SFPE Fire Exposures

The SFPE Task Group on Fire Exposures to Structural Elements

Chairman James G. Quintiere, Ph.D., FSFPE University of Maryland

Members

Farid Alfawakhiri, Ph.D. Stephen Hill, P.E. James Mehaffey, Ph.D. American Institute of ATF Fire Research Laboratory Forintek Canada Corp. Steel Construction Barbara Lane, Ph.D. Amal Tamim Andrew Buchanan, Ph.D. ARUP Fire University of Canterbury James Milke, P.E., Ph.D., Sean Hunt, P.E. FSFPE Vytenis Babrauskas, Ph.D. Hughes Associates, Inc. University of Maryland Fire Science & Technology Inc. Brian Lattimer, Ph.D. Ian Thomas, Ph.D. Jonathan Barnett, Ph.D., Hughes Associates, Inc. Victoria University FSFPE Worcester Polytechnic Institute Rodney McPhee Christopher Wieczorek, Ph.D. Canadian Wood Council FM Global Thomas Izbicki, P.E. Dallas Fire Department Harold Nelson, P.E., FSFPE

Staff

Morgan J. Hurley, P.E. Society of Fire Protection Engineers

The SFPE Task Group on Fire Exposures to this information was not included in this guide, the Structural Elements began its work in March 1998. work of Margaret Law, “Design Guide for Fire The purpose of this guide is to provide the infor- Safety of Bare Exterior Structural Steel,” Technical mation and methodology needed to predict the Reports and Designer’s Manual (London, Ove thermal boundary condition for a fire over time. Arup & Partners, 1977), is recommended for such The methods contained herein are based on experi- fire scenarios. mental measurements and correlations, and mostly The work in completing this guide was mostly give global rather than local results. Eventually, done voluntarily. All contributions, no matter how “CFD” methods for fire must be subjected to some small, are appreciated and enabled this guide to of the same tests used here and judged accordingly come to closure. for accuracy and application. This guide is written for those with an under- On September 11, 2001, the world changed, and standing of fire and heat transfer, but should be edu- this task took on a new life and significance. Issues cational and informative to a structural engineer. It identified during examination of the collapse of the includes some theoretical background for orientation, World Trade Center buildings raised questions and examples to appreciate the process of calcula- regarding the design of fire protection of structures. tion. It is the sixth engineering practice guide pub- Indeed, the role of the fire protection engineer lished by the Society of Fire Protection Engineers. (FPE) in structural fire-resistance design may I take responsibility for the “theory” on compart- change and embrace more of these calculations. ment fires, and for the general approach of the Presently, the architect is generally responsible for guide. But the guide could not have been completed the fire protection of the structure. An engineered without the dedicated contribution of Morgan design method would involve: Hurley, Technical Director of SFPE. He performed the role of technical editor and personally per- 1. A prediction of the fire over time formed the analyses and evaluations of the various 2. Heat transfer analysis of the structural member methods for predicting the temperature–time curves 3. Response of the structural system for fully developed fires. That comparison had never been done before, and it was imperative to Such full calculations will have to be dealt with conduct in order to make judgment on the methods. by the fire protection engineer in conjunction with In making those comparisons, we decided to use the the structural engineer. Items 1 and 2 are more in CIB and Carrington data sets to serve as a bench- the domain of the FPE. Note, however, that item 2 mark. While the CIB data are of scales no more that is not addressed here. 1.5 m in height, the Carrington tests are much more This guide was originally divided into three realistic in scale. However, the theory section should areas. The first included fully developed fires in offset any issues of the relevance of small scale. compartments. Since it was an “old” area of study The section on fire plumes was developed by with many contributors, care was required to sort Brian Lattimer with the assistance of Sean Hunt. out the key pieces. The second area was fire plumes, That was a significant contribution and had never or the exposure of discrete fires to elements. Since been assembled before. Christopher Wieczorek it was more recent in exposition, this work could be organized the material describing the various evaluated more easily. A third area intended for this approaches. Barbara Lane presented a thorough guide included the effect of window flames on the review of the time-equivalent method and drafted façade and external structural elements. While material on parametric equations for estimating

ii compartment fire temperatures and durations. The Buchanan, Thomas Izbicki, Rodney McPhee, Amal time-equivalent method is limited but well known. Tamim, and James Milke, were critical readers, and We included this material to explicitly explain its Vytenis “Vyto” Babrauskas continually provided basis and limitations. useful comments and critiques. Readers outside the Others made noteworthy contributions. Jonathan Committee included Ulf Wickstrom, Takeyoshi Barnett and his students got us started on the Tanaka, Tibor Harmathy, and T.T. Lie, and for this literature of fully developed fire, and Stephen Hill we are greatly appreciative. brought this to the production point in a presentation for SFPE. James Mehaffey, Ian Thomas, and Harold “Bud” Nelson were early contributors. James G. Quintiere Others, including Farid Alfawakhiri, Andrew November 10, 2003

The Society of Fire Protection Engineers wishes to acknowledge and thank the American Institute of Steel Construction, the National Fire Protection Association, the American Forest and Paper Association, and the Canadian Wood Council for their generous support of this project.

iii

Contents

Foreword...... ii

Executive Summary ...... xii

Introduction ...... 1 Model Inputs ...... 1 Basis of Fire Resistance...... 2 Accounting for Suppression...... 2 Heat Transfer Boundary Conditions ...... 3 Computer Modeling ...... 3

Fully Developed Enclosure Fires ...... 4 Theory ...... 5 Theoretical Development ...... 5 Wall Heat Transfer...... 7 General Form of Correlations...... 12 Methods for Predicting Fire Exposures ...... 16 Eurocode Parametric Fire Exposure Method ...... 16 Lie’s Parametric Method ...... 19 Tanaka...... 21 Magnusson and Thelandersson Parametric Curves...... 22 Harmathy ...... 24 Babrauskas...... 26 Ma and Mäkeläinen ...... 29 CIB...... 31 Law ...... 33 Simple Decay Rates...... 34 Recommendations...... 34

Fire Exposures from Plumes ...... 40 Axisymmetric Fire Plumes ...... 41 Heat Flux Boundary Condition...... 44 Bounding Heat Flux: Objects Immersed in Flames ...... 45 Heat Fluxes for Specific Geometries...... 48 Flat Vertical Walls...... 48 Fires in a Corner ...... 52 Fires Impinging on Unbounded Ceilings ...... 58 Fire Impinging on a Horizontal I-Beam Mounted Below a Ceiling...... 63 Summary and Recommendations ...... 68

v Appendix A – Theoretical Examination of Methods...... 69 Results by Harmathy for Wood Cribs...... 69 Results by Bullen and Thomas for Pool Fires...... 70 CIB Data ...... 71 Eurocode ...... 71 Lie ...... 71 Magnusson, Thelandersson, and Petersson...... 71 Babrauskas ...... 71 Law...... 72 Ma and Mäkeläinen ...... 72

Appendix B – Comparisons of Enclosure Fire Predictions with Data...... 73 CIB Data ...... 74 Cardington Data ...... 74 Eurocode ...... 76 Lie ...... 83 Tanaka ...... 89 Magnusson and Thelandersson...... 95 Harmathy...... 101 Babrauskas ...... 106 Ma and Mäkeläinen...... 113 CIB ...... 118 Law...... 122

Appendix C – Time-Equivalent Methods ...... 129 Real Structural Response ...... 129 Discussion of Methods...... 130 Fire Load Concept ...... 130 Kawagoe and Sekine ...... 131 Law ...... 131 Pettersson...... 132 Normalized Heat Load Concept ...... 133 Eurocode Time-Equivalent Method ...... 133 New Zealand Code ...... 136 Comparisons...... 136 Limitations and Assumptions...... 137

Appendix D – Examples...... 139

Glossary Nomenclature Used in the Enclosure Fires Section ...... 143 Nomenclature Used in the Plumes Section...... 145

References ...... 147

vi Illustrations

FIGURE 1 Phases of Fire Development...... 4 2Model for the Fully Developed Fire ...... 6 3Wall Heat Transfer...... 7 4MQH Correlation for Fuel-Controlled Fires ...... 11 5 Approximate Theoretical Behavior for Fuel Burning Rate ...... 15 6 Approximate Theoretical Behavior of Compartment Temperature ...... 15 7 Schematic Illustration of the Heat Balance Equation Terms ...... 23 8 Examples of Temperature–Time Curves...... 23 9 Non-Dimensionalized Temperature–Time Curves Developed by Ma and Mäkeläinen ...... 29 10 Average Temperature During Fully Developed Burning ...... 31 11 Normalized Burning Rate During Fully Developed Burning...... 32 12 Comparison of CIB Temperature Data to Predictions Using Law’s Method ...... 35 13 Comparison of Burning Rate Data to Predictions Using Law’s Method ...... 35 14 Comparison of Predictions Using Law’s Modified Method for Cardington Test #1 ...... 36 15 Comparison of Predictions Using Law’s Modified Method for Cardington Test #2 ...... 36 16 Comparison of Predictions Using Law’s Modified Method for Cardington Test #8 ...... 37 17 Comparison of Predictions Using Law’s Modified Method for Cardington Test #9 ...... 37 18 Comparison of Predictions from Magnusson and Telandersson’s Method (Type C) to Data for Cardington Test #3...... 38 19 Comparison of Predictions from Magnusson and Telandersson’s Method (Type C) to Data for Cardington Test #4...... 39 20 Comparison of Predictions from Magnusson and Telandersson’s Method (Type C) to Data for Cardington Test #5...... 39 21 Comparison of Predictions from Lie for Cardington Test #6 ...... 40 22 Axisymmetric Fire Plume ...... 41 23 Maximum Turbulent Fire Plume Temperatures from Various Sources ...... 42 24 Heat Balance at the Material Surface...... 44 25 Magnitude of Surface Temperature Corrections on the Measured Total Heat Flux Using a Cooled Gauge...... 45 26 Averaged Peak Heat Flux as a Function of Angular Position...... 46 27 Fire Against a Flat Vertical Wall ...... 48 28 Peak Heat Release Rates Measured in Square Propane Burner Fires Against a Flat Wall ...... 49 29 Vertical Heat Flux Distribution Along the Centerline of a Square Propane Burner Fire Adjacent to a Flat Wall...... 50 30 Horizontal Heat Flux Distribution (a) Below the Flame Height and (b) Above the Flame Height with Distance from the Centerline of the Fire...... 50 31 Fire in a Corner Configuration...... 52 32 Corner with a Ceiling Configuration Showing the Three Regions Where Incident Heat Flux Correlations Were Developed in the Study of Latimer et al...... 53 33 Peak Heat Flux Along the Height of the Walls in the Corner...... 53 34 Maximum Heat Fluxes to the Walls Near the Corner with Square Burner Sides of ●-0.17 m, ▲-0.30 m, ▼-0.30 m (Elevated), and ■-0.50 m and Fires Sizes from 50 to 300 kW...... 54 35 Heat Flux Distribution Horizontally out from the Corner on the Lower Part of the Corner Walls .....55

vii 36 Maximum Heat Flux Along the Top of the Walls During Corner Fire Tests with Square Burner Sides of ●-0.17 m, ▲-0.30 m, ▼-0.30 m (Elevated), and ■-0.50 m and Fires Sizes from 50 to 300 kW ...... 56 37 Heat Flux Along the Ceiling Above a Fire in a Corner During Tests with Square Burner Sides of ●-0.17 m, ▲-0.30 m, ▼-0.30 m (Elevated), and ■-0.50 m and Fires Sizes from 50 to 300 kW...... 57 38 Unbounded Ceiling Configuration ...... 59 39 Stagnation Point Heat Fluxes on an Unbounded Ceiling with a Fire Impinging on It ...... 60 40 Heat Fluxes to a Ceiling Due to a Propane Fire Impinging on the Surface ...... 61 41 Comparison of the Best Fit Curve Proposed by Wakamatsu and a Bounding Fit to the Data...... 62 42 I-Beam Mounted Below an Unbounded Ceiling...... 64 43 Heat Flux Measured onto the Surfaces of an I-Beam Mounted Below an Unbounded Ceiling for Fires 95 to 900 kW ...... 66 44 Heat Flux Measured on the ●-Bottom Flange, ■-Web, and ▲-Upper Flange of an I-Beam Mounted Below and Unbounded Ceiling for Fires 565 to 3,870 kW ...... 67

A.1 Comparison of Burning Rate Predictions ...... 69 A.2 Wood Crib and Liquid Pool Fires ...... 70

B.1 Histogram of Ratio of Fuel Surface Area to Enclosure Surface Area for the CIB Experiments ...... 74 B.2 Comparison of CIB Temperature Data to Predictions Made Using Eurocode, 2 Buchanan, and Franssen Methods, qt,d = 100 MJ/m ...... 77 B.3 Comparison of CIB Temperature Data to Predictions Made Using Eurocode, 2 Buchanan, and Franssen Methods, qt,d = 50 MJ/m ...... 77 B.4 Comparison of CIB Burning Rate Data to Predictions Made Using the Eurocode Method...... 78 B.5 Comparison of Predictions Made Using Eurocode, Buchanan, and Franssen Methods to Data from Cardington Test #1...... 79 B.6 Comparison of Predictions Made Using Eurocode, Buchanan, and Franssen Methods to Data from Cardington Test #2...... 79 B.7 Comparison of Predictions Made Using Eurocode, Buchanan, and Franssen Methods to Data from Cardington Test #3...... 80 B.8 Comparison of Predictions Made Using Eurocode, Buchanan, and Franssen Methods to Data from Cardington Test #4...... 80 B.9 Comparison of Predictions Made Using Eurocode, Buchanan, and Franssen Methods to Data from Cardington Test #5...... 81 B.10 Comparison of Predictions Made Using Eurocode, Buchanan, and Franssen Methods to Data from Cardington Test #6...... 81 B.11 Comparison of Predictions Made Using Eurocode, Buchanan, and Franssen Methods to Data from Cardington Test #7...... 82 B.12 Comparison of Predictions Made Using Eurocode, Buchanan, and Franssen Methods to Data from Cardington Test #8...... 82 B.13 Comparison of Predictions Made Using Eurocode, Buchanan, and Franssen Methods to Data from Cardington Test #9...... 83 B.14 Comparison of CIB Temperature Data to Predictions Made Using Lie’s Method...... 84 B.15 Comparison of CIB Burning Rate Data to Predictions Made Using Lie’s Method ...... 84 B.16 Comparison of Predictions Made Using Lie’s Method to Data from Cardington Test #1...... 85 B.17 Comparison of Predictions Made Using Lie’s Method to Data from Cardington Test #2...... 85 B.18 Comparison of Predictions Made Using Lie’s Method to Data from Cardington Test #3...... 86 B.19 Comparison of Predictions Made Using Lie’s Method to Data from Cardington Test #4...... 86 B.20 Comparison of Predictions Made Using Lie’s Method to Data from Cardington Test #5...... 87

viii B.21 Comparison of Predictions Made Using Lie’s Method to Data from Cardington Test #6...... 87 B.22 Comparison of Predictions Made Using Lie’s Method to Data from Cardington Test #7...... 88 B.23 Comparison of Predictions Made Using Lie’s Method to Data from Cardington Test #8...... 88 B.24 Comparison of Predictions Made Using Lie’s Method to Data from Cardington Test #9...... 89 B.25 Comparison of CIB Temperature Data to Predictions Made Using Tanaka’s Methods ...... 90 B.26 Comparison of CIB Burning Rate Data to Predictions Made Using Tanaka’s Methods...... 90 B.27 Comparison of Predictions Made Using Tanaka’s Methods to Data from Cardington Test #1 ...... 91 B.28 Comparison of Predictions Made Using Tanaka’s Methods to Data from Cardington Test #2 ...... 91 B.29 Comparison of Predictions Made Using Tanaka’s Methods to Data from Cardington Test #3 ...... 92 B.30 Comparison of Predictions Made Using Tanaka’s Methods to Data from Cardington Test #4 ...... 92 B.31 Comparison of Predictions Made Using Tanaka’s Methods to Data from Cardington Test #5 ...... 93 B.32 Comparison of Predictions Made Using Tanaka’s Methods to Data from Cardington Test #6 ...... 93 B.33 Comparison of Predictions Made Using Tanaka’s Methods to Data from Cardington Test #7 ...... 94 B.34 Comparison of Predictions Made Using Tanaka’s Methods to Data from Cardington Test #8 ...... 94 B.35 Comparison of Predictions Made Using Tanaka’s Methods to Data from Cardington Test #9 ...... 95 B.36 Comparison of CIB Temperature Data to Predictions Made Using Magnusson and Thelandersson’s Method...... 96 B.37 Comparison of CIB Burning Rate Data to Predictions Made Using Magnusson and Thelandersson’s Method...... 96 B.38 Comparison of Predictions Made Using Magnusson and Thelandersson’s Method (Type C) to Data from Cardington Test #1...... 97 B.39 Comparison of Predictions Made Using Magnusson and Thelandersson’s Method (Type C) to Data from Cardington Test #2...... 97 B.40 Comparison of Predictions Made Using Magnusson and Thelandersson’s Method (Type C) to Data from Cardington Test #3...... 90 B.41 Comparison of Predictions Made Using Magnusson and Thelandersson’s Method (Type C) to Data from Cardington Test #4...... 90 B.42 Comparison of Predictions Made Using Magnusson and Thelandersson’s Method (Type C) to Data from Cardington Test #5...... 99 B.43 Comparison of Predictions Made Using Magnusson and Thelandersson’s Method (Type C) to Data from Cardington Test #7...... 99 B.44 Comparison of Predictions Made Using Magnusson and Thelandersson’s Method (Type C) to Data from Cardington Test #8...... 100 B.45 Comparison of Predictions Made Using Magnusson and Thelandersson’s Method (Type C) to Data from Cardington Test #9...... 100 B.46 Comparison of CIB Burning Rate Data to Predictions Made Using Harmathy’s Method ...... 101 B.47 Comparison of Predictions Made Using Harmathy’s Method to Data from Cardington Test #1 ...... 102 B.48 Comparison of Predictions Made Using Harmathy’s Method to Data from Cardington Test #2 ...... 102 B.49 Comparison of Predictions Made Using Harmathy’s Method to Data from Cardington Test #3 ...... 103 B.50 Comparison of Predictions Made Using Harmathy’s Method to Data from Cardington Test #4 ...... 103 B.51 Comparison of Predictions Made Using Harmathy’s Method to Data from Cardington Test #5 ...... 104 B.52 Comparison of Predictions Made Using Harmathy’s Method to Data from Cardington Test #6 ...... 104 B.53 Comparison of Predictions Made Using Harmathy’s Method to Data from Cardington Test #7 ...... 105 B.54 Comparison of Predictions Made Using Harmathy’s Method to Data from Cardington Test #8 ...... 105 B.55 Comparison of Predictions Made Using Harmathy’s Method to Data from Cardington Test #9 ...... 106 B.56 Comparison of CIB Temperature Data to Predictions Made Using Babrauskas’ Method ...... 107 B.57 Comparison of CIB Burning Rate Data to Predictions Made Using Babrauskas’ Method...... 108 B.58 Comparison of Predictions Made Using Babrauskas’ Method to Data from Cardington Test #1 .....108 B.59 Comparison of Predictions Made Using Babrauskas’ Method to Data from Cardington Test #2 .....109

ix B.60 Comparison of Predictions Made Using Babrauskas’ Method to Data from Cardington Test #3 .....109 B.61 Comparison of Predictions Made Using Babrauskas’ Method to Data from Cardington Test #4...... 110 B.62 Comparison of Predictions Made Using Babrauskas’ Method to Data from Cardington Test #5...... 110 B.63 Comparison of Predictions Made Using Babrauskas’ Method to Data from Cardington Test #6...... 111 B.64 Comparison of Predictions Made Using Babrauskas’ Method to Data from Cardington Test #7...... 111 B.65 Comparison of Predictions Made Using Babrauskas’ Method to Data from Cardington Test #8...... 112 B.66 Comparison of Predictions Made Using Babrauskas’ Method to Data from Cardington Test #9...... 112 B.67 Comparison of CIB Burning Rate Data to Predictions Made Using Ma and Mäkeläinen’s Method ....113 B.68 Comparison of Predictions Made Using Ma and Mäkeläinen’s Method to Data from Cardington Test #1...... 114 B.69 Comparison of Predictions Made Using Ma and Mäkeläinen’s Method to Data from Cardington Test #2...... 114 B.70 Comparison of Predictions Made Using Ma and Mäkeläinen’s Method to Data from Cardington Test #3...... 115 B.71 Comparison of Predictions Made Using Ma and Mäkeläinen’s Method to Data from Cardington Test #4...... 115 B.72 Comparison of Predictions Made Using Ma and Mäkeläinen’s Method to Data from Cardington Test #5...... 116 B.73 Comparison of Predictions Made Using Ma and Mäkeläinen’s Method to Data from Cardington Test #7...... 116 B.74 Comparison of Predictions Made Using Ma and Mäkeläinen’s Method to Data from Cardington Test #8...... 117 B.75 Comparison of Predictions Made Using Ma and Mäkeläinen’s Method to Data from Cardington Test #9...... 117 B.76 Comparison of Cardington and CIB Temperature Data...... 118 B.77 Comparison of Predictions Made Using the CIB Data to Cardington Test #1...... 119 B.78 Comparison of Predictions Made Using the CIB Data to Cardington Test #2...... 119 B.79 Comparison of Predictions Made Using the CIB Data to Cardington Test #3...... 120 B.80 Comparison of Predictions Made Using the CIB Data to Cardington Test #4...... 120 B.81 Comparison of Predictions Made Using the CIB Data to Cardington Test #7...... 121 B.82 Comparison of Predictions Made Using the CIB Data to Cardington Test #8...... 121 B.83 Comparison of Predictions Made Using the CIB Data to Cardington Test #9...... 122 B.84 Comparison of CIB Temperature Data to Predictions Made Using Law’s Method...... 122 B.85 Comparison of CIB Burning Rate Data to Predictions Made Using Law’s Method ...... 123 B.86 Comparison of Predictions Made Using Law’s Method to Data from Cardington Test #1 ...... 124 B.87 Comparison of Predictions Made Using Law’s Method to Data from Cardington Test #2 ...... 124 B.88 Comparison of Predictions Made Using Law’s Method to Data from Cardington Test #3 ...... 125 B.89 Comparison of Predictions Made Using Law’s Method to Data from Cardington Test #4 ...... 125 B.90 Comparison of Predictions Made Using Law’s Method to Data from Cardington Test #5 ...... 126 B.91 Comparison of Predictions Made Using Law’s Method to Data from Cardington Test #6 ...... 126 B.92 Comparison of Predictions Made Using Law’s Method to Data from Cardington Test #7 ...... 127 B.93 Comparison of Predictions Made Using Law’s Method to Data from Cardington Test #8 ...... 127 B.94 Comparison of Predictions Made Using Law’s Method to Data from Cardington Test #9 ...... 128

C.1 Fire Severity Concept...... 130 1/2 C.2 Law’s Correlation Between Fire Resistance Requirements (tf ) and L/(AwAt ) ...... 137

D.1 Temperature–Time Curve for Burning Duration of 1.5 Hours and Opening Factor of 0.02 m1/2...... 141

x Tables TABLE 1 Estimates of Conduction for Common Materials ...... 8 2 Range of Values for Key Parameters from the 25 Data Sets Used to Develop the Shape Function....30 3 Rate of Decrease in Temperature ...... 34 4 Selected Heat Fluxes to Objects Immersed in Large Pool Fires ...... 47

B.1 Compartment Dimensions of the Cardington Tests ...... 75 B.2 Opening Dimensions of the Cardington Tests ...... 75 B.3 Properties of Enclosure Materials ...... 75 B.4 Fuel Loading for the Cardington Tests...... 75

C.1 Fuel Load Density Determined from a Fuel Load Classification of Occupancies...... 134 C.2 Safety Factor Taking Account of the Risk of a Fire Starting Due to the Size of Compartment ...... 134 C.3 Safety Factor Taking Account of the Risk of a Fire Starting Due to the Type of Occupancy ...... 134 C.4 A Factor Taking Account of the Different Active Fire-Fighting Measures ...... 135 C.5 Relationship Between kb and the Thermal Property b...... 135 C.6 Values for kb Recommended by the New Zealand Fire Engineering Design Guide ...... 136

xi Executive Summary

Designing fire resistance on a performance basis of temperatures during the decay stage is desired, a requires three steps: decay rate of 7ºC/min can be used for fires with a predicted duration of 60 minutes or more, and a 1. Estimating the fire boundary conditions decay rate of 10°C/min can be used for fires with a 2. Determining the thermal response of the structure predicted duration of less than 60 minutes. 3. Determining the structural response For long, narrow spaces in which is in

This guide provides information relevant to esti- the range of 45 to 85 m–1/2, Magnusson and mating the fire boundary conditions resulting from a Thelandersson provide reasonable predictions of fully developed fire. Methods are provided for fully temperature and duration. For long, narrow spaces developed enclosure fires and for fire plumes. Fully in which is approximately 345 m–1/2, Lie’s developed enclosure fires can be expected in compartments with fuel uniformly distributed over their method is recommended. interiors. For situations where a fire would not be For ranges of that fall outside the ranges enclosed or for enclosures with sparse distributions or concentrated fuel packets, the methods identified identified above, the calculations should be per- in the fire plumes section should be used. formed using the methods identified for the ranges Several methods are evaluated for fully developed of that bound the situation of interest, and enclosure fires. Law’s method is recommended for all roughly cubic compartments and in long, narrow the most conservative results should be used. For fire plumes, methods are presented for compartments where does not exceed conducting a bounding analysis and for specific ≈ 18 m–1/2. To ensure that predictions are sufficiently geometries. These geometries include flat vertical conservative in design situations, the predicted walls, corners with a ceiling, unbounded flat burning rate should be reduced by a factor of 1.4 ceilings, and an I-beam mounted below a ceiling. and the temperature adjustment should not be Additionally, correlations are provided for axisym- reduced by Law’s Ψ factor. metric plumes for those wishing to conduct a heat Law’s method does not predict temperatures transfer analysis from first principles. during the decay stage. For cases where a prediction

xii Engineering Fire Exposures to Guide Structural Elements

Introduction situations where a fire would not be enclosed or for enclosures with sparse distributions or concentrated An engineering analysis to evaluate the response fuel packets, the methods identified in the fire of a structure during a fire must consider both the plumes section should be used. heat transfer from the fire to the structural members and the structural response of these members under the defined threat. The focus of this guide is to define MODEL INPUTS the heat flux boundary condition due to the fire used For fully developed enclosure fires, predictive in the heat transfer analysis portion of this problem. methods require as input one or more of the following: Guidance is provided for two potential fire threats: fully developed enclosure fires and local fire plumes. 1. Fuel load In fully developed enclosure fires, the conditions 2. Dimensions of windows, doors, and other similar (gas temperatures, velocities, and smoke levels) are horizontal openings assumed to be uniform throughout the entire enclo- 3. Wall thermal properties sure, and all combustible contents are generally considered to be contributing to the fire size and Thermal properties of walls are generally fixed duration. Historically, conditions inside fully devel- very early in the design of a building. They typically oped enclosure fires have been defined by the gas do not change much during a building’s lifetime. temperatures inside the enclosure, and the enclosure Furthermore, this is the least critical of the three fire section includes a review of the most widely variables in its effect on the fire temperature–time used methods for predicting gas temperatures. history. Thus, it is generally acceptable to use Local fire plumes may be confined to a single normal design values for the thermal properties. fuel package in intimate contact with a structural Ventilation is usually handled by simply deter- member. The thermal exposure from local fires is mining the potential window and door openings spatially variable and is dependent on the geometry from the building’s architectural drawings. This being considered. Though local fires may not may not be a robust strategy since these openings expose as large an area as enclosure fires, the heat may vary as a consequence of alteration of a build- fluxes from local fires can be considerable and ing. Some serious fire losses have occurred during should not be neglected in an analysis. Heat fluxes construction or remodeling. Two examples are the from reasonable-size local fires can easily exceed One Meridian Plaza fire1 and the Broadgate fire.2 120 kW/m2 and have been measured as high as During construction or remodeling, the geometric 220 kW/m2 in very large pool fires. Due to the aspects of a building can vary from what they are spatially and geometric dependence, the thermal intended to be during ultimate occupancy. Uncer- exposure from local fire plumes has historically tainty in ventilation characteristics can be addressed been measured directly using heat flux gauges. by a variety of techniques.3 For example, analyses Therefore, the boundary condition for local fire could be conducted using the range of ventilation plumes will be provided as a measured heat flux characteristics that could reasonably be expected to with guidance on correcting this measurement based occur. The ventilation characteristics that result in on the actual structural element temperature. the most severe exposure could then be used as The methods applicable to fully developed en- the basis for design. If uncertainty in ventilation closure fires should be used for compartments with characteristics is not addressed during the design, fuel uniformly distributed over their interiors. For then any change that affects ventilation openings

1 would require reanalysis to confirm that the build- used as part of a strategy to achieve life safety, ing is still within its design basis. property protection, mission continuity, or environ- Similarly, fuel loads may vary during the life of a mental protection goals.3 More specific objectives building. During construction, periods of work may can be developed from these generic goals. exist where the fuel load is great. Such construction Structural fire resistance has historically been fuel (and debris) may often be much greater than specified as ratings for individual structural ele- projected for the ultimate occupancy. Furthermore, ments based on a number of building characteristics at these times normal fire defense mechanisms— such as occupancy type and building height. Given sprinklers, detectors, pull-stations, etc.—are often that the fire resistance and permissible materials of inoperable. construction vary with building use and building An example may be a building lobby. During height and area, a uniform level of performance does normal occupancy, the expected fuel load can be not result from compliance with prescriptive codes. trivial: perhaps a single guard’s desk. Yet during In the case of performance-based codes, the per- construction or renovation, the lobby may hold the formance intended also may vary. The International highest concentration of combustible building and Code Council Performance Code4 states that some packing materials. Another example is special events risk of loss of life may be acceptable, depending upon (e.g., school fair exhibits) that are sometimes staged the magnitude of the event and performance group of in lobbies that are generally otherwise fuel free. the building. Similarly, the serviceability expected of Fuel load statistics obtained from building surveys a building varies with the event size and performance are typically used by designers to derive their input group. The National Fire Protection Association’s data on fuel load. First, these statistics are “typical” Building Construction and Safety Code5 states that values, such as 50% or 80% occurrence values. As structural integrity must be maintained for a suffi- “typical” values, these statistics would not provide cient time to protect occupants and enable fire bounding or conservative estimates of fire severity. fighters to perform search and rescue operations. Additionally, all available fuel load surveys focus This guide provides a methodology to estimate solely on normal occupancy characteristics. the thermal aspects of a fire as they impact exposed Methods of predicting fire exposures from fire structural members. Given those heat transfer condi- plumes also require input values such as heat release tions, a structural engineer can compute the effect rate or dimension of the fire source. When selecting on the structure. input values for these methods, it is recommended Prior to designing or analyzing structural fire that bounding or reasonably conservative input resistance, it is necessary to determine the objec- values be used. tives that the structural fire resistance is intended to Whatever input values are used, designers should meet. Guidance on determining goals and objectives clearly communicate the limits of the design to can be found in the SFPE Engineering Guide to project stakeholders such as enforcement officials Performance-Based Fire Protection Analysis and and building owners and operators. Design of Buildings.3

BASIS OF FIRE RESISTANCE ACCOUNTING FOR SUPPRESSION Engineered fire protection design is typically Many building codes and design guides permit performed to meet a set of goals and objectives. a reduction in fire resistance when active fire pro- These goals and objectives may come from a tection systems, such as sprinklers, are used. For performance-based code, from a desire to establish example, the Eurocode6 contains an approach for equivalency with a prescriptive code, or from a accounting for interventions where the design fire building owner, insurer, or other stakeholder who load is reduced by a factor (0.0 to 1.0). This results desires to have added safety beyond compliance in a design fire load that is less than the actual with a code or standard. Fire resistance might be fire load.

