particles are factors that promote or resist resuspension. Wu et al. indicate that the rate of desorption is dependent on many variables including wind speed, particle size, relative humidity, and the amount of time that the particles dwell on the surface [139].

As discussed above, adsorption/desorption processes are based upon many factors that are difficult to quantify. Not mentioned among these factors is the growth of corrosion products on surfaces and the change in surface morphology that results. Addressing such complexity would require additional data (assuming it were available) and mathematical algorithms that would further increase model complexity and the computational requirements imposed by the Monte

Carlo simulation process. Thus, an approach to infer the approximate amount of gaseous contaminants present on material surfaces was needed.

Rather than consider adsorption/desorption mechanisms explicitly, the cumulative damage model was calibrated by simply considering hourly levels of SO2 and ozone present at the calibration and validation sites. An evaluation of accuracy following model calibration and

65 validation was used to determine whether such an approach was suitable. These pollutant levels were applied to the model by using Linear Temperature-Linear Contaminant shape functions.

The temperature-contaminant shape functions do not have a threshold condition such as exists for relative humidity. Thus, the two boundary conditions used in its formulation concern the assumption that the corrosion rate equals zero when either the temperature drops to 273.15K

(as before) or the contaminant level falls to zero.

As specified by Equations 2.14 and 2.15, three temperature contaminant shape functions were used, one for each of the contaminants considered by the model. Nonlinear temperature- contaminant functions could possibly improve upon model accuracy but they were not considered due to the extensive time needed to conduct the necessary simulations using the available computer hardware. As will be seen later, the combination of the optimal nonlinear temperature-nonlinear relative humidity shape functions with linear temperature-linear contaminant shape functions appears to possess sufficient accuracy. The linear temperature function defined by Equation 2.22 (shown below as Equation 2.48) was used as the basis for the linear temperature-linear contaminant shape function.

Tf )( max  4 Tf )(  T 15320  Tf )().( max (2.48)  .. 1527315320

2.5.1 Linear Temperature-Linear Chloride Shape Function Constructing the linear temperature-linear chloride shape function followed the same general process as used during the derivation of the linear temperature-linear RH function. Constructing the function began by first determining the range of atmospheric contaminant values that was considered by the model. Chloride deposition data used for this research program was measured using chemical analysis of rainwater. As will be discussed later, there were initially

66 four different locations where chloride deposition data was obtained to calibrate the model being developed here. Based upon an analysis of this data, the highest deposition rate within range of the model was 3.2 mg/L. This value was selected so that all data used during the calibration process was within model parameters.

Figure 2-17 illustrates the linear chloride function. Like before, the minimum value of this function equals -4 when the contaminant level equals zero, whether or not such a low value is physically possible.

f(Cl,T)

Cl=0 f(T)

0 Cl (mg/L)

-4 Cl=3.2

Figure 2-17 Linear Chloride Function

The line shown by Figure 2-17 is defined by Equation 2.49. The slope of this linear function is calculated using Equation 2.50 and the vertical axis intercept is as shown on the figure.

Tf Cl),(  bmCl (2.49)

Tf  4)()( Tf )(  4 m   (2.50) .  023 .23

67

Substituting Equation 2.50 and the f(T,Cl) axis intercept of -4 into Equation 2.49 results in the linear temperature-linear chloride shape function shown by Equation 2.51. As before, f(T) was calculated using Equation 2.48. Figure 2-18 illustrates this shape function.

Tf  4)( Tf Cl),(  Cl  4 (2.51) 2.3

f(T)

f(T)max 0 -0.5 -1 -1.5 -2 T,Cl)

f( -2.5 -3 3.2 2.4 -3.5 -4 1.6 0.8

0 Temperature Boundary Condition Temperature (K) Chloride Boundary Condition

Figure 2-18 Linear Temperature-Linear Chloride Shape Function - f(T)max=-2

2.5.2 Linear Temperature-Linear SO2 Shape Function

The linear temperature-linear SO2 shape function has the exact form as the chloride function showed by Equation 2.51 with the exception that it is scaled to the maximum SO2 levels possible under exposure conditions. Gaseous analyzers are used to measure SO2 pollution levels, which are reported in units of parts-per-million (ppm) or parts per billion (ppb). An analysis of the databases reflecting environmental conditions at the 14 locations used to calibrate and validate the model reveals that a value of 0.15 ppm exceeded all measurements. Thus, Equation 2.51

68 was updated as shown by Equation 2.52. This function, when combined with Equation 2.48, defines the linear temperature-linear SO2 shape function.

Tf  4)( Tf SO ),(  SO  4 (2.52) 2 .0 15 2

2.6 Linear Temperature-Linear Ozone Shape Function The linear temperature-linear ozone shape function was developed in the same way as the SO2 function. In this case, the maximum ozone value, which was determined via an analysis of the calibration and validation data, was specified as 0.38 ppm. Equation 2.53 along with Equation

2.48 defines this shape function.

Tf  4)( OTf ),(  O  4 (2.53) 3 .0 38 3

2.7 Identification of Optimal Shape Functions The previous sections describe the various candidate Temperature-Relative Humidity and

Temperature-Contaminant shape functions that were considered for use in the final model.

These shape functions were used to create a series of candidate models by replacing the notional functions seen in Equations 2.14 and 2.15. When employing the aforementioned shape functions in the model, there were numerous unknown coefficients that needed to be quantified using simulations. Three of these coefficients pertain to the maximum values of each

Temperature-Relative Humidity shape function employed by the model while another three are the maximum values for the three related Temperature-Contaminant shape functions. As seen in Equation 2.14 and 2.15, the unknown coefficients also include three scaling factors (ACl, ASO2, and AO3) and three temperature adjustment terms (CL, SO2, and O3). The remaining unknown is the activation energy, H. This coefficient was systematically set at various constant values while Monte Carlo simulations were used to apply random numbers in place of the twelve other

69 unknown coefficients. Eventually, acceptable values for each unknown were determined so that each candidate model achieved optimal accuracy.

70

CHAPTER 3

COLLECTION, ESTIMATION, FILTERING, AND SYNTHESIS OF PROXY DATA

Characterization methods that scavenge contaminants from the atmosphere, such as those discussed in Section 1.2.5, are incapable of quantifying short-term variability, which is assumed under this effort to be a contributing factor responsible for variable corrosion rates. In addition, such techniques are not routinely employed under large-scale environmental monitoring programs. As a result, the environmental characterization data measured using these techniques are generally unavailable. Constructing the cumulative corrosion damage model described here required a different approach to measuring atmospheric contaminant levels.

The intent of this project is to lay the groundwork for a future design methodology capable of providing affordable, timely, and accurate corrosion predictions for locations where a structure might be built. In order to meet this objective, the model cannot be based upon costly and time-intensive environmental characterization tests conducted at a specific location of interest.

In comparison to past approaches, this current effort takes an entirely different tack by assuming that accurate corrosion rate predictions can be made using a model that considers

(proxy) characterization data measured at other locations with similar environmental conditions. The cumulative damage model being developed here is specifically intended to

71 consider the impact of diurnal temperature cycling in combination with associated changes in relative humidity as well as stochastic atmospheric contaminant levels. Thus, hourly measurements of gaseous contaminants combined with longer-term deposition of chloride aerosols were used to calibrate/validate the various candidate models. It should be noted that the types of data used to quantify gaseous contaminants (ppm or ppb) are quite different than the rainwater concentration measurements (mg/liter) used to describe wet chloride deposition.

Such differences in data types are accounted for in the modeling coefficients calculated during the model calibration process.

3.1 Calibration and Validation Data Two different sets of data were used to develop and evaluate the atmospheric corrosion model described here. One of these sets was employed to calibrate the candidate models while the other was used to independently validate their accuracy. Both data sets contained corrosion test results and proxy weather/atmospheric contaminant measurements.

3.1.1 Selection of Calibration and Validation Test Sites In order to properly quantify interactions between the different acceleration factors considered by the model, calibration data must contain high and low values for each particular environmental acceleration factor. Such diversity in environmental conditions is needed so the final calibrated model can make accurate predictions for a wide variety of locations. An analysis of the sites where corrosion tests were conducted under the SERDP and earlier Battelle programs was used to identify locations suitable for calibration purposes. It was recognized early on that high chloride deposition rates in areas adjacent to coastal surf zones was a factor that would significantly complicate the development of the proof-of-concept corrosion model.

As such, the location chosen to provide high chloride deposition rates during the calibration process was a site five miles inland at Kennedy Space Center, FL. Therefore, the model

72 developed here is limited to making predictions five miles or more away from surf zones, which of course represents the vast majority of the area in the United States. Further (follow-on) work is needed to enhance the model developed here so that it is capable of making predictions for areas with extremely high chloride deposition rates.

As shown on Table 3-1, four different calibration sites were initially chosen so that high and low values for relative humidity and temperature would be considered during the model development effort. These same four sites also provided a diverse range of values for the atmospheric contaminants.

Table 3-1 Calibration Test Sites and Associated Climate Zones Corrosion Test Climate Zone Conditions Site Hot, infrequent high humidity, low SO China Lake, CA Hot - Dry 2 and chlorides Warm summer, cool winters, humid, high Dobbins ARB, GA Mixed - Humid SO2, medium chlorides Cold, dry winters (low absolute humidity), Ft. Drum, NY Cold cool and humid summers, low SO2 and chlorides Kennedy Space Warm, humid, coastal (very high chloride Hot - Humid Center, FL levels)

Under earlier work, Battelle conducted corrosion tests at three different locations at Kennedy

Space Center: on the beach, ¼ mile inland, and five miles inland [25]. The test data used for this effort came from the five mile inland site. This was done to avoid the effect of larger chloride aerosol sizes that are present at sites closer to the beach [73, 74]. The additional complexity resulting from highly variable chloride deposition near surf zones could be addressed in an enhanced model based upon this work.

73

After each of the initial candidate models were calibrated, they were applied to environmental characterization data reflecting conditions at ten other locations. Predictions made using this data were then statistically compared to corrosion test data in order to independently validate model accuracy. Table 3-2 identifies these additional locations and associated climate zones.

Figure 3-1 illustrates the locations of the calibration and validation sites in comparison to various climate zones in the United States [142].

Table 3-2 Validation Test Sites and Associated Climate Zones Corrosion Test Site Climate Zone Kirtland AFB, NM Mixed Dry Ft. Hood, TX Hot – Humid Tyndall AFB, FL Hot – Humid Daytona Beach, FL Hot – Humid Ft. Rucker, AL Hot – Humid Ft. Campbell, KY Mixed - Humid Rock Island Arsenal, IL Cold Wright Patterson AFB, OH Cold West Jefferson, OH Cold Point Judith, RI Cold

Calibration Sites Validation Sites

Figure 3-1 Model Calibration and Validation Test Sites in Comparison to Climate Zones [142]

74

3.1.2 Corrosion Rate Measurements Constructing an accurate corrosion prediction methodology requires the use of corrosion rate measurements (weight loss per increment of time) to both calibrate and validate the model. To eliminate the time and expense associated with conducting corrosion tests at numerous locations, corrosion rates measured under other programs were used for this effort. One source of such test data was an AFRL program sponsored by SERDP and supported by the University of

Dayton and Battelle [136]. Data from this ongoing effort was used to supplement test data measured earlier by Battelle [25]. As will be discussed below, common test protocols and materials were used during all of the tests, thus providing a consistent set of weight loss data for modeling purposes.

3.1.2.1 Corrosion Test Data

Figure 3-2 Battelle Corrosion Test Fixture [Photo Courtesy of Battelle]

The cumulative damage corrosion model developed here was calibrated and later validated using weight loss measurements obtained from exposure tests. As seen in Figure 3-2, these tests

75 involved the use of small, inexpensive test fixtures, which were used to hold four cards. Figure

3-3 displays one such card, which contained specimens of 99.99% silver, 99.99% copper, three different alloys of aluminum (2024, 6061, and 7075), and one specimen of steel (AISI 1010).

Figure 3-3 Battelle Corrosion Test Specimen Card. (L-R): 99.99% silver, 99.99% copper, three different alloys of aluminum (2024, 6061, and 7075), and one specimen of steel (AISI 1010). [Photo Courtesy of Battelle]

Battelle’s test protocol began by cleaning and precisely weighing each specimen before installing it on a nonconductive card using nonconductive fasteners. Four cards were placed in each test fixture, which were then installed in the field. Single cards were later removed from each fixture every three months (in most cases) and sent to Battelle’s laboratory in Columbus,

Ohio. The individual specimens were then removed from the card, cleaned of corrosion products, and precisely weighed to obtain the specific corrosion rates (in g/cm2 weight loss) for each particular exposure period. The results are a series of weight loss measurements for each alloy. Tables B-1 and B-2 in Appendix B contains the complete set of corrosion weight loss measurements and exposure intervals for the 14 different locations used to calibrate and validate the candidate models developed under this research effort.

76

3.1.3 Environmental Characterization The environmental data used during this effort was measured under existing government funded efforts. An important thing to note is this information should be considered as “dirty data” due to the fact that each of the individual atmospheric contaminant and weather datasets had missing data points. As will be discussed later, some of this missing information was extensive. Weather data had the added problem that in many cases, there were numerous multiple observations for particular hourly increments of time. Prior to conducting simulations during the calibration process or making predictions to validate model accuracy, a detailed analysis of each dataset was conducted to identify anomalies, estimate missing data, and filter out duplicate observations.

Ideally, the proxy data used to calibrate the models and later validate their accuracy would be complete and accurate. However, this was not possible when using available data measured for other purposes. Thus, it is assumed that even though individual data files have known inaccuracies, when they are combined with large amounts of other data (also with some inaccuracies), the composite information is sufficiently accurate to infer the environmental severity at locations of interest. Statistical testing of the predictions for the calibration and validation locations was used to determine whether such an assumption has merit.

3.1.3.1 Atmospheric Characterization Data The cumulative damage model developed here makes hourly incremental predictions for the first year of exposure. Thus, hourly atmospheric contaminant measurements for the same time and location of the calibration and validation corrosion tests were needed. As mentioned previously, the contaminants considered by the corrosion model include chloride aerosols and two gaseous substances: sulfur dioxide and ozone. Most of the hourly SO2 and ozone measurements were obtained from the Air Quality System (AQS) database provided by the U.S.

77

Environmental Protection Agency (EPA) [143] but some data came directly from other federal and state environmental organizations.

3.1.3.1.1 Sulfur Dioxide Data

Figure 3-4 Illustration of SO2 Monitoring Sites across the United States [144]

As seen on Figure 3-4, there are literally hundreds of locations across the United States where hourly SO2 levels are measured in support of pollution monitoring programs. Gaseous analyzers are placed at each of these locations, where they sample air to quantify pollution levels. The resultant data is typically reported as either ppm or ppb. Data measured at most of these sites is uploaded into the EPA AQS system, which contains a nationwide historical record of hourly monitor readings that date back to 1993. However, there are many locations where SO2 levels are monitored (using AQS standards and nomenclature) but whose measurements are not uploaded to the AQS by the sponsoring agencies. For this effort, data from the AQS was supplemented with data (and in some cases supporting information and advice) obtained

78 directly from the Florida Department of Environmental Protection, the Georgia Department of

Natural Resources, the Kentucky Department for Environmental Protection, the Massachusetts

Department of Environmental Protection, the National Park Service, the New Mexico

Environmental Department, and the New York State Department of Environmental

Conservation.

Table 3-3 Proxy SO2 Monitor Locations Type of Site Locations

Calibration Trona, CA (for China Lake, CA) Atlanta and Stilesboro, GA (for Dobbins, GA) Old Forge, NY (for Fort Drum, NY) Winter Park, FL (for Kennedy Space Center, FL) Validation Winter Park, FL (for Daytona Beach, FL) Mammoth Cave National Park, KY (for Fort Campbell, KY) Longview, TX (for Fort Hood, TX) Columbus, GA (for Fort Rucker, AL) Shiprock, NM (for Kirtland AFB, NM) Fall River, MA (for Point Judith, RI) Davenport, IA (for Rock Island Arsenal, IL) Columbus, GA (for Tyndall AFB, FL) Fairborn, OH (for West Jefferson, OH) Fairborn, OH (for Wright Patterson AFB, OH)

Using the aforementioned information sources, an analysis of available SO2 data was conducted to identify specific monitor sites to serve as suitable proxies for each of the 14 corrosion test

79 sites identified in Tables 3-1 and 3-2. The locations for the SO2 monitor sites used to construct the proxy records are identified in Table 3-3. Specific details for each of these sites are found in

Appendix B (Tables B-2 through B-15 and Figures B-1 through B-14).

3.1.3.1.1.1 Problems with SO2 Proxy Data

Automated pollution monitoring equipment can go off-line for numerous reasons such as power failures, vandalism, and maintenance (among many other things). In addition, monitoring programs at some locations have been ongoing for years while other efforts come and go in response to changing priorities at the individual monitoring agencies. As a result, there were numerous gaps of various sizes in the SO2 data records used to calibrate and validate the models. Each candidate model was specifically constructed to make predictions based upon an entire year of hourly observations. As a result, the missing data had to be estimated and merged with actual observations to complete the proxies used for this research.

Each gap in the SO2 records was individually analyzed to develop an approach to estimate the missing data. The processes used to conduct these estimations were based upon the size of the data holes. Estimating values for the unknown data certainly induced some error that influenced model accuracy. It was assumed that that if the estimates for the missing data were based upon consideration of actual data adjacent to these “gaps”, then such errors could be minimized. The specific procedures that were employed to make these estimates are discussed below. The data analysis and reconstruction process used for this effort was conducted by hand, which required a detailed analysis of the 14 individual proxy records.

SHORT PERIODS OF MISSING DATA: Inspection of the various SO2 databases used in this effort revealed that small periods of observations (e.g., ranging from an hour to a few hours) were commonly missing. In such cases, the missing data was replaced with estimates based upon

80 adjacent measurements. For example, if one data point was missing, it could be replaced with a value approximately equal to the average value of the two points adjacent to the hole or it could have the same value of one of these points. The selection of which approach to take was decided upon on an individual basis.

In more isolated cases, data pertaining to longer periods of time ranging from a few hours to a few weeks were missing. In such cases, blocks of hours from adjacent periods of time with complete observations were used to replace the missing data. In some instances, data on the ends of the replacement blocks were adjusted to be of consistent magnitude with the actual adjacent measurements in the complete proxy.

LONG PERIODS OF MISSING DATA: As will be discussed later, point sources of pollution can markedly affect the accuracy of proxy data. The problem is that it can be difficult to know whether such pollution sources are near monitoring equipment. Midway through the model development effort, it was discovered that the monitor site at Pensacola, Florida was very near a coal fired power plant and other potential sources of SO2 emissions. As such, the measurements at this site were too high to be a reasonable proxy for Fort Rucker, Alabama.

Columbus, Georgia was selected as the alternate proxy site but an analysis of the data measured there revealed that nothing was recorded in the database for the last month (January) of the corrosion tests at Fort Rucker. An inquiry was sent to the Georgia Department of Natural

Resources to ascertain whether the missing data might be available elsewhere. Their response indicated that the monitoring program at that site had ended one month prior to the end of the corrosion test program and that no other data was available. It was then decided that since the complete proxy database containing weather and the other contaminant measurements had already been constructed for Fort Rucker, the Columbus, Georgia data would be used rather

81 than undertake the time intensive task of constructing a complete proxy record for another corrosion test site. Thus, a way to estimate the missing data was needed.

The method used to replace the missing data for the Columbus, Georgia monitoring site was to simply use the data measured for the previous January and to then plug any holes in this data using the aforementioned processes. This seems like a reasonable approach since the replacement data was measured at the same location and time of year. The result was a complete proxy database, which was then used to infer the variable levels of SO2 at Fort Rucker.

3.1.3.1.1.2 Point Sources of SO2 Emissions

In order to validate the accuracy of candidate models, they were applied to proxy data for ten additional corrosion test locations, none of which were used for model calibrations. This was done in order to make independent predictions that were then compared to test measurements in an effort to statistically validate the accuracy of the models. Figure 3-5 illustrates a scatter plot that was used to evaluate a preliminary model. On this plot, the actual test measurement values are shown on the x-axis in comparison to predictions on the y-axis. These values represent weight loss over the exposure period and are in units of micrograms per square centimeter (g/cm2). Linear regression is then used to fit a trend line to the data. As can be seen on this figure, four data points appear to be outliers in comparison to other data shown.

These four data points pertain to Fort Campbell, Kentucky and the difference between these points and the others indicates the strong possibility that something was wrong with the proxy data used for this site.

The metadata provided by the EPA to describe individual monitor characteristics fails to state their intended purpose. Initially it was believed that all such monitors are used as part of general pollution monitoring programs. However, through a combination of internet queries,

82 dialog with state environmental officials, and a review of satellite imagery, it was found that some specific monitoring programs were specifically established to monitor pollution levels near point sources of SO2.

200000 R² = 0.2824 180000 Fort Campbell, KY 160000

) 140000 2

120000 g/cm  100000

80000

Predictions ( Predictions 60000

40000

20000

0 0 10000 20000 30000 40000 50000 60000 70000 80000 90000 100000 Test Results (g/cm2)

Figure 3-5 Scatter Plot Indicating Problems with the Predictions for Fort Campbell (monthly chloride model)

Upon observing that the Fort Campbell predictions were far too high, an analysis of the proxy

SO2 monitoring site near Clarksville, Tennessee was conducted. It was then determined that this site was approximately 1.5 km from a large zinc processing facility, which is known emitter of

SO2 [145, 146]. In addition, the monitor was also just 26.3 km from a 2,386 MW coal-fired power plant, which is the largest of the 11 coal fired power plants operated by the Tennessee Valley

Authority. Figure 3-6 displays the proximity of these two large point sources of SO2 in comparison to the monitor at Clarksville, Tennessee. Because of these observations, an alternate SO2 proxy site was selected to provide the data used for the Fort Campbell predictions.

83

Nyrstar Zinc Refinery

SO2 Monitor

Site Coordinates Approximate Distance from Monitor Cumberland Monitor 36.505185, -87.397708 - Power Plant Zinc 36.51631, -87.4064 1.5km/0.9miles Refinery Power 36.39123, -87.65468 26.3km/16.3miles Plant

Figure 3-6 Proximity of Point Sources of SO2 to the Monitor at Clarksville, TN [147]

Table 3-4 Proximity of SO2 Emission Sites to Pensacola Proxy Data Monitoring Location AQS Site ID Coordinates Point Source Coordinates Distance Number and (latitude, (latitude, Address longitude) longitude) 12-033-0004 30.525367, International Paper 30.605708 14.4 km Ellyson -87.203550 Pensacola Coal -87.321911, (9.0 miles) Industrial Park, Power Plant. (83 Copter Road, MW capacity) and Ferry Pass, FL collocated paper mill Crist Coal Power 30.565328, 4.9 km Plant (1229 MW -87.224751 (3.0 miles) capacity) Ellyson Industrial Same as 0 km Park AQS (0 miles)

(possible SO2 Monitor source)

84

After the problem concerning the Fort Campbell proxy was resolved, a similar analysis was used to determine whether any of the remaining sites had similar problems with proxy data. This led to the discovery that predictions for Tyndall Air Force Base, Florida appeared to be somewhat higher than they should have been. As discussed earlier, the proxy monitoring site at Pensacola,

Florida was also determined to be near SO2 point sources. Table 3-4 identifies the location of this monitoring site in comparison to nearby point source emitters. An alternate proxy SO2 monitoring site was found for the Tyndall AFB corrosion test site location in order to improve prediction accuracy. It should be noted that the problems associated with the initial proxy site at Pensacola were minimal in comparison to the problems with the initial Fort Campbell proxy.

Changing the proxy site for Fort Campbell significantly reduced the magnitude of the predictions, which were originally much higher than the associated test measurements. Making similar changes to the proxy sites for Tyndall AFB and Fort Rucker also reduced the magnitude of the predictions, which were also higher than the test measurements. However, these adjustments were far smaller than was seen for Fort Campbell. It is possible that predictions for the remaining 11 corrosion test sites could also benefit from improved SO2 proxies. However, because no obvious over-predictions were noted when evaluating scatter plots, such actions were not taken for the remainder of this study.

3.1.3.1.2 Ozone Data

Similar to SO2 monitoring, there are hundreds of sites across the United States where ozone levels are monitored (see Figure 3-7). Some of these locations are collocated with SO2 monitors.

Unlike SO2, large quantities of ozone are not directly emitted as a pollutant but instead are formed through the interaction between UV light and airborne pollutants [71, 101, 102]. As a result, ozone monitoring equipment is typically placed in locations where levels are highest such as in or near urban areas. However, in some instances, they are also placed in rural areas since

85 ozone pollution is known to be detrimental to plant growth [148]. Like before, data measured at ozone monitoring sites is freely available through on-line resources such as the EPA AQS database.

Figure 3-7 Illustration of Ozone Monitoring Sites across the United States [143]

Ozone monitoring suffers from the same maladies as SO2 monitoring. Specifically, there are small blocks of time when ozone readings were not made due to equipment problems or other issues. As such, estimates were made to fill in the missing data using the same processes as described earlier. However, there is a far bigger problem with ozone that had to be addressed in order to complete the proxy databases used to calibrate and validate the cumulative damage model.

Tropospheric ozone is primarily created by UV light, which results in pollutant levels that experience diurnal cycling. As a result, levels fall during the night time hours and begin to rise

86 again at sunrise. An even larger effect occurs during the late fall and winter months when UV intensity decreases due to reduced daylight hours and the lower inclination angle of the sun.

An analysis of ozone data used for this research effort revealed that many monitors, including some near the corrosion test sites, are not operated during months of the year when ozone levels are low.

Table 3-5 Proxy Ozone Monitor Locations and Time Periods for Observations Type of Site Locations Period of Ozone Observation

Calibration Trona, CA complete annual (for China Lake, CA) Atlanta, GA March 1st - November 1st (for Dobbins, GA) Lafargeville, NY April 1st - November 8th (for Fort Drum, NY) Melbourne, FL complete annual (for Kennedy Space Center, FL) Validation Port Orange, FL complete annual (for Daytona Beach, FL) Hopkinsville, KY March 1st - October 31st (for Fort Campbell, KY) Leander, TX complete annual (for Fort Hood, TX) Dothan, Al March 14th - October 31st (for Fort Rucker, AL) Albuquerque, NM complete annual (for Kirtland AFB, NM) Narragansett, RI March 1st - Sept 30th (for Point Judith, RI) Rock Island Arsenal, IL complete annual (for Rock Island Arsenal, IL) Panama City Beach, FL complete annual (for Tyndall AFB, FL) London, OH April 1st - October 31st (for West Jefferson, OH) Fairborn, OH April 1st - October 31st (for Wright Patterson AFB, OH)

Table 3-5 displays the locations of the ozone monitor sites whose data was used to construct the proxy records for the 14 corrosion test sites considered under this research program. Specific

87 details on these sites are found in Appendix B (Tables B-2 through B-15 and Figures B-1 through

B-14).

3.1.3.1.2.1 Estimating Data to Fill Gaps in the Proxy Ozone Records

Table 3-5 contains the ozone observation periods for each proxy site. Inspection of the information on the table reveals that half of the monitors did not collect ozone levels throughout the year. Thus, an approach was created to estimate ozone levels to fill the gaps and thus facilitate complete ozone proxies. While estimating levels over such lengthy periods of time is certainly not optimum, the lack of complete data records for some locations necessitated this approach. Based upon the accuracy of the predictions that will be discussed later, the approach described here seems to be sufficient for modeling purposes.

ESTIMATING VARIABLE OZONE LEVELS OVER MONTHS-LONG PERIODS OF TIME: A multistep process was used to estimate hourly ozone levels over the time periods when observations were not made. The process is based upon a mathematical comparison between the partial ozone data for one location with data measured at Trona, California (proxy for China Lake, CA), which has a complete annual record. Trona was selected because of its location in the Mojave Desert, which is known for frequent sunshine. Thus, UV levels at that site exhibit a Gaussian-like behavior throughout the year that is more consistent than other locations with highly variable weather (see Figure 3-8).

To illustrate the estimation of missing data, the ozone record for Perch River (proxy for Fort

Drum, New York) was completed as seen below. Figure 3-8 displays the monthly average ozone levels at Perch River in comparison with those measured at Trona. As can be seen in this plot, the Trona data shows a remarkable amount of symmetry. The Gaussian-like behavior of the

88 data used to construct the Trona plot was exploited in order to create estimates for the missing hourly ozone levels at Perch River.

0.06

0.05

0.04

0.03

0.02

0.01

Average Ozone Level (ppm) Level Ozone Average 0 0 2 4 6 8 10 12 Month of Year

Trona Perch River

Figure 3-8 Monthly Average Proxy Ozone Levels for Fort Drum and China Lake

The first step in the process was to calculate monthly averages for the ozone levels at Trona and

Perch River. Once the monthly averages were calculated, comparison ratios between the endpoints of the Perch River data and the Trona data were calculated as was the ratio between the peak month at Trona (June) and the associated month at Perch River. Table 3-6 presents this information.

89

Table 3-6 Average Monthly Proxy Ozone Levels and Comparison Ratios (Fort Drum vs. China Lake) Month of Year Average Ozone Levels (ppm)

Trona Perch River

January 0.02398167

February 0.02730506

March 0.03739785

April 0.04055972 0.031857

May 0.04961559 0.040382

June 0.05565278 0.042933

July 0.05253091 0.035174

August 0.05195833 0.033052

September 0.040625 0.029665

October 0.03163844 0.021099

November 0.02598194

December 0.01987903

Comparison Ratios

April June October 0.7854314 0.77144672 0.666871

The next step was based upon the assumption that the data shown in Table 3-6 is representative of every year, which allows Table 3-7 to be constructed. As shown below, making such an assumption enables a symmetric relationship that can be used to further the analysis. Table 3-7 illustrates the periodic nature of the ratios that results from this assumption. After this table was constructed, the ratios within it were plotted and a 3rd order polynomial calculated to describe their periodic nature. This is shown by Figure 3-9.

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Table 3-7 Periodic Relationship of Comparison Ratios (Fort Drum vs. China Lake Proxies) Month Cumulative Number of Ratio Months

October 1 0.666871

November 2

December 3

January 4

February 5

March 6

April 7 0.7854314

May 8

June 9 0.77144672

July 10

August 11

September 12

October 13 0.666871

November 14

December 15

January 16

February 17

March 18

April 19 0.7854314

The independent variables used by the cubic equation shown on Figure 3-9 pertain to the numeric value of the month starting with a value of 1.0 for October. Table 3-8 displays the application of the cubic equation to calculate ratios for each month of the year. The ratios indicated by the shaded blocks are for the months where proxy data is missing in Table 3-6.

91

0.9 0.8 0.7 0.6 0.5 0.4 y = 0.0004x3 - 0.0114x2 + 0.0915x + 0.5856 0.3 0.2

Ratio (nondimensional) Ratio 0.1 0 0 5 10 15 20 Months

Figure 3-9 Cubic Function Fit to Monthly Ratios that Compare Fort Drum and China Lake Proxies

Table 3-8 Comparison Ratios Calculated Using Cubic Equation from Figure 3-9 Month Cumulative Ratio Number of Months October 1 0.6661

November 2 0.7262

December 3 0.7683

January 4 0.7948

February 5 0.8081

March 6 0.8106

April 7 0.8047

May 8 0.7928

June 9 0.7773

July 10 0.7606

August 11 0.7451

September 12 0.7332

92

0.06

0.05

0.04

0.03

0.02

0.01

Average Ozone Level (ppm) Level Ozone Average 0 0 2 4 6 8 10 12 Month of Year

Trona Perch River

Figure 3-10 Complete Monthly Average Proxy Ozone Levels for Fort Drum and China Lake

The comparison ratios shown on Table 3-8 can then be multiplied by the appropriate monthly averages for Trona as shown on Table 3-6, thus providing an estimate for the missing monthly averages for Perch River. This data is plotted in Figure 3-10.

Section B.2.1 contains the results of the analyses for the remaining six proxy ozone data sites that required construction of estimated hourly ozone levels for months where they were not measured. The analysis for Fort Campbell, Kentucky; Dobbins Air Reserve Base, Georgia, and

Fort Rucker, Alabama required one additional step before the aforementioned processes could be employed. As can be seen in Figure3-11, there was an unusual dip in the average ozone levels at the Fort Campbell (Hopkinsville, KY) proxy site during the summer months that was not seen in the China Lake (Trona, CA) plot or the Fort Drum plot shown previously. Dobbins and

Fort Rucker ozone plots have a similar appearance to Fort Campbell, which is assumed to be related to weather conditions during the respective monitoring periods. As described in the appendix, adjustments were made to the data used to construct these plots in order to give them a more Gaussian-like appearance (see Figure 3-11). This was done to enable a more

93 representative comparison of the likely peak average ozone levels at these sites (under normal weather conditions) and the peak at China Lake. Details on how these adjustments were conducted are seen in Appendix B.

The final step towards estimating hourly ozone levels for Perch River is to multiply the hourly ozone measurements for Trona, California with the appropriate monthly ratios shown on Table

3-8 (e.g., all October hourly values for Trona are multiplied by 0.6661). The results from this process are estimated hourly ozone values for the period at Perch River where levels were not measured. While employing such estimates induces some level of uncertainty into predictions based upon them, there is no alternative other than to conduct monitoring for a full year, which is not economically feasible for a method intended to be routinely used to make corrosion predictions in support of design analyses.

0.06

0.05

0.04

0.03

0.02

0.01 Average Average Monthof(ppm) Year 0 0 2 4 6 8 10 12 Month of Year

Trona Hopkinsville Adjusted

Figure 3-11 Average Monthly Ozone Levels at Trona and Hopkinsville (includes adjustments)

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3.1.3.1.3 Chloride Deposition Data Chloride aerosol particles cannot be measured via gaseous analyzers such as those used to measure SO2 and ozone levels. Thus, longer-term chloride deposition rates were used instead of the preferred hourly measurements. Chemical analysis of rainwater is one common way to determine longer-term chloride wet deposition rates. The National Atmospheric Deposition

Program (NADP) is a cooperative program where various local, state and federal agencies collect rainwater at approximately 250 sites across the United States (illustrated by Figure 3-12).

Samples of the collected water are sent each week to a centralized laboratory for chemical analysis to determine the concentration of various deposited atmospheric contaminant species including chlorides [149]. Concentration measurements, which include weekly, monthly, and annual averages, are placed in a database that is available on-line to the general public [150].

Figure 3-12 Map Illustrating the Location of NADP Monitor Locations [137]

Figure 3-13 illustrates the information available through the NADP database. While this figure is limited to showing the wet deposition of chloride at Sequoia National Park, California (proxy site

95 for China Lake), numerous other atmospheric contaminants are also measured including Ca, K,

Na, NH4, NO3, and SO4. During an analysis of chloride data, it was observed that some individual weekly and monthly data points were missing, especially for arid locations in the summer time.

This was likely due to the lack of rainfall during these monitoring periods. Missing data on the table shown in the figure is indicated by a value of -9 (null reading).

Examples of Monthly Deposition Averages (mg/L)

A value of “9” is a null reading (no data collected)

Figure 3-13 Example of Monthly Chloride Deposition Measurements [137]

Since chloride aerosols deposit both by both wet and dry mechanisms, the absence of wet measurement data does not mean that chlorides were not present during particular increments of time. Since dry deposition measurements are generally unavailable, it was assumed that the average wet deposition measurements are proportional to the total amount of chlorides present in the atmosphere over a period of time. This assumption allows for the estimation of data needed for those months where wet deposition data was unavailable. The estimation process simply used the average of the months adjacent to the missing month(s). For example,

96 the value for May in Figure 3-13 would be estimated by taking the average of 0.139 mg/L and

0.104 mg/L. Longer periods of missing data would be estimated in the same way with each of the missing monthly data points being assigned the same value.

Table 3-9 Proxy Chloride Monitor Locations Type of Site Locations

Calibration Average of Sequoia National Park and Yosemite National Park, CA (for China Lake, CA) Georgia Experiment Station, Pike County, GA (for Dobbins, GA) Bennett Bridge, NY (for Fort Drum, NY) Kennedy Space Center, FL (for Kennedy Space Center, FL) Validation Kennedy Space Center, FL (for Daytona Beach, FL) Mulberry Flat, KY (for Fort Campbell, KY) Sonora, TX (for Fort Hood, TX) Quincy, FL (for Fort Rucker, AL) Bandolier National Monument, NM (for Kirtland AFB, NM) Cedar Beach, NY (for Point Judith, RI) Monmouth, IL (for Rock Island Arsenal, IL) Sumatra, FL (for Tyndall AFB, FL) Deer Creek State Park, OH (for West Jefferson, OH) Oxford, OH (for Wright Patterson AFB, OH)

Rainfall and wind as well as wash cycles can remove deposited materials from surfaces. This has a lesser effect on gaseous pollutants such as SO2 or ozone since these substances will adsorb and desorb in response to rapidly changing atmospheric conditions. However, chloride aerosols deposit as very small quantities of solid matter. Thus, they will dwell on surfaces until removed

97 by some physical process. From a modeling standpoint, the length of time where it is assumed that the amount of deposited material is the optimum constant value was determined through analysis of model accuracy. When conducting simulations during the model calibration process, weekly, monthly, and annual averages for wet chloride deposition were examined to determine which approach led to the most accurate model.

Table 3-9 displays the locations of the chloride monitor sites whose data was used to construct the proxy records for the 14 corrosion test sites considered under this research program.

Specific details on these sites are found in Appendix B (Tables B-2 through B-15 and Figures B-1 through B-14).

3.1.3.1.4 Weather Data The cumulative damage corrosion model includes both temperature and relative humidity in its formulation. Hourly measurements of these two variables were needed in order for the model to consider both diurnal and seasonal cycling effects. Such data could not be directly found on- line so help was provided by Mr. Paul Gehred from Detachment 3 of the US Air Force’s 16th

Weather Squadron. Mr. Gehred sought and obtained the requisite weather data from the US

Air Force’s 14th Weather Squadron. This organization, which is affiliated with the National

Climate Data Center (NCDC), maintains databases of world-wide climatology data including the hourly data used for this effort.

As mentioned previously, weather databases suffer from the same missing data as does the atmospheric contaminant databases. However, such instances of missing data are far more limited due to the fact that the weather observations used for this modeling effort came from either civilian or military weather offices that were manned 24 hours a day/7 days a week. For the small amount of missing data, estimates were made based upon adjacent temperature and

98 relative humidity and/or dew point readings. A bigger problem concerned the filtering of data to remove extraneous data points. The weather offices that measured the data used under this effort were all affiliated with airports. As such, they often make multiple observations per hour when visibility becomes impaired due to lowering of cloud ceilings [151]. For those cases, the extra readings were filtered out so that only the first observation during a particular hour was considered during the modeling process.

Table 3-10 Proxy Weather Office Locations Type of Site Locations

Calibration Naval Air Warfare Center-China Lake, CA (for China Lake, CA) Atlanta Hartsfield International Airport, GA Pike County, GA (for Dobbins, GA) Wheeler-Sack Army Airfield, NY (for Fort Drum, NY) Melbourne International Airport, FL (for Kennedy Space Center, FL) Validation Daytona Beach International Airport, FL (for Daytona Beach, FL) Fort Campbell Army Airfield, KY (for Fort Campbell, KY) Robert Gray Army Airfield, TX (for Fort Hood, TX) Cairns Army Airfield, AL (for Fort Rucker, AL) Albuquerque International Airport, NM (for Kirtland AFB, NM) Newport State Airport, RI (for Point Judith, RI) Quad-City International Airport, IL (for Rock Island Arsenal, IL) Tyndall AFB, FL (for Tyndall AFB, FL) Rickenbacker International Airport, OH (for West Jefferson, OH) Wright Patterson AFB, OH (for Wright Patterson AFB, OH)

99

Table 3-10 displays the locations of the weather offices whose data was used to construct the proxy records for the 14 corrosion test sites considered under this research program. Specific details on these sites are found in Appendix B (Tables B-2 through B-15 and Figures B-1 through

B-14).

3.1.4 Composite Data File Used for Model Calibration and Validation

Table 3-11 Illustration of Composite Proxy Data File Fort Drum Data Temp RH SO2 Ozone Chloride Year Month Day Hour (K) (%) (ppm) (ppm) (mg/L) 2005 5 1 0 281.15 94 0.0005 0.030 0.09 2005 5 1 1 280.15 98 0.0005 0.028 0.09 2005 5 1 2 280.15 98 0.0006 0.028 0.09 2005 5 1 3 279.15 100 0.0006 0.028 0.09 2005 5 1 4 279.15 100 0.0007 0.031 0.09 2005 5 1 5 279.15 100 0.0005 0.030 0.09 2005 5 1 6 279.15 100 0.0004 0.029 0.09 2005 5 1 7 279.15 100 0.0004 0.031 0.09 2005 5 1 8 279.15 100 0.0004 0.036 0.09 2005 5 1 9 278.15 100 0.0004 0.037 0.09 2005 5 1 10 278.15 100 0.0006 0.034 0.09 2005 5 1 11 278.15 100 0.0005 0.037 0.09 2005 5 1 12 278.15 100 0.0008 0.040 0.09 2005 5 1 13 280.15 96 0.0009 0.048 0.09

Table 3-11 illustrates the information contained within the composite proxy data files. Included in these files are the year, date, and time associated with individual weather observations. Also included are the hourly measurements (and estimates) for ozone and SO2 as well as longer term average values for chloride deposition, which are assumed to apply on an hourly basis. As discussed earlier, three different chloride deposition averages including weekly, monthly, and annual values were considered during the Monte Carlo simulation processes to determine which average provided the optimal predictions. This required separate proxy databases for each.

100

Proxy data files were constructed for each of the fourteen sites (four calibration and ten validation sites) considered during this research effort. Each file was checked line-by-line to identify and correct anomalies.

101

CHAPTER 4

RESULTS FROM NUMERICAL EXPERIMENTS

Table 4-1 Fifteen Types of Initial Models that were Constructed and Tested under this Effort Calibration T-RH Shape Functions Chloride Activation RH Sites Deposition Energies Threshold Linear Temperature-Linear 4 annual 1 60% RH Relative Humidity Linear Temperature-Convex 4 annual 1 60% RH Relative Humidity Convex Temperature-Convex 4 annual 1 60% RH Relative Humidity Convex Temperature-Convex 4 monthly 1 60% RH Relative Humidity Convex Temperature-Convex 4 Weekly 1 60% RH Relative Humidity Convex Temperature-Convex 3 annual 1 60% RH Relative Humidity Convex Temperature-Convex 3 monthly 1 60% RH Relative Humidity Convex Temperature-Convex 3 Weekly 1 60% RH Relative Humidity Convex Temperature-Convex 1 annual 1 60% RH Relative Humidity Convex Temperature-Convex 1 monthly 1 60% RH Relative Humidity Convex Temperature-Convex 1 Weekly 1 60% RH Relative Humidity Convex Temperature-Convex 3 monthly 3 60% RH Relative Humidity Convex Temperature-Convex Linear RH 3 monthly 1 Relative Humidity Function Convex Temperature-Convex Nonlinear 3 monthly 1 Relative Humidity RH Function Convex Temperature-Convex 3* monthly 1 60% RH Relative Humidity *This is a model calibrated using a different data set

102

Table 2-2 identifies nine different Temperature-Relative Humidity shape functions that were initially considered as the basis for the proof-of-concept model. The analysis shown in Appendix

E.1 was used to discount those shape functions that were less likely to provide optimal results.

