Mathematical Modeling of Ammonia Electro-Oxidation on Polycrystalline Pt Deposited

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A dissertation presented to

the faculty of

the Russ College of Engineering and Technology of Ohio University

In partial fulfillment

of the requirements for the degree

Doctor of Philosophy

Luis A. Diaz Aldana

May 2014

This dissertation titled

Mathematical Modeling of Ammonia Electro-Oxidation on Polycrystalline Pt Deposited

Electrodes

LUIS A. DIAZ ALDANA

has been approved for

the Department of Chemical and Biomolecular Engineering

and the Russ College of Engineering and Technology by

Gerardine G. Botte

Professor of Chemical and Biomolecular Engineering

Dennis Irwin

Dean, Russ College of Engineering and Technology 3

ABSTRACT

DIAZ ALDANA, LUIS A., Ph.D., May 2014, Chemical Engineering

Mathematical Modeling of Ammonia Electro-Oxidation on Polycrystalline Pt Deposited

Electrodes

Director of Dissertation: Gerardine G. Botte

The ammonia electrolysis process has been proposed as a feasible way for electrochemical generation of fuel grade hydrogen (H2). Ammonia is identified as one of the most suitable energy carriers due to its high hydrogen density, and its safe and efficient distribution chain. Moreover, the fact that this process can be applied even at low ammonia concentration feedstock opens its application to wastewater treatment along with H2 co-generation.

In the ammonia electrolysis process, ammonia is electro-oxidized in the anode side to produce N2 while H2 is evolved from water reduction in the cathode. A thermodynamic energy requirement of just five percent of the energy used in hydrogen production from water electrolysis is expected from ammonia electrolysis. However, the absence of a complete understanding of the reaction mechanism and kinetics involved in the ammonia electro-oxidation has not yet allowed the full commercialization of this process. For that reason, a kinetic model that can be trusted in the design and scale up of the ammonia electrolyzer needs to be developed.

This research focused on the elucidation of the reaction mechanism and kinetic parameters for the ammonia electro-oxidation. The definition of the most relevant elementary reactions steps was obtained through the parallel analysis of experimental 4 data and the development of a mathematical model of the ammonia electro-oxidation in a well defined hydrodynamic system, such as the rotating disk electrode (RDE). Ammonia electro-oxidation to N2 as final product was concluded to be a slow surface confined process where parallel reactions leading to the deactivation of the catalyst are present.

Through the development of this work it was possible to define a reaction mechanism and values for the kinetic parameters for ammonia electro-oxidation that allow an accurate representation of the experimental observations on a RDE system. Additionally, the validity of the reaction mechanism and kinetic parameters were supplemented by means of process scale up, performance evaluation, and hydrodynamic analysis in a flow cell electrolyzer. An adequate simulation of the flow electrolyzer performance was accomplished using the obtained kinetic parameters.

DEDICATION

To my support and pride (my beloved wife Sandra), my family, and the recovery of my

brother Nelson.

ACKNOWLEDGMENTS

I acknowledge the financial and technical support of the Center for

Electrochemical Engineering Research and the Department of Chemical and

Biomolecular Engineering at Ohio University.

I would like to thank the support and teaching of my advisor Dr. Gerardine G.

Botte. I will always be grateful for her support. I want to thank and the suggestions and recommendations of the committee members Dr. Savas Kaya, Dr. Gang Chen and specially Dr. Howard Dewald and Dr. Valerie Young from whom I got very important lessons about technical aspects related with the dissertation as well as for the future of my career.

I wish to thank the contribution of Jim Caesar and John Goettge, who were fundamental for the successful completion of the experiments performed. Thank you all

CEER members and staff, Shannon Bruce, Dr. Dan Wang, and Dr. Damilola Daramola, all the students, and special thank you to my friends and mentors Dr. Ana Valenzuela and

Dr. Madhivanan Muthuvel.

Thank you to Vedasri Vedharathinam, Santosh Vijapur, and Samy Palaniappan for being such good friends and make this work place completely enjoyable. I acknowledge your significant contributions to the successful completion of this work.

Finally I would like to thank God and the support of my family, my loved wife

Sandra and our little bundle of joy; you make my life so rich and happy making possible to get the best of every moment as Ph.D. student. 7

TABLE OF CONTENTS

Page

Abstract ...... 3

Dedication ...... 5

Acknowledgments...... 6

List of Tables ...... 11

List of Figures ...... 12

Chapter 1: Introduction ...... 19

1.1. Project Significance ...... 19

1.2. Statement of Objectives ...... 20

1.3. Project Overview ...... 21

Chapter 2: Literature Review ...... 24

2.1. Importance of Ammonia Electrolysis ...... 24

2.1.1. Ammonia electro-oxidation for fuel cell applications ...... 25

2.1.2. Ammonia electrolysis for hydrogen production ...... 25

2.1.3. Ammonia electrolysis for water remediation ...... 27

2.2. Ammonia Electro-Oxidation Kinetics ...... 27

2.2.1. Oswin and Salomon’s mechanism ...... 28

2.2.2. Gerisher and Maurer mechanism ...... 31 8

2.2.3. Studies of reaction intermediates ...... 32

2.2.4. Differential electrochemical mass spectrometry (DEMS) ...... 34

2.2.5. Spectro-electrochemical studies ...... 35

2.2.6. Hydrodynamic electrochemistry studies ...... 36

Chapter 3: Analysis of Ammonia Electro-Oxidation Kinetics Using a Rotating Disk

Electrode ...... 38

3.2. Introduction ...... 38

3.2. Experimental ...... 41

3.2.1. Electrode preparation ...... 41

3.2.2. Morphological and surface characterization ...... 43

3.2.3. Electrochemical measurements ...... 44

3.3. Results and Discussions ...... 46

3.3.1. Morphological characterization ...... 46

3.3.2. Electrochemical characterization ...... 50

3.3.3. RDE experiments ...... 54

3.3.4. Mathematical approach to the RDE experimental results ...... 60

3.4. Conclusions ...... 63

Chapter 4: Mathematical Modeling of Ammonia Electro-Oxidation Kinetics in a

Platinum Deposited Nickel Rotating Disk Electrode (RDE) System ...... 66 9

3.2. Introduction ...... 66

4.2. Ammonia Electro-Oxidation...... 68

4.3. Mathematical Model of Ammonia Electro-Oxidation ...... 75

4.3.1. Combined surface-diffusion controlled process (SDCP) model in the RDE

system ...... 76

4.3.2. Surface confined process (SCP) model in the RDE system...... 84

4.3.3. Solution methodology ...... 86

4.4. Results ...... 87

4.5. Conclusions ...... 96

Chapter 5: Hydrodynamic Analysis of a Flow Cell Ammonia Electrolyzer ...... 97

5.1. Introduction ...... 97

5.2. Experimental Methods ...... 98

5.2.1. Electrode preparation ...... 98

5.2.2. Electrochemical measurements ...... 99

5.2.3. Residence time distribution (RTD) ...... 99

5.3. Results and Discussions...... 101

5.3.2. RTD results ...... 106

5.3.3. Electrolyzer simulation ...... 109

5.4. Conclusions ...... 115 10

Chapter 6: Conclusions and Future Scope ...... 116

6.1. Conclusions ...... 116

6.2. Future Scope and Recommendations ...... 117

References ...... 118

Appendix A: SDCP Model Codes For Fortran and PEST ...... 131

Fortran Code ...... 131

PEST Code ...... 159

Appendix B: SCP Model Codes for Fortran and PEST ...... 167

Fortran Code ...... 167

PEST Code ...... 204

LIST OF TABLES

Page

Table 1. Summary of catalysts used for ammonia electro-oxidation ...... 30

Table 2. Experimental matrix for Pt deposition over Ni disk ...... 43

Table 3. Surface characteristics of the Pt electrodeposited over Ni disk electrodes ...... 48