2 The methods presented in this guide for predict- uniform conditions throughout the compartment. ing fire exposures are based on conditions where Indeed, even the computer model referenced above7 there is no mitigation of a fully developed fire. assumes a uniform temperature in the enclosure. Analyses of fire exposures to structures in which Many computer models exist that predict fire active mitigation is considered are outside the scope temperatures for user-defined heat release rates. Use of this guide. of most computer fire models for predicting post- flashover fire boundary conditions requires the HEAT TRANSFER BOUNDARY modeler to estimate the burning rate in the compart- CONDITIONS ment using other methods. Given that the heat release rate in a post-flashover compartment fire is Analyzing the thermal response of a structure a function of the characteristics of the enclosure, it requires prediction of the heat flux boundary con- is difficult to apply these models without making ditions. For fire plumes, methods are provided for additional simplifying assumptions. For example, estimating the heat flux boundary conditions directly, by assuming that burning in the compartment is although basic plume correlations are provided for stoichiometric or ventilation limited, a burning rate those who wish to conduct a heat transfer analysis could be estimated as a constant multiplied by the from first principles. ventilation characteristics of the enclosure. Pool For enclosure fires, most of the predictive fires could be modeled using burning rate correla- methods contained in this guide provide just the tions that were developed for open-air burning; temperature boundary conditions. Determining the however, these correlations neglect thermal feed- heat flux boundary conditions of a structure requires back to the fuel from the enclosure. prediction of the gas emissivity, the absorbtivity Field models such as NIST’s Fire Dynamics of the element, and the convective heat transfer Simulator (FDS) allow abandoning the assumption coefficient. The absorbtivity for a surface in a fully that compartment gasses are well stirred.8 Instead of developed enclosure fire can be assumed to be 1.0 modeling the enclosure as one zone, field models since the surface will become covered in soot. The model an enclosure as many rectangular prisms and gas emissivity will also approach 1.0 for large fires.* assume the conditions are uniform throughout each Assuming natural convection, the convective heat of these cells. transfer coefficient, hc, will generally be approxi- FDS contains pyrolysis models for solid and liquid 2 mately 10 W/m K, although it could be as high as fuels. The pyrolysis rate of the fuel is predicted by 2 30 W/m K.* For conservative predictions, a con- FDS as a function of the modeled heat transfer to 2 vective heat transfer coefficient of 30 W/m K the fuel, and thermally thick, thermally thin, and should be used. liquid fuels can be treated. Combustion is modeled For insulated materials, such as concrete or insu- by FDS using a mixture fraction model. lated steel, a bounding estimate of the heat transfer While FDS holds promise in calculating heat boundary condition would be to assume that the release rates in fires, it presently must be used with temperature of the exposed surface is equal to the caution since a number of simplifications are used surrounding gas temperature.* as a result of computational, resolution, and knowledge limitations. As stated in the FDS User’s Guide, COMPUTER MODELING “The various phenomena [associated with modeling With one exception,7 all the methods identified combustion] are still subjects of active research; above for calculating the temperature–time history thus the user ought to be aware of the potential 9 for a fire in a compartment are relatively simple, errors introduced into the calculation.” Any errors closed-form equations. Simple, closed-form equa- that are present with pool-like or slab-like fuels tions are possible because of the assumptions made would likely be magnified when considering crib- to solve the fundamental conservation equations, e.g., like fuels such as furniture. ______*See the “Theory” section beginning on page 5 for a derivation of this value.

3 Fully Developed Enclosure Fires 2. Methods that determine an equivalent exposure to the standard temperature–time relationship Fire in enclosures may be characterized in three phases. The first phase is fire growth, when a fire The former is the only true engineering method of grows in size and heat release rate from a small designing structural fire resistance. The latter is incipient fire. If there are no actions taken to sup- based on determining the “equivalent” fire exposure press the fire, it will eventually grow to a maximum to the “standard” temperature–time relationship, size, which is a function of the amount of fuel pres- which carries an implicit assumption that the fire ent or the amount of air available through ventila- resistance requirements contained in prescriptive tion openings. As all of the fuel is consumed, the codes provide a firm design basis. While the stan- fire will decrease in size (decay). These stages of dard temperature–time relationship provides an fire development can be seen in Figure 1. hourly rating, this rating is only intended to be a The size (magnitude) of the fire and the relative relative measure and does not necessarily reflect importance of these phases (growth, fully devel- structural performance in a fire. Time-equivalent oped, and decay) are affected by the size and shape methods are further discussed only in Appendix C. of the enclosure; the amount, distribution, form, and With the exception of Babrauskas’ method, type of fuel in the enclosure; the amount, distribu- which allows for the consideration of pool fires, all tion, and form of ventilation of the enclosure; and the methods summarized in this guide have their the form and type of materials forming the roof (or basis in fires involving wood cribs. Although many ceiling), walls, and floor of the enclosure. hydrocarbon-based materials, such as plastics, have The significance of each phase of an enclosure approximately twice the heat of combustion of fire depends on the fire safety system component cellulosic materials, such as wood (in other words, under consideration. For components such as detec- burning 1 kg of a plastic can liberate twice the tors or sprinklers, the fire growth part is likely to be energy as burning an equal mass of wood), use of the most significant because it will have a great influence on the time at which they activate. The fire growth stage usually proves no threat to Fire Fully Developed Decay the structure, but if it can (for Growth example, if concentrated fuel packets are located close to an element), the direct heating by flames must be considered in accordance Flashover Cooling Phase with the section on fire plumes. The threat of fire to the structure is primarily during the fully developed and decay phases.10,11 There are two methods of Temperature Development design based on fully developed compartment fires:

1. Methods that predict the boundary conditions to which Time the structure will be exposed, Significant effect on structure from which a thermal analysis and structural analysis of the structure may be performed FIGURE 1. Phases of Fire Development

4 the methods contained in this guide should be rea- 3. Only natural ventilation is considered as would sonable for most design scenarios. occur through the wall vents. (The effect of This statement is made for two reasons. First, forced ventilations and wind and stack-effect while real fuels are not wood cribs, cribs might flows in tall buildings are not included.) approximate structural wood furniture such as desks 4. Large fires are considered whose heating effects and chairs. Other furnishings are mostly composed are felt uniformly through the compartment. of large flat surfaces that would more easily vaporize fuel in a fire. These flat surfaces might be classi- Concern has been expressed that fires in long, fied as “pools” since they represent a surface fully narrow enclosures exhibit different burning behav- exposed to the fire. On the other hand, cribs burn ior than fires in other types of enclosures13 and, from within and feel very little of the surrounding hence, predictive methods that were developed heat of the fire. The heat flux of the fire will based on fires in compartments that are not long increase vaporization over the ambient level. This and narrow may not accurately predict burning depends on the fuel’s heat of gasification (typically behavior in long, narrow enclosures. Specifically, L = 0.5 to 1 kJ/g for liquids, 2 to 3 for non-charring these long, narrow compartments with a uniformly solids, and 5 to 10 for charring solids). distributed fuel load can exhibit non-uniform heat- Since the fuel volatilization rate is the heat trans- ing in ventilation-limited fires. To address this con- fer to the fuel divided by the heat of gasification of cern, the methods presented in this guide have been the fuel12 and woods tend to have higher heats of evaluated using data from fires in long, narrow gasification, wood cribs will tend to result in fires enclosures in addition to compartments in which the of longer duration than other fuels. In ventilation- ratio of length to width is nearly one. limited fires involving non-charring fuels, the rate of airflow into the enclosure will govern the heat THEORY release rate into the enclosure, and fuels that cannot burn inside the enclosure will burn outside once It would appear that geographical reasons explain they encounter fresh air. the proliferation of many models for fire resistance. Secondly, the primary fuel in many design or Most of the work on fire resistance took place before analysis situations is typically cellulosic in nature 1970, when communication and dissemination of (wood, paper, etc.). While many compartments con- research in fire was limited. This might explain the tain other fuels, the total mass of non-cellulosic existence of the different models. However, their fuels could be a small fraction of the mass of cellu- differences are superficial for the most part, clouded losic materials. Design or analysis situations in by notation or parameters that might appear as which the fuels are not predominantly cellulosic and different. For that reason, it was felt important to the burning is not expected to be ventilation limited develop a theoretical base for the models. So doing may require special attention. might appear to be establishing yet another model. Additionally, each of the methods presented in Indeed, the contrary is intended. The purpose of this this guide is subject to the following limitations: theoretical exposition is to present a rationale for the physics of the models and to show their simi- 1. The methods are only applicable to compart- larities and deficiencies. It is in this context that a ments with fuel uniformly distributed over their theoretical introduction is provided to the models interior. (Sparse distributions or concentrated fuel that exist in the literature. packets should be considered using the methods identified in the fire plumes section.) Theoretical Development 2. The methods presented in this guide are only The purpose of this theoretical development is to: applicable to compartments having vents in walls. (Ceiling and floor vents require a special 1. Present the governing equations formulation, as would underground compart- 2. Explain and justify typical approximations ments having only roof vents.)

5 3. Present the equations in dimensionless terms elements absorb a small amount of heat relative to to show heat loss into the wall or ceiling surfaces together a. Their generality with the energy loss out of the vents. These vents b. Independence of scale include the windows broken by the thermal stress c. Relationship to variables used in the of the impinging flames and heat. The model is established methods depicted in Figure 2. The conservation of mass and energy for the con- The common objective of all the models has been trol volume (CV), which follows, also applies. to predict the following: Mass: (Eq. 1) 1. Compartment gas temperature 2. Burning rate of the fire Energy: 3. Duration of the fire

The purpose of the studies considered has been to (Eq. 2) predict the thermal effects of fully developed building fires so that their impact on the structural mem- The Equation of State: (Eq. 3) bers could be assessed. Fully developed fires with considerable fuel will tend to produce a fairly uni- The volume, V, is constant. The pressure, p, is form temperature smoke layer that will descend to nearly constant and at the ambient condition for the floor. This will particularly occur for a large fire vents that are even very small, e.g., those in the and relatively small vents. The radiation effects of leakage category. Only for abrupt changes in the fire such a fire will further tend to cause uniform heat- will pressure pulses above or below ambient occur. ing of the contents. Consequently, the model for the The temperature slowly varies during the fully fully developed fire has been an enclosure with uni- developed fire state. As a consequence, steady-state form smoke or gas properties. The bounding wall conditions can be justified. surfaces are also considered uniform. The structural

FIGURE 2. Model for the Fully Developed Fire

6 (Eq. 4) The conductances, hi, can be computed as follows from standard heat transfer estimates: The mass flow rate from the vent (m• ) equals the • • air supply (mo) and the fuel gases produced (mF ). Convection The energy equation can be written as Convection can be estimated from natural convection.14 (Eq. 5a)

The heat losses ( q• ) consist of the heat transfer into the boundary surfaces and the radiation loss out of the vent. Some simplification can be made since 2 , so that the second term on the It gives hc of about 10 W/m K. Under some right may be neglected. other flow conditions, it is possible hc might be as high as 30 W/m2K. (Eq. 5b) Conduction Wall Heat Transfer Conduction might be represented as steady or The heat transfer into the boundary surface is unsteady. The latter is more likely. Only a finite by convection and radiation from the enclosure, difference numerical solution can give exact results. then conduction through the walls. The boundary Most often the following approximate analysis is element will be represented as a uniform material used for the unsteady case assuming a semi-infinite of properties: wall under a constant heat flux. The exact solution for constant heat flux gives: • Thickness, δ • Thermal conductivity, k (Eq. 6a) • Specific heat, c • Density, ρ or

(Eq. 6b) It conducts to a sink at To. The heat transfer can be represented as an equivalent electric circuit as shown in Figure 3. This result for hk can be used as an approximation for variable heat flux. For steady conduction, the exact result is

(Eq. 6c)

The steady-state result would be considered to hold for14

FIGURE 3. Wall Heat Transfer

7 Some estimations for common TABLE 1. Estimates of Conduction for Common Materials materials are given in Table 1. For Approximate Properties a wall 6" thick, δ ≈ 0.15 m, then Concrete/Brick Gypsum Mineral Wool

k (W/mK) 1 0.5 0.05 kρc (W2s/m4K2)106 105 103 Hence, most boundaries might be k/ρc (m2/s) 5 × 10-7 4 × 10-7 5 × 10-7 approximated as thermally thick since most fires would have a duration of The absorption coefficient κ, can range from less than 3 hours. about 0.4 to 1.2 m-1 for typical flames (see The thermally thick case will predominate under Karlsson and Quintiere,15 p. 167). Experimental most fire and construction conditions: fires might use H ≈ 1 m, while buildings generally have H ≈ 3 m. For the smoke conditions in fully developed fires, κ =1 m-1 is reasonable in the least. Hence, ε ranges from about 0.6 for a small experi- 3 6 Based on kρc of 10 to 10 , it is estimated mental enclosure to 0.95 for realistic fires. It follows that: 2 t (min) hk (W/m k) 10 0.8-26 (Eq. 9) 30 0.3-10 where ε is generally nearly 1. It can be estimated 120 0.2-5 ε for = 1, and T = Tw, that

h = 104 – 725 W/m2K Radiation r Radiation heat transfer can be derived from the for T = 500 to 1200°C. method presented in Karlsson and Quintiere15 From the circuit in Figure 3, the equivalent con- (p. 170) for enclosures. It can be shown as14 ductance, h, allows

(Eq. 7) (Eq. 10a)

Where: Where: ε = Emissivity of the enclosure gas (flames (Eq. 10b) and smoke) ε w = Emissivity of the boundary surface It follows from the estimates that h ≈ hk, which implies T ≈ T for fully developed fires. This result Since the boundary surface will become soot w applies to structural elements that are insulated, covered in a fully developed fire, ε = 1. w including unprotected concrete elements. Hence, The gas emissivity can be represented as predicting the fire temperature provides a simple (Eq. 8) boundary condition for the corresponding computation for the structural element. Its surface tempera- Where: ture can be taken as the fire temperature. H =Acharacteristic dimension of the enclosure, This result is very important and helps to explain its height why most of the methods only present the fire temperature without any detailed consideration of the

8 heat transfer in representing the fully developed The Fire—Firepower and Burning Rate fire. From the estimates made here, the gas phase To complete the energy equation in order to solve radiation and convection heat transfer have negligi- for the temperature, the fire must be described. The ble thermal resistance compared to conduction into heat of the flames and smoke causes the fuel to the boundary. As a consequence, the fire tempera- • vaporize, supplying a mass flow rate, m . While all ture is approximately the surface temperature. This F the fuel may eventually burn, it may not necessarily boundary condition is “conservative” in that it gives burn completely in the compartment. This depends the maximum possible heat transfer from the fire. on the air supply rate. Either all the fuel is burned, or all the oxygen in the incoming air is burned. Radiation Loss from the Vent What burns inside gives the firepower within From Karlson and Quintiere15 (p.170), an analy- the enclosure. ε Thus, sis of an enclosure with blackbody surfaces ( w = 1) gives the radiation heat transfer rate out of the vent of area Ao as (Eq. 15)

The equivalence ratio, φ, determines if the com- (Eq. 11) bustion is fuel-lean (<1), or fuel-rich (>1).

ε Since is also near 1 and Tw ≈ T, it follows that (Eq. 16)

(Eq. 12) Where: s = Stoichiometric air-to-fuel ratio This blackbody behavior for the vents has been ∆Hc = Heat of combustion (chemical heats of verified.16 combustion according to Tewarson17) The total heat losses can be written as ∆Hair = Heat of combustion per unit mass of air ≈ 3kJ/g, which holds for most fuels

Note: (Eq. 13) (Eq. 17)

Vent Mass Flow Rate Air • The mass supply rate of the fuel, m F , depends on The mass flow rate of air can be approximated the fuel properties, its configuration, and the heat for small ventilation as (Karlsson and Quintiere,15 transfer. Most studies have been done using wood p.100) cribs. These are composed of ordered layers of square sticks of side b. Gross18 and Heskestad19 have developed correlations to describe how they or in general burn. For cribs that have sufficient air supply, their (Eq. 14) burning rate per unit area is found as

ρ 3 where ko = 0.145 (for 0 = 1.1 kg/m ). This result is prevalent in all analyses, and the parameter ( ) (Eq. 18) shows up in many experimental correlations. where C depends on the wood (approximately 1 mg/cm1.5s).

9 For a range of crib experiments in compartments, radiant heating. The radiation geometric view factor Harmathy20 gives F is, in the limits, 0 and 1, respectively, for crib-like and pool-like fuels. This expression is the governing equation for the mass loss rate. Together with the while Tewarson17 gives 11 g/m2s. These values give energy equation, there are two equations and two • an approximation for wood, but it should be noted unknowns: T and mF that, in general, it depends on the stick size. Real fuels are not wood cribs, although cribs Development of a Solution and might approximate structural wood furniture such as Dimensionless Groups desks and chairs. Other furnishings are mostly composed of large flat surfaces that would more easily The equations will be examined to achieve vaporize fuel in a fire. These flat surfaces might be insight into the form of a solution. They are not classified as “pools” since they represent a surface difficult to solve by iteration using a computer. fully exposed to the fire. On the other hand, cribs However, analytical approximations can be of burn from within and feel very little of the sur- value. A dimensionless form of the equations will rounding heat of the fire. be presented to demonstrate the important variables. In general, the mass flux of fuel produced in a These variables will be used to explain the theoreti- fire can be represented as cal and experimental results presented in this guide in terms of the methods available in the literature. (Eq. 19) Compartment Temperature The fire “free”-burning flux is how the fuel Substituting for the heat loss rate from would burn in ambient air. In a fire, this would be Equation 13 into the energy equation (5b) yields: modified by the oxygen concentration the fuel expe- riences. Also, the heat flux of the fire will increase vaporization over the ambient level. This depends on the fuel’s heat of gasification (typically (Eq. 21a) L = 0.5 to 1 kJ/g for liquids, 2 to 3 for non-charring solids, and 5 to 10 for charring solids). It is known Dividing the numerator and denominator by that large fires, burning in air, reach an asymptotic and representing burning flux as their flames reach an emissivity of 1. Such values are tabulated (see Tewarson17 or Babrauskas21). Since the radiant heat transfer domi- nates, the fuel mass loss rate in typical building gives compartments, where the fire is large, can be approximated as

(Eq. 21b) (Eq. 20)

Here, it is assumed that for φ < 1, the “fuel- By substituting for , the controlled” fire, the fire burns as a large fire with following dimensionless groups emerge. The sufficient air. Such “large” fires need only achieve a dimensionless variables are presented in terms of a burning diameter of greater than about 1 to 2 m. In frequently used Q* factor. the “ventilation-controlled” fire, φ > 1, the fuel mass loss rate is composed of all that burns inside with the available airflow plus what is vaporized by (Eq. 22)

10 The correlation by McCaffrey, Quintiere, and (Eq. 23) Harkleroad (MQH)22 is

(Eq. 27)

(Eq. 24) This result has only been developed from data where φ < 1. But Tanaka, Sato, and Wakamatsu23 have applied it for φ > 1 with some success.

(Eq. 25) Maximum Possible Temperature Examine the limit of the stoichiometric adiabatic state that would yield the maximum temperature. Here * * Qw = Qr = 0 (Eq. 26a) And from Equations 15 and 22 or .

With φ = 1, the adiabatic stoichiometric fire (Eq. 26b) temperature is

(Eq. 28) 700 ) (K)

0 600 T

– The experimental results for an adia-

T batic turbulent fire plume24 suggest 500 (T– To)ad ≈ 1500°C at most. This might represent as well the maximum possible 400 temperatures attainable in a compartment fire. The plume adibaticity occurs 300 due to smoke preventing the radiation loss. This occurs as the diameter of the fire becomes large. Large compartment 200 fires can act similarly as the floor area becomes large, and only smoke is seen 100 from the windows, particularly in an mperature Rise Under Ceiling ( φ

Te over-ventilated state, < 1. 0 0 0.3 0.6 0.9 1.2 1.5 1.8 N M X1 X2

FIGURE 4. MQH Correlation for Fuel-Controlled Fires. ≡ = X1 Q*, X2 Qw*

11 Burning Rate The form of Equation 26 suggests a corresponding dimensionless form for Equation 20:

(Eq. 29)

The last term suggests another dimensionless Therefore, all terms can be significant under group governing compartment feedback. some circumstances. Define General Form of Correlations The dimensionless variables developed here can (Eq. 30) be used to explain the methods presented in this guide. From Equations 26 and 29, the approximate following solutions, in general, can be derived: Significant Relationships Now examine the values of the dimensionless variables. Estimating values are as follows: For typical building compartments, the geometric (Eq. 31a) compartment parameter is ≈ 1 m–1/2 for full windows, ≈ 10 m–1/2 for typical windows, and ≈ 100 m–1/2 for very small vents. Since the fuel surface area is similar and related to the room area, has a similar range.

The burning rate term can be estimated as ≈ 10-3 – 1 for wood and ≈ 10-2 – 10 for liquid fuels from very large to very small vents, respectively. (Eq. 31b) The heating terms can be estimated as follows:

* × -5 Qw ≈3 10 – 90 for large to small vents, from estimates of hk

* × -4 × -4 Qr ≈1 10 – 2 10 for Ho ≈ 3 m

* × -4 × QF ≈ 1.3 10 for wood,

× -3 × 1.3 10 for liquid fuels (Eq. 31c)

12 A functional form of these equations is given The temperature, from Equation 27, can be from the theoretical approximation given here, but written as complete analytical solutions cannot be determined. Only limiting analytical solutions are possible, but these still depend on empirical factors, e.g., , etc. Some limiting cases are as follows: (Eq. 34)

Large Ventilation Small Ventilation Large ventilation, Small ventilation,

In this case, ko is not a constant (Equation 14), From Equation 31b, it can be estimated for wood cribs and for large pool fires where the radiation but depends on due to the effect of feedback is small: temperature difference on the buoyancy velocity, i.e., and . (Eq. 35)

For the case of large vents (φ < 1), Equation 26a The radiation feedback is negligible for cribs can be rewritten as because of the stick blockage and for large pool fires because of obscuration by smoke. For small- scale pool fires in compartments, there can be a considerable enhancement in the burning rate due to radiation feedback. The corresponding temperature can be estimated This suggests that as follows, neglecting the vent radiation, since the vent is small.

(Eq. 32) (Eq. 36) This is consistent with the MQH correlation for φ < 1 given by Equation 27. But Q* depends on the airflow, so, by Equation 31c, The mass loss rate for large ventilation (φ < 1) is given directly by Equation 31a.

(Eq. 33a) or alternatively or

(Eq. 33b)

Both forms of are used in the experimental correlations; however, the ratio has not (Eq. 37) generally been included in their results. It should be recalled that, for well-ventilated wood cribs, For small-scale pool fires in compartments, the , where b is the stick thickness. effect of heat feedback from the compartment is large and cannot be neglected as above.

13 Summary This results in the following trends, as shown in Figure 6. The theory suggests that the correlations be of In the theoretical development, the dimensionless the following form: variables that should show up in the literature correlations have been identified. The dimensionless • Large ventilation, variables contain the scaling factors that allow for the extrapolation of results over geometric scales. In addition, the dimensionless groups exhibit the proper combination of other vari- (Eq. 38a) ables including time and material properties. The theoretical results give the following (Eq. 38b) functional behavior:

• Small ventilation,

These dimensionless variables are not usually represented in the literature correlations in the (Eq. 39a) same manner. They have equivalent surrogates. For example:

•, Maximum Gas Temperature, is usually given as T only. (Eq. 39b) •, Burning Rate/Vent Flow, is Usual forms of the correlations have been usually given as .

• Q*, Fire Power or heat release rate; usually only ventilation-limited fire states are considered, and, for wood and liquid pool fires. This would lead to consequently, this variable does not explicitly results as shown in Figure 5. show up; however, in general, A typical form for temperature is

From Equation 38a, it follows that

Note that in the latter case (φ > 1) Q* is constant. The former, or fuel- controlled, state contains the effect of fuel.

14 •,

Wall Heat Loss, is usually represented as a scaling factor for time that allows for the temperature to be represented over dimensionless time,

•,

Vent Radiation Loss, usually does not appear in the correlations since likely has a small variation over the range of data considered.

•, FIGURE 5. Approximate Theoretical Behavior for Fuel Burning Rate Enhanced Fuel Vaporiza-

tion; for wood cribs this 1200 term is small, but for other forms of fuel in the form Increases as of flat surfaces it can be fuel mass flux, 1000 heat of combustion, considerable. Compared to fuel area increase wood cribs, it will reduce the duration of the fire, Increases as heat loss to walls 800 making the wood crib decreases model conservative in design since it would give a longer duration. 600 Temperature °C

400

Fuel lean Fuel rich Well-ventilated Ventilation-limited 200 Φ < 1 Φ > 1

0 01020304050

1/2 –1/2 A/AoHo m

FIGURE 6. Approximate Theoretical Behavior of Compartment Temperature

15 METHODS FOR PREDICTING Where: FIRE EXPOSURES T =Temperature (°C) t* = tΓ (hours) Several methods are available for predicting t =Time (hours) temperatures and duration of fire exposure in a compartment. These methods are presented in an arbitrary order.