As a result, three shape functions were used to construct the initial candidate models considered under this effort. These include the Linear Temperature-Linear Relative Humidity,

Linear Temperature-Convex Relative Humidity, and Convex Temperature- Convex Relative

Humidity shape functions.

As seen by Table 4-1, fifteen different types of initial models were considered during this research effort. Numerous specific models for each type were constructed by incrementally varying the activation energy and then calibrating the associated coefficients using Monte Carlo simulations. The various processes used to calibrate the candidate models and subsequently analyze the results are discussed in Appendix C. The results are discussed in the following sections. Supporting details on the results shown here are found in Appendix E.

When calibrating individual models, the Temperature-Relative Humidity shape function of interest (Chapter 2) was combined with the notional cumulative damage model (Equation 2.14 or 2.15) and the appropriate calibration dataset, thus leading to the specific types of models.

Individual activation energies were then selected and Monte Carlo simulations were conducted whereby random values for the unknown coefficients were systematically selected and input into the model, which was then applied to the entire set of (annual) hourly environmental data pertaining to the calibration sites where corrosion rates were conducted. Cumulative quarterly predictions for each site were then calculated based upon the summation of hourly predictions and then statistically compared to the associated quarterly test measurements. The set of coefficients that lead to the lowest overall error was deemed optimal for the combination of a

103 particular type of model and the specific activation energy. Models were calibrated for a series of activation energies to find the combination that provided the most accurate results.

4.1 Linear Temperature-Linear Relative Humidity Shape Function (Initial Baseline)

400000000

350000000

300000000

Error 250000000

200000000

150000000 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 Activation Energy (eV)

Figure 4-1 Error Evaluation to Identify the Most Accurate Model Calibrated from Data Pertaining to Four Test Sites (annual chloride deposition, optimum model: H=0.81 eV)

Figure 2-6 illustrates the Linear Temperature-Linear Relative Humidity shape function. Results from the simulations used to calibrate the initial models based upon this shape function are shown in Figure 4-1. This figure compares twelve candidate models that were calibrated using annual average chloride deposition rates and other environmental data from four different corrosion test sites (China Lake, CA; Dobbins ARB, GA; Fort Drum, NY; and Kennedy Space

Center, FL). The results shown on this figure were used to identify the specific model that provided the lowest error.

Figure 4-1 compares the errors for twelve different models, each with their own unique set of twelve calibrated coefficients. An activation energy of 0.81 electron volts (eV), which is

104 equivalent to 75.15 KJ/mol, corresponds to the model that had the lowest error, and thus the best fit to the calibration data.

The error for this calibrated model is 192,176,163. While this number seems very large, it should be remembered that it is calculated using the residual sum of squares method, which employs the summation of the square of the differences between each of the sixteen test points

(four for each calibration test site) and their associated predictions. Because the test points and predictions both use units in thousands of micrograms, then the difference between them also uses the same units. Squaring such numbers and adding sixteen of them together will thus lead to a very large number. Errors could have been calculated using units of grams. However, the differences of weight loss between specimens using such units would have been imperceptible.

Regardless, it is not the specific value of the error that is important, but rather its relative ranking when compared to other candidate models.

Table 4-2 Coefficients Determined for the Model Based Upon Linear Temperature-Linear Relative Humidity Shape Functions Maximum Values for Shape Functions Values for other Modeling Constants

2 fCl(T,RH)max -3.9993 ACl 0.133154489611 g/cm

2 fSO2(T,RH)max -3.99157831465 ASO2 0.092333854192 g/cm

2 fO3(T,RH)max -3.96046466678 AO3 0.000411888552 g/cm

f(T,Cl)max -3.9999 Cl -0.210225475346

f(T,SO2)max -3.99815760476 SO2 -0.241886468226

f(T,O3)max -3.9999 O3 0.161993508629

The coefficients for the optimal model as determined by the Monte Carlo process are shown in

Table 4-2. Those on the left of the table correspond to the six shape functions used by the model while those on the right are the six other constants.

105

Figure 4-2 is a scatterplot used to evaluate the accuracy of the model. Each of the sixteen points in this figure (four quarterly points for each of the four calibration test locations) defines the relationship between individual corrosion test data points (x-axis) and their corresponding model predictions (y-axis). Section C.4 contains a discussion on evaluating such plots. In this case, the trend line shown on the figure was fit to the data points using the automated linear regression tool available in Microsoft Excel. This line has a slope approximately equal to 1.0 and it extends close to the origin, both of which are good model attributes. In addition, the scatter of the data around this trend line does not exhibit a funnel-shape, thus indicating homoscedastic behavior, another positive attribute. A final consideration is that the R2 of 0.973 indicates that the model has a very high degree of fit to the data used during calibration.

80000 y=0.9811x+452.4 70000 R² = 0.973 ) 2 60000

g/cm 50000  40000 30000 20000 Predictions ( Predictions 10000 0 0 10000 20000 30000 40000 50000 60000 70000 80000 Test Results (g/cm2)

Figure 4-2 Comparison of Corrosion Test Data and Associated Predictions for the Annual Chloride Model Applied to Calibration Data from Four Test Sites (optimum H=0.81 eV)

Based upon the results shown on Figure 4-2, the initial model constructed from Linear

Temperature-Linear Relative Humidity shape functions is capable of making predictions that are

106 highly correlated to the test measurements used during the calibration process. As seen in the next two sections, the results (R2 value) for the Linear Temperature – Linear Relative Humidity model described here was compared to models using the other two candidate shape functions in order to identify which shape function should form the basis for further model development efforts.

4.2 Linear Temperature-Convex Relative Humidity Shape Function Figure 2-10 illustrates the Linear Temperature-Convex Relative Humidity shape function. Results from the simulations used to calibrate the initial model based upon this shape function (using the same calibration data) are shown below by Figure 4-3. Like before, each point on the plot corresponds to an individual model based upon its own activation energy. The lowest error

(171,866,771) corresponds to the model with an activation energy of 0.86 eV (82.98 KJ/mol).

This error is slightly less than the error calculated for the model that employed linear- temperature-linear relative humidity shape functions.

175500000 175000000 174500000 174000000 173500000 Error 173000000 172500000 172000000 171500000 0.8 0.82 0.84 0.86 0.88 0.9 Activation Energy (eV)

Figure 4-3 Error Evaluation to Identify the Most Accurate Model Calibrated from Data Pertaining to Four Test Sites (annual chloride deposition, optimum model: H=0.86 eV)

107

The coefficients determined by the Monte Carlo process are shown in Table 4-3. Those on the left of the table correspond to the six shape functions used by the model (Equation 2.14) while those on the right are the six remaining constants.

Table 4-3 Coefficients Determined for the Model Based Upon Linear Temperature-Convex Relative Humidity Shape Functions Maximum Values for Shape Functions Values for other Modeling Constants

2 fCl(T,RH)max -3.994777857 ACl 0.00093333333333 g/cm

2 fSO2(T,RH)max -3.977748306 ASO2 0.005439847 g/cm

2 fO3(T,RH)max -3.98855413 AO3 0.00026666666667 g/cm

f(T,Cl)max -3.99971333333 Cl -0.290665507

f(T,SO2)max -3.9984 SO2 -0.295757643

f(T,O3) -3.99873333333 O3 -0.283298482

90000 y=0.9758x+648.48 80000

) R² = 0.9758 2 70000 60000 g/cm  ( 50000 40000 30000 20000 Predictions 10000 0 0 20000 40000 60000 80000 Test Results (g/cm2)

Figure 4-4 Comparison of Corrosion Test Data and Associated Predictions for the Annual Chloride Model Applied to Calibration Data from Four Test Sites (optimum H=0.86 eV)

108

Figure 4-4 is the scatterplot that relates the test data points to their associated predictions. Like before, the slope of the trend line is nearly equal to 1.0, the trend line extends to the origin, and the residuals appear to be normally distributed around the trend line. The R2 value is higher than the previous case, thus indicating a model with a higher degree of fit to the data.

4.3 Convex Temperature-Convex Relative Humidity Shape Function Figure 2-16 in Section 2 illustrates the Convex Temperature- Convex Relative Humidity shape function. Results from the simulations used to calibrate the initial model based upon this shape function are shown below by Figure 4-5. Each point on the plot corresponds to an individual model based upon its own activation energy. The lowest error (167,324,659) corresponds to the model with an activation energy of 0.66 eV (63.68 KJ/mol). This error is less than the error calculated for either of the two previous models.

175000000 174000000 173000000 172000000 171000000

Error 170000000 169000000 168000000 167000000 166000000 0.55 0.6 0.65 0.7 0.75 Activation Energy (eV)

Figure 4-5 Error Evaluation to Identify the Most Accurate Model Calibrated from Data Pertaining to Four Test Sites (annual chloride deposition, optimum model: H=0.66 eV)

109

The coefficients determined by the Monte Carlo process are shown in Table 4-4. Those on the left of the table correspond to the six shape functions used by the model (Equation 2.14) while those on the right are the six remaining constants.

Table 4-4 Coefficients Determined for the Model Based Upon Convex Temperature- Convex Relative Humidity Shape Functions Maximum Values for Shape Functions Values for other Modeling Constants

2 fCl(T,RH)max -3.99487997448 ACl 0.0673118397366 g/cm

2 fSO2(T,RH)max -3.99425483353 ASO2 0.0421306411041 g/cm

2 fO3(T,RH)max -3.99669813604 AO3 0.0505055847112 g/cm

f(T,Cl)max -3.99566119953 Cl -0.194823188432

f(T,SO2)max -3.82087786712 SO2 0.0640668860481

f(T,O3)max -3.98562926972 O3 -0.130420633443

80000 y=0.9752x+598.17 70000 )

2 R² = 0.9765 60000

g/cm 50000  ( 40000 30000 20000 Predictions 10000 0 0 10000 20000 30000 40000 50000 60000 70000 80000 Test Results (g/cm2)

Figure 4-6 Comparison of Corrosion Test Data and Associated Predictions for the Annual Chloride Model Applied to Calibration Data from Four Test Sites (optimum: H=0.66 eV)

110

Figure 4-6 is the scatterplot that relates the test data points to their associated predictions. Like before, the slope of the trend line is nearly equal to 1.0, the trend line extends towards the origin, the residuals appear to be normally distributed around the trend line, and the R2 value is the highest of the three candidate models considered during this initial assessment.

4.4 Selection of the Most Accurate Base Model Inspection of Table 4-5 enables a comparison of the accuracy of the three different models that were calibrated during the initial screening simulations. Evaluation of this data leads to the conclusion that the model constructed from Convex Temperature-Convex Relative Humidity shape functions has both the lowest error and the highest R2 value. Inspection of the scatterplot pertaining to this model also indicates positive results. Specifically, the slope of the linear trend line shown on Figure 4-6 appears to be close to 1:1 (the optimum ratio) and it extends directly towards the origin. In addition, the scatter of the data points around the trend line does not exhibit a “funnel shape”, this providing evidence of homoscedastic behavior. As such, the model constructed from Convex Temperature-Convex Relative Humidity shape functions was used as the basis for further model development efforts.

Table 4-5 Accuracy of Models that were Calibrated during Initial Screening Simulations (annual chlorides) Model Type Error R2 Value (fit of model to calibration data)

Linear Temperature-Linear Relative 192,176,163 0.973 Humidity Shape Functions

Linear Temperature-Convex Relative 171,866,771 0.9758 Humidity Shape Functions Convex Temperature-Convex Relative Humidity Shape Function 167,324,659 0.9765

111

4.5 Revision of the Linear Temperature-Linear Chloride Shape Function After conducting the initial screening simulations that identified the Convex Temperature-

Convex Relative Humidity Shape Functions as the most accurate of the three candidate model types, environmental characterization databases were constructed for ten additional (model validation) sites where corrosion tests were conducted but not used for calibration (see Table

3.2). These sites were chosen so that diverse environmental conditions were considered when validating the accuracy of the candidate models.

Once the validation databases were constructed, an analysis was undertaken to determine the maximum values for each contaminant at each location. At this point, it was determined that the maximum chloride deposition value found in two of the validation site databases exceeded the maximum value used during the initial simulations. Thus, the Linear Temperature-Linear

Chloride shape function described by Equation 2.51 had to be revised, as seen by Equation 4.1.

Tf  4)( Tf Cl),(  Cl  4 (4.1) 5.7

A revised model based upon the new Linear Temperature-Linear Chloride shape function in combination with the initial Convex Temperature-Convex Relative Humidity shape functions and the initial Linear Temperature-Linear SO2 and Linear Temperature-Linear Ozone shape functions was constructed and calibrated in order to later apply it to the validation data. As can be seen by Figure 4-7, eleven different models, each based upon different activation energies, were then calibrated via simulations in order to identify the one with the lowest error.

112

169500000

169000000

168500000

Error 168000000

167500000

167000000 0.6 0.62 0.64 0.66 0.68 0.7 Activation Energy (eV)

Figure 4-7 Error Evaluation to Identify the Most Accurate Model Calibrated from Data Pertaining to Four Test Sites (annual chloride deposition, optimum model: H=0.66 eV)

Figure 4-8 is a scatterplot that compares the predictions made using this new model with the associated (calibration) test measurements. Like the earlier models, the residuals related to the trend line shown on this plot exhibit homoscedastic behavior, which is the desired trait.

80000 y=0.9751x+609.49 70000 )

2 R² = 0.9765 60000 50000 (mg/cm 40000 30000 20000 Predictions 10000 0 0 10000 20000 30000 40000 50000 60000 70000 80000 Test Results (g/cm2)

Figure 4-8 Comparison of Corrosion Test Data and Associated Predictions for the Annual Chloride Model Applied to Calibration Data from Four Test Sites (optimum H=0.66 eV)

113

200000 y=1.1163x+29632 180000

) R² = 0.3057

2 160000 140000 g/cm

 120000 ( 100000 80000 60000

Predictions 40000 20000 0 0 20000 40000 60000 80000 100000 Test Results (g/cm2)

Figure 4-9 Comparison of Corrosion Test Data and Associated Predictions for the Annual Chloride Model Applied to Validation Data from Ten Independent Locations (H=0.66 eV)

Figure 4-9 is a scatterplot that compares predictions and associated test measurement for the ten independent locations not used to calibrate the model. As can be seen, data plotted on this figure exhibits a funnel shape illustrative of heteroscedastic behavior, which is an undesirable attribute (see Section C.4). This indicates that something is wrong with either the model, the data, or both. Further analysis was needed to determine the specific reasons for this behavior.

Table 4-6 lists all of the parameters pertaining to the updated Convex Temperature-Convex

Relative Humidity shape function model calibrated using annual average chloride deposition data and corrosion test results measured at four different locations. Comparison of the data on this table with the data seen on Table 4-5 indicates that changing the Linear Temperature-Linear

Chloride shape function had little difference on the error, which is very close to the original value. In addition, the R2 value for the model applied to the calibration data is identical to

114 before. As such, it seemed reasonable to select this model as the baseline to compare with the candidate models that were subsequently developed.

Table 4-6 Modeling Overview and Coefficients for the Optimum Convex Temperature- Convex RH Shape Function Model Calibrated using Data from Four Test Sites (annual chlorides) Variable Value Number of Models Calibrated 11 (see Figure 4-7) Computing Time (hours: minutes) 176:38 Simulations Conducted 2,689,500,000 Optimum H 0.66 eV/63.68 KJ/mol R2 (calibration) 0.9765 R2 (validation) 0.3057 Error 167,319,440

fCl(T,RH)max -3.99644404646 fSO2(T,RH)max -3.99723388 fO3(T,RH)max -3.99645279497 f(T,Cl)max -3.98874397632 f(T,SO2)max -3.78379406207 f(T,O3)max -3.95326540076 2 ACl 0.0605068846143 g/cm 2 ASO2 0.0531961448821 g/cm 2 AO3 0.00879548 g/cm

Cl -0.12982504708

SO2 0.11868400896

O3 -0.0425769439962

In addition to the error and R2 values, Table 4-6 contains a wide variety of other data including metrics pertaining to the simulation processes as well as the coefficients identified for the optimum model. In order to make predictions, these coefficients are input into spreadsheets containing the environmental data and algorithms used to make the hourly and cumulative predictions.

For the remainder of this section, the various figures and tables such as those shown above will not be shown. The reader is referred to Appendix E for such detail. Instead, summary

115 information limited to the optimum activation energy, model error, and R2 values will be presented to compare candidate models.

4.6 Simulations to Determine the Optimum Chloride Deposition Interval

Table 4-7 Comparison of Metrics Pertaining to Models that were Calibrated Using Different Time Spans for Chloride Deposition Model Type Activation Error R2 Value R2 Value Energy (calibration) (validation)

Annual Chlorides 0.66 167,319,440 0.9765 0.3057

Monthly Chlorides 0.64 148,424,895 0.9791 0.2824

Weekly Chlorides 0.66 155,259,104 0.9782 0.3102

All of the previous simulation scenarios were conducted using annual average chloride deposition measurements for each of the four initial calibration sites. However, weekly and monthly average measurements are also available from the NADP website. Thus simulations were conducted to ascertain which exposure interval would provide the best results. Table 4-7 compares the model calibrated by considering annual average chloride deposition measurements at the four initial calibration locations with models calibrated using monthly and weekly deposition averages (see Section E.3.1 for details). As can be seen from this table, the model constructed from monthly chloride deposition levels has a significantly lower error than either the annual or weekly chloride deposition models. In addition, the R2 value for the monthly model fit to the calibration data is higher. However, the R2 value of the model fit to the independent data is somewhat lower than the other two. Since the model calibrated using monthly chlorides had the lowest error and best fit to the calibration data, it was chosen to be

116 the focus of further enhancement and refinement efforts in an attempt to improve the R2 value pertaining to the fit to the independent validation data.

4.7 Candidate Models Calibrated Using Data from Three Corrosion Test Sites

The initial simulations shown above were all conducted using proxy characterization data and corrosion test results for four locations with diverse environmental conditions. Upon determining that the initial candidate model based upon monthly chlorides and Convex

Temperature-Convex RH Shape Functions has a low R2 value when applied to independent validation data, an analysis of the China Lake data was conducted to determine if this could be a contributing factor.

Table 4-8 Evaluation of Parameters Relating to China Lake, CA Total Weight Weight Loss Per Hours of Time Period Loss (g/cm2) Quarter (g/cm2) Exposure Above 60% RH Threshold 1/19/2006-4/16/2006 1593 1593 209

4/17/2006-7/17/2006 3563 1970 5

7/18/2006-10/17/2006 4428 865 44

10/18/2006-1/25/2007 5485 1057 49

China Lake, CA is in the Mojave Desert, and thus has a climate that is very hot and dry for much of the year with infrequent bouts of rainfall and humidity levels that exceed the relative humidity threshold. This location was specifically chosen for these climatic features so that the simulations used to calibrate the model would consider low humidity and high temperature conditions. Table 4-8 compares the weight loss to the number of hours of exposure that exceed

117 the relative humidity threshold for each of the four quarters over which corrosion tests were conducted.

Table 4-9 Comparison of Metrics Pertaining to Models that were Calibrated Using Data from Three Calibration Sites and Different Time Spans for Chloride Deposition R2 Value R2 Value Model Type Activation Energy Error (calibration) (validation)

Annual Chlorides 0.65 eV/62.72 KJ/mol 133,534,628 0.9798 0.305

Monthly 0.62 eV/59.82 KJ/mol 114,924,809 0.9832 0.2807 Chlorides

Weekly Chlorides 0.63 eV/60.79 KJ/mol 121,418,584 0.9821 0.3061

As can be seen by inspection of the data on Table 4-8, the quarter with the largest weight loss was the second quarter, which only had five hours of exposure where the 60% RH threshold was exceeded. This fact seems to be inconsistent with the corrosion rates and number of hours of exposure for the other quarters that the test was conducted. An analysis of the published test protocols indicated that the specimens were not vacuum packaged or packed in a container with desiccant during shipping [25]. Thus, the hours of exposure to RH levels above the threshold after the specimens were cleaned and weighed but before installation in the field likely had a significant effect on the corrosion measurement. Likewise, the time spent in shipping and on the bench awaiting cleaning and precision weighing after exposure could also have skewed the results. As a result of the uncertainty associated with the China Lake data, it was decided to eliminate this location from consideration and instead calibrate the model using environmental characterization data from the three remaining locations (Dobbins Air Reserve

Base, GA; Fort Drum, NY, and Kennedy Space Center, FL). Summary modeling results are shown on Table 4-9 and supporting details can be found in Section E.3.2.

118

Like before, the model based upon monthly chloride deposition rates has the lowest error.

Similarly, it has the highest R2 value when applied to calibration data and the lowest value when applied to the independent validation data. Like before, the errors shown on Table 4-9 were calculated using the residual sum of squares method, which considers the sum of the differences between each prediction and its related test measurement. Since there were more combinations of quarterly predictions/test measurements for the model calibrated from data measured at four locations in comparison to the three site model, then the statistical errors for the two different scenarios cannot be directly compared. However, because the calibration R2 value for the model constructed using monthly chloride data and three calibration locations

(Table 4-9) is higher than the value for the associated model from four locations (see Table 4-7), it appears as if removing China Lake data from consideration led to a slightly improved model. It should be noted that this new model did not improve the R2 value for the model applied to the independent validation data. Thus, further analysis was needed to identify the reasons for this low number and identify ways to increase its value.

4.8 Candidate Models Calibrated Using Data from Rock Island, IL Another possibility concerning why the candidate models do not have a high degree of fit with the validation data concerns the proxy data used for the analysis. As discussed earlier, the ozone measurements used as the proxy for Dobbins Air Reserve Base, GA were not measured year-round so that a model had to be constructed to estimate hourly ozone levels for those months of the year when ozone was not measured. If this estimation process was insufficiently accurate, it could lead to erroneous predictions at locations other than calibration sites. One way to determine whether this could be an issue was to calibrate a model using complete annual data measured very close to the corrosion test site. Details from this analysis are found in Section E.3.3.

119

Table 4-10 Rock Island Arsenal Corrosion Test Site and Associated Environmental Characterization Sites Approximate Approximate Coordinates Distance Identification Available Location from Test Code Data Latitude Longitude Site

Rock Island NA – Weight Loss 41.51821 -90.537058 - Arsenal, IL Corrosion Test Site

SO2 Monitoring Site th 10 Street & 19-163-0015 SO2 41.53001 -90.587611 4.4 km Vine Street, (2.8 miles) Davenport, IA

Ozone Monitoring Site 32 Rodman, Ave., 17-161-3002 O3 41.51473 -90.51735 1.7 km Rock Island (1.1 miles) Arsenal, IL

Chloride Monitoring Site Monmouth, IL78 Chloride 40.9333 -90.7231 66.8 km Warren County, (41.5 miles) IL Weather Monitoring Site Quad-City KMLI/72544 Temperature 41.45 -90.517 7.8 km International Dewpoint (4.8 miles) Airport

Rock Island Arsenal, IL is one of the sites used as part of the initial validation dataset. As seen in

Table B-15 (seen again here as Table 4-10), each of the proxy environmental characterization sites are very close to the corrosion test site with the exception of the chloride deposition measurement site. Since Rock Island is far from coastal areas, chloride aerosol levels should be low. Thus, it was assumed that differences in chloride deposition between Rock Island and the chloride measurement site in Monmouth, IL were minimal. A model was constructed specifically for this site in order to develop some insight concerning the use of proxy data and possible inaccuracies in the model formulation.

120

Table 4-11 Comparison of Metrics Pertaining to Models that were Calibrated Using Data from Rock Island, Illinois and Different Time Spans for Chloride Deposition R2 Value R2 Value Model Type Activation Energy Error (calibration) (validation)

Annual Chlorides 0.91 eV/87.80 KJ/mol 38,141,437 0.9309 0.6298

Monthly 0.93 eV/89.73 KJ/mol 44,305,051 0.9783 0.6268 Chlorides

Weekly Chlorides 0.96 eV/92.63 KJ/mol 28,561,646 0.9773 0.7687

1000000 900000 y=8.6078x-49479

) R² = 0.7687 2 800000 700000 g/cm

 600000 ( 500000 400000 300000

Predictions 200000 100000 0 0 20000 40000 60000 80000 100000 Test Results (g/cm2)

Figure 4-10 Comparison of Corrosion Test Data and Associated Predictions for the Weekly Chloride Model Applied to Validation Data from Nine Independent Locations (H=0.96 eV)

Table 4-11 compares the three models for Rock Island that were calibrated using annual, monthly, and weekly averages for chloride deposition measurements. As can be seen, the weekly model had the lowest error, nearly the highest R2 value for the model applied to calibration data, and the highest R2 value when it was applied to the independent validation data. The R2 value of 0.7687 for the model applied to the independent validation data is much

121 larger than any of the previous models. However, inspection of the data plotted on Figure 4-10 indicates the presence of a funnel type shape, especially at lower corrosion levels. This indication of heteroscedastic behavior provides evidence of problems with the model constructed only from proxy data for Rock Island Arsenal.

Equation 2.14 (shown again here as Equation 4.2) displays the notional cumulative damage corrosion model. This equation employs three scaling factors (ACl, ASO2, and AO3) which are constants used to adjust the three components of the model that represent the kinetics associated with the three contaminants considered by the model. The other coefficients determined via Monte Carlo simulations include three terms used to adjust the influence of

temperature (Cl, SO2, and O3) and six other coefficients, fCl(T,RH)max, fSO2(T,RH) max, fO3(T,RH) max, f(T,Cl) max, f(T,SO2) max, and f(T,O3) max, that dictate the maximum value of the six shape functions employed by the model.

 H  CL Ki   [exp CL Cl TfTA RH  Tf ,(exp,((exp Cl   kT  (4.2) SO2 O3 SO2 SO2 TfTA RH  Tf ,(exp,(exp SO2  O3 O3 TfTA RH  OTf 3 ],(exp,(exp

Table 4-12 displays the values for the twelve unknown coefficients for the weekly average chloride deposition model that were determined via Monte Carlo simulations. As can be seen from this table, two of the scaling factors, ASO2, and AO3, have values of zero, indicating that this particular model only considers corrosion rates based upon chloride deposition. The reason this model focuses upon one contaminant is likely due to the lack of diversity in the environmental characterization data considered during the calibration process. Since the model calibrated using Rock Island data alone ignores the other contaminants, which are known to influence corrosion rates, and the plotted results exhibit heteroscedastic behavior (see Figure 4-10) when

122 applied to make predictions for independent validation locations, it is only capable of making accurate predictions at the calibration locations.

Table 4-12 Coefficients for the Optimum Convex Temperature-Convex RH Shape Function Model Calibrated using Data from Rock Island (weekly chlorides) Variable Value

fCl(T,RH) max -3.9998 fSO2(T,RH) max -3.979235029

fO3(T,RH) max -3.970863416 f(T,Cl) max -3.9999 f(T,SO2) max -3.812354978 f(T,O3) max -3.992123149 2 ACl 0.150208111 g/cm 2 ASO2 0 g/cm 2 AO3 0 g/cm

Cl -0.528380672

SO2 -0.040513304

O3 -0.099388648

4.9 Candidate Models Based Upon Three Activation Energies In a further attempt to improve the accuracy of the model (as indicated by the R2 value) when it is applied to locations other than those used for calibration, a new model was constructed based upon the use of Convex Temperature-Convex Relative Humidity shape functions, monthly average chloride deposition rates, and three different activation energies, one for each of the contaminants considered by the model. Similar to the best fitting model yet (monthly chlorides for three different locations), this new model was also calibrated using data from Ft. Drum, NY;

Dobbins Air Reserve Base, GA; and Kennedy Space Center, Florida. Details for this analysis are found in Section E.3.4. Table 4-13 compares the new model with the associated one based upon a single activation energy. As can be seen, the new model has a lower error. However, both of its R2 values (for calibration and validation) were also lower, indicating mixed results.

123

Table 4-13 Comparison of Metrics Pertaining to Monthly Chloride Models that were Calibrated Using Data from Three Calibration Sites R2 Value R2 Value Model Activation Energy Error (calibration) (validation) Type

Single 0.62 eV/59.82 KJ/mol 114,924,809 0.9832 0.2807 Activation Energy

HCl: 0.97eV/93.59 KJ/mol Three H : 0.58 eV/55.96 KJ/mol 105,093,938 0.9827 0.2697 Activation SO2

Energies HO3: 0.84 eV/81.05 KJ/mol

114000000 113000000 112000000 111000000 110000000 109000000 Error 108000000 107000000 106000000 105000000 104000000 0.9 0.92 0.94 0.96 0.98 1 1.02 1.04 1.06 Activation Energy (eV)

Figure 4-11 Error Evaluation Used to Identify the Most Accurate Model Calibrated from Data Pertaining to Three Test Sites (monthly chloride deposition, optimum HCl=0.97 eV)

The process of calibrating this new model was more difficult than before because the addition of two more unknown coefficients, which brings the total to fourteen. The increased number of unknown coefficients resulted in the need to conduct far more simulations in order to identify an acceptable model. Figure 4-11 compares the fourteen different models based upon the use

124 of three activation energies that were subjected to the calibration process. For each of the

points shown on the figure, one of the activation energies (HCl) was held constant while the other two were allowed to vary along with the other twelve unknowns. Values for each of the activation energies are seen in Table 4-13. Despite the fact that far more simulations (in comparison to the earlier models) were run in an attempt to calibrate this new model, the number conducted were still insufficient to converge upon the best solution. This lack of convergence is exemplified by the oscillations seen in the curve shown on Figure 4-11. Thus, the results shown for the three activation energy model have not been fully optimized.

4.10 Evaluation of Data Used for Modeling and Simulations

y=1.0782x+30213 250000 Ft. Campbell, KY R² = 0.2807

) 200000 2 Daytona Beach, FL Point Judith, RI g/cm 150000  (

100000

Predictions 50000

0 0 20000 40000 60000 80000 100000 Test Results (g/cm2)

Figure 4-12 Illustration of the Impact of Applying Model Using Data Measured Near Point Sources of SO2 Pollution and Locations Adjacent to Surf Zones (ten independent locations, optimum H=0.62 eV)

It is possible that conducting far more simulations would smooth out the curve seen on Figure 4-

11 and that the final optimum set of coefficients would improve the accuracy of the model and

125 improve its R2 value when applied to independent validation data. However, doing so in a reasonable amount of time exceeded the capabilities of the computer workstation used for this effort.

The next attempt to ascertain why the R2 values pertaining to the model applied to independent validation data were too low focused upon the environmental characterization data used to make predictions. Figure 4-12 is the scatterplot for the single activation energy model constructed from Convex Temperature-Convex Relative Humidity shape functions and calibrated using monthly average chloride deposition rates and other proxy environmental data from three locations. As can be seen, the majority of data points clustered near the origin of the coordinate system appear to follow the same general trend. Had a trend line been fit to these points alone, that line would extend upward at roughly a 45o angle. The data points for Fort Campbell, KY;

Daytona Beach, FL; and Point Judith, RI are significantly different from the clustered data and clearly fall far outside the parameters of this hypothetical trend line. An investigation of the data used to make the predictions for these three sites was conducted in order to determine whether it might be causing the unexpected results.

During the calibration process, the corrosion test site nearest the coastline was five miles from the surf zone at Kennedy Space Center (KSC), Florida. The chloride deposition measurement site used to make predictions for this location (during model calibration) was also located approximately five miles inland at KSC. Both the Daytona Beach, Florida and Point Judith, Rhode

Island sites are on/adjacent to a beach where chloride deposition levels are known to be much higher than five miles inland. The proxy chloride measurement sites used to help infer the environmental severity for these two locations were located miles inland from the coastal test sites. Thus, the chloride deposition rates used by the model to make predictions were much

126 lower than what the actual rates were at the beach-side test sites. As a result, it should not be surprising that their predicted corrosion rates were not consistent with the majority of the other data shown on the figure. Due to the inability of obtaining chloride deposition measurements that were more reflective of conditions at the beach-side test sites (i.e., chloride data measured adjacent to surf zones), these two sites were dropped from the validation dataset.

Inspection of Figure 4-12 and the data used to construct it indicated that the predicted corrosion rates for Fort Campbell, KY were much higher than would be expected based upon other predictions shown in the figure. Upon identifying this perceived anomaly, a review of the proxy data and sites where they were measured was conducted in an attempt to determine the specific reasons why the predictions were so high.

As discussed earlier, chloride aerosols found in the atmosphere come largely from the ocean and are blown inland on the prevailing winds. Since aerosols are known to fall from airborne suspension as a function of the distance from coastlines, and Fort Campbell is far inland, this factor was quickly discounted as the reason why corrosion rates at this location were too high.

An analysis was then conducted to determine if there were point sources of contaminant emissions (e.g., a coal fired power plant or paper mill) nearby the proxy environmental characterization sites where the data used to make predictions were originally measured.

Ozone was eliminated from consideration since its presence in the troposphere is primarily due to the interaction between unburned hydrocarbons and UV radiation, which discounts the likelihood of point sources of pollution. Thus, it rapidly became apparent that the SO2 proxy for

Fort Campbell may not have been reflective of the conditions present at the corrosion test site.

The SO2 data used for this effort was obtained from the EPA’s AQS database. Metadata in this database pertaining to the individual sites where SO2 (and ozone) are measured is limited to the

127 address and coordinates as well as some information on the measurement technique. There is no indication given as to whether any monitoring site is intended for general pollution monitoring or conversely, whether its purpose is to monitor a nearby point source of emissions.

As a result, an investigation was conducted to determine whether SO2 levels near Clarksville, TN

(proxy site for Fort Campbell) were reflective of rural conditions or if there was a nearby source of emissions.

Internet searches in combination with a review of satellite imagery using Google Earth revealed that the Clarksville monitor site was very close to a zinc refinery and in the general area downwind from the largest coal fired power plant in the Tennessee Valley Authority (TVA) system (see Figure 3-6). Both of these sites are known to produce significant SO2 emissions.

Once it was discovered that SO2 emission sources were near the Clarksville monitor, this site was deemed unfit for predicting corrosion rates at Fort Campbell. An alternate proxy site at

Mammoth Cave National Park, KY was used as the source to replace the proxy data measured at

Clarksville.

After determining that point sources of emissions could lead to the use of SO2 data that is not representative of levels at the sites used for this model development effort, a review of all the

SO2 proxy sites was conducted. Two more corrosion test sites, each using the same proxy site to make their associated predictions, were identified during this review. These sites included

Tyndall AFB, FL and Fort Rucker, AL. Alternate proxy data were found for these locations and the monthly chloride models calibrated using data from three locations was applied. The results are shown on Figure 4-13 and Figure 4-14.

128

100000 y=2.8632x-772.15 90000 R² = 0.8373 )

2 80000 70000 g/cm

 60000 ( 50000 40000 30000

Predictions 20000 10000 0 0 5000 10000 15000 20000 25000 30000 35000 Test Results (g/cm2)

Figure 4-13 Improvement in R2 Value Results when Data Proxies are Revised and Model is not Applied to Locations Near Surf Zones (single activation energy formulation, validation data from eight independent locations, optimum H=0.62 eV)

100000 y=2.8458x-657.73 90000

) R² = 0.8296

2 80000 70000 g/cm

 60000 ( 50000 40000 30000

Predictions 20000 10000 0 0 5000 10000 15000 20000 25000 30000 35000 Test Results (g/cm2)

Figure 4-14 Results Obtained when the Monthly Chloride Model Using Three Activation Energies is Applied to Validation Data from Eight Independent Locations

As seen by Figure 4-13, eliminating the two beach sites that were outside of the calibration range and improving the proxies by replacing data measured near point sources of SO2 markedly

129 increased the R2 value for the model with a single activation energy. The same can be seen for

Figure 4-14, which pertains to the monthly chloride model with three activation energies. A comparison of the two models is shown by Table 4-14. Had it been possible to conduct significantly more simulations, the R2 value for the model with three activation energies might actually be higher than was calculated for the model with a single activation energy.

Regardless, the single activation energy formulation was used for all subsequent modeling efforts.

Table 4-14 Comparison of Metrics Pertaining to Monthly Chloride Models that were Calibrated Using Different Numbers of Activation Energies R2 Value R2 Value Model Activation Energy Error (calibration) (validation) Type

Single 0.62 eV/59.82 KJ/mol 114,924,809 0.9832 0.8373 Activation Energy

HCl: 0.97eV/93.59 KJ/mol Three 105,093,938 0.9827 0.8296 Activation HSO2: 0.58 eV/55.96 KJ/mol

Energies HO3: 0.84 eV/81.05 KJ/mol

Evaluation of Figures 4-13 and 4-14 reveals that both models exhibit homoscedastic behavior, as exemplified by the apparent normal distribution of statistical residuals pertaining to the data points and the associated trend lines. The lack of a funnel shape on the scatterplot provides an indication that variability of these data points is primarily due to random variability, which is the desired attribute. However, the slopes of the trend lines shown on these two figures indicate

130 that these models over-predict corrosion rates. Thus, further study was needed to ascertain the reasons for such over-predictions.

4.11 Refinement of the Monthly Chloride Model Simulations were conducted to ascertain whether using reduced ranges for the uniform distributions from which the twelve unknown coefficients were selected from would improve the accuracy of the monthly chloride model calibrated using characterization data from three locations. The earlier simulations discussed previously had range sizes (width of uniform distributions) that started with a value of 0.005 for each coefficient. During the simulation process, they were systematically reduced a minimum of seven times by successively multiplying each range by a factor of 0.9 to determine the size of the following range (see Section C.2.1.3 for more details). The final distribution range sizes were thus 0.00239148, although many were smaller since the code did not have provisions to stop simulations once the final range conditions were met. The use of a minimum of eight increasingly smaller range sizes was an arbitrary decision that appeared to provide a sufficient number of simulations to converge upon acceptable coefficient values (for most models). However, to determine whether accuracy could be improved further, smaller beginning and ending range sizes were used to refine coefficient values.

In order to determine whether refining the coefficients would have a significant effect on changing model accuracy, new simulations were conducted using the initial (eight distribution range size) model coefficients as a starting point. The initial ranges for the twelve distributions were set at a starting value of 0.0025 (close to the final range width for the initial simulations) and again sequentially reduced by a factor of 0.9 until at least fifteen different range sizes were considered during the simulation process. The final ranges were a maximum of 0.00057192.

Systematically narrowing the ranges while conducting simulations to identify the best fitting

131 coefficients enabled an approach that more effectively converged upon coefficient values that provided optimum accuracy for the number of simulations conducted. As can be seen by Table

4-15, this process led to a model with slight improvements in the error and R2 values in comparison to the model calibrated using wider ranges.

Table 4-15 Comparison of Metrics Pertaining to the Monthly Chloride Models that were Calibrated Using Different Starting and Ending Range Sizes Model Type Activation Energy Error R2 Value R2 Value (calibration) (validation)

Starting Range 0.005, Minimum of 0.62 eV/59.82 114,924,809 0.9832 0.8373 Eight Range Sizes KJ/mol

Starting Range 0.0025, Minimum of 0.638 eV/61.56 114,659,126 0.9833 0.8375 Fifteen Range Sizes KJ/mol

4.12 Simulations to Investigate the RH Threshold

-3.84 Constant RH Threshold -3.86 -3.88 -3.9 -3.92 -3.94 f(RH,T) -3.96 100 -3.98 90 -4 80 -4.02 70

60

Temperature Axis (K)

Figure 4-15 Illustration of the Constant Temperature-Relative Humidity Shape Function

132

All model development efforts up to this point were based upon using a constant relative humidity threshold of 60% RH, whereby nonzero corrosion predictions only result at times when humidity levels exceed this point (see Figure 4-15). The three Temperature-Relative Humidity shape functions employed by the model were specifically designed to accommodate this boundary condition. It should be noted, however, that this threshold value, which was determined by Vernon in the 1930’s [66], appears to pertain to observations made under ambient laboratory conditions.

50

) 45 3 40 35 30 25 20 15 10

Absolute humidity (g/mhumidity Absolute 5 0 270 280 290 300 310 320 330 Temperature (K)

Figure 4-16 Temperature Dependence of Absolute Humidity at a Constant RH of 60%

Figure D-1 (shown again here as Figure 4-16) is a plot that illustrates the role that temperature has upon the ability of air to absorb moisture. This specific curve was constructed for the case when the relative humidity was held at a constant 60% and the temperature range shown pertains to the model’s range. As can be seen, the amount of moisture in the air (in terms of absolute humidity – mass per unit volume) at the highest temperature considered by the model

(320.15K) is an order of magnitude higher than at the freezing point.

133

Since atmospheric water vapor at a constant 60% RH shows such a strong temperature dependence, it seems reasonable to conclude that the amount of moisture that adsorbs onto surfaces at a constant RH also shows a temperature dependence. Thus, it does not seem likely that sufficient adsorbed moisture for corrosive reactions to occur would be available on surfaces under situations where the RH is 60% and the temperature is lower than ambient laboratory conditions. Conversely, at temperatures above those present during Vernon’s laboratory experiments, it seems likely that the amount of adsorbed moisture at a constant 60% RH would exceed the minimum needed for corrosion to occur. Considering these two scenarios leads to the conclusion that the relative humidity threshold is likely not a constant 60% RH as postulated by Vernon but instead is a temperature-dependent function that includes Vernon’s observation.

The sure way to determine whether or not a temperature-dependent RH threshold exists would be to conduct controlled temperature/relative humidity testing using an environmental chamber. Because such testing was beyond the scope of this current effort, simulations were conducted to ascertain whether using a variable temperature-dependent RH threshold would improve model accuracy, thus providing some evidence that a variable threshold does exist.

Equations D.6 and D.16 (seen again here as Equations 4.3 and 4.4) display the linear and nonlinear threshold functions that were considered under this research program. Details concerning the derivation of these equations are found in Section D.2. The negative root for the nonlinear function provided the best results and was used in this current analysis. Both equations contain the variables RHcal and Tcal, which are the calibration points pertaining to

Vernon’s laboratory conditions (i.e., 60% RH at an assumed ambient temperature of 298.15K).

These variables were retained in the equations to facilitate possible future simulations to investigate different approaches to implementing RH thresholds. The remaining variable seen in

134 the equations, RHvar, corresponds to the unknown relative humidity threshold at the freezing point. Optimal values for this variable were determined by conducting simulations.