Table 4. Fixed model parameters ...... 77

Table 5. Estimated model kinetic parameters ...... 89

Table 6. Parameters used for the electrolyzer simulation as a CSTR reactor ...... 112

LIST OF FIGURES

Page

Figure 1. Gerisher and Maurer mechanism scheme [52]...... 31

Figure 2. CVs of polycrystalline Pt electrodes, in presence of ammonia (solid lines), and in absence of ammonia (dotted lines): a) Shows the CVs for 1 M KOH and 0.1 M NH3

-3 [17]. b) Shows the CVs in 0.1 M NaOH and 1x10 M NH3 [42]...... 32

Figure 3. XRD spectra of the Pt electrodeposited Ni disk electrodes at different conditions: a) Teflon holder, b) plain Ni, c) 50 ºC for 20 minutes, d) 50 ºC for 40 minutes, e) 50 ºC for 80 minutes, f) 60 ºC for 80 minutes, and g) 70 ºC for 80 minutes.

The patterns confirmed the presence of polycrystalline Pt deposited on the Ni disk...... 47

Figure 4. SEM and AFM images of: a) plain Ni, b) Pt electrodeposited Ni disk electrodes at 50 ºC for 20 minutes, c) 40 minutes, and d) 80 minutes, e) at 60 ºC for 80 minutes, and f) at 70 ºC for 80 minutes. Nucleation growth of Pt was observed with the increase in deposition time (a) to d)). Pt agglomerates and bigger clusters are formed at temperature higher than 50 °C especially at 70 ºC...... 49

Figure 5. CO stripping voltammetry for EASA quantification of the Pt deposited disk electrode at 50 ºC for 80 minutes. In the first cycle, the electrochemical oxidation of CO occurred between -0.45 and -0.05 V vs. Hg/HgO. This feature was not found in the subsequent cycles (i.e. 9th cycle)...... 51

Figure 6. LSV scans for the Pt deposited disk electrodes in a 0.01 M NH4Cl + 0.1 M

KOH solution at a scan rate of 10 mV s-1. The fourth oxidation scan is displayed. The ammonia oxidation peak is observed from ca. -0.17 to -0.14 V vs. Hg/HgO, while the 13 peak observed ca. 0.63 V vs. Hg/HgO is attributed to the reversible adsorption on ammonia [44]. The highest catalytic activity was presented by the electrode deposited at

50 °C for 80 minutes...... 52

Figure 7. Histogram of ammonia oxidation peak current and peak current densities for the

Pt deposited electrodes in a 0.01 M NH4Cl + 0.1 M KOH solution. The electrode deposited at 50 ºC for 80 minutes had the highest catalytic activity towards ammonia electro-oxidation...... 53

Figure 8. Oxidative LSV scan for the Pt deposited electrode at 50 °C for 80 minutes in a

-1 0.01 M NH4Cl + 0.1 M KOH solution at a scan rate of 50 mV s for different rotational speeds (0, 500, 1000, and 2000 rpm). Insert: LSV profiles of the electrode in 0.1 M KOH

-1 (blank) solution for different rotational speeds at 50 mV s . The peak current (ip) for ammonia oxidation was calculated from the base line to the top of the oxidation peak. . 55

Figure 9. a) Plot of the peak potential (Ep) for the ammonia electro-oxidation versus the logarithm of the scan rate [log(υ)] in a 0.1 M KOH and 0.01 M NH4Cl solution at 0 rpm.

1/2 Three different zones with different Tafel slope values were observed. b) iP vs. υ plot at different rotational speeds (0, 500, 1000 and 2000 rpm). These plots suggest that the ammonia electro-oxidation could be a diffusion controlled process...... 58

Figure 10. Comparison of the Levich equation with the RDE experimental results at different scan rates (5, 10, 30, and 50 mV s-1). The slopes of the experimental data are in parallel with the theoretical Levich curve, which indicates that the diffusion coefficient calculated from the experimental data at 0 rpm can be used to determine the peak currents at different rotational speeds...... 60 14

Figure 11. Plot of the normalized ammonia oxidation peak current ip/iL versus the square root of the dimensionless scan rate σ1/2. The shape of the curve strongly suggests that the ammonia electro-oxidation found during the LSV in the RDE experiments is not taking place under steady state conditions...... 64

Figure 12. Open window cyclic voltammetry for ammonia electro oxidation in 0.1 M

-1 KOH + 0.01 M NH4Cl at 10 mV s (continuous lines) and 0.1 M KOH (dashed line). The peaks A1 and C1 are attributed to reversible NH3 adsorption. Peak A2 is recognized to ammonia electro-oxidation. Peak C2 is observed when the applied potential overcomes the ammonia electro-oxidation onset potential, therefore recognized as the reduction of surface intermediates. Another peak C3 is observed when high upper limit potentials are reached and is attributed to the reduction of NO-kind components [36]...... 72

Figure 13. Variaton of the current function represented by the scan rate normalized peak

1/2 current ip with scan rate. Current function shapes suggest that the ammonia electro- oxidation mechanism proceeds as a catalytic reaction following an electrochemical reaction...... 75

Figure 14. Comparison of the LSV curves obtained by the SDCP model (solid thin line), the SCP model (solid thick line), and the experimental results (dotted line) obtained at 10

-1 mV s and 1000 rpm in a 0.1 M KOH + 0.01 M NH4Cl solution. A better representation of the experimental data was obtained with the surface confined model...... 88

Figure 15. Uncertainty intervals with the 95% confidence level for the fitting parameters presented in Equation (4.41). The plot was obtained through the parameter estimation process performed in Win PEST 3.0.0...... 90 15

Figure 16. Results obtained for the SCP model of ammonia electro-oxidation LSV at different scan rates. Experimental values (dotted lines), Predicted values (continuous

-4 4 -1 -1 lines). Parameter values are as reported in Table 2 except for k1 f =1.46x10 cm s mol

-10 -2 -1 -1 -4 4 -1 -1 and k2 f =4.16x10 mol cm s at  =0.01 V s , =1.97x10 cm s mol and

=5.34x10-10 mol cm-2 s-1 at =0.03 V s-1, and =2.30x10-4 cm4 s-1 mol-1 and

=6.00x10-10 mol cm-2 s-1 at =0.05 V s-1. The SCP model can predict the dependence of the LSV profile with the scan rate validating the application of the model...... 91