Eurocode Parametric Fire Exposure Method Where: The Eurocode 1, Part 2.2,6 provides three “stan- The opening factor has limits of dard” fire curves and a parametric fire exposure. The standard fire curves include the ISO 834 curve, an external fire curve, and a hydrocarbon fire curve; these standard curves are not addressed further in A = Area of vertical openings (m2) this guide. The parametric fire exposure in the Euro- o H = Height of vertical openings (m) code was originally developed by Wickstrom.25 o A =Total area of enclosures (walls, ceilings, Wickstrom stated25 that this method assumes that and floor including openings) (m2) the fire is ventilation controlled and all fuel burns b = (J/m2 s1/2 K) and has the limits within the compartment. 1000 ≤ b ≤ 2000 Wickstrom modified an approximation of the k = Thermal conductivity of enclosure lining ISO 834 standard fire curve by altering the time (W/m-K) scale based on the ventilation characteristics and ρ = Density of enclosure lining (kg/m3) enclosure thermal properties. The modified c = Specific heat of enclosure lining (J/kg-K) time scale compares the enclosure of interest to Magnusson and Thelandersson’s “type A” enclosure For enclosures with different layers of material, with an opening factor of 0.04 m1/2. Wickstrom b = is calculated as follows: found that the resulting curve approximated the ISO 834 standard fire curve. The Eurocode states that this parametric exposure b = (J/m2 s1/2 K) may be used for fire compartments up to 100 m2 only, without openings in the roof, and for a maximum compartment height of 4 m. The Eurocode Where: δ does not provide any basis for these limits. i = Thickness of layer i (m) The Eurocode provides the following tempera- ci = Specific heat of layer i (J/kg K) ture–time curve for a natural fire (also known as a ki = Thermal conductivity of layer i (W/m K) 2 1/2 parametric curve): bi = (J/m s K)

To account for different materials in walls, ceiling, and floor, b = should be calculated as follows:

Where:

Atj = Area of enclosure including openings with 2 the thermal property bj (m )

16 The temperature–time curves in the cooling Franssen26 noted two shortcomings of the phase are given by: Eurocode procedure for accounting for layers of different materials:

1. The Eurocode procedure does not distinguish which material is on the side exposed to a fire. 2. The contribution of each material to the b factor is weighted by thickness, so Where: the adjusted b factor for an enclosure with Tmax = Maximum temperature (°C) in the a nominal thickness of an insulating * * heating phase for t = td material over a much thicker, heavier material will be biased towards the b * td = (hours) factor of the thicker, heavier material.

with: Franssen therefore suggests the following qt,d = Design value of fuel load alternative method of accounting for layers of dif- density related to surface area A ferent materials: of the enclosure whereby qt,d = 2 qf,d Afloor/A (MJ/m ). The limits 1. If a heavy material is insulated by a lighter 2 50 ≤ qt,d ≤ 1000 (MJ/m ) should material, the b factor for the lighter material be observed. should be used. qf,d = Design value of the fuel load 2. If a light material is covered by a heavier material, density related to the surface area for example in a sandwich panel, then a limit 2 Afloor of the floor (MJ/m ). thickness should be calculated according to:

* By making simple substitutions, td can also be expressed as: where the subscript 1 indicates the properties of the material on the side exposed to the fire and t is the duration of the heating phase of the fire in seconds, which can be calculated as Where: E =Total energy content of the fuel in the compartment, expressed by δ δ If 1 > lim, then the b factor for the heavier material should be used; otherwise, Buchanan10 suggested that the temperatures in the Eurocode are often too low and that it would be more accurate to scale based on a reference of 1900 J/m2 s1/2 K. This would result in the following modified equation for Γ : Franssen observed26 that, as the ratio between the fuel load and the ventilation factor decreases, the Eurocode predicts unrealistically short burning durations. Therefore, Franssen suggests that if

17 is less than 20 minutes, then the following proce- Data Sources dure should be used: 1. Thermal properties: SFPE Handbook of Fire Protection Engineering27 or manufacturer’s data. 1. The opening factor should be set 2. Several surveys have been published of mass of combustible materials per unit area for different equal to , Γ should be set equal occupancies.28,29,30,31 Given that fire loading can vary significantly over the life of a building, uncertainty should be carefully considered. Heats of combustion are available in the SFPE * 32,33 to , and td should be Handbook of Fire Protection Engineering or other sources. To determine qf,d, sum the products of the heat of combustion and the total mass of each material and divide this sum by the set equal to , total floor surface area. Given the uncertainty that is expected in estimating the mass of materials, 40 MJ/kg is a reasonable estimate of where 0.33 is 20 minutes, expressed in hours. the heat of combustion of plastics and other hydrocarbon-based materials, and 15 MJ/kg is a 2. If > 0.04 m1/2 (calculated based reasonable estimate of the heat of combustion of on actual compartment geometry, not as wood and other cellulosic materials. 2 modified above) and qt,d < 75 MJ/m and b < 3. Building characteristics can be obtained from 1160 J/m2 s1/2 K, then Γ should be set equal to surveys of existing buildings or architectural plans of new buildings.

Validation and Limitations

where is calculated based on actual See Appendix B for compartment geometry. comparisons of predictions with test data. The Eurocode method, without modifications, 2 bounds all CIB temperature data for qt,d = 50 MJ/m Data Requirements 2 and most data for qt,d = 100 MJ/m . The Eurocode, 1. Enclosure thermal properties, k, ρ, and c. If the without modification, overpredicted the burning rate lining is not the same over the entire surface, the of all the CIB data and, hence, underpredicted the percentage of the enclosure area composed of burning duration. In Cardington tests #1, 2, 8, and 9 each material is required. If multiple layers of , the Eurocode, without material are present in the enclosure, the thickness of each layer is required. For thermally modifications, bounds average temperatures, but thick enclosure materials, it should be sufficient underpredicted burning duration. In tests #3, 4, 5, to account only for the innermost layer. and 6 , the Eurocode, 2. The fuel load density present in the enclosure, qf,d. without modifications, reasonably predicted the 3. The area and height of the enclosure opening(s), burning duration but underpredicted temperature. In

Ao and Ho. test #7, which was square in plan view, the Eurocode, 4. The interior surface total area of the enclosure, without modification, underpredicted temperature including the floor and openings, A. but predicted the burning duration; however, a faster decay was predicted than was observed.

18 Predictions for CIB data using the Buchanan Lie’s Parametric Method modification bound all temperature data, more Lie suggested that, if the objective is to develop a so that the Eurocode method without modifica- method of calculating fire resistance requirements, tion, for q = 50 MJ/m2 and q = 100 MJ/m2. t,d t,d then it is necessary only to find a fire temperature– In Cardington tests #1, 2, 8, and 9 time curve “whose effect, with reasonable proba- , Buchanan’s modi- bility, will not be exceeded during the use of the building.”34 Lie developed an expression based on fication bounds peak temperature and under- the series of temperature–time curves computed by predicts burning duration. In tests #3, 4, 5, and 6 Kawagoe and Sekine35 for ventilation-controlled , Buchanan’s modifi- fires, which he proposed could be used as an approximation for the most severe fire that is likely cation reasonably predicted average temperatures to occur in a particular compartment.36 and the burning duration; however, peak tempera- He describes the opening factor tures were underpredicted. In test #7, Buchanan’s A H modification underpredicted temperature but F = o o predicted the duration of peak burning; however, A Buchanan’s modification predicted a faster decay Where: 2 than was observed. Ao = Area of vertical openings (m ) The Franssen modification fell within the scatter Ho = Height of vertical openings (m) A =Total area of enclosures (walls, ceilings, of temperature data for values of between and floor including openings) (m2) 0 m–1/2 and approximately 15 m–1/2 for The rate of burning of the combustible materials q = 50 MJ/m2 and for values of t,d in the enclosures is given by: between 0 m–1/2 and approximately 20 m–1/2

2 for qt,d = 100 MJ/m . For values of Where: between 20 and 50 m–1/2, Franssen’s modification =Mass burning rate of fuel bounds all temperature data. Franssen’s modification reasonably predicts peak temperatures and Thus, if is the fuel load per unit area of the underpredicted the burning duration in Cardington surfaces bounding the enclosure, the duration of the fire, τ, is: tests #1, 2, 8, and 9 . In tests #3 and 4 , Franssen Where: τ reasonably predicts average temperatures and burn- = Duration of fire (hours) ing duration; however, Franssen’s modification predicts a faster decay than was observed in test #4 For given thermal properties of the material (where the fire load was 40 kg/m2). In tests #5 and 6 bounding the enclosure, the heat balance can be solved for the temperature as a function of the Franssen’s modification opening factor F. Besides depending on F, the tem- slightly underpredicted average temperatures. perature course is also a function of the thermal Franssen’s modification reasonably predicted burn- properties of the material bounding the enclosure. ing duration in tests #5 and 6. In test #7, Franssen’s Lie derived a series of temperature–time curves modification reasonably predicted burning duration for ventilation-controlled fires in two types of but underpredicted temperature data. enclosures: “dominantly heavy materials” and “dominantly light materials.”

19 He found these curves could be reasonably described by the expression

Where: T = Time in hours C = Constant taking into account influence of 3. Building characteristics can be obtained from the properties of the boundary material on surveys of existing buildings or architectural the temperature: plans of new buildings. C = 0 for heavy material with a density ρ 2 ≥ 1600kg/m Validation and Limitations C = 1 for light materials ρ < 1600kg/m2 See Appendix B for comparisons of predictions Lie states that the expression is valid for with test data. Lie’s method bounded almost all the CIB temperature data. Lie’s method generally overpredicted burning rate and underpredicted burning duration If t > (0.08/F) + 1 a value of t = (0.08/F) + 1 should be used. for . For If F > 0.15 a value of F = 0.15 should be used. Lie also derived an expression to define the tem- predictions using Lie’s method fell within the perature course in the decay period, over time: scatter of points. The data in the ventilation- controlled regime can be bounded by multiplying and dividing Lie’s burning with the condition T = 20 if T < 20°C. rate prediction by a factor of 1.8. In Cardington tests #1, 2, 8, and 9 Where: Tτ = Temperature at time τ (°C) , Lie’s method predicted or slightly underpredicted average temperatures and Data Requirements underpredicted peak temperatures. The burning 1. Enclosure density, ρ duration was underpredicted in these experiments.

2. The mass of fuel in the enclosure, mf In test #7 , Lie underpredicted 3. The area and height of the enclosure opening(s), temperature and duration. Lie’s method underpre- Ao and Ho 4. The interior surface total area of the enclosure, dicted temperatures in tests #3, 4, and 5 including the floor and openings, A ; however, predictions

Data Sources improved as increased. Lie’s method

1. Density: SFPE Handbook of Fire Protection reasonably predicted burning duration in these Engineering27 or manufacturer’s data. 2. Several surveys have been published of mass of experiments. In test #6, , combustible materials per unit area for different Lie’s method reasonably predicted both temperature 28,29,30,31 occupancies. Given that fire loading can and duration. vary significantly over the life of a building, uncertainty should be carefully considered.

20 Tanaka room of origin and the connected corridor and can be used for predicting the temperature of a Tanaka extended the equation for pre-flashover single fire room. In this case becomes room fire temperature developed by McCaffrey et al.22 to obtain equations for ventilation-controlled fire temperatures of the room of origin and the 37 corridor connected to the room. The temperature and substituting rise in a compartment can be predicted by the following equation according to McCaffrey et al. Tanaka’s method performs all calculations in Kelvin; the equation for temperature in degrees Celsius follows. where the effective heat transfer coefficient defined as

Tanaka uses Kawagoe and Sekine’s method of ρ Substituting hk and the values of g, c0 , 0, and T∞, predicting the mass burning rate as follows: the equation reduces to

Where: =Mass burning rate of fuel Where: Upon comparison of the results of the simple g = Gravity, 9.81 m/s2 equations to results of a more detailed computer c = Specific heat of air, 1.15 kJ/kg K 0 model, Tanaka refined the equations to improve ρ = Density of air, 1.2 kg/m3 0 accuracy. Tanaka defined the parameter as = Heat release rate (kW) T =Temperature (K) and the equations for T∞ = 300 K temperature of the fire room are 2 A0 = Area of opening (m ) H0 = Height of opening (m) A =Total surface area of room, excluding or opening (m2) t =Time (s) k = Thermal conductivity of enclosure lining (kW/m K) Where: ρ = Density of enclosure lining (kg/m3) c = Specific heat of enclosure lining (kJ/kg K)

Tanaka studied the effect of an opening between the corridor and the outdoors when the corridor was connected to the room of origin. His equations can be reduced where there is no opening between the and KF reduces to 1. can be simplified to

21 The equation for temperature must be re-dimen- The simple method overpredicts temperature and sionalized and converted to degrees Celsius in the reasonably predicts duration for test #7 same manner as before. , while the refined method

Data Requirements reasonably predicts both values. The simple method greatly overpredicts temperature, and the refined ρ 1. Enclosure thermal properties, k, , and c method reasonably predicts average temperature for 2. The height and area of the enclosure opening(s), tests #3, 4, and 5 , while Ao and Ho 3. The interior total surface area of the enclosure, both underpredict duration. For test #6, Tanaka’s including the floor, but excluding the simple method overpredicts temperature, and the opening(s), A refined method underpredicts temperature, yet both 4. The mass of fuel in the enclosure, mf reasonably predict duration. The quality of temperature predictions using Tanaka’s refined method Data Sources decreases as increases. 1. Thermal properties: SFPE Handbook of Fire Protection Engineering27 or manufacturer’s data. 2. Several surveys have been published of mass of Magnusson and Thelandersson combustible materials per unit area for different Parametric Curves occupancies.28,29,30,31 Given that fire loading can Magnusson and Thelandersson38 studied the vary significantly over the life of a building, variations in the development of energy, the effects uncertainty should be carefully considered. of air supply, and the resulting evolution of gases 3. Building characteristics can be obtained from with time in the course of a fire. They determined surveys of existing buildings or architectural the temperature of the combustion gases from wood plans of new buildings. fuel fires, in an enclosed space as a function of time, under different conditions. Validation and Limitations Magnusson and Thelandersson made adjustments See Appendix B for comparisons of predictions to Kawagoe’s work to accommodate the effect of with test data. a cooling phase since Kawagoe and Sekine’s work Both of Tanaka’s methods bounded all the CIB is more applicable to the flame phase process of temperature data; however, the refined method more fire development. closely approximates the values. Both Tanaka’s They used the equation of energy balance derived simple and refined methods use the same correlation by Kawagoe and Sekine35: for burning rate. Tanaka’s methods overpredicted burning rate and underpredicted burning duration Where: for . For = Rate of heat energy released per unit time Tanaka’s methods fell within the scatter of points. during combustion Burning rate for those tests in the ventilation- = Rate of heat energy withdrawn per unit time from the enclosed space owing to controlled regime can be replacement of hot gases by cold air bounded by multiplying Tanaka’s prediction by a = Rate of heat energy withdrawn per unit factor of 1.6 and dividing by a factor of 1.9. time from enclosed space through the Tanaka’s simple and refined methods overpredict wall floor or ceiling and roof structures temperatures but underpredict duration for Carding- = Rate of heat energy withdrawn per unit time from the enclosed space by radiation ton tests #1, 2, 8, and 9 . through the openings in the enclosed space

22 = Rate of the heat energy stored per unit For practical design, they suggest that the designer time in the gas volume that is contained choose the type of enclosed space most similar to in the enclosed space one of the eight types with respect to the thermal properties of the bounding structure. The designer Magnusson and Thelandersson also use the should then determine the opening factor and the opening factor, fuel load for his/her case, and finally interpolate linearly, if necessary. Alternatively, the designer can choose a curve Where: that is determined without interpolation so as to be 2 Ao = Area of opening (m ) on the safe side; the designer chooses the next Ho = Height of opening (m) higher value of opening factor and fuel load. A =Total surface area of room, excluding opening (m2)

Magnusson and Thelandersson evaluated eight specific types of enclosures and developed temperature–time curves for each, assuming wood fuel. The opening factor and the fuel load were varied for each of the eight types of enclosures, and temperature as a function of time was presented in both graphic and tabular formats. Figure 8 shows examples of temperature–time curves developed by Magnusson and Thelandersson. FIGURE 7. Schematic Illustration of the Heat Balance Equation Terms38

FIGURE 8. Examples of Temperature–Time Curves

23 Data Requirements Magnusson and Thelandersson’s predictions fell within the scatter of points. Those tests in the 1. Construction materials of the enclosure 2. The fuel load density (related to the surface area ventilation-controlled regime of the enclosure), q can be bounded by multiplying Magnusson and 3. The area and height of the enclosure opening(s), Thelandersson’s prediction by a factor of 1.3 A and H o o and dividing by a factor of 2.3. 4. The interior surface total area of the enclosure, Magnusson and Thelandersson’s method including the floor and openings, A predicts peak temperatures, but underpredicts duration, for Cardington tests #1, 2, 8, and 9 Data Sources . Magnusson and 1. Several surveys have been published of mass of combustible materials per unit area for different Thelandersson reasonably predict average tempera- occupancies.28,29,30,31 Given that fire loading can tures and duration for Cardington tests #3 and 4 vary significantly over the life of a building, . For test #5 , uncertainty should be carefully considered. Heats of combustion are available in the SFPE Magnusson and Thelandersson reasonably predict Handbook of Fire Protection Engineering27,33 duration but slightly underpredict temperature. In and other sources. (Note that values expressed in Cardington Test #7 , which was MJ/kg must be converted to Mcal/kg by multiplying by 0.239.) To determine q, sum the square in plan view, predictions made using products of the heat of combustion and the Magnusson and Thelandersson’s method almost total mass of each material and divide this sum coincided with the data. by the total enclosure surface area. Given the uncertainty that is expected in estimating the Harmathy mass of materials, 40 MJ/kg (10 Mcal/kg) is a Harmathy published a method for predicting reasonable estimate of the heat of combustion of burning rates and heat fluxes in compartment fires plastics and other hydrocarbon-based materials, with cellulosic fuels.39,40 Harmathy’s method is and 15 MJ/kg (4 Mcal/kg) is a reasonable based on theory, with a number of simplifications estimate of the heat of combustion of wood and comparisons of data to define constants. The and other cellulosic materials. methods that Harmathy presented are applicable to 2. Building characteristics can be obtained from fully developed fires in compartments that are surveys of existing buildings or architectural ventilation limited or fuel bed controlled. plans of new buildings. Harmathy developed a method for calculating the burning rate as follows: Validation and Limitations See Appendix B for comparisons of predictions with test data. For values of for which Magnusson and Thelandersson provide predictions, Magnusson and Where: Thelandersson’s predictions bounded the tempera- =Mass burning rate of fuel (kg/s) ture data from the CIB tests. Magnusson and ρ = Density of air (kg/m3) Thelandersson’s predictions overpredicted burning 0 g = Gravitational constant (9.81 m/s2) rate and underpredicted burning duration for 2 Ao = Area of ventilation opening (m ) . However, for Ho = Height of ventilation opening (m) 2 Af = Surface area of fuel (m )

24 Harmathy notes that a “critical regime” exists To apply the temperature for heat flux, it is where the burning rate is poorly predicted using the necessary to determine T. To do this, it is first above equations. This regime is the range necessary to determine the surface temperature of boundary elements in the compartment. Harmathy recommends the following equation to determine the surface temperature of Harmathy established the duration of the fully boundary elements: developed burning period as the time that the combustible mass remaining in the compartment is 80% or more of the initial mass. Using this definition, Harmathy established the following expressions for the duration of the fully developed fire exposure: Where: Tw = Surface temperature of boundary elements (K) κ = k/ρc k = Thermal conductivity of enclosure lining (W/m-K) ρ = Density of enclosure lining (kg/m3) c = Specific heat of enclosure lining (J/kg-K) Where: t =Time (s) τ =Time of primary (fully developed) burning (s) Harmathy states that, where boundary materials Harmathy provides a method of computing the are not homogeneous, a weighted average can be effective heat flux from the compartment fire to used. Also, Harmathy suggests that, where lining objects within the compartment as follows: materials are layered, the properties of the inner layer may be used. Where:

Where: σ = Stefan-Boltzmann constant, 5.67 x 10–8 W/m2 T4 η = Factor (-) (0.9) τ = Burning duration (s)

This results in two equations and two unknowns. Harmathy suggests selecting a value for T and inserting it into the equation for determining the effective heat flux. The calculated value for can then be substituted into the equation for determining T, which can be substituted back into the equation for determining . This process of iteration can be repeated until the changes in calculated values are small.

25 Decay Due to the iterative nature of Harmathy’s method, it was not possible to compare predictions to the CIB Harmathy suggests that during the decay period temperature data. For ventilation-limited fires, pre- the temperature can be calculated as follows: dictions made using Harmathy’s method fell within the scatter of the test points. The burning rate data can be bounded by multiplying and dividing predictions made using Harmathy’s method by a factor of 1.8. In the CIB tests, for fuel-controlled fires, Data Requirements fell within a range of approximately 1. Enclosure thermal properties, k, ρ and c ρ 2. The density and specific heat of air, 0 and c0 0.003 to 0.012 . Since occurs 3. The total mass of fuel, mf 4. The total free surface area of the fuel, Af in the denominator of both terms, ranged from 5. The area and height of the enclosure opening(s), approximately 0.003 to 0.012 A. In the CIB tests, A and H o o the average value of AF /A was approximately 6. The interior surface total area of the enclosure, 0.75. Substituting, ranged from approximately 2 including the floor but not including openings, A, 0.002 to 0.009 (kg/m s) AF. Therefore, multiplying and the height of the interior of the enclosure, H Harmathy’s burning rate prediction for fuel-controlled 7. Heat of combustion of the volatiles and char, fires by 1.5 and dividing it by 2.8 bounds most of ∆ ∆ Hv and Hc the data. Harmathy’s method underpredicted temperature Data Sources and duration in Cardington tests #1, 2, 8, and 9 1. Thermal properties: SFPE Handbook of Fire , and in tests #3, 4, and 5 Protection Engineering27 or manufacturer’s data. 2. Density and specific heat of air: 1.2 kg/m3 and overpredicted tempera- 1150 J/kg-K, respectively. tures but underpredicted duration. Harmathy 3. The surface area-to-mass ratio of the fuel typi- reasonably predicted duration in test #6 cally varies between 0.1 and 0.4 m2/kg for larger wood cribs and conventional furniture, and more but overpredicted temperature. often varies between 0.12 and 0.18 m2/kg.40 4. For wood products, the heat of combustion of In test #7 , which was square in × 6 volatiles can be assumed to be 16.7 10 J/kg, plan view, Harmathy’s method predicted duration and the heat of combustion of char can be taken well but overpredicted temperature. as 33.4 × 106 J/kg.39 5. Several surveys have been published of mass of combustible materials per unit area for different Babrauskas occupancies.28,29,30,31 Given that fire loading can The software program COMPF was completed vary significantly over the life of a building, and released to the public in 1975.7 The documenta- uncertainty should be carefully considered. tion of the program comprised a user’s guide and a 6. Building characteristics can be obtained from complete source code listing of the program. A surveys of existing buildings or architectural comprehensive presentation of the theory was then plans of new buildings. presented as part of Babrauskas’ Ph.D. dissertation.41 The portions of the dissertation pertinent to Validation and Limitations COMPF theory were subsequently made available as a pair of journal articles.42,43 See Appendix B for comparisons of predictions with test data.

26 The original COMPF program treated only wood Where: 2 crib fuels, or else arbitrary fuels for which burning Ao = Area of ventilation opening (m ) rate data were known and could be inputted. A Ho = Height of ventilation opening (m) second version, COMPF2,44 allowed treatment of =Mass burning rate of fuel (kg/s) liquid and thermoplastic pools. =Mass burning rate of fuel at During the development of COMPF, it was real- stoichiometry (kg/s) ized that not all the input data that might be desired φ = Equivalence ratio (-) would necessarily be available to the designer. s = Ratio such that Thus, the idea of “pessimization” was introduced. 1 kg fuel + s kg air = (1 + s) kg products ∆ In addition to running in a purely deterministic Hc = Heat of combustion (MJ/kg) mode, two other modes of computation were avail- σ = Stefan-Boltzmann constant able. In one case, the fuel mass loss rate would be (5.67 × 10–11 kW/m–2-K–4) computed as usual, but window ventilation would not be set to the maximum open area. Instead, the For pool fires, instantaneous open area was computed by the program to always be a value that would lead to the highest room temperature (up to the maximum full- opening size). In a second pessimization mode, the window ventilation would have a fixed value, but the fuel mass loss rate would be instantaneously Where: adjusted to give the highest room temperature. T = Fuel boiling point (K) Babrauskas used COMPF2 to create a series of b A = Surface area of fuel (m2) closed-form algebraic equations that can be used to f ∆H = Heat of vaporization of liquid (kJ/kg) estimate temperatures resulting from fully devel- p oped fires. According to Babrauskas, estimations Additionally, the heat release rate may be used in made using the closed-form equations are accurate place of the mass loss rate according to the follow- to within 3% to 5% of COMPF2 predictions, typi- ing equation: cally closer to 3%.45 The general equation follows:

Where: Where: T =Temperature in compartment (°C) = Heat release rate (kW)

To = Ambient temperature (°C) θ T* = Constant = 1452°C The second variable, 2, accounts for wall steady-state losses and is determined using the θ The first variable, 1, known as the burning following equation: rate stoichiometry, is found for two separate regimes using:

Where: A = Interior surface area of the enclosure, excluding the floor and openings δ = Thickness of wall surface (m) k = Thermal conductivity of enclosure lining (W/m-K)

27 θ Transient wall losses are incorporated into 3 Data Sources as follows: 1. For ventilation-controlled fires, the mass pyrolysis rate of fuel can be calculated from .44 For fuel-controlled fires, 39 Harmathy suggests , where Af is the free surface area of the fuel. The surface area- Where: to-mass ratio of the fuel typically varies between t =Time (hours) 0.1 and 0.4 m2/kg for larger wood cribs and c = Specific heat of enclosure lining (J/kg-K) conventional furniture, and more often varies ρ = Density of enclosure lining (kg/m3) between 0.12 and 0.18 m2/kg.40 2. For hydrocarbon-based fuels, s can be calculated If only steady-state temperatures need to be θ as follows: evaluated, 3 = 1. θ ep051 03 The variable 4 accounts for the effect that the height of a vent in relation to the total vent size can have on a compartment’s radiative losses and is given as follows: where

θ The final variable, 5, describes the effect of and . combustion efficiency on the compartment temperature. This variable takes into account the fact that Babrauskas46 suggests that for wood fuels the gases in the compartment may not be complete- s = 5.7. Harmathy39 notes that a typical wood ly mixed, and is found using: would have the chemical formula

CH1.455O0.645•0.233H2O, which would result in a value of s of 6.0. Where: 3. Several surveys have been published of mass of bp = Maximum combustion efficiency combustible materials per unit area for different (ranges from 0.5 to 0.9) occupancies.28,29,30,31 Given that fire loading can vary significantly over the life of a building, Data Requirements uncertainty should be carefully considered. 4. Properties of liquid fuels: SFPE Handbook of 1. Mass pyrolysis rate of fuel, , or heat release Fire Protection Engineering.32 rate, 5. Thermal properties: SFPE Handbook of Fire 2. The ratio s where 1 kg fuel + s kg air = (1+s) kg Protection Engineering27 or manufacturer’s data. products or the chemical formula of the fuel 6. Building characteristics can be obtained from 3. The area and height of the enclosure opening(s), surveys of existing buildings or architectural A and H o o plans of new buildings. 4. The interior surface total area of the enclosure, A, 7. For design purposes, a value of 0.9 should be not including the floor or openings assumed for b since this would result in the 5. For liquid fuels, the heat of vaporization of the p most conservative prediction of T. θ is only liquid, ∆H , the fuel boiling point, T , and the 5 p b relevant if the theoretical heat of combustion is pool area, A f used. If an effective heat of combustion is used, 6. Enclosure thermal properties, k, ρ, and c, and the e.g., “chemical” heats of combustion from δ thickness of the enclosure, 33 θ Tewarson, 5 = 1.0. 7. The combustion efficiency, bp

28 Validation and Limitations The shape of the curve is determined using the following equation, and an appropriate value for the See Appendix B for comparisons of predictions shape constant, δ. The recommended values for the with test data. shape constant are 0.5 for the ascending phase and Predictions using Babrauskas’ method bounded 1.0 for the decay phase. These values produce a the average temperatures measured in the CIB tests curve that encompasses a majority of the experimen- for ventilation-controlled fires but underpredicted tal data. It is reported, however, that values for the average temperatures for fuel-controlled fires. For shape factor of 0.8 for the ascending phase and 1.6 ventilation-controlled fires, Babrauskas’ burning for the descending phase provided a best-fit curve to rate prediction falls in the scatter of points. The the data.47 Both curves are shown in Figure 9. burning rate data can be bounded by multiplying Babrauskas’ prediction by a factor of 1.3 and by dividing by a factor of 2.3. Babrauskas’ method reasonably predicted peak temperatures but underpredicted burning duration in Where: all of the Cardington tests; however, predictions of T =Temperature in compartment (°C) burning rate improved as increased. To = Ambient temperature (°C) Tgm = Maximum temperature in compartment (°C) t =Time (min) Ma and Mäkeläinen tm =Time corresponding to maximum gas Ma and Mäkeläinen developed a parametric tem- temperature (min) perature–time curve for compartments that are small t = or medium in size (floor area < 100 m2). The method m was developed for use mainly with cellulosic fires. mf = Mass of fuel (kg) Their aims were to develop a simple calculation pro- =Mass burning rate of fuel (kg/min) cedure that would reasonably estimate the tempera- δ = Appropriate shape constant of the ture, with time, of a fully developed compartment fire. temperature–time curve discussed above Ma and Mäkeläinen noted that fires generally only impact the structures during the fully developed and decay stages. They developed a general shape 1.2 function to define the tempera-

) δ = 0.5, 1.0 ture history of a compartment 1.0

gm δ = 0.8, 1.6

fire that is a function of fuel /T g loading, ventilation conditions, 0.8 and geometry and material properties of the compartment. 0.6 The general shape function was developed by non-dimen- 0.4 sionalizing temperature–time data from 25 different data sets 0.2 and was based on the maxi- Temperature Ratio (T mum gas temperature, Tgm, 0.0 and the time to reach the maxi- 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 mum temperature, t . The m Time Ratio (t / t m) non-dimensionalized data col- lapses to the general shape FIGURE 9. Non-Dimensionalized Temperature–Time Curves shown in Figure 9. Developed by Ma and Mäkeläinen47

29 For ventilation-controlled fires, Ma and Mäkeläinen use Law’s correlation to describe the = Ratio of floor area to the total duration of the fully developed stage: compartment surface area 2 m"f =Mass of fuel per unit area (kg/m ) ρ 3 0 = Density of air (kg/m )

The shape function is based on 25 experimental data sets whose key parameters, fuel load density, ventilation factor, thermal boundary properties, and room dimensions varied between experimental Where: studies. The ranges for each of these parameters are A = Surface area of interior of enclosure (m2) 2 listed in Table 2. Ao = Area of ventilation opening (m ) Ho = Height of ventilation opening (m) D = Depth of compartment (m) TABLE 2. Range of Values for Key Parameters from the 25 Data Sets Used to Develop the W =Width of wall containing ventilation Shape Function opening (m) Property Range Units For fuel-controlled fires, Ma and Mäkeläinen 2 Fuel load density, m"f 10 – 40 kg/m use Harmathy’s correlation for the burning rate of Ventilation Factor, 5 – 16 m5/2 fuel-controlled fires:

555 – 1800 J/m2 s1/2 K Compartment floor < 100 m2 Where: area, A 2 floor Af = Surface area of fuel (m ) Maximum height, H < 4.5 m

For furniture, the value for Af /mf is generally Shape of 0.5 – 2.0 between 0.1 and 0.4 m2/kg; however, the most compartment, (W/D) common value is between 0.12 and 0.18 m2/kg, and 0.131 represents the value obtained from a series of Japanese tests.47 The maximum gas tem- Data Requirements perature is determined using 1. Ratio of floor area to total surface area

2. The mass of fuel per unit area, m"f 3. The area and height of the enclosure opening(s),

Ao and Ho with the maximum fire temperature in the critical 4. The interior surface total area of the enclosure, region, Tgmcr, determined by including the floor and openings, A, and the width, W, and depth, D, of the enclosure 5. The surface area-to-mass ratio of the fuel, A m η f / f and the value of cr determined using Data Sources 1. The surface area-to-mass ratio of the fuel typically varies between 0.1 and 0.4 m2/kg for larger Where: wood cribs and conventional furniture, and more 2 2 40 Af = Surface area of fuel (m ) often varies between 0.12 and 0.18 m /kg. g = Gravitational constant (9.81 m/s2)

30 2. Several surveys have been published of mass of the test data. See the conclusions regarding combustible materials per unit area for different Harmathy’s and Law’s methods for an evaluation occupancies.28,29,30,31 Given that fire loading of burning rate predictions. can vary significantly over the life of a building, uncertainty should be carefully considered. CIB 3. Building characteristics can be obtained from surveys of existing buildings or architectural In 1958, under the auspices of CIB W014, labora- plans of new buildings. tories from several countries agreed to investigate the factors that influence the development of enclosure fires.48 Compartments with dimension ratios of 211, Validation and Limitations 121, 221, and 441 (where the first number denotes See Appendix B for comparisons of predictions compartment width, the second number denotes com- with test data. partment depth, and the last number denotes com- Predictions made using Ma and Mäkeläinken’s partment height) with length scales of 0.5 m, 1.0 m, method’s maximum temperature predictions bounded and 1.5 m were analyzed. A total of 321 experiments the average temperatures measured in the CIB tests were conducted in still air conditions. The fuel load- for ventilation-limited fires but underpredicted aver- ing (m"f ) in the compartments ranged from 10 to age temperatures for fuel-limited fires. Given that 40 kg/m2 of wood cribs with stick spacing to stick maximum temperature predictions using Ma and width ratios of 1/3, 1, and 3. Test data was modified Mäkeläinken’s method were compared to the CIB through statistical analysis to account for systematic data, which represented the average temperatures differences between test laboratories. measured during the fully developed stage, and Average temperature and normalized burning rate predictions of average temperature would be lower were presented as a function of in graphical than average temperatures, Ma and Mäkeläinken’s method would underpredict much of the CIB form (A was defined to exclude the area of the temperature data. ventilation opening and the floor area). Separate Ma and Mäkeläinken’s method reasonably graphs were presented for cribs with 20 mm thick predicted average temperatures and duration wood sticks spaced 20 mm apart, and for cribs with in Cardington tests #1, 2, 8, and 9 20 mm wide sticks spaced 60 mm apart, or with 10 mm wide sticks spaced 30 mm apart. Because ; however, as the cribs with 20 mm thick wood sticks spaced increased, predictions increasingly deviated from 20 mm apart resulted in higher compartment temperatures and lower normalized burning rates (and, hence, 1200 longer predicted burning dura- CIB Data tions), these graphs are recom- 1000 CIB Curve mended for design analysis 800 and are presented here. Figure 10 shows the average 600 compartment temperature dur- 400 ing the fully developed burning stage, where “fully developed Temperature (°C) 200 burning” was defined as the 0 period where the mass of fuel 01020304050was between 80% and 30% of 1/2 –1/2 A/AoHo (m ) the original, unburned fuel mass. The line represents a FIGURE 10. Average Temperature During Fully Developed Burning best-fit through the data.