푅퐻푐푎푙−푅퐻푣푎푟 푅퐻푇퐻 = (푇 − 273.+15) 푅퐻푣푎푟 (4.3) 푇푐푎푙−273.15

푇−273.15 푅퐻 = ± + 푅퐻 (4.4) 푇퐻 √ 푇푐푎푙−273.15 푣푎푟 2 (푅퐻푐푎푙−푅퐻푣푎푟)

-3.84 -3.86 Laboratory -3.88 Conditions -3.9 -3.92 f(RH,T) -3.94 RHvar -3.96 -3.98 320.15 305 -4Linear 20 25

30 290 35 40

RH Threshold 45 50 55 60 65 70

75 273.15 80 85 90 95 Relative Humidity (%) 100

Figure 4-17 Illustration of the Temperature-Relative Humidity Shape Function Based Upon a Linear RH Threshold Function

Figures 4-17 and 4-18 illustrates the linear and nonlinear RH threshold functions described by

Equations 4.3 and 4.4. During the Monte Carlo simulation process, the variable “RHvar“ was randomly varied (along with the twelve other unknown coefficients) in order to identify the specific threshold functions that provided the greatest accuracy. The results shown here pertain to the RH threshold functions applied to the monthly chloride model with a single activation energy.

135

Parabolic RH Threshold

0 Laboratory -0.5 Conditions -1 -1.5 RHvar -2 f(RH,T) -2.5

-3 315 -3.5 305 -4 295 45 50 285 55 60 65 70 75 80 273.15 85 90 95 100 Relative Humidity (% RH)

Figure 4-18 Illustration of the Temperature-Relative Humidity Shape Function Based Upon a Parabolic RH Threshold Function (RHvar=80%RH)

Table 4-16 Metrics Pertaining to the Monthly Chloride Model (three calibration sites) and Models Using Variable Thresholds (refined simulations as described above) R2 Value R2 Value Model Type Activation Energy RH Error var (calibration) (validation)

Refined 0.638 eV/61.56 - 114,659,126 0.9833 0.8375 (Section 4.11) KJ/mol

Linear RHTH 0.352 eV/33.96 0 86,693,508 0.9873 0.8327 KJ/mol 95,621,664 Nonlinear 0.449 eV/43.32 0 0.9864 0.84

RHTH KJ/mol

Table 4-16 compares the refined monthly chloride model calibrated using data from three locations (the best previous model) in comparison to the two models constructed using variable

136 threshold functions (see Section E.3.6 for details). Both of the variable threshold models have total errors that are significantly lower than the refined model and the R2 values pertaining to their fit to the calibration data are slightly higher. When applied to the validation data, the R2 value of the nonlinear model is also slightly higher than the refined model while the value for the linear threshold model is slightly lower.

A surprising result seen on Table 4-16 concerns the value of RHvar, which pertains to the RH threshold at the freezing point. For both variable threshold models, the lowest error occurred when RHvar had a value of zero. Such a value does not seem possible since it implies that corrosion starts when the RH is only slightly higher than 0% at the freezing point. Thus, a problem with the mathematical form of the notional model (Equation 2.14) seems likely.

4.13 Simulations to Calibrate a New Basic Model with Improved Characterization Data

Upon nearing conclusion of this research program, a technical review was held at Wright

Patterson AFB, OH. Discussions revealed that the corrosion test measurements for Dobbins Air

Reserve Base, GA were in error [152]. Data from this site was used to calibrate all models previously discussed in this dissertation, which means that all calibrated models to this point were based upon erroneous data.

Because of the length of time it would take to recalibrate all of the models discussed previously, it was decided to initially focus upon one model to get an indication of the impact that a more accurate dataset had upon accuracy. To accomplish this, the data for Dobbins was replaced with data from Rock Island, IL. As seen in Table 4-10 (shown again here as Table 4-17) , the proxy ozone monitor site, which was operated year-round, was very close (less than 2 km) to the test site at Rock Island while the SO2 monitor was only slightly farther (approximately 4 km)

137 away. The weather station was also quite close (approximately 8 km). The only site that was somewhat far away was the chloride deposition measurement site, which was approximately 67 km from Rock Island. Since there are no local sources of chlorides (e.g., salt flats) and Rock

Island is located far from the ocean, the chloride proxy data used should be an adequate reflection of deposition rates at the corrosion test site. As will be discussed in the following sections, the simulations made using this new dataset initially focused on calibrating a Convex

Temperature-Convex Relative Humidity model. These efforts were followed by calibrating a

Linear Temperature-Linear Relative Humidity model.

Table 4-17 Rock Island Arsenal Corrosion Test Site and Associated Environmental Characterization Sites Location Identification Available Approximate Approximate Code Data Coordinates Distance Latitude Longitude from Test Site Rock Island NA – Corrosion Weight Loss 41.51821 -90.5371 - Arsenal, IL Test Site

SO2 Monitoring Site th 10 Street & 19-163-0015 SO2 41.53001 -90.58761 4.4 km Vine Street, 1 (2.8 miles) Davenport, IA

Ozone Monitoring Site 32 Rodman, 17-161-3002 O3 41.51473 -90.51735 1.7 km Ave., Rock (1.1 miles) Island Arsenal, IL

Chloride Monitoring Site Monmouth, IL78 Chloride 40.9333 -90.7231 66.8 km Warren (41.5 miles) County, IL Weather Monitoring Site Quad-City KMLI/72544 Temperature 41.45 -90.517 7.8 km International Dewpoint (4.8 miles) Airport

4.13.1 Convex Temperature-Convex Relative Humidity Model After the Dobbins data was replaced with Rock Island data, simulations were conducted to calibrate the subsequent model, which was based upon the same parabolic Temperature-

138

Relative Humidity shape functions used to develop the previous models. Figure 4-19 compares four such shape functions. The data points within the ellipse illustrate values of f(T)max (for each of the illustrated functions) and the dashed portions of each parabola (and the associated

-f(T)max values) provide the symmetry used to establish the simultaneous equations that led to the calculation of the model’s coefficient values. The y-axis intercept value of -4.0 is the boundary condition value that applies when the temperature is at the freezing point and/or the

RH is equal to the threshold value (RHTH).

-3.5 -3.6 -3.7 -3.8 -3.9 -4 f(T) -4.1 -4.2 -4.3 -4.4 -4.5 270 280 290 300 310 320 330 Temperature (K)

Figure 4-19 Parabolas of Different Widths

When conducting simulations using the revised environmental characterization data (including

Rock Island data), it was observed that the hourly corrosion rate predictions were often output as “NAN”, which is the computerized notation for “not a number”. Computer programs output such results after conducting mathematical operations such as dividing the number zero by zero or by taking the square root of a negative number. An analysis of the individual coefficients calculated during simulations that resulted in the output of NANs indicated that such results occurred when at least one of the three f(T)max values were very near to the -4.0 boundary

139 condition. As illustrated by Figure 4-19, such a condition corresponds to a very narrow parabola.

To determine why NANs were being calculated, the algorithms used to calculate the parabolic function coefficients were thoroughly tested and found to be working correctly. Since the mathematical operations applied by the simulation code were correct, the only other possible reason for the NAN output concerned the numerical accuracy of the calculations made during the simulation process. As mentioned above, NANs only resulted when the optimal parabolic functions were very narrow. Mathematically describing such a parabola requires highly accurate coefficient values. In an attempt to eliminate the NANs by improving coefficient accuracy, the algorithm was modified to convert the calculations from single precision into double precision floating-point format. However, this modification still did not provide the necessary precision.

4.13.2 Linear Temperature-Linear Relative Humidity Model As a result of the problem identified above, it was decided to replace the parabolic

Temperature-Relative Humidity shape functions with Linear Temperature-Linear Relative

Humidity shape functions. Linear functions were chosen because they could be used without having to solve simultaneous equations. Table 4-18 displays the coefficients for the Linear

Temperature-Linear RH shape function model constructed using monthly chlorides and proxy data for Rock Island, IL; Fort Drum, NY; and Kennedy Space Center, FL. As before, the set of coefficients that provided the lowest error values were determined using Monte Carlo simulations.

Figure 4-20 compares test measurements vs. predictions for the three calibration locations including Rock Island. As can be seen on the figure, the slope of the trend line and the associated R2 value are both close to 1.0, which indicates the model suitably fits the calibration

140 data. However, the y-intercept of 2567.2 combined with the scatter of the data points in relation to the trend line does indicate some imprecision.

Table 4-18 Optimum Coefficients for New Model Calibrated Using Rock Island Data Coefficient Value H 0.859

fCL(T,RH)max -3.99995 fSO2(T,RH)max -3.99999572 fO3(T,RH)max -3.998016623 f(T,CL)max -3.987494399 f(T,SO2)max -3.708761292 f(T,O3)max -3.995014098 2 ACL 0.00628469 g/cm 2 ASO2 0.00095 g/cm 2 AO3 0.000462654 g/cm

CL -0.281917975

SO2 0.199865098

O3 -0.094026503 Error 128480965.9

60000 y = 0.9022x + 2567.2 R² = 0.9527 ) 50000 2

40000 (mg/cm 30000

20000

Predictions 10000

0 0 10000 20000 30000 40000 50000 60000 Test Results (g/cm2)

Figure 4-20 Comparison of Corrosion Test Data and Associated Predictions for the Monthly Chloride Model Applied to Calibration Data (includes Rock Island Data, optimum H=0.81 eV)

Figure 4-21 visually validates the accuracy of the new linear model by comparing test measurements vs. predictions for seven independent locations where corrosion tests were conducted but not used for calibration. As can be seen on the figure, the slope of the trend line

141 is very near the optimal value of 1.0. However, the R2 value of 0.6067 is significantly lower than the value of 0.9527 that was calculated for the same model applied to the calibration data

(Figure 4-20). An additional indication of the poor degree of the model fit relates to the increasing size of the statistical residuals (the distance between the trend line and the data points) as the corrosion rates increase. Such funnel-shaped behavior indicates a coupling of error to model features/construction details rather than an indication of random variability of the data being plotted, which is the desirable case.

50000 y = 1.0644x + 2806.5 45000 R² = 0.6067 )

2 40000 35000 g/cm

 30000 25000 20000 15000

Predictions ( Predictions 10000 5000 0 0 5000 10000 15000 20000 25000 30000 35000 Test Measurement (g/cm2)

Figure 4-21 Comparison of Corrosion Test Data and Associated Predictions for the Monthly Chloride Model Applied to Validation Data (includes Rock Island Data, optimum H=0.81 eV)

4.13.3 Development of Final Proof-of-Concept Model As shown above, neither of the two new models (constructed from linear and quadratic

Temperature-Relative Humidity shape functions) applied to the revised environmental characterization data file possessed the desired accuracy. Thus, it was decided to revise the actual mathematical formulation of the model to find a form that improved upon the accuracy of the models already discussed.

142

H   K  exp[ ]{ Cl TfTA ,(exp[ RH )] Tf ,(exp[ Cl )] i kT Cl Cl  SO2 TfTA ,(exp[ RH )] Tf ,(exp[ SO )] SO2 SO2 2  O3 TfTA ,(exp[ RH )] ,(exp[ OTf )]} O3 O3 3

NOTE: Exponential functions are applied to values obtained from the six shape functions used by the model

Figure 4-22 Original Model Employs Exponential Functions to Calculate Hourly Interactions between the Acceleration Factors

As can be seen by Figure 4-22, the original base model employed six exponential functions, three of which were applied to hourly numerical values calculated using the three different

Temperature-Relative Humidity shape functions and three more that were similarly applied to the Temperature-Contaminant shape functions. All of the previously discussed models were constructed in such a fashion to be consistent with the NIST/SEMATECH model upon which they were based [135]. Applying exponential functions to the numerical values output from the individual shape functions increased the amount of the calculated interactions, especially at higher temperatures (see Figures 2-6 and 2-7 for illustration). Therefore, it was decided to revise the model formulation by removing the exponential functions and constructing new shape functions that would directly calculate the numerical interactions between the acceleration factors that form the basis for the corrosion rate predictions.

As discussed above, modifying the model as described above required the development of new shape functions to meet the boundary conditions. Specifically, these new functions were constructed to have a value of zero at the freezing point and/or the RH threshold (instead of a value of -4.0 as before). This was required so the subsequent corrosion rate predictions had a

143 value of zero at the boundary conditions. Like before, the new Temperature – Relative Humidity shape functions had a maximum value at the point corresponding to the combination of 100%

RH and the highest temperature considered by the model (i.e., f(T)max). Similarly, the

Temperature – Contaminant shape function had its maximum value at the combination of the maximum contaminant level considered by the model and the highest temperature considered by the model.

H    Cl  Moisture and contaminants adsorb Ki exp[ ]{ Cl Cl TfTA RH Tf Cl ),(),( as “monolayers” on surfaces kT  SO2 TfTA RH Tf SO ),(),(  Insufficient Moisture for Corrosion SO2 SO2 2  O3 TfTA RH ,(),( OTf )} O3 O3 3

Threshold Condition f(T) f(T)max

f(T,RH)

Above Threshold (higher rates)

H2O SO 2 Temperature O3 Cl- Boundary Condition Relative Humidity (= zero) Boundary Condition (= zero)

Figure 4-23 Notional Illustration of the Final Proof-of-Concept Model

In addition to displaying the new model, Figure 4-23 displays a graphical illustration of the adsorption of water vapor and contaminants on material surfaces and how the related new nonlinear shape functions were constructed to address the temperature and RH threshold boundary conditions. Equation 4.5 represents the new temperature function (illustrated by f(T) on the figure) while Equation 4.6 shows the related Temperature-Relative Humidity shape

144 function. The complete derivation of the functions is found in Section A.3.1. These equations were used to construct three similar functions, one for each of the contaminants considered by the model.

Final Temperature Function (to construct Temperature - Relative Humidity shape functions)

188 [ ](273.15 T) )f(T)( 2 Tf )(  max (4.5) 94 2 max )f(T)(

Form of the Final Temperature - Relative Humidity Shape Functions

1 RH  [4 TH ](  RHRH ) f(T)2 TH Tf RH ),(  (4.6) 1 RH [2 TH ] f(T)2

During the 2013 Meeting of the Electrochemical Society, Dr. Ivan Cole from Australia’s

Commonwealth Scientific and Industrial Research Organisation (CSIRO) described an approach to nonlinear corrosion modeling that was being developed by his organization. Rather than using exponential functions such as used in the preliminary models developed here, CSIRO was constructing a model based directly upon quadratic functions [153]. Replacing exponential functions with quadratic functions still provides the ability to consider nonlinear effects. An added benefit with respect to this current effort is that employing such an approach simplifies the computational processes by eliminating tens of thousands of exponentials and associated computations per simulation, thus enabling faster model development efforts. It was therefore decided to revise the approach to constructing Temperature-Contaminant shape functions by eliminating the exponential functions and directly replacing them with quadratic functions.

145

The new Temperature-Contaminant shape functions for use in the revised model shown on

Figure 4-23 are derived in Section A.3.2. Equations A.196, A.203, A.212, and A.221 (seen below as Equations 4.7 through 4.10) were used to construct the temperature functions that form the basis for the new Temperature-Contaminant shape functions (Equation A.240, seen here as

Equation 4.11). Obviously Equations 4.8 through 4.10 could be simplified by conducting the elementary mathematical operations contained within them. However, they were retained in the forms shown here since the computer algorithms used to calculate the “a”, “b”, and “c” coefficients during the simulation process used the exact same forms so that computer machine precision made all calculations to reduce round-off error.

Final Temperature Function (to construct Temperature - Contaminant shape functions)

Tf )( 2 bTaT  c (4.7)

)(  .15320 bcTf a  max (4.8) 15320 ).( 2

15226 2 ).( (4.9) [ 1] 15320 ).( 2 b   Tfc max ))(( 15226 2 ).( .[ 15226  ] .15320

2 2 2 (226.15) (226.15) (273.15 [)  Tf )(]1 max [  Tf )(]1 max (273.15 2 Tf )() (320.15)2 (320.15)2  max [ ]  273.15 (320.15)2 (226.15)2 (226.15)2 [226.15  ](320.15) 226.15  c  320.15 320.15 (4.10) (226.15)2 (226.15)2 (273.15 2[)  ]1 [  ]1 (273.15)2 (320.15)2 (320.15)2 [ [ ]  273.15  ]1 (320.15)2 (226.15)2 (226.15)2 [226.15  ](320.15) 226.15  320.15 320.15

146

Form of the Final Temperature – Contaminant Shape Functions

Tf )( 2  2 CTfC ),( (4.11) Cmax

Similar to above, three different Temperature-Contaminant shape functions were employed by the model, one for each of the contaminants. These equations were constructed by replacing

2 the 퐶푚푎푥 value in Equation 4.11 with the maximum chloride, SO2, and ozone values used to construct the previous models. As proposed in Chapter 7 (“Recommendations for Future

Work”), the approach used to construct the Temperature-Contaminant shape functions employed by the final proof-of-concept model can be revisited later to determine which nonlinear approach (there are many others that could also be considered) will ultimately provide the best results.

The revised model as described above makes more physical sense than the exponential models based upon the form of the Eyring equation published by NIST [135]. As can be seen by the notional graphic seen on Figure 4-23, the new Temperature – Relative Humidity shape functions have a value of zero at the boundary conditions. As the temperature and relative humidity levels increase from these boundaries, the functional values rapidly increase. However, at higher temperature and relative humidity levels, the shape function values begin to flatten out.

Thus, corrosion rates calculated from such shape functions will consider the effects of moisture adsorption on surfaces by accounting for the initiation, growth, and equilibration of corrosion rates based upon increases to temperature and relative humidity levels. The revised

Temperature – Contaminant shape functions based upon a parabolic function with upward curvature ensures that corrosion rates also increase with increasing temperature and contaminate levels. Thus, when properly constructed and calibrated, the three sets of shape

147 functions (as seen in the Equation in Figure 4-23), one set for each of the contaminants, should provide a way to calculate accurate corrosion rate estimates resulting from the combined effects of temperature, relative humidity, chlorides, SO2, and ozone levels.

Table 4-19 Coefficients Determined via Simulations for the Final Cumulative Damage Model Material 1010 Steel Activation 1.68 Energy (H)

fCl(T,RH) 4.02E-11

fSO2(T,RH) 6.25E-07

fO3(T,RH) 8.77E-06 f(T,CL) 1.33E-13

f(T,SO2) 7.00E-05

f(T,O3) 9.90E-06 2 ACl 1.80E-15 g/cm 2 ASO2 3.32E-09 g/cm 2 AO3 6.95E-10 g/cm

Cl 4.776592

SO2 -2.3335

O3 -2.4272

The optimum coefficients for the final proof-of-concept model, which were identified via simulations, are seen in Table 4-19. Figure 4-24 compares predictions made with this new nonlinear model with the test measurements used during the calibration process. As seen on the figure, the trend line’s slope of 0.9512 is close to 1.0, which indicates the magnitude of each prediction is closely related to the associated test measurement. The y-intercept of 1036.1 indicates some inaccuracy of the model but since this value is small in comparison to the maximum range of predictions (over 50,000), then it should not unduly affect the accuracy of predictions, especially for predictions for locations with more severe (i.e., other than benign) environmental conditions. The R2 value of 0.9569 is very high, thus indicating the model has an excellent degree of fit to the calibration data. The final observation concerns the proximity of

148 the data points to the trend line. All of these points lie close to the line, which also indicates that the model has a high degree of fit.

60000 y=0.9512x+1036.1

) 50000 R² = 0.9569 2

40000 g/cm  ( 30000

20000

Predictions Predictions 10000

0 0 10000 20000 30000 40000 50000 60000 Test Results (g/cm2)

Figure 4-24 Comparison of 1010 Steel Test Points and Associated Predictions (Calibration Data)

Figure 4-25 illustrates the results when the new model was applied to environmental data for seven of the eight independent corrosion test locations used to validate the previous candidate models. Data from the eighth site, Rock Island, was not considered during validation since it was used to help calibrate the current model. As can be seen, the slope of the trend line on this figure is even closer to the optimum value of 1.0 than was seen above for the model applied to calibration data (Figure 4-24). The y-intercept is over twice as large as seen on the calibration plot but this should not be problematic since the magnitude is still small in comparison to the maximum value of the plot. The R2 value is smaller (0.8628 vs. 0.9569) but this too is not surprising since a model should always fit calibration data more closely than independent validation data. In comparison to the calibration plot, there is more scatter of the data points in

149 relation to the trend line but these points do not exhibit a pronounced “funnel shape”, which is an undesirable attribute (see Section C.4 “Model Attributes”).

40000 y=1.0702x+2502.7 35000

) R² = 0.8628 2 30000

g/cm 25000  ( 20000 15000 10000 Predictions 5000 0 0 5000 10000 15000 20000 25000 30000 35000 Test Results (g/cm2)

Figure 4-25 Comparison of 1010 Steel Test Points and Associated Predictions (Validation Data)

The current model is the most accurate of those developed under this research program. Like the earlier models, a comparison of model predictions to the associated calibration data reveals that the R2 value is very high, the trend line has a slope near 1.0, and the scatter of data points is uniformly distributed very near to the trend line. These facts indicate that the calibration process using Monte Carlo simulations is robust and repeatable. However, the far more interesting thing about the current model concerns the validation plot. Figure 4-25 illustrates that the new model described here has the highest validation R2 value of those developed under this current research effort. In addition, it is the first model with a validation trend line slope near 1.0, thus representing a significant improvement over all earlier models (see Figure 4-14 for an illustration of results from the preliminary model with the highest R2 value). More work can be done to improve this model (see Chapter 7 for recommendations) but the results shown here

150 demonstrate the utility of employing the cumulative damage approach to predict the atmospheric corrosion rates of steel.

151

CHAPTER 5

DISCUSSION

5.1 Discussion of Preliminary Models Tables 5-1 through 5-3 compare the results from the various models that were described in the previous chapter. Please note that the order of the models shown on these tables and discussed in the following sections is slightly different from the chronological order in which they were developed in Chapter 4. The tables shown here were constructed to organize the earlier results in a way that facilitates direct analytical comparisons. This will enable the identification of the best model attributes while also providing clues as to approaches that could lead to future models with even greater accuracy.

5.1.1 Initial Modeling Efforts

5.1.1.1 Simulations Used to Identify the Optimum Temperature-Relative Humidity Shape Function Table 5-1 compares a series of different models pertaining to the initial attempts to formulate a predictive methodology based upon cumulative damage. The first efforts, which are shown in shaded area at the top of the table, examined three different shape functions. These functions were the ones deemed most promising by the analysis described in Section E.1. As discussed in

Section E.1.1, the nonlinear shape functions used during these initial efforts, and those that follow, were limited to parabolic upward curvatures. They are indicated on the table as

“convex”. It is possible, if not likely, that shape functions constructed from some other nonlinear function could lead to improved model accuracy. Due to the large number of

152

Table 5-1 Comparison of Metrics for Initial Models Number of Number of Temperature- Chloride Activation Total R2 Value for R2 Value for Calibration Data Calibration Validation RH Shape Data Energy Calibration Calibration Validation Sources Sites Sites Function Types (eV) Model Error Data Data

Linear Temp Annual H=0.81 192,176,163 0.973 China Lake, CA; Linear RH - Ft. Drum, NY, Linear Temp Annual H=0.86 171,866,771 0.9758 4 - Dobbins, GA; Convex RH - Kennedy Space Convex Temp Center, FL Annual H=0.66 167,324,659 0.9765 Convex RH -

China Lake, CA; Annual H=0.66 167,319,440 0.9765 0.3057 Ft. Drum, NY, Convex Temp 4 10 Dobbins, GA; Monthly H=0.64 148,424,895 0.9791 0.2824 Convex RH Kennedy Space Center, FL Weekly H=0.66 155,259,104 0.9782 0.3102

Ft. Drum, NY, Annual H=0.65 133,534,628 0.9798 0.3050 Dobbins, GA; Convex Temp 3 10 Monthly H=0.62 114,924,809 0.9832 0.2807 Kennedy Space Convex RH Center, FL Weekly H=0.63 121,418,584 0.9821 0.3061

Annual H=0.91 38,141,437 0.9309 0.6298 Convex Temp 1 9 Rock Island, IL Monthly H=0.93 44,305,051 0.9783 0.6268 Convex RH Weekly H=0.96 28,561,646 0.9773 0.7687

153 simulations and associated time needed to calibrate each model, the preliminary work under this research effort was limited to examining convex Temperature-Relative Humidity shape functions.

The twelve different models described in Table 5-1 are arranged into four different groups. The first group of three (seen in the shaded area at the top) pertains to models that were constructed using different shape function formulations. These models were calibrated using proxy environmental characterization data from four calibration sites. Details concerning the models are discussed in Sections 4.1 through 4.3 and they are all based upon annual chloride deposition rates. The model calibrated using Convex Temperature - Convex Relative Humidity shape functions has a lower error and a higher R2 value (for calibration data) than the other two models in this group. As a result, models constructed from these shape functions were used for most of the remaining development efforts.

5.1.1.2 Simulations Used to Evaluate Different Chloride Deposition Intervals Immediately below the shaded area on Table 5-1 are metrics addressing the second group of three models, which were developed from the same proxy data used for the first group. The first model in this new group is mathematically similar to the last model in the first group

(Convex Temperature - Convex Relative Humidity shape functions combined with annual chloride deposition). This new model is discussed in Section 4.5. The difference between the two models relates to the maximum value considered by the Linear Temperature-Linear

Chloride shape function (see Equations 2.51 and 4.1). The maximum chloride level for each of the three initial models in the shaded rows was established via an analysis of the calibration data. This maximum level was reevaluated after the validation dataset (ten additional locations) was constructed, which occurred after the initial three models had been constructed. When this expanded dataset was analyzed, it was found that the maximum value for chloride deposition

154 exceeded the range established for the initial Temperature-Chloride shape function. Thus, the model formulation had to be slightly revised to accommodate the full range of chloride values found in the combined calibration/validation dataset. When the new annual chloride model constructed from Convex Temperature-Convex Relative Humidity shape functions was compared with the initial one (bottom row of shaded area in Table 5-1), it was observed that they had very similar calculated errors and identical R2 values (with respect to the calibration data). This illustrates the robustness and repeatability of the inverse Monte Carlo model development process. The Linear Temperature-Linear Chloride shape function based upon the revised maximum chloride level was used for all but the last of the models.

The second group of three models was developed in order to evaluate which chloride deposition interval would provide the greatest accuracy. Annual (Section 4.5), monthly (Sections 4.6 and

E.3.1.1), and weekly (Sections 4.6 and E.3.1.2) average chloride deposition rates were used to develop these models. A comparison of the results shown on the Table 5-1 indicates that the model constructed from monthly chloride deposition rates has the lowest error (148,424,895) and the highest R2 value (0.9791) for the calibration data. It is also seen that the R2 values pertaining to predictions made by applying the three calibrated models to the independent validation data for all ten sites are very low, which indicated that something was wrong with the model formulation, the environmental characterization data, or both. Additional simulations were conducted to identify the cause of this problem.

5.1.1.3 Simulations Used to Evaluate a Revised Calibration Dataset and Different Chloride Deposition Intervals The third group of models seen on Table 5-1 uses the same shape function formulation as the second group but revises the calibration dataset by removing data for China Lake, California, which limits the proxy environmental characterization data sites to three. This was done

155 because an analysis of the environmental data and associated corrosion rates for China Lake provided some evidence that the rates reported for this site might be erroneous (see discussion in Section 4.7). Details pertaining to the new annual, monthly, and weekly chloride deposition models are found in Sections 4.7 and E.3.2. Results shown on the table indicate this new monthly chloride model had a lower error (114,924,809) and a higher calibration R2 value

(0.9832) in comparison to the previous monthly chloride model calibrated using data from four sites. However, the R2 value pertaining to the monthly chloride model applied to the validation data is still quite low (0.2807), which indicates the need for further analysis.

Please note that the errors determined using the statistical Residual Sum of Squares method

(Section C.3.1) for the monthly chloride model calibrated using data from three calibration sites cannot be directly compared with the error calculated for the model developed using data from four sites. This is because there were 16 statistical residuals used in the calculation of the errors for models calibrated using data from four sites and only 12 used for the new three site model.

5.1.1.4 Simulations Used to Evaluate a New Model Calibrated Using High-Quality Data from a Single Site An analysis of all of the calibration and validation site proxy data was conducted in order to find a corrosion test site with nearby proxy environmental characterization sites. This was done to ascertain whether the accuracy of the proxy data could be responsible for the low validation R2 value. This analysis led to the identification of Rock Island, Illinois as the site with the highest quality data. Details for this analysis are found in Sections 4.8 and E.3.3.

Analysis of the Rock Island models shown in Table 5-1 reveals that the monthly chloride model has the largest error (44,305,051) but also the highest calibration R2 value (0.9783), which is a contradictory set of metrics since larger errors are bad while larger R2 values are good. The

156 weekly chloride model has a significant improvement in the validation R2 value (0.7687) in comparison to the other Rock Island models as well as all other previous models.

It should be noted that in comparison to the models discussed previously, the validation sites used for this analysis were limited to nine instead of ten as before. This is because Rock Island was among the original ten validation sites and in this case it was extracted from these sites in order to become the single calibration location.

Despite the much higher R2 values that result when the models calibrated from data for Rock

Island is applied to the validation data, the best Rock Island model has problems that make it unsuitable for making accurate predictions. Inspection of the scatterplot for this model (Figure

4-10) reveals that the model exhibits heteroscedastic behavior (see Section C.4), which is an undesirable trait that indicates the errors attributed to the model are due to something other than random variability. It is unknown why the model calibrated using high-quality data from a single location exhibits heteroscedastic behavior, but it could be due to the lack of diversity (i.e., insufficient high and low values for each acceleration factor) in the calibration dataset.

157

Table 5-2 Metrics for Best Initial Model Compared to Models with Revised Validation Data and Refined Simulation Parameters Number of Number of Calibration Data Temperature- Chloride Specific Attributes Activation Total R2 Value for R2 Value for Calibration Validation Sources RH Shape Data Energy Calibration Calibration Validation Sites Sites Function Types (eV) Model Error Data Data

From Table 5-1. Ft. Drum, NY, Initial model with Dobbins, GA; Convex Temp 3 10 Monthly lowest error, H=0.62 114,924,809 0.9832 0.2807 Kennedy Space Convex RH highest R2 for Center, FL calibration data No Beach Sites in Ft. Drum, NY, Validation Data. Updated SO Dobbins, GA; Convex Temp Monthly 2 3 8 proxy. Significant H=0.62 114,924,809 0.9832 0.8373 Kennedy Space Convex RH improvement in Center, FL R2 for validation data No Beach Sites in Validation Data. Ft. Drum, NY, Updated SO2 proxy. Refined Dobbins, GA; Convex Temp Monthly 3 8 Range Sizes. H=0.638 114,659,126 0.9833 0.8375 Kennedy Space Convex RH Lower error and Center, FL higher R2 values. Best refined initial model

158

5.1.1.5 Selection of the Most Accurate Initial Model The initial model with the highest calibration R2 value thus far is the monthly chloride model developed using data from three proxy sites. This was the model on which all subsequent initial models were based.

5.1.2 Refinement of Initial Models Table 5-2 compares metrics for the best initial model (shaded row at top of the table) in comparison with two new scenarios. The first new scenario is the same aforementioned optimum model (Convex Temperature-Convex RH shape function calibrated using data from three locations) but in this case, the model was applied to a revised validation dataset (see

Section 4.10). The second scenario has the same mathematical formulation for the model but in this new case, which is discussed in Sections 4.11 and E.3.5, far more simulations were conducted to further refine the coefficients and improve correlation. The resultant model was then applied to the revised validation dataset developed for the first new scenario.

5.1.2.1 Best Initial Model Applied to Revised Validation Data With the exception of the Rock Island Models, all models up to this point were applied to validation data from ten independent sites (i.e., sites not used to calibrate the models). Analysis of the scatterplots such as seen in Figure 4-12 revealed that predictions for two of these sites

Daytona Beach, FL and Point Judith, RI) did not follow the trends for most of the other locations

(see discussion in Section 4.10). Both of these sites were immediately adjacent to surf zones with very high chloride deposition rates. The National Atmospheric Deposition Program does not measure chloride deposition in such locations so in these two cases, predictions were made using rates that were measured miles inland from the coast. Thus, it is not surprising that the model results for these two locations do not follow the trends seen for other sites. Due to the problem with the chloride proxies, these two sites were removed from the validation dataset.

159

Another problem with the validation data concerns SO2 proxy measurement sites. As described in Section 4-10, an investigation was conducted to determine whether any of the proxy sites were near point sources of SO2 emissions. During this investigation, which involved internet searches and evaluation of satellite imagery, it was discovered that several SO2 proxy sites were near point sources of SO2 emissions. Thus, the proxy database had to be revised to replace the suspect data with SO2 data that was more reflective of the conditions at the corrosion test sites.

The second row of Table 5-2 contains metrics for the best initial model (same formulation of the model shown in the first row) but in this case, it was applied to validation data that was revised by eliminating the two locations adjacent to the surf zones and by replacing the SO2 data that was measured near point sources of emissions. The R2 value of 0.8373 indicates a significant improvement over the initial value of 0.2807. These results indicate that the model formulation was not the primary cause of the low validation R2 value. Thus, care must be taken when both selecting proxy environmental data to use when making predictions and also to ensure the model is used for locations consistent with calibration parameters.

While a validation R2 value of 0.8373 indicates a high degree of fit to the data, the fact that it is less than the value obtained when the same model is fit to the calibration data indicates that there may be an opportunity to further improve the formulation. This issue will be discussed later.

5.1.2.2 Assessment of a Best Initial Model Developed Using Refined Distribution Ranges The next step in the model development process was to examine the effect of using smaller ranges for the uniform distributions from which the random numbers were selected (see

Sections 4.11 and E.3.5). All models to this point were calibrated using starting ranges of 0.005

(applied to each of the twelve coefficients). Through the process described in Section 4.11 and

160

Section C.2.1.3, the range sizes used during simulations for the previous models were reduced a minimum of seven different times with each new range being 90% of the size of the previous range. Thus, after (at least) eight total ranges, the uniform distributions each had a final

(maximum) range size of 0.00239148.

Initial range sizes of 0.0025 were used when conducting simulations to ascertain whether refining the distribution ranges would improve model accuracy. In most cases, the starting value for each unknown coefficient was the value determined using results from earlier simulations that started with 0.005 range sizes. Under the refinement process, the range adjustment process was completed a minimum of fourteen additional times so that the final ranges were a maximum of 0.00057192. Due to the lack of automation, some of the final ranges were even smaller if more than fourteen range reductions were used. The intent of this process was to ascertain whether reducing the final range sizes so they were an order of magnitude smaller than those used in the earlier simulations would lead to coefficients values that provided improved convergence and more accurate models.

As can be seen in the last row on Table 5-2, refining the uniform distribution ranges from which random numbers were selected slightly improved model accuracy, as indicated by the lowest error yet and the highest R2 values. The particular model described in this row is the most accurate one yet. Thus, it was used for comparison to the following models.

From a simulation standpoint, it makes little sense to immediately start with very small range sizes as used in the refinement process described here. This is because it took a lengthy period of time to conduct enough simulations to converge upon the best fit coefficients. Instead, a process starting with larger ranges was used to hunt for the approximate activation energy and associated coefficients. These values were then used as a starting point for subsequent

161 simulations using the refined range process described here. The activation energy and related coefficients identified through this process resulted in a model with acceptable accuracy.

5.1.3 Revised Mathematical Formulations of Initial Models

5.1.3.1 Assessment of a Model Based Upon Three Activation Energies The three activation energy model is fully described in Sections 4.9, 4.10 (improved validation data), and E.3.4. This model was constructed to determine whether explicitly considering three independent chemical reactions rather than a single reaction based upon a mixture of contaminants would provide better accuracy. As seen in the second row of Table 5-3, this model has a slightly lower error but also slightly lower R2 values in comparison to the top

(shaded) row. Inspection of Figure 4-11 indicates there were significant perturbations in the error vs. activation energy curve. Such behavior provides evidence that the optimal activation energies and associated coefficients had not yet been identified. This particular model has two additional unknown coefficients (14 total vs. 12) that were not found in any of the previous models. Adding additional unknowns significantly increases the difficulty in converging upon acceptable solutions due to the increased number of combinations of variables. Three million simulations per run were conducted when attempting to calibrate this model (versus 1.5 million for the other models) but the perturbations seen in the error vs. activation energy plot imply that far more simulations are need to establish convergence. Establishing such convergence should decrease error while increasing R2 values. A more powerful computer is needed to research this approach more fully.

162

Table 5-3 Metrics for Best Initial Model Compared to Models with Revised Mathematical Forms or Revised Calibration Data Number of Number of Temperature- Chloride Total Model R2 for R2 for Calibration Specific Model Activation Calibration Validation RH Shape Data Error Calibration Validation Data Sources Attributes Energy (eV) Sites Sites Function Types (calibration) Data Data Dobbins, GA; From Table 5.2. Fort Drum, NY; Convex Temp Best refined 3 8 Monthly 0.638 114,659,809 0.9833 0.8375 Kennedy Space Convex RH initial model Center, FL Three activation Dobbins, GA; energies. 0.97 Fort Drum, NY; Convex Temp Reduced error 3 8 Monthly 0.58 105,093,938 0.9827 0.8296 Kennedy Space Convex RH compared to 0.62 Center, FL above. Slightly lower R2 values. Dobbins, GA; Linear RH Fort Drum, NY; Convex Temp threshold 3 8 Monthly 0.352 86,693,508 0.9873 0.8327 Kennedy Space Convex RH modification to Center, FL best initial model Dobbins, GA; Nonlinear RH Fort Drum, NY; Convex Temp threshold 3 8 Monthly 0.449 95,621,664 0.9864 0.84 Kennedy Space Convex RH modification to Center, FL best initial model Fort Drum, NY; Revised Kennedy Space cumulative Center, FL; Convex Temp damage model Rock Island, IL Convex RH formulation 3 7 Monthly 1.68 106,746,708 0.9569 0.8628 (revised constructed from formulation) a new characterization data set

163

5.1.3.2 Assessment of Models Based Upon Temperature-Dependent RH Threshold Functions A revised model that includes a linear temperature-dependent RH threshold rather than the constant 60% RH threshold used in all previous models is fully described in Sections 4.12, D.2.1, and E.3.6.1. In addition to this linear function, a nonlinear RH threshold function based upon a variable parabolic function was also constructed. This second model is fully described in

Sections 4.12, D.2.2 and E.3.6.2. Both of these RH threshold functions were specifically designed to accommodate (as a calibration point) a 60% RH threshold at ambient laboratory conditions

(i.e., 298.15K). Thus, they are consistent with Vernon’s laboratory observations [66].

As described in the referenced sections, simulations were conducted to construct functions so they include both Vernon’s observed RH threshold and a variable RH threshold at the freezing point. This variable threshold point was determined using Monte Carlo simulations at the same time that the 12 other coefficients were being determined. Thus, there were 13 total unknowns for these formulations. The intent of the simulations was to identify specific threshold functions that improved the Temperature-Relative Humidity shape function formulations so the resultant models provided better fit to the calibration data.

Table 5-3 compares metrics for the two variable RH threshold function models (3rd and 4th rows) with the results from the best refined initial model (first row). As can be seen, both variable threshold models have errors that are significantly lower than the baseline, with the linear model being the best by a significant margin. They also have better R2 values, with the linear threshold model having the highest value pertaining to the calibration data (0.9873) while the nonlinear model has the highest value (of all models yet considered) when applied to the validation data (0.84). As seen here, models based upon RH threshold functions appear to have

164 the best accuracy in comparison to all other models. However, further analysis reveals problems that must be addressed.

5.1.3.2.1 Analysis of RH Threshold Functions

0 -0.5 -1 -1.5 -2

f(RH,T) -2.5

-3 312.5 -3.5 302.5 -4 292.5 0 282.5 10 20 30 40 50 60 273.15 70 80 90 100 Relative Humidity (% RH)

Figure 5-1 Convex Temperature-Convex RH Shape Function Modified Using a Nonlinear RH Threshold Function

The results from the variable RH threshold models shown on Table 5-3 are encouraging.

However, further analysis of just how these functions modify the Temperature-Relative

Humidity shape functions provides indications of problems with the current mathematical foundation of the model.

Figure 5-1 illustrates the nonlinear RH threshold function applied to the Convex Temperature-

Convex Relative Humidity shape function. A figure based upon the linear RH threshold would look somewhat similar. The temperature dependent threshold value, RHvar, is the point where the curved surface intersects the relative humidity axis on the horizontal plane established by a value of -4 on the vertical axis. As discussed previously, the numerical results from the

165 simulation give a value of zero for RHvar at the freezing point. Evaluation of the numerical results from the model indicate that this was the case. However, because of the coarseness of the mesh used to plot the data shown in Figure 1, it does not appear to attain this value. This perceived problem is an artifact of the plotting program rather than a problem with the calculations.

The temperature dependent threshold is indicated by the intersection of the back side of the curved surface with the horizontal plane seen on Figure 5-1. The plot indicates that as the temperature increases, the critical amount of relative humidity needed for corrosion to occur also increases. For example, this implies that the RH threshold above the calibration temperature (298.15K) is higher than 60% RH, which is highly unlikely since the combination of higher relative humidity levels and higher temperatures is favorable for increasing the kinetic reaction rate of electrochemical reactions.

1

0.8

0.6

0.4 exp[f(RH,T)] 0.2 312.5 302.5 0 292.5 0 282.5 10 20 30 40 50 60 273.15 70 80 90 100 Relative Humidity (% RH)

Figure 5-2 Exponential Function Applied to the Values Obtained from the Convex Temperature-Convex RH Shape Function Modified by a Nonlinear RH Threshold Function (adjusted values)

166

Figure 5-2 was constructed by applying the exponential function to the full range of values calculated by the Convex Temperature-Convex RH shape function modified by a nonlinear RH threshold function (as illustrated by Figure 5-1). As mentioned above, results using the linear threshold function would be similar. As can be seen, at temperatures towards the midpoint of the range considered by the model (e.g., 290K and 80% RH), the value calculated by the exponential function is greater than for the same relative humidity level at higher temperatures. This implies that the calculated corrosion rate at the lower temperature point is higher than the rate for the same humidity level at a higher temperature. This too makes little physical sense and thus indicates a problem with the mathematical formulation of the

Temperature-Relative Humidity shape function.

Despite the fact that shape functions employed by the models discussed here are likely to be in need of revision, predictions made by using them are surprisingly accurate, especially considering that the best model has an R2 value of 0.84 when applied to environmental characterization data for eight independent validation locations. This accuracy is likely due to the robustness of the inverse Monte Carlo modeling approach and its ability to construct models that compensate for poor mathematical formulations through the identification and employment of optimal modeling coefficients. When combined with an improved future mathematical formulation, this robustness should lead to a model with even better accuracy.

5.2 Discussion of the Final Proof-of-Concept Model

5.2.1 Accuracy of Legacy Models Table 5-4 describes numerous atmospheric corrosion models that were developed and published over the past forty years. Descriptions of these models can be found in Section 1.2.4.