Figure 17. Dimensionless rate parameters  ( ) obtained from the fitted kinetic

parameters a) and b) at different scan rates. b n F RT . The linear dependence of the dimensionless rate parameter with the scan rate was used to calculate

0 0 the scan rate independent rate constants k1 f , and k2 f ...... 93

Figure 18. Current potential profiles for the different reactions taking place in the ammonia electro oxidation mechanism at different scan rates. a) reaction 1, b) reaction 2-

3, c) reaction 4, and d) reaction 5. The highest contribution toward the total corrent density in the potential range from -0.2 to -0.1 V vs. Hg/HgO is due to the formation of

N2. However the current contribution due to the successive dehydrogenation of adsorbed intermediates increases along with the applied potential...... 94

Figure 19. Fractional coverage of the predicted adsorbed intermediates at different scan

    rates. a) NH2 , b) NH , c) N , and d) free space . NH2 is the predominant adsorbed intermediate within the potential region studied...... 95 16

Figure 20. Ammonia electrolysis set up used for the RTD experiments. A constant level of fluid was kept in the upper chamber of the gas collection system to maintain constant pressure inside the system. Insert parallel plate electrolyzer parts and configuration.

Dimension units are in cm...... 100

Figure 21. Calibration curve of the tracer (Yellow 5) concentration measured in the UV-

Vis spectrophotometer at 400 nm...... 101

Figure 22. Sustained cyclic voltammograms obtained at 10 mV s-1 with the Pt deposited on FNG electrode in 0.1 M KOH (solid thin line) and 0.1 M KOH + 0.01 M NH4Cl (solid thick line). Sustained CVs obtained at 10 mV s-1 after the addition of 150 µL of Yellow 5 dye in the 0.1 M KOH (dashed thin line) and 0.1 M KOH + 0.01 M NH4Cl (dashed thick line), are also presented, which shows no significant effects of the Yellow 5 dye in the current potential profiles for ammonia electro-oxidation. HUPD region is observed from -

0.75 to -0.4 V vs. Hg/HgO while NH3 electro-oxidation can be observed in the ammonia containing solutions ca. -0.1 V vs. Hg/HgO...... 102

Figure 23. Polarization curves of Pt deposited on framed Ni gauze electrode in 0.1 M

KOH + 0.01 M NH4Cl using Platinum foil (♦), and expanded Ni (■) as auxiliary electrodes. a) Working electrode potential controlled versus reference electrode Hg/HgO. b) Measured cell voltage. Same current density can be obtained with both auxiliary electrodes under the same polarization of the working electrodes. However, the cell voltage required for the expanded Ni electrode is higher than the Pt foil. Error bars show the standard deviation of the averaged measurements during the last five minutes of the potential step...... 104 17

Figure 24. Results for the scan hold potential experiments performed to the deposited Pt on framed Ni gauze electrode in 0.1 MKOH + 0.01 M NH4Cl using Expanded Ni as auxiliary electrode. Oxidative scan rate a) and reductive scan rate c) were performed at

10 mV s-1. Upper limit potentials - .2 ( ), - .1 ( ), and - .1 vs. g gO ( ) were held for one hour. Then current and measured cell voltage profiles are shown in b) and d), respectively. Insert in figure b) zoom in the current profiles. Showing a more stable current obtained at -0.2 V vs. Hg/HgO...... 106

Figure 25. Residence time distribution in the ammonia electrolysis system obtained at a

-1 -1 flow rate of 18 mL min of 200 µL L Yellow 5 dye + 0.1 M KOH + 0.01 M NH4Cl solution and an applied potential of -0.2 V vs. Hg/HgO. The continuous line shows the fitting obtained with the axial dispersion model...... 107

Figure 26. Residence time distribution in the ammonia electrolysis system obtained at a

Figure 27. Current profiles of the fluid electrolyzer with a volumetric flow rate of 1.8 mL

-1 min of 0.1 M KOH + 0.01 M NH4Cl solution and a constant applied potential of -0.2 V vs. Hg/HgO. The dotted line shows the experimental data, while the continuous solid line shows the results obtained with the simulation of the electrolyzer as a CSTR with a correlation coefficient R2=0.94...... 113

Figure 28. Ammonia fractional conversion profiles for the fluid cell ammonia electrolyzer. Error bars on the experimental measurement (▲) represent the standard 18 deviation of two samples measurements. Solid continuous line show the results obtained by mean of the electrolyzer simulation as a CSTR reactor...... 114

CHAPTER 1: INTRODUCTION

1.1. Project Significance

Ammonia electrolysis is a developing technology that has the potential to be applied in the deammonification of wastewater. Taking into account that wastewater plants are net energy consumers [1], the co-generation of hydrogen from ammonia electrolysis for subsequent energy production motivates the efforts in the development, optimization, and implementation of the ammonia electrolysis technology.

The application of technologies, such as ammonia electrolysis, is driven by the need to develop more efficient ways to recover water for human use. The actual energy consumption in wastewater plants represents about three percent of the total energy consumption in the U.S. [2]. Furthermore, the growth of the population in the future will cause an increase in the amount of wastewater, where the energy consumption of wastewater treatment plants is expected to increase at least 20% in the next 15 years [1].

Ammonia electrolysis has already been proved with synthetic wastewater in a bench scale process, obtaining ammonia conversions above 80%. If the process is compared with current technologies as Nitrification-Denitrification process where 5.4 kWh are required to remove 1 kg of ammonia [3], the application of ammonia electrolysis can reduce energy consumption below 3 kWh per kilogram of NH3 removed in wastewater [4]. These results open the window toward the prompt implementation of the ammonia electrolysis, initially in small municipal wastewater plants, but with a prominent horizon for the treatment of industrial and agricultural wastewater. 20

However, despite advances in technology, the process design, by means of a jump from the actual state of the ammonia electrolysis technology (lab and bench scales) to industrial process sizes, represents a big risk toward the successful implementation of the process.

The importance of this project relies on the development of a mathematical model which is the most powerful tool for the equipment size design, simulation and therefore prediction of the ammonia electrolysis process behavior [5] .

1.2. Statement of Objectives

The objective of this project was to obtain a design equation for the ammonia electrolyzer using electrodeposited polycrystalline platinum electrodes. Therefore, the following specific objectives have been proposed:

1) To identify feasible reactions pathways, based on experimental and theoretical

studies. The validity of a reaction mechanism will be considered based on how

well the kinetic model represents the real behavior of the ammonia electrolysis.

2) To supplement the design equation for ammonia electro-oxidation with an

accounting of mass transfer effects.

Mathematical modeling along with hydrodynamic electrochemical experiments will be used to determine the intrinsic kinetic of the reaction. Effects of flow non-uniformity will be identified in a flow single cell system studied with residence time distribution analysis. 21

A kinetic expression, which has the capacity to predict and interpret the behavior of the ammonia electrolyzer will be useful in basic and detail process engineering. Not just equipment sizes can be calculated out from an appropriated kinetic expression. Also, more accurate mass, energy, and cost balance can be accomplished.

1.3. Project Overview

The ammonia electrolysis process, developed in the Center for Electrochemical

Engineering Research at Ohio University (CEER) under the direction of Dr. Gerardine G.