31 Figure 11 shows the burning rate, , during ) 0.18 5/2 the fully developed burn- 0.16 ing stage, normalized by 0.14 the ventilation factor 0.12 and the square (kg/s – m root of the ratio of com- 1/2 0.1 partment depth to width 0.08 D/W)

( 121 (where the width is the 0.06

1/2 221 dimension of the wall o

H 0.04 211 containing the ventilation o 441 /A

f 0.02 Curve Fit opening). . m 0 To apply these graphs 0102030405060 in a design context, A/A H 1/2 (m–1/2) first calculate the factor o o and use FIGURE 11. Normalized Burning Rate During Fully Developed Burning Figure 10 to determine the average gas temperature. Then use Figure 11 to determine the normalized Validation and Limitations burning rate. This normalized burning rate can be See Appendix B for comparisons of predictions re-dimensionalized by multiplying by the ventila- with test data. tion factor and dividing by the square root The averaged Cardington temperature data falls of the ratio of compartment width to depth. The in the same range as the CIB temperature data for duration of burning can be determined by dividing values of less than 30. However, once the the total mass of fuel, mf , by the burning rate. opening factor exceeds 30 m–1/2 the CIB tempera- Data Requirements ture graph underpredicts temperature, and the CIB data has much lower values than the Cardington data. 1. The total mass of fuel, mf 2. The area and height of the enclosure opening(s), As a curve fit through data, the CIB temperature graph reasonably predicted the aggregate of all CIB Ao and Ho 3. The interior surface total area of the enclosure, temperature and burning rate data but underpredicted excluding the floor and openings, A, and the some experiments and overpredicted others. width, W, and depth, D, of the enclosure Using the CIB graphs resulted in reasonable predictions of average temperature and duration in Cardington tests #1, 2, 8, and 9, and reasonable pre- Data Sources diction of duration but underprediction of tempera- 1. Several surveys have been published of mass of ture in Cardington tests #3 and 4. In test #7, using combustible materials per unit area for different the CIB graphs resulted in reasonable predictions of occupancies.28,29,30,31 Given that fire loading can duration but underprediction of temperatures. Due to vary significantly over the life of a building, the large values of in Cardington tests #5 uncertainty should be carefully considered. 2. Building characteristics can be obtained from and 6, predictions were not possible using the surveys of existing buildings or architectural CIB graphs. plans of new buildings.

32 Law Law derived a method of predicting compartment temperatures resulting from fully developed fires Where: based on data from tests conducted under the =Mass burning rate of fuel (kg/s) auspices of CIB. Law’s method takes into account W = Length of wall containing ventilation the geometry of the compartment. The area of the opening (m) compartment’s lining surface through which heat is D = Depth of compartment (m) lost is expressed by subtracting the vent area from the total interior compartment surface area (A – A ); o The duration of burning can be calculated by the temperature in the compartment is therefore dividing the total mass of combustibles by the burn- dependent on A, as well as variables incorporated in ing rate as follows: the ventilation factor, Ao, and H. Law derived the following equation to determine the maximum temperature of the compartment with natural ventilation49: Where: τ = Burning duration (s)

Data Requirements

1. The total mass of fuel, mf 2. The area and height of the enclosure opening(s), A and H Where: o o 3. The interior surface total area of the enclosure, T = Maximum compartment temperature (°C) gm including the floor and openings, A, and the A = Surface area of interior of enclosure (m2) 2 width, W, and depth, D, of the enclosure Ao = Area of ventilation opening (m ) Ho = Height of ventilation opening (m) Data Sources This equation does not account for the effects on 1. Several surveys have been published of mass of compartment temperature due to fuel loading. It combustible materials per unit area for different simply represents the maximum temperature occupancies.28,29,30,31 Given that fire loading can achieved in a compartment for a given geometry vary significantly over the life of a building, and ventilation. The following equation incorporates uncertainty should be carefully considered. the effect of fuel loading on the temperature and is 2. Building characteristics can be obtained from valid for wood-based fuels: surveys of existing buildings or architectural plans of new buildings.

Where: Validation and Limitations See Appendix B for comparisons of predictions with test data. Without applying the adjustment factor Ψ, Law’s temperature predictions bounded all the CIB data. The mass loss rate is correlated as Law reasonably predicted the CIB burning rate data, and, if the burning rate was adjusted by a factor of 1.4, Law bounded all the CIB burning rate data.

33 Law reasonably predicted average temperature temperature was a function of the fire duration and and duration for Cardington tests #1, 2, 8, and 9 opening factor.51,52,53 Magnusson and Thelandersson reported that for shorter duration fires the rate of . temperature decrease was higher than 10°C/min, For Cardington tests #3, 4, 5, and 6 while for longer duration fires the rate of temperature decrease was lower than 10°C/min.38 Based on , a series of short-duration fires, Harmathy reported Law reasonably predicted duration but under- decay rates between 15 and 20°C/min, which are predicted temperature. In test #7 consistent with the results presented by Magnusson and Thelandersson. Typical rates found in the litera- , ture are listed in Table 3. which was square in plan view, Law’s method In the absence of better information, it would be underpredicted temperatures but predicted the appropriate to select a decay rate of 7°C/min for burning duration. fires with a predicted duration of 60 minutes or more, and a decay rate of 10°C/min for fires with a predicted duration of less than 60 minutes, since Simple Decay Rates these rates would result in the slowest decay rates Many of the methods cited according to the above. previously do not contain a method of estimating the TABLE 3. Rate of Decrease in Temperature compartment temperature Temperature during the decay stage. For Decay these methods, a number of (°C/min) Restrictions Reference simple decay rates can be 10 τ < 60 min Kawagoe applied if the engineer wishes τ to account for heating that 7 > 60 min Kawagoe occurs during the decay phase. >10 τ < 60 min Magnusson and Thelandersson Decay cannot be modeled <10 τ > 60 min Magnusson and Thelandersson by basic physics because the “decay rate” is actually the 10 No restrictions Swedish Building Regulations heat transfer from the compart- 15 – 20 Short-duration fires Harmathy ment and the heat release rate of combustibles that have charred and collapsed onto the floor, with poor access of RECOMMENDATIONS oxygen and therefore limited heat release rate. All Based on comparison of predictions to the data methods are wholly empirical. from the CIB and Cardington tests, Law’s method is The simplest way to determine the temperature– recommended for use in all roughly cubic compart- time profile during the decay phase is to use a fixed ments (compartment width to depth ratio within the rate of temperature decay. Originally, the tempera- range of 0.5 to 2.0) and in long, narrow compart- ture decay during the cooling phase was selected arbitrarily. Kawagoe first suggested that the rate of ments where does not exceed ≈ 18 m–1/2. temperature decrease during the cooling period was To ensure that predictions are sufficiently conserva- a function of the fire duration, reporting values of tive in design situations, the predicted burning rate 7°C/min for fire durations greater than 60 minutes should be reduced by a factor of 1.4, and the tem- and 10°C/min for fire durations less than 60 min- perature adjustment should not be reduced by the utes.50 The pioneering work of Magnusson and factor Ψ. See Figures 12 through 17, which show Thelandersson indicated that the rate of decrease in

34 comparisons made using Law’s method to the Law’s method does not predict temperatures dur- CIB data and to data for Cardington tests ing the decay stage. For cases where a prediction of temperatures during the decay stage is desired, a #1 , #2 , decay rate of 7°C/min can be used for fires with a predicted duration of 60 minutes or more, and a #8 , and #9 . decay rate of 10°C/min can be used for fires with a predicted duration of less than 60 minutes.

1400

1200

1000

800

600

400 Temperature (°C) CIB Data 200 Law (max)

0 01020304050

1/2 –1/2 A/AoHo (m )

FIGURE 12. Comparison of CIB Temperature Data to Predictions Using Law’s Method

) 0.25 5/2

0.2 /s – m /s –

kg 121 ( 0.15 221 1/2 211 441

D/W) 0.1 ( Law X 1.4 1/2 o H

o 0.05 /A f . m

0 0102030405060

1/2 –1/2 A/AoHo (m )

FIGURE 13. Comparison of CIB Burning Rate Data to Predictions Using Law’s Method

35 Cardington Test #1

1400

1200

1000

800 Measured Law Adjusted 600

Temperature (°C) 400

200

0 00.511.522.5 Time (h)

FIGURE 14. Comparison of Predictions Using Law’s Modified Method for Cardington Test #1

Cardington Test #2 1400

1200

1000

800 Measured Law Adjusted 600

Temperature (°C) 400

200

0 00.511.5 2 Time (h)

FIGURE 15. Comparison of Predictions Using Law’s Modified Method for Cardington Test #2

36 1400

1200

1000

Measured 800 Law Adjusted

600

Temperature (°C) 400

200

0 00.511.522.53 Time (h)

FIGURE 16. Comparison of Predictions Using Law’s Modified Method for Cardington Test #8

1400

1200

1000

800 Measured Law Adjusted

600

Temperature (°C) 400

200

0 00.511.5 2 Time (h)

FIGURE 17. Comparison of Predictions Using Law’s Modified Method for Cardington Test #9

37 For long, narrow spaces in which for Cardington tests #3 ,

is in the range of 45 to 85 m–1/2, Magnusson and #4 , and Thelandersson provide reasonable predictions of temperature and duration. See Figures 18 through 20, which show comparisons made using #5 . Magnusson and Thelandersson’s method to data

1400

1200

1000

800 Measured Magnusson 600 (Type C)

Temperature (°C) 400

200

0 00.511.522.5 Time (h)

FIGURE 18. Comparison of Predictions from Magnusson and Thelandersson’s Method (Type C) to Data for Cardington Test #3

38 1400

1200

1000

800 Measured 600 Magnusson (Type C)

400 Temperature (°C)

200

0 01234 Time (h)

FIGURE 19. Comparison of Predictions from Magnusson and Thelandersson’s Method (Type C) to Data for Cardington Test #4

1400

1200

1000

800 Measured 600 Magnusson (Type C)

400 Temperature (°C)

200

0 00.511.522.53 Time (h)

FIGURE 20. Comparison of Predictions from Magnusson and Thelandersson’s Method (Type C) to Data for Cardington Test #5

39 on comparison of predictions to the Cardington data, For long, narrow spaces in which is its use is still recommended. See Figure 21, which approximately 345 m–1/2, Lie’s method is recom- shows comparisons made using Lie’s method to mended. Note that this value of is outside data for Cardington test #6 . Lie’s stated range of applicability. However, based

900

800

700

600

Measured 500 Lie 400

300 Temperature (°C)

200

100

0 012345678 Time (h)

FIGURE 21. Comparison of Predictions from Lie for Cardington Test #6

Fire Exposures from Plumes develops heat transfer boundary conditions for two different types of exposure: This section of the guide focuses on predicting the heat transfer from area exposure fire plumes to 1. Bounding, or elements immersed in a fire plume adjacent surfaces. Area exposure fires are burning 2. Specific geometries, or specific element shapes objects or fuel located adjacent to or near the sur- and orientations face being heated. For certain scenarios, the local fire exposure may produce a more extreme expo- Detailed modeling of the fire from first principles sure than the hot gas layer that develops in the area can also be conducted to predict the boundary condi- of consideration. Some examples are open parking tion; however, this type of analysis is not addressed garages, large warehouses, and bridges and over- in this guide. If detailed modeling is conducted, passes. To analyze these scenarios, one needs to the model should be verified with existing data for have knowledge of the incident heat flux levels pro- similar configurations to validate predicted heat duced by local fire plumes. fluxes. Some additional data on gas temperatures The boundary condition between the fire plume and velocities generated by a fire plume are also and the structural element needs to be properly included to aid in this type of modeling effort. defined in order to predict the temperature of the The fire exposure recommended for a bounding structural element with time. This part of the guide analysis will consist of a constant fire exposure. If a

40 more refined analysis is required, guidance is pro- can be calculated using the following equation for 55 vided on how to predict the boundary condition regions above the average flame height, Lf: with the fire in specific geometries. These geometries include the following: (Eq. 40) • Flat vertical walls • Corners with a ceiling Where: • Unbounded flat ceilings Um,c(Z) = Centerline plume velocity (m/s) χ • I-beam mounted below a ceiling r = Fraction of energy released as radiation in the fire The boundary condition in these configurations is = Fire heat release rate (kW) based on experimental data and may be limited to Z =Target elevation above the base of the conditions tested in the study. the fire (m) Studies have also been conducted to measure zo = Elevation of the virtual origin relative the heat flux boundary condition with fires in other to the base of the fire (m) configurations. Lattimer54 provides a review of existing incident heat flux data and correlations for The centerline plume velocity for regions below exposure fires and burning surfaces in a variety of the average flame height may be determined using 56 configurations including flat walls, corners, corners Equation 41 where all terms have been defined. with a ceiling, parallel flat walls, walls above a window containing a fire plume, unbounded ceiling, and an I-beam under a ceiling. (Eq. 41)

AXISYMMETRIC FIRE PLUMES The simplest fire plume is the unconfined Plume bu axisymmetric fire plume, shown in Region Figure 22. Correlations for velocity and r temperature produced by an axisymmetric plume are provided in this section to aid those in modeling the heat flux to elements from first principles. Unconfined axisymmetric fire plumes are typically approxi- Flame Region mated as point heat sources when estimating the local velocity and temperature profile. This section describes how to estimate the location of the virtual point source relative to Lf the base of the fire, the flame height, and the velocity and temperature distribution within the fire plume.

Z Velocity Profile o Z D The velocity profile of the fire plume is a function of the elevation above the virtual origin and the distance from the plume centerline. The velocity at the plume centerline FIGURE 22. Axisymmetric Fire Plume

41 The virtual origin may be calculated using Equa- The velocity distribution within a fire plume tion 42 where D is the effective fire diameter (m)57: has been found to fit a Gaussian profile, though no theoretical grounds exist for this.55,59 The following (Eq. 42) equation may be used to determine the velocity as a function of the distance from the plume For noncircular fuel packages with a length to width centerline55: ratio of near one, the equivalent diameter of the fuel package can be estimated using the surface area, A, (Eq. 45) of the noncircular fuel package: Where: (Eq. 43) U(r) =Velocity in plume at a distance r (m) from the centerline (m/s) Where: b = Plume width parameter (m) A = Surface area of the fuel package (m2) u

The plume width parameter is found via The average flame height can be calculated using Equation 46 where all terms have been defined.55 the relation developed by Heskestad58:

(Eq. 44) (Eq. 46)

Where: Temperature Profile = Heat release rate of the fire (kW) The temperature profile is also a function of the D = Diameter of the fuel package (m) elevation above the plume virtual origin and the

1600

1500

1400

1300 Nat. Gas 0.3 m Heptane 1.7, 6 1200 Methanol 1.7 Kerosene 30 JP-4 15 1100

1000

Max. Turbulent Flame Temperature (°C) 900

800 00.10.20.30.40.5

Fire Plume Radiation Fraction, Xr

FIGURE 23. Maximum Turbulent Fire Plume Temperatures from Various Sources61,62,63,64

42 distance from the plume centerline. The centerline As the pool fire diameter is increased, flames temperature may calculated using Equation 47 for produce more soot, reducing the flame radiation elevations above the average flame height55: being emitted to the surroundings. From the SFPE Engineering Guide on Assessing Flame Radiation to External Targets from Pool Fires66 and Beyler,67 (Eq. 47) radiative fraction will decrease linearly from an average radiative fraction of 0.22 for a small- Where: (~0 m) diameter pool fire to approximately 0.04 for Tm,c(Z) = Centerline plume temperature (K) a 50 m diameter pool fire. Baum and McCaffrey68 T∞ = Ambient temperature (K) clearly showed the dependence of gas temperature on diameter, with measured gas temperatures as The centerline plume temperature for elevations high as 1000°C for 6 m diameter fires and 1250°C below the average flame height may be determined for 30 m diameter fires. These data are represented using the following where all terms have been in Figure 23. 56 defined : The temperature distribution as a function of the distance from the (Eq. 48) plume centerline also fits a Gaussian profile.55 Equation 51 can be used to determine the temperature (Eq. 49) at any distance r (m) from the plume centerline55: These temperatures represent average temperatures in the flaming and plume regions, and they will tend to be higher when the radiative fraction, χ r, of the fire is decreased. For turbulent fire χ (Eq. 51) plumes, having a radiative loss fraction r, the turbulent flame (centerline) temperature follows the Where: relationship60 bt = Thermal plume width parameter (m) (Eq. 50) The thermal plume width parameter may be calculated using Equation 52 where all terms have From the best available data,61,62,63 the turbulent been defined55: mixing parameter, kT is found to be about 0.5 for cp = 1 kJ/kg-K. As the fire diameter increases, the (Eq. 52) radiative fraction falls due to soot blockage.64 Figure 23 shows flame temperature data for turbu- χ Data Requirements lent plumes as a function of r. The extrapolated adiabatic temperature is about 1500°C. 1. Source fire heat release rate, (kW) χ Temperatures have been measured to be as low 2. Radiative fraction, r as 820°C for flames produced by fuels with a radia- 3. Elevation above source fire, Z (m) χ 65 tive fraction of r ~ 0.20. Thus, Equations 48 and 4. Radial (horizontal) separation from centerline of χ 49 correspond to fires of r ≈ 0.3. source fire, r (m)

43 Data Sources HEAT FLUX BOUNDARY CONDITION 1. Heat release rate data may be obtained from The governing boundary condition for a fire heat- Babrauskas,69 Hoglander and Sundstrum,70 or ing an adjacent surface is determined using the heat Mudan and Croce.71 balance shown in Figure 24 to be 2. Radiant fraction data may be obtained from Tewarson.72 (Eq. 53) Assumptions assuming negligible heating from the surrounding 1. The axisymmetric fire plume may be environment (i.e., no hot gas layer heating). To approximated as a point heat source. This apply this relation directly, the local gas tempera- assumption is valid for many types of fires ture, Tf, local heat transfer coefficient, h, and the ε including pool fires, but may yield poor results emissivity of the gases, f, must be known. The sur- α ε for three-dimensional burning objects (i.e., sofa), face absorbtivity, s, and emissivity, s, must also momentum-driven plumes (jets), or regions near be known, but approach 1.0 as they become soot the base of the fire. covered. All these parameters are scenario depend- 2. The effect of a hot smoke layer formation in a ent, and all are not readily known or predicted. As a compartment on the temperature and velocity result, several research efforts have been conducted profiles in a fire plume is ignored. Refer to to measure the total incident heat flux to a surface Evans73 and Cooper74 for a discussion of hot in a variety of configurations. This is typically done layer–plume interactions. using cooled total heat flux gauges. These gauges 3. There is no air movement (wind, vent flows) in are cooled so that their surface temperature remains the vicinity of the plume. Such air motions may near ambient and are coated with a high-emissivity cause a plume to deflect. paint to maximize the absorbed radiation. By setting the surface temperature to the ambient in Equation 53, the boundary condition at the total Validation heat flux gauge is represented by Equation 54: There have been numerous experiments on the centerline temperature and velocity in fire plumes. (Eq. 54) The form of the correlations is generally identical; however, there is some variation among the correlated constants.55,75 Those presented in this section tend to be conservative in terms of predicting the greatest velocity and centerline temperature for a given heat release rate and target elevation.

Limitations The fire plume equations in this section are limited to open, axisymmetric thermal plumes in a quiescent environment. The source fire should have a relatively square plan area, though fuel packages or source fires with aspect ratios on the order of two or three may be acceptable. Larger aspect ratios could result in a line fire. Refer to Quintiere and Grove76 FIGURE 24. Heat Balance at the for a discussion of line fire thermal plumes. Material Surface

44 Cooling the gauge surface maximizes the convec- diffusion flame impinges on a ceiling is on the order tive heat transfer and minimizes the radiative losses; of 0.050 kW/(m-K). Figure 25 contains plots of the thus, the cooled heat flux gauges measure the maxi- radiative correction for different element surface mum total incident heat flux. Assuming that the temperatures along with convective correction for surface absorbtivity and emissivity are identical, convective heat transfer coefficients of 0.015 and and the emissivity of the heat flux gauge is similar 0.050 kW/(m K). From Equation 56 and Figure 25, ε ≈ ε to that of the material surface ( s,hfg s), the total overestimating the convective correction will result incident heat flux measured using the heat flux in a non-conservative boundary condition. There- gauge, Equation 54, is related to the actual heat flux fore, a convective heat transfer correction through the following relation, is only recommended in simple configurations where local heat transfer coefficients can be calculated (e.g., flat walls). (Eq. 55) or BOUNDING HEAT FLUX: (Eq. 56) OBJECTS IMMERSED IN FLAMES The simplest and most conservative way to treat Therefore, measuring the heat flux has removed the fire exposure boundary condition would be to the need to predict both the gas temperature and the apply a constant, bounding heat flux to all structural emissivity of the gases. To get the actual net heat elements in the area of interest. The bounding heat flux into the surface from the measured heat flux, a flux boundary condition was developed from data surface temperature correction needs to be applied on objects immersed in large hydrocarbon pool as done in Equation 56. fires. The heat flux data for objects immersed in A conservative estimate of the net heat flux into the structural element can be determined by either not applying any surface Surface Temperature (°F) temperature correction or only 200 400 600 800 1000 1200 1400 1600 1800 applying the radiative correction. 60 A closer estimate of the actual net heat flux into the surface 50 )

would include both radiative and 2 convective corrections. Applying 40 a convective correction involves estimating the local heat transfer 30 coefficient, h, which is dependent on the local velocity and 20 Heat Flux [kW/m gas temperature. Heat Flux (kW/m Local heat transfer coeffi- 10 cients may range from 0.015 to 0.030 kW/(m K) for hot gas flow 0 up a wall or along a ceiling. At 0100200300 400 500 600 700 800 900 1000 points where hot gases impinge Surface Temperature (°C) on a surface, this value may be higher. Based on data from FIGURE 25. Magnitude of the Surface Temperature Corrections Kokkala77,78 and You and on the Measured Total Heat Flux Using a Cooled Gauge (see Faeth,79,80 the local convective Equation 56). Radiation (—), Convection with h =0.015 kW/(m K) heat transfer coefficient where a (– ..–), and Convection with h =0.050 kW/(m K) (– – –).