Some of these models were calibrated from data measured in single locations (e.g., three models developed by Kobus) or a limited geographical region while others were calibrated from

167

Table 5-4 Legacy Atmospheric Corrosion Models Developed Over the Past Forty Years Coefficient of Year Researchers Corrosion Test Sites Determination (R2) 2000 Kobus [28] 3 sites in Poland 0.96-0.99* 2002 Lien and San [36] 7 sites in Vietnam 0.88 1985 Kucera [10] 7 in Sweden and 0.96 Czechoslovakia 1985 Haagenrud [11] 32 sites in Scandinavia 0.58 1986 Haagenrud, 15 in 26 x 31 km region 0.86 Henrickson, Gram in Norway [12] 1988 Duncan and Balance 18 sites in New Zealand 0.94-0.95* [13] 1988 Kucera, Haagenrud, 32 sites in Scandinavia 0.89 Atteraas, and Gullman [16] 2002 Dean and Reiser [29] 52 sites, 14 nations 0.83** 2002 Roberge, Klassen, 52 sites, 14 nations 0.65 and Haberecht [26] 2002 Morcillo, Almeida, 52 sites, 14 nations 0.58 Chico, and de la 2002 Fuente [33] 75 sites, 14 nations 0.46 2001 Tidblad, Kucera, 39 sites, 12 nations NR Mikhailov, et al [23] 2007 Klinesmith, McQuen, 52 sites, 14 nations 0.67 and Albrecht [27] 2008 Abbott [25] Numerous sites 0.75-0.80* internationally 1986 Benarie and Lipfert 41 world-wide 0.45 [15] 1980 Barton and Czerny 7 sites: 0.76 [18] 3 USSR, 2 Sweden/Czechoslovakia, Bulgaria 1974 Haynie and Upham 57 sites in US 0.77 [20] 1982 Mikhailovsky [22] 8 in Russia NR * Multiple models were developed, some for different materials and others for specific exposure times ** More than half of the initial data points were thrown out to improve R2 NR: No R2 values reported for individual models far more diverse data measured in multiple locations within large geographical regions or in multiple countries spread around the world. It should be noted that some of the competing

168 models shown on the table were developed using the exact same data sets, which were developed under large international programs.

None of the papers presenting the legacy models identified in Table 5-4 discussed model validation. Thus, it seems reasonable to assume the R2 values (index-of-fit) seen on the table relate to the fit of the individual models to their calibration data. This implies that the accuracy of these existing models when applied to independent validation data is unknown. Therefore, the values seen on the table cannot be used to either directly falsify or support the cumulative corrosion damage hypothesis.

Despite not being useful for directly evaluating the (validation) accuracy of candidate models developed under this current research program, the R2 values and supporting information seen on Table 5-4 were used to help evaluate the fit of the final cumulative corrosion damage models to their calibration data. As seen on the table, when legacy models were calibrated from data measured under narrow environmental conditions (i.e., a single location or small geographical region), they tended to have very high calibration R2 values. Conversely, models calibrated using data measured in multiple countries or large geographical regions with diverse environmental conditions had lower calibration R2 values. Thus, a scenario where diverse data was used to create new models with very high calibration R2 values (comparable to the high values seen on

Table 5-4) could provide an indication that constructing a model based upon cumulative damage improves upon the legacy approaches.

5.2.2 Metrics Used to Evaluate Final Models Consistent with earlier efforts, four numerical metrics were used to evaluate the accuracy of the final candidate models developed under this research program. These include the error, R2 values, the slope of the trend line pertaining to the data points that relate individual predictions

169 to their associated test measurements, and the y-intercept value of the line. In addition, a visual inspection of the scatter of the data points in proximity to the trend line was used as a qualitative measure of model fit.

Like all earlier efforts, the residual sum of squares method was used to calculate errors for each of the billions of candidate models that were constructed and evaluated via computer simulations. This was done to identify the specific candidate with the lowest error. However, error was not used as a metric to compare the accuracy of the final model to legacy models because none of the past efforts calculated such errors. Comparisons were thus limited to using

R2 values.

As discussed previously, accurate models have an R2 value and a trend line slope both near 1.0, a y-intercept that is small in comparison to maximum value of data seen on the plot, and a statistically uniform distribution of data points in proximity to the trend line. Accuracy can only be ascertained when these metrics are evaluated for models applied to both calibration and independent validation data.

5.2.3 Evaluation of the Best Model Calibrated Using Data from Dobbins, GA Figure E-47 (shown again here as Figure 5-3) illustrates the aforementioned metrics as they pertain to the most accurate monthly chloride model (based upon the R2 value) calibrated using data from Kennedy Space Center, FL; Fort Drum, NY; and Dobbins, Air Reserve Base, GA. As can be seen on this figure, the R2 value of 0.9864, trend line slope of 0.9257, y-intercept value of

2994.6 (in comparison to a maximum plotted value of over 70,000), and a narrow distribution of data points in relation to the trend line all indicate that this candidate model has a high degree of fit to the calibration data. However, evaluating these metrics for the same model applied to independent validation data reveals an entirely different story.

170

80000 y=0.9257x+2994.6 70000

) R² = 0.9864 2 60000

g/cm 50000  ( 40000 30000 20000 Predictions 10000 0 0 10000 20000 30000 40000 50000 60000 70000 80000 Test Results (g/cm2)

Figure 5-3 Model Applied to Calibration Data for Kennedy Space Center, FL; Fort Drum, NY; and Dobbins, Air Reserve Base, GA (Most Accurate Initial Model)

Figure E-48 (shown again here as Figure 5-4) illustrates the same model used to construct Figure

5-3 but in this case, it was applied to independent validation data. As can be seen, the R2 value for the validation plot is somewhat smaller than the calibration plot (0.84 vs. 0.9864) but this is not too surprising since a model should always fit its calibration data more closely. The combination of this R2 value with a y-intercept value of 337.53 and uniformly (i.e., non-funnel shaped) distributed data points appears to indicate an accurate model. However, further inspection of the figure indicates that the model is quite inaccurate. Specifically, the slope of the validation trend line is 2.6976, which indicates that the model substantially overestimates corrosion rates. These results show that an inaccurate model can have a very high calibration R2 value. Thus, the necessary and sufficient conditions for ascertaining model accuracy are to evaluate the R2 value, y-intercept, trend line slope, and scatter of data points that result when candidate models are applied both to calibration and independent validation data.

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90000 y = 2.6976x + 337.53 80000 R² = 0.84 ) 2 70000 60000 g/cm  50000 40000 30000 20000 Predictions ( Predictions 10000 0 0 5000 10000 15000 20000 25000 30000 35000 Test Results (g/cm2)

Figure 5-4 Model Applied to Validation Data for Kennedy Space Center, FL; Fort Drum, NY; and Dobbins, Air Reserve Base, GA (Most Accurate Initial Model)

There is no specific numerical metric pertaining to the y-intercept that can be used to identify a good model other than it should be a small in comparison to the maximum value of predictions shown on a plot. As the accuracy of candidate models improves during the inverse modeling process, the y-intercept value should converge to a value of zero while the trend line slope and

R2 values both converge towards a value of one.

5.2.4 Evaluation of the Final Proof-of-Concept Model After the problem with the calibration data for Dobbins Air Reserve Base, GA, was discovered

(see Section 4.13), a revised set of diverse data was constructed, which led to the development of a new series of models. This new set included data from Kennedy Space Center (KSC), Florida, which is in the humid subtropical climate zone; Rock Island, Illinois (humid-continental-warm summer climate zone); and Fort Drum, NY (humid-continental-cool summer climate zone). The site at KSC is located five miles from the coastal surf zone, thus the chloride levels present during the corrosion tests at this location were significantly higher than seen at the other two

172 sites. In comparison, Fort Drum is a rural inland location subject to low pollution and chloride levels combined with severe winter weather including low temperatures and low absolute humidity levels. The last site, Rock Island, is located in an inland urban area with a total population of nearly 500,000. Thus, the levels of anthropogenic pollutants at this site are higher than seen at KSC or Fort Drum. Despite being limited to just three locations, the calibration data used to develop the final models contains a diverse combination of temperature, humidity, chloride, SO2 and ozone levels. Such diversity was needed so that the resultant final model was robust enough to accurately predict corrosion rates at the independent validation locations.

60000 y=0.9512x+1036.1

) 50000 R² = 0.9569 2

40000 g/cm  ( 30000

20000

Predictions 10000

0 0 10000 20000 30000 40000 50000 60000 Test Results (g/cm2)

Figure 5-5 Comparison of 1010 Steel Test Points and Associated Predictions (Calibration Data)

As described in Section 4.13.3 and Section E.3.7, the use of new calibration data led to a significant revision to the mathematical formulation of the model. Figure E-50 (seen again here as Figure 5-5) displays the final proof-of-concept model as it is applied to the new calibration data. As can be seen, the calibration R2 value of 0.9569 is very high, the trend line slope of

173

0.9512 is close to the optimum value of 1.0, and the distribution of the data points adjacent to the trend line appear to be uniformly distributed. If there is one problem, it pertains to the value of the y-intercept. This value (1036.1) appears to be somewhat high, which could indicate that the proxy data used to calibrate the model does not completely describe the environmental conditions at the test sites and/or the model formulation was not precise enough. Regardless, the intercept value is small in comparison to the maximum value (greater than 50,000) seen on the plot. Thus, the metrics seen on Figure 5-5 indicate that the final model developed under this research effort accurately reflects the calibration data.

40000 y=1.0702x+2502.7 35000

) R² = 0.8628 2 30000

g/cm 25000  ( 20000 15000 10000 Predictions 5000 0 0 5000 10000 15000 20000 25000 30000 35000 Test Results (g/cm2)

Figure 5-6 Comparison of 1010 Steel Test Points and Associated Predictions (Validation Data)

As discussed previously, before the final cumulative damage model could be deemed to accurately reflect the corrosion process, it had to be applied to independent validation data.

Figure E-51 (shown again here as Figure 5-6) illustrates the model applied to the validation data for seven independent corrosion test locations. These sites include Fort Campbell, KY (humid- subtropical climate zone), Fort Hood, TX (humid-subtropical climate zone), Fort Rucker, AL

174

(humid-subtropical climate zone), Kirtland AFB, NM (semiarid-steppe climate zone), Tyndall

AFB, FL (humid-subtropical climate zone), West Jefferson, OH (humid continental-warm summer climate zone), and Wright Patterson AF, OH (humid continental-warm summer climate zone).

The locations contain a diverse combination of climatic conditions, distances from the oceans, and proximity to anthropogenic sources of pollutants. Such environmental diversity should be sufficient to evaluate the accuracy of the final model.

5.2.5 Evidence Supporting the Cumulative Corrosion Damage Hypothesis The principal hypothesis of this research program is that atmospheric corrosion of steel is a cumulative damage process analogous to variable amplitude metal fatigue. Numerous models were calibrated and tested in order to determine whether directly considering environmental variability when making predictions would improve accuracy in comparison to past modeling approaches. As discussed above and in Section C.3, a variety of approaches were employed to evaluate the candidate models.

Figure 5-5 illustrates that the R2 value for the final model applied to the calibration data was very high. Under legacy modeling efforts (see Table 5-4), only those models calibrated from narrow characterization data (i.e., data measured at a single location or within a small geographical region) attained a high calibration R2 value. As seen on Figure 5-6, the validation R2 value of 0.8628 is quite high. In fact, this value is the highest yet for any candidate cumulative corrosion damage model applied to independent validation data. Another positive metric concerns the slope of the trend line on the figure (1.0702), which is very close to the optimum value of 1.0. This metric is also better than any previous model. In addition, the data points do not exhibit pronounced heteroscedastic (funnel-shaped) behavior but instead appear to be narrowly distributed adjacent to the trend line. Taken together, these factors all indicate an accurate model. Similar to the plot of the model applied to calibration data, the y-intercept for

175 the validation plot is not optimum. In fact, the value of 2502.7 is over twice as large as the associated value for the calibration data plot. Future efforts through additional iterative improvements to the model formulation via simulations are needed to reduce y-intercept values.

In their entirety, the results shown for the final proof-of-concept model are better than any of the others developed under this research program or any of the previous efforts. The fit of the final model to its calibration data combined with the high degree of prediction accuracy seen when applying the same model to independent validation data provides evidence supporting the cumulative corrosion damage hypothesis. In addition, the ability of the final model to make accurate predictions for independent locations with diverse environmental conditions, something not possible with legacy models, provides evidence disproving the null hypothesis.

5.3 Model Limitations Because of the calibration procedures and the locations where chloride deposition rates are measured under existing monitoring programs, all models developed under this effort are limited to making predictions for sites that are five or more miles from coastal surf zones. In addition, the models are limited to predicting corrosion rates for the first year of exposure. This is because the corrosion test measurements used to calibrate and validate the models were (in most cases) measured during one-year test periods. Research by de la Fuente et al. indicates that independent of location, corrosion rates for mild steel are highest during the first year of exposure [154]. Furthermore, they state that due to the formation of corrosion products, the long-term rates continually decrease until they become constant at approximately the seven year point. Thus, longer-term exposures are needed to calibrate enhanced model algorithms that consider how the build-up of corrosion products influences corrosion rates. It should be noted, however, that chloride deposition rates can vary significantly from year-to-year. Thus, it

176 is possible that under some conditions, corrosion rates could be higher during the later years of exposure.

5.4 Application of Cumulative Damage Model The model presented here can be used when designing structures that will be located anywhere within the United States subject to the limitations discussed in Section 5.3. Prior to making a prediction, a proxy environmental characterization dataset must first be constructed using the processes discussed in Chapter 3. Such data is then applied to the model to make a cumulative prediction.

Under atmospheric conditions, AISI 1010 steel corrodes via the mechanism of uniform attack.

Thus, location-specific mass loss predictions can be used along with the material density to calculate how fast a structure will uniformly thin as a function of time, a phenomenon commonly referred to as a uniform penetration rate, which is usually expressed as inches per year (IPY) or thousandths of an inch (or mils) per year (mpy). Such information can be used during product design to specify a corrosion allowance, which represents an increase in the structural thickness over and above that needed for strength or stiffness purposes. In other words, accurate predictions would enable the optimization of design features/thicknesses to meet both safety and lifespan requirements.

There are other possible ways that the cumulative damage model presented here could be employed to support engineering analyses. Using existing processes, a corroded structure could be cleaned of corrosion products and inspected in order to measure remaining material thickness. Such data could then be compared with the design specification in order to ascertain the remaining corrosion allowance. Model predictions made using the model presented here

177 could then be compared to this allowance in order to make a more accurate estimate of the remaining life.

Since the model presented here is based upon the concept of cumulative corrosion damage, it is not limited to making predictions for stationary structures. Thus, the approach could also be employed to ascertain the cumulative damage to moveable structures subjected to short-term exposures at multiple locations. Should this model be extended to other materials such as aluminum alloys, it could be used to predict the cumulative damage that occurs when aircraft are flown between different airports or military operating bases. Of course, aluminum aircraft structures are typically painted so that actual corrosion rates would be far lower than the predictions. However, knowledge of how individual aircraft are exposed to specific environmental conditions could enable a new paradigm in maintenance planning. Such actions as aircraft washing and repainting intervals could be based, in-part, on the predicted exposures.

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CHAPTER 6

CONCLUSIONS

Atmospheric corrosion is a physical process that is dependent upon the environmental severity upon which a material is exposed. A common feature of past modeling efforts was to quantify environmental severity using long-term average values for individual acceleration factors. For instance, many of the statistically based (i.e., regression or power-law) models were calibrated using annual deposition rates for chlorides and/or SO2. Relative humidity is either not considered by such models or it is used as a metric to define the number of hours per year when humidity levels are high enough for corrosion to occur. Most statistical models ignore temperature effects and the few that do consider it employ long-term (e.g., annual) average values. Corrosion rates calculated using such models are typically depicted as linear or simple nonlinear curves.

Mechanistic corrosion modeling approaches assume that the complex reactions known to consume materials can be quantified using algorithms that mathematically describe the predominant physical processes and the manner in which they are coupled. Such models are typically based upon laboratory testing results and they can be very useful for understanding specific mechanisms and their effects on overall corrosion rates. However, there have been no mechanistic models developed that are capable of accurately predicting corrosion rates in actual

(i.e., highly variable) atmospheric environments. This is likely due to the complexity of reactions

179 resulting from rapidly changing values of often competing parameters in combination with other complex factors that are very difficult, if not impossible, to quantify and model.

Overcoming the inability of statistical and mechanistic models to accurately consider stochastic environmental conditions requires an entirely different modeling approach. The cumulative corrosion damage model described here was developed using a tailored form of the Eyring equation, which was originally developed to predict the kinetics of chemical reactions.

Candidate models based upon Equation 2.14 were constructed and calibrated using an inverse approach based upon the use of the Monte Carlo simulation method. Using this approach, simulations were conducted that employed candidate values for the unknown modeling coefficients randomly selected from probability distributions. Through a systematic process, billions of candidate models were fit to training data. The training datasets used to calibrate models were large and diverse with high and low values for each environmental parameter.

Using large amounts of diverse data ensured the calibrated models were based upon the full range and possible combinations of physical parameters likely to be seen at locations where predictions are desired.

The intent of this research effort was to lay the ground work for a future analysis method that design engineers with little or no corrosion experience could use to make realistic predictions for specific locations where structures might be built or systems employed. Thus, it was important that the environmental characterization data used to both calibrate models and later make predictions using them was available from existing sources. For this current effort, it was assumed that hourly SO2 and ozone data from government pollution monitoring programs combined with longer-term (e.g., monthly average) wet chloride deposition measurements from agricultural/forestry environmental monitoring programs and hourly temperature/relative

180 humidity readings measured by weather offices could adequately quantify the stochastic environmental severity at locations of interest such as the calibration and validation sites.

Despite the fact that environmental monitoring sites are spatially distributed at hundreds of locations across the United States, they are rarely located at the exact locations where corrosion predictions are desired. Thus, a proxy approach was employed to infer environmental severity at such locations based upon actual measurements made at monitoring sites elsewhere in the same geographical region. This was done to avoid the need to conduct costly environmental characterization tests. While this approach appears to result in accurate predictions, it is important to note that the proxy data does not exactly reflect the variable environmental conditions at the corrosion test sites.

The simulation process employed an optimization approach, which resulted in the best fit of the model to its proxy calibration data. As a result, the values of the modeling coefficients determined via simulations were slightly different from what they would have been had the environmental data been measured directly at the calibration test sites. Thus, when a model based upon slightly inaccurate coefficients is applied to proxy validation data, which itself is a slightly inaccurate representation of the conditions at the actual test sites, then the resultant predictions would have greater inaccuracy (i.e., more scatter) than when the same model was applied to the proxy calibration data on which it is based. The above description can be seen when evaluating scatterplots that compare individual predictions to their associated test measurements. For instance, Figure 5-5 compares predictions made from proxy calibration data compared to associated test measurements while Figure 5-6 makes a similar comparison for the validation data. Consistent with the above description, the data scatter on the validation plot is greater than what is seen on the calibration plot. Calibrating new models using environmental

181 data measured directly at the corrosion test sites will presumably reduce the scatter seen on the validation plots.

Atmospheric contaminant levels vary quickly over very short periods of time. When combined with weather events (e.g., storms, cold fronts, etc.) and cyclical changes in temperature and humidity levels due to diurnal/seasonal cycles, it is easy to conclude that environmental severity is far from a static entity. Cumulative damage models that make incremental predictions for very short periods of time and then add the increments together seems to be the only way such variability can be addressed. The challenge, of course, is to develop an approach to construct and statistically test such a model.

The candidate models developed and evaluated under this research effort employed the use of shape functions to quantify how the highly variable (hourly) atmospheric contaminant and weather variables combine to influence corrosion rates. It should be noted that consistent with other inverse models fit to data, there is currently no direct physical meaning (determined via testing) for the shape functions illustrated notionally by Equation 2.14. Instead, they are constructed and calibrated to provide numerical values representing interactions between acceleration factors. Two different types of shape functions are employed. These include

(three) Temperature-Relative Humidity shape functions and (three) Temperature-Contaminant shape functions (one each for chlorides, SO2, and ozone). These functions were arranged into three sets that pair individual Temperature-Relative Humidity shape functions with individual

Temperature-Contaminant shape functions. All functions were calibrated simultaneously.

When making predictions with a calibrated model, three numerical values were calculated from the paired shape functions. These values represent the interaction between hourly values of the three contaminants in combination with the associated temperature and RH measurements

182 for the same time period. Hourly corrosion rate predictions (weight loss per unit area) result when these values were appropriately combined with the kinetic terms considered by the model

(i.e., the activation energy and the Boltzmann constant) as well as the associated scaling factors.

It should be noted that the activation energy, H, determined during the simulation process does not represent the actual kinetics and thermodynamics of the reaction. Instead, the value represents a curve fitting parameter that enables the model to make the most accurate predictions. Additional simulations combined with experimental work can lead to an improved representation of H, and thus an improved model.

An iterative method whereby refined coefficients obtained via additional simulations combined with revised mathematical formulations was used to develop the candidate models. These simulations calibrated the actual shape functions and also determined values for the associated scaling factors. Billions of scenarios were run in order to identify the model formulation and associated set of coefficients that provided the lowest error.

The proof-of-concept models developed and evaluated under this research program are limited to predicting AISI 1010 steel corrosion rates for the first year of exposure. They are remarkably accurate, which is likely due to the way they were constructed using an inverse approach employing Monte Carlo simulations to fit models to large amounts of diverse environmental characterization data. The R2 value of 0.9569 calculated for the final model applied to large amounts of diverse calibration data demonstrates a very high degree of fit. Perhaps more importantly, the value of 0.8628 that results when the best fitting model was applied to independent validation data indicates the robustness of the inverse Monte Carlo development process and its ability to iteratively develop and refine a model capable of making accurate predictions for general environmental conditions. None of the past modeling efforts evaluated

183 for this research effort addressed validation. Thus, it is likely that the cumulative damage approach developed here already exceeds the accuracy of the best existing model.

It should be noted that the models presented here are based upon proxy data for the exact same timeframe when corrosion tests were underway at the calibration and validation sites.

However, environmental conditions are never the same from year-to-year. Thus, when employing the model to make predictions for design purposes, a statistical approach employing either hourly averages or probabilistic profiles based upon historical environmental measurements must be employed. Such an approach would result in proxies that reasonably represent typical conditions at locations where a structure might be built.

As discussed in Section 5.2.5, the results from this proof-of-concept effort provide evidence to support the hypothesis that atmospheric corrosion of steel is a cumulative damage process. It should be noted that the model presented here was calibrated using corrosion test results obtained from single specimens exposed to each test condition per time interval. As such, measurement error is not statistically accounted for in the model formulation. In addition, slight variations in material properties and/or processing could lead to slight differences between predicted and actual corrosion rates. Regardless, it appears as if the model predicts corrosion rates with a high degree of fidelity based upon its calibration and validation R2 values, slopes, and y-intercepts.

When considering that the number of mathematical formulations of the final proof-of-concept model were limited, it seems likely that future improvements could be made to further increase accuracy. Chapter 7 describes numerous approaches that could lead to such improvements.

Efficiently developing new models based upon these recommendations requires a computing capability far more powerful than the one used under the initial efforts shown here. A

184 supercomputer would be the preferred choice but an acceptable option would be to construct a

“cluster” of GPU computers that are networked into a GPU computer/server like the one used for this effort. Such a cluster could easily be 5-10 times faster than the current system. Such a system could facilitate the development of the enhanced predictive methodology identified as a

Corrosion Grand Challenge in the 2011 National Research Council report [8].

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CHAPTER 7

RECOMMENDATIONS FOR FUTURE WORK

The proof-of-concept cumulative corrosion damage model has a very high degree of fit with the calibration data but when applied to other independent locations, it is less accurate. As was shown during the discussion on variable threshold functions, there are problems with the mathematical formulation of the current model that could be corrected in order to further improve accuracy. The following sections discuss some recommendations for future work that if employed could lead to a model that even more accurately reflects the atmospheric corrosion process as it relates to AISI 1010 steel and potentially other materials as well.

7.1 Conduct Simulations Using High-Performance Computing to Overcome Limitations of Current Hardware During this research effort, large numbers of candidate models were constructed and calibrated.

Calibrating these models required over 100 billion simulations, which took place over thousands of hours of computing time. These figures do not include extensive preliminary simulations that were conducted during the preliminary model development efforts. Despite being a very capable massively parallel workstation, the GPU computer used for the initial model development efforts was not fast enough to examine other models in a timely fashion. Thus, improving model accuracy by further evolving the model’s mathematical formulation requires a new computational approach. Either a supercomputer or a cluster of networked GPU

186 computers is needed to significantly increase the speed of computations over the single GPU computer used during this proof-of-concept study. Such a capability will significantly accelerate the speed at which new candidate models can be calibrated and statistically tested.

7.2 Apply Variable RH Threshold Functions to the Final Proof-of-concept Model As shown earlier, RH threshold functions improved the accuracy of candidate models. Thus, such an approach is likely to improve the accuracy of the final proof-of-concept model.

7.3 Examine Models that Combine Variable RH Threshold Functions and Multiple Activation Energies As shown during this current effort, using both variable RH threshold functions and multiple activation energy formulations improved model accuracy. Thus, it seems reasonable to see whether combining both of these approaches into a single model will provide even greater accuracy. Such an approach would be based upon the final proof-of-concept model. Because of the added complexity, which increases the number of unknown coefficients, it would only be possible to calibrate such a model by using a very high performance computer.

7.4 Examine Other Temperature-Contaminant Shape Functions The most accurate proof of concept model examined under this current research program was constructed using parabolic Temperature-Contaminant shape functions. Additional nonlinear functions can be evaluated via simulations to determine whether they might improve accuracy further. Such an effort would follow the development of the new model discussed previously.

7.5 Investigate the Relative Humidity Threshold via Environmental Chamber Tests Simulations can provide insight into the RH threshold but conducting chamber tests to determine the actual relationship between temperature and relative humidity could lead to an empirical RH threshold function that could be used to evolve the current inverse model into one that more accurately reflects the physical processes occurring on material surfaces.

187

7.6 Conduct New Corrosion and Environmental Characterization Tests The proof-of-concept cumulative damage model was calibrated using proxy environmental data.

As such, the conditions represented by the data were not identical to the actual conditions present at the corrosion test sites. In addition, the corrosion test data used during calibration was point data (i.e., one test point per condition). The combination of these factors induces uncertainty into the modeling coefficients identified during the simulation process. Improved accuracy should result if improved calibration data is obtained from new corrosion tests (with multiple specimens per test condition). Under such testing, it would be preferred to characterize the environment directly at the test sites. Environmental characterization protocols should be the same as those used for this proof-of-concept effort.

7.7 Develop Statistical Representations for Stochastic Variables Since environmental conditions are never the same from year-to-year, simply using environmental data from a single year is not appropriate for making a design prediction. Thus, a different approach is needed to facilitate the employment of a practical design tool. One approach would be to average hourly conditions for multiple years into a single proxy containing average hourly conditions. Another approach that would still enable consideration of high and low values of each variable would be to develop statistical probabilistic models for each atmospheric contaminant and weather variable. These two approaches should be considered under a future research effort.

7.8 Expand Model to Consider Longer Periods of Time The proof-of-concept model was developed to predict corrosion rates for the first year of exposure. However, it is well known that corrosion rates change as a result of the growth of oxide layers on surfaces [155]. As such, the model will need to be modified to consider longer exposure times. This could be done by using data measured over longer periods of time, incorporating data measured under accelerated testing conditions, or a mixture of the two

188 approaches. Approximations could also be made using a scaling process based upon published results.

7.9 Expand Model to make Predictions at Locations with Very High Chloride Deposition Rates The proof-of-concept models described here are limited to making predictions for locations that are five or more miles from coastal surf zones. To improve the utility of these models, new versions must be calibrated using deposition measurements that were obtained at environmental characterization sites closer to the ocean. Alternatively, functions can be developed that consider Abbott’s silver witness specimen test results. The chloride deposition rates measured using these specimens could be used to construct functions to scale estimates of chloride deposition based upon proximity to surf zones [25].

7.10 Consider Additional Atmospheric Contaminants The proof-of-concept model is limited to considering gaseous SO2 and ozone as well as chloride aerosols. However, there are other atmospheric contaminants such as nitrogen oxides (NOx) that also contribute to corrosive reactions (see Section 1.2.1.3). The model could be expanded to consider such additional substances.

7.11 Construct Models for Other Materials The approach used to create the proof-of-concept model described here could form the basis for creating cumulative corrosion damage models for other materials.

189

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202

APPENDIX A

DERIVATION OF NONLINEAR TEMPERATURE-RELATIVE HUMIDITY SHAPE FUNCTIONS AND AN IMPROVED MODEL FORMULATION

A.1 Derivation of Nonlinear Temperature Functions The general methodology for developing temperature-relative humidity shape functions is described in Section 2. This same methodology is used here to derive the more complex nonlinear shape functions that were considered during model development efforts. As before, the words “concave” and “convex” are used to describe the shapes of these functions.

A.1.1 Concave Temperature Function A concave temperature function can be derived using a quadratic equation. Similar to the linear temperature equation derived in Section 2, a variable value for f(T)max must be incorporated into the nonlinear function to enable variability of the resultant exponential function values used during model convergence studies. Like before, this new function has a minimum value of -4.

Equation A.1 displays the general form of the equation that was developed.

2 bTaT  Tfc )( (A.1)

The coefficients “a”, “b”, and “c” seen in Equation A.1 were determined based upon the temperature boundary conditions shown by Equations A.2 through A.4 and Figure A-1. Note

that for this figure, an f(T)max value equal to -2.0 was chosen for illustration purposes only.

203

0

-0.5 f(T) used for f(T) used for -1 symmetry modeling -1.5

-2 f(T) -2.5

-3 Boundary -3.5 Conditions -4 200 220 240 260 280 300 320 340 Temperature (K)

Figure A-1 Illustration of the Concave Temperature Function

f(T)@T=320.15=f(T)max (A.2)

f(T)@T=273.15=-4 (A.3)

f(T)@T=226.15= f(T)max (A.4)

From a corrosion standpoint, there is no physical meaning to the third boundary condition

(Equation A.4) since the temperature of 226.15 is well below the freezing point. The sole purpose of this condition is to provide a functional symmetry with the condition shown in

Equation A.2 and illustrated by Figure A-1. As shown on the figure, once the functional equation was derived, the model only considered the region from the freezing point to the maximum temperature of 320.15K.

204

The conditions shown in Equations A.2 through A.4 were inserted into Equation A.1, thus resulting in Equations A.5 through A.7. These three equations were then solved simultaneously in order to determine expressions to solve for values of the unknown coefficients.

2 a 1532015320  Tfcb )(.).( max (A.5)

a 2 .).( 1527315273 cb  4 (A.6)

2 a 1522615226  Tfcb )(.).( max (A.7)

The first step was to solve Equation A.5 in terms of the coefficient “a.”

)(  .15320 bcTf a  max (A.8) 15320 ).( 2

The next step was to substitute Equation A.8 into Equation A.7 and solve for the coefficient “b.”

)( max  .15320 bcTf 2 [ ]( 1522615226  Tfcb )(.). max (A.9) 15320 ).( 2

Equation A.9 was expanded and simplified as shown by the steps contained in Equations A.10 through A.17. The final equation in this sequence describes the coefficient “b” in terms of the

coefficient “c” and the variable f(T)max.

2 2 2 15226 Tf )().( max 15226 ).( c 1522615320 ).(. b   15226  Tfcb )(. max (A.10) 15320 ).( 2 15320 ).( 2 15320 ).( 2

2 2 2  15226 ).( c 15226 ).( b 15226 Tf )().( max  15226 Tfcb )(. max  (A.11) 15320 ).( 2 .15320 15320 ).( 2

205

2 2 2 15226 ).( b  15226 ).( c 15226 Tf )().( max .15226 b c  Tf )( max  (A.12) .15320 15320 ).( 2 15320 ).( 2

2 2 2 15226 ).( b 15226 ).( c 15226 Tf )().( max .15226 b  Tfc )( max  (A.13) .15320 15320 ).( 2 15320 ).( 2

15226 2 ).( 15226 2 ).( 15226 2 ).( .[ 15226  b  [] c [] 11  Tf )(] max (A.14) .15320 15320 ).( 2 15320 ).( 2

15226 2).( 15226 2 ).( 15226 2).( .[ 15226  b [] 1 c  [] 1 Tf )(] max (A.15) .15320 15320 ).( 2 15320 ).( 2

15226 2 ).( 15226 2 ).( .[ 15226  b  [] 1](  Tfc max ))( (A.16) .15320 15320 ).( 2

15226 2 ).( [ 1] 15320 ).( 2 b   Tfc max ))(( (A.17) 15226 2 ).( .[ 15226  ] .15320

The next step was to substitute Equations A.8 and Equation A. 17 into Equation A.6, which were then solved to provide the explicit expression needed to calculate values for “c.”

15226 ).( 2 [ 1](  Tfc max ))( 15320 ).( 2 )( max cTf  15320 [. ] 15226 ).( 2 .15226  .15320 15273 ).( 2  15320 ).( 2 (A.18) 15226 ).( 2 [ 1](  Tfc max ))( 15320 ).( 2 .15273 c  4 15226 ).( 2 .15226  .15320

206

Equation A.18 was then expanded and reorganized as seen in Equations A.19 through A.26 in order to obtain the desired explicit relationship for the coefficient “c”.

15226 ).( 2  [ 2 1]( Tfc max ))( Tf )( c 15320 ).( 2 max   15320 [. ]( 15273 ).  2 1532015320 ).().( 2 15226 ).( 2 .[ 15226  ]( 15320 ). 2 .15320 (A.19) 15226 ).( 2 [ 1](  Tfc max ))( 15320 ).( 2 .15273 c  4 15226 ).( 2 .15226  .15320

15226 ).( 2 15273 2[).( 1]c 15273 2 Tf )().( 15273 ).( 2 c ).( 2 max  [ 15320 ] 15320 ).( 2 15320 ).( 2 15226 ).( 2 .[ 15226  ]( 15320 ). .15320 2 2 (A.20) 2 15226 ).( 15226 ).( 15273 [).( 1]( Tf max ))( [ 1](  Tfc max ))( 15320 ).( 2 15320 ).( 2 [  .] 15273 c  4 15226 ).( 2 15226 ).( 2 .[ 15226  ]( 15320 ). .15226  .15320 .15320

2 2 2 15226 ).( 15226 ).( 15273 [).( 1]c [ 1](  Tfc max ))( 15273 ).( 2 c 15320 ).( 2 15320 ).( 2  [  .] 15273 c  15320 ).( 2 15226 ).( 2 15226 ).( 2 .[ 15226  ]( 15320 ). .15226  .15320 .15320 2 (A.21) 2 15226 ).( 15273 [).( 1 Tf )(] max 15273 2 Tf )().( 15320 ).( 2 4  max [ ] 15320 ).( 2 15226 ).( 2 .[ 15226  ]( 15320 ). .15320

207

15226 ).( 2 15226 ).( 2 15273 2[).( 1]c [ 1]c 15273 ).( 2 c 15320 ).( 2 15320 ).( 2  [  .] 15273  15320 ).( 2 15226 ).( 2 15226 ).( 2 .[ 15226  ]( 15320 ). .15226  .15320 .15320 15226 ).( 2 [ 1]( Tf max ))( 15320 ).( 2 15273 2 Tf )().( .15273 c 4  max  (A.22) 15226 ).( 2 15320 ).( 2 .15226  .15320 2 2 15226 ).( 15273 [).( 1 Tf )(] max 15320 ).( 2 [ ] 15226 ).( 2 .[ 15226  ]( 15320 ). .15320

15226 ).( 2 15226 ).( 2 15273 2[).( 1]c [ 1]c 15273 ).( 2 c 15320 ).( 2 15320 ).( 2  [  .] 15273  15320 ).( 2 15226 ).( 2 15226 ).( 2 .[ 15226  ]( 15320 ). .15226  .15320 .15320 15226 ).( 2 [ 1 Tf )(] max 15320 ).( 2 15273 2 Tf )().( .15273 c 4  max  (A.23) 15226 ).( 2 15320 ).( 2 .15226  .15320 2 2 15226 ).( 15273 [).( 1 Tf )(] max 15320 ).( 2 [ ] 15226 ).( 2 .[ 15226  ]( 15320 ). .15320

15226 ).( 2 15226 ).( 2 15273 2[).( 1]c [ 1]c 15273 ).( 2 c 15320 ).( 2 15320 ).( 2  [  .] 15273 c  15320 ).( 2 15226 ).( 2 15226 ).( 2 .[ 15226  ]( 15320 ). .15226  .15320 .15320 2 2 (A.24) 2 15226 ).( 15226 ).( 15273 [).( 1 Tf )(] max [ 1 Tf )(] max 15273 2 Tf )().( 15320 ).( 2 15320 ).( 2 4 max [  .] 15273 15320 ).( 2 15226 ).( 2 15226 ).( 2 .[ 15226  ]( 15320 ). .15226  .15320 .15320

208

15226 ).( 2 15226 ).( 2 15273 2[).( 1] [ 1] 15273 ).( 2 15320 ).( 2 15320 ).( 2 c[ [  .] 15273 1]  15320 ).( 2 15226 ).( 2 15226 ).( 2 .[ 15226  ]( 15320 ). .15226  .15320 .15320 2 2 (A.25) 2 15226 ).( 15226 ).( 15273 [).( 1 Tf )(] max [ 1 Tf )(] max 15273 2 Tf )().( 15320 ).( 2 15320 ).( 2 4  max [  .] 15273 15320 ).( 2 15226 ).( 2 15226 ).( 2 .[ 15226  ]( 15320 ). .15226  .15320 .15320

2 2 2 15226 ).( 15226 ).( 15273 [).( 1 Tf )(] max [ 1 Tf )(] max 15273 2 Tf )().( 15320 ).( 2 15320 ).( 2 4  max [  .] 15273 15320 ).( 2 15226 ).( 2 15226 ).( 2 .[ 15226  ]( 15320 ). .15226  c  .15320 .15320 (A.26) 15226 ).( 2 15226 ).( 2 15273 2[).( 1] [ 1] 15273 ).( 2 15320 ).( 2 15320 ).( 2 [ [  .] 15273 1] 15320 ).( 2 15226 ).( 2 15226 ).( 2 .[ 15226  ]( 15320 ). .15226  .15320 .15320

Obviously Equation A.26 could be simplified by solving the many elementary mathematical expressions seen in its formulation. However, this full form was retained so that all calculations were made using the numerical precision of a computer in order to preclude round-off error.

Thus, this equation was solved during each simulation run using a computer algorithm to obtain a numerical value for “c”, which was then be used by an algorithm implementing Equation A.17 to obtain a value for “b”. The numerical values for “b” and “c” were then used to calculate a numerical value for “a” based upon Equation A.8.

Equations A.8, A.17, and A.26 were all constructed to consider a variable entitled f(T)max. If for

instance, f(T)max has a value of -2, then the associated numerical values for the coefficients “a”,

“b”, and “c” are shown by Equations A.27, A.28, and A.29. When these values are input into

209

Equation A.1 (shown again here as Equation A.30), a concave plot for a range of temperatures from 273.15 through 320.15 can be developed, as seen in Figure A-2.

a= 0.00091 (A.27)

b= -0.49461 (A.28)

c= 63.55176 (A.29)

2 bTaT  Tfc )( (A.30)

0 -0.5 -1 f(T)max=-2 -1.5 -2

f(T) -2.5 -3 -3.5 -4 -4.5 270 280 290 300 310 320 330 Temperature (K)

Figure A-2 Concave Temperature Function

A.1.2 Convex Temperature Function Equation A.31 provides the general expression for the convex temperature function.

af T 2 bf )()(  TcT (A.31)

210

Like before, the coefficients for this equation were determined by establishing boundary conditions and solving a set of simultaneous equations. The major difference of this case in comparison to the concave temperature function was that the convex curve was symmetric around the “T” axis rather than the “f(T)” axis as before. This implies that there were two possible values of f(T) for each value for “T.” Equations A.32 through A.34 describe the boundary conditions that were used to establish the three equations needed to determine values for the unknown coefficients. Figure A-3 illustrates the implementation of the boundary conditions for the specific case where f(T)max equals -2.0. Please note that the expression “-f(T)max -8” used in

Equation A.34 was necessary to appropriately implement the symmetry of the parabolic function.

f(T)= f(T)max @T=320.15 (A.32)

f(T)= -4 @T=273.15 (A.33)

f(T)= -f(T)max-8 @T=320.15 (A.34)

211

0

-1 Boundary Conditions -2 f(T) used for -3 modeling

f(T) -4

-5 f(T) used for symmetry -6

-7 270 280 290 300 310 320 330 Temperature (K)

Figure A-3 Illustration of the Convex Temperature Function

Equations A.35 through A.37 are the three equations formed by applying the conditions seen in

Equations A.32 through A.34 into Equation A.31.

2 a max max cb  .]f(T)[]f(T)[ 15320 (A.35)

2 44 cba  .)()( 15273 (A.36)

2 a max 8 b max 8 c  .]f(T)[)f(T)( 15320 (A.37)

Equations A.36 and A.37 were then rewritten as shown in Equations A.38 and A.39.

16 4 cba  .15273 (A.38)

2 a max 16 max 64 b max 8 c  .]f(T)[)f(T))f(T)( 15320 (A.39)

Equation A.38 was then solved for “c.”

212

16 4bac  .15273 (A.40)

The next step was to substitute Equation A.40 into Equations A.35 and A.39, as seen in

Equations A.41 and A.42.

2 a max b max ]f(T)[]f(T)[ 16 4ba  .. 1532015273 (A.41)

2 a max 16 max 64 b max 8]f(T)[]f(T))[f(T) 16 4ba  .. 1532015273 (A.42)

Equation A.41 was then rewritten to solve for “a” in terms of “b.”

2 a max 16 b max 4  ..]f(T)[])f(T)[ 1527315320 (A.43)

2 a max 152731532016 b max  4]f(T)[)..(])f(T)[ (A.44)

1527315320 b  4]f(T)[)..( a  max 2 (A.45) max )f(T) 16

Equation A.42 was simplified by combining terms and reorganizing.

2 a max 16f(T))[f(T) max 1664 b f(T)[] max 48  ..] 1527315320 (A.46)

2 a max 16 max 48 b max 4  ..]f(T)[]f(T))[f(T) 1527315320 (A.47)

2 a max 16 max 48 b max 4  ..]f(T)[]f(T))[f(T) 1527315320 (A.48)

Equation A.45 was then substituted into Equation A.48 in order to develop an explicit expression for “b.”