Botte [6], has proved to be a feasible way for electrochemical generation of hydrogen and subsequent integration with a fuel cell to produce clean energy. Ammonia electrolysis presents several advantages for the production of energy. Ammonia is identified as one of the most suitable energy carriers due to its high hydrogen density, safety, and efficient distribution chain [7]. Fuel grade hydrogen can be produced from ammonia electrolysis with a Farady’s efficiency above 99.9% [8, 9], in a process that is not limited to low ammonia concentration feedstock [10], which opens its application to wastewater treatment along with hydrogen co-generation.

In the ammonia electrolysis process, ammonia is electro-oxidized in the anode side to produce N2 and H2O while H2 is evolved from water in the cathode side. This process requires, in theory, just five percent of the energy used in water electrolysis for hydrogen production. However, the reaction mechanism and kinetics involved in the ammonia electro-oxidation have not yet been fully elucidated, requiring the development of a 22 kinetic model that can be trustfully applied in the design and scaled up of the ammonia electrolyzer.

Within this context, the CEER has attempted to understand the ammonia electro- oxidation mechanism by means of experimental electrochemical analysis with different catalysts and support materials [8, 9, 11, 12]. On the other hand, theoretical studies of the ammonia electro-oxidation mechanism based on molecular modeling have allowed the study of possible interactions among the catalyst, reactants, and products [13, 14].

However, a comprehensive understanding of the reactions kinetics, elucidation of reaction intermediates, and quantification of the transport phenomena associated with the ammonia electro-oxidation process are required for the design and optimization of the catalyst. Therefore, in this work the microkinetic approach is proposed to develop a kinetic expression that describes the behavior of ammonia electro-oxidation on electrodeposited polycrystalline platinum electrodes.

A reaction mechanism of the ammonia electro-oxidation will be developed from the mathematical manipulation of the most relevant elementary reactions defined from the literature review presented in Chapter 2, and electrochemical tests in a RDE system

(Chapter 3). Hence, the Chapter 4 describes the way in which the parameters will be adjusted with the experimental results and the mathematical modeling of the ammonia electro-oxidation in an RDE system at different potential scan rates. This iterative process provides a valid ammonia electro-oxidation mechanism on polycrystalline platinum electrode along with the kinetic constants. 23

The final kinetic expression should take into account the effect of applied potential and electrochemical active surface area in the observed current density and reaction selectivity. To consider the effects of non-uniform flow patterns in a real electrolyzer, the Pt deposited electro-catalyst is scaled up in single flow cell system for hydrodynamic analysis (Chapter 5). The residence time distribution function, along with the kinetic expression provide the tools for the development of an appropriate electrolyzer model that can be used for simulation and scale up.

As formerly mentioned, the aim of this work is to develop a valid kinetic expression for the ammonia electro-oxidation. The success of this project will allow using this kinetic expression in the design and scaling up of the ammonia electrolyzer. Its application with the mass and energy balances will be significant in the analysis of process cost and feasibility, thus obtaining the best process design according to the requirements.

CHAPTER 2: LITERATURE REVIEW

2.1. Importance of Ammonia Electrolysis

The ammonia electro-oxidation reaction has been studied for more than four decades due to its relevance in analytical tests [15, 16], electro-chemical remediation of wastewater [10, 16], and as a renewable energy source either as a hydrogen carrier or directly as fuel for fuel cells [17].

Ammonia, besides its industrial production from natural gas and more recently from coal [18], is one of the main contaminants in industrial, residential, and agricultural wastewaters [10, 15, 19]. Moreover, water contamination, which leads to eutrophication of rivers, lakes, and seas, is not the only concern regarding ammonia emissions.

Ammonia is a gas at normal conditions of pressure and temperature and its presence in the atmosphere leads to the formation of particulate matter (ammonia aerosols), affects human health, and determines the formation of acid rain which causes soils acidification

[19, 20]. These factors along with the continuous increase of wastewater production associated with the growing human population [21], draw attention to the electro- oxidation of ammonia as a feasible alternative to solve the future environmental and energy problems. In the next sections a more detailed description of the role of ammonia electro-oxidation in fuel cell applications, hydrogen production, and wastewater remediation will be presented.

2.1.1. Ammonia electro-oxidation for fuel cell applications

Among different possible fuels, for solid oxide fuel cells (SOFCs), ammonia presents several advantages over hydrogen and some hydrocarbons in terms of safety, availability, energy density, and carbon free combustion [7, 22]. Although early research indicate that NOx may be obtained as by-products at high operation temperatures, recent developments in medium and low temperature SOFCs have allowed obtaining similar performance with ammonia as that obtained with hydrogen as fuel without NOX generation [23]. Nevertheless, the operation of these fuel cells requires high concentrations of ammonia as feed [23]; therefore, no diluted or low ammonia concentration feedstock can be used for fuel cell applications.

2.1.2. Ammonia electrolysis for hydrogen production

Hydrogen is recognized as the main fuel source for both SOFCs and polymer electrolyte membrane fuel cells (PEMFC) [8, 18]. However, different challenges associated with production, distribution, and storage have drawn the attention towards the search for hydrogen carriers such as hydrocarbons, metal hydride complexes, and ammonia compounds [8, 9, 18]. Ammonia is a promising alternative as a hydrogen carrier due to its availability, existing distribution system, volumetric hydrogen density

(45% more volumetric hydrogen for liquid ammonia in comparison with liquid hydrogen

[7]), and 50% higher hydrogen density for liquid ammonia than liquid hydrogen [8, 9,

12]. 26

Ammonia can be decomposed to hydrogen and nitrogen through either ammonia cracking or ammonia electrolysis [7]. Ammonia cracking takes place on nickel-alumina or ruthenium catalysts at temperatures ca. 300-500 °C [18]. To integrate this process with fuel cells, hydrogen needs to be separated from nitrogen and the non reacted ammonia.

Therefore, this process was classified as not economically feasible by the United States

Department of Energy (DOE) [24].

On the other hand, ammonia electro-oxidation reaction (Equation (2.1)) can be coupled with the hydrogen evolution reaction in alkaline media (Equation (2.2)). This process known as ammonia electrolysis (Equation (2.3)), can proceed at room temperature generating hydrogen with just five per cent of the theoretical potential required for the water electrolysis reaction [25].

   2NH3(aq )  6 OH  N2 6 H 2 O  6 e E = -0.77 V vs. SHE (2.1)

   6H22 O 6 e  3 H  6 OH E = -0.83 V vs. SHE (2.2)

2NH3 (aq ) N 23 H 2 Ecell = 0.06 V (2.3)

Since hydrogen is produced in the cathode and ammonia is electro-oxidized to nitrogen and water in the anode, pure hydrogen and pure nitrogen are obtained as main products in a two compartment-cell set up with a Faraday efficiency >99.99% [8, 9]. The ammonia electrolysis technology presents as a main advantage that diluted ammonia can be used as a feedstock, which opens its applicability from industrially produced ammonia to the ammonia present in wastewater [10]. 27

2.1.3. Ammonia electrolysis for water remediation

Ammonia electrolysis for effluents with concentrations below 30 ppm of ammonia, which is the expected ammonia concentration in municipal wastewater plants, have shown to be technically feasible on platinum electrodes [26]. However, the capital investments, mainly associated with the catalysts, are high, thus making the process economically unfeasible [27]. Botte et al. [25] suggested that with the optimization of the catalysts, ammonia electrolysis can be used for the purification of wastewaters along with hydrogen generation. The hydrogen generated becomes the main revenue required for the cost effective implementation of ammonia electrolysis in wastewater treatments [28].