45 fires are presented in this section and used to deter- measurements were taken at various elevations and mine the magnitude of the bounding heat flux. The angular positions on the calorimeters. The cold-wall information in this section is derived primarily from (i.e., peak) heat fluxes to the large calorimeter varied direct or indirect measurements of heat flux taken in between 100 kW/m2 and 160 kW/m2 at any one loca- open hydrocarbon pool fires with optically thick tion, with the largest peak heat fluxes observed on flames. There is insufficient data available at this the underside and the lowest on the top. Figure 26 time to adequately address the impact of a boundary shows the average peak heat flux at various angular such as a wall or ceiling on the heat flux conditions positions as a function of the external surface tem- to an immersed object. It is expected that the data perature of the large calorimeter, which increases as obtained from optically thick flames in unconfined a function of time, and the angular position. pool fires is bounding. The cold-wall fluxes to the small calorimeter varied between 150 kW/m2 and 220 kW/m2. As with Test Data the large calorimeter data, the maximum heat fluxes were observed on the bottom of the calorimeter and A series of 30-minute, 9.1 m by 18.3 m hydrocar- the minimum were observed on the top. There was bon pool fires (JP-4) conducted by Gregory, Mata, no decrease in the cold-wall heat flux detected over 81 and Keltner provided useful temperature and heat the elevation range (1 to 11 m) sampled. flux data at various elevations above the base of the Russell and Canfield82 immersed a steel cylinder fire. Steel cylinders filled or lined with insulation in a 2.4 m by 4.9 m JP-5 pool fire in windy condi- (referred to as small or large calorimeters, respec- tions. The inside surface temperature of the cylinder tively) at several locations were used to indirectly was directly measured, and the exposure heat flux measure the net heat flux for objects immersed in was determined in the same manner as Gregory, the fire. The temperature inside the cylinder was Mata, and Keltner.81 The peak heat fluxes to the recorded, and the net heat flux was extracted using surface of the cylinder were measured at various the inside temperature as a boundary condition. The angular positions. The peak heat fluxes ranged from

140

120 ) 2 100

40 Bottom

Average Heat Flux (kW/m Top 20 Left Side Right Side

0 400 600 800 1000 External Surface Temperature of Large Calorimeter (K)

FIGURE 26. Averaged Peak Heat Flux as a Function of Angular Position

46 18 kW/m2 on the windward side to 144 kW/m2 on Data Sources the leeward side. The heat fluxes on the top and For pool fires, the radiative fraction can be deter- 2 bottom of the cylinder were 48 kW/m and mined as a function of pool diameter from the SFPE 2 103 kW/m , respectively. Engineering Guide to Assessing Flame Radiation to 83 Cowley summarized the peak heat fluxes External Targets from Pool Fires. This radiative measured directly or indirectly to objects immersed fraction can be substituted into Figure 23 to esti- in various large-scale pool fires. The values range mate the flame temperature. For noncircular pools 2 2 between 80 kW/m and 270 kW/m . Table 4 sum- with a length-to-width ratio of near one, the equiva- marizes some of this information. Cowley speculates lent diameter of the pool can be estimated using the differences between low- and high-volatile fuels surface area, A, of the noncircular pool: with heat fluxes as high as 300 kw/m2 are possible in the latter. (Eq. 56b) Most of the heat flux test data suggest a bound- 2 ing cold-wall heat flux between 150 kW/m and Where: 2 170 kW/m . Although some data (small calorimeter) A = Surface area of the fuel package (m2) indicate that the peak may be as high as 220 kW/m2, these appear to be exceptional. The heat flux in a flame increases with fire Assumptions diameter and where the object or flame impinge- 1. The flame emissivity and surface absorbtivity are ment is located. The upper bound of heat flux can equal to 1.0. be calculated as follows: 2. The impact of a compartment on the heat fluxes at the surface of an immersed object can be (Eq. 56a) ignored. 3. Reduction in net heat flux due to heating of the Data Requirements target is not considered. The flame temperature is needed to perform this calculation. Validation Equation 56a is based on first TABLE 4. Selected Heat Fluxes to Objects Immersed in principles. Heat fluxes calculated Large Pool Fires83 using Equation 56a are much larger Peak than measured heat fluxes. For 68 Heat Flux example, Baum and McCaffrey Test Pool Size Fuel (kW/m2) reported gas temperatures as high as 1250°C in 30 m diameter pool AEA Winfrith84 0.49 x 9.4 m Kerosene 150 fires. Assuming that the gases US DOT84 Not listed Kerosene 138 are optically thick, emissivity USCG84 Not listed Kerosene 110-142 of 1.0, the cold-wall heat flux is 305 kW/m2. As seen in Table 4, 84 US DOT Not listed Kerosene 136-159 measured values are less than this Sandia84 Not listed Kerosene 113-150 value, indicating that the assumed HSE Buxton84 Not listed Kerosene 130 emissivity may be significantly less than 1.0 or the effective gas 84 Shell Research 4.0 x 7.0 m Kerosene 94-112 temperatures providing the radia- Large cylinder82 9 x 18 m JP-4 100-150 tion are lower than measured or reported temperatures. Large cylinder82 9 x 18 m JP-4 150-220 Russell and Canfield83 2.4 x 4.9 m JP-5 144

47 Limitations or the flame tip length. Therefore, heat flux correlations should be applied using either the flame length The results of this section are limited to Class A correlation developed in the study or with one that (plastic or wood-based) combustible material fires has been demonstrated to predict the flame length in or hydrocarbon pool fires. Gaseous jet flames are that study. beyond the scope of this section because they may produce larger cold-wall (200 to 270 kW/m2) heat fluxes to immersed objects.83 Flat Vertical Walls The results are also not applicable to objects that The simplest geometry is with the fire directly are located near (collocated), but not in, the burning against a flat wall as shown in Figure 27. Correla- region. Methods of estimating the incident heat flux tions are developed in this section to estimate the to collocated objects are available in another vertical and horizontal variation in the heat flux to 66 Engineering Guide. the wall due to a fire in this configuration. Correlations to estimate the incident heat flux HEAT FLUXES FOR from an exposure fire against a flat wall have been SPECIFIC GEOMETRIES developed through an experimental study performed by Back et al.84 In this study, fires were generated The incident heat flux from a fire plume to a using square propane sand burners with edge lengths surface is dependent on: of 0.28, 0.37, 0.48, 0.57, and 0.70 m. Heat flux fields were measured for fires ranging from 50 to 520 kW. • Geometry The flame height to burner diameter aspect ratio • Dimensions of the fire ranged from approximately 1 to 3 in these tests. • Fire heat release rate •Effective radiative path length • Soot production rate

Research has been conducted to evaluate the effects of each of these variables on the incident heat flux Wall from a fire. However, a general engineering approach has not been developed for predicting the incident heat flux from a fire to an adjacent surface. This section provides empirical correlations for estimating the heat flux boundary condition in some specific geometries. These correlations were developed over a specific range of fire source size, heat release rate, and geometry, which limits their general applicability. Lf The heat transfer from a flame to an adjacent surface or object has historically been characterized Z with respect to the flame length. Many of the heat flux correlations developed in the literature are based on flame length data taken in a particular study. Measured flame lengths can vary depending on the measurement technique, definition, and surrounding geometry. For the studies considered in A this section, the data were nondimensionalized with either the average (50% intermittent) flame length FIGURE 27. Fire Against a Flat Vertical Wall

48 140 ) 2 120

100 (kW/m peak

" 80 q

. 60

40 Aspect Ratio ~1 Aspect Ratio ~2 Aspect Ratio ~3 20 Peak Heat Flux,

0 0100200300 400 500 600 Heat Release Rate (kW)

FIGURE 28. Peak Heat Release Rates Measured in Square Propane Burner Fires Against a Flat Wall84

The average flame length of fires against a flat Where: wall was determined to be equal to the average A = Surface area of the fuel package (m2) flame length of unconfined fire plumes. Flame lengths can be calculated using the relation devel- A plot of the peak heat fluxes measured for each oped by Heskestad58: of the fires considered in the study is shown in Figure 28. Peak heat fluxes for the different fires (m) (Eq. 57) evaluated were determined to be a function of the fire heat release rate. This dependence was attributed Where: to the larger size fires resulting in thicker boundary = Heat release rate of the fire (kW) layers on the wall, thus increasing the radiation path D = Diameter of the fuel package (m) length. Based on gray-gas radiation theory, the authors found the following relation adequately Flame lengths are taken relative to the base of the represented the data: fire. For noncircular fuel packages with a length to width ratio of near one, the equivalent diameter of the fuel package can be estimated using the surface (Eq. 59) area, A, of the noncircular fuel package: These peak heat fluxes were measured in the

(Eq. 58) lower part of the fire (z/Lf ≤ 0.4) along the centerline. Above this region, the heat fluxes were

49 measured to decrease with distance above the fire, are shown in Figure 29. Lines in this plot are a z. The heat flux data measured along the centerline general correlation of the centerline data: (Eq. 60a)

(Eq. 60b)

(Eq. 60c)

1000 ) 2

(kW/m 100 cl " q

Q ≈ 59 kW . 10 Q ≈ 121 kW Q ≈ 212 kW Q = 313 kW Q = 523 kW Correlation for Q = 59 kW Centerline Heat Flux, Correlation for Q = 523 kW

1 0.01 0.1 1 10

z/Lf

FIGURE 29. Vertical Heat Flux Distribution Along the Centerline of a Square Propane Burner Fire Adjacent to a Flat Wall84

1.4 1.4

1.2 1.2

1.0 1.0

0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2 Heat Flux/CL Flux Heat Flux/CL Flux 0 0 0 0.5 1.0 1.5 2.0 2.0 3.0 3.5 0 0.5 1.0 1.5 2.0 2.0 3.0 3.5 Distance/Burner Half Length Distance/Burner Half Length (a) (b)

FIGURE 30. Horizontal Heat Flux Distribution (a) Below the Flame Height and (b) Above the Flame Height with Distance from the Centerline of the Fire84

50 Heat fluxes were measured by Back et al.84 to of the incident heat flux levels measured in the decrease with horizontal distance from the center- propane burner experiments is consistent with fire line as shown in Figure 30. Significant heat fluxes produced by burning items. In tests with propane gas were measured as far as twice the burner radius burners against a non-combustible boundary, similar from the centerline. Conservatively, it can be heat flux levels have been measured by other inves- assumed that the heat flux is equal to the centerline tigators for limited conditions.87,88 heat flux at distances as far as twice the fire radius from the centerline. Limitations Correlations for incident heat fluxes were devel- Data Requirements oped using luminous flames in an open environment 1. Diameter of the fuel package, D. For noncircular with the fire directly against a flat vertical wall. fuel packages, the equivalent diameter may be Using these relations inherently assumes: calculated using Equation 58 and the surface area of the fuel package. • There is negligible heating from a hot gas layer 2. Heat release rate of the fire, . in the surroundings. 3. Elevation along the flame length, z. • The fire is against the wall. • The flames are luminous. Data Sources • The wall is vertical.

1. Heat release rate data may be obtained from The experimental study considered fire diameters as 69 70 Babrauskas, Hoglander and Sundstrum, or large as 0.70 m and heat release rates as large as 71 Mudan and Croce. 520 kW. No data was available to validate the correlations against fires with larger diameters or higher Assumptions heat release rates. The presence of a hot gas layer This analysis assumes that the fire is attached to may increase the total incident flux onto the wall, the wall and that the wall is vertical. Walls that are and if significant in the area of interest adding this not vertical may result in different total incident contribution to the total incident heat flux from the 89 heat flux levels due to the flame’s becoming sepa- fire plume may be warranted. Moving the fire rated from the wall or the difference in entrainment away from the wall will eventually cause the inci- into the plume. dent heat fluxes to become lower, largely because the flame becomes detached from the wall.90 Thus, the use of correlations in this section for fires that Validation may be slightly spaced from the wall will yield Some studies have made measurements of incident conservative results. Flames less luminous than heat fluxes from various burning objects to walls, but those produced by the propane fires (i.e., natural the data is sparse. Incident heat fluxes at the rim of gas) may transmit lower total incident heat fluxes to wastebasket fires were reported by Gross and Fang.85 the wall because the radiative heat flux to the wall At the rim, heat fluxes as high as 50 kW/m2 were will be lower.87,88 Propane fuel fires used to develop measured; however, the authors noted that peak heat the heat flux data presented in this section produce fluxes for these fires occurred approximately 0.22 m a moderate amount of soot; therefore, heat flux above the rim. Mizuno and Kawagoe86 performed levels presented in this section should be considered experiments with upholstered chair fires against a to be average but not bounding for all different flat wall. In these tests, Mizuno and Kawagoe meas- fuels. Propane burners are also used extensively in ured heat fluxes to the wall of 40 to 100 kW/m2 over standard fire tests as an exposure fire that is repre- the continuous flaming region (~z/Lf < 0.4). All sentative of real fires. Therefore, the incident heat these tests were performed using foam-padded chairs. fluxes from these flames are considered to be repre- These data do provide evidence that the magnitude sentative of those produced by most fires.

51 Fires in a Corner were placed directly in the corner. The study included fires with heat release rates ranging from Fires located in a corner geometry as shown in 50 to 300 kW. Figure 31 produce a more complicated flow field, Correlations were developed for the three regions particularly when a ceiling is present. As indicated in the corner shown in Figure 32. The regions were in Figure 31, fires in a corner rise vertically in the the corner walls on the lower part of the walls, the corner until the gases impinge on the ceiling, at top portion of the walls near the ceiling, and along which point the fire will be redirected along the the ceiling. The corner walls region extended from ceiling and the top of the walls. Near the top of the the fire to approximately 1.8 m above the floor. walls, flaming vortices will flow out from the cor- Above this region, the incident heat flux onto the ner resulting in elevated heat fluxes along the top of walls was measured to be affected by the hot gases the wall as much as twice the ceiling jet thickness. flowing along the ceiling. The distance of 1.8 m Incident heat flux correlations in a corner with a is approximately twice the ceiling jet thickness ceiling were developed by Lattimer et al.91 The below the ceiling or H – 2δ where H = 2.2 m and study was conducted using a 2.4 m high open cor- δ = 0.1H.92 Correlations for the top part of the walls, ner constructed of two walls and a ceiling. Fires which are heated by the ceiling jet, were developed were produced using square propane burners having using data from 1.8 m to 2.2 m or H – 2δ < z < H. single side lengths of 0.17, 0.30, and 0.50 m. Fires The flame length in the corner with a ceiling was taken to be the flame length in the corner plus any flame extension along the ceiling. The following relation can be used to calculate the flame tip length Ceiling with the fire in the corner: (Eq. 61)

Lf,c r Where:

(Eq. 62)

Wall = Heat release rate of the fire (kW) Wall D = Diameter of the fuel package (m) ρ 0 = Density of air at initial ambient conditions (1.2 kg/m3) L H f,w cp = Specific heat capacity of air at initial ambient conditions [1.0 kJ/(kg K)]

To =Temperature at initial ambient conditions X (293 K) g = Gravitational acceleration (9.81 m/s2)

Flame lengths are taken relative to the base of the Z fire. This correlation can be used to estimate flame lengths in a corner with or without a ceiling. For noncircular fuel packages with a length to width A ratio of near one, the equivalent diameter of the fuel package can be estimated using the surface area, A, of the noncircular fuel package:

(Eq. 63) FIGURE 31. Fire in a Corner Configuration

52 Top of Walls Region X

Corner Walls Region

H 2D

r Ceiling Region Z Corner with Fire D Exposure Fire Ceiling Above Corner

FIGURE 32. Corner with a Ceiling Configuration Showing the Three Regions Where Incident Heat Flux Correlations Were Developed in the Study of Lattimer et al.91

Where: Where: A = Surface area of the fuel package (m2) = Peak heat flux in the corner (kW/m2) D = Diameter of the fuel package (m) Walls at Corner Correlations in this section can be used to estimate the incident heat flux in the 120 corner with the fire. These correlations can

) 110 2 be used to estimate the incident heat flux 100 in a corner configuration with or without 90 a ceiling. When a ceiling is present, the (kW/m 80 peak correlations are valid up to an elevation of " 70 z = H – 2δ, where δ = 0.1H.92 Along the q 60 height of the walls in the corner, the peak . 50 heat fluxes were typically measured near 40 the base of the fire. These peak heat fluxes 30 were measured to be a function of the fire 20 Peak Heat Flux, 10 diameter as shown in Figure 33. The curve 0 in Figure 33 is a correlation to the data 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 and is expressed using Equation 64: Length of Area Burner Side, D (m)

FIGURE 33. Peak Heat Flux Along the Height of the (Eq. 64) Walls in the Corner. Data from Lattimer et al.91

53 The vertical distribution in the maximum heat length. Peak heat flux levels were measured in the flux along the walls near the corner is shown in lower part of the flame (z/Lf,tip ≤ 0.4) and decreased Figure 34 plotted with the elevation above the with distance above z/Lf,tip = 0.4. A general correla- fire, z, normalized with respect to the flame tip tion to represent this behavior is as follows:

(Eq. 65a)

(Eq. 65b)

(Eq. 65c)

Where: z = Elevation along the flame height in the =Maximum heat flux at a particular corner (m) 2 elevation in the corner (kW/m ) Lf,tip = Flame tip length calculated using = Peak heat flux in the corner (kW/m2) Equations 61 and 62 (m)

1000 ) 2

(kW/m 100 peak " q .

10 Maximum Heat Flux,

1 0.01 0.1 1 10

z/Lf,tip

FIGURE 34. Maximum Heat Fluxes to the Walls Near the Corner with Square Burner Sides of ●-0.17 m, ▲-0.30 m, ▼-0.30 m (Elevated), and ■-0.50 m and Fire Sizes from 50 to 300 kW. Data from Lattimer et al.91

54 1.25

D = 0.17 m D = 0.30 m D = 0.50 m 1.00 Correlation

0.75 max " /q " . . . q 0.50

0.25

0.00 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 x/D

FIGURE 35. Heat Flux Distribution Horizontally out from the Corner on the Lower Part of the Corner Walls

Heat fluxes will decay with distance away from below the ceiling. The maximum heat fluxes are the corner as shown in Figure 35. Significant heat shown in Figure 36 plotted against the dimensionless fluxes can exist as far as two fire diameters horizon- distance along the flame, (x + H)/Lf,tip. These heat tally out from the corner. For a conservative analy- fluxes can be estimated using the following relations: sis, the maximum vertical heat flux distribution measured in the corner should be assumed from the corner (Eq. 66a) to two fire diameters horizontally out from the corner. (Eq. 66b) Top of Walls Where: This section provides correlations to estimate x = Distance horizontally out from the heat fluxes along the top of the walls in a corner corner (m) configuration with a ceiling. These incident heat H = Distance between the base of the fire and flux correlations apply to the top of the walls the ceiling (m) approximately twice the ceiling jet thickness below L = Flame tip length calculated using δ δ 92 f,tip the ceiling or H – 2 < z < H where = 0.1H. Equations 61 and 62 (m) Along the top part of the wall, the maximum heat fluxes were measured at locations less than 0.15 m

55 The assumed plateau in the correlation was based on the 1000 maximum heat flux levels measured in larger fire tests 91 with burning boundaries. ) 2 Heat fluxes will decrease with distance below the ceiling. Conservatively, it can be (kW/m 100 max assumed that incident flux " q along the top of the walls is constant with distance below

the ceiling and is equal to the . maximum incident flux pre- 10 dicted through Equation 66.

Ceiling Above a Corner Maximum Heat Flux, Correlations in this section can be used to predict the incident heat flux distribution radi- 1 0.1 1 10 ally out from a corner along the (x+H) / L ceiling. The heat fluxes to the f,tip ceiling were determined to be a function of dimensionless dis- FIGURE 36. Maximum Heat Flux Along the Top of the Walls tance along the flame length, During Corner Fire Tests with Square Burner Sides of ●-0.17 m, ▲-0.30 m, ▼-0.30 m (Elevated), and ■-0.50 m and Fire Sizes from (r + H)/Lf,tip. A plot of the heat fluxes measured along the ceil- 50 to 300 kW. Data from Lattimer et al.91 ing out from the corner is shown in Figure 37. A correlation to predict the heat flux distribution along the ceiling is as follows:

(Eq. 67a)

(Eq. 67b)

Where: boundaries.91 The heat flux at the impingement point r = Radial distance from the corner (m) can be estimated using Equation 67 with r = 0. H = Distance between the base of the fire and the ceiling (m) Data Requirements Lf,tip = Flame tip length calculated using Equations 61 and 62 (m) 1. Diameter of the fuel package, D. For noncircular fuel packages, the equivalent diameter may be This correlation is similar to the one developed calculated using Equation 63 and the surface area for predicting the maximum heat flux along the top of the fuel package. of the walls, Equation 66, except the length scale 2. Heat release rate of the fire, . here is r instead of x. Again, the assumed plateau in 3. Distance between the base of the fire and the the correlation was based upon the maximum heat ceiling, H. flux levels measured in larger fire tests with burning

56 1000 ) 2

100

10 Heat Flux to Ceiling (kW/m

1 0.1 1 10

(r+H) / L f,tip

FIGURE 37. Heat Flux Along the Ceiling Above a Fire in a Corner During Tests with Square Burner Sides of ●-0.17 m, ▲-0.30 m, ▼-0.30 m (Elevated), and ■-0.50 m and Fire Sizes from 50 to 300 kW. Data from Lattimer et al.91

4. Location along the surface where incident heat vertical may result in different total incident heat flux level is needed. This could be the elevation flux levels as a result of the flame’s becoming sepa- along the height of the corner, z, horizontal rated from the wall or the difference in entrainment distance from the corner along the top of the into the plume. Incident heat fluxes to the corner walls, x, or radially out from the corner along walls across the width of the fire are constant and are the ceiling, r. equal to the maximum vertical heat flux distribution in the corner. Heat fluxes along the top of the walls Data Sources are constant and equal to the maximum horizontal heat flux distribution along the top of the walls. 1. Heat release rate data may be obtained from 69 70 Babrauskas, Hoglander and Sundstrum, or Validation Mudan and Croce.71 Other studies have been conducted with propane fires in a corner configuration with and without a Assumptions ceiling. Corner heat flux data with no ceiling93 This analysis assumes that the fire is attached to agree well with the heat flux data in Figure 34 when the corner walls, the corner walls are vertical and at considered relative to the flame tip. In a study with a 90° angle, and the ceiling is horizontal and at a fires in a corner and a ceiling, Hasemi et al.94 90° angle with the corner walls. Walls that are not measured incident heat flux levels on the walls and

57 ceiling from both exposure fires and simulated may be slightly spaced from the corner will yield burning boundaries. Trends in incident heat flux conservative results. Flames less luminous than levels measured by Hasemi et al.94 along the top of those produced by the propane fires (i.e., natural the walls and the ceiling agree well with the data in gas) may transmit lower total incident heat fluxes to Figures 36 and 37 when using the dimensionless the surfaces because the radiative heat flux to the distances used in these figures. In tests with propane wall will be lower.87,88,96 The propane fuel fires gas burners against a non-combustible boundary, used to develop the heat flux data presented in this similar heat flux levels have been measured by section produce a moderate amount of soot; there- other investigators for limited conditions.87,88 fore, heat flux levels presented in this section should Ohlemiller, Cleary, and Shields95 measured peak be considered to be average but not bounding for all heat fluxes approximately 10% to 20% higher using different fuels. Propane burners are also used exten- similar size propane square burners. sively in standard fire tests as an exposure fire that Lattimer et al.91 also demonstrated that the corre- is representative of real fires. Therefore, the incident lations for incident heat fluxes in the three regions heat fluxes from these flames are considered to be of the corner configuration also hold when the representative of those produced by most fires. boundary is combustible and burning. For this case, a modified length scale is required to correctly pre- Fires Impinging on Unbounded Ceilings dict flame length. Fires that impinge onto an unbounded ceiling as shown in Figure 38 have flames that are redirected Limitations radially out from the impingement point. The Correlations for incident heat fluxes were devel- highest heat fluxes onto the ceiling will be at the oped using luminous flames in an open environment impingement or stagnation point. Heat fluxes will with the fire directly in the corner. Using these tend to decrease with radial distance away from the relations inherently assumes: stagnation point. Correlations are provided in this section to estimate the heat fluxes from such a fire • There is negligible heating from a hot gas layer to the ceiling. in the surroundings. The incident heat flux due to a fire impinging • The fire is against the wall. onto an unbounded flat ceiling has been experimen- • The flames are luminous. tally characterized by Hasemi et al.97 In this study, • The corner walls are vertical and at a 90° angle. Hasemi et al.97 conducted a series of fire tests using • The ceiling is horizontal and at a 90° angle propane gas burners located at different distances with the corner walls. beneath a non-combustible unbounded ceiling. The test configuration is shown in Figure 38 along with The experimental study considered fire diameters as important variables. Fires as large as approximately large as 0.50 m and heat release rates as large as 400 kW were considered in the study. Heat flux 300 kW. No data were available to validate the cor- gauges were used to measure the incident heat flux relations against fires with larger diameters or high- along the ceiling both directly above the centerline er heat release rates. The presence of a hot gas layer of the fire (i.e., stagnation point) and radially out may increase the total incident flux onto the wall, from the stagnation point. and if significant in the area of interest adding this A plot of the heat flux levels at the stagnation contribution to the total incident heat flux from the point is shown in Figure 39. Heat fluxes at the fire plume may be warranted.89 Moving the fire stagnation point are shown in this figure to plateau away from the corner will eventually cause the inci- at approximately 90 kW/m2. In order to collapse dent heat fluxes to become lower, largely because the data, the unconfined flame tip length was the flame becomes detached from the wall.90 Thus, normalized with respect the distance between the the use of correlations in this section for fires that ceiling and fire, H, plus the virtual source origin

58 Stagnation Point

Ceiling

Lf (Unconfined Flame H Length)

Exposure Fire

Virtual Point Z' Source Correction

FIGURE 38. Unbounded Ceiling Configuration correction, z'. The unconfined fire flame tip length = Heat release rate of the fire (kW) was calculated using the following relation: D = Diameter of the fuel package (m) ρ 0 = Density of air at initial ambient conditions (Eq. 68) (1.2 kg/m3)

cp = Specific heat capacity of air at initial Where: ambient conditions [1.0 kJ/(kg K)] 2 * n = /5 for Q D > 1.0 T =Temperature at initial ambient conditions 2 * 0 n = /3 for Q D < 1.0 (293 K) g = Gravitational acceleration (9.81 m/s2) (Eq. 69)

59 100

) 60 2 D = 1.0m H = 1.0m H = 1.2m (kW/m

" H = 0.8m s

. q 40 H = 0.6m H = 0.4m D = 1.0m H = 0.64m 20 H = 0.8m H = 1.0m D = 0.3m H = 1.0m H = 0.8m 0 0 1 2 3 4 5 6 7 8 9 10

Lf /(H + z') (-)

FIGURE 39. Stagnation Point Heat Fluxes on an Unbounded Ceiling with a Fire Impinging on It. Data from Hasemi et al.97

For noncircular fuel packages with a length to Where: * width ratio of near one, the equivalent diameter of Q D = Dimensionless quantity defined in the fuel package can be estimated using the surface Equation 69 area, A, of the noncircular fuel package: D = Diameter of the fuel package (m)

(Eq. 70) The radial distribution in the incident heat flux decays with distance from the stagnation point as Where: shown in Figure 40. The length of the flame used to A = Surface area of the fuel package (m2) correlate this data was the measured flame extension plus a virtual origin correction. The measured flame The virtual point source correction for this geom- extension was defined as the distance between the etry was determined using the following relations: fire and the ceiling, H, plus the radial extension

of the flame out from the center of the fire, LH. The location of the flame tip in this geometry was (Eq. 71a) * found to correlate with Q H, which is defined the same as in Equation 69 except the length scale is H instead of D. The flame tip correlation was deter- (Eq. 71b) mined to be

(Eq. 72)

60 100 D = 0.5m H = 1.0m H = 1.2m H = 0.8m H = 0.6m H = 0.4m D = 1.0m H = 0.64m H = 0.8m H = 1.0m ) 2 D = 0.3m H = 1.0m 10 H = 0.8m (kW/m . q"

0 0.1 1.0 10.0

(r + H + z')/(LH + H + z') (-)

FIGURE 40. Heat Fluxes to a Ceiling Due to a Propane Fire Impinging on the Surface. Data from Hasemi et al.97

Where: The radial heat flux distribution along the ceiling at w > 0.45 can be estimated using the correlation (Eq. 73) recommended by Wakamatsu98:

LH = Flame extension along ceiling from the (Eq. 74a) stagnation point to the flame tip (m) H = Distance between the base of the fire and Where: the ceiling (m) w = (-) (Eq. 74b) = Heat release rate of the fire (kW) ρ 0 = Density of air at initial ambient conditions r = Radial distance along the ceiling from the (1.2 kg/m3) stagnation point (m) cp = Specific heat capacity of air at initial H = Distance between the base of the fire and ambient conditions [1.0 kJ/(kg K)] the ceiling (m) T0 =Temperature at initial ambient conditions z' =Virtual source origin correction (m)

(293 K) LH = Flame extension along ceiling from the g = Gravitational acceleration (9.81 m/s2) stagnation point to the flame tip (m)

61 Figure 41 contains a plot of Equation 74 (dashed Data Requirements line) along with a representation of the data of 1. Diameter of the fuel package, D. For noncircular Hasemi et al.97 for a flat unbounded ceiling. As fuel packages, the equivalent diameter may be noted in Equation 74, this correlation adequately calculated using Equation 70 and the surface area estimates the data when w > 0.45, but significantly of the fuel package. overestimates heat flux levels for smaller values 2. Heat release rate of the fire, . of w. Based on the data from Hasemi et al.97 and 3. Distance between the base of the fire and the other data from fires impinging on I-beams mounted ceiling, H. to a ceiling,98 a correlation was developed to 4. Radial location out from the centerline of the predict the bounding heat flux levels where w is fire, r, where the incident heat flux level defined in Equation 74b: is needed. (Eq. 75a) Data Sources (Eq. 75b) 1. Heat release rate data may be obtained from This correlation is shown in Figure 41 as the solid Babrauskas,69 Hoglander and Sundstrum,70 line. The peak heat flux of 120 kW/m2 at w ≤ 0.5 or Mudan and Croce.70 bounds nearly all the heat flux measurements made in this range for the studies of Hasemi et al.97 and Myllymaki and Kokkala.98

1000 ) 2 100 (kW/m " q .

10 Heat Flux,

1 0.1 1 10

w = (r + H + z')/(LH + H + z')

FIGURE 41. Comparison of the Best Fit Curve Proposed by Wakamatsu (– –) and a Bounding Fit to the Data (—). The unbounded ceiling data of Hasemi et al.97 is represented as the outlined area.