213

1527315320 b  4]f(T)[)..( Max 2 b ]f(T)[]f(T))[f(T)  2 max 16 max 48 max 4 max )f(T) 16 (A.49)  .. 1527315320

2 2  1527315320 max 16 max 48 b max  4 max 16 max  48]f(T))f(T)([]f(T)[]f(T))f(T)([)..( 2 max )f(T) 16 (A.50)

b max 4  ..]f(T)[ 1527315320

2 b max  4 max 16 max  48]f(T))f(T)([]f(T)[  b max 4]f(T)[  )f(T)( 2 16 max (A.51)  1527315320 2 16  48]f(T))f(T)([)..( )..(  max max 1527315320 2 max )f(T) 16

2 max  4 max 16 max  48]f(T))f(T)([]f(T)[ b[ f(T)[ max 4]]  )f(T)( 2 16 max (A.52)  1527315320 2 16  48]f(T))f(T)([)..( )..(  max max 1527315320 2 max )f(T) 16

 4 2 16 48 f(T)[]f(T))f(T)([]f(T)[  4][( 2 16])f(T) b[ max max max max max ]  )f(T)( 2 16 max (A.53)  1527315320 2 16  48]f(T))f(T)([)..( )..(  max max 1527315320 2 max )f(T) 16

Equation A.53 is quite messy but was simplified in parts as seen below in order to obtain the most concise form for calculating values for “b”. Equation A.54 shows the numerator for the left hand side of Equation A.53. This part of the equation was simplified as shown below.

214

2 2 max  4 max 16 max 48 f(T)[]f(T))f(T)([]f(T)[ max  4][( max 16])f(T) (A.54)

2 16 48  4 2 16 48]f(T))f(T)([]f(T))f(T)([f(T)  max max max max max (A.55) 2 2 f(T)max [( max 16])f(T)  4[( max 16])f(T)

3 16 2  48  4 2  64f(T))f(T)(f(T))f(T)()f(T)( 192 max max max max max (A.56) 3 2 max 16 max  4 max 64])f(T)(f(T))f(T)(

3  3 16 2  4 2  4 )f(T)()f(T)()f(T)()f(T)()f(T)( 2 max max max max max (A.57)  48f(T)max  64f(T)max 16f(T)max  64192 ]

3 2 2 max  24 max 96f(T))f(T)()f(T)( max  128 (A.58)

The next step was to replace the numerator for the left hand side of Equation A.53 with the expression seen in Equation A.58 and then solve for “b”.

2 3  24 2  96f(T))f(T)()f(T)( 128 b[ max max max 1527315320 )..(]  )f(T) 2 16 max 2 (A.59)  1527315320 max 16 max  48]f(T))f(T)([)..( 2 max )f(T) 16

b 2 3  24 2 96f(T))f(T)()f(T)([ 128  ..(] 1527315320 )( 2 16))f(T)  max max max max (A.60) 2  1527315320 max 16 max  48]f(T))f(T)([)..(

 ..( 1527315320 )( 2  152731532016 2 16  48]f(T))f(T)([)..())f(T) b  max max max 3 2 (A.61) 2 max  24 max 96f(T))f(T)()f(T)( max  128

215

 152731532016   152731532064 )..(][f(T))..( b  max 3 2 (A.62) 2 max  24 max 96f(T))f(T)()f(T)( max  128

Equation A.62 was used to obtain an explicit value for “b” based upon the variable f(T)max. As seen below, this equation was then substituted into Equation A.45 to develop an explicit solution for the unknown coefficient “a”.

 152731532016 max   152731532064 )..(][f(T))..( 1527315320  [)..( ][ max  4]f(T) 2 3  24 2 96f(T))f(T)()f(T)( 128 (A.63) a  max max max 2 max )f(T) 16

 152731532016 max   152731532064 )..(][f(T))..( [ ][ max  4]f(T)  1527315320 )..( 2 3  24 2 96f(T))f(T)()f(T)( 128 (A.64) a   max max max 2 2 max )f(T) 16 max )f(T) 16

 152731532016 max   152731532064 )..(][f(T))..( [ ][ max  4]f(T)  1527315320 )..( 2 3  24 2 96f(T))f(T)()f(T)( 128 (A.65) a   max max max 2 max )f(T) 16 max  4 max  4))(f(T)))(f(T)

 152731532016   152731532064 )..(][f(T))..( [ max ]  1527315320 )..( 2 3  24 2 96f(T))f(T)()f(T)( 128 (A.66) a   max max max 2 max )f(T) 16 max  4))(f(T)

 1527315320 )..(  152731532016   152731532064 )..(][f(T))..( a   max 2 3 2 (A.67) max )f(T) 16 2 max  24 max 96f(T))f(T)()f(T)([ max  128][ max )(f(T) 4)]

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Equation A.67 was used to calculate values for the coefficient “a.” This equation along with

Equation A.62 was then substituted into Equation A.40 to develop an expression for the coefficient “c.”

 1527315320 )..(  152731532016   152731532064 )..(][f(T))..( c  16[  max ] )f(T) 2 16 2 3  24 2 96f(T))f(T)()f(T)([ 128][ )(f(T) 2  4)] max max max max max (A.68)  152731532016   152731532064 )..(][f(T))..(  [ max  .] 4 3 2 15273 2 Max  24 max 96f(T))f(T)()f(T)( max 128

Equation A.31 (repeated here as Equation A.69) was used to calculate the value of f(T) when a temperature is known. As derived above, Equations A.62, A.67, and A.68 were used to calculate values of the coefficients used in this equation.

af T 2 bf )()(  TcT (A.69)

There are two roots to Equation A.69. The Quadratic Equation was used to solve for the positive root in terms of the temperature as shown by Equation A.70.

2 4  Tcabb )( Tf )(  (A.70) 2a

Consistent with the previous discussion following the derivation of the equations for the concave temperature function, it is obvious that a significant amount of simplification could have been accomplished earlier in the process of developing these new equations. However, to minimize any round-off error, the complete formulations were carried entirely through the derivation.

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Equations A.71 through A.73 display coefficient values calculated using Equations A.62, A.67, and A.68 for the specific example of when the maximum value of the variable temperature

function, f(T)max, equals -2. Figure A-4 displays the resultant convex temperature function.

Please note that the equations and figure displayed here simply illustrate a possible temperature function that could be used to construct a shape function to describe the interaction between temperature and another corrosion acceleration factor (e.g., RH or

contaminant levels). The optimal value of f(T)max used in this function was determined through convergence studies during the modeling and simulation process.

a=11.75 (A.71)

b=94.0 (A.72)

c=461.15 (A.73)

0 -0.5 -1 -1.5 -2

f(T) -2.5 -3 -3.5 f(T)max=-2 -4 -4.5 270 280 290 300 310 320 330 Temperature (K)

Figure A-4 Convex Temperature Function

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A.2 Derivation of Nonlinear Relative Humidity Functions As shown in Section 2, relative humidity functions were also needed before the actual temperature-relative humidity shape functions could be constructed. Below are the derivations for the concave and convex relative humidity functions that were considered under this current effort.

A.2.1 Concave Relative Humidity Function

Equation A.74 describes the general form of the concave relative humidity function. This equation was used to calculate the value of the Temperature-Relative Humidity shape function, which applies when the temperature is between 273.15K and 320.15K and the relative humidity is between the threshold and 100%.

2  TfcbRHaRH RH),( (A.74)

-3 -3.1 Boundary -3.2 Conditions -3.3 -3.4 -3.5 f(T) used for f(T) used for f(T) -3.6 symmetry modeling -3.7 -3.8 -3.9 f(RH,T)= f(T) -4 0 20 40 60 80 100 Relative Humidity (%RH)

Figure A-5 Illustration of the Concave Temperature-Relative Humidity Function

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Three boundary conditions were needed to create the three equations that were solved simultaneously to determine values for the coefficients “a”, “b”, and “c.” One condition concerns the maximum value of the function, which occurs at 100% RH. At this point, the value of f(T,RH) is the same as the value of f(T) for the same temperature. The second boundary condition concerns the minimum value for f(T,RH), which was defined as -4 (the same as the other functions). This boundary condition applies when the RH equals a threshold value

(RH=RHTH). As before, the symmetry of a parabola was used to create the third boundary

condition. Therefore, when the RH=2RHTH-100%, f(T,RH,T)=f(T). Please note that this third relationship has no meaning with respect to corrosion. It was only used to define the full parabolic shape, half of which was used for the corrosion model. All three boundary conditions are recorded below as Equations A.75, A.76, and A.77. Figure A-5 illustrates the application of the boundary conditions with respect to the desired parabolic shape. This figure was based

upon an RHTH value of 60%.

RH=100%, f(T,RH)=f(T) (A.75)

RH=RHTH, f(T,RH)=-4 (A.76)

RH=RHTH-(100%-RHTH)=2RHTH-100%, f(T,RH)=f(T) (A.77)

Please note that as before, the variable “RHTH” was carried through the subsequent derivations.

This was being done so that later, the relative humidity threshold could be considered as a function of temperature to determine if such an approach would improve model accuracy. As previously discussed, the relative humidity values was considered as a decimal (e.g., 100% equals 1.0).

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Equations A.78 through A.80 result when Equations A.75 through A.77 were applied to Equation

A.74.

2 11  Tfcba )()()( (A.78)

2 a RHTH b RHTH )()( c  4 (A.79)

2 a RHTH b RHTH 1212  Tfc )()()( (A.80)

These equations were then reduced to the following forms.

 Tfcba )( (A.81)

2 aRHTH TH cbRH  4 (A.82)

(A.83)

Equation A.82 was then rewritten to solve for “a”, as shown in Equation A.84.

4  cbRH  TH a 2 (A.84) RHTH

Equation A.84 was then substituted into Equation A.81, which was rewritten in terms of “b.”

4  cbRH TH  2 Tfcb )( (A.85) RHTH

4 bRH c TH  2 2 2 Tfcb )( (A.86) RHTH RHTH RHTH

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b c 4  2 Tfcb )( 2 (A.87) RHTH RHTH RHTH

b c 4  b c 2 Tf )( 2 (A.88) RHTH RHTH RHTH

1 1 4  b 1 c 1 2 Tf )()()( 2 (A.89) RHTH RHTH RHTH

1 4 1  b 1 Tf )()( 2 c 1 2 )( (A.90) RHTH RHTH RHTH

4 1 Tf )( c 1 )( RH 2 RH 2 b  TH TH (A.91) 1 1 )( RHTH

Equation A.91 was then substituted into Equation A.84 to solve for “a” in terms of “c.”

4 1 Tf )( c 1 )( RH 2 RH 2 4  TH TH RH  c 1 TH (A.92) 1 )( RH  TH a 2 RHTH

4 1 Tf )( c 1 )( RH 2 RH 2 ( TH TH )RH 4  c 1 TH 1 )( (A.93) RH  TH a 2 RHTH

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4 1 Tf )( c 1 )( RH 2 RH 2 4 c  TH TH  a ( )RHTH 2 2 (A.94) 2 1 RHTH RHTH RHTH 1 )( RHTH

4 1 Tf )( c 1 )( RH 2 RH 2 4 c  TH TH  a ( ) 2 2 (A.95) 1 RHTH RHTH RHTH 1 )( RHTH

The next step was to substitute Equation A.91 and A.95 into Equation A.83 and solve explicitly for the coefficient “c.”

4 1 Tf )( c 1 )( RH 2 RH 2 4 c ([ TH TH )  ]( RH 12 )2  1 2 2 TH RH 1 )( RHTH RHTH TH RH TH (A.96) 4 1 Tf )( c 1 )( RH 2 RH 2 TH TH RH 12  Tfc )()( 1 TH 1 )( RHTH

4 1 Tf )( c 1 )( RH 2 RH 2 4 c ( TH TH )( RH 12 2  RH 12 2  RH 12 )()() 2  1 TH 2 TH 2 TH RH 1 )( RHTH RHTH TH RH TH (A.97) 4 1 Tf )( c 1 )( RH 2 RH 2 TH TH RH 12  Tfc )()( 1 TH 1 )( RHTH

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2 4 2 1 2  (Tf )( RHTH 12  RHTH c()() 112  )( RHTH 12 ) RH 2 RH 2 4 TH TH  RH 12 )( 2  1 2 TH RH 1 )( RHTH TH RH TH (A.98) 4 1 (Tf )( RH 12  RH c()() 112  )( RH 12 ) c TH RH 2 TH RH 2 TH RH 12 )( 2  TH TH  Tfc )( 2 TH 1 RHTH 1 )( RHTH

2 4 2 1 2  (Tf )( RHTH 12  RHTH 12 )() c(1 )( RHTH 12 ) RH 2 RH 2 4 TH  TH  RH )( 2  2 TH 12 1 1 RHTH RHTH 1 )( RHTH 1 )( RHTH RHTH (A.99)

4 1 (Tf )( RH  RH c()()  )( RH  ) TH 12 2 TH 112 2 TH 12 c 2 RHTH RHTH RHTH 12 )(   Tfc )( RH 2 1 TH 1 )( RHTH

4 2 1 2 Tf )((  )( RHTH 12 ) c(1 )( RHTH 12 ) RH 2 RH 2 4 TH  TH  RH 12 )( 2  1 1 2 TH RH 1 )( RH 1 )( RHTH TH RH TH RH TH TH (A.100) 4 1 (Tf )( RH 12  RH c()() 112  )( RH 12 ) c TH RH 2 TH RH 2 TH RH 12 )( 2  TH TH  Tfc )( 2 TH 1 RHTH 1 )( RHTH

4 2 1 2 Tf )((  )( RHTH 12 ) c(1 )( RHTH 12 ) RH 2 RH 2 4 TH  TH  RH 12 )( 2  1 1 2 TH RH 1 )( RH 1 )( RHTH TH RH TH RH TH TH (A.101) 4 1 (Tf )( RHTH 12  RHTH 12 )() c(1 )( RHTH 12 ) c RH 2 RH 2 RH 12 )( 2  TH  TH  Tfc )( 2 TH 1 1 RHTH 1 )( 1 )( RHTH RHTH

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1 1 c(1 )( RH 12 )2 c(1 )( RH 12 ) RH 2 TH c RH 2 TH TH  RH 12 )( 2  TH Tfc )(  1 2 TH 1 RH 1 )( RHTH 1 )( TH RH RH TH TH (A.102) 4 2 4 Tf )((  )( RHTH 12 ) Tf )((  )( RHTH 12 ) RH 2 4 RH 2 TH  2  TH 2 RHTH 12 )( 1 RHTH 1 RHTH 1 )( 1 )( RHTH RHTH

1 2 1 (1 )( RHTH 12 ) (1 )( RHTH 12 ) RH 2 1 RH 2 c[ TH  RH 12 )( 2  TH 1 Tf )(]  1 2 TH 1 RH 1 )( RHTH 1 )( TH RH RH TH TH (A.103) 4 2 4 Tf )((  )( RHTH 12 ) Tf )((  )( RHTH 12 ) RH 2 4 RH 2 TH  2  TH 2 RHTH 12 )( 1 RHTH 1 RHTH 1 )( 1 )( RHTH RHTH

4 2 4 Tf )((  )( RHTH 12 ) Tf )((  )( RHTH 12 ) RH 2 4 RH 2 Tf )(  TH  RH 12 )( 2  TH 1 2 TH 1 RH 1 )( RHTH 1 )( TH RH RH c  TH TH (A.104) 1 2 1 (1 )( RHTH 12 ) (1 )( RHTH 12 ) RH 2 1 RH 2 TH  2  TH  [ 2 RHTH 12 )( 1] 1 RHTH 1 RHTH 1 )( 1 )( RHTH RHTH

Equation A.104 presents an explicit expression for the coefficient “c”. Because this equation is so complicated, it was not used to develop explicit expressions for the coefficients “a” and “b”.

Instead, a computer algorithm based upon Equation A.104 was used to obtain numerical values for this coefficient (for each simulation run and associated value of f(T)max), which were then substituted into Equation A.95 to solve for the coefficient “a” and Equation A.91 to solve for “b.”

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A.2.2 Convex Relative Humidity Function Equation A.105 provides the general expression for the convex temperature-RH function.

af T RH 2 bf T RH),(),( c  RH (A.105)

Like before, the coefficients for this equation were determined by establishing boundary conditions and solving a set of simultaneous equations. Similar to the convex temperature function derived earlier, this function has two possible values of f(T,RH) for each value for “RH.”

Equations A.106 through A.108 describe the boundary conditions that were used to establish the three equations needed to determine expressions for the coefficients “a”, “b”, and “c” seen in Equation A.105. Please note that the boundary condition “-f(T)-8” shown in Equation A.108 was used to implement the parabolic symmetry necessary to construct the desired function.

RH=100%, f(T,RH)=f(T) (A.106)

RH=RHTH, f(T,RH)=-4 (A.107)

RH=100% (2nd root), f(T,RH)=-f(T)-8 (A.108)

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-2 Boundary

-2.5 Conditions

-3 f(T,RH)=f(T) f(T) used for -3.5 modeling -4 f(T,RH) f(T) used for -4.5 symmetry -5

-5.5 50 60 70 80 90 100 f(T,RH)= Relative Humidity (%RH) -f(T)-8

Figure A-6 Illustration of the Convex Temperature-Relative Humidity Function

Equations A.109 through A.111 were formed by taking the boundary conditions defined by

Equations A.106 through A.108 and inserting them into Equation A.105. These boundary conditions are illustrated in Figure A-6. As before, the decimal form for relative humidity was used (e.g. 100% RH=1.0).

2 ]f(T)[]f(T)[ cba  1 (A.109)

2 44 )()( cba  TH (A.110)

a 8 2 b ]f(T)[)f(T)( c  18 (A.111)

Equations A.109 through A.111 were rewritten as shown in Equations A.112 through A.114.

2 f(T)f(T) cba  1 (A.112)

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16 4 cba  RHTH (A.113)

a 2 6416 ]f(T)[]f(T)f(T)[ cb  18 (A.114)

The next step was to solve Equation A.113 for “c” in terms of “a” and “b.”

16 4bac  RHTH (A.115)

Equation A.115 was then substituted into Equations A.112 and A.114, thus resulting in Equations

A.116 and A.117.

2 f(T)f(T) 16 4baba RHTH  1 (A.116)

2 a 6416 b 8]f(T)[]f(T))[f(T) 16 4ba RHTH  1 (A.117)

Equation A-116 was then rewritten to solve for “a” in terms of “b.”

2 a 16 b ]f(T)[]f(T)[ 14  RHTH (A.118)

2 a 16 1 RHTH b  4]f(T)[)(])f(T)[ (A.119)

1 RH b  4]f(T)[)( a  TH (A.120) f(T) 2 16

Equation A.117 was then simplified by reorganizing and combining terms.

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2 a f(T))[f(T) 166416 b ]f(T)[] 148  RHTH (A.121)

2 a 4816 b ]f(T)[]f(T))[f(T) 14  RHTH (A.122)

Equation A.120 was then substituted into Equation A.122 to solve for “b.”

1 RHTH b  4]f(T)[)( 2 4816 b ]f(T)[]f(T))[f(T) 14  RHTH (A.123) f(T) 2 16

2 2 1RHTH [f(T))( 4816 b 4 [f(T)]f(T)[]f(T)  4816 ]f(T) b ]f(T)[ 14  RHTH (A.124) )f(T 2 16

2 2 b 4  4816 ]f(T))[f(T]f(T)[ 1RHTH  4816 ]f(T))[f(T)(  b 14 RHTH )(]f(T)[  (A.125) )f(T)( 2 16 )f(T 2 16

2 2 4  4816 ]f(T))[f(T]f(T)[ 1RHTH [f(T))(  4816 ]f(T) b[ f(T)[ ]] 14 RHTH )(  (A.126) )f(T 2 16 f(T) 2 16

2 2 2 4 [f(T)]f(T)[ 4816 f(T)[]f(T) 4][  16]f(T) 1RH TH [f(T))(  4816 ]f(T) b[ 1 RH TH )(]  (A.127) f(T)2 16 f(T)2 16

b[[ 4 [f(T)]f(T) 2 4816 f(T)[]f(T) 4][f(T) 2 16]]  (A.128) 2 2 (1 RHTH )[ 16 1 RHTH [f(T))(]f(T)  4816 ]f(T)

(1 RH )[ 2 16 1 RH [f(T))(]f(T) 2  4816 ]f(T) b  TH TH (A.129) 4 [f(T)]f(T)[ 2 4816 f(T)[]f(T) 4][ 2  16]f(T)

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1 RH 2 16 ((f(T))( 1 RH )) 1 RH 2 1 RH 4816 1 RH )(f(T))(f(T))( b  TH TH TH TH TH (A.130) 4 [f(T)]f(T)[ 2 4816 f(T)[]f(T) 4][ 2  16]f(T)

16 RH 11  RH 4816 1 RH )(f(T))()( b  TH TH TH (A.131) 4 [f(T)]f(T)[ 2 4816 f(T)[]f(T) 4][ 2  16]f(T)

161 RH 641 RH )(f(T))( b  TH TH (A.132) 4 [f(T)]f(T)[ 2 4816 f(T)[]f(T) 4][ 2  16]f(T)

16f(T)( 64)1 RH )( b  TH (A.133) 4 [f(T)]f(T)[ 2 4816 f(T)[]f(T) 4][ 2  16]f(T)

16f(T)( 64)1 RH )( b  TH (A.134) 4 2  16 4 48 44 2 16  4]f(T)[f(T)]f(T)[]f(T)[]f(T)[f(T)f(T)]f(T)[

16f(T)( 64)1 RH )( b  TH (A.135) f(T)f(T) 2 4f(T) 2 16f(T)f(T) 16 4 48 4 48 f(T)f(T))(f(T)f(T) 2 4 2 16  4 16)(f(T)f(T)

16f(T)( 64)1 RH )( b  TH (A.136) 3 4 2 2 f(T)f(T)f(T)f(T)f(T) 192486416 3 4 2 f(T)f(T)f(T)  6416

16f(T)( 64)1 RH )( b  TH (A.137) 3 4 2 16 2 4 2 f(T)f(T)f(T)f(T)f(T)f(T)2f(T)  64192164864

16f(T)( 64)1 RH )( b  TH (A.138) 3 4 16 4 2 [f(T)][2f(T) f(T)]  64192164864

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16f(T)( 64)1 RH )( b  TH (A.139) 2f(T)3 2 f(T)f(T)  1289624

Equation A.139 represents an explicit expression to solve for the unknown coefficient “b.” This equation was substituted into Equation A.120 to develop an explicit solution for “a.”

16f(T)( 64)1 RH )( 1 RH )(  TH  4]f(T)[ TH 2f(T)3 2 f(T)f(T)  (A.140) a  1289624 f(T) 2 16

16f(T)( 64)1 RH )( TH  4]f(T)[ 1RH )( 2f(T)3 2 f(T)f(T)  (A.141) a  TH  1289624 f(T) 2 16 f(T) 2 16

16f(T)( 64)1 RH )( TH  4]f(T)[ 1RH )( 2f(T)3 2 f(T)f(T)  (A.142) a  TH  1289624 f(T) 2 16 f(T)[ ][  44 ]f(T)

16f(T)( 64)1 RHTH )( 1RH )( 2f(T)3 2 f(T)f(T)  (A.143) a  TH  1289624 f(T) 2 16  4]f(T)[ 1RH )( 16f(T)[ 64]1 RH )( a  TH  TH (A.144) f(T) 2 16 [2f(T) 3 2 f(T)f(T) 1289624 ][  4]f(T)

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Equation A.144 was used to explicitly calculate a value for the coefficient “a.” This equation along with Equation A.139 was then substituted into Equation A.115 to develop the explicit expression needed to determine values for the coefficient “c.”

1RH )( 16f(T)[ 64]1 RH )( c  16[ TH  TH ] f(T) 2 16 [2f(T) 3 2 f(T)f(T) 1289624 ][  4]f(T) (A.145) 16f(T)( 64)1 RHTH )( 4  RHTH 2f(T)3 2 f(T)f(T)  1289624

Equations A.139, A.144, and A.145 were used with Equation A.105 (repeated here as Equation

A.146) to calculate values for f(T,RH) when an RH value is known.

af T RH 2 bf T RH),(),( c  RH (A.146)

Equation A.146 was then reorganized as shown by Equation A.147.

af T RH 2 bf T RH),(),( c RH  0 (A.147)

There are two roots to Equation A.147. The positive root, which is the one used for modeling and simulation purposes, was determined in terms of the relative humidity as shown by

Equation A.148.

2 4 cabb  RH)( Tf RH),(  (A.148) 2a

A.3 Derivation of Shape Functions for Use with a New Modeling Approach Applied to a Revised Environmental Characterization Dataset

A.3.1 Derivation of Revised Nonlinear Temperature-Nonlinear Relative Humidity Shape Functions As discussed in Section 4.13.3, the inability to create an accurate model using a revised environmental characterization dataset led to the decision to modify the formulation of the

232 base model by eliminating the three exponential functions applied to the Temperature –

Relative Humidity shape functions. This was done to alter the way that numerical interactions between temperature and relative humidity were calculated. All of the Temperature – Relative

Humidity shape functions used to this point had a minimum value of -4.0. Eliminating the exponentials as described above required the construction of new Temperature – Relative

Humidity shape functions that had a value of zero at both the freezing point and the RH threshold.

A.3.1.1 Revised Convex Temperature Function Equation A.149 provides the general expression for the convex temperature function.

af T 2 bf )()(  TcT (A.149)

The coefficients for this equation were determined by establishing boundary conditions and solving a set of simultaneous equations. The convex curve is symmetric around the temperature axis. This implies that there were two possible values of f(T) for each value for “T.” Equations

A.150 through A.151 describe the boundary conditions that were used to establish the three equations needed to determine values for the unknown coefficients in Equation A.149. Please note that the expression “-f(T)max” used in Equation A.152 is necessary to appropriately implement the symmetry of the parabolic function.

f(T)= f(T)max @T=320.15 (A.150)

f(T)= 0 @T=273.15 (A.151)

f(T)= -f(T)max @T=320.15 (A.152)

233

Equations A.153 through A.155 are the three equations formed by applying the conditions seen in Equations A.150 through A.152 into Equation A.149.

2 a max max cb  .]f(T)[]f(T)[ 15320 (A.153)

2 )0()0( cba  273.15 (A.154)

2 a max b max ]f(T)[)f(T)( c  320.15 (A.155)

Equations A.156 and A.157 were obtained by substituting Equation A.154 into Equation A.153.

2 a max  b max ]f(T)[]f(T)[ 273.  32015 .15 (A.156)

2 (A.157) a max  b max ]f(T)[]f(T)[  47

Equation A.157 was then solved for the coefficient “b” in terms of “a” as seen by Equation A.158.

47  a ]f(T)[ 2 (A.158) b  max f(T) max

Equations A.154 and A.158 was then substituted into Equation A.155 and solved for the coefficient “a”.

2 2 47  a max ]f(T)[ a  max )f(T)(  max ]f(T)[ 273.  32015 .15 (A.159) f(T) max

2 2 a max [)f(T)( 47  a max ]f(T)[ ][ ]1 273.  32015 .15 (A.160)

2 2 a max [)f(T)( 47  a max ]]f(T)[ 273.  32015 .15 (A.161)

234

2 2 a max )f(T)( 47  a max ]f(T)[ 273.  32015 .15 (A.162)

2 a max )f(T)(2  4747 (A.163)

(A.164)

2 a max )f(T)(2  94 (A.165)

94  a 2 (A.166) max )f(T)(2

Equation A.166 was inserted into Equation A.158 to solve explicitly for the coefficient “b”. 94 2 (A.167) 47 [ ][ max ]f(T) )f(T)(2 2 b  max f(T) max

94 (A.168) 47 [ ] b  2 f(T) max

 4747 (A.169) b  f(T) max

b  0 (A.170)

The next steps were to insert Equations A.154, A.166, and A.170 into Equation A.149 as seen in

Equations A.171 and A.172.

94 2  [ 2 Tf )(] 273.15 T (A.171) max )f(T)(2

235

94 2  [ 2 Tf ()(] 273.15 T 0) (A.172) max )f(T)(2

The final step was to solve Equation A.172 for f(T) using the form of the Quadratic equation seen in Equation A.173, which resulted in Equations A.174 and A.175.

2  Tcabb )(4 Tf )(  (A.173) 2a

94  [4 ](273.15  T) )f(T)(2 2 Tf )(  max (A.174) 94 2 2 max )f(T)(2

188  [ ](273.15  T) )f(T)( 2 Tf )(  max (A.175) 94 2 max )f(T)(

A.3.1.2 Revised Convex Temperature-Convex Relative Humidity Shape Function Equation A.175 provides the general expression for the convex temperature-RH shape function.

This function works with the temperature function derived in Section A.3.1.1 to numerically determine interactions between temperature and relative humidity.

af T RH 2 bf T RH),(),( c  RH (A.175)

Like before, the coefficients for this equation were determined by establishing boundary conditions and solving a set of simultaneous equations. Similar to the convex temperature function derived earlier, this function had two possible values of f(T,RH) for each value for “RH.”

Equations A.176 through A.178 describe the boundary conditions that were used to establish

236 the three equations needed to determine expressions for the coefficients “a”, “b”, and “c” seen in Equation A.175.

RH=100%, f(T,RH)=f(T) (A.176)

RH=RHTH, f(T,RH)=0 (A.177)

RH=100% (2nd root), f(T,RH)=-f(T) (A.178)

Equations A.179 through A.181 were formed by taking the boundary conditions defined by

Equations A.176 through A.178 and inserting them into Equation A.175. As before, the decimal form for relative humidity was used (e.g. 100% RH=1.0).

2 ]f(T)[]f(T)[ cba  1 (A.179)

2 )0()0( cba  RHTH (A.180)

a 2 cb  1]f(T)[)f(T)( (A.181)

Equations A.179 through A.181 were rewritten as shown in Equations A.182 through A.184.

2 f(T)f(T) cba  1 (A.182)

c  RHTH (A.183)

2 cba  1f(T)f(T) (A.184)

237

The next step was to solve Equation A.182 for “a” in terms of “b” and “c.”

f(T)1  cb a  (A.185) f(T) 2

Equation A.183 was inserted into Equation A.185 and solved for “a” in terms of “b”.

bf(T)1  RH a  TH (A.186) f(T) 2

Equations A.183 and A.186 were then inserted into Equation A.184 and solved for “b”.

bf(T)1  RHTH 2 [ bf(T)f(T)] RHTH  1 (A.187) f(T) 2

bf(T)1 RHTH bf(T) RHTH  1 (A.188)

b  0f(T)2 (A.189)

b  0 (A.190)

Equation A.190 was then inserted into Equation A.186 and solved for “a”.

1 RH a  TH (A.191) f(T) 2

Equations A.183, A.190, and A.191 were then inserted into Equation A.175 to calculate values for f(T,RH) when an RH value is known.

1 RHTH 2 Tf ),( TH  RHRHRH (A.192) f(T) 2

238

Equation A.192 was then reorganized as shown by Equation A.193.

1 RHTH 2 Tf ),( TH RHRHRH  0 (A.193) f(T) 2

There are two roots to Equation A.193. The negative root was used for symmetry during the derivation of the model and is of no further interest. The positive root is used for modeling and simulation purposes and is determined in terms of the relative humidity as shown by Equation

A.194.

2 4 cabb  RH)( Tf RH),(  (A.194) 2a

Equations A.183, A.190, A.191 were then inserted into Equation A.194, thus resulting in the desired Temperature - Relative Humidity shape function seen by Equation A.195.

1 RHTH  [4 ]( TH  RHRH ) f(T) 2 Tf RH ),(  (A.195) 1 RH [2 TH ] f(T) 2

It should be noted that the shape function, f(T,RH), only applies when f(T) is greater than zero.

A.3.2 Derivation of Revised Nonlinear Temperature-Nonlinear Contaminant Shape Functions

A.3.2.1 Revised Concave Temperature Function

Equation A.196 displays the concave temperature function that was used by the final proof-of- concept model to determine the effect of contaminant levels on corrosion rates.

239

2 bTaT  Tfc )( (A.196)

The coefficients “a”, “b”, and “c” seen in Equation A.196 were determined based upon the temperature boundary conditions shown by Equations A.197 through A.199.

f(T)@T=320.15=f(T)max (A.197)

f(T)@T=273.15=0 (A.198)

f(T)@T=226.15= f(T)max (A.199)

From a corrosion standpoint, there is no physical meaning to the third boundary condition

(Equation A.199) since the temperature of 226.15 is well below the freezing point. The sole purpose of this condition was to provide symmetry with the condition shown in Equation A.197.

Once the functional temperature equation was derived, the model only considered the region from the freezing point to the maximum temperature of 320.15K.

The conditions shown in Equations A.197 through A.199 were inserted into Equation A.196, thus resulting in Equations A.200 through A.202. These three equations were then solved simultaneously in order to determine expressions to solve for values of the unknown coefficients.

2 a 1532015320  Tfcb )(.).( max (A.200)

a(273.15)2 273.15 cb  0 (A.201)

240

2 a 1522615226  Tfcb )(.).( max (A.202)

The first step was to solve Equation A.200 in terms of the coefficient “a.”

)(  .15320 bcTf a  max (A.203) 15320 ).( 2

The next step was to substitute Equation A.203 into Equation A.202 and solve for the coefficient “b.”

)( max  .15320 bcTf 2 [ ]( 1522615226  Tfcb )(.). max (A.204) 15320 ).( 2 Equation A.204 was expanded and simplified as shown by the steps contained in Equations

A.205 through A.212. The final equation in this sequence describes the coefficient “b” in terms

of the coefficient “c” and the variable f(T)max.

2 2 2 15226 Tf )().( max 15226 ).( c 1522615320 ).(. b   15226  Tfcb )(. max (A.205) 15320 ).( 2 15320 ).( 2 15320 ).( 2 2 2 2  15226 ).( c 15226 ).( b 15226 Tf )().( max  15226 Tfcb )(. max  (A.206) 15320 ).( 2 .15320 15320 ).( 2

2 2 2 15226 ).( b  15226 ).( c 15226 Tf )().( max .15226 b c  Tf )( max  (A.207) .15320 15320 ).( 2 15320 ).( 2

2 2 2 15226 ).( b 15226 ).( c 15226 Tf )().( max .15226 b  Tfc )( max  (A.208) .15320 15320 ).( 2 15320 ).( 2

15226 2 ).( 15226 2 ).( 15226 2 ).( .[ 15226  b  [] c [] 11  Tf )(] max (A.209) .15320 15320 ).( 2 15320 ).( 2

15226 2).( 15226 2 ).( 15226 2).( .[ 15226  b [] 1 c  [] 1 Tf )(] max (A.210) .15320 15320 ).( 2 15320 ).( 2

241

15226 2 ).( 15226 2 ).( .[ 15226  b  [] 1](  Tfc max ))( (A.211) .15320 15320 ).( 2

15226 2 ).( [ 1] 15320 ).( 2 b   Tfc max ))(( (A.212) 15226 2 ).( .[ 15226  ] .15320

The next step was to substitute Equations A.203 and Equation A.212 into Equation A.201, which was then solved to provide the explicit expression needed to calculate values for “c.”

(226.15)2 [ 1](  Tfc max ))( (320.15)2 )( max cTf  320.15[ ] (226.15)2 226.15  320.15 (273.15)2  (320.15)2 (A.213) (226.15)2 [ 1](  Tfc max ))( (320.15)2 273.15 c  0 (226.15)2 226.15  320.15 Equation A.213 was then expanded and reorganized as seen in Equations A.214 through A.221 in order to obtain the desired explicit relationship for the coefficient “c”.

(226.15)2 [ 1](  Tfc max ))( Tf )( c (320.15)2 max   320.15[ ](273.15)2  (320.15 2 () 320.15)2 (226.15)2 [226.15  ](320.15)2 320.15 (A.214) (226.15)2 [ 1](  Tfc max ))( (320.15)2 273.15 c  0 (226.15)2 226.15  320.15

242

(226.15)2 (273.15 2[)  ]1 c (273.15 2 Tf )() (273.15)2 c (320.15)2 max  [ ]  (320.15)2 (320.15)2 (226.15)2 [226.15  ](320.15) 320.15 (A.215) 2 2 2 (226.15) (226.15) (273.15 [) 1]( Tf max ))( [ 1](  Tfc max ))( (320.15)2 (320.15)2 [ ]  273.15 c  0 (226.15)2 (226.15)2 [226.15  ](320.15) 226.15  320.15 320.15

2 2 2 (226.15) (226.15) (273.15 [)  ]1 c [ 1](  Tfc max ))( (273.15)2 c (320.15)2 (320.15)2  [ ]  273.15 c  (320.15)2 (226.15)2 (226.15)2 [226.15  ](320.15) 226.15  320.15 320.15 (A.216) 2 2 (226.15) (273.15 [)  Tf )(]1 max (273.15 2 Tf )() (320.15)2  max [ ] (320.15)2 (226.15)2 [226.15  ](320.15) 320.15

)15.226( 2 )15.226( 2 2[)15.273(  ]1 c [  ]1 c )15.273( 2 c )15.320( 2 )15.320( 2  [  15.273]  )15.320( 2 )15.226( 2 )15.226( 2 15.226[  ]( )15.320 15.226  15.320 15.320 )15.226( 2 [ 1]( Tf max ))( )15.320( 2 2 Tf )()15.273( 15.273 c  max  (A.217) )15.226( 2 )15.320( 2 15.226  15.320 2 2 )15.226( [)15.273(  Tf )(]1 max )15.320( 2 [ ] )15.226( 2 15.226[  ]( )15.320 15.320

243

)15.226( 2 )15.226( 2 2[)15.273(  ]1 c [  ]1 c )15.273( 2 c )15.320( 2 )15.320( 2  [  15.273]  )15.320( 2 )15.226( 2 )15.226( 2 15.226[  ]( )15.320 15.226  15.320 15.320 )15.226( 2 [  Tf )(]1 max )15.320( 2 2 Tf )()15.273( 15.273 c  max  (A.218) )15.226( 2 )15.320( 2 15.226  15.320 2 2 )15.226( [)15.273(  Tf )(]1 max )15.320( 2 [ ] )15.226( 2 15.226[  ]( )15.320 15.320

(226.15)2 (226.15)2 (273.15 2[)  ]1 c [  ]1 c (273.15)2 c (320.15)2 (320.15)2  [ ]  273.15 c  (320.15)2 (226.15)2 (226.15)2 [226.15  ](320.15) 226.15  320.15 320.15 (A.219) 2 2 2 (226.15) (226.15) (273.15 [)  Tf )(]1 max [  Tf )(]1 max (273.15 2 Tf )() (320.15)2 (320.15)2  max [ ]  273.15 (320.15)2 (226.15)2 (226.15)2 [226.15  ](320.15) 226.15  320.15 320.15

(226.15)2 (226.15)2 (273.15 2[)  ]1 [  ]1 (273.15)2 (320.15)2 (320.15)2 c[ [ ]  273.15 ]1  (320.15)2 (226.15)2 (226.15)2 [226.15  ](320.15) 226.15  320.15 320.15 (A.220) 2 2 2 (226.15) (226.15) (273.15 [)  Tf )(]1 max [  Tf )(]1 max (273.15 2 Tf )() (320.15)2 (320.15)2  max [ ]  273.15 (320.15)2 (226.15)2 (226.15)2 [226.15  ](320.15) 226.15  320.15 320.15

244

2 2 2 (226.15) (226.15) (273.15 [)  Tf )(]1 max [  Tf )(]1 max (273.15 2 Tf )() (320.15)2 (320.15)2  max [ ]  273.15 (320.15)2 (226.15)2 (226.15)2 [226.15  ](320.15) 226.15  c  320.15 320.15 (A.221) (226.15)2 (226.15)2 (273.15 2[)  ]1 [  ]1 (273.15)2 (320.15)2 (320.15)2 [ [ ]  273.15  ]1 (320.15)2 (226.15)2 (226.15)2 [226.15  ](320.15) 226.15  320.15 320.15

Obviously Equation A.221 could be simplified by solving the many elementary mathematical expressions seen in its formulation. However, this full form was retained so that all calculations were made using the numerical precision of a computer in order to preclude round-off error.

Thus, this equation was solved during each simulation run using a computer algorithm to obtain a numerical value for “c”, which was then used by an algorithm implementing Equation A.212 to obtain a value for “b”. The numerical values for “b” and “c” were then used to calculate a numerical value for “a” based upon Equation A.203. These numerical values were then used with Equation A.196 to complete the concave temperature function.

A.3.2.2 Revised Concave Temperature-Contaminant Shape Function Equation A.222 describes the general form of the concave Temperature - Contaminant shape function. This function works with the temperature function derived in Section A.3.2.1 to numerically determine interactions between temperature and contaminant levels. The equation was used to calculate values when the temperature is between 273.15K and 320.15K and the contaminant level (C) is between zero and the maximum value possible.

2 bCaC  CTfc ),( (A.222)

Three boundary conditions were needed to create the three equations that were solved simultaneously to determine values for the coefficients “a”, “b”, and “c.” One condition

245 concerns the maximum value of the function, which occurs at the highest contaminant level,

Cmax, within the range considered by the model. At this point, the value of f(T,C) was the same as the value of f(T) for the same temperature. The second boundary condition concerns the minimum value for f(T,RH), which was defined as 0 (the same as the other functions). This boundary condition applies when the level of any particular atmospheric contaminant is undetectable. As before, the symmetry of a parabola can be used to create the third boundary

condition. Therefore, when C=-Cmax, f(T,C)=f(T). Please note that this third relationship has no meaning with respect to corrosion. It was only used to define the full parabolic shape, half of which was used for the corrosion model. All three boundary conditions are recorded below as

Equations A.223 to A.225.

C=Cmax, f(T,C)=f(T) (A.223)

(A.224) C=0, f(T,C)=0

C=-Cmax, f(T,C)=f(T) (A.225)

Equations A.226 through A.228 result when Equations A.223 through A.225 were applied to

Equation A.222.

2 max max  TfcCbCa )()()( (A.226)

2 cba  0)0()0( (A.227)

2 max max  TfcCbCa )()()( (A.228)

246

These equations were then reduced to the following forms.

2 max bCaC max  Tfc )( (A.229)

c  0 (A.230)

2 max bCaC max  Tfc )( (A.231)

Inserting Equation A.230 into Equations A.229 and A.231 leads to Equations A.232 and A.233.

2 max bCaC max  Tf )( (A.232)

2 max bCaC max  Tf )( (A.233)

Equation A.232 was then rewritten to solve for “a” in terms of “b”, as shown in Equation A.234.

Tf )(  bC  max a 2 (A.234) Cmax

Equation A.234 was now substituted into Equation A.233, which was then rewritten in terms of “b.”

Tf )(  bC max 2  [ 2 ]Cmax bCmax Tf )( (A.235) Cmax

(A.236) Tf )( max bCbC max  Tf )(

(A.237) bCmax  02 (A.238) b  0

Equation A.238 was then substituted into Equation A.234 to solve for “a” in terms of “c.” Tf )(  a 2 (A.239) Cmax

247

The next step was to substitute Equations A.230, A.238, and A.239 into Equation A.222 to obtain the desired parabolic equation, as seen by Equation A.240.