Different substrates and metals (Table 1) have been tested with the aim to improve the performance of platinum electrodes in the electro-oxidation of ammonia using less expensive catalysts [10, 12, 29]. Nevertheless, the lack of understanding of the ammonia electro-oxidation mechanism has not allowed yet for the design of a better catalyst [30].

2.2. Ammonia Electro-Oxidation Kinetics

Ammonia in aqueous media can be found either as ammonia (NH3(aq) in alkaline

+ media) or ammonium (NH4 in acid media). However, the oxidation in acid media has shown to be slow and almost non active [31-35]. Hence, ammonia (NH3) is recognized as the active molecule toward electro-oxidation, where N2 should be the most favorable product from the thermodynamic point of view [34]. For this reason, the mechanism and kinetics of the ammonia electro-oxidation in liquid phase has been approached from its 28 analogue reaction in the gas phase [33, 36], where several transition and noble metals have been tested [18, 33]. Table 1 shows a summary of the different catalysts and conditions tested for ammonia electro-oxidation.

De Vooys et al. [37] studied the role of the adsorbed intermediates in the ammonia electro-oxidation over different polycrystalline transition metals. De Vooys et al. [37] found that 4d transition metals, such as Rh, Pd, and Ru, have low activity toward

N2 formation, while Ir and Pt were found as the most active metals for ammonia electro- oxidation to N2. Among them, Ir provides a lower onset potential for the ammonia oxidation but less activity in terms of current density compared with Pt [17].

Bi-metallic catalysts, such as Pt-Ir, Pt-Rh, and Pt-Ru, have shown better performance than plain polycrystalline Pt electrodes [8, 9, 11, 38, 39]. Whereas, some authors have observed that catalytic activity is strongly associated with the crystalline face [40-42] where single oriented Pt(100) has shown the highest catalytic activity towards ammonia electro-oxidation to N2 [40-42]. The fact that the role of the different studied metals have not been defined yet support the analysis of ammonia oxidation in Pt electrodes as a valid starting point to develop a kinetic model for ammonia electro- oxidation.

2.2.1. Oswin and Salomon’s mechanism

The first reaction mechanism for ammonia electro-oxidation on Pt was proposed by Oswin and Salomon [33]. In this mechanism ammonia electro-oxidation in alkaline media takes place in four consecutive steps (Equations (2.4)-(2.7)). Ammonia is initially 29 adsorbed in the active sites (Pt), dissociating to adsorbed NH2 and H2O (Equation (2.4)).

The next two steps correspond to consecutive dehydrogenation of -NH2 and =NH

(Equations (2.5)-(2.6)), followed by the association of N(ads) to form molecular nitrogen.

 2NH3(aq )  Pt  OH Pt  NH2  H 2 O  e (2.4)

 Pt NH22  OH Pt  NH  H O  e (2.5)

 Pt NH  OH Pt  N  H2 O  e (2.6)

Pt N  Pt  N2 Pt  N2 (2.7)

The analysis of the ammonia electro-oxidation kinetics mechanism with the Tafel slope, performed by Oswin and Salomon [33], established that the Langmuir isotherm model is the most suitable assumption to describe the adsorption of the intermediates over the active sites. Furthermore, it was concluded that the second electron transfer reaction (Equation (2.5)) or the atomic nitrogen association (Equation (2.7)) could be the rate determining steps, at low and high currents, respectively. In this mechanism, Oswin and Salomon did not take into consideration the adsorption of OH- in the active sites for the development of the kinetic model.

Table 1. Summary of catalysts used for ammonia electro-oxidation

Summary of catalysts used for ammonia electro-oxidation

Catalyst Substrate Electrolyte Temperature Reference Pt 0.5 M HClO4 [35] 0.1 M KOH [36, 44] 5 M KOH [8, 45] Raney Ni 1 M, 5 M KOH 25 °C [12] Carbon fiber/paper 1 M, 5 M KOH 25 °C [10, 11] 0.5 M KOH [46] Au, Pt 1 M KOH [15, 43] Ni 1 MKOH 25, 40, 50 °C [43] [47, 48] Pt(100) 0.5 M NaOH [41] 0.1 M NaOH [40, 42] Pt(111) 0.5 M NaOH [41] 0.1 M NaOH [40, 42] Pt(110) 0.5 M NaOH [41] 0.1 M NaOH [40, 42] Ir 0.1 M KOH [36] Carbon fiber 1 M KOH 25 °C [10] Pt 1 M KOH [49] Ru 0.1 M KOH [36] Carbon paper 5 M KOH 25 °C [11] Rh 0.1 M KOH [36] Raney Ni 1 M, 5 M KOH 25 °C [12] Carbon fiber/paper 1 M, 5 M KOH 25 °C [10, 11] Pd 0.1 M KOH [36] Ag, Cu, Au 0.1 M KOH [36] Ni Carbon paper 5 M KOH 25 °C [11] 1 M NaClO4 + NaOH [31] Pt-Ir Pt 5 M KOH [8] Carbon fiber/paper 1 M, 5 M KOH 25 °C [10, 11] Au, Pt 1 M, 0.1 M KOH [39, 49] Pt-Ru Pt 5 M KOH [8] Carbon paper 5 M KOH 25 °C [11] Pt-Rh Raney Ni 1 M, 5 M KOH 25 °C [12] Carbon fiber/paper 1 M, 5 M KOH 25 °C [10, 11] Pt-Cu 1 M KOH [17] Pt-Rh-Ir Carbon fiber/paper 1 M, 5 M KOH 25 °C [10, 11] IrO2 1 M NaClO4 + NaOH 25 °C [50] Ti 0.1 M Na2SO4 [51] RuO2 Ti 1 M, 0.1 M Na2SO4 23 °C [16, 51] TiO2 Ti 1 M Na2SO4 23 °C [16]

2.2.2. Gerisher and Maurer mechanism

Figure 1. Gerisher and Maurer mechanism scheme [52].

The ammonia electro-oxidation reaction mechanism, most accepted to date, was proposed by Gerisher and Maurer [52]. In this mechanism, the dimerization of NHx

(x=1,2), followed by the dehydrogenation of the dimerized species was presented as the most feasible way to get N2 as the main product from the ammonia electro-oxidation [34,

37]. This conclusion was reached after ex-situ analysis, where the electrode was heated ca. 400-600 °C and the gases produced were followed by gas chromatography. It was observed that just atomic nitrogen N(ads) remains on the surface and is not able to form molecular nitrogen and, therefore, it becomes a poison for the catalyst. Additionally, this

- mechanism suggested that adsorbed oxygen in the form of adsorbed OH (ads) plays an important role in the ammonia electro-oxidation mechanism [34]. Figure 1 describes the 32

Gerisher and Maurer’s Mechanism, where the second electron transfer reaction was identified as the rate determining step [44, 52]

Although, there is no indisputable evidence for the presence of the intermediates proposed in Gerisher and Maurer’s mechanism, evidence of its validity has been obtained by several authors applying different electrochemical techniques [30, 34, 36-41, 44, 46,

53]. In the next section, a summary of the main electrochemical studies and their conclusions will be presented.