62 Assumptions Fire Impinging on a Horizontal I-Beam Mounted Below a Ceiling The fire is assumed to be impinging on a horizontal, flat ceiling far from walls or any The final geometry considered is an I-beam that other obstructions. is mounted below a ceiling as shown in Figure 42, with the fire impinging on the lower flange of the Validation I-beam. The focus here is the heat fluxes from the fire onto the I-beam. This case turns out to be quite Several experimental and theoretical studies have similar to a fire impinging onto an unbounded ceiling. been performed on fires impinging on an unbounded Two separate studies have been conducted to ceiling.77,78,79,80,97,99,100,101 Total heat fluxes from evaluate the heat flux incident onto an I-beam fires and fire plumes impinging on the ceiling were mounted below a ceiling with an exposure fire measured by Hasemi et al.97, You and Faeth,79,80 impinging upon the beam (Hasemi et al.,97 and Kokkala.77,78 Due to the fuel type and size of Wakamatsu et al.,102 and Myllymaki and fires evaluated, heat flux levels measured by Hasemi Kokkala98). In these studies, the heat flux was et al.97 were higher than those measured in other measured along the four surfaces of the I-beam studies. Therefore, the correlations developed using noted in Figure 42: the data of Hasemi et al. are considered conservative. 1. Downward face of the lower flange Limitations 2. Upward face of the lower flange 3. The web Correlations for incident heat fluxes were devel- 4. Downward face of the upper flange oped using luminous flames in an open environment with the fire beneath an unbounded flat ceiling. The I-beam evaluated in these studies was 3.6 m Using these relations inherently assumes negligible long, a web 150 mm high and 5 mm thick, and heating from a hot gas layer in the surroundings, the flanges 75 mm wide and 6 mm thick. For each of flames are luminous, and the ceiling is horizontal. these surfaces, heat fluxes were measured from the The presence of a hot gas layer may increase the stagnation point of the fire (centerline of the fire) total incident flux onto the wall, and if significant in along the length of the I-beam. the area of interest adding this contribution to the Results from these studies have demonstrated total incident heat flux from the fire plume may be that the incident heat flux onto all surfaces of the warranted. Flames less luminous than those pro- beam will be equal to or less than the heat flux duced by the propane fires (i.e., natural gas) may levels measured with a fire impinging onto a flat transmit lower total incident heat fluxes to the wall unbounded ceiling. Wakamatsu et al.102 measured because the radiative heat flux to the wall will be this for fires up to 900 kW. Flame lengths were lower. Propane flames do not have the highest soot observed to be different along the lower flange, production of any fuel, and, therefore, incident heat upper flange, and center of the web of the I-beam. fluxes may not be bounding. However, propane Correlations to predict these flame lengths were burners are used extensively in standard fire tests as developed for the lower flange,102 an exposure fire that is representative of real fires. Therefore, the incident heat fluxes from these (Eq. 76) flames are considered to be representative of those produced by most fires. Where: (Eq. 77)

63 the upper flange,102 and for the center of the web,98 (Eq. 78) (Eq. 80)

Where: Where: (Eq. 79) (Eq. 81)

Ceiling

Downward Face of Upper Flange Web Upward Face of Lower Flange

Downward Face of Lower Flange

Ceiling Stagnation Point LC

LW I-Beam

HC HW HB

Exposure Fire

FIGURE 42. I-Beam Mounted Below an Unbounded Ceiling

64 LB = Flame extension along lower flange from z' =Virtual source origin correction (m) the stagnation point to the flame tip (m) LC = Flame extension along upper flange from LC = Flame extension along upper flange from the stagnation point to the flame tip (m) the stagnation point to the flame tip (m)

LW = Flame extension along the web center from The dimensionless distance for the web on the the stagnation point to the flame tip (m) I-beam was taken to be

HB = Distance between the base of the fire and bottom of the lower flange (m) (Eq. 84)

HC = Distance between the base of the fire and the ceiling (m) Where:

HW = Distance between the base of the fire and r = Radial distance along the I-beam from the the center of the web (m) stagnation point (m)

= Heat release rate of the fire (kW) HW = Distance between the base of the fire and ρ 0 = Density of air at initial ambient conditions the center of the web (m) (1.2 kg/m3) z' =Virtual source origin correction (m)

cp = Specific heat capacity of air at initial LW = Flame extension along web center from ambient conditions [1.0 kJ/(kg K)] the stagnation point to the flame tip (m)

T0 =Temperature at initial ambient conditions (293 K) The incident heat flux levels measured by g = Gravitational acceleration (9.81 m/s2) Wakamatsu et al.102 on the different faces of the I-beam are shown in Figure 43. On the downward The form of these correlations is similar to that face of the lower flange (where the fire was directly for the unbounded ceiling flame length correlation impinging), heat flux levels along the flame length given in Equation 72. The dimensionless distance were measured to be similar to the incident heat along the flame beneath the downward face of the fluxes measured along a flame under an unbounded lower flange was taken to be ceiling. However, all other surfaces of the I-beam had heat fluxes somewhat lower than those meas- (Eq. 82) ured along a flame under an unbounded ceiling. The study of Myllymaki and Kokkala98 con- Where: sidered the effects of larger fires (up to 3.9 MW) on r = Radial distance along the I-beam from the the heat flux incident on the different faces of the stagnation point (m) I-beam. Some of the heat flux measurements made

HB = Distance between the base of the fire and in this study are shown in Figure 44. In this study, the lower flange (m) Myllymaki and Kokkala98 found that, for fires over z' =Virtual source origin correction (m) 2.0 MW, the incident heat fluxes onto all faces of

LB = Flame extension along lower flange from the I-beam were equivalent to or slightly higher the stagnation point to the flame tip (m) than those measured along an unbounded ceiling. Data from these studies demonstrate that the The dimensionless distance for the upper flange heat flux to the I-beam can be conservatively esti- on the I-beam was taken to be mated using the bounding heat flux correlation in Equation 85 using the appropriate expression for w (Eq. 83) provided in Equations 82 through 84:

Where: (Eq. 85a) r = Radial distance along the I-beam from the (Eq. 85b) stagnation point (m)

HC = Distance between the base of the fire and the upper flange (m)

65 H = 1.0m Q = 100 kW H = 0.6m Q = 95 kW H = 1.2m Q = 540 kW Flat Ceiling Maximum Q = 150k Q = 130k Q = 750k Flat Ceiling Minimum Q = 200k Q = 160k Q = 900k

Lower Flange Downward Web 100 100 ) ) 2 2

10 10 (kW/m (kW/m

" " . q . q

flame tips

0 0 0.1 1.0 10.0 0.1 1.0 10.0 (r + H + z')/(L + H + z') (-) B B B (r + HC + z')/(LC + HC + z') (-)

Lower Flange Upward Upper Flange Downward 100 100 ) ) 2 2

10 10 (kW/m (kW/m

" " . q . q

0 0 0.1 1.0 10.0 0.1 1.0 10.0 (r + H + z')/(L + H + z') (-) (r + HC + z')/(LC + HC + z') (-) C C C

FIGURE 43. Heat Flux Measured Onto the Surfaces of an I-Beam Mounted Below an Unbounded Ceiling for Fires 95 to 900 kW102

Data Requirements Data Sources 1. Diameter of the fuel package, D. For noncircular 1. Heat release rate data may be obtained from fuel packages, the equivalent diameter may be Babrauskas,69 Hoglander and Sundstrum,70 or calculated using Equation 70 and the surface area Mudan and Croce.71 of the fuel package. 2. Heat release rate of the fire, . Assumptions 3. Distance between the base of the fire and the bottom flange, center of the web, and the top of The I-beam being analyzed should have similar the flange. dimensions to the one considered in these two 4. Distance out from impingement point on the studies (3.6 m long, a web 150 mm high and 5 mm I-beam where the heat flux is needed, r. thick, and flanges 75 mm wide and 6 mm thick), and the fire is assumed to be impinging directly onto the bottom flange of the I-beam. The I-beam is

66 1000

100 ) 2 (kW/m "

q 10 . Heat Flux, 1

0.1 0.1 1 10 w (- -)

FIGURE 44. Heat Flux Measured on the ●-Bottom Flange, ■-Web, and ▲-Upper Flange of an I-Beam Mounted Below an Unbounded Ceiling for Fires 565 to 3,870 kW.98 The line in the plot is the curve given in Equation 85.

also assumed to be located remote from any walls the fire directly impinging on the I-beam. Using or ceiling obstructions. these relations inherently assumes negligible heating from a hot gas layer in the surroundings and that Validation the I-beam is not located near any boundaries. The presence of a hot gas layer may increase the total These two studies provide a good validation of incident flux onto the I-beam, and, if significant, this the heat fluxes experienced by the particular I-beam contribution should be added to the total incident tested. Results produced using propane fuel fires heat flux from the fire plume.90 Moving the fire agreed well with the larger liquid heptane pool away from the I-beam so that it does not impinge on fire tests. the lower flange will change the heat flux distribution on the I-beam. These test data were developed

Limitations with 0.48 < QH*< 1.27, fire distance below the lower flange of 0.6 < H < 1.9, fire diameters up to The height of the webbing and the width of the B 1.6 m, and heat release rates up to 3.9 MW. Though flanges may affect the heat fluxes to the I-beam. results in this section indicate the heat flux is Other size I-beams have not been tested to evaluate bounded by the correlation in Equation 85, heat the impact of I-beam dimensions on heat flux. Cor- fluxes from large pool fires (D > 1.6 m) impinging relations for incident heat fluxes were developed on an I-beam may be higher due to the changes in using luminous flames in an open environment with gas emissivity and flame temperature.67,68

67 SUMMARY AND RECOMMENDATIONS while for the smaller fires in the section on specific geometries, the upper limit of the heat flux measure- The motivation for the work in the Bounding ments is more like 120 kW/m2. Therefore, the user Heat Flux section has been the effect of the fire on of this information must take into account the size objects in flames. Those studies were interested in and configuration of the fire. The type of fuel is less the ability of nuclear waste casks or structural ele- likely to be a factor. ments in offshore drilling facilities to withstand fire. Another issue that should be recognized in apply- On the other hand, the motivation for the work ing these results is that they are presented in terms reported in the section on Heat Fluxes for Specific of incident heat flux, or the heat flux as measured to Geometries was primarily the effect of fire on igni- a cold target. In a design application, the heat flux tion and fire growth (except for the I-beam studies). that is absorbed into the structural element will As a consequence, smaller exposure fires are con- decrease as the surface temperature increases. The sidered in the latter section. For example, in the for- boundary condition that should be used for the mer, fires of up to 9 by 18 m were used (more than structure should account for the radiation loss for 300 MW) as compared to fires of up to 1 m at most elements impacted by a fire plume: or about 500 kW for the latter section. For the I- beam study, data include larger fires of 3.9 MW at most. The differences in the two sections are pro- found, and the reader should be aware of these dis- Where: tinctions in using the correlations. It is clear that = Incident heat flux given herein pool-like fires exhibit higher temperatures and ε = Surface emissivity therefore higher heat fluxes as they become bigger. To = Cold target temperature For example, a flame temperature of 1200°C corre- 2 sponds to a radiant heat flux of 267 kW/m . Yet in No factor of safety is addressed, and the user the Bounding section (Table 4), most measurements must be aware that that is not implicit in any of 2 are more generally in the range of 150 kW/m , these results.

68 Appendix A Theoretical Examination of Methods

As can be seen in Figure A.1, predictions of Results by Harmathy burning rate vary markedly among the different for Wood Cribs methods. Some of the methods assume stoichiometric or ventilation-limited burning, while others account for fuel-controlled burning.

0.18

0.16

0.14 ) 5/2

0.12 Tanaka Eurocode

(kg/s – m Lie

0.10

1/2 Harmathy o H

o Magnusson 0.08 /A Babrauskas 1/2 Law 0.06 Ma (D/W) F . m 0.04

0.02

0 05101520 25 30 35 40 45 50 55 1/2 –1/2 A/AoHo (m )

FIGURE A.1. Comparison of Burning Rate Predictions

69 From Equation 20 similar results can be derived:

Results by Bullen and Thomas for Pool Fires

, s ≈ 4 for wood, The locus of s ≈ 7-10 for liquids. The pool fire results are explained by the stoichiometry and thermal feedback.

s and both are larger for liquids compared to wood.

FIGURE A.2. Wood Crib and Liquid Pool Fires

70 CIB Data In the CIB experiments the fuel is placed over the entire floor; therefore, AF ≈ A. Here the theory gives

From Equation 26, for and For temperature, T= f(F, t) and C, a constant which takes into account the properties of the bounding materials of the enclosure. for which agrees with the trends in Figure A.2.

This represents kρc since k ~ ρ. Eurocode Magnusson, Thelandersson, The Eurocode prescribes and Petersson Magnusson, Thelandersson, and Petersson compute a result for temperature based on a similar theory. They augment it with a rate of rise for developing fire and a prescribed cooling phase. They use only the ventilation-controlled fire for cribs from Kawagoe and Sekine:

* *–2 . Here t is essentially Qw (The theoretical development gives or only. .)

This specification must assume a ventilation- They compute results for and kρc for limited fire and ignores the other variables. various fuel loads. based on a distribution of fuel over Lie the entire compartment surface area A. (Normally Only a ventilation-limited fire is assumed. fuel loading is based on floor area, i.e. .) Lie gives . The theoretical development gives Babrauskas

. A computer solution was correlated to give an analytical result:

71 The mass loss rate is correlated as

for

Where: D = Compartment depth W = Compartment width

Ma and Mäkeläinen These authors develop a correlation based on the CIB and other data. Its novel feature is that it θ θ * Both 2 and 3 correspond to Qw , but not includes a prediction of temperature over time θ exactly, since the powers are different in 3 for starting at the onset of the fully developed stage. each term. Since the dimensionalization of the They use Harmathy’s result for the burning rate in equations must be consistent, it suggests that there the fuel-controlled regime, and his demarcation of θ is an inconsistency in 3. the regime change to ventilation-limited:

This corresponds to . They use Law’s correlation for the ventilation- θ 5 pertains to combustion efficiency and is only limited burning rate. relevant if the theoretical heat of combustion is used. The temperature is given as It is interesting that the maximum temperature given by the correlation is 1425ºC. The theory suggests this is 1500ºC at most. Where: Law δ = 0.5 for the ascending phase and 1.0 for the descending phase Law developed a correlation based on the CIB 16 data. A fit giving the maximum or upper values The maximum temperature is given as linear fits of data is to the CIB and other data in terms of . The time at the maximum temperature is selected as . This model does not include the effect of the wall thermal properties. where A is the heat transfer area of the boundary surfaces, not including the vents (as used in the theory). An adjustment is made if the fuel load is low.

72 Appendix B Comparisons of Enclosure Fire Predictions with Data

Predictions of compartment fire temperature and the CIB data was not normalized by the square root duration are compared to two sets of data. The first of the ratio of compartment depth to width, there set of data is from 321 experiments conducted was more scatter in the data. under the auspices of CIB.48 See the section entitled The methods presented in this guide were evalu- CIB beginning on page 31 for more information ated by plotting predictions of average temperature on these experiments. The compartments in these during the fully developed stage along with the experiments were roughly cubic, although some of CIB data. When comparing predictions to data, the compartments had aspect ratios (length to averages were taken of what appeared to be the width) of 1/2 or 2. fully developed stage from the temperature data. In these experiments, the stage of fully devel- Similarly, predictions of duration were compared to oped burning was defined as the period from when the CIB data by dividing the initial mass of fuel, the mass of fuel was between 80% and 30% of the mf , by the predicted duration, τ, and plotting this original, unburned fuel mass. Average temperatures quantity along with the CIB data. during the period of fully developed burning from Some of the predictive methods required as input these experiments were presented as a function the surface area of the fuel. The ratio of fuel surface of . area to total room surface area (defined as including the area of the ceiling and walls, but not the area of Average burning rate data during the fully the ventilation opening or the floor) was calculated developed stage was presented as for each of the CIB experiments. The average ratio of fuel surface area to total room surface area in as a function of . Data was these experiments was 0.75, with a standard devia- tion of 0.90. Figure B.1 shows a histogram of the also included where the average burning rate during ratio of fuel surface area to the enclosure surface the fully developed burning stage was presented in area for the CIB experiments. For methods that tables of as a function of . require as input the fuel surface area, the value of 0.75A was used for comparing predictions to the 48 Although both the CIB report and the Cardington CIB data. 103 data show that the aspect ratio of a compartment To explicitly analyze the effect of long, narrow can influence the burning rate for fully developed, compartments, temperature data as a function of ventilation-limited fires, most predictive methods time from a series of experiments that were con- do not explicitly account for this effect. Therefore, ducted in a compartment that was approximately predictive methods that do not account for compart- 23 meters long, 2.7 meters high, and 5.5 meters ment aspect ratio were evaluated using the CIB wide103 were compared to predictions. In these burning rate data, which was normalized by the area experiments, the ventilation opening ranged from and square root of the height of the ventilation 1/8 to 1/1 of the small side of the compartment. opening, but not by the square root of the ratio of The fuel loading consisted of wood cribs with a compartment depth to width. Methods that do total density of 20 or 40 kg/m2. Additionally, for specifically account for the compartment aspect one experiment, the compartment size was reduced ratio were evaluated using the CIB data that was to approximately 5.6 x 5.6 x 2.75 meters (high). The normalized by both the area and square root of the full details of the experiments may be found in height of the ventilation opening and the square root reference 103. of the ratio of compartment depth to width. When

73 Distribution of Area Ratio 100

40 Frequency 30

0 0.04 0.40 0.76 1.11 1.47 1.83 2.18 2.54 2.90 3.25 3.61 3.97 4.32 4.68 Area of Fuel/Total Area

FIGURE B.1. Histogram of Ratio of Fuel Surface Area to Enclosure Surface Area for the CIB Experiments

CIB Data Cardington Data The experiments in the CIB study were con- A total of nine experiments were conducted ducted in a variety of enclosures since multiple under a collaborative project between British Steel laboratories participated. Statistical means were and the British Research Establishment’s Fire used to overcome systematic differences between Research Station. The experiments were conducted the laboratories. The majority of the laboratories in a purpose-built compartment within the British used a test enclosure constructed of 10 mm thick Research Establishment’s ex-airship hanger. asbestos millboard with a reported thermal conduc- The floor of the compartment was made of tivity of 0.15 W/m°C, and this is the value that was 75 mm thick concrete covered with sand. The walls used for methods that required specific heat as an were made of lightweight concrete blocks that input. The density of the asbestos millboard and measured 440 x 215 x 215 mm. In most tests the the specific heat were not reported, so values of walls were lined with a 50 mm thick ceramic fiber 816 J/kg°C and 1100 kg/m3 were selected.27,104 blanket. However, in one of the tests (test #8) the In the CIB study, separate graphs of temperature walls were lined with two 12.5 mm thick plaster- and burning rate data were presented for cribs with board sheets affixed onto 47 x 47 mm wood studs 20 mm thick wood sticks spaced 20 mm apart, and spaced 600 mm apart. The ceiling was constructed for cribs with 20 mm wide sticks spaced 60 mm of 200 mm thick aerated concrete slabs and was apart, or with 10 mm wide sticks spaced 30 mm lined in the same manner as the walls. apart. However, for purposes of comparing predic- The opening of the compartment was located on tions with the CIB data, all temperature and burning one of the smaller walls, and concrete blocks were rate data was aggregated into single graphs. used to restrict the opening to 100%, 50%, 25%, or 12.5% of the wall size. Additionally, in some of the tests a 400 mm insulated steel column was placed flush with the opening, which further reduced the opening size.

74 TABLE B.1. Compartment Dimensions of the TABLE B.4. Fuel Loading for the Cardington Tests Cardington Tests

Test # Length (m) Width (m) Height (m) Test # Fuel Load (kg/m2) 1 22.855 5.595 2.750 140 2 22.855 5.595 2.750 220 3 22.855 5.595 2.750 320 4 22.855 5.595 2.750 440 5 22.855 5.595 2.750 520 6 22.855 5.595 2.750 620 7 5.595 5.595 2.750 720 8 22.780 5.465 22.780 8 20.6 9 22.855 5.595 2.750 920

TABLE B.2. Opening Dimensions of the The dimensions of the enclosure are provided in Cardington Tests Table B.1,103 the dimensions of the opening are listed in Table B.2,103 and the properties of the Test # Total Width (mm) Height (mm) enclosure materials are listed in Table B.3.103 1 5595 2750 The fuel for the Cardington tests was wood cribs, 2 5595 2750 constructed of 1 m long sticks of 50 x 50 mm western hemlock spaced 50 mm apart. The heat of combustion 3 5195 1470 of the wood was reported as 19.0 MJ/kg. The fuel 4 5195 1470 loading for each of the tests can be found in Table B.4. 5 2139 1730 In all but tests #7 and #9, the fires were ignited at the rear of the compartment (opposite the end with 6 5195 375 the ventilation opening). In tests #7 and #9, all cribs 7 1370 2750 were ignited simultaneously. In all the tests, the fire spread to the cribs nearest the ventilation opening, 8 5065 2680 and, once the fire reached the cribs nearest the ven- 9 5195 2750 tilation opening, the cribs further away from the ventilation opening ceased burning. The cribs TABLE B.3. Properties of Enclosure Materials nearest the ventilation opening continued Thermal burning, and, as the fuel Density Specific Heat Conductivity Structure Material (kg/m3) (J/kg K) (W/m K) was depleted, the fires progressed toward the Walls Lightweight 1375 753 0.42 rear of the enclosure. concrete blocks As a result, the temper- Roof Aerated 450 1050 0.16 atures were not hori- concrete slabs zontally homogeneous, Floor Sand 1750 800 1.0 and higher temperatures at any given time were Fiber lining Ceramic fiber 128 1130 0.02 measured above the Plasterboard Fireline 900 1250 0.24 location where the fire lining plasterboard was burning.

75 The temperature data from the Cardington tests averages of the temperature predictions during the * * was compared to predictions made using the time in which t < td were compared to the CIB methods identified in this guide by comparing the data. The Eurocode method was evaluated as measured temperatures to predictions. Temperatures presented, and the modifications suggested by were measured at locations approximately 3, 11, Buchannan and Franssen also were evaluated. A and 19 m (measured horizontally) from the ventila- graph of Eurocode predictions and the CIB data is tion opening. In the graphs, averages of the thermo- presented in Figures B.2 and B.3. couple measurements are plotted, with error bars The predicted duration of the fully developed * * indicating the range of the measured temperatures. burning stage is when t = td . Given that Predictions were made using each of the methods , and t* = tΓ, identified in this guide at 3-minute intervals for tests #1, 2, 3, 7, and 9; at 6-minute intervals for tests #4, the predicted duration in hours would be 5, and 8; and at 25-minute intervals for test #6. For predictive methods that have distinct correla- , tions for fuel-controlled and ventilation-controlled burning, the fire was assumed to be ventilation con- where τ is in hours and can be rewritten as trolled. Given the behavior of the burning, this is a . reasonable assumption.

Eurocode Substituting ,

CIB DATA In the CIB experiments, the mass of fuel per unit Since , area ranged from 20 to 40 kg/m2. (A few tests used a mass of fuel per unit area of 10 kg/m2 but, since . the CIB report indicated that only a “few” tests were conducted at this density, this value was not modeled.) For an effective heat of combustion for 33 pine of 12.4 MJ/kg, qt,f would range from 248 to 496 MJ/m2, and multiplying this by the ratio of Since Afloor/A in the CIB compartments results in a range 2 of qt,d of approximately 50 to 100 MJ/m . , which can be Predictions of temperature as a function of time were made using the Eurocode method for values of ranging from 5 to 50 m–1/2. Predictions were made at time increments ranging from rearranged as (kg/h) or 0.005 hours to 5 hours, depending on the values of ∆ qt,d and used. For each value of , (kg/s). Substituting Hc = 12.4 MJ/kg, the predicted burning rate would be . This is compared to the CIB burning rate data in Figure B.4.

76 1400

1200

1000 CIB Data 800 Eurocode

600 Buchanan

Temperature (°C) 400 Franssen

200

0 01020304050

1/2 –1/2 A/AoHo (m )

FIGURE B.2. Comparison of CIB Temperature Data to Predictions Made Using Eurocode, 2 Buchanan, and Franssen Methods, qt,d = 100 MJ/m

1400

1200

1000 CIB Data 800 Eurocode

600 Buchanan

Temperature (°C) 400 Franssen

200

0 01020304050

1/2 –1/2 A/AoHo (m )

FIGURE B.3. Comparison of CIB Temperature Data to Predictions Made Using Eurocode, 2 Buchanan, and Franssen Methods, qt,d = 50 MJ/m

77 0.18

0.16

) 0.14 5/2

0.12 121 221 0.10 211

(kg/s – m 441

1/2 0.08

o Eurocode H o 0.06 /A f

. m 0.04

0.02

0 01020304050 1/2 –1/2 A/AoHo (m )

FIGURE B.4. Comparison of CIB Burning Rate Data to Predictions Made Using the Eurocode Method

Franssen’s modification results in a calculated CARDINGTON DATA burning duration of 20 minutes when t* /Γ is d Inputs were created in accordance with the rec- less than 20 minutes. For the CIB data and 2 * ommendations of the Eurocode. When calculating qt,d = 50 MJ/m , t d /Γ is less than 20 minutes qt,d, the area of the ventilation opening was not for cases where was less than or equal to included in the calculation of the total surface area; however, the area of the openings was included in 10 m–1/2. With q = 100 MJ/m2, t* /Γ is less than t,d d calculations of the total surface area of the enclo- 20 minutes for cases where was less than sure. Predictions less than 20°C were assumed to indicate that the decay period had completed and or equal to 30 m–1/2. the temperature in the compartment was ambient. The results of the comparisons of predictions using the Eurocode to the Cardington data are presented in Figures B.5 through B.13.

78 1400

1200

1000

800 Measured Eurocode 600 Buchanan Franssen

Temperature (°C) 400

200

0 00.51.01.5 2.0 Time (h)

FIGURE B.5. Comparison of Predictions Made Using Eurocode, Buchanan, and Franssen Methods to Data from Cardington Test #1

1400

1200

1000

800 Measured Eurocode 600 Buchanan Franssen

Temperature (°C) 400

200

0 00.20.40.6 0.8 1.0 1.2 1.4 1.6 Time (h)

FIGURE B.6. Comparison of Predictions Made Using Eurocode, Buchanan, and Franssen Methods to Data from Cardington Test #2

79 1400

1200

1000

800 Measured Eurocode 600 Buchanan Franssen

Temperature (°C) 400

200

0 00.511.522.5 Time (h)

FIGURE B.7. Comparison of Predictions Made Using Eurocode, Buchanan, and Franssen Methods to Data from Cardington Test #3

1400

1200

1000

800 Measured Eurocode 600 Buchanan Franssen

Temperature (°C) 400

200

0 00.511.5 2 2.5 3 3.5 4 Time (h)

FIGURE B.8. Comparison of Predictions Made Using Eurocode, Buchanan, and Franssen Methods to Data from Cardington Test #4

80 1400

1200

1000

800 Measured Eurocode 600 Buchanan Franssen

Temperature (°C) 400

200

0 00.511.522.53 Time (h)

FIGURE B.9. Comparison of Predictions Made Using Eurocode, Buchanan, and Franssen Methods to Data from Cardington Test #5

900

800

700

600 Measured 500 Eurocode Buchanan 400 Franssen

300 Temperature (°C)

200

100

0 012345678 Time (h)

FIGURE B.10. Comparison of Predictions Made Using Eurocode, Buchanan, and Franssen Methods to Data from Cardington Test #6

81 1400

1200

1000 Measured 800 Eurocode Buchanan 600 Franssen

Temperature (°C) 400

200

0 00.20.40.60.8 1.0 1.2 Time (h)

FIGURE B.11. Comparison of Predictions Made Using Eurocode, Buchanan, and Franssen Methods to Data from Cardington Test #7

1200

1000

800

Measured 600 Eurocode & Franssen Buchanan 400 Temperature (°C)

200

0 00.511.5 2 2.5 3 Time (h)

FIGURE B.12. Comparison of Predictions Made Using Eurocode, Buchanan, and Franssen Methods to Data from Cardington Test #8

82 1400

1200

1000

Measured 800 Eurocode Buchanan 600 Franssen

Temperature (°C) 400

200

0 00.51.01.5 2 Time (h)

FIGURE B.13. Comparison of Predictions Made Using Eurocode, Buchanan, and Franssen Methods to Data from Cardington Test #9

Lie enclosures used in the CIB tests was assumed to be 1100 kg/m3, the C factor used in Lie’s method Since it was not possible to determine the dura- would equal 1. A comparison of Lie’s predictions tion of burning for each data point in the CIB data and the CIB data can be found in Figure B.14. in a straightforward manner, to compare predictions Lie gives (kg/s). This is com- using Lie’s method to the CIB data average temper- pared to the CIB burning rate data in Figure B.15. ature predictions were made for a fire of 2 hours’ Comparisons of predictions using Lie’s method duration with opening factors F = to the Cardington data can be found in Figures B.16 through B.24. ranging from 0.02 to 1. Because the density of the

83 1400

1200

1000

800 CIB Data 600 Lie

Temperature (°C) 400

200

0 01020304050

1/2 –1/2 A/AoHo (m )

FIGURE B.14. Comparison of CIB Temperature Data to Predictions Made Using Lie’s Method

0.18

0.16

) 0.14 5/2 121 0.12 221 211 0.10 441 (kg/s – m Lie 1/2

o 0.08 Lie * 1.8 H

o Lie / 1.8 0.06 /A f

. m 0.04

0.02

0 01020304050

1/2 –1/2 A/AoHo (m )

FIGURE B.15. Comparison of CIB Burning Rate Data to Predictions Made Using Lie’s Method

84 1400

1200

1000

800 Measured Lie 600

Temperature (°C) 400

200

0 00.511.5 2 Time (h)

FIGURE B.16. Comparison of Predictions Made Using Lie’s Method to Data from Cardington Test #1

1400

1200

1000

800 Measured Lie 600

Temperature (°C) 400

200

0 00.20.40.6 0.8 1 1.2 1.4 1.6 Time (h)

FIGURE B.17. Comparison of Predictions Made Using Lie’s Method to Data from Cardington Test #2

85 1400

1200

1000

800 Measured Lie 600

Temperature (°C) 400

200

0 00.511.5 2 2.5 Time (h)

FIGURE B.18. Comparison of Predictions Made Using Lie’s Method to Data from Cardington Test #3

1400

1200

1000

800 Measured Lie 600

Temperature (°C) 400

200

0 00.511.522.5 3 3.5 4 Time (h)

FIGURE B.19. Comparison of Predictions Made Using Lie’s Method to Data from Cardington Test #4

86 1400

1200

1000

800 Measured Lie 600

Temperature (°C) 400

200

0 00.511.522.5 3 Time (h)

FIGURE B.20. Comparison of Predictions Made Using Lie’s Method to Data from Cardington Test #5

900

800

700

600

Measured 500 Lie 400

300 Temperature (°C)

200

100

0 012345678 Time (h)

FIGURE B.21. Comparison of Predictions Made Using Lie’s Method to Data from Cardington Test #6

87 1400

1200

1000

Measured 800 Lie

600

Temperature (°C) 400

200

0 00.20.40.6 0.8 1 Time (h)

FIGURE B.22. Comparison of Predictions Made Using Lie’s Method to Data from Cardington Test #7

1200

1000

800

Measured 600 Lie

400 Temperature (°C)

200

0 00.511.5 2 2.5 3 Time (h)

FIGURE B.23. Comparison of Predictions Made Using Lie’s Method to Data from Cardington Test #8

88 1200

1000

800

Measured 600 Lie

Temperature (°C) 400

200

00.511.5 2 Time (h)