Tf )( 2  2 CTfC ),( (A.240) Cmax

248

APPENDIX B

DATA COLLECTION AND ESTIMATION

B.1 Corrosion Test Measurements and Environmental Characterization at Proxy Locations

Table B-1 Weight Loss Measurement Data Used to Calibrate Candidate Models Corrosion Test Site Date 2) Calibration Site Data (legacy weight loss data measured by Battelle [25]) China Lake, CA 4/16/2006 1593 7/18/2006 3563 Start - 1/19/2006 10/18/2006 4428 1/25/2007 5485 Dobbins ARB, GA 11/1/2003 9466 3/1/2004 23897 Start - 9/1/2003 6/1/2004 47225 8/31/2004 75748 Ft. Drum, NY 8/1/2005 5513 11/1/2005 12104 Start - 5/1/2005 2/1/2006 17310 4/30/2006 23541 Kennedy Space 10/1/2005 10708 Center, FL 1/1/2006 32940 4/1/2006 43876 Start - 7/1/2005 6/30/2006 53447

The cumulative damage model was calibrated using published test data measured by Battelle at four different locations including China Lake, California; Dobbins Air Reserve Base, Georgia; Fort

Drum, New York; and Kennedy Space Center, Florida [25]. The accuracy of the calibrated model was then validated by comparing predictions to test measurements made at ten other locations.

249

Table B-2 Weight Loss Measurement Data Used to Validate Candidate Models Corrosion Test Site Date 2) Validation Site Data (data measured Under SERDP Sponsored Program [136] Daytona Beach, FL 8/6/2009 28626 11/6/2009 68157 Start – 5/6/2009 2/6/2010 90335 5/5/2010 End of Life Kirtland AFB, NM 11/20/2009 674 2/20/2010 1575 Start – 8/20/2009 6/10/2010 1849 8/19/2010 3256 Point Judith, RI 9/10/2009 24767 12/10/2009 47885 Start – 6/10/2009 3/10/2010 67116 7/6/2010 81671 Tyndall AFB, FL 2/1/2010 7970 5/11/2010 12652 Start – 10/29/2009 7/29/2010 16972 11/1/2010 21001 Wright Patterson AFB, 8/6/2010 5833 OH 11/12/2010 9893 3/2/2011 10111 Start – 5/6/2010 5/5/2011 14450 Validation Site Data (legacy weight loss data measured by Battelle [25]) Ft. Campbell, KY 8/1/2005 7637 11/1/2005 14353 Start – 5/1/2005 2/1/2006 20805 4/30/2006 26949 Ft. Hood, TX 7/1/2005 4130 10/1/2005 6938 Start – 4/1/2005 1/1/2006 9246 3/31/2006 13454 Ft. Rucker, AL 5/1/2005 3825 8/1/2005 10222 Start – 2/1/2005 11/1/2005 15707 1/31/2006 21782 Rock Island Arsenal, IL 7/1/2005 5168 10/1/2005 11536 Start – 4/1/2005 1/1/2006 19870 3/31/2006 26735 West Jefferson, OH 3/1/2005 8006 6/1/2005 15924 Start - 1/1/2005 9/1/2005 24182 21/31/2005 31514

250

Five of these locations were sites affiliated with an AFRL program sponsored by SERDP [136].

These sites include Daytona Beach, Florida; Kirtland Air Force Base, New Mexico; Point Judith,

Rhode Island; Tyndall Air Force Base, Florida;, and Wright Patterson Air Force Base, Ohio. The remaining five sites were part of earlier efforts conducted by Battelle [25]. These sites include

Fort Campbell, Kentucky; Fort Hood, Texas; Fort Rucker, Alabama; Rock Island Arsenal, Illinois, and West Jefferson, Ohio. All 14 testing efforts were conducted using the same protocol.

Tables B-1 and B-2 contain the test measurements (and related metadata) that was used during this research program.

B.2 Descriptions of Corrosion Test and Environmental Characterization Sites The following tables provide a variety of information pertaining to the corrosion test and environmental characterization sites including locations/addresses, identification numbers

(where appropriate), type of data measured, geographical coordinates of sites, and distances from the actual corrosion test sites. The identification codes for ozone and SO2 are those used by the EPA AQS database. Chloride sites are identified using the NADP identification codes and weather offices are identified using their International Civil Aviation Organization (ICAO)/World

Meteorological Organization (WMO) designations. A map for each of the fourteen test locations is also included in order to visually illustrate the location of the environmental characterization sites in comparison to the corrosion test sites. It should be noted that in some cases, the corrosion test site was located between two contaminant measurement sites. In these cases, the average of the two measurements was used as an estimate for the contaminant levels at the test site.

251

Table B-3 China Lake Corrosion Test Site and Associated Environmental Characterization Sites Location Identification Available Approximate Coordinates Approximate Distance from Code Data Latitude Longitude Test Site Naval Air Warfare Center, China NA – Corrosion Weight Loss 35.68 -117.68 - Lake, CA Test Site

SO2 Monitoring Site Corner of Athol and Telescope, 06-071-1234 SO2 35.76387 -117.397 27.2 km Trona, CA (16.9 miles)

Ozone Monitoring Site Corner of Athol and Telescope, 06-071-1234 O3 35.76387 -117.397 27.2 km Trona, CA (16.9 miles)

Chloride Monitoring Sites Sequoia National Park-Giant Forest, CA75 Chloride 36.5661 -118.778 139.4 km Tulare County, CA (86.6 miles) Yosemite National Park-Hodgdon CA99 Chloride 37.7961 -119.8581 304.9 km Meadow, (189.5 miles) Weather Monitoring Site Naval Air Warfare Center, China KNID/74612 Temperature 35.68 -117.68 0 km Lake, CA Dewpoint (0 miles)

252

Chloride Site Chloride Site

Ozone and SO2 Site

Test and Weather Sites

Calibration Data Site

Figure B-1 Illustration of Corrosion Test Site at China Lake and Proximity of Associated Environmental Characterization Sites [147]

253

Table B-4 Dobbins Air Reserve Base Corrosion Test Site and Associated Environmental Characterization Sites Location Identification Available Approximate Coordinates Approximate Code Data Latitude Longitude Distance from Test Site Dobbins Air Reserve Base (formerly NA – Corrosion Weight Loss 33.9153 -84.5163 - Dobbins AFB), Marietta, GA Test Site

SO2 Monitoring Sites 311 Ferst St., Atlanta, Georgia 13-121-0048 SO2 33.779330 -84.395760 18.8 km (11.7 miles)

HWY. 113, Stilesboro, GA 13-015-0002 SO2 34.103333 -84.915278 42.4 km (26.3 miles) Ozone Monitoring Site

Confederate Ave., Atlanta, GA 13-121-0055 O3 33.72057 -84.3574 26.1 km (16.2 miles) Chloride Monitoring Site Georgia Experiment Station, Pike GA41 Chloride 33.1805 -84.4103 82.1 km County, GA (51.0 miles) Weather Monitoring Site National Weather Service, Atlanta KATL/72219 Temperature 33.64028 -84.42694 31.6 km Hartsfield International Airport Dewpoint (19.6 miles)

254

Test Site

SO2 Site

SO2 Site

Ozone Site

Weather Site

Chloride Site

Calibration Data Site

Figure B-2 Illustration of Corrosion Test Site at Dobbins Air Reserve Base and Proximity of Associated Environmental Characterization Sites [147]

255

Table B-5 Fort Drum Corrosion Test Site and Associated Environmental Characterization Sites Location Identification Available Approximate Coordinates Approximate Code Data Latitude Longitude Distance from Test Site Ft. Drum, NY NA – Corrosion Weight Loss 44.05 -75.73 - Test Site

SO2 Monitoring Site 278 Bisby Road, Old Forge, NY 36-043-0005 SO2 43.685780 -74.985380 72.3 km (44.9 miles)

Ozone Monitoring Site Perch River Game Management 36-045-0002 O3 44.08747 -75.97316 19.9 km Area, LaFargeville, NY (12.4 miles)

Chloride Monitoring Site Bennett Bridge, Oswego NY52 Chloride 43.5282 -75.9492 60.6 km County, NY (37.7 miles) Weather Monitoring Site Wheeler-Sack Army Airfield, Ft. KGTB/74370 Temperature 44.05 -75.73 0 km Drum, NY Dewpoint (0 miles)

256

Ozone Site Test and Weather Sites

SO2 Site

Chloride Site

Calibration Data Site

Figure B-3 Illustration of Corrosion Test Site at Fort Drum and Proximity of Associated Environmental Characterization Sites [147]

257

Table B-6 Kennedy Space Center Corrosion Test Site and Associated Environmental Characterization Sites Location Identification Available Approximate Coordinates Approximate Code Data Latitude Longitude Distance from Test Site Kennedy Space Center, NA – Corrosion Weight Loss 28.659019 -80.743619 - Titusville, FL Test Site

SO2 Monitoring Site Morris Blvd., Winter Park, FL 12-095-2002 SO2 28.599444 -81.363056 60.9 km (37.9 miles)

Ozone Monitoring Site

401 Florida Ave., Melbourne, FL 12-009-0007 O3 28.053889 -80.628611 68.0 km (42.3 miles)

Chloride Monitoring Site Kennedy Space Center, Brevard FL99 Chloride 28.5428 -80.644 16.2 km County, FL (10.0 miles) Weather Monitoring Site National Weather Service, KMLB/72204 Temperature 28.08 -80.62 65.3 km Melbourne International Dewpoint (40.6 miles) Airport

258

Test Site

SO2 Site

Chloride Site

Weather Site

Ozone Site Calibration Data Site

Figure B-4 Illustration of Corrosion Test Site at Kennedy Space Center and Proximity of Associated Environmental Characterization Sites [147]

259

Table B-7 Daytona Beach Corrosion Test Site and Associated Environmental Characterization Sites Location Identification Available Approximate Coordinates Approximate Code Data Latitude Longitude Distance from Test Site Daytona Beach, FL NA – Corrosion Weight Loss 29.08229 -80.9242 - Test Site

SO2 Monitoring Site Morris Blvd., Winter Park, FL 12-095-2002 SO2 28.599444 -81.363056 68.5 km (42.6 miles)

Ozone Monitoring Site

5200 Spruce Creek Rd., 12-127-2001 O3 29.10889 -80.99389 7.4 km Port Orange, FL (4.6 miles)

Chloride Monitoring Site Kennedy Space Center, Brevard FL99 Chloride 28.5428 -80.644 65.8 km County, FL (40.9 miles) Weather Monitoring Site Daytona Beach International KDAB/none Temperature 29.167 -81.05 15.4 km Airport Dewpoint (9.6 miles)

260

Weather Site

Ozone Site

Test Site

SO2 Site

Chloride Site

SERDP Program Test Site

Figure B-5 Illustration of Corrosion Test Site at Daytona Beach and Proximity of Associated Environmental Characterization Sites [147]

261

Table B-8 Kirtland Air Force Base Corrosion Test Site and Associated Environmental Characterization Sites Location Identification Available Approximate Coordinates Approximate Code Data Latitude Longitude Distance from Test Site Kirtland AFB, NM NA – Corrosion Weight Loss 35.040278 -106.60917 - Test Site

SO2 Monitoring Site Dine College, Shiprock, NM 35-045-1233 SO2 36.807100 -108.69523 271.8 km (168.9 miles)

Ozone Monitoring Site

6000 Anderson Ave SE, 35-001-0024 O3 35.0631 -106.57879 3.8 km Albuquerque, NM (2.3 miles)

Chloride Monitoring Site Bandolier National Monument, NM07 Chloride 35.7788 -106.266 87.7 km Los Alamos County NM (54.5 miles) Weather Monitoring Site Albuquerque International KABQ/72365 Temperature 35.033 -106.617 1.1 km Airport Dewpoint (0.7 miles)

262

Chloride Site

SO2 Site

Test, Ozone, and Weather Sites

SERDP Program Test Site

Figure B-6 Illustration of Corrosion Test Site at Kirtland AFB and Proximity of Associated Environmental Characterization Sites [147]

263

Table B-9 Point Judith Corrosion Test Site and Associated Environmental Characterization Sites Location Identification Available Approximate Coordinates Approximate Code Data Latitude Longitude Distance from Test Site Point Judith, RI NA – Corrosion Weight Loss 41.361028 -71.481389 - Test Site

SO2 Monitoring Site 659 Globe Street, Fall River, 25-005-1004 SO2 41.683279 -71.169171 44.3 km MA (27.5 miles)

Ozone Monitoring Site

Tarzwell Rd., Narragansett, RI 44-009-0007 O3 41.495110 -71.423705 15.7 km (9.7 miles)

Chloride Monitoring Site Cedar Beach, Suffolk County, NY96 Chloride 41.0347 -72.3891 84.3 km NY (52.4 miles) Weather Monitoring Site Newport State Airport KUUU/none Temperature 41.532 -71.282 25.3 km Dewpoint (15.7 miles)

264

SO2 Site

Weather Site

Ozone Site

Test Site

Chloride Site

SERDP Program Test Site

Figure B-7 Illustration of Corrosion Test Site at Point Judith and Proximity of Associated Environmental Characterization Sites [147]

265

Table B-10 Tyndall Air Force Base Corrosion Test Site and Associated Environmental Characterization Sites Location Identification Available Approximate Coordinates Approximate Code Data Latitude Longitude Distance from Test Site Tyndall AFB, FL NA – Corrosion Weight Loss 30.067 -85.583 - Test Site

SO2 Monitoring Site 3100 Airport Thruway Road, 13-215-0008 SO2 32.521271 -84.944630 278.8 km Columbus, Georgia (173.3 miles)

Ozone Monitoring Site

St. Andrews State Park, Panama 12-005-0006 O3 30.130433 -85.731517 16.0 km City Beach, FL (9.9 miles)

Chloride Monitoring Site Sumatra, Liberty County, FL FL23 Chloride 30.1106 -84.9902 57.4 km (35.6 miles) Weather Monitoring Site Tyndall AFB KPAM/74775 Temperature 30.067 -85.583 0 km Dewpoint (0 miles)

266

SO2 Site

Test and Weather Sites Ozone Site

Chloride Site

SERDP Program Test Site

Figure B-8 Illustration of Corrosion Test Site at Tyndall AFB and Proximity of Associated Environmental Characterization Sites [147]

267

Table B-11 Wright Patterson Air Force Base Corrosion Test Site and Associated Environmental Characterization Sites Location Identification Available Approximate Coordinates Approximate Code Data Latitude Longitude Distance from Test Site Wright Patterson AFB, OH NA – Corrosion Weight Loss 39.82306 -84.049444 - Test Site

SO2 Monitoring Site 5400 Spangler, Fairborn, OH 39-023-0003 SO2 39.855670 -83.997730 5.7 km (3.6 miles)

Ozone Monitoring Site

5400 Spangler, Fairborn, OH 39-023-0003 O3 39.855670 -83.997730 5.7 km (3.6 miles)

Chloride Monitoring Site Oxford, Butler County, OH OH09 Chloride 39.5309 -84.7238 66.3 km (41.2 miles) Weather Monitoring Site Wright Patterson AFB, OH KFFO/74570 Temperature 39.833 -84.05 1.1 km Dewpoint (0.7 miles)

268

Ozone and SO2 Site

Test and Weather Sites

Chloride Site

SERDP Program Test Site

Figure B-9 Illustration of Corrosion Test Site at Wright Patterson AFB and Proximity of Associated Environmental Characterization Site [147]

269

Table B-12 Fort Campbell Corrosion Test Site and Associated Environmental Characterization Sites Location Identification Available Approximate Coordinates Approximate Code Data Latitude Longitude Distance from Test Site Ft. Campbell, KY NA – Corrosion Weight Loss 36.667 -87.5 - Test Site

SO2 Monitoring Site Mammoth Cave National Park, 21-061-9000 SO2 37.131874 -86.147869 131.1 km KY (81.5 miles)

Ozone Monitoring Site

10800 Pilot Rock Road, 21-047-0006 O3 36.911710 -87.323337 31.4 km Hopkinsville, KY (19.5 miles)

Chloride Monitoring Site Mulberry Flat, Trigg County, KY KY99 Chloride 36.9029 -88.0121 52.7 km (32.7 miles) Weather Monitoring Site Ft. Campbell Army Airfield KHOP/74671 Temperature 36.667 -87.5 0 km (AAF) Dewpoint (0 miles)

270

Chloride Site Ozone Site SO2 Site

Test and Weather Sites

Legacy Battelle Test Site

Figure B-10 Illustration of Corrosion Test Site at Fort Campbell and Proximity of Associated Environmental Characterization Sites [147]

271

Table B-13 Fort Hood Corrosion Test Site and Associated Environmental Characterization Sites Location Identification Available Approximate Coordinates Approximate Code Data Latitude Longitude Distance from Test Site Ft. Hood, TX NA – Corrosion Weight Loss 31.13 -97.78 - Test Site

SO2 Monitoring Site Gregg Co. Airport, Longview, TX 48-183-0001 SO2 32.378682 -94.711811 322.0 km (200.1 miles)

Ozone Monitoring Site

12200 Lime Creek Rd. 48-453-0020 O3 30.483168 -97.872301 72.3 km Leander, TX (44.9 miles)

Chloride Monitoring Site Sonora, Edwards County, TX TX16 Chloride 30.2613 -100.5551 282.8 km (175.7 miles) Weather Monitoring Site Robert Gray Army Airfield, FT. KGRK/none Temperature 31.06722 -97.82889 8.4 km Hood, TX Dewpoint (5.2 miles)

272

SO2 Site

Test Site

Weather Site

Chloride Site Ozone Site

Legacy Battelle Test Site

Figure B-11 Illustration of Corrosion Test Site at Fort Hood and Proximity of Associated Environmental Characterization Sites [147]

273

Table B-14 Fort Rucker Corrosion Test Site and Associated Environmental Characterization Sites Location Identification Available Approximate Coordinates Approximate Code Data Latitude Longitude Distance from Test Site Ft. Rucker, AL NA – Corrosion Weight Loss 31.267 -85.7 - Test Site

SO2 Monitoring Site 3100 Airport Thruway Road, 13-215-0008 SO2 32.521271 -84.944630 156.4 km Columbus, Georgia (97.2 miles)

Ozone Monitoring Site

161 Buford Lane, Dothan, AL 01-069-0004 O3 31.19066 -85.4231 27.7 km (17.2 miles)

Chloride Monitoring Site Quincy, Gadsden County, FL FL14 Chloride 30.5486 -84.6004 131.9 km (82.0 miles) Weather Monitoring Site Cairns AAF, Ft. Rucker, AL KOZR/none Temperature 31.267 -85.7 0 km Dewpoint (0 miles)

274

SO2 Site

Test and Weather Sites

Ozone Site

Chloride Site

Legacy Battelle Test Site

Figure B-12 Illustration of Corrosion Test Site at Fort Rucker and Proximity of Associated Environmental Characterization Sites [147]

275

Table B-15 Rock Island Arsenal Corrosion Test Site and Associated Environmental Characterization Sites Location Identification Available Approximate Coordinates Approximate Code Data Latitude Longitude Distance from Test Site Rock Island Arsenal, IL NA – Corrosion Weight Loss 41.51821 -90.537058 - Test Site

SO2 Monitoring Site th 10 Street & Vine Street, 19-163-0015 SO2 41.530011 -90.587611 4.4 km Davenport, IA (2.8 miles)

Ozone Monitoring Site

32 Rodman, Ave., Rock Island 17-161-3002 O3 41.51473 -90.51735 1.7 km Arsenal, IL (1.1 miles)

Chloride Monitoring Site Monmouth, Warren County, IL IL78 Chloride 40.9333 -90.7231 66.8 km (41.5 miles) Weather Monitoring Site Quad-City International Airport KMLI/72544 Temperature 41.45 -90.517 7.8 km Dewpoint (4.8 miles)

276

Ozone Site

SO2 Site

Test Site

Weather Site

Chloride Site

Legacy BattelleTest Site

Figure B-13 Illustration of Corrosion Test Site at Rock Island Arsenal and Proximity of Associated Environmental Characterization Sites [147]

277

Table B-16 West Jefferson Corrosion Test Site and Associated Environmental Characterization Sites Location Identification Available Approximate Coordinates Approximate Code Data Latitude Longitude Distance from Test Site West Jefferson, OH NA – Corrosion Weight Loss 39.96817 -83.25059 - Test Site

SO2 Monitoring Site 5400 Spangler, Fairborn, OH 39-023-0003 SO2 39.855670 -83.997730 65.1 km (40.5 miles)

Ozone Monitoring Site

940 State Route 38 SW, 39-097-0007 O3 39.788190 -83.476060 27.8 km London, OH (17.3 miles)

Chloride Monitoring Site Deer Creek State Park, OH54 Chloride 39.6359 -83.2606 36.9 km Pickaway County, OH (22.9 miles) Weather Monitoring Site Rickenbacker International KLCK/none Temperature 39.814 -82.928 32.5 km Airport Dewpoint (20.2 miles)

278

Test Site

SO2 Site

Weather Site Ozone Site

Chloride Site

Legacy Battelle Test Site

Figure B-14 Illustration of Corrosion Test Site at West Jefferson and Proximity of Associated Environmental Characterization Sites [147]

279

B.2.1 Estimation of Hourly Ozone Levels to Replace Missing Data As discussed in Section 3.1.3.1.2, many ozone monitoring stations only measured levels during times of the year when UV light intensity was sufficiently high to react with hydrocarbons to produce the pollutant. Thus, hourly ozone levels had to be estimated for the remaining periods of the year for those locations that do not have annual coverage. Of the 14 monitoring sites used to produce proxy ozone data for this modeling effort, only seven of them had complete coverage throughout the year. The processes used to create hourly estimates for the remaining seven sites were described in Section 3.1.3.1.2 for the specific example of Fort Drum, New York.

Following these processes, details pertaining to the estimates for Fort Campbell, KY; Dobbins,

GA; Point Judith, RI; Fort Rucker, AL; West Jefferson, OH; and Wright Patterson, OH are shown below.

B.2.1.1 Estimates for Missing Ozone Readings at Fort Campbell, Kentucky

0.06

0.05

0.04

0.03

0.02

0.01

Average Ozone Level (ppm) Level Ozone Average 0 0 2 4 6 8 10 12 Month of Year

Trona Hopkinsville

Figure B-15 Monthly Average Proxy Ozone Levels for Fort Campbell and China Lake

280

Table B-17 Average Monthly Ozone Levels/Comparison Ratios (Ft Campbell vs. China Lake) Month of Year Average Ozone Levels

Trona Hopkinsville, KY Adjusted Hopkinsville Data

January 0.02398167

February 0.02730506

March 0.03739785 0.03686

April 0.04055972 0.044972

May 0.04961559 0.04838

June 0.05565278 0.04189 0.05

July 0.05253091 0.039942 0.049

August 0.05195833 0.041106 0.046

September 0.040625 0.043031

October 0.03163844 0.032173

November 0.02598194

December 0.01987903

Comparison Ratios

March June October 0.9856239 0.89842775 1.016908

In the same manner as used to estimate the missing data for Fort Drum, ozone levels measured at Trona, California (proxy for China Lake) were used as the basis for completing the proxy for the Hopkinsville, KY site (proxy for Fort Campbell) as well as the other sites needing such estimation. As seen on Figure B-15, this plot differs from the plot for Fort Drum (Figure 3-8) in that it has two separate peaks. The reason for this behavior is unknown but it is possible that

281

UV light intensity at Hopkinsville KY was altered during the time period between the two peaks by the weather during that time period (e.g., a cold, wet summer). Thus, it was assumed that the data behind this plot could be adjusted to give it a more Gaussian-like appearance that would be more representative of a typical year. The actual data for both sites and the adjusted data for Hopkinsville are shown on Table B-17. Also shown on the bottom of this table are comparison ratios that were obtained by dividing select monthly averages (the adjusted peak month and the beginning and end of observations) for the Hopkinsville site by the associated monthly average for Trona. These ratios were used later to calculate ratios for the missing months.

Figure B-16 illustrates the original and adjusted data for Fort Campbell in comparison to the monthly averages that were calculated for China Lake.

0.06

0.05

0.04

0.03

0.02

0.01 Average Average Monthof(ppm) Year 0 0 2 4 6 8 10 12 Month of Year

Trona Hopkinsville Adjusted

Figure B-16 Monthly Ozone Levels for Fort Campbell and China Lake (includes adjustments)

282

Table B-18 Periodic Relationship of Comparison Ratios (Fort Campbell vs. China Lake Proxies) Month Cumulative Number of Comparison Ratio Months

October 1 1.016908

November 2

December 3

January 4

February 5

March 6 0.9856239

April 7

May 8

June 9 0.89842775

July 10

August 11

September 12

October 13 1.016908

November 14

December 15

January 16

February 17

March 18

April 19

The comparison ratios shown in Table B-17 were applied to a numerical listing of sequential months as shown in Table B-18. When constructing this new table, it was assumed that the

283 ratios from one year to the next were exactly the same. While such an assumption is unrealistic from a physical standpoint, it was necessary to create the symmetric point, which was then used to create a function for the period of time in question. The next step was to plot the comparison ratios shown on Table B-18 and then fit a cubic polynomial as seen in Figure B-17.

This equation, which is shown on the figure, was then used to calculate ratios for the months requiring estimated data. The independent variable “x” seen on this equation corresponds to the numerical value of the month (e.g., January has a value of 1.0).

2.5 y = 0.0009x3 - 0.0206x2 + 0.1168x + 0.8584 2

1.5

1

0.5 Ratios (nondimensional) Ratios

0 0 5 10 15 20 Months

Figure B-17 Cubic Function Fit to Monthly Ratios Comparing Fort Campbell and China Lake

Table B-19 shows all of the monthly comparison ratios that were calculated using the equation shown on Figure B-17. Those ratios shown in the shaded blocks of this table pertain to the months where ozone measurements were not made at Hopkinsville, KY. The values shown for the shaded blocks were multiplied against the corresponding monthly averages for Trona,

California in order to provide estimates of the averages for the missing months. Figure B-18 displays the complete monthly average proxy for Fort Campbell in comparison to the monthly

284 proxy averages for China Lake. Please note that these monthly data points were only calculated and plotted to compare the Fort Campbell monthly averages with those of China Lake. Hourly estimates were needed for the model.

Table B-19 Comparison Ratios Calculated Using Cubic Equation from Figure B-17 Month Cumulative Comparison Ratio Number of Months September 1 0.9555

October 2 1.0168

November 3 1.0477

December 4 1.0536

January 5 1.0399

February 6 1.012

March 7 0.9753

April 8 0.9352

May 9 0.8971

June 10 0.8664

July 11 0.8485

August 12 0.8488

September 13 0.8727

The comparison ratios shown in Table B-19 were used to estimate the missing hourly ozone measurements by multiplying each hour of the China Lake proxy corresponding to the missing hours at Hopkinsville by the appropriate ratio shown in the shaded blocks on the table (e.g., all hourly Trona measurements for the month of November were multiplied by 1.0477 in order to provide estimates for the missing data for Hopkinsville/Fort Campbell). While certainly not

285 perfect by any stretch of the imagination, this process does provide estimates for the missing data that are based upon both diurnal cycling and the seasonal magnitudes as calculated by ratios between the two sites.

0.06

0.05

0.04

0.03

0.02

0.01 Average Ozone Level (ppm) 0 0 2 4 6 8 10 12 Month of Year

Trona Hopkinsville Estimates

Figure B-18 Complete Monthly Average Proxy Ozone Levels for Fort Campbell and China Lake

B.2.1.2 Estimates for Missing Ozone Readings at Dobbins Air Reserve Base (ARB), Georgia

0.06

0.05

0.04

0.03

0.02

0.01

Average Ozone Level (ppm) Level Ozone Average 0 0 2 4 6 8 10 12 Month of Year

Trona Confederate Ave.

Figure B-19 Monthly Average Proxy Ozone Levels for Dobbins and China Lake

286

Figure B-19 compares the monthly average ozone levels for the site on Confederate Avenue in

Atlanta, Georgia (proxy for Dobbins Air Reserve Base, Georgia) in comparison to levels measured at Trona, California (proxy for China Lake, California). Table B-20 displays the average monthly ozone levels for both locations as well as the comparison ratios used during the estimation process. This case, like the above case of Fort Campbell, employed adjustments to make the data behave more in accordance with a Gaussian distribution.

Table B-20 Average Monthly Proxy Ozone Levels/Comparison Ratios (Dobbins vs. China Lake) Month of Year Average Ozone Levels

Trona Confederate Avenue Adjusted Confederate Avenue Data

January 0.02398167

February 0.02730506

March 0.03739785 0.029215

April 0.04055972 0.031936 0.032

May 0.04961559 0.028918 0.036

June 0.05565278 0.0263 0.04

July 0.05253091 0.032716 0.036

August 0.05195833 0.032668 0.032

September 0.040625 0.029276

October 0.03163844 0.020468

November 0.02598194

December 0.01987903 Comparison Ratios

March June October 0.7811961 0.7187422 0.646926

287

Figure B-20 illustrates the original and adjusted data for Dobbins in comparison to the monthly averages that were calculated for China Lake.

0.06

0.05

0.04

0.03

0.02

0.01

Average Ozone Level (ppm) Level Ozone Average 0 0 2 4 6 8 10 12 Month of Year

Trona Confederate Ave. Adjusted

Figure B-20 Monthly Average Ozone Levels for Dobbins and China Lake (includes adjustments)

The next step was to take the comparison ratios shown in Table B-20 and apply them to a numerical listing of sequential months as shown in Table B-21. When constructing this new table, it was assumed that the ratios from one year to the next were exactly the same. While such an assumption is unrealistic from a physical standpoint, it was necessary to create the symmetric point, which was then used to create a function for the period of time in question.

Once the comparison ratios had been determined as shown in Table B-21, they were plotted along with the associated numeric value for the month, which is also shown on the table. After this data was plotted, as seen in Figure B-21, a cubic polynomial was fit to the four points under consideration. The equation shown on the figure was then used to calculate comparison ratios

288 for the months requiring estimated data. The independent variable “x” seen on this equation corresponds to the numerical value of the month (e.g., January has a value of 1.0).

Table B-21 Periodic Relationship of Comparison Ratios (Dobbins vs. China Lake Proxies) Month Cumulative Number Comparison Ratio of Months

October 1 0.646926

November 2

December 3

January 4

February 5

March 6 0.7811961

April 7

May 8

June 9 0.718742

July 10

August 11

September 12

October 13 0.646926

November 14

December 15

January 16

February 17

March 18

April 19

289

0.9 0.8 0.7 0.6 y = 0.0005x3 - 0.0144x2 + 0.1052x + 0.5557 0.5 0.4 0.3

Ratios (nondimensional) 0.2 0.1 0 0 2 4 6 8 10 12 14 16 Months

Figure B-21 Cubic Function Fit to Monthly Ratios Comparing Dobbins and China Lake Proxies

Table B-22 shows all of the monthly comparison ratios that were calculated using the equation shown on Figure B-21. Those ratios shown in the shaded blocks of this table pertain to the months where ozone measurements were not made at Confederate Avenue.

The values shown in the shaded blocks in Table B-22 were multiplied against the corresponding monthly averages for Trona, California in order to provide estimates of the needed averages for the missing months. Figure B-22 displays the complete monthly average proxy for Dobbins in comparison to the monthly proxy averages for China Lake.

290

Table B-22 Ratios Calculated Using Cubic Equation from Figure B-21 Month Cumulative Number Ratio of Months October 1 0.647

November 2 0.7125

December 3 0.7552

January 4 0.7781

February 5 0.7842

March 6 0.7765

April 7 0.758

May 8 0.7317

June 9 0.7006

July 10 0.6677

August 11 0.636

September 12 0.6085

October 13 0.5882

November 14 0.5781

December 15 0.5812

As before, the comparison ratios shown in Table B-22 were also used to estimate the missing hourly ozone measurements by multiplying each hour of the China Lake proxy corresponding to the missing hours of the Dobbins proxy by the ratios for the appropriate months, as shown in the shaded blocks.

291

0.06

0.05

0.04

0.03

0.02

0.01

Average Ozone Level (ppm) Level Ozone Average 0 0 2 4 6 8 10 12 Month of Year

Trona Confederate Ave.

Figure B-22 Complete Monthly Average Proxy Ozone Levels for Dobbins and China Lake

B.2.1.3 Estimates for Missing Ozone Readings at Point Judith, Rhode Island

0.06

0.05

0.04

0.03

0.02

0.01

Average Ozone Level (ppm) Level Ozone Average 0 0 2 4 6 8 10 12 Month of Year

Trona Narragansett

Figure B-23 Monthly Average Proxy Ozone Levels for Point Judith and China Lake

Figure B-23 compares the monthly average ozone levels for the site at Narragansett, Rhode

Island (proxy for Point Judith Rhode Island) in comparison to levels measured at Trona,

292

California (proxy for China Lake, California). Table B-23 displays the average monthly ozone levels for both locations as well as the comparison ratios used during the estimation process.

Table B-23 Average Monthly Ozone Levels and Comparison Ratios (Point Judith vs. China Lake) Month of Year Average Ozone Levels

Trona Narragansett

January 0.02398167

February 0.02730506

March 0.03739785 0.04116

April 0.04055972 0.042625

May 0.04961559 0.039289

June 0.05565278 0.039231

July 0.05253091 0.037155

August 0.05195833 0.03454

September 0.040625 0.026119

October 0.03163844

November 0.02598194

December 0.01987903

Comparison Ratios

March June September 1.1005966 0.7049164 0.64294

293

Table B-24 Periodic Relationship of Comparison Ratios (Point Judith vs. China Lake Proxies) Month Cumulative Number of Comparison Ratio Months

September 1 0.64294

October 2

November 3

December 4

January 5

February 6

March 7 1.1005966

April 8

May 9

June 10 0.7049164

July 11

August 12

September 13 0.64294

October 14

November 15

December 16

January 17

February 18

March 19 1.1005966

The comparison ratios shown in Table B-23 were then applied to a numerical listing of sequential months as shown in Table B-24. When constructing this new table, it was assumed

294 that the ratios from one year to the next were exactly the same. While such an assumption is unrealistic from a physical standpoint, it was necessary to create the symmetric point, which was then be used to create a function for the period of time in question.

The next step was to plot the ratios shown on Table B-24 in comparison to the associated numeric value for the month, which was also shown on the table. After these were plotted, as seen in Figure B-24, a cubic polynomial was fit to the four points under consideration. The equation shown on the figure was then used to calculate comparison ratios for the months requiring estimated data. The independent variable “x” seen on this equation corresponds to the numerical value of the month (e.g., January has a value of 1.0).

1.2

1

0.8

0.6

0.4 y = 0.0014x3 - 0.0415x2 + 0.3171x + 0.3723

Ratio (nondimensional) Ratio 0.2

0 0 5 10 15 20 Months

Figure B-24 Cubic Function Fit to Monthly Ratios Comparing Point Judith and China Lake

Table B-25 shows all of the monthly comparison ratios that were calculated using the equation shown on Figure B-24. Those ratios shown in the shaded blocks of this table pertain to the months where ozone measurements were not made at the Narragansett site.

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Table B-25 Comparison Ratios Calculated Using Cubic Equation from Figure B-24 Month Cumulative Comparison Ratios Number of Months September 1 0.6493

October 2 0.8517

November 3 0.9879

December 4 1.0663

January 5 1.0953

February 6 1.0833

March 7 1.0387

April 8 0.9699

May 9 0.8853

June 10 0.7933

July 11 0.7023

August 12 0.6207

September 13 0.5569

The values shown in the shaded blocks in Table B-25 were multiplied against the corresponding monthly averages for Trona, California in order to provide estimates of the needed averages for the missing months. Figure B-25 displays the complete monthly average proxy for Point Judith in comparison to the monthly proxy averages for China Lake.

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0.06

0.05

0.04

0.03

0.02

0.01

Average Ozone Level (ppm) Level Ozone Average 0 0 2 4 6 8 10 12 Month of Year

Trona Narragansett

Figure B-25 Complete Monthly Average Proxy Ozone Levels for Point Judith and China Lake

The comparison ratios in the shaded blocks shown in Table B-25 were also used to estimate the missing Point Judith proxy hourly ozone measurements by appropriately multiplying them against the requisite hours of the China Lake proxy (corresponding to the missing hours at Point

Judith proxy).

B.2.1.4 Estimates for Missing Ozone Readings at Fort Rucker, Alabama 0.06

0.05

0.04

0.03

0.02

0.01

Average Ozone Level (ppm) Level Ozone Average 0 0 2 4 6 8 10 12 Month of Year

Trona Dothan

Figure B-26 Monthly Average Proxy Ozone Levels Fort Rucker and China Lake

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Table B-26 Average Monthly Ozone Levels and Comparison Ratios (Ft Rucker vs. China Lake) Month of Year Average Ozone Levels

Trona Dothan, AL Adjusted Dothan Data

January 0.02398167

February 0.02730506

March 0.03739785

April 0.04055972 0.036236

May 0.04961559 0.03621

June 0.05565278 0.026689 0.04

July 0.05253091 0.020914 0.038

August 0.05195833 0.022978 0.034

September 0.040625 0.026599

October 0.03163844 0.02305

November 0.02598194

December 0.01987903

Comparison Ratios

April June October 0.8934014 0.7187422 0.728536

Figure B-26 compares the monthly average ozone levels for the monitor site at Dothan, Alabama

(proxy for Fort Rucker, AL) in comparison to levels measured at Trona, California (proxy for

China Lake, California). Table B-26 displays the average monthly ozone levels for both locations as well as the comparison ratios used during the estimation process. This case, like some previous cases, employed adjustments to make the data behave more in accordance with a

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Gaussian distribution. Figure B-27 illustrates the original and adjusted data for Fort Rucker in comparison to the monthly averages that were calculated for China Lake.

0.06

0.05

0.04

0.03

0.02

0.01

Average Ozone Level (ppm) Level Ozone Average 0 0 2 4 6 8 10 12 Month of Year

Trona Dothan Adjusted

Figure B-27 Monthly Ozone Levels for Fort Rucker and China Lake (includes adjustments)

The comparison ratios shown in Table B-26 were then applied to a numerical listing of sequential months as shown in Table B-27. When constructing this new table, it was assumed that the ratios from one year to the next were exactly the same. While such an assumption is unrealistic from a physical standpoint, it was necessary to create the symmetric point, which was then used to create a function for the period of time in question. The next step was to plot the ratios shown on Table B-27 in comparison to the associated numeric value for the month, which is also shown on the table. After these were plotted, as seen in Figure B-28, a cubic polynomial was fit to the four points under consideration. The equation shown on the figure was then used to calculate ratios for the months requiring estimated data. The independent variable “x” seen on this equation corresponds to the numerical value of the month (e.g.,

January has a value of 1.0).

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Table B-27 Periodic Relationship of Comparison Ratios (Fort Rucker vs. China Lake Proxies) Month Cumulative Number of Comparison Ratio Months

September 1

October 2 0.728536

November 3

December 4

January 5

February 6

March 7

April 8 0.8934014

May 9

June 10 0.7187422

July 11

August 12

September 13

October 14 0.728536

November 15

December 16

January 17

February 18

March 19

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3 y = 0.0024x3 - 0.0632x2 + 0.4544x + 0.0531 2.5

2

1.5

1

Ratio (nondimensional) Ratio 0.5

0 0 5 10 15 20 Months

Figure B-28 Cubic Function Fit to Monthly Ratios Comparing Fort Rucker and China Lake

Table B-28 shows all of the monthly comparison ratios that were calculated using the equation shown on Figure B-28. Those ratios shown in the shaded blocks of this table pertain to the months where ozone measurements were not made at the Dothan site.

The values shown in the shaded blocks in Table B-28 were multiplied against the corresponding monthly averages for Trona, California in order to provide estimates of the needed averages for the missing months. Figure B-29 displays the complete monthly average proxy for Fort Rucker in comparison to the monthly proxy averages for China Lake.

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Table B-28 Comparison Ratios Calculated Using Cubic Equation from Figure B-28 Month Cumulative Comparison Ratio Number of Months September 1 0.4467

October 2 0.7283

November 3 0.9123

December 4 1.0131

January 5 1.0451

February 6 1.0227

March 7 0.9603

April 8 0.8723

May 9 0.7731

June 10 0.6771

July 11 0.5987

August 12 0.5523

September 13 0.5523

The comparison ratios shown in Table B-28 were also used to estimate the missing hourly ozone measurements by appropriately multiplying the ratios in the shaded blocks against the hours of the China Lake proxy that correspond to the missing data in the Fort Rucker proxy.

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0.06

0.05

0.04

0.03

0.02

0.01

Average Ozone Level (ppm) Level Ozone Average 0 0 2 4 6 8 10 12 14 Month of Year

Trona Dothan

Figure B-29 Complete Monthly Average Proxy Ozone Levels for Fort Rucker and China Lake

B.2.1.5 Estimates for Missing Ozone Readings at West Jefferson, Ohio Figure B-30 compares the monthly average ozone levels for the site at London, Ohio (proxy for

West Jefferson, OH) in comparison to levels measured at Trona, California (proxy for China Lake,

California). Table B-29 displays the average monthly ozone levels for both locations as well as the comparison ratios used during the estimation process.

0.06

0.05

0.04

0.03

0.02

0.01

Average Ozone Level (ppm) Level Ozone Average 0 0 2 4 6 8 10 12 Month of Year

Trona London

Figure B-30 Monthly Average Proxy Ozone Levels for West Jefferson and China Lake

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Table B-29 Average Monthly Proxy Ozone Levels and Comparison Ratios (West Jefferson vs. China Lake) Month of Year Average Ozone Levels

Trona London, OH

January 0.02398167

February 0.02730506

March 0.03739785

April 0.04055972 0.038556

May 0.04961559 0.038884

June 0.05565278 0.042015

July 0.05253091 0.040441

August 0.05195833 0.036027

September 0.040625 0.036361

October 0.03163844 0.022273

November 0.02598194

December 0.01987903

Comparison Ratios

April June October 0.9505873 0.75495383 0.703981

The comparison ratios shown in Table B-29 were then applied to a numerical listing of sequential months as shown in Table B-30. When constructing this new table, it was assumed that the ratios from one year to the next were exactly the same. While such an assumption is unrealistic from a physical standpoint, it was necessary to create the symmetric point, which was then used to create a function for the period of time in question.

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Table B-30 Periodic Relationship of Comparison Ratios (West Jefferson vs. China Lake Proxies) Month Cumulative Number of Comparison Ratio Months

September 1

October 2 0.703981

November 3

December 4

January 5

February 6

March 7

April 8 0.9505873

May 9

June 10 0.75495383

July 11

August 12

September 13

October 14 0.703981

November 15

December 16

January 17

February 18

March 19

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The next step was to plot the ratios shown on Table B-30 in comparison to the associated numeric value for the month, which is also shown on the table. After these were plotted, as seen in Figure B-31, a cubic polynomial was fit to the four points under consideration. The equation shown on the figure was then used to calculate ratios for the months requiring estimated data. The independent variable “x” seen on this equation corresponds to the numerical value of the month (e.g., January has a value of 1.0).