2.2.3. Studies of reaction intermediates

Typical cyclic voltammograms (CVs) of poly crystalline Pt electrode are presented in Figure 2.

b) a)

Figure 2. CVs of polycrystalline Pt electrodes, in presence of ammonia (solid lines), and in absence of ammonia (dotted lines): a) Shows the CVs for 1 M KOH and 0.1 M NH3

-3 [17]. b) Shows the CVs in 0.1 M NaOH and 1x10 M NH3 [42].

When comparing the cyclic voltammograms (CV), obtained by measuring the current response produced by a linear sweep potential perturbation, of the Pt electrodes in ammonia containing solutions with those obtained in ammonia free solutions, some important differences can be pointed out:

1) Although the CV profiles for ammonia, and ammonia free electrolytes in the

1 hydrogen under potential deposition (H (UPD)) region are different (ca. 0.1 to 0.45

V in Figure 2b), the calculated charge under both profiles is the same [37]. De

Vooys et al. [37] indicated that ammonia is more preferentially adsorbed on Pt

than water. Therefore, under this region, it can be considered that one atom of

ammonia is adsorbed per active site.

2) Ammonia electro-oxidation onset potential is observed before the butterfly region

where Pt surface gets partially oxidized [41, 44, 46].

3) In the reductive potential scan, no cathodic peaks that can be clearly attributed to

the reversion of ammonia electro-oxidation are observed [37, 44, 46]. Moreover,

the cathodic peak associated with the reduction of platinum oxide, which can be

observed in absence of ammonia, is not noticeable when ammonia oxidation takes

place [44, 46].

4) When the upper switch potential of the CV, in presence of ammonia, overcomes

potentials above 0.8 V vs. reversible hydrogen electrode (RHE), some additional

reduction peaks can be observed in the H(UPD) region [37, 44, 46]. These were

attributed to the reduction of adsorbed oxidized nitrogen species [46].

1 H(UPD) region in Pt electrodes is commonly observed between 0 to ca. 0.4 V vs RHE [41, 54]. 34

Despite the features observed in ammonia electro-oxidation CVs, compared with those obtained in the absence of ammonia, it is not possible to get conclusive evidence about the nature of the process from this electrochemical technique by itself. Therefore, more advanced electrochemical techniques have been applied.

2.2.4. Differential electrochemical mass spectrometry (DEMS)

Mass spectrometry has been coupled with the study of electrochemical reactions by means of the in-situ analysis of the volatile products from a cyclic voltammetry test

[46]. N2 was observed as the only product (100% selectivity) during the ammonia electro- oxidation in DEMS experiments from the onset potential to the peak potential observed in the CV [44, 46]. Some authors reported the formation of other products, such as nitrogen oxides, and N2O at potential higher than the peak potential along with the decrease of the N2 signal in the mass spectrometer [39, 46]. Whereas other authors reported that no other products rather than N2 could be detected through DEMS [37, 41,

44]. Molecular nitrogen as product was also detected in the DEMS when the potential was scanned in the reductive direction following an oxidative scan, regardless that a neat cathodic current was observed [44, 46].

The nature of the adsorbed intermediates was analyzed isolating the adsorbed species at different potentials. Thereafter, the solution with ammonia was changed for an ammonia free solution “base electrolyte”, and CVs were performed starting with either an oxidative or a reductive scan [44, 46]. Through these experiments, ammonia was identified as the main product from the reduction scan confirming the presence of NHx 35

(x=0-2) adsorbates [44, 46]. Similar electrochemical tests consisted in moving the electrodes from an ammonia containing solution, once the adsorbed intermediates have been formed, to an ammonia free solution [37]. In both cases, ammonia was identified as product after the reduction of the adsorbed intermediates and subsequent oxidation. When the adsorbed potential was established at potentials higher than the peak potential in the ammonia electro-oxidation CV, the nitrogen production either decreased considerably

[44], or was not detected [37]. Gootzen et al. [44] analyzed the difference between adsorption and reduction charge of the adsorbed intermediates, concluding that Nads is the most likely adsorbed species. Additionally, it was demonstrated that the production of N2 in the reductive scan can be explained by parallel redox reactions taking place. This behavior can be better explained by means of the Gerisher and Maurer’s mechanism rather than Oswin and Salomon’s [37, 44].

2.2.5. Spectro-electrochemical studies

Spectroscopic techniques, such as infra red (IR), and Raman, have been approached for in-situ studies of ammonia electro-oxidation. However, the overlap in the spectral regions presented among ammonia, water, and OH- ions, have not allowed identifying the intermediates presented in Pt electrodes [30, 41]. Nevertheless, a few conclusions have been reached.

De Vooys et al. [36] studied the role of the adsorbates in ammonia electro- oxidation in gold and platinum by means of surface enhanced Raman Spectroscopy

(SERS). This study demonstrated that Nads does not participate in the formation of N2. 36

Furthermore, it was concluded, along with the DEMS studied presented in [37] that the activity of Pt and Ir, toward N2 as the main product of ammonia electro-oxidation, is due to their lower affinity to Nads than the other transition metals combined with their capacity to dissociate ammonia.

Vidal-Iglesias et al. [30] reported the presence of azide ions by means of in-situ

SERS during ammonia electro-oxidation on Pt plated over gold electrodes. The production of azide ions in the potential dependent spectrums can be only explained from the presence of hydrazine, which reacts with ammonia in alkali media to form azide ions

(Equation (2.8)). Thereafter, azide ions can also react to form N2 through the reaction presented in Equation (2.9) [30]. This result is the strongest evidence for the formation of hydrazine as a reaction intermediate and as the reaction pathway to produce N2 from the electro-oxidation of NH3.

   NH3(aq ) PtNH 2 4  7 OH PtN3 7 HO 2  6 e (2.8)

   Pt N3 22 H 2 O  e Pt  NH2  OH  N2 (2.9)

2.2.6. Hydrodynamic electrochemistry studies

Ammonia electro-oxidation studies in rotating disk electrode (RDE) systems were presented by Endo et al. [38]. This study observed that increasing the rotation speed of the Pt deposited disk electrode between 0 to 1000 rpm does not show any increase in the current density. A decrease in the peak current was contrariwise observed. The decrease 37 in the peak current value was then attributed to the formation and escape of hydroxylamine out of the electrode surface at higher rotation speeds.

To analyze the intermediates escaping out of the electrode surface, the authors performed collection experiments in a rotating ring disk electrode. In these experiments, the disc electrode is polarized in a cyclic voltammetry scan while the ring electrode is kept at a constant potential <0.2 V vs. RHE. A small anodic current was detected in the ring at disk potentials < 0.8 V vs. RHE while a cathodic current was observed at disk potential > 0.8 V vs. RHE. The authors pointed out that the small current signal obtained in the ring electrode can be attributed to the low amount of intermediate coming out of the disk electrode surface, indicating that the cathodic current could be attributed to the reduction of NOX species, and the anodic current can be attributed either to hydroxylamine or hydrazine going out of the electrode [34, 38]. However the results obtained were not conclusive.