FIGURE B.24. Comparison of Predictions Made Using Lie’s Method to Data from Cardington Test #9

TANAKA produced rapidly declining temperatures, and any temperature below 600°C was neglected. The result For Tanaka’s methods, it was not possible to of this comparison can be seen in Figure B.25. determine the duration of burning for each point in Both Tanaka’s method and Tanaka’s refined the CIB data in a straightforward manner. To com- method predict the mass loss rate as . pare predictions using Tanaka’s method and his This is compared with the CIB data in Figure B.26. refined method to the CIB data, average tempera- Comparisons of predictions using Tanaka’s ture predictions were made for a fire of 2 hours’ method, both the simple and refined versions, to the –1/2 duration with ranging from 1 to 50 m . Cardington data can be found in Figures B.27 through B.35. For = 1 m–1/2, Tanaka’s refined method

89 5000

4000

3000 CIB Data Tanaka Refined Tanaka 2000 Temperature (°C)

1000

0 01020304050 1/2 –1/2 A/AoHo (m )

FIGURE B.25. Comparison of CIB Temperature Data to Predictions Made Using Tanaka’s Methods

0.18

0.16

) 0.14 5/2

0.12 121 221 0.1 211

(kg/s – m 441

1/2 0.08 o Tanaka H

o Tanaka * 1.6 0.06 /A Tanaka / 1.9 f

. m 0.04

0.02

0 01020304050 1/2 –1/2 A/AoHo (m )

FIGURE B.26. Comparison of CIB Burning Rate Data to Predictions Made Using Tanaka’s Methods

90 1600

1400

1200

1000 Measured 800 Tanaka Refined Tanaka 600 Temperature (°C) 400

200

0 00.511.5 2 Time (h)

FIGURE B.27. Comparison of Predictions Made Using Tanaka’s Methods to Data from Cardington Test #1

3500

3000

2500

2000 Measured Tanaka 1500 Refined Tanaka

Temperature (°C) 1000

500

00.20.40.60.8 1.0 1.2 1.4 1.6 Time (h)

FIGURE B.28. Comparison of Predictions Made Using Tanaka’s Methods to Data from Cardington Test #2

91 3000

2500

2000

Measured 1500 Tanaka Refined Tanaka

1000 Temperature (°C)

500

0 00.511.5 2 2.5 Time (h)

FIGURE B.29. Comparison of Predictions Made Using Tanaka’s Methods to Data from Cardington Test #3

3500

3000

2500

2000 Measured Tanaka 1500 Refined Tanaka

Temperature (°C) 1000

500

0 01234 Time (h)

FIGURE B.30. Comparison of Predictions Made Using Tanaka’s Methods to Data from Cardington Test #4

92 2500

2000

1500 Measured Tanaka Refined Tanaka 1000 Temperature (°C)

500

0 00.511.522.53 Time (h)

FIGURE B.31. Comparison of Predictions Made Using Tanaka’s Methods to Data from Cardington Test #5

2000

1800

1600

1400

1200 Measured 1000 Tanaka Refined Tanaka 800

Temperature (°C) 600

400

200

0 012345678910 Time (h)

FIGURE B.32. Comparison of Predictions Made Using Tanaka’s Methods to Data from Cardington Test #6

93 3000

2500

2000

Measured 1500 Tanaka Refined Tanaka

1000 Temperature (°C)

500

0 00.20.40.6 0.8 1 Time (h)

FIGURE B.33. Comparison of Predictions Made Using Tanaka’s Methods to Data from Cardington Test #7

1600

1400

1200

1000 Measured 800 Tanaka Refined Tanaka 600 Temperature (°C) 400

200

0 00.511.522.53 Time (h)

FIGURE B.34. Comparison of Predictions Made Using Tanaka’s Methods to Data from Cardington Test #8

94 3500

3000

2500

2000 Measured Tanaka

1500 Refined Tanaka

Temperature (°C) 1000

500

0 00.511.52 Time (h)

FIGURE B.35. Comparison of Predictions Made Using Tanaka’s Methods to Data from Cardington Test #9

MAGNUSSON AND THELANDERSSON Since , the burning rate predicted using The enclosures that were used in the CIB tests Magnusson and Thelandersson’s method would were modeled as Type C (as defined by Magnusson and Thelandersson38) since the Type C enclosure be , which is identical to the most closely represents the material properties of method that Babrauskas recommends for ventila- the CIB enclosures. tion-controlled burning. A comparison of predic- Given that it was not possible to estimate the tions of burning rate made using Magnusson and burning rates applicable to the CIB data in a Thelandersson’s method to the CIB data is shown in straightforward manner, a duration of 2 hours was Figure B.37. arbitrarily selected. This selection should have only With the exception of test #8, which was modeled a minor influence on the comparison with the CIB as Type G, the Cardington enclosure was modeled data since only the average temperature during the as Type C. The area of the ventilation opening was fully developed stage is of interest. A comparison of not included in calculations of the surface area of predictions made in this manner with the CIB data the enclosure. Where values of or the is shown in Figure B.36. Magnusson and Thelandersson’s method predicts burning duration were not sufficiently close to burning duration as follows: the values presented in the tables, linear interpolation was performed. It was not possible to model test #6 using Magnusson and Thelandersson’s method since no table or graph was provided that where q is the fuel load in Mcal/m2 related to the resembled the conditions associated with test #6. surface area of the enclosure. Using a heat of com- Comparisons of predictions made using Magnusson bustion of 12.4 MJ/kg and converting units, this can and Thelandersson’s method to the Cardington data be reduced to . can be found in Figures B.38 through B.45.

95 1400

1200

1000

800 CIB Data Magnusson 600

Temperature (°C) 400

200

0 01020304050 1/2 –1/2 A/AoHo (m )

FIGURE B.36. Comparison of CIB Temperature Data to Predictions Made Using Magnusson and Thelandersson’s Method

0.18

0.16 )

5/2 0.14

0.12 121 221 0.10 211 (kg/s – m 441 1/2 0.08 o Magnusson H o 0.06 Magnusson * 1.3 /A

f Magnusson / 2.3

. m 0.04

0.02

0 01020304050 1/2 –1/2 A/AoHo (m )

FIGURE B.37. Comparison of CIB Burning Rate Data to Predictions Made Using Magnusson and Thelandersson’s Method

96 1400

1200

1000

800 Measured Magnusson 600 (Type C)

Temperature (°C) 400

200

0 00.511.5 2 Time (h)

FIGURE B.38. Comparison of Predictions Made Using Magnusson and Thelandersson’s Method (Type C) to Data from Cardington Test #1

1400

1200

1000

800 Measured Magnusson 600 (Type C)

Temperature (°C) 400

200

0 00.20.40.60.811.21.41.6 Time (h)

FIGURE B.39. Comparison of Predictions Made Using Magnusson and Thelandersson’s Method (Type C) to Data from Cardington Test #2

97 1400

1200

1000

800 Measured Magnusson 600 (Type C)

Temperature (°C) 400

200

0 00.511.522.5 Time (h)

FIGURE B.40. Comparison of Predictions Made Using Magnusson and Thelandersson’s Method (Type C) to Data from Cardington Test #3

1400

1200

1000

800 Measured Magnusson 600 (Type C)

Temperature (°C) 400

200

0 01234 Time (h)

FIGURE B.41. Comparison of Predictions Made Using Magnusson and Thelandersson’s Method (Type C) to Data from Cardington Test #4

98 1400

1200

1000

800

Measured 600 Magnusson (Type C) 400 Temperature (°C) 200

0 00.511.5 22.5 3 Time (h)

FIGURE B.42. Comparison of Predictions Made Using Magnusson and Thelandersson’s Method (Type C) to Data from Cardington Test #5

1400

1200

1000

800

600 Measured Magnusson (Type C) Temperature (°C) 400

200

0 00.20.40.6 0.8 1 Time (h)

FIGURE B.43. Comparison of Predictions Made Using Magnusson and Thelandersson’s Method (Type C) to Data from Cardington Test #7

99 1200

1000

800

600

Measured 400 Magnusson Temperature (°C) (Type G)

200

0 00.511.522.53 Time (h)

FIGURE B.44. Comparison of Predictions Made Using Magnusson and Thelandersson’s Method (Type C) to Data from Cardington Test #8

1400

1200

1000

800

600 Measured Magnusson Temperature (°C) 400 (Type C)

200

0 00.511.5 2 Time (h)

FIGURE B.45. Comparison of Predictions Made Using Magnusson and Thelandersson’s Method (Type C) to Data from Cardington Test #9

100 HARMATHY For ventilation-limited burning, Harmathy gives: Because of the iterative nature of Harmathy’s . method for predicting compartment fire tempera- ρ 3 2 tures, it is not possible to compare predictions using Substituting 0 = 1.2 kg/m and g = 9.8 m/s ,

Harmathy’s method to the CIB data in a straight- . Substituting this into = mf /τ forward manner. Harmathy distinguishes fuel-limited burning yields . This is compared to the from ventilation-limited burning as the point where CIB data in Figure B.46. 3 Comparisons of predictions using Harmathy’s = 0.263. Substituting ρ0 = 1.2 kg/m method to the Cardington data are presented in Figures B.47 through B.55. Predictions for times and g = 9.8 m/s2, = 0.07. In the CIB tests, less than the burning duration were created by using the average value of AF/A was approximately 0.75. the iterative method recommend by Harmathy, and Substituting and inverting, the threshold between a minimum resolution of 1°C was required for the fuel-limited and ventilation-limited burning would prediction to be accepted. be = 19.0.

For fuel-limited burning Harmathy gives:

. Substituting Af = 0.75A and

= mf /τ yields = 0.00465A.

0.20

0.18

0.16 )

5/2 121 0.14 221 0.12 211 441 0.10 Harmathy (kg/s – m Harmathy * 1.8 1/2 o 0.08 Harmathy / 1.8 H o Harmathy / 1.5

/A 0.06 f Harmathy * 2.8 . m 0.04

0.02

0 01020304050 1/2 –1/2 A/AoHo (m )

FIGURE B.46. Comparison of CIB Burning Rate Data to Predictions Made Using Harmathy’s Method

101 1400

1200

1000

800 Measured Harmathy 600

Temperature (°C) 400

200

0 00.511.5 2 Time (h)

FIGURE B.47. Comparison of Predictions Made Using Harmathy’s Method to Data from Cardington Test #1

1400

1200

1000

800 Measured Harmathy 600

Temperature (°C) 400

200

0 00.20.40.6 0.8 1 1.2 1.4 1.6 Time (h)

FIGURE B.48. Comparison of Predictions Made Using Harmathy’s Method to Data from Cardington Test #2

102 1400

1200

1000

800 Measured Harmathy 600

Temperature (°C) 400

200

0 00.511.5 2 2.5 Time (h)

FIGURE B.49. Comparison of Predictions Made Using Harmathy’s Method to Data from Cardington Test #3

1400

1200

1000

800 Measured Harmathy 600

Temperature (°C) 400

200

0 00.511.5 2 2.5 3 3.5 4 Time (h)

FIGURE B.50. Comparison of Predictions Made Using Harmathy’s Method to Data from Cardington Test #4

103 1600

1400

1200

1000 Measured 800 Harmathy

600 Temperature (°C) 400

200

0 00.511.522.53 Time (h)

FIGURE B.51. Comparison of Predictions Made Using Harmathy’s Method to Data from Cardington Test #5

1600

1400

1200

1000 Measured 800 Harmathy

600 Temperature (°C) 400

200

0 012345678 Time (h)

FIGURE B.52. Comparison of Predictions Made Using Harmathy’s Method to Data from Cardington Test #6

104 1600

1400

1200

1000 Measured 800 Harmathy

600 Temperature (°C) 400

200

0 00.20.40.6 0.8 1 Time (h)

FIGURE B.53. Comparison of Predictions Made Using Harmathy’s Method to Data from Cardington Test #7

1200

1000

800

600 Measured Harmathy

400 Temperature (°C)

200

0 00.511.522.53 Time (h)

FIGURE B.54. Comparison of Predictions Made Using Harmathy’s Method to Data from Cardington Test #8

105 1400

1200

1000

800 Measured Harmathy 600

Temperature (°C) 400

200

0 00.511.5 2 Time (h)

FIGURE B.55. Comparison of Predictions Made Using Harmathy’s Method to Data from Cardington Test #9

BABRAUSKAS For fuel-controlled burning, Harmathy estimates the burning rate as = 0.0062A . Substituting this Babrauskas provides the equivalence ratio as f into the above yields: where and s is the ratio such . For stoichiometric burning, φ = 1. that 1 kg fuel + s kg air = (1 + s) kg products. In the CIB tests, the average value of A /A was Harmathy39 notes that a typical wood would have F approximately 0.75. Substituting and solving for the chemical formula CH1.455O0.645•0.233H2O, which would result in a value of s of 6.0, which is , the threshold between fuel-limited and slightly larger than the value of 5.7 proposed by Babrauskas.46 Using . ventilation-limited burning would be = 18.0. Substituting this into the correlation for the Substituting in the relevant values for enclosure equivalence ratio yields . properties from the CIB tests and assuming that Ho ≈ 1 m (in the CIB tests, Ho ranged from 0.5 m to Babrauskas provides methods for modeling 1.5 m, but, given that Babrauskas’ method varies burning rate for ventilation-controlled burning, and –0.3 with Ho , predictions are not highly sensitive to for fuel-controlled burning, for wood cribs, and this parameter) and bp = 0.9 results in the predic- 45 thermoplastic or liquid pools. Babrauskas’ model tions of the CIB temperatures shown in Figure B.56. for calculating the burning rate of ventilation-con- For ventilation-controlled burning, Babrauskas trolled fires is used here; however, in most design estimates the burning rate as45: situations, the input data needed to use Babrauskas’ models for fuel-controlled burning is not available. Therefore, Harmathy’s model for the burning rate of over-ventilated fires was used for the present analysis.

106 Given that Harmathy’s method of estimating immediately return to ambient. Thus, the only time- burning rate for fuel-controlled burning was used, dependent variable remaining was θ3, which very the evaluation of that method is applicable to the quickly equaled one. Therefore, compartment fire assumption made here. A comparison of burning temperatures were modeled as a square wave. rate predictions using Babrauskas’ method to the The value of s was calculated as 6.0, based on CIB data for ventilation-controlled fires is presented the chemical formula for typical wood provided by 39 in Figure B.57. Harmathy of CH1.455O0.645•0.233H2O. The closed form approximation was used to Calculations of the wall area did not include create predictions of compartment fire temperatures either the area of the floor or the area of the ventila- for the Cardington tests. In these tests, it was appar- tion opening. The lining properties used were those ent that the fires were ventilation controlled from of the ceramic fiber lining. For calculation of θ5, a the observed burning behavior. While Babrauskas’ value of 0.9 was used for bp. The burning duration method is capable of predicting burning rate and was calculated by dividing the mass of unburned compartment fire temperatures during the growth fuel by the burning rate. and decay stages of a fire, these stages were Comparisons of predictions using Babrauskas’ neglected. The burning rate was calculated as45: method to the Cardington data are presented in Figures B.58 through B.66.

Once the fuel was depleted, the fire was considered to cease, and the temperature assumed to

1400

1200

1000

800 CIB Data Babrauskas 600

Temperature (°C) 400

200

0 01020304050 1/2 –1/2 A/AoHo (m )

FIGURE B.56. Comparison of CIB Temperature Data to Predictions Made Using Babrauskas’ Method

107 0.18

0.16

0.14 ) 5/2 0.12 121 221 0.10 211 441 (kg/s – m Babrauskas

1/2 0.08 o Babrauskas * 1.3 H o 0.06 Babrauskas / 2.3 /A f

. m 0.04

0.02

0 01020304050 1/2 –1/2 A/AoHo (m )

FIGURE B.57. Comparison of CIB Burning Rate Data to Predictions Made Using Babrauskas’ Method

1400

1200

1000

800 Measured Babrauskas

600

Temperature (°C) 400

200

0 0 0.5 1 1.5 2 Time (h)

FIGURE B.58. Comparison of Predictions Made Using Babrauskas’ Method to Data from Cardington Test #1

108 1400

1200

1000

800 Measured Babrauskas 600

Temperature (°C) 400

200

0 00.20.40.6 0.8 1 1.2 1.4 1.6 Time (h)

FIGURE B.59. Comparison of Predictions Made Using Babrauskas’ Method to Data from Cardington Test #2

1400

1200

1000

800 Measured Babrauskas

600

Temperature (°C) 400

200

0 00.511.5 2 2.5 Time (h)

FIGURE B.60. Comparison of Predictions Made Using Babrauskas’ Method to Data from Cardington Test #3

109 1400

1200

1000

800 Measured Babrauskas

600

Temperature (°C) 400

200

0 00.511.522.5 3 3.5 4 Time (h)

FIGURE B.61. Comparison of Predictions Made Using Babrauskas’ Method to Data from Cardington Test #4

1400

1200

1000

800 Measured Babrauskas 600

Temperature (°C) 400

200

0 00.511.522.53 Time (h)

FIGURE B.62. Comparison of Predictions Made Using Babrauskas’ Method to Data from Cardington Test #5

110 900

800

700

600 Measured 500 Babrauskas 400

300 Temperature (°C)

200

100

0 012345678 Time (h)

FIGURE B.63. Comparison of Predictions Made Using Babrauskas’ Method to Data from Cardington Test #6

1400

1200

1000

800 Measured Babrauskas

600

Temperature (°C) 400

200

0 00.20.40.6 0.8 1 Time (h)

FIGURE B.64. Comparison of Predictions Made Using Babrauskas’ Method to Data from Cardington Test #7

111 1200

1000

800

Measured 600 Babrauskas

400 Temperature (°C)

200

0 00.511.522.53 Time (h)

FIGURE B.65. Comparison of Predictions Made Using Babrauskas’ Method to Data from Cardington Test #8

1400

1200

1000

800 Measured Babrauskas 600

Temperature (°C) 400

200

0 00.511.5 2 Time (h)

FIGURE B.66. Comparison of Predictions Made Using Babrauskas’ Method to Data from Cardington Test #9

112 Ma and Mäkeläinen For fuel-controlled fires, Ma and Mäkeläinen state that the maximum temperature would be Ma and Mäkeläinen define the critical value of where ηcr is the value that separates the fuel-controlled and ventilation-controlled regimes as of that differentiates between fuel- and ventilation-controlled burning (for the CIB data, –1/2 ηcr was calculated as 13.68 m ) and Tgmcr is the η η In the CIB tests, the ratio Afloor/A ranged from value of Tgm for = cr. It should be noted that 0.18 to 0.25. Ma and Mäkeläinen noted that Af /mf the above temperature correlations provide an esti- typically ranges from 0.1 to 0.4 m2/kg, and that in a mation of the maximum temperature that would be 2 series of Japanese tests Af /mf = 0.131 m /kg. attained during a fire; for the majority of the fire 2 Substituting Afloor/A = 0.2, Af /mf = 0.131 m /kg, duration the temperature would be lower, and, hence, the average temperature during the fire would be 2 and m"f = 40 kg/m , the critical value of lower. Figure B.67 provides a comparison of pre- that separates the fuel-controlled and ventilation- dicted maximum temperatures with the CIB data. Ma and Mäkeläinen use Harmathy’s correlation controlled regimes would be = 13.68. to predict the burning rate for fuel-controlled burn- Ma and Mäkeläinen estimate the maximum ing and Law’s correlation to predict the burning rate temperature that would be achieved for ventilation- for ventilation-controlled burning. See the discus- controlled fires would be: sion of those methods for an evaluation of their burning rate predictions. Comparisons of predictions to the Cardington data are presented in Figures B.68 through B.75. For test #6, Ma and Mäkeläinen’s method predicted temperatures below ambient.

1400

1200

1000

800 CIB Data Ma (Max) 600

Temperature (°C) 400

200

0 01020304050 1/2 –1/2 A/AoHo (m )

FIGURE B.67. Comparison of CIB Burning Rate Data to Predictions Made Using Ma and Mäkeläinen’s Method

113 1400

1200

1000

800 Measured Ma

600

Temperature (°C) 400

200

0 00.511.5 2 Time (h)

FIGURE B.68. Comparison of Predictions Made Using Ma and Mäkeläinen’s Method to Data from Cardington Test #1

1400

1200

1000

800 Measured Ma

600

Temperature (°C) 400

200

0 00.20.40.6 0.8 1 1.2 1.4 1.6 Time (h)

FIGURE B.69. Comparison of Predictions Made Using Ma and Mäkeläinen’s Method to Data from Cardington Test #2

114 1400

1200

1000

800 Measured Ma

600

Temperature (°C) 400

200

0 00.511.5 2 2.5 Time (h)

FIGURE B.70. Comparison of Predictions Made Using Ma and Mäkeläinen’s Method to Data from Cardington Test #3

1400

1200

1000

800 Measured Ma 600

Temperature (°C) 400

200

0 00.511.5 2 2.5 3 3.5 4 Time (h)

FIGURE B.71. Comparison of Predictions Made Using Ma and Mäkeläinen’s Method to Data from Cardington Test #4

115 1400

1200

1000

800 Measured Ma 600

Temperature (°C) 400

200

0 00.511.522.53 Time (h)

FIGURE B.72. Comparison of Predictions Made Using Ma and Mäkeläinen’s Method to Data from Cardington Test #5

1400

1200

1000

800 Measured Ma

600 Temperature (°C) 400

200

0 00.20.40.6 0.8 1 Time (h)

FIGURE B.73. Comparison of Predictions Made Using Ma and Mäkeläinen’s Method to Data from Cardington Test #7

116 1200

1000

800

Measured 600 Ma

400 Temperature (°C)

200

0 00.511.5 2 2.5 3 Time (h)

FIGURE B.74. Comparison of Predictions Made Using Ma and Mäkeläinen’s Method to Data from Cardington Test #8

1400

1200

1000

800 Measured Ma 600

Temperature (°C) 400

200

0 00.511.5 2 Time (h)

FIGURE B.75. Comparison of Predictions Made Using Ma and Mäkeläinen’s Method to Data from Cardington Test #9

117 CIB Predictions using the CIB method are compared to data from the Cardington tests in Figures B.77 The temperature data from the Cardington tests through B.83. The compartment temperature and was compared to the temperature data from the CIB burning duration were predicted using the graphs tests by averaging the temperatures measured at dif- presented earlier in this guide for cribs with 20 mm ferent horizontal locations in the Cardington tests. thick wood sticks spaced 20 mm apart. No decay These average temperatures were averaged over the rate was imposed, and for times greater than the duration of maximum burning and plotted along duration the compartment temperature was assumed with the CIB data. Error bars on the Cardington to be ambient. data are included to show the range of temperatures measured during the period of maximum burning. The results are shown in Figure B.76, with the abscissa plotted in logarithmic scale.

1200

1000

800

CIB Data 600 Cardington CIB Curve

400 Temperature (°C)

200

0 1101001000 1/2 –1/2 A/AoHo (m )

FIGURE B.76. Comparison of Cardington and CIB Temperature Data

118 1400

1200

1000

800 Measured CIB 600

Temperature (°C) 400

200

0 00.511.5 2 Time (h)

FIGURE B.77. Comparison of Predictions Made Using the CIB Data to Cardington Test #1

1400

1200

1000

800 Measured CIB 600 Temperature (°C) 400

200

0 00.20.40.6 0.8 1 1.2 1.4 1.6 Time (h)

FIGURE B.78. Comparison of Predictions Made Using the CIB Data to Cardington Test #2

119 1400

1200

1000

800 Measured CIB 600

Temperature (°C) 400

200

0 00.511.5 2 2.5 Time (h)

FIGURE B.79. Comparison of Predictions Made Using the CIB Data to Cardington Test #3

1400

1200

1000

800 Measured CIB

600

Temperature (°C) 400

200

0 00.511.522.5 3 3.5 4 Time (h)

FIGURE B.80. Comparison of Predictions Made Using the CIB Data to Cardington Test #4

120 1400

1200

1000

800 Measured CIB 600

Temperature (°C) 400

200

0 00.20.40.6 0.8 1 Time (h)

FIGURE B.81. Comparison of Predictions Made Using the CIB Data to Cardington Test #7

1200

1000

800

Measured 600 CIB

400 Temperature (°C)

200

0 00.511.522.53 Time (h)

FIGURE B.82. Comparison of Predictions Made Using the CIB Data to Cardington Test #8

121 1400

1200

1000

800 Measured CIB 600

Temperature (°C) 400

200

0 00.511.5 2 Time (h)

FIGURE B.83. Comparison of Predictions Made Using the CIB Data to Cardington Test #9

Law CIB data. Note that, because Law’s method con- siders the effect of compartment depth and width, Figure B.84 shows predictions of maximum tem- the CIB burning rate data that was normalized perature using Law’s method compared to the CIB by was utilized. data. Law’s method includes a means of reducing Comparisons of predictions made using Law’s the predicted temperature based on the fuel loading. method to the Cardington data are shown in However, for the range of conditions in the tests Figures B.86 through B.94. For times less than the from which the CIB data were collected, utilizing calculated burning duration, the temperature was this factor would result in unrealistically low tem- calculated using Law’s adjustment for fuel load. peratures for some combinations of scale, opening No decay rate was imposed, and for times greater factor, and ventilation area. Therefore, this method than the duration the compartment temperature was of reducing the temperature was not utilized. assumed to be ambient. Figure B.85 shows a comparison of burning rate predictions made using Law’s method to the

122 1400

1200

1000

800 CIB Data Law (max) 600

Temperature (°C) 400

200

0 01020304050 1/2 –1/2 A/AoHo (m )

FIGURE B.84. Comparison of CIB Temperature Data to Predictions Made Using Law’s Method

0.25 ) 5/2 0.20 121 221

(kg/s – m 0.15 211

1/2 441 Law Law X 1.4 (D/W) 0.10 Law / 1.4 1/2 o H o /A f 0.05 . m

0 0102030405060

–1/2 A/AoHo (m )

FIGURE B.85. Comparison of CIB Burning Rate Data to Predictions Made Using Law’s Method

123 1400

1200

1000

800 Measured Law 600

Temperature (°C) 400

200

0 00.511.5 2 Time (h)

FIGURE B.86. Comparison of Predictions Made Using Law’s Method to Data from Cardington Test #1

1400

1200

1000

800 Measured Law

600

Temperature (°C) 400

200

0 00.20.40.6 0.8 1 1.2 1.4 1.6 Time (h)

FIGURE B.87. Comparison of Predictions Made Using Law’s Method to Data from Cardington Test #2

124 1400

1200

1000

800 Measured Law

600

Temperature (°C) 400

200

0 00.511.5 2 2.5 Time (h)

FIGURE B.88. Comparison of Predictions Made Using Law’s Method to Data from Cardington Test #3

1400

1200

1000

800 Measured Law 600

Temperature (°C) 400

200

0 00.511.522.5 3 3.5 4 Time (h)

FIGURE B.89. Comparison of Predictions Made Using Law’s Method to Data from Cardington Test #4

125 1400

1200

1000

800 Measured Law 600

Temperature (°C) 400

200

0 00.511.522.53 Time (h)

FIGURE B.90. Comparison of Predictions Made Using Law’s Method to Data from Cardington Test #5

900

800

700

600 Measured 500 Law 400

Temperature (°C) 300

200

100

0 012345678 Time (h)

FIGURE B.91. Comparison of Predictions Made Using Law’s Method to Data from Cardington Test #6

126 1400

1200

1000

800 Measured Law

600 Temperature (°C) 400

200

0 00.20.40.6 0.8 1 Time (h)

FIGURE B.92. Comparison of Predictions Made Using Law’s Method to Data from Cardington Test #7

1200

1000

800

Measured 600 Law

400 Temperature (°C)

200

0 00.511.522.53 Time (h)

FIGURE B.93. Comparison of Predictions Made Using Law’s Method to Data from Cardington Test #8

127 1400

1200

1000

800 Measured Law 600

Temperature (°C) 400

200

0 00.511.5 2 Time (h)

FIGURE B.94. Comparison of Predictions Made Using Law’s Method to Data from Cardington Test #9

128 Appendix C Time-Equivalent Methods

As stated in ASTM E119,105 standard furnace described. Pettersson’s method is put forward as tests such as ASTME119; BS 476, Part 20106; and the preferred time-equivalent method, and its range ISO 834107 provide a relative measure of the fire of use is outlined. test response of comparable assemblies under standardized fire exposure conditions. The exposure Real Structural Response is not representative of all fire conditions because conditions vary with changes in the amount, nature, It is important to note that time-equivalent meth- and distribution of fire loading; ventilation; com- ods do not assess local or global structural response. partment size and configuration; and thermal char- They relate only to heating effects and their rela- acteristics of the compartment. Real fires can be tionship to the standard furnace test. more or less severe in terms of duration, rate of The t-equivalent methods do not address transient heating, and peak temperature than the standard temperature gradients or associated load-bearing temperature–time relationship in a furnace test. capacities. The ratings derived do not relate to Real fires are a function of fuel load, compartment actual frame performance in fire. These methods are dimensions, thermal properties of the compartment simply refined versions of performance of a single boundaries, and the quantity of unprotected open- element in fire, but only relative to the standard fur- ings that allow ventilation in a post-flashover fire. nace test. They normally assume insulated struc- Also of importance is that the standard furnace tures only (protected steel or reinforced concrete). test does not assess real structural response in fire Pettersson’s work, however, does address uninsu- conditions because single elements of structure are lated steel also. In the work carried out for the tested in the furnace even though they form compo- Natural Fire Safety Concept,108 good correlation nent parts of complex three-dimensional frames in was achieved when the t-equivalent results were real buildings. compared to real fire test data for insulated steel Various methods exist for designers to derive structures. The results for uninsulated steel struc- more realistic temperature–time relationships for tures gave very poor correlation, as would be compartments. For example, as a result of concerns expected. Bare steel tends to follow the furnace test with the standard furnace test temperature–time curve, so use of bare steel elements, in terms of relationship, work was carried out by Ingberg, standard fire resistance, would not be expected Law, and Pettersson, among others, to determine beyond 20 to 30 minutes depending on section size. what is known as an equivalent fire resistance. For A time-equivalent calculation does not apply if these methods, the heating effect in a compartment the pre-flashover calculations show that flashover is based on real compartment fire behavior and will not occur, i.e., the calculation is no longer therefore takes into account fuel load density, venti- relevant if flashover has not occurred. Then, local lation openings, compartment dimensions, and heating effects are relevant, not temperatures in a enclosure thermal properties. This allows some uniformly heated compartment, as is assumed in improvement in the grading method based on the time-equivalent analysis methods. standard furnace test that is currently assumed in Time-equivalent methods are empirical formulae building codes worldwide. developed by regression analysis using a selected This section describes various calculation pro- number of tests or calculations. Therefore, they cedures for these time-equivalent methods. Limi- have been developed for a certain range of struc- tations and assumptions for each method are tural steel sizes and thicknesses of insulation and so may not be appropriate outside this range.