3 y = 0.0026x3 - 0.0699x2 + 0.5197x - 0.0766 2.5

2

1.5

1

Ratios (nondimensional) Ratios 0.5

0 0 5 10 15 20 Months

Figure B-31 Cubic Function Fit to Monthly Ratios Comparing West Jefferson and China Lake

Table B-31 shows all of the monthly comparison ratios that were calculated using the equation shown on Figure B-31. Those ratios shown in the shaded blocks of this table pertain to the months where ozone measurements were not made at the London site.

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Table B-31 Comparison Ratios Calculated Using Cubic Equation from Figure B-31 Month Cumulative Number Comparison Ratio of Months September 1 0.3758

October 2 0.704

November 3 0.9236

December 4 1.0502

January 5 1.0994

February 6 1.0868

March 7 1.028

April 8 0.9386

May 9 0.8342

June 10 0.7304

July 11 0.6428

August 12 0.587

September 13 0.5786

0.06

0.05

0.04

0.03

0.02

0.01

Average Ozone Level (ppm) Level Ozone Average 0 0 2 4 6 8 10 12 Month of Year

Trona London

Figure B-32 Complete Monthly Average Proxy Ozone Levels for West Jefferson and China Lake

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The values shown in the shaded blocks in Table B-31 were multiplied against the corresponding monthly averages for Trona, California in order to provide estimates of the needed averages for the missing months. Figure B-32 displays the complete monthly average proxy for West

Jefferson in comparison to the monthly proxy averages for China Lake. The comparison ratios shown in Table B-31 were also used to estimate the missing hourly ozone measurements by appropriately multiplying them against each hour of the China Lake proxy that corresponded to the missing hours at London.

B.2.1.6 Estimates for Missing Ozone Readings at Wright Patterson AFB, Ohio

0.06

0.05

0.04

0.03

0.02

0.01

Average Ozone Level (ppm) Level Ozone Average 0 0 2 4 6 8 10 12 Month of Year

Trona Fairborn

Figure B-33 Monthly Average Proxy Ozone Levels for Wright Patterson AFB and China Lake

Figure B-33 compares the monthly average ozone levels for the site at Fairborn, Ohio (proxy for

Wright Patterson AFB, OH) in comparison to levels measured at Trona, California (proxy for

China Lake, California). Table B-32 displays the average monthly ozone levels for both locations as well as the comparison ratios used during the estimation process.

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Table B-32 Average Monthly Proxy Ozone Levels and Comparison Ratios (Wright Patterson vs. China Lake) Month of Year Average Ozone Levels

Trona Fairborn

January 0.02398167

February 0.02730506

March 0.03739785

April 0.04055972 0.032422

May 0.04961559 0.030776

June 0.05565278 0.030947

July 0.05253091 0.028847

August 0.05195833 0.027281

September 0.040625 0.0269

October 0.03163844 0.022336

November 0.02598194

December 0.01987903

Comparison Ratios

April June October 0.7993699 0.55607687 0.705977

The next step was to take the comparison ratios shown in Table B-32 and apply them to a numerical listing of sequential months as shown in Table B-33. When constructing this new table, it was assumed that the ratios from one year to the next were exactly the same. While such an assumption is unrealistic from a physical standpoint, it was necessary to create the symmetric point, which will then be used to create a function for the period of time in question.

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Table B-33 Periodic Relationship of Comparison Ratios (Wright Patterson vs. China Lake) Month Cumulative Number of Comparison Ratio Months

September 1

October 2 0.705977

November 3

December 4

January 5

February 6

March 7

April 8 0.7993699

May 9

June 10 0.55607687

July 11

August 12

September 13

October 14 0.705977

November 15

December 16

January 17

February 18

March 19

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The next step was to plot the ratios shown on Table B-33 in comparison to the associated numeric value for the month, which is also shown on the table. After these were plotted, as seen in Figure B-34, a cubic polynomial was fit to the four points under consideration. The equation shown on the figure was then used to calculate ratios for the months requiring estimated data. The independent variable “x” seen on this equation corresponds to the numerical value of the month (e.g., January has a value of 1.0).

4.5 4 y = 0.0036x3 - 0.0899x2 + 0.6092x - 0.1819 3.5 3 2.5 2 1.5 1

Ratios (nondimensional) Ratios 0.5 0 0 5 10 15 20 Months

Figure B-34 Cubic Function Fit to Monthly Ratios that Compare Wright Patterson and China Lake Proxies

Table B-34 shows all of the monthly comparison ratios that were calculated using the equation shown on Figure B-34. Those ratios shown in the shaded blocks of this table pertain to the months where ozone measurements were not made at the Fairborn site.

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Table B-34 Comparison Ratios Calculated Using Cubic Equation from Figure B-34 Month Cumulative Number Comparison Ratio of Months September 1 0.341

October 2 0.7057

November 3 0.9338

December 4 1.0469

January 5 1.0666

February 6 1.0145

March 7 0.9122

April 8 0.7813

May 9 0.6434

June 10 0.5201

July 11 0.433

August 12 0.4037

September 13 0.4538

The values shown in the shaded blocks in Table B-34 were multiplied against the corresponding monthly averages for Trona, California in order to provide estimates of the needed averages for the missing months. Figure B-35 displays the complete monthly average proxy for Wright

Patterson AFB in comparison to the monthly proxy averages for China Lake. The comparison ratios shown in Table B-34 were also used to estimate the missing hourly ozone measurements by appropriately multiplying them against each hour of the China Lake proxy that corresponds to missing hours of the Fairborn proxy.

312

0.06

0.05

0.04

0.03

0.02

0.01

Average Ozone Level (ppm) Level Ozone Average 0 0 2 4 6 8 10 12 Month of Year

Trona Fairborn

Figure B-35 Complete Average Proxy Ozone Levels for Wright Patterson AFB and China Lake

313

APPENDIX C

PROCESSES USED TO DEVELOP THE CUMULATIVE DAMAGE MODEL

An Eyring equation-based corrosion model enables predictions that consider kinetics of the electrochemical reactions responsible for oxidation of metal surfaces. As shown earlier, the model developed here to predict atmospheric corrosion rates is quite complex due to the large number of modeling coefficients used to consider the effects of temperature, relative humidity, and three different atmospheric contaminants. Since the new model employs assumed relationships to define interactions between acceleration factors, there is no physical meaning to any of these coefficients. Thus, their values cannot be determined through testing or other direct means.

As described below, suitable coefficient values were determined using an inverse modeling approach based upon simulations that implement the Monte Carlo method. A partially automated process was used to reduce the amount of manual intervention needed when conducting the vast numbers of simulations used to identify the sets of coefficient values that provided the most accurate model.

C.1 Inverse Modeling Using Monte Carlo Simulations

Soetaert states that inverse predictive models are often developed by using simulations that consider environmental or experimental data. During such simulations, coefficients are fitted to

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the data so that the model obtains the ability to make predictions [156]. Mosegaard and

Sambridge identify the Monte Carlo method as an important tool to help develop nonlinear inverse models “where no analytical expression between data and model parameters is available, and where linearization is unsuccessful.” [157] This definition succinctly describes the problem of atmospheric corrosion, which is a complex phenomenon resulting from unmeasureable interactions between competing acceleration factors. The Monte Carlo method was originally developed to predict neutron diffusion rates during nuclear reactions [158].

Example: Evaluating the Effect of Input Uncertainty Monte Carlo simulations can be used to determine the variability of the model output in response to input uncertainty Known Simulations are conducted by repetitively applying random y=af(x) numbers from a distribution as the input to a model in order to develop a distribution that represents the variability of model outputs.

In such cases as shown in this example, the model is already known and calibrated

Figure C-1 Simplistic Illustration of Monte Carlo Simulations as Applied to a Forward Problem

Monte Carlo simulations can be used in both forward and inverse modeling approaches. When used to support a forward problem, values obtained from a probability distribution representing

315 input parameters are applied to a known model and the results are calculated and recorded. By repetitively accomplishing this task, a distribution of model responses based upon the probabilistic inputs can be determined. This type of modeling is illustrated in Figure C-1.

Example: Fitting the Model to Data Monte Carlo simulation can be used to calibrate a model when the input data (e.g., f(x) and Unknown Unknown g(x)) and the output Coefficient Coefficient value (target) is Value Value y=a1f(x) + a2g(x) known Simulations are conducted by Target (e.g., test repetitively applying random measurement) numbers from distributions until a combination of coefficients are found that allow the model to predict the test measurement (target) with sufficient accuracy. EXAMPLE: • f(x) and g(x) are variables representing data that has been measured • The “target” is explicitly known • Coefficients a1 and a2 are randomly varied until model output approximates the target

Figure C-2 Simplistic Illustration of Monte Carlo Simulations as Applied to an Inverse Problem

When employed for an inverse modeling effort, the Monte Carlo method is used to fit the coefficients of an assumed model to data and thus obtain a “near-optimal solution (measured in terms of data fit and adherence to given constraints)” [157]. In this approach, there is a clearly defined result (e.g., a test measurement) that is known, which represents the target for the desired model output. During the simulation process, such models are iteratively developed in order to obtain the optimum coefficients so that the prediction approaches the actual target.

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Figure C-2 illustrates the inverse modeling approach. It should be noted that this figure is a gross simplification for illustration purposes. In reality, the coefficients a1 and a2 on the figure would be assumed functions that include coefficients while f(x) and g(x) would be actual data.

The values for the coefficients would be determined using an iterative process employing Monte

Carlo simulations. Numerous iterations of the model would be constructed and statistically tested until an acceptable model is found.

Different types of statistical distributions can be used when conducting Monte Carlo simulations

[159]. Oftentimes the normal distribution, such as illustrated in Figures C-1 and C-2, are used.

However, if there is little or no information available concerning the distribution parameters and its correlation to the problem, then a uniform distribution is used [159]. In such cases, the range of the distribution, which is defined by its boundaries, must be broad enough so that the simulations can delineate between different potential solutions. However, such ranges cannot be too broad or sparse sampling (defined later) will prohibit the ability to converge on the unknown value. Uniform distributions were employed under this research effort.

C.2 Computational Approaches to Running Simulations Conducting simulations using random numbers selected from unknown distributions required an approach to first assume a distribution, conduct simulations, quantify the accuracy of model predictions by statistically comparing them to test measurements, and then adjusting and refining the distribution parameters for use in subsequent simulations. These processes were repeated billions of times in order to converge upon acceptable values for the unknown coefficients needed to calibrate the candidate models.

C.2.1 Limitations on Random Numbers Selected from Distributions Numerous candidate models were constructed and calibrated in order to find one that provided the best results. Each of these different models was calibrated using uniform distributions from

317 which candidate values for each unknown coefficient were randomly selected. The number of distributions used during each simulation ranged from twelve to fourteen depending upon the complexity of the model. This resulted in twelve or more random numbers, each from a different probability distribution, which were simultaneously input into the model and statistically tested during each simulation in an attempt to identify the complete set of coefficients that provided suitable accuracy.

A critical factor that dictates the amount of simulations needed to converge upon a solution concerns the number of possible values that exist within a particular distribution. There are many different approaches that can be used to generate (pseudo) random numbers when conducting Monte Carlo simulations. If only one distribution is used, then an algorithm that picks any possible value within the distribution can be employed. However, this approach becomes problematic when numerous distributions are employed simultaneously because of the massive number of combinations of coefficients that can result. In other words, if large quantities of random numbers are selected from each distribution, then the possible combinations of candidate coefficients becomes an extremely large number. From a computational efficiency standpoint, such large numbers of combinations becomes problematic because it leads to very large, if not impossible, numbers of simulations that need to be conducted in order to establish convergence. Limiting the number of possible coefficient values within each distribution enables the use of reasonable numbers of simulations during the Monte

Carlo process.

For this research, a random number generator function was selected from the Python programming language library. This function allows the user to specify the number of equally spaced numbers that exist within each of the specified uniform distributions. Simulations were

318 conducted to examine a variety of different numbers of coefficient values within each range in order to find an acceptable number that provided good results within a reasonable number of simulations. Based upon this analysis, it was decided that limiting the number to fifty uniformly spaced values within each distribution would be sufficient. Despite this limitation, a massive number of combinations of coefficients still results. For example, when fifty possible values are applied to twelve different distributions, the number equals 5012 or 2.44 x 1020 combinations.

Even with the power of the massively parallel processing workstation used for this research, it was not possible to consider every combination of coefficients. As with all inverse modeling efforts, the intent of this current effort was to find a set of coefficients that provided acceptable prediction accuracy within a reasonable amount of processing time. Fewer quantities of random numbers from within each range could have been employed, which certainly would have reduced the possible combinations. However, it was decided to use more so the difference between each number within the range was smaller, thus offering better resolution

(more significant figures) to define the magnitude of each potential coefficient.

C.2.1.1 Iterative Model Refinement

A factor that determines how quickly the simulation process converges upon a set of coefficient values concerns the actual range of values within each distribution. If the range (numerical difference between endpoints) is too broad, then it makes it very difficult, if not impossible, to converge upon solutions since an optimum coefficient value for one unknown will more often than not be paired with suboptimum values for the others, thus resulting in a high error calculation. This is the issue related to the expression “sparse sampling” as described by

Wagener [159]. During this research, it was determined that using small distribution ranges from which coefficients were randomly selected and then systematically shifting the midpoint of the range when conducting simulations resulted in better convergence. When combined with a

319 process that systematically narrowed the ranges (i.e., reduced the difference between the minimum and maximum values) as the simulation process continued, the result was an approach that effectively allowed convergence on the multiple unknown coefficients.

C.2.1.2 Distribution Adjustment When conducting simulation runs to calibrate the initial cumulative corrosion damage model, there was no advance knowledge concerning the expected values of the individual modeling coefficients. As a result, the specific lower and upper endpoints (the limits) for each distribution were unknown. To address this issue, the initial lower endpoints for the distributions associated with the coefficients used to describe the six shape functions used in the model were assumed to be the same as the boundary conditions. Lower boundary values for the model’s “alpha” coefficients were determined by trial and error while the scaling factors (the three “A” coefficients) were assigned an initial lower value of zero because these coefficients must always have a positive value so the predicted corrosion rates are always positive. An algorithm used when conducting simulations was developed to incrementally adjust the endpoints while retaining the overall width of the distribution.

During the model development process, distributions for each new run of simulations (e.g.,

1.5M individual simulations per run) were established by adjusting the distributions used in the previous run. This adjustment process was based upon consideration of the error for each simulation, which was calculated by conducting a statistical comparison (using the residual sum of squares method) between quarterly (3 month) predictions and the associated test measurements made at each of the calibration sites. The coefficients identified for the specific simulation within the run that had the lowest error were then used as the basis for the distributions used during the next run of simulations.

320

-4 -3 -2 -1 0

Value for the unknown coefficient

Best Fit 1st Run

Endpoint of one 2nd Run distribution rd becomes the 3 Run midpoint for the 4th Run next 5th Run

Figure C-3 Shifting of Uniform Distributions until an Acceptable Coefficient Value is Found

Figure C-3 simplistically illustrates the process used to iteratively readjust a distribution until the optimum coefficient value is within a range and thus can be identified. In this example, the value of the coefficient associated with the lowest error hits the right hand limit in each of the first four simulation runs (the first run is on the upper left). This implies that the optimum coefficient value is larger than the largest value of the distribution. In each case, the best fit coefficient for one particular run (i.e., the maximum value of the distribution) is used to calculate the midpoint of the range of the following distribution. This process continues until the value of the near-optimal coefficient is found to be within a distribution, as seen for the fifth simulation run indicated in the example. In practice, the shifting process was far more difficult because there were twelve to fourteen distributions (depending on the complexity of the model) that were simultaneously adjusted. Thus, the lowest calculated error for a particular simulation run might occur when a single distribution limit is hit while the others are not. Each

321 distribution was adjusted based upon the best results from the previous simulation and the process continued until the entire set of coefficients were somewhere within the bounds of their individual distributions. Extremely large numbers of simulations were employed during this process.

C.2.1.3 Distribution Refinement

The final process used to facilitate convergence towards optimum coefficient values was to incrementally reduce the ranges of each uniform distribution while keeping the number of possible values within them constant. This process was only employed after there was some evidence that the unknown coefficients were actually within their associated distributions. The specific process is described below. When combined with the adjustment process described earlier, the result was an effective mechanism to hunt for and converge upon the unknown coefficients.

The refinement (narrowing) of each distribution was only done after five successive runs (7.5M total simulations) were conducted and none of the distribution limits were hit. When this condition was met, the range of each new distribution was established by multiplying the previous ranges by a factor of 0.9. The aforementioned shifting process continued in the succeeding simulations until five more runs were conducted where none of the limits were hit, at which time the range was again narrowed by multiplying it by a factor of 0.9. During the initial simulations to screen possible models, this process continued until at least eight distribution ranges with increasing smaller sizes were examined. There was no automation to stop the simulation upon finding the best result for the eighth range, hence many simulations went further.

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For screening of initial coefficient values, the process started with distribution ranges of 0.005 for each unknown coefficient. These initial simulation runs were conducted to determine the candidate models that showed the most promise. Some of the more promising candidate models with the lowest errors determined during the screening process were further refined by using the final coefficients as a starting point for new simulations. In these cases, the new beginning ranges were set at 0.0025 for each distribution, and the distribution adjustment/refinement processes were employed until a minimum of fifteen distributions of increasingly smaller sizes was used. The target final range was 0.000572, which is 11.4% of the beginning range (0.005) used for the screening simulations. Like before, the lack of automated control resulted in some final ranges being reduced even further. While the selection of the number of simulations and range reductions used when conducting these processes may appear arbitrary, the results obtained as a result of employing these processes appear to validate their usage. That being said, more simulations using increasingly smaller distribution ranges could possibly provide slightly better results.

The distribution adjustment and refinement processes described earlier are illustrated by Figure

C-4. As shown by the figure, the initial lower endpoint of the distributions for each of the six initial shape functions employed in the model was set at -4.0, which is the same value as the boundary conditions used when constructing the functions (these were changed to a value of zero for the shape functions used for the final proof-of-concept model). The lower boundary for the remaining coefficients (the “A” and “” coefficients in Equation 2.14), which were discussed previously, were subjected to the same adjustment and refinement processes in order to converge on near-optimal solutions.

As previously discussed and also shown on Figure C-4, 1.5M simulations were conducted during each run. The results were then automatically sorted to identify the set of coefficients that

323 when input into the model resulted in the lowest error as calculated by the residual sum of squares method. These coefficients were then used as the basis for the next run of simulations.

While 1.5M simulations per run may seem like a large number, it is very small in comparison to the number of combinations of coefficient values that are possible (as discussed earlier).

However, this number of simulations did appear to be sufficient for converging on a set of suitable coefficients for most models, especially when the distribution ranges were reduced to small sizes. It should be noted that some models required far more simulations (e.g., 10M) to achieve convergence.

Adjustment of Ranges used for Shape Functions

-3.995

-4

Cycle 1 2 1 2 3 4 5 1 1 2 3 4 5 1 Total Simulation Runs 1 2 43 5 6 7 8 9 10 11 12 13 14 “Optimal” Solution

Distribution Ranges 1.5M simulations per run

Best fit coefficient value for each simulation Starting Range =0.005 2nd Range =0.9 x Starting Range 3rd Range =0.9 x 2nd Range

Figure C-4 Illustration of the Distribution Adjustment and Refinement Process

C.2.2 Advanced Computing An entire year’s worth of hourly environmental data for multiple sites was needed to calibrate the various candidate models developed and evaluated under this research program. This data was combined into a large file that was used to support the simulation processes. A large number of simulations (e.g., 1.5M) were made during each run, which required the use of a

324 computer capable of completing all calculations in a reasonable amount of time. A special purpose workstation designed specifically for number crunching was constructed for this effort.

This computer, which was based on server architecture, contained dual CPUs (eight cores each), solid-state hard drives, and a state-of-the-art Graphics Processing Unit (GPU). The CPUs were used to conduct pre and post-processing of data and the solid state drives provided extremely fast read/write speed. As will be discussed later, the GPU, which is a card most commonly associated with computer graphics, was used for its capability in performing parallel calculations.

In the early days of personal computers, CPUs were used primarily for control and logic purposes while special purpose chips called “math coprocessors” were used to efficiently perform floating point calculations. As integrated circuit manufacturing processes advanced and allowed for more complex CPU designs, the functionality provided by math coprocessors migrated to the CPU, thus negating the need for a separate processor. For normal mathematical operations, the CPUs used today have more than enough power to handle most tasks.

However, computationally intensive scientific computing problems often require more power than is provided by even the fastest CPUs. For certain classes of problems that can be solved using parallel processing, GPUs can provide an enormous increase in computational speed and capability.

GPUs were originally developed to enable the rapid construction of complex graphical images for video games and other video intensive applications. Such applications require hardware that can make rapid mathematical calculations related to the vectors and shapes that are the foundational elements of graphical images. GPUs employ large numbers of relatively simple (in comparison to CPUs) “stream processors” that rapidly make the calculations needed to

325 construct graphical elements. They are called stream processors because the data they use to make calculations comes in a continuous stream, all of which require similar computations

[160]. So in a way, GPUs are the modern day equivalent to the math coprocessors of the past in that they can be used to make floating point calculations on a device separate from the CPU.

The large numbers of stream processors found in high performance GPUs can be exploited to solve complex mathematical problems of a parallel nature, whereby a large problem can be broken into small pieces, each of which is solved independently in a continuous process involving streams of data. Afterwards the solutions to the individual pieces are properly combined to give the final solution to the large problem. It should be noted that not every complex problem can be solved using parallel processing. Because independent hourly predictions were made during the simulation process, the situation was ideally suited for parallel processing.

There are a wide variety of GPUs on the market to support the spectrum of graphical needs ranging from office computing to complex video gaming. Higher-end devices have greater amounts of stream processors and even on-board memory. The GPU employed under this effort was an AMD Radeon HD 6990, which for well over a year was the most powerful card available on the market. This card employs 3072 stream processors, which enables a massively parallel approach to scientific computing on a computer workstation.

The downside of using GPUs is that programming requirements are far more complex in comparison to programs written specifically to run on a single CPU. In GPU programming, the

CPU and GPU perform separate functions that are dependent upon on another. Because of the differences in how they operate, CPUs and GPUs must be programmed using different languages. For this current effort, Python, which is a high-level programming language, was

326 used to write the program used to control the CPU processes. Conversely, OpenCL [161], which is an open source language used to develop parallel processing applications, was used write the code for the GPU. A Microsoft Windows program known as an HTML Application (HTA) was used to generate the windows needed to input and output simulation parameters from the computer and also to monitor the status while simulation runs were being conducted. GPU computing using OpenCL has been used previously to solve difficult problems including those requiring complex Monte Carlo simulations [162-164].

GPU Process CPU Process

Make calculations Solution for Data: Block # 1 Sequentially for Block #1 Block #1 Combine Make calculations Solution for Results Data: Block # 2 for Block #2 Block #2

Data Make calculations Solution for Data: Block # 3 (e.g., blocks 1-5) for Block #3 Block #3 Analyze Results and Make calculations Solution for Identify Coefficients Import Data: Block # 4 for Block #4 Block #4 that Provide the environmental data Make calculations Solution for Lowest Error and1.5M sets of Data: Block # 5 random coefficients for Block #5 Block #5 from CPU. Parse Use results to set Break Data Set Develop parameters for the environmental data Apply Model to into Multiple Solution for next simulation. into 3072 streams. Each Block Blocks Each Block Construct 1.5M sets of random coefficients

Repeat

Figure C-5 Illustration of the GPU and CPU Processes

When conducting simulations to calibrate candidate corrosion models, the environmental data was ported from the CPU to the GPU memory for use in computations. The CPU also calculated sets of random numbers for each simulation run (e.g., 1.5M sets of coefficients per run) and ported them over to the GPU, where they were applied to the candidate model and the environmental data. All of the calculations were sequentially run and the results temporarily housed in the GPUs memory until the entire run was finished. After all the simulations in the run were completed, the entire output file was ported back to the CPU for analysis, calculation

327 of the statistical error, and calculation of the adjusted statistical distribution parameters needed for the next simulation run. This process is illustrated by Figure C-5.

C.3 Statistical Analysis of Results During the simulation process, models based upon certain assumed coefficient values were statistically compared with others in a repetitive process to identify the specific set of coefficients that led to the most accurate model. Two statistical methodologies were used to evaluate accuracy. These included the residual sum of squares (RSS) method and the calculation of the coefficient of determination.

C.3.1 Residual Sum of Squares Method The residual sum of squares method (RSS) is a process used to calculate a measure of error based upon a comparison of model predictions and associated test measurements [165].

Equation C-1 is a tailored form of the RSS method that was used to calculate the error of candidate corrosion models.

n 2 RSS (P i  Ti ) (C-1) i1

When applying this equation to the corrosion model, the variable indicated as “Pi “ pertains to individual predictions made by the model, “Ti” refer to the associated test data points (corrosion weight loss measurements), and “n” is the number of data points involved in the calculation.

Because corrosion test measurements were taken at three month intervals, hourly predictions made by the model could not be directly considered when calculating error. Instead, cumulative predictions corresponding to each corrosion test point were used. In addition, error calculations were not made for the individual calibration sites. Instead, test data and associated predictions for all locations were pooled so the resultant error calculation indicated how a particular model

328 fit all of the calibration site data. This approach was used to optimize the coefficients so the model would provide the best overall predictions.

It should be noted the error calculated using RSS is not considered when making actual predictions but instead is limited to comparing the accuracy of candidate models during the calibration process. As such, there is no physical meaning concerning its numerical value.

Corrosion rates used during the simulation processes and the subsequent spreadsheet analyses were in units of micrograms. Thus, when calculating the RSS using such units, the error, which is based upon the square of the difference between the test measurements and the predictions, will appear to be very large. Had these errors been calculated using units of grams, then of course the calculated errors would be very small. Regardless, it is not the magnitude of the error that is important but instead the relative error of one candidate model to the others, with the most accurate model having the smallest error.

C.3.2 Coefficient of Determination Method The second method used to evaluate model accuracy was to calculate the coefficient of determination, which is more commonly referred to as the R2 value. This statistic provides a metric that indicates how much of the data variability is explained by the model. If the R2 value has a value of 1.0, it indicates a perfect model while a value of zero indicates that none of the variability is explained by the model. Thus, the R2 value is an indication of the goodness-of-fit of the model to the data. Figure C-6 illustrates the linear trend line and associated R2 value as it is fit to plotted data.

There are no universally accepted criteria as to what a good R2 value is other than the fact that models with a perfect correlation to data have a value of 1.0, models with no correlation have a value of zero, and for others in between, higher values are better. An additional metric that can

329 be used to gain a qualitative appreciation for the degree of fit between model predictions and test data is to consider the slope of the trend line fit to scatter plots such as shown in Figure C-6.

A trend line for a perfect model will have a slope of 1.0 and will extend to the origin while models with lessor accuracy will not.

Figure C-6 Illustration of a Trend Line and R2 Value in Relation to Plotted Data

C.4 Model Attributes As seen in the following discussions, a number of different approaches were used to evaluate the accuracy of various models. These include calculating the error using the residual sum of squares method as well as creating scatterplots that compare individual predictions and their associated test measurements. Once the scatterplots were created, a trend line was fit through them using linear regression and the aforementioned coefficient of determination (R2) value was calculated to determine the index of fit. A visual comparison of the trend line and the data it was based upon was also used as a metric to describe model accuracy. For example, if the model perfectly fit the test data, then the trend line should have a slope of one and extend to

330 the origin of the scatterplot. A trend line failing to meet these conditions indicates a model of lesser accuracy.

60 -Linear trend line has 50 a slope near 1:1 and 40 extends towards the origin 30 -Statistical “residuals” Predictions 20 appear to be normally distributed around 10 the linear trend line 0 0 10 20 30 40 50 60 Test Measurements

Figure C-7 Illustration of Attributes of a Model that Accurately Reflects Test Data

Another issue relating to model accuracy concerns the data points on the scatterplot and their relationship to the trend line. Ideally, the data points should appear randomly oriented on either side of the trend line so that the residuals (the distances from the trend line to the individual points) are normally distributed. Such a case, as illustrated in Figure C-7, is referred to as homoscedasticity and is the desired attribute [166]. However, if the data points shown on the scatterplot appear to be funnel-shaped in comparison to the trend line (i.e., the residuals are small near the origin and grow larger as the test data/predictions get larger), then this situation is referred to as heteroscedastic behavior (see Figure C-8). Such behavior is undesirable as it indicates the errors are correlated, which implies that that something other than random variability of environmental characterization or corrosion test data measurements is at play.

331

60 -Linear trend line 50 does not have a slope 40 near 1:1 and does not extend to the origin 30 -Statistical “residuals” Predictions 20 exhibit a “funnel shape” 10

0 0 10 20 30 40 50 60 Test Measurements

Figure C-8 Illustration of Attributes of a Model that Poorly Reflects Test Data

332

APPENDIX D

VARIABLE THRESHOLD FUNCTIONS

D.1 Examination of the RH Threshold As mentioned previously, Vernon conducted a series of experiments to determine how steel corrosion rates change in response to changing relative humidity and atmospheric contaminant levels [66]. There were two principal conclusions that resulted from this early work.

 Corrosion rates increase with increasing RH

 The critical value of relative humidity required to initiate and sustain corrosive reactions of steel in atmospheres containing small amounts of SO2 corresponds to approximately 60% RH.

Vernon examined two different alloys in his research, both of which were reported by chemical compositions. Table D-1 displays the compositions of Vernon’s material “G” and material “T” in comparison to the SAE-AISI 1010 alloy used to calibrate the model developed here. As can be seen on the table, the constituents found in each of the alloys are similar with the exception that 1010 steel does not contain either copper or silicon. While it seems possible that the presence of these constituents in Vernon’s alloys could have influenced the observed RH threshold value for those materials, the absence of any published RH threshold information for

1010 steel led to the use of the 60% RH threshold when developing the new cumulative damage model.

Closer inspection of Vernon’s work reveals two potential issues that could impact the accuracy of corrosion predictions made using the cumulative damage model. The first of these concerns

333 the 60% RH threshold, which may not be a precise enough value and thus could impact prediction accuracy. The second concern pertains to the fact that temperature effects were not examined under Vernon’s efforts. The following discussions will focus upon these issues and postulate approaches to investigate possible deficiencies using simulations.

Table D-1 Comparison of Steel Alloy Compositions (weight percent) Constituents Material Carbon Silicon Manganese Sulfur Phosphorus Copper “G” 0.12 0.01 0.39 0.052 0.029 0.12 “T” 0.081 0.01 0.39 0.04 0.035 0.10 AISI 1010 0.08 – - 0.3 – 0.6 0.05 (max) 0.04 (max) - 0.13

The experiments conducted by Vernon involved the precise weighing and optical examination of specimens exposed to various humidity levels in order to ascertain when corrosive reactions began. During these experiments, the specimens were subjected to relative humidity levels that were increased in 10% RH increments. It was observed that the weight of specimens markedly increased when the relative humidity was changed from 50% to 60%. This observation forms the basis for the assumed 60% threshold. However, since the relative humidity was increased in

10% RH increments, it is possible and indeed likely that weight gain began at levels greater than

50% RH but less than 60%. Another complicating factor concerns just what such weight gain means. Visual inspection of the specimens exposed to 60% RH failed to reveal any corrosion products on the surface. In fact, it wasn’t until specimens were exposed to the next increment of 70% RH that corrosion products were visually observed. It is possible that the weight gain measured at 60% RH was due to water filling the pores and capillaries on the material surface, similar to what Lee and Staehle had postulated for gold [52]. Another possibility is that because of the 10% RH increments used during the test program, corrosion products began to form at an

RH level above 60% but below 70%. A final possibility concerns the fact that oxidation could

334 have been present at 60% RH but not to the degree that it could be seen through optical microscopes. Modern surface analysis and characterization equipment (e.g., scanning electron microscopy, energy dispersive X-ray spectroscopy, etc.) had not yet been invented at the time of

Vernon’s experiments. Had they been, perhaps morphological and chemical changes to specimen surfaces could have been detected at the 60% RH level.

2165푒 푥 = 푑 (D.1) 푇+273.16

ln(푅퐻/100) 푇 237.3( + ) 17.27 237.3+푇 푇푑 = ln(푅퐻/100) 푇 (D.2) 1−( + ) 17.27 237.3+푇

17.27푇푑 푒푑 = 0.6108exp [ ] (D.3) 푇푑+237.3

As mentioned above, a concern about the published RH threshold value relates to the fact

Vernon’s work did not investigate the effects of temperature. Thus, it seems reasonable to assume that all of his experiments occurred at ambient conditions within the laboratory. It is well known that the absolute humidity (the mass of water per unit volume of air) is highly dependent upon temperature. Equation D.1 is used to determine the absolute humidity based upon the Dew Point (Equation D.2) and the Vapor Pressure (Equation D.3) [167]. These equations can be used to illustrate the large changes in moisture present in the air over a range of temperatures at a constant value of 60% RH (see Figure D-1).

Atmospheric corrosion results when sufficient quantities of adsorbed moisture combine with adsorbed/deposited ionic contaminants to form an electrolytic solution (i.e., a weak acid) on metallic surfaces [104]. As seen in Figure D-1, the atmospheric moisture concentration at a constant 60% RH increases by an order of magnitude when the temperature increases from the freezing point to the maximum temperature considered by the corrosion model (320.15K). It seems likely that the large increase in available moisture would affect the degree that water

335 molecules adsorb onto surfaces, with more “monolayers” of water present at higher temperatures. Lee and Staehle concluded that the number of monolayers of adsorbed moisture increased markedly at higher temperatures and RH levels, which supports the above hypothesis

[52]. However, their results at lower temperatures and RH values were not so clear due to a large degree of scatter in the data.

50

) 45 3 40 35 30 25 20 15 10

Absolute humidity (g/mhumidity Absolute 5 0 270 280 290 300 310 320 330 Temperature (K)

Figure D-1 Temperature Dependence of Absolute Humidity at a Constant RH of 60%

When considering the additional moisture available at higher temperatures (and by definition, lower amounts at colder temperatures), it seems plausible to assume the number of monolayers adsorbed onto surfaces will be a function of temperature. Whether this function has a strong or weak temperature dependence is a question that can only be conclusively answered through carefully conducted laboratory testing using modern surface characterization and analysis equipment. However, numerical experiments using Monte Carlo simulations can provide an indication of how the RH threshold might vary in response to changing temperatures. Such experiments might also lead to an improved empirical model that can more accurately predict the corrosion rate of steel.

336

Figure D-2 graphically illustrates the initial shape functions that are used to quantify interactions between relative humidity and temperature (see Section 2.4). This figure is based upon a constant RH threshold of 60% for the full range of temperatures considered by the model.

Figure D-2 Illustration of the Constant Temperature-Relative Humidity Shape Function

D.2 Variable RH Threshold Functions As described below, there are several different ways that variable RH threshold functions can be applied to determine whether anything other than a constant value of 60% RH will enable a more accurate model. The simplest approach was to look at other constant values of RH to see if a value could be found that would increase prediction accuracy. Simulations were conducted and quickly converged on an optimum threshold of 0% RH, which implies that corrosion occurs at all levels of RH. Obviously this makes little physical sense and suggests that something could be wrong with the model. Additional simulations were conducted to investigate this issue.

There are two more complex approaches that were considered, both of which included the employment of assumed temperature dependent RH threshold functions. Numerical

337 experiments using Monte Carlo simulations were used to calibrate these functions (along with the other model coefficients) to determine whether either of these approaches provided improved model accuracy.

D.2.1 Linear RH Threshold Function A linear RH threshold function was derived by tailoring the general equation for a straight line,

Equation D.4, thus resulting in Equation D.5. The term RHTH seen in this new equation provides the linear RH Threshold function.

푦 = 푚푥 + 푏 (D.4)

푅퐻푇퐻 = 푚∆푇 + 푅퐻푣푎푟 (D.5)

Slope (m) = (RHcal-RHvar)/(298.15-273.15) RHvar RH intercept (@273.15) = RHvar T = T-273.15

RHTH

RHcal Relative Humidity (%) Relative 273.15 T 298.15 320.15 Temperature (K)

Figure D-3 Illustration of the Linear RH Temperature Function

Figure D-3 illustrates the conditions used to calibrate the linear threshold RH model. Included on this figure are expressions to replace the independent variables used in Equation D.5. As discussed below, the equation was calibrated so that Vernon’s RH threshold at ambient conditions (e.g., 60% RH at 298.15K) was met. This condition is addressed through the calibration variable “RHcal” shown on the figure. Also shown is the variable RHvar, which is the assumed RH threshold required for corrosion to occur at the freezing point. The optimum value

338 for RHvar was also determined via Monte Carlo simulations. It should be noted that the

approach for calculating RHTH was not limited to situations where RHvar was greater than RHcal, which is the condition indicated by the figure.

When applied to Equation D.5, the information on Figure D-3 leads to Equation D.6. This new equation displays the complete linear temperature dependent RH threshold function.

푅퐻 −푅퐻 푅퐻 = 푐푎푙 푣푎푟 (푇 − 273.+15) 푅퐻 (D.6) 푇퐻 298.15−273.15 푣푎푟

Equation D.6 was used during the Monte Carlo simulation process in order to determine whether a weak or strong temperature dependence was likely. For example, if during simulations, it was found that RHvar was close but not identical to RHcal, then weak temperature dependence was indicated. Conversely, if the two variables were significantly different, then a strong dependence was likely. Ultimately, the best way to determine whether temperature dependence exists would be to run some carefully controlled experiments, which is beyond the scope of this present work. The simulations as described here could be used to help plan such a test program.

Figure D-4 illustrates a revised Temperature-Relative Humidity shape function that results from employing a linear threshold RH function as a boundary condition. Please note that the value of

80% RH at the freezing point shown on the figure was used only to illustrate how this shape function differs from the constant RH threshold shape function seen on Figure D-2. It is obvious when comparing Figure D-2 and Figure D- 4, however, that the employment of a linear RH threshold function in place of a constant RH threshold will significantly alter the way that the

339 corrosion model considers relative humidity when making incremental predictions based upon hourly conditions. Such a difference may help reduce model error.

Figure D-4 Illustration of the Temperature-Relative Humidity Shape Function Based Upon a Functional Linear RH Threshold Boundary Condition

D.2.2 Parabolic RH Threshold Function Equation D.7 displays the vertex form of a parabolic equation with a horizontal axis of symmetry.

X=푎(푦 − 푘)2 + ℎ (D.7)

When tailored to the case where the x-axis corresponds to temperature and the y-axis is relative humidity, Equation D.7 was rewritten to the form seen by Equation D.8.

2 푇 = 푎(푅퐻푇퐻 − 푘) + ℎ (D.8)

As was the case for the linear RH threshold function, RHTH shown in this current equation is defined as the variable threshold value of relative humidity, which was examined during the

Monte Carlo simulation process. The coordinates (h,k) used in Equation D.8 pertain to the

340 vertex of the parabola, with h corresponding to the temperature of the freezing point (273.15K) while k is the associated temperature dependent variable threshold RH (RHvar). Thus, Equation

D.8 was rewritten as Equation D.9.

2 푇 = 푎(푅퐻푇퐻 − 푅퐻푣푎푟) + 273. 15 (D. 9)

A functional relationship that determines the parabolic temperature dependent threshold value of relative humidity, RHTH, was determined by rewriting Equation D.9, as shown in Equations

D.10 through D.12.

푇−273.15 (푅퐻 − 푅퐻 )2 = (D.10) 푇퐻 푣푎푟 푎

푇−ℎ 푅퐻 − 푅퐻 = ±√ (D.11) 푇퐻 푣푎푟 푎

푇−273.15 푅퐻 = ±√ + 푅퐻 (D.12) 푇퐻 푎 푣푎푟

The two roots of Equation 12 pertain to whether the threshold RH is above (-root) or below

(+root) the 60%RH threshold value at ambient laboratory conditions (298.15K). The two possible scenarios are illustrated by Figures D- 5 and D-6.

341

1.4

RHvar 1.2

1 RHTH Curve

0.8 Calibration Point (+) root 0.6 (-) root

0.4 Relative Humidity Relative RH Tcal 0.2 cal

0 270 280 290 300 310 320 330 Temperature (K)

Figure D-5 Illustration of the Parabolic RH Temperature Function (- root)

0.8 RHTH Curve RHcal 0.6

0.4 Calibration Point (+) root (-) root 0.2 Tcal Relative Humidity Relative RHvar 0 270 280 290 300 310 320 330

-0.2 Temperature (K)

Figure D-6 Illustration of the Parabolic RH Temperature Function (+root)

342

The unknown coefficient “a” seen in Equation D.12 was determined by solving Equation D.9, as seen in Equations D.13 through D.15. During this process, the RH threshold, RHTH, seen in

Equation D.9 was replaced by the calibration conditions in a manner analogous to that employed when developing the linear RH threshold function.

2 푇푐푎푙 = 푎(푅퐻푐푎푙 − 푅퐻푣푎푟) + 273. 15 (D. 13)

The calibration conditions corresponding to those from Vernon’s experiments (RH=0.6 at a temperature of 298.15K) were input into this equation in order to obtain the explicit expression used to solve for the constant “a”, as seen in Equations D.14 and D.15.

2 298.15=푎(0.6 − 푅퐻푣푎푟) + 273. 15 ( D.14)

298.15−273.15 푎 = 2 (D.15) (0.6−푅퐻푣푎푟)

Equation D.15 was inserted into Equation D.12 to complete the derivation of the expression needed to calculate the RH threshold at any temperature based upon a variable threshold at the freezing point. Equation D.16 displays this functional relationship. Analyses were conducted that indicated the negative root of Equation D.16 provided the best results.

푇−273.15 푅퐻푇퐻 = ±√ 298.15−273.15 + 푅퐻푣푎푟 (D.16) 2 (0.6−푅퐻푣푎푟)

The value of “a” calculated using Equation D.15 is dependent upon the RH threshold value at the freezing point, RHvar, which was varied during each Monte Carlo simulation run. To illustrate the impact that the parabolic RH threshold function has upon the Temperature -Relative Humidity shape function that forms the basis for the corrosion model, Equation D.17 was used to

343 calculate the value of “a” for the case when RHvar equals 0.8. This value was then input into

Equation D.15, which was subsequently used to create the illustration shown by Figure D-7.

298.15−273.15 푎 = = 277.7778 ( D.17) (0.6−0.9)2

Figure D-7 Illustration of the Temperature-Relative Humidity Shape Function Based Upon a Parabolic RH Threshold Boundary Condition

D.3 Evaluation of RH Threshold Functions The RH threshold functions discussed above hold the potential for improving the accuracy of model predictions by tailoring the Temperature-Relative Humidity shape functions to more accurately reflect moisture adsorption on surfaces. However, it is also possible that the variable

RH threshold functions will simply help compensate for a poor Temperature-Relative Humidity shape function. If such a result is indicated, then a different shape function would be more appropriate and could lead to even better results.