CHAPTER 3: ANALYSIS OF AMMONIA ELECTRO-OXIDATION KINETICS

USING A ROTATING DISK ELECTRODE

Equation Chapter (Next) Section 1

Parts of this chapter have been published in a peer reviewed journal: L.A. Diaz,

A. Valenzuela-Muniz, M. Muthuvel, G.G. Botte. 2013. Electrochim. Acta. 89, 413-421

[55].

3.2. Introduction

The electro-oxidation of ammonia has been studied for more than four decades due to its relevance in analytical chemistry [15, 16], electrochemical remediation of wastewater [4, 10, 16], and as a renewable energy source either as hydrogen carrier or directly as fuel for fuel cells [17].

The electrochemical oxidation of ammonia (NH3), in Equation (2.1), when coupled with the hydrogen evolution reaction (Equation (2.2)), in an alkaline electrolytic cell results in a process known as ammonia electrolysis [6, 25]. The ammonia oxidation potential (Equation (2.1)) is reported to be -0.77 V at 25 ºC versus standard hydrogen electrode (SHE) [8, 9, 56]. Therefore, in comparison with water electrolysis (theoretical cell voltage = 1.23 V), ammonia electrolysis (Equation (2.3)) requires less than 5% of the cell voltage needed for the electrolysis of water [12]. Moreover, pure hydrogen and nitrogen gases are obtained as main products in a two compartment-cell set up with a

Faraday efficiency greater than 99.99% for the production of hydrogen [8, 9]. 39

To date, the most accepted mechanism for the ammonia electro-oxidation was proposed by Gerisher and Maurer [52], which is described in section 2.2.2. Although most of the experimental evidence, published in the literature so far, supports this mechanism [34, 37, 44], it has not been possible to identify the intermediates. The limitation in understanding the role of intermediates in the reaction mechanism, and their interactions with the media and catalysts has not yet permitted to gather full knowledge of the ammonia electro-oxidation process [30]. On the other hand, optimization of the ammonia electrolysis process depends on appropriate electrode design, which will be responsible for both, operational and capital costs associated with the process. Electrode design is not limited to structure of the electrode but also to the kind of catalyst required for the electro-oxidation of ammonia and the corresponding support material. Therefore, a complete understanding of the mechanism for the ammonia electrolysis is necessary to develop and design a better catalyst.

As presented in Table 1 in Chapter 2, attempts to understand the mechanism for the ammonia electro-oxidation by means of experimental analysis with different catalysts and support materials have been performed by several researchers [11, 17, 32, 36, 37, 39-

41, 44, 50]. Platinum (Pt) and iridium (Ir) have been identified as the most active catalyst towards ammonia electro-oxidation, based on their ability to electrochemically convert ammonia to nitrogen (N2) as the final product [37]. However, a comprehensive study of the reactions, intermediates, and transport phenomena associated with the electro- oxidation of ammonia has not been performed yet. Therefore, combining the analysis of ammonia electro-oxidation reaction and transport properties of the reactants, products, 40 and intermediates, will provide a better understanding of the mechanism for the ammonia electro-oxidation. In this study, we will focus on using Pt as the electrocatalyst for the oxidation of ammonia in order to minimize the complexity of the system.

One way to achieve this objective is to perform experiments using a rotating disk electrode (RDE). RDE experiments maintain control over hydrodynamic behavior of the electrochemical system, thereby keeping the mass transfer uniform towards the electrode surface and outwards from it [57]. Endo et al. [38] studied the electro-oxidation of ammonia using Pt rotating disk electrodes. They observed that increasing the rotational speed of the Pt disk electrode from 0 to 1000 rpm did not increase the current density for the electrochemical oxidation of ammonia, as expected from the Levich equation

(Equation (3.1)). Contrarily, a decrease in the peak current of the ammonia oxidation peak was observed. This peak current decrease, at higher rotational speed, was attributed to the feasible formation and escape of hydroxylamine at the electrode surface. The authors’ attempt to identify the formation of hydroxylamine by employing rotating ring disk electrode (RRDE) experiments was not successful. Therefore, no conclusive information was obtained to support the escape of hydroxylamine from the electrode surface [38].

The Levich equation is given by [57]

2/3 1/2 -1/6 b iL  0.62 nFAD CNH (3.1) NH3 3

41 where iL is the diffusion limited current in amperes (A), n is the total number of electrons transferred, F is the Faraday’s constant (9648 C mol-1), A is the electrode surface area in

2 2 -1 cm , DNH3 is the diffusion coefficient of the NH3 in cm s , ω is the rotational speed in rad

-1 2 -1 b -3 s , ν is the kinematic viscosity in cm s , and C NH3 in mol cm is the concentration of

NH3 in the bulk solution.

Previous studies have reported Pt deposited on nickel (Ni) as a promising electrode for ammonia electro-oxidation [47, 48]. Moreover, Ni, as the catalyst support material, is a low cost substrate that can be easily scaled up from small disk size to large size electrodes for commercial electrolytic cells. In this investigation, linear sweep voltammetry (LSV) will be coupled with RDE experiments on well characterized Pt deposited Ni electrodes. The aim of this study is to elucidate the intrinsic kinetic mechanism of the ammonia electro-oxidation by combining the mathematical description of RDE systems with LSV coupled RDE experimental data. It is anticipated that the final outcome of this investigation will pave the way for the future development of a mathematical model for the electro-oxidation of ammonia in alkaline media.

3.2. Experimental

3.2.1. Electrode preparation

Ni disks of 3 mm thick were cut from a Puratronic® Ni rod (5 mm diameter and

99.999 % metal basis) for usage as the electrode for the RDE system. The Ni disks were inserted into Teflon disk holders (Pine Instruments ET4Q series) and the following pretreatment was performed before Pt deposition: 42

1) The Ni disk held in the Teflon-disk holder was polished with sand papers, starting

from grade # 400 and followed by grade #1500, with the aim to assure a clean and

plain surface.

2) The Ni surface was sandblasted using 27. μm polishing alumina, in a Crystal

Mark. Inc. micro sandblaster, to provide surface roughness.

3) The Ni disk electrode was rinsed and immersed in acetone (Fisher scientific,

99+%), for sonication with a Zenith Ultrasonic bath at 40 kHz for 20 minutes.

4) Finally, the Ni disk electrode was rinsed with ultrapure water and dried at room

temperature until no changes in the electrode weight were observed.

Pt was electrodeposited over the pretreated Ni disk electrodes using a three electrode cell in a 250 mL glass jacketed reactor. The auxiliary electrode was a Pt foil (6 cm x 0.8 cm), which was placed as a ring around the RDE electrode. A Hg/HgO electrode (Koslow Scientific, Inc.) was used as the reference electrode. The reference electrode was immersed in a Luggin capillary filled with 6 M sodium hydroxide (NaOH)

(Fisher Scientific®, 99.5%). The tip of the Luggin capillary was positioned as close as possible to the surface of the disk electrode. The Pt solution used for the deposition was 6

-1 M NaOH (Fisher Scientific®, 99.5%) and 2.4 g L H2Cl6Pt*6H2O (Alfa Aesar ®,

99.9%). Ultrapure water (>18 MΩ) was used to prepare all the solutions. The platinum electro deposition was performed at -0.6 V vs. Hg/HgO using a Solartron 1470E potentiostat.