129 They are used for other materials, but beyond FIRE LOAD CONCEPT protected steel and reinforced concrete very little is By 1918 there was a concern in the fire protec- known of the accuracy in applying this method to tion and code enforcement communities that there other materials. was no accepted method for establishing appropri- Note that all t-equivalent methods described here ate levels of fire endurance for buildings of different involve combustible solids only. sizes and occupancies.109 The original work had been based on “fireproof” large commercial build- Discussion of Methods ings. It was recognized that these differed significantly from residential fires, but it was not under- Time-equivalent methods can be described as stood how their severity related to the conditions in methods that define the thermal exposure of a par- the now-formulated standard fire resistance test. ticular compartment fire in terms of the duration of To develop a solution to this problem, in 1922 the equivalent standard fire. the National Bureau of Standards investigated the Equivalence of thermal exposure has been nature of building fires under the direction of Simon defined in two ways: Ingberg.110 The main aim was to determine the intensity and duration of uncontrolled fires in par- 1. Equal areas under the temperature–time curves ticular occupancies resulting from different levels 2. Equal temperatures at the critical part of a of fuel load. Ingberg was also to investigate the structural element validity of the standard temperature–time curve. Ingberg investigated office and record storage- The two methods give similar results where the type occupancies. The effects of the building size and element selected has a fire resistance of the order of fuel load, combustible and noncombustible flooring, half an hour or more. plus wood and steel furniture were investigated.

2400 1200 The "fire severity" is 2000 considered to be the same 1000 when Area 1 = Area 2 1600 Area 1 800

1200 600

800 Area 2 400 Temperature (°C) Temperature (°F)

400 200

0 0 0.5 1.0 1.5 2.0 2.5 3.0 Time (h)

FIGURE C.1. Fire Severity Concept109

130 As a result of these tests, Ingberg established a This time, τ is approximated as: simple relationship between the average weight of combustible material within a room and the fire (min) (Eq. C.2) endurance necessary to withstand a complete burnout of the contents. This is known as the “fuel Where: load concept.” It assumes that the area under any τ =Time (min) 2 temperature–time curve from ignition through decay m"f = Fuel load (kg/m ) 2 provides a comparative measure of fire severity, and Afloor = Floor area (m ) that fire severity is a function of the fuel load only. H = Height of the window (m) 2 Ingberg compared the area under the tempera- Ao = Area of the windows (m ) ture–time curves generated in the burnout tests to an equivalent area under the standard temperature–time LAW curve. The areas below a threshold temperature of 112,113 about 300°C were not taken into account. The graph Law developed a t-equivalent formula 114 in Figure C.1 shows the basis for Ingberg’s work. from the results of the CIB test program. Ingberg developed the following relationship for The maximum temperature that would be time-equivalence: attained by a protected steel element in a real fire compartment was chosen as a basis for comparison te = k1m"f (Eq. C.1) with the heating effect in a standard fire. Where: For a temperature–time curve, the maximum temperature obtained by a protected steel element in te = t-equivalent (min) a compartment fire is calculated as: m"f = Fuel load (wood) per unit floor area 2 k1 = Unity when m"f is in units of kg/m ; 2 (Eq. C.3) k1 = 5 when m"f is in units of lb/ft Where: Ingberg’s work became widely accepted as the T = Steel temperature (K) general basis for establishing fire endurance s t =Time (s) requirements. T = Fire temperature (K) R = δi /(kiPH) KAWAGOE AND SEKINE δi = Thickness of insulating material (m) In 1963, Kawagoe and Sekine111 went on to ki = Thermal conductivity of insulating material show the importance of the ventilation parameter: (kW/m-K) P = Heated perimeter of steel member (m) H = Height (or length) of steel member (m) C = AHρscs Where: A = Cross-sectional area of steel member (m2) H = Window height (m) 3 o ρs = Density of steel (kg/m ) A = Total area of openings (m2) o cs = Specific heat of steel (kJ/kg-K) A = Total area of inside surfaces including 2 opening area (m ) The temperature of the heated surface of the pro- tective material is assumed to be the same as the Kawagoe and Sekine also developed a formula fire temperature. The heat transfer through the steel for fire duration and defined it as the period from section can then be calculated. the beginning of temperature rise until the time the For a given temperature–time curve, the value RC temperature drops after most of the combustible was determined so that the maximum temperature material is burnt. of the protected member was 550°C.

131 The time for the protected member to attain PETTERSSON 550°C when exposed to the standard temperature– In 1976, Pettersson117 adopted Law’s approach to time curve gives the value of t-equivalent. t-equivalent, but, instead of the experimental curves The best correlation was obtained from the on which her work was based, used the family of product (m /A ) and a term taking into account A f o o calculated temperature–time curves for particular and the solid surface to which heat is lost: compartments as derived by Magnusson and Thelandersson.118 (Eq. C.4) When the fuel load is expressed in mass (kg) of wood instead of “effective calorific value” (MJ), Where: 2 Pettersson’s expression for t-equivalent is as follows: Afloor = Floor area of the compartment (m ) mf = Fuel load (wood equivalent) (kg) 2 (Eq. C.6) Ao = Area of ventilation opening (m ) k = 1.3 to 1.5, depending on the stick 3 Where: spacing in the cribs used as fuel H = Height of vertical opening (m) (min m2/kg) o A =Total area of internal envelope (walls, A = Surface area of interior of enclosure floor, ceiling, and openings). (Note that in (walls, floor, ceiling, and openings) (m2) his original heat balance work he excluded A but for an unstated reason does not In this correlation, A was not included in the o floor in his final equations presented in his evaluation of solid surfaces because the floors were design guide.) very well insulated. In all experiments the openings were the full This equation includes because of the input compartment height. parameters in the method for calculating the The values of t-equivalent were found to be inde- temperature–time curves on which this equation pendent of scale and height of ventilation openings. is based. Law then analyzed temperature–time data from a Equation C.6 can be modified to take into number of fires in larger brick and concrete com- account the thermal properties of the compartment partments (approximately 3 m high)115,116 with enclosure by applying the factor k to each input fuels consisting of wood cribs, furniture, and liquid f parameter. fuels, and developed This yields: (Eq. C.5) (Eq. C7) where k is 1.0. This was due to the little effect 4 where k = factor applied to input parameters to take fuel arrangement appeared to have in these larger f account of the thermal properties kρc of the com- scale tests. partment enclosure expressed as a proportion of In this correlation, the floor area was included in the kρc for Pettersson’s “standard” compartment. the evaluation of solid surfaces to which heat is lost. This is the compartment, defined in the Swedish The larger scale data also showed no significant Building Regulations in 1967, as where the sur- effect of ventilation opening height on t . e rounding structure has the thermal properties of an Law concluded this equation (C.5) was most suit- average of concrete, brick, and lightweight concrete able for engineering purposes for protected steel with a thickness of 20 cm. (Note also that the fire is columns and went on to demonstrate that it gave ventilation controlled and with a cooling phase of good results for reinforced concrete also. She dis- 10°C/min.) covered that it overestimates the time prediction for tightly baled paper and cloth.

132 NORMALIZED HEAT LOAD CONCEPT EUROCODE TIME-EQUIVALENT METHOD In 1983, Harmathy and Mehaffey119 developed The Eurocode120 defines t-equivalent as the “normalized heat load” concept. The total heat described in the German standard DIN 18230, ver- penetrating the compartment boundaries is calcu- sion 94, method.121 The derivation of this formula lated taking into account and the proportion has never been published, but it is understood to of heat evolution in the compartment, χ. When no have come from an empirical analysis of calculated unburnt gases emerge from the compartment, χ = 1. steel temperatures in a large number of simulated Based on the results of many experiments and fires computed by the German program “Multi tests using the Division of Building Research/ Room Fire Code.”121 Though this reference refers National Research Council of Canada floor test to an earlier published version of the Eurocode, the furnace,119 they derived the following relationship basic formulations remain, and therefore this origin for t-equivalent: is believed to still apply. This method is dependent on ceiling height of the (s) (Eq. 8) compartment but not the opening height. The fuel type assumed in the original work is unknown, for though it is widely believed to be cellulosic. 4 0 < HN < 9 × 10 The t-equivalent is defined in the Eurocode as: Where: te,d = qf,dkbwf kc (Eq. C.11)

(Eq. 9) Where: qf,d = Fuel load density related to the floor area (MJ/m2), which can be calculated χ = or 1, according to whichever is less H = Compartment height (m) qf,d = qf,kmδq1δq2δn (Eq. C.12) k = Thermal conductivity (kW/m K) Where: ρ = Density (kg/m3) qf,k = Fuel load density determined from a fuel c=Specific heat (kJ/kg K) load classification of occupancies (see Table C.1) te from Equation C.9 is then given m = Combustion factor, which for cellulosic approximately by: materials is defined as 0.8 δ te = 0.0016HN (Eq. C.10) q1 = Safety factor taking account of the risk of a fire starting due to the size of for compartment (see Table C.2) 4 HN ≤ 9 × 10 δq2 = Safety factor taking account of the risk of a fire starting due to the type of occupancy (see Table C.3) δn = Factor taking account of the different active fire-fighting measures such as sprinklers, detection, fire fighters, etc.) (see Table C.4)

133 TABLE C.1. Fuel Load Density Determined from a Fuel Load Classification of Occupancies120

80% Occupancy Average Fractile Dwelling 780 948 Hospital (room) 230 280 Hotel (room) 310 377 Library 1500 1824 Office 420 511 Classroom of a school 285 347 Shopping center 600 730 Theater (cinema) 300 365 Transport (public space) 100 122 Gumbel distribution is assumed for the 80% fractile

TABLE C.2. Safety Factor Taking Account of the Risk of a Fire Starting Due to the Size of Compartment120

Danger of Fire Starting (δq2) Examples of Occupancies 0.78 Art gallery, swimming pool 1.00 Offices, hotel, residential 1.22 Manufacturing for machinery and engines 1.44 Chemical lab, panting workshop 1.66 Manufacturing of fireworks or paints

TABLE C.3. Safety Factor Taking Account of the Risk of a Fire Starting Due to the Type of Occupancy120 Compartment Danger of Floor Area Fire Activation 2 Af (m )(δq1) 25 1.1 250 1.5 2500 1.9 5000 2.0 10000 2.13

134 TABLE C.4. A Factor Taking Account of the Different Active Fire-Fighting Measures (Sprinklers, Detection, Fire Fighters, Etc.)120

δni Function of Active Fire-Fighting Measures Automatic Fire Suppression Automatic Fire Detection Manual Fire Suppression

Independent Auto Work Off-site Safe Fire- Smoke Auto water water Auto fire transmission fire fire access fighting exhaust extinguishing supplies detection to fire brigade brigade routes devices system system (δq2) and alarm brigade

(δn1) 0 12 By By (δn5)(δn6)(δn7)(δn8)(δn9)(δn10) heat smoke (δn3)(δn4) 0.61 1 0.87 0.7 0.87 or 0.73 0.87 0.61 or 0.78 0.9/1/1.5 1/1.5 1/1.5 Note: According to the Eurocode, for “normal fire-fighting measures” such as safe access routes, firefighting devices, and smoke exhaust systems in staircases, the factors should be taken as 1.0, and if these measures have not been foreseen but provided, then the values can be taken as 1.5.

2 α kb is a conversion factor = 0.07 (min m /MJ) h = Ah /Afloor = Area of horizontal opening in when no detailed assessment of the thermal proper- the roof related to the floor area of the ties of the boundary is pursued, and when qd is compartment 2 given in MJ/m2. bv = 12.5(1+10α v – α v ) ≥ 10 Otherwise kb may be related to the thermal H = Height of the compartment (m) property in accordance with Table C.5: For small fire compartments (defined in the 2 Eurocode as Afloor < 100 m ) without openings in TABLE C.5. Relationship Between kb and the the roof, the factor wf may also be calculated as: Thermal Property b

Kb (Eq. C.14) J/m2 s1/2 Kmin m2 /MJ b > 2500 0.04 Where: 720 ≤ b ≤ 2500 0.055 0.02 ≤ ≤ 0.20 with the default value b < 720 0.07 kb = 0.07 and assuming 18 MJ/kg for wood, Equation C.14 becomes the same as Equation C.7. wf is calculated as: Kc = A correction factor that is a function of the material composing structural (Eq. C.13) cross sections and is defined as 13.7 for unprotected steel. Where: Reinforced concrete and protected α = A /A = Area of vertical openings A v o floor 0 concrete remain as 1. in the façade related to the floor area of the compartment where the limit 0.025 ≤ αv ≤ 0.25 should be observed

135 TABLE C.6. Values for kb Recommended by the New Zealand Fire Engineering Design Guide

2 1/2 (J/m Ks ) Construction Materials kb Value 400 Very light insulating materials 0.10 700 Plasterboard ceiling and walls, timber floor 0.09 1100 Lightweight concrete ceiling and floor, plasterboard walls 0.09 1700 Normal concrete ceiling and floor, plasterboard walls 0.065 2500 Thin sheet steel roof 0.045

The basis of this method is that it should be veri- account for compartment geometry, ventilation fied that te,d < tfi,d where tfi,d is the design value of openings, fuel load density, and compartment the standard fire resistance of the members, assessed boundary materials in addition to fuel load density, according to the relevant parts of the Eurocode. the key factors that affect full-scale fire develop- This method could therefore be used for other ment. However, the temperatures calculated on defined periods of fire resistance such as in these principles are then related back to the standard U.S. codes. temperature–time relationship. It is also important to note that they are based on specific compartment NEW ZEALAND CODE test data rather than generalized heat balance solutions. Time-equivalent methods are therefore unlike The New Zealand Fire Engineering Design natural temperature time relationships, which repre- 122 Guide gives the same empirical expression for sent a real temperature–time relationship and are equivalent fire severity te (min) as the Eurocode. used as such, independent of the standard fire resist- The upper and lower kb values have been ance test formulation. increased by a factor of 1.3 compared to the Euro- Drysdale14 describes a comparison Harmathy code due to what it declares inherent uncertainties carried out where the Ingberg, Law, Pettersson, and in the Eurocode formula, the use of fuels other than Harmathy equations for te were compared. Drysdale wood, structures other than steel, and deep compart- rejects Ingberg’s method since radiative heat flux ment effects. The values for kb recommended by the varies with T4, which makes simple scaling impos- New Zealand Fire Engineering Design Guide are sible when heat transfer is dominated by radiation. shown in Table C.6. If the properties of the linings He concluded that Law and Harmathy provided are not known, a value of kb = 0.09 is suggested. more conservative solutions than the others. Note This formula is based on cellulosic-type fuels. that Ingberg’s method ignores ventilation, unlike the The ventilation factor limits of use are retained, other methods presented here. though the small compartment formula in the Law compared results using the time-equivalent Eurocode does not form part of the New Zealand relationships by Ingberg, Kawagoe, Law, guidance. Pettersson, Harmathy, and Mehaffey, plus the 1993 Eurocode formula with experimental data from Comparisons post-flashover fires in full-scale compartments.115 These consisted of small insulated compartments, Time-equivalent methods are an improvement on 30 m2 area, 2.5 to 3 m high, with brick or concrete the grading method in building codes worldwide, enclosures,113 and larger, deeper rooms 128 m2 in which is based on the standard fire temperature– area (depth to width ratio 4:1).123 Law concluded time relationship (such as ASTM E119, BS 476 Law, Pettersson, Harmathy, and Mehaffey were the Part 20, or ISO834). This is because they attempt to most promising methods.

136 200 180 160 140 L = Fuel load (kg) 120 AW = Area of the ventilation 2 AT =Total internal surface area (m ) e

t 100 of the compartment 80 60 40 20 0 0 10 20 30 40 50 60 70 80 90 100 L" Af/[Av(At – Av )]1/2

Small Standard Deep Insulated Small Insulated Compartment Compartment Compartment

1/2 115 FIGURE C.2. Law’s Correlation Between Fire Resistance Requirements (tf) and L/(AW AT)

Limitations and Assumptions THE EUROCODE The Eurocode formulae do not reference the THE DEEP COMPARTMENT EFFECT source of the equation derivations, particularly the ventilation factor needed in the time-equivalent cal- Law examined deep compartments further since culation, as well as the correction factor to take all her derived time-equivalent formulae gave odd account of cross section material types, plus the results when deep compartments were studied. In other factors of safety recommended for application deep compartments, temperatures and local burning to the calculated time-equivalent value. rates are not uniform, but rather progress from the k is defined for unprotected steel as 13.7 times opening toward the back of the enclosure as fuel is c depleted.103 Law also discovered that the 1993 the opening factor. Since 0.02 < < 0.2, this Eurocode t-equivalent method gives poor correla- gives 0.27 < k < 2.7 for unprotected steel. tion for both small and deep compartments.115 c For small compartments (A < 100 m2) She concluded that the depth of the compartment floor has an effect on time-equivalent over and above wf = , which can be written for what can be allowed for by increase in insulation and in internal surface area A. Thomas and unprotected steel in small compartments: Heselden114 had already shown that the ventilation- controlled rate of burning is affected by the com- Ted = 13.7 qfd × kb × partment depth to width ratio. Recent research on this phenomenon has also (fuel load given as per unit floor area) resulted in the New Zealand code’s recommending Pettersson calculated Te (h) for unprotected steel factors of safety that have been increased by 30% to for values of ranging from 0.2 to 0.12 m1/2. account for this effect in its time-equivalent formula.

137 Pettersson’s time-equivalent formula is compared Eurocode with the Eurocode formula for resultant emmisivity, Qt,d Te (h) εr, of 0.5 for unprotected steel as follows. = 0.02 0.04 0.08 0.12 Pettersson Section factor = 50 m-1 42 0.07 0.11 0.15 0.18 84 0.15 0.21 0.30 0.37 Qt,d Te (h) 126 0.22 0.32 0.45 0.55 = 0.02 0.04 0.08 0.12 The trends in the Eurocode are different, and no 42 0.29 0.25 0.21 0.16 explanation as to why has been provided. For unprotected steel, if the steel is assumed to 84 0.42 0.43 0.36 0.29 be at a uniform temperature, the post-flashover fire 126 0.51 0.58 0.525 0.41 temperatures are also perfectly stirred, and the temperature of the exposed surface is the same as Section factor = 150 m-1 the fire temperatures, this implies the steel temperature is always the same as the fire temperature. –1 Qt,d Te (h) Therefore, section factors of 100 m or more would not be expected to survive a post-flashover = 0.02 0.04 0.08 0.12 fire, but a localized fire. Until suitable justification of such difference 42 0.21 0.23 0.19 0.16 is made, it seems the original work of Pettersson or Law is best suited to this type of time- 84 0.27 0.37 0.38 0.30 equivalent calculation. 126 0.30

It can be seen that, while the values of Te tend to increase with increasing fuel load, they tend to decrease with ventilation factor. They are not independent of section factor. Yet when the Eurocode formula with kb = 0.055 for Pettersson’s compartment type A, for all sections, is used to do a similar check, the following data is produced.

138 Appendix D Examples

Example 1 BURNING DURATION A room 5 m wide, 4 m deep, and 2.5 m high has one vent that is 2 m high and 3 m wide. The fuel load is 10 lb/ft2. The enclosure is made of gypsum plaster on metal with the following properties: Where: k = 0.47 W/m°C mf = m"f × Afloor 2 2 ρ = 1440 kg/m3 m"f = 10 lb/ft or 49 kg/m 2 c = 0.84 kJ/kg°C Afloor = 20 m mf = 49 × 20 = 980 kg Find the maximum temperature of the fire and its burning duration. and Law’s method is recommended for all roughly cubic compartments and in long, narrow compartments where is approximately less than or where W is width of the room and D is depth of equal to 18 m–1/2. Since this room is roughly cubic, the room. Law’s method is applicable.

2 A = 2(5 * 4) + 2(5 * 2.5) + 2(4 * 2.5) = 85 m This equation is valid for Ao =2 * 3 = 6 m Ho =2 m

and in this case

MAXIMUM TEMPERATURE To ensure that predictions are sufficiently conservative using Law’s method, the predicted burning rate should be reduced by a factor of 1.4. The adjusted burning rate is then

Thus, The burning duration can be found by

139 Example 2 These characteristics normally belong to concrete, brick, and lightweight concrete. A room 7 m wide, 28 m deep, and 4 m high has one vent that is 4 m high and 3.5 m wide in one Thermal conductivity: k = 0.7 kcal/m-h-°C of the small end walls. The fuel load is 35 kg/m2. Product of the specific heat and the density, The enclosure is made of brick with the following c * ρ = 400 kcal/ m3°C properties: In this case k = 0.5937 kcal/mh°C and c * ρ = k = 0.69 W/m°C 321.22 kcal/ m3°C. Thus, Type A enclosed space ρ = 1600 kg/m3 is used. c = 0.84 kJ/kg°C The next step is to calculate the burning duration, τ. Find the burning duration, and plot the temperature–time curve. For long, narrow spaces in which the value q, the fire load per bounding surface area, is of is in the range of 45 to 85 m–1/2, calculated using the fuel load and the heat of combustion, ∆Hc. A heat of combustion of cellulosic Magnusson and Thelandersson’s method materials, 15 MJ/kg, is used for this example. is recommended.

A = 2(7 * 4) + 2(7 * 28) + 2(4 * 28) = 672 m2 2 Ao = 1.9 * 3.7 = 7 m q = 35 kg/m *(7 * 28)/672 * 15 MJ/kg 2 2 Ho = 3.7 m = 153 MJ/m = 37 Mcal/m Therefore, In this case

and Magnusson and Thelandersson’s method The type of enclosure, opening factor, and the is used. burning duration can be used to reference The first step is to decide which of the seven Magnusson and Thelandersson’s tables. The tables models in Magnusson and Thelandersson’s method give the temperature at 0.05-, 0.10-, and 0.20-hour is applicable to the problem. intervals up to 6.00 hours for various burning Type A enclosed spaces consist of a material, durations. The temperatures for a burning duration 20 cm in thickness, whose thermal properties are of 1.5 hours and an opening factor of 0.02 m1/2 characterized by the following average values. were used to create the temperature–time curve in Figure D.1.

140 800

700

600

500

400

300 Temperature (°C) 200

100

0 0.00 1.00 2.00 3.00 4.00 5.00 6.00 Time (h)

FIGURE D.1. Temperature–Time Curve for Burning Duration of 1.5 Hours and Opening Factor of 0.02 m1/2

141

Glossary

Nomenclature Used in the Enclosure Fires Section

A Surface area of interior of enclosure (m2) 2 Af Surface area of fuel (m ) 2 Afloor Surface area of floor (m ) 2 Ao Area of ventilation opening (m ) b Stick width (m) bp Factor (-) C Wood constant (g/m1.5-s) c Specific heat of enclosure lining (J/kg-K) cp Specific heat of air (J/kg-K) D Depth of compartment (m) F View factor (-) or opening factor (m1/2) G Gravitational constant (9.81 m/s2) h Equivalent conductance (W/m2-K) 2 hc Convection coefficient (W/m -K) 2 hk Conduction coefficient (W/m -K) 2 hr Radiation coefficient (W/m -K) H Height of compartment (m) Ho Height of ventilation opening (m) k Thermal conductivity of enclosure lining (W/m-K) ko Coefficient (-) L Latent heat of vaporization (kJ/g) mf Mass of fuel (kg) Mass burning rate of fuel (kg/s) 2 m"f Mass of fuel per unit area (kg/m ) Mass burning rate of fuel per unit area (kg/m2-s) Free burning mass loss rate of fuel per unit area (kg/m2-s) Asymptotic mass loss rate of fuel per unit area (kg/m2-s) Mass flow rate of air (kg/s) p Pressure (Pa) q Fuel load density (Mcal/m2) Heat loss rate (kW) Heat loss through walls (kW) Heat flux from fire (kW/m2) Effective heat flux (W/m2) Heat release rate (kW) Rate of the heat energy stored in the gas volume Rate of heat energy withdrawn from the enclosed space due to air flow Rate of heat energy withdrawn from the enclosed space by radiation Rate of heat energy withdrawn from enclosed space through the wall, floor, or ceiling Q* Dimensionless heat release rate (-)

143 * Qf Dimensionless radiation rate to fuel (-) * Qr Dimensionless radiation loss rate (-) * Qw Dimensionless heat loss rate to walls (-) R Universal gas constant (8.31 J/kMol-K) s Atoichiometric air to fuel ratio (-) t Time (units as stated) tm Time corresponding to maximum temperature (units as stated) T Temperature in compartment (units as stated) Tb Fuel boiling point (units as stated) Tf Flame temperature (units as stated) Tgm Maximum temperature (units as stated) Tgmcr Maximum temperature in the critical region (units as stated) To Ambient temperature (units as stated) Tw Wall temperature (units as stated) V Volume (m3) W Width of wall containing ventilation opening (m)

YO2 Mass fraction of O2 (-)

GREEK β Factor (-) Γ Scaling factor φ Equivalence ratio (-) δ Thickness (m) or shape factor (-) ∆Hp Heat of vaporization of liquid (kJ/kg) ∆Hc Heat of combustion (MJ/kg) ∆Hair Heat of combustion per unit mass of air (MJ/kg) ε Gas emissivity (-) εw Wall emissivity (-) κ Absorbsion coefficient (m–1) or factor (-) θ1-θ5 Variable (-) η Factor (-) ηcr Factor (-) ρ Density of enclosure lining (kg/m3) 3 ρ0 Density of air (kg/m ) σ Stefan-Boltzmann Constant [5.67 × 10–11 kW/(m2 K4)] τ Duration of fully developed fire (units as stated) ν Kinematic viscosity (m2/s) ς Factor (-) Ψ Factor (kg/m2)

144 Nomenclature Used in the Plumes Section

A Surface area of noncircular fuel package (m2) bu Plume width (m) bt Thermal plume width (m) Cp Specific heat capacity of air at 300 K [1.0 kJ/(kg K)] D Length of single side of square burner, diameter (m) g Acceleration of gravity (9.81 m/s2) h Convective heat transfer coefficient [kW/(m2 K)] H Distance between base of fire and ceiling (m) HB Distance between base of fire and lower flange of I-beam (m) HC Distance between base of fire and upper flange of I-beam (m) HW Distance between base of fire and center of web on I-beam (m) h Convective heat transfer coefficient [kW/(m K)] LB Distance from fire centerline to flame tip along lower flange of an I-beam (m) LC Distance from fire centerline to flame tip along upper flange of an I-beam (m) LH Distance from fire centerline to flame tip length along ceiling or upper flange of an I-beam (m) LW Distance from fire centerline to flame tip length along the web center of an I-beam (m) Lf Average flame length or unconfined flame tip length (m) Lf,tip Flame tip length (m) Lf,tipB Flame tip length along lower flange of I-beam (m) Lf,tipC Flame tip length along upper flange of I-beam (m) Lf,tipW Flame tip length along web center of an I-beam (m) Q Fire heat release rate (kW) Q* Dimensionless parameter, , with D being length scale r Distance from corner or stagnation point to measurement location or radial distance for plume centerline (m) Heat flux (kW/m2) T Temperature (K) Tm,c Centerline plume temperature (K) Tg Room gas temperature (K) Ts Material surface temperature (K) T∞ Ambient temperature (300 K) U Plume velocity (m/s) Um,c Centerline plume velocity (m/s) w Dimensionless distance along ceiling or I-beam, w = (r + H + z' )/(LH + H + z' ) x Horizontal distance from corner or fire centerline or width distance into the material thickness (m) y Horizontal distance from corner (m) z Vertical distance above base of fire (m) z' Virtual source origin correction in tests with fires impinging on ceilings and I-beams (m) zo Virtual source origin correction for plumes (m)

145 GREEK α Absorbtivity (- -) χ r Radiative fraction (- -) ε Emissivity (- -) ρ∞ Ambient density of air (1.2 kg/m3) π Constant (3.14159) σ Stefan-Boltzmann constant [5.67 × 10–11 kW/(m2 K4)]

SUBSCRIPTS cl Centerline conv Convective D Defined using D as length scale f Flame hfg Heat flux gauge H Defined using H as length scale HB Defined using HB as length scale HC Defined using HC as length scale inc Incident m Measured max Maximum level net Net peak Peak rad Radiative rr Reradiated s Material surface w Centerline of web

146 References

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