344

APPENDIX E

RESULT DETAILS

Prior to conducting simulations to calibrate candidate models, the specific shape functions used to represent the interaction between acceleration factors had to be selected. Afterwards, these functions were input into the notional model and calibrated during the simulation process by applying them to large amounts of diverse environmental characterization data. Statistical testing was conducted to identify the models most capable of making accurate hourly predictions. The overall results are discussed in Chapters 4 and 5 while the detailed results are seen below.

E.1 Shape Functions The final notional equations representing the cumulative damage model (Equations 2.14 and

2.15) each contain six different shape functions used by the model. Three of these describe the interaction between temperature and relative humidity while the other three describe the interaction between temperature and contaminant levels. Numerous candidate shape functions could be constructed and calibrated via Monte Carlo simulations in order to make predictions.

However, conducting simulations on such a complex model, especially when considering the amount of characterization data used during each simulation, is a very time intensive process.

To reduce the amount of simulation time, an analysis was conducted to eliminate some of the

345 candidate shape functions so that simulations focused on those likely to lead to the most accurate models

E.1.1 Down-Selection of Temperature-Relative Humidity Shape Functions As shown in Section 2, a number of different temperature-relative humidity shape functions were formulated as candidates for this model development effort (see Table 2-2, shown again here as Table E-1)). Parabolic equations were used to create both the concave and convex functions. Obviously there are numerous other nonlinear functions that could have been used for this proof-of-concept effort. However, the lengthy time needed to conduct the simulations used to calibrate each model and its associated shape functions precluded consideration of other nonlinear approaches. The results from this current research should be sufficient to determine the general parameters (linear or nonlinear) of the shape function that will lead to acceptable model results. This information could provide valuable clues that in the future might lead to the selection of other nonlinear functions that could provide even better results than shown here.

Table E-1 Nine Candidate Temperature-Relative Humidity Shape Functions RH Functions Temperature Functions Linear Concave Convex Linear Linear-Linear Concave -Linear Convex -Linear Concave Linear- Concave Concave - Concave Convex - Concave Convex Linear- Convex Concave - Convex Convex - Convex

Because of the lengthy periods of time needed to calibrate each of the candidate models, it was not possible to numerically examine each of the derived shape functions shown in Table E-1.

Instead, an analysis was conducted to identify those that were more likely to lead to acceptable predictions.

346

-1.5

-2

-2.5 f(T) -3

-3.5

-4 270 280 290 300 310 320 330 Temperature (K)

concave convex linear

Figure E-1 Temperature Functions Used to Create Shape Functions

0 -0.5 -1 -1.5 -2

f(T, RH) f(T, -2.5 100 -3 90 -3.5 80 -4 70

60

Temperature (K)

Figure E-2 Illustration of the Convex Temperature-Convex Relative Humidity Shape Function

Figure E-1 illustrates the temperature functions from which all temperature-relative humidity shape functions were based. For illustration purposes, this figure was based upon a maximum value (f(T)max) equal to zero. Different values of f(T)max were considered during the Monte Carlo

347 simulation process in order to determine the specific coefficients that provide the most accurate predictions. It should be noted that while the value of f(T) for each temperature function is the same at the maximum temperature considered by the model (320.15K), they are markedly different everywhere else except the origin.

As shown by Table E-1, three relative humidity functions were combined with the three temperature functions to provide the nine different Temperature-Relative Humidity Shape

Functions. Figure E-2 illustrates one such shape function, which was constructed using the combination of a convex temperature function and a convex relative humidity function.

Obviously the curvature, or lack thereof, pertaining to each candidate shape function dictates the interaction between temperature and relative humidity that was considered by the model.

This becomes evident when considering that for all models (except the last and most accurate proof-of-concept model described in Sections 4.13, 5.2, and A.3), it is not the shape function itself that was considered by the model, but rather the results obtained when applying the exponential function to values determined via the shape function (see Section 2.3). Figure E-3 illustrates the adjusted exponential surface (based upon Equations 2.38 and 2.39) pertaining to the data used to construct Figure E-2. As seen, applying exponential functions generally flattened the surface over a large part of the total area, which indicates the importance of shape function curvature to provide reasonable values considered by the model, especially at low temperatures.

In comparison to the other shape functions identified in Table E-1, the convex temperature- convex relative humidity shape function provided the largest amount of interaction between the two acceleration factors, as indicated by the numerical values of the surfaces illustrated by

Figures E-2 and E-3. A comparison of the three different types of shape functions at the

348 maximum temperature considered by the model and at a lower temperature was used to determine whether the convex temperature-convex relative humidity function was optimal or whether any of the other eight shape functions might be expected to provide as good or better results.

1

0.8

0.6

0.4

0.2 90

Adjusted exp[f(T, RH)] exp[f(T, Adjusted 0 75

60

Temperature (K)

Figure E-3 Illustration of the Adjusted Exponential Function Applied to the Convex Temperature-Convex Relative Humidity Shape Function

0 -0.5 -1 -1.5 f(T)=f(T)max -2

f(T,RH) -2.5 -3 -3.5 -4 60 65 70 75 80 85 90 95 100 Relative Humidity (%RH)

convex linear concave

Figure E-4 Comparison of Three Different Temperature-Relative Humidity Shape Functions at 320.15K

349

Figure E-4 compares the shape functions pertaining to the linear, convex, and concave relative humidity functions identified in Table E-1. This comparison was limited to a temperature of

320.15K, which is the highest temperature within the model calibration range. At this point, the value of f(T) used to construct the shape functions equals f(T)max, which for this example has a value of zero. As long as the values of the three different temperature functions upon which shape functions were based are the same, then an analysis could be conducted to discount the associated relative humidity functions not likely to be suitable for modeling purposes. This is illustrated by Figure E-4, which enabled a direct comparison of how the differences in curvature affected the numerical values obtained from the functions and thus their influence on predicted corrosion rates.

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3

Adjusted exp[f(T,RH)] 0.2 0.1 0 60 65 70 75 80 85 90 95 100 Relative Humidity (%RH)

convex linear concave

Figure E-5 Adjusted Exponential Function Applied to the Three Different Temperature- Relative Humidity Shape Functions at 320.15K

As previously discussed, it was not the shape function itself that was considered by the model, but the values obtained when the exponential function was applied to the shape function output for a specific set of hourly environmental parameters. Figure E-5 illustrates this

350 condition using the data used to construct Figure E-4. Please note that the exponential functions shown here were adjusted using the approach shown by Equations 2.38 and 2.39. This was done so that the shape function was identically equal to zero at the threshold (i.e., 60% RH).

The curve shown on Figure E-5 corresponding to the exponential function applied to the concave relative humidity function has values nearly equal to zero until approximately 75% relative humidity, at which point it slowly begins to increase. If used by the model, this shape function would result in corrosion rate predictions with the same characteristics for a wide variety of RH values, which does not seem to be reasonable, especially at 320.15K.

0 -0.5 f(T) -1 -1.5 -2

f(T,RH) -2.5 -3 -3.5 -4 60 65 70 75 80 85 90 95 100 Relative Humidity (%RH)

convex linear concave

Figure E-6 Comparison of Three Different Temperature-Relative Humidity Shape Functions at 285K

Figure E-6 shows a comparison of the three shape functions at a temperature of 285K. The only way that each of these shape functions could have the same value of f(T) as shown in the figure is if they were constructed using linear temperature functions, each with the same value of f(T)max., which for this case was a value of zero (as before). Figure E-7 displays the adjusted exponential form of Figure E-6. Inspection of this new figure leads to the same conclusions as

351 the case at 320.15K in that the concave shape function has very small values, especially at lower

RH levels. Thus, it seems likely this function is incapable of adequately describing the interaction between temperature and RH as required to make accurate predictions. Similarly, the use of the concave temperature function depresses values for all related shape functions so that corrosion predictions at other than high temperatures will be very small, thus it is unlikely that the use of this function could lead to an accurate corrosion model.

0.14

0.12

0.1

0.08

0.06

0.04

Adjusted exp[f(T,H)] Adjusted 0.02

0 60 65 70 75 80 85 90 95 100 Relative Humidity (%RH)

convex linear concave

Figure E-7 Adjusted Exponential Function Applied to the Three Different Temperature- Relative Humidity Shape Functions at 285K

Because of the likelihood that any shape function based upon a concave function was deemed unlikely to be sufficient for modeling purposes, the number of shape functions to be considered during the model development effort, as described in Table E-1, was reduced to four. These include the Linear Temperature-Linear Relative Humidity, Linear Temperature-Convex Relative

Humidity, Convex Temperature-Linear Relative Humidity, and Convex Temperature-Convex

Relative Humidity shape functions.

352

A comparison of the four remaining candidate shape functions was conducted to determine if any of the remaining forms could be eliminated from consideration. Because of the minimal curvature at lower temperatures that results when the exponential is applied to linear relative humidity functions, the ability of linear functions to make predictions at such temperatures seems limited. Thus, the convex temperature-linear relative humidity shape function was dropped from consideration. This decision may be worth revisiting in a follow-on effort.

At this point, it was thought that either the Linear Temperature-Convex Relative Humidity or the

Convex Temperature-Convex Relative Humidity shape functions would provide the best results.

However, the linear temperature-linear relative humidity shape function was retained as a baseline.

E.1.2 Temperature-Contaminant Shape Functions Like the Temperature-Relative Humidity shape functions, numerous types of Temperature-

Contaminant shape functions could be tested to ascertain the impact on model accuracy.

However, because of the extensive amount of time needed to conduct simulations to calibrate each candidate model, only the Linear Temperature-Linear Contaminant shape functions described in Section 2.5.1 were considered during this proof-of-concept effort.

E.2 Screening of Initial Models to Identify the Most Accurate Shape Functions As described earlier, six different shape functions (three Temperature-Relative Humidity and three Temperature-Contaminant functions) were used to construct the initial models to evaluate whether the cumulative damage approach to modeling atmospheric corrosion has merit. Monte Carlo simulations were used to randomly change the maximum value of these functions, as well as the other model variables while statistical analyses were conducted to benchmark the accuracy of each model to enable a comparison with other candidates.

353

Since chloride aerosols cannot be determined using gaseous analyzers such as the equipment used to measure ozone and SO2 levels, longer-term deposition measurements were employed both to calibrate the model via simulations and to later validate their accuracy. Annual average chloride deposition measurements were used during initial screening simulations to determine which of the aforementioned Temperature-Relative Humidity shape functions provide the best fit to data. After the optimal shape function was identified, later simulations were conducted to determine whether using weekly or monthly average chloride deposition rates improved accuracy in comparison to predictions made from annual average deposition rates.

Calibration data for the initial screening simulations came from the four different sites as mentioned in the earlier discussion on data collection. These sites included China Lake, CA; Ft

Drum, NY; Dobbins Air Reserve Base, GA; and Kennedy Space Center, Florida. Because of concerns about the corrosion test protocol as it applies to desert locations that typically have infrequent relative humidity levels above the corrosion threshold value, later simulations evaluated whether eliminating use of the China Lake, CA data during calibration simulations would improve accuracy. Near the end of this research effort, it was determined that the published data for Dobbins, GA was in error (see Section 4.13), which necessitated the use of replacement data. Data from Rock Island, IL was used to calibrate new models to compare with those constructed using data from Dobbins.

Before presenting the results from various simulation runs on candidate models, it is important to note that the errors calculated when conducting simulations are slightly different from those calculated when the results of the simulation (i.e., the calibrated coefficients) are input into the associated spreadsheet model used to analyze and plot results. This difference is likely due to the differences in precision used during the calculation. The simulation code uses double

354 precision for some variables while others are used in single precision. This was done to ensure that the calculated coefficients were as accurate as possible while minimizing the impact on processing speed. In comparison, the spreadsheet-based models all operate using their native precision. Considering the numbers of calculations required to make 8760 hourly predictions for a single location and that the error is based upon either three or four locations (depending upon the type of model), it is not surprising that the errors calculated by two separate approaches were slightly different.

E.3 Detailed Results from Simulations Discussion on each of the models pertaining to the figures and tables that follow can be found in the appropriate sections of Chapter 4.

E.3.1 Simulations to Determine Optimum Chloride Deposition Interval – Candidate Models Calibrated Using Data from Four Corrosion Test Sites

E.3.1.1 Monthly Chloride Deposition Measurements

151000000

150500000

150000000

149500000 Error 149000000

148500000

148000000 0.6 0.62 0.64 0.66 0.68 0.7 Activation Energy (eV)

Figure E-8 Error Evaluation to Identify the Most Accurate Model Calibrated from Data Pertaining to Four Test Sites (monthly chloride deposition, optimum model:H=0.64 eV)

355

80000 y=0.9788x+500.73 70000

) R² = 0.9791 2 60000

g/cm 50000  ( 40000 30000 20000 Predictions 10000 0 0 10000 20000 30000 40000 50000 60000 70000 80000 Test Results (g/cm2)

Figure E-9 Comparison of Corrosion Test Data and Associated Predictions for the Monthly Chloride Model Applied to Calibration Data from Four Test Sites (optimum H=0.64 eV)

200000 180000 )

2 160000 140000 g/cm

 120000 ( 100000 80000 60000 40000 Predictions R² = 0.2824 20000 y=1.077x+30117 0 0 20000 40000 60000 80000 100000 Test Results (g/cm2)

Figure E-10 Comparison of Corrosion Test Data and Associated Predictions for the Monthly Chloride Model Applied to Validation Data from Ten Independent Locations (H=0.64 eV)

356

Table E-2 Modeling Overview and Coefficients for the Optimum Convex Temperature- Convex RH Shape Function Model Calibrated using Monthly Chloride Data from Four Test Sites Variable Value Number of Models Calibrated 11 (see Figure E-8) Computing Time (hours: minutes) 145:24 Simulations Conducted 2,685,000,000 Optimum H 0.64 eV/61.75 KJ/mol R2 (calibration) 0.9791 R2 (validation) 0.2824 Error 148,424,895

fCl(T,RH) -3.9942101515 fSO2(T,RH) -3.99759566496

fO3(T,RH) -3.99799762679 f(T,Cl) -3.99335127885

f(T,SO2) -3.77200165525 f(T,O3) -3.95154883737 2 ACl 0.0949333802058 g/cm 2 ASO2 0.116073871448 g/cm 2 AO3 0.032813027348 g/cm

Cl -0.0883512753998

SO2 0.138361465177

O3 -0.0220807547314

E.3.1.2 Weekly Chloride Deposition Measurements

157000000 156800000 156600000 156400000 156200000 156000000 Error 155800000 155600000 155400000 155200000 155000000 0.6 0.62 0.64 0.66 0.68 0.7 Activation Energy (eV)

Figure E-11 Error Evaluation to Identify the Most Accurate Model Calibrated from Data Pertaining to Four Test Sites (weekly chloride deposition, optimum model:H=0.66 eV)

357

80000 y=0.9771x+542.16 70000

) R² = 0.9782 2 60000

g/cm 50000  ( 40000 30000 20000 Predictions 10000 0 0 20000 40000 60000 80000 Test Results (g/cm2)

Figure E-12 Comparison of Corrosion Test Data and Associated Predictions for the Weekly Chloride Model Applied to Calibration Data from Four Test Sites (optimum H=0.66 eV)

200000 y=1.127x+29108 180000

) R² = 0.3102 2 160000 140000 g/cm

 120000 ( 100000 80000 60000 40000 Predictions 20000 0 0 20000 40000 60000 80000 100000 Test Results (g/cm2)

Figure E-13 Comparison of Corrosion Test Data and Associated Predictions for the Weekly Chloride Model Applied to Validation Data from Ten Independent Locations (H=0.66 eV)

358

Table E-3 Modeling Overview and Coefficients for the Optimum Convex Temperature- Convex RH Shape Function Model Calibrated using Weekly Chloride Data from Four Test Sites Variable Value Number of Models Calibrated 11 (see Figure E-11) Computing Time (hours: minutes) 116:47 Simulations Conducted 1,929,000,000 Optimum H 0.66 eV/63.68 KJ/mol R2 (calibration) 0.9782 R2 (validation) 0.3102 Error 155,259,104

fCl(T,RH) -3.992079751 fSO2(T,RH) -3.9991

fO3(T,RH) -3.999 f(T,Cl) -3.982544152

f(T,SO2) -3.651870738 f(T,O3) -3.963382218 2 ACl 0.14256496 g/cm 2 ASO2 0.115855799 g/cm 2 AO3 0.107433957 g/cm

Cl -0.296542124

SO2 0.094905711

O3 -0.202069589

E.3.2 Candidate Models Calibrated Using Data from Three Corrosion Test Sites

E.3.2.1 Annual Chloride Deposition Measurements

135400000 135200000 135000000 134800000 134600000 134400000 Error 134200000 134000000 133800000 133600000 133400000 0.6 0.62 0.64 0.66 0.68 0.7 Activation Energy (eV)

Figure E-14 Error Evaluation to Identify the Most Accurate Model Calibrated from Data Pertaining to Three Test Sites (annual chloride deposition, optimum model:H=0.65 eV)

359

80000 y=0.9144x+3375.4 70000

) R² = 0.9798 2 60000

g/cm 50000  ( 40000 30000 20000 Predictions 10000 0 0 10000 20000 30000 40000 50000 60000 70000 80000 Test Results (g/cm2)

Figure E-15 Comparison of Corrosion Test Data and Associated Predictions for the Annual Chloride Model Applied to Calibration Data from Three Test Sites (optimum H=0.65 eV)

200000 y=1.1183x+29655 180000

) R² = 0.305

2 160000 140000 g/cm

 120000 ( 100000 80000 60000

Predictions 40000 20000 0 0 20000 40000 60000 80000 100000 Test Results (g/cm2)

Figure E-16 Comparison of Corrosion Test Data and Associated Predictions for the Annual Chloride Model Applied to Validation Data from Ten Independent Locations (H=0.65 eV)

360

Table E-4 Modeling Overview and Coefficients for the Optimum Convex Temperature- Convex RH Shape Function Model Calibrated using Data from Three Test Sites Variable Value Number of Models Calibrated 11 (see Figure E-14 ) Computing Time (hours: minutes) 132:25 Simulations Conducted 2,706,000,000 Optimum H 0.65 eV/62.72 KJ/mol R2 (calibration) 0.9798 R2 (validation) 0.305 Error 133,534,628

fCl(T,RH) -3.97750687337 fSO2(T,RH) -3.9988

fO3(T,RH) -3.99554055151 f(T,Cl) -3.99697310527

f(T,SO2) -3.71630514002 f(T,O3) -3.98281268155 2 ACl 0.126595831715 g/cm 2 ASO2 0.0645734584818 g/cm 2 AO3 0.0414624637462 g/cm

Cl -0.28554786291

SO2 0.254358279809

O3 -0.109198728175

E.3.2.2 Monthly Chloride Deposition Measurements

118500000

118000000

117500000

117000000

116500000 Error 116000000

115500000

115000000

114500000 0.6 0.62 0.64 0.66 0.68 0.7 Activation Energy (eV)

Figure E-17 Error Evaluation to Identify the Most Accurate Model Calibrated from Data Pertaining to Three Test Sites (monthly chloride deposition, optimum model:H=0.62 eV)

361

80000 y=0.9194x+3215.3 70000

) R² = 0.9832 2 60000

g/cm 50000  ( 40000 30000 20000 Predictions 10000 0 0 10000 20000 30000 40000 50000 60000 70000 80000 Test Results (g/cm2)

Figure E-18 Comparison of Corrosion Test Data and Associated Predictions for the Monthly Chloride Model Applied to Calibration Data from Three Test Sites (optimum H=0.62 eV)

250000 y=1.0782x+30213 )

2 200000 R² = 0.2807 g/cm

 150000 (

100000

50000 Predictions

0 0 20000 40000 60000 80000 100000 Test Results (g/cm2)

Figure E-19 Comparison of Corrosion Test Data and Associated Predictions for the Monthly Chloride Model Applied to Validation Data from Ten Independent Locations (H=0.62 eV)

362

Table E-5 Modeling Overview and Coefficients for the Optimum Convex Temperature- Convex RH Shape Function Model Calibrated using Data from Three Test Sites Variable Value Number of Models Calibrated 11 (see Figure E-17) Computing Time (hours: minutes) 134:26 Simulations Conducted 2,739,000,000 Optimum H 0.62 eV/59.82 KJ/mol R2 (calibration) 0.9832 R2 (validation) 0.2807 Error 114,924,809

fCl(T,RH) -3.98986304843 fSO2(T,RH) -3.99421269483

fO3(T,RH) -3.99616532231 f(T,Cl) -3.98595674658

f(T,SO2) -3.74195578705 f(T,O3) -3.9447137936 2 ACl 0.0901061707643 g/cm 2 ASO2 0.0624184539396 g/cm 2 AO3 0.0353806725127 g/cm

Cl -0.173185506121

SO2 0.211892262004

O3 -0.0296903546528

E.3.2.3 Weekly Chloride Deposition Measurements

124500000

124000000

123500000

123000000

Error 122500000

122000000

121500000

121000000 0.6 0.62 0.64 0.66 0.68 0.7 Activation Energy (eV)

Figure E-20 Error Evaluation to Identify the Most Accurate Model Calibrated from Data Pertaining to Three Test Sites (weekly chloride deposition, optimum model:H=0.63 eV)

363

80000 y=0.9171x+3304.2 70000

) R² = 0.9821 2 60000

g/cm 50000  ( 40000 30000 20000 Predictions 10000 0 0 10000 20000 30000 40000 50000 60000 70000 80000 Test Results (g/cm2)

Figure E-21 Comparison of Corrosion Test Data and Associated Predictions for the Weekly Chloride Model Applied to Calibration Data from Three Test Sites (optimum H=0.63 eV)

200000 y=1.1272x+29299 180000 R² = 0.3061 )

2 160000 140000 g/cm

 120000 ( 100000 80000 60000

Predictions 40000 20000 0 0 20000 40000 60000 80000 100000 Test Results (g/cm2)

Figure E-22 Comparison of Corrosion Test Data and Associated Predictions for the Weekly Chloride Model Applied to Validation Data from Ten Independent Locations (H=0.63 eV)

364

Table E-6 Modeling Overview and Coefficients for the Optimum Convex Temperature- Convex RH Shape Function Model Calibrated using Data from Three Test Sites Variable Value Number of Models Calibrated 11 (see Figure E-20) Computing Time (hours: minutes) 93:30 Simulations Conducted 2,058,000,000 Optimum H 0.63 eV/60.79 KJ/mol R2 (calibration) 0.9821 R2 (validation) 0.3061 Error 121,418,584

fCl(T,RH) -3.98436211584 fSO2(T,RH) -3.99749510231

fO3(T,RH) -3.99572574492 f(T,Cl) -3.95605334658

f(T,SO2) -3.73301918072 f(T,O3) -3.94711716766 2 ACl 0.0621896736816 g/cm 2 ASO2 0.0624990995484 g/cm 2 AO3 0.0255408037941 g/cm

Cl -0.229592508861

SO2 0.281847624793

O3 -0.052850903874

E.3.3 Candidate Models Calibrated Using Data from Rock Island, Illinois

E.3.3.1 Annual Chloride Deposition Measurements

65000000

60000000

55000000

50000000 Error 45000000

40000000

35000000 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 Activation Energy (eV)

Figure E-23 Error Evaluation to Identify the Most Accurate Model Calibrated from Data Pertaining to Rock Island (annual chloride deposition, optimum model: H=0.91 eV)

365

30000 y=0.6801x+5711.7 25000 R² = 0.9309 ) 2

20000 g/cm  ( 15000

10000

Predictions 5000

0 0 5000 10000 15000 20000 25000 30000 Test Results g/cm2)

Figure E-24 Comparison of Corrosion Test Data and Associated Predictions for the Annual Chloride Model Applied to Calibration Data from Rock Island (optimum H=0.91 eV)

1200000 y=7.2716x-24902

) 1000000 R² = 0.6298 2

800000 g/cm  ( 600000

400000

Predictions 200000

0 0 20000 40000 60000 80000 100000

Test Results (g/cm2)

Figure E-25 Comparison of Corrosion Test Data and Associated Predictions for the Annual Chloride Model Applied to Validation Data from Nine Independent Locations (H=0.91 eV)

366

200000 y=2.4839x+21081 180000 R² = 0.1666 )

2 160000 140000 g/cm

 120000 ( 100000 80000 60000

Predictions 40000 20000 0 0 5000 10000 15000 20000 25000 30000 35000 Test Results (g/cm2)

Figure E-26 Comparison of Corrosion Test Data and Associated Predictions for the Annual Chloride Model Applied to Validation Data from Seven Independent Locations (no beach sites, H=0.91 eV)

Table E-7 Modeling Overview and Coefficients for the Optimum Convex Temperature- Convex RH Shape Function Model Calibrated using Data from Rock Island Variable Value Number of Models Calibrated 12 (see Figure E-23) Computing Time (hours: minutes) 74:04 Simulations Conducted 2,619,000,000 Optimum H 0.91 eV/87.80 KJ/mol R2 (calibration) 0.9309 R2 (validation) 0.6298 2 R (validation-updated SO2, no coastal sites) 0.1666 Error 38,141,437

fCl(T,RH) -3.9998

fSO2(T,RH) -3.976482646 fO3(T,RH) -3.988425 f(T,Cl) -3.9998

f(T,SO2) -3.70652592

f(T,O3) -4 2 ACl 0.165455499 g/cm 2 ASO2 0 g/cm 2 AO3 0.006139093 g/cm

Cl -0.296152

SO2 0.308553988

O3 -0.087460211

367

E.3.3.2 Monthly Chloride Deposition Measurements

65000000

60000000

55000000

Error 50000000

45000000

40000000 0.65 0.7 0.75 0.8 0.85 0.9 0.95 Activation Energy (eV)

Figure E-27 Error Evaluation to Identify the Most Accurate Model Calibrated from Data Pertaining to Rock Island (monthly chloride deposition, optimum model:H=0.93 eV)

25000 y=0.6203x+6953.7

20000 R² = 0.9783

15000

Error 10000

5000

0 0 5000 10000 15000 20000 25000 30000 Test Results

Figure E-28 Comparison of Corrosion Test Data and Associated Predictions for the Monthly Chloride Model Applied to Calibration Data from Rock Island (optimum H=0.93 eV)

368

1000000 y=7.9108x-34414 900000 R² = 0.6268 )

2 800000 700000 g/cm

 600000 ( 500000 400000 300000

Predictions 200000 100000 0 0 20000 40000 60000 80000 100000 Test Results (g/cm2)

Figure E-29 Comparison of Corrosion Test Data and Associated Predictions for the Monthly Chloride Model Applied to Validation Data from Nine Independent Locations (H=0.93 eV)

250000 y=2.8375x+15959

) R² = 0.146 2 200000

g/cm 150000  (

100000

Predictions 50000

0 0 5000 10000 15000 20000 25000 30000 35000 Test Results (g/cm2)

Figure E-30 Comparison of Corrosion Test Data and Associated Predictions for the Monthly Chloride Model Applied to Validation Data from Seven Independent Locations (no beach sites, H=0.93 eV)

369

Table E-8 Modeling Overview and Coefficients for the Optimum Convex Temperature- Convex RH Shape Function Model Calibrated using Data from Rock Island Variable Value Number of Models Calibrated 13 (11 shown in Figure E-27) Computing Time (hours: minutes) 147:06 Simulations Conducted 4,870,500,000 Optimum H 0.93 eV/89.73 KJ/mol R2 (calibration) 0.9783 R2 (validation) 0.6268 2 R (validation-updated SO2, no coastal sites) 0.146 Error 44,305,051

fCl(T,RH) -3.9998

fSO2(T,RH) -3.990775067

fO3(T,RH) -3.990579257 f(T,Cl) -3.9999

f(T,SO2) -3.781738892 f(T,O3) -4 2 ACl 0.147363189 g/cm 2 ASO2 0 g/cm 2 AO3 0.040320623 g/cm

Cl -0.281368957

SO2 -0.044745755

O3 -0.089753855

E.3.3.3 Weekly Chloride Deposition Measurements

140000000

120000000

100000000

80000000

Error 60000000

40000000

20000000

0 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 Activation Energy (eV)

Figure E-31 Error Evaluation to Identify the Most Accurate Model Calibrated from Data Pertaining to Rock Island (weekly chloride deposition, optimum model:H=0.96 eV)

370

30000 y=0.7191x+5487.8

) 25000 R² = 0.9773 2

20000 g/cm  ( 15000

10000

Predictions 5000

0 0 5000 10000 15000 20000 25000 30000 Test Results (g/cm2)

Figure E-32 Comparison of Corrosion Test Data and Associated Predictions for the Weekly Chloride Model Applied to Calibration Data from Rock Island (optimum H=0.96 eV)

1000000 900000 y=8.6078x-49479

) R² = 0.7687 2 800000 700000 g/cm

 600000 ( 500000 400000 300000

Predictions 200000 100000 0 0 20000 40000 60000 80000 100000 Test Results (g/cm2)

Figure E-33 Comparison of Corrosion Test Data and Associated Predictions for the Weekly Chloride Model Applied to Validation Data from Nine Independent Locations (H=0.96 eV)

371

200000 )

2 180000 y=2.3458x+17514 160000 R² = 0.1365 140000 g/cm 

( 120000 100000 80000 60000 40000

Predictions 20000 0 0 10000 20000 30000 40000 Test Results (g/cm2)

Figure E-34 Comparison of Corrosion Test Data and Associated Predictions for the Weekly Chloride Model Applied to Validation Data from Seven Independent Locations (no beach sites, H=0.96 eV)

Table E-9 Modeling Overview and Coefficients for the Optimum Convex Temperature- Convex RH Shape Function Model Calibrated using Data from Rock Island Variable Value Number of Models Calibrated 13 (see Figure E-31) Computing Time (hours: minutes) 104:00 Simulations Conducted 3,879,000,000 Optimum H 0.96 eV/92.63 KJ/mol R2 (calibration) 0.9773 R2 (validation) 0.7687 2 R (validation-updated SO2, no coastal sites) 0.1365 Error 28,561,646

fCl(T,RH) -3.9998 fSO2(T,RH) -3.979235029 fO3(T,RH) -3.970863416 f(T,Cl) -3.9999

f(T,SO2) -3.812354978 f(T,O3) -3.992123149 2 ACl 0.150208111 g/cm 2 ASO2 0 g/cm 2 AO3 0 g/cm

Cl -0.528380672

SO2 -0.040513304

O3 -0.099388648

372

E.3.4 Candidate Models Based Upon Three Activation Energies

114000000 113000000 112000000 111000000 110000000 109000000 Error 108000000 107000000 106000000 105000000 104000000 0.9 0.92 0.94 0.96 0.98 1 1.02 1.04 1.06 Activation Energy (eV)

Figure E-35 Error Evaluation to Identify the Most Accurate Three Activation Energy Model Calibrated using Data Pertaining to Three Test Sites (monthly chloride deposition, optimum model:HCl=0.97 eV)

106400000

106200000

106000000

105800000

Error 105600000

105400000

105200000

105000000 0.9 0.92 0.94 0.96 0.98 1 1.02 1.04 Activation Energy (eV)

Figure E-36 Error Evaluation to Identify the Most Accurate Three Activation Energy Model Calibrated using Data Pertaining to Three Test Sites (narrowed range, monthly chloride deposition, optimum model:HCl=0.97 eV)

373

80000 y=0.9339x+2578.3 70000

) R² = 0.9827 2 60000

g/cm 50000  ( 40000 30000 20000 Predictions 10000 0 0 10000 20000 30000 40000 50000 60000 70000 80000 Test Results (g/cm2)

Figure E-37 Comparison of Corrosion Test Data and Associated Predictions for the Three Activation Energy Monthly Chloride Model Applied to Calibration Data from Three Test Sites (optimum HCl=0.97 eV)

140000 y=0.7169x+23716 120000 ) R² = 0.2697 2 100000 g/cm 

( 80000

60000

40000

Predictions 20000

0 0 20000 40000 60000 80000 100000 Test Results (g/cm2)

Figure E-38 Comparison of Corrosion Test Data and Associated Predictions for the Three Activation Energy Monthly Chloride Model Applied to Validation Data from Ten Independent Locations (HCl =0.97 eV)

374

100000 y=2.8458x-657.73 90000

) R² = 0.8296

2 80000 70000 g/cm

 60000 ( 50000 40000 30000

Predictions 20000 10000 0 0 5000 10000 15000 20000 25000 30000 35000 Test Results (g/cm2)

Figure E-39 Comparison of Corrosion Test Data and Associated Predictions for the Annual Chloride Model Applied to Validation Data from Eight Independent Locations (updated SO2 proxies, no beach sites, HCl =0.97 eV) Table E-10 Modeling Overview and Coefficients for the Optimum Convex Temperature- Convex RH Shape Function Model Calibrated using Three Activation Energies and Data from Three Test Sites Variable Value Number of Models Calibrated 14 (see Figure E-35) Computing Time (hours: minutes) 315:42 Simulations Conducted 7,231,000,000

Optimum HCl 0.97 eV/93.59 KJ/mol

Optimum HSO2 0.58 eV/55.96 KJ/mol

Optimum HO3 0.84 eV/81.05 KJ/mol R2 (calibration) 0.9827 R2 (validation) 0.2697 2 R (validation-updated SO2, no coastal sites) 0.8296 Error 105,093,938

fCl(T,RH) -3.99936428571 fSO2(T,RH) -3.94887582572 fO3(T,RH) -3.98843032114 f(T,Cl) -3.99985714286

f(T,SO2) -3.76664114757 f(T,O3) -3.99992857143 2 ACl 0.000285714285714 g/cm 2 ASO2 0.0247689441429 g/cm 2 AO3 0.00175714285714 g/cm

Cl -0.295285714286

SO2 0.289038002143

O3 -0.0930234225715

375

E.3.5 Refinement of the Monthly Chloride Model

115400000 115300000 115200000 115100000 115000000 Error 114900000 114800000 114700000 114600000 0.59 0.6 0.61 0.62 0.63 0.64 0.65 0.66 Activation Energy (eV)

Figure E-40 Error Evaluation to Identify the Most Accurate Model Calibrated from Data Pertaining to Three Test Sites (refined model with monthly chloride deposition, optimum model:H=0.638 eV)

80000 y=0.9194x+3215.3 70000 0.9833 ) 2 60000 50000 g/cm  ( 40000 30000 20000

Predictions 10000 0 0 20000 40000 60000 80000 Test Results (g/cm2)

Figure E-41 Comparison of Corrosion Test Data and Associated Predictions for the Monthly Chloride Model Applied to Calibration Data from Three Test Sites (refined model with optimum H=0.638 eV)

376

100000 y=2.8609x-754.21 90000

) R² = 0.8375

2 80000 70000 g/cm

 60000 ( 50000 40000 30000

Predictions Predictions 20000 10000 0 0 5000 10000 15000 20000 25000 30000 35000 Test Results (g/cm2)

Figure E-42 Comparison of Corrosion Test Data and Associated Predictions for the Monthly Chloride Model Applied to Validation Data from Eight Independent Locations (refined model, updated SO2 proxies, no beach sites, H=0.638 eV)

Table E-11 Modeling Overview and Coefficients for the Optimum Convex Temperature- Convex RH Shape Function Model Calibrated using Data from Three Locations (refined model) Variable Value Number of Models Calibrated 11 (see Figure E-40) Computing Time (hours: minutes) 243:14 Simulations Conducted 4,989,000,000 Optimum H 0.638 eV/61.56 KJ/mol R2 (calibration) 0.9833 2 R (validation-updated SO2, no coastal sites) 0.8375 Error 114,659,126

fCl(T,RH) -3.99163778735 fSO2(T,RH) -3.99839524064

fO3(T,RH) -3.99416362832 f(T,Cl) -3.99170840535

f(T,SO2) -3.69264748599 f(T,O3) -3.93969754274 2 ACl 0.123344645914 g/cm 2 ASO2 0.0529918607033 g/cm 2 AO3 0.0111714415572 g/cm

Cl -0.224509442513

SO2 0.309182862859

O3 -0.0455423093879

377

E.3.6 Simulations to Investigate the Relative Humidity Threshold

E.3.6.1 Linear Relative Humidity Threshold

87200000

87100000

87000000

86900000 Error 86800000

86700000

86600000 0.32 0.33 0.34 0.35 0.36 0.37 0.38 Activation Energy (eV)

Figure E-43 Error Evaluation to Identify the Most Accurate Model Calibrated from Data Pertaining to Three Test Sites (refined model with the linear RH threshold function and monthly chloride deposition, optimum model:H=0.352 eV)

80000 y=0.9312x+2760.7 70000

) R² = 0.9873 2 60000

g/cm 50000  ( 40000 30000 20000 Predictions 10000 0 0 10000 20000 30000 40000 50000 60000 70000 80000 Test Results (g/cm2)

Figure E-44 Comparison of Corrosion Test Data and Associated Predictions for the Monthly Chloride Model Applied to Calibration Data from Three Test Sites (refined model, linear RH threshold function, H=0.352 eV)

378

90000 y=2.6119x+1038.5 80000

) R² = 0.8327 2 70000 60000 g/cm  ( 50000 40000 30000 20000 Predictions 10000 0 0 5000 10000 15000 20000 25000 30000 35000 Test Results (g/cm2)

Figure E-45 Comparison of Corrosion Test Data and Associated Predictions for the Monthly Chloride Model Applied to Validation Data from Eight Independent Locations (refined model, updated SO2 proxies, no beach sites, linear RH threshold function, monthly chloride deposition, H=0.352 eV)

Table E-12 Modeling Overview and Coefficients for the Optimum Convex Temperature- Convex RH Shape Function Model Calibrated using Data from Three Locations (refined model using the linear RH threshold function) Variable Value Number of Models Calibrated 12 (11 shown by Figure E-43) Computing Time (hours: minutes) 307:00 Simulations Conducted 5,299,500,000 Optimum H 0.352 eV/33.96 KJ/mol R2 (calibration) 0.9873 2 R (validation-updated SO2, no coastal sites) 0.8327 Error 86,693,508

fCl(T,RH) -3.89674208183 fSO2(T,RH) -3.99738313583 fO3(T,RH) -3.96538249172 f(T,Cl) -3.80728738715

f(T,SO2) -2.69098444646 f(T,O3) -3.44197449203 2 ACl 0.713806897211 g/cm 2 ASO2 0.0419278563451 g/cm 2 AO3 0.641234501657 g/cm

Cl 0.421553706564

SO2 1.97018688436

O3 0.586622264537 RHvar 0

379

E.3.6.2 Nonlinear Relative Humidity Threshold

95800000 95780000 95760000 95740000 95720000 95700000 Error 95680000 95660000 95640000 95620000 95600000 0.42 0.43 0.44 0.45 0.46 0.47 0.48 Activation Energy (eV)

Figure E-46 Error Evaluation to Identify the Most Accurate Model Calibrated from Data Pertaining to Three Test Sites (refined model with the nonlinear RH threshold function and monthly chloride deposition, optimum model:H=0.449 eV)

80000 y=0.9257x+2994.6 70000

) R² = 0.9864 2 60000

g/cm 50000  ( 40000 30000 20000 Predictions 10000 0 0 10000 20000 30000 40000 50000 60000 70000 80000 Test Results (g/cm2)

Figure E-47 Comparison of Corrosion Test Data and Associated Predictions for the Monthly Chloride Model Applied to Calibration Data from Three Test Sites (refined model, nonlinear RH threshold function, H=0.449 eV)

380

90000 y = 2.6976x + 337.53 80000 R² = 0.84 ) 2 70000 60000 g/cm  50000 40000 30000 20000 Predictions ( Predictions 10000 0 0 5000 10000 15000 20000 25000 30000 35000 Test Results (g/cm2)

Figure E-48 Comparison of Corrosion Test Data and Associated Predictions for the Monthly Chloride Model Applied to Validation Data from Eight Independent Locations (refined model, updated SO2 proxies, no beach sites, nonlinear RH threshold functionH=0.449 eV)

Table E-13 Modeling Overview and Coefficients for the Optimum Convex Temperature- Convex RH Shape Function Model Calibrated using Data from Three Locations (refined model using the nonlinear RH threshold function) Variable Value Number of Models Calibrated 8 (see Figure E-46) Computing Time (hours: minutes) 257:18 Simulations Conducted 4,701,000,000 Optimum H 0.449 eV/43.32 KJ/mol R2 (calibration) 0.9864 2 R (validation-updated SO2, no coastal sites) 0.84 Error 95,621,664

fCl(T,RH) -3.997298554 fSO2(T,RH) -3.9996 fO3(T,RH) -3.99945 f(T,Cl) -3.915479776

f(T,SO2) -3.016172723

f(T,O3) -3.391019963 2 ACl 0.751063597 g/cm 2 ASO2 0.34239196 g/cm 2 AO3 0.724733819 g/cm

Cl 0.537608525

SO2 1.313924816

O3 0.590848205 RHvar 0

381

E.3.7 Simulations to Calibrate the Final Proof-of-Concept Model with Improved Characterization Data

130000000

125000000

120000000

Error 115000000

110000000

105000000 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 Activation Energy (eV)

Figure E-49 Error Evaluation to Identify the Most Accurate Model Calibrated from Data Pertaining to the Final Proof-of-Concept Model , optimum model:H=1.68 eV)

60000 y=0.9512x+1036.1

) 50000 R² = 0.9569 2

40000 g/cm  ( 30000

20000

Predictions 10000

0 0 10000 20000 30000 40000 50000 60000 Test Results (g/cm2)

Figure E-50 Comparison of Corrosion Test Data and Associated Predictions for the Final Proof-of-Concept Model Applied to Calibration Data (monthly chlorides, H=1.68 eV)

382

40000 y=1.0702x+2502.7 35000

) R² = 0.8628 2 30000

g/cm 25000  ( 20000 15000 10000 Predictions 5000 0 0 5000 10000 15000 20000 25000 30000 35000 Test Results (g/cm2)

Figure E-51 Comparison of Corrosion Test Data and Associated Predictions for the Final Proof-of-Concept Model Applied to Validation Data (monthly chlorides, H=1.68 eV)

Table E-14 Modeling Overview and Coefficients for Final Proof-of-Concept Model Calibrated using Data from Three Locations Including Rock Island, IL Variable Value Number of Models Calibrated 24 (see Figure E-49) Computing Time (hours: minutes) 240:10 Simulations Conducted 8,022,500,000 Optimum H 1.68 eV/162.1 KJ/mol R2 (calibration) 0.9569 2 R (validation-updated SO2, no coastal sites) 0.8628 Error 106,746,708

fCl(T,RH) 4.02E-11 fSO2(T,RH) 6.25E-07 fO3(T,RH) 8.77E-06 f(T,Cl) 1.33E-13

f(T,SO2) 7.00E-05 f(T,O3) 9.90E-06 2 ACl 1.80E-15 g/cm 2 ASO2 3.32E-09 g/cm 2 AO3 6.95E-10 g/cm

Cl 4.776591579

SO2 -2.3335

O3 -2.4272

383