Three different electrolyte temperatures and deposition times were studied (Table

2) to identify the better conditions to deposit Pt over the Ni disk electrodes. The electro 43 deposition solution temperature was controlled with an Isotemp 3013HD heated recirculating bath. During the electro deposition process, the Pt solution was continuously stirred at 60 rpm with a 1 cm long stir bar. The electrodes were deposited in duplicates using the conditions described on Table 2. One of the plated disks was left on the Teflon holder for the electrochemical studies or analysis, whereas the second disk was removed from the Teflon holder to perform the morphological characterization.

Table 2. Experimental matrix for Pt deposition over Ni disk

Experimental matrix for Pt deposition over Ni disk Deposition Deposition Deposition Theoretical Pt Temperature / Time / min charge / C ± SD+ loading / mg °C Faraday’s law 50 20 -0.12 ±0.03 0.06 50 40 -0.31 ±0.06 0.16 50 80 -0.65 ±0.12 0.33 60 80 -0.94 ±0.24 0.47 70 80 -1.29 ±0.23 0.65 +Standard deviation

3.2.2. Morphological and surface characterization

The morphology of the Pt deposited electrode surface was characterized using a

JEOL-JSM-639OLV scanning electron microscope (SEM) operated at 10 kV with a magnification of 2500x. Additionally, the electrode surface was analyzed using an

Agilent 5500 Atomic Force microscope (AFM).The AFM was operated in AC mode with a resonance frequency of 325 kHz using a 10 nm size cantilever. Atomic force microscopy images of 1 μm x 1 μm were taken at a scan rate of .2 lines s-1 with pixel density of 512 x 512.The chemical composition of the Pt deposited electrode surface was 44 determined using energy dispersive X-ray spectroscopy (EDX) with a Genesis 2000

HX1622 instrument attached to the SEM equipment, which was operated at 15 kV. The morphological analysis for the deposited electrode was completed with X-ray diffraction

(XRD) spectroscopy in a Rigaku Ultima IV diffractometer operated at 40 kV and 44 mA, with a copper (Cu) target (Kα = 1.5419 Å) . The diffractograms or XRD spectra were obtained for the Pt deposited Ni disk electrodes by scanning from to 8 º (2θ) with a step size of 0.05º and a scan rate of 2º min-1. In order to avoid a strong interference from the holders (made of aluminum (Al)) on the diffractograms, the disk electrodes were located in the center of a Teflon plate.

3.2.3. Electrochemical measurements

The electrochemical active surface area (EASA) of the electrode was determined using the carbon monoxide (CO) stripping method, where the CO molecules adsorbed on the electrode surface were electrochemically oxidized [58, 59]. The CO stripping experiments were performed at room temperature in the formerly described three electrode cell system (see Section 3.2.1, Electrode preparation). The electrolyte, 0.1 M

KO (Fisher Scientific®, >8 %) solution prepared with ultrapure water (>18 MΩ), was initially bubbled with argon (Ar) gas for 10 minutes to remove dissolved oxygen from the electrolyte. Then CO gas was bubbled through the cell while a potential of -0.72 V vs.

Hg/HgO was applied to the Pt deposited disk electrode using the Solartron 1470E potentiostat. The potential was maintained for 15 minutes but the CO flow through the electrolyte was stopped after five minutes and substituted by Ar in order to remove the 45 excess CO from the electrolyte. After this step, the Pt deposited Ni disk electrode was subjected to 10 scans of cyclic voltammetry (CV) between -0.8 and 0 V vs. Hg/HgO at a scan rate of 50 mV s-1. The EASA was calculated using the CO oxidation charge as well as the charge from the so called hydrogen underpotential deposition (HUPD) region. In order to obtain an accurate electrochemical active surface area value, three repetitions of the CO stripping experiments were performed for each electrode using fresh electrolyte solution every time.

The electro-oxidation of ammonia was performed in a 0.1 M potassium hydroxide

(KOH) and 0.01 M ammonium chloride (NH4Cl) solution prepared from granular NH4Cl

(Alfa Aesar® 99.5%). The experiments were carried out using a three electrode cell system in a 125 mL glass cell for rotating electrodes (Pine Instruments). The Pt deposited disk electrodes were tested for the catalytic activity towards ammonia electro-oxidation using cyclic voltammetry between -0.95 and 0 V vs. Hg/HgO. The auxiliary electrode was a Pt ring electrode as described earlier (3.2.1. Electrode preparation) and the reference electrode was Hg/HgO placed in a Luggin capillary filled with 0.1 M KOH solution. Five cycles of cyclic voltammetry were recorded for each Pt deposited disk electrode at both scan rates of 10 and 50 mV s-1. The value of the oxidation peak current was used to compare the activity of each electrode.

Using the same electrolytic cell configuration and the electrolyte solution composition the rotating disk electrode (RDE) experiments were investigated. The RDE experiments were performed using a Pine Instrument MSR electrode rotator with a

MSRX speed controller. A factorial experimental design was performed for two factors 46

(rotational speed and scan rate), with five levels of variations in the rotational speed: 0,

500, 1000, 1500, and 2000 rpm; and seven different levels in the scan rate: 1, 5, 10, 30,

50, 70, 90 mV s-1, resulting in a total of 35 experiments.

3.3. Results and Discussions

3.3.1. Morphological characterization

The amount of Pt deposited over the Ni disks was intended to be obtained by tracking the weight changes of the Ni disks. However, due to the low loadings expected based on the deposition charge (in the order of 10-1 mg), it was not possible to obtain a reliable and consistent measurement of the amount of Pt deposited.

Table 2 shows the deposition charge obtained for each electro deposition conditions. Derived from these values, it can be inferred that the Pt loading increases with the increase in the deposition time and the temperature. Theoretical loadings calculated using Faraday’s law are also presented in Table 2.

Figure 3 displays the XRD patterns of the electrodes deposited with Pt at different conditions of time and temperature. Additionally, the XRD spectra of the Teflon plate as well as the plain Ni disk are included in Figure 3. The XRD patterns for the Pt deposited electrode evince the presence of polycrystalline Pt on top of the Ni disk. On combining the Pt(111) crystal plane with the values of the lattice parameters (not shown in this paper), the formation of face-centered cubic (fcc) crystal structures for the deposited Pt can be implied. The observed peak intensity ratio for the crystallographic planes of the deposited Pt indicated the presence of polycrystalline Pt with no preferential orientation. 47

XRD analysis also corroborates the increment in the deposition of Pt with the increase in deposition time. The intensities of the Pt peaks increased while a clear decrement in the intensity of the Ni (220) was observed; this should be attributed to the deposition of a thicker Pt layer with increasing deposition time.

Pt (1 1 1 ) Ni (1 1 1 ) Ni (2 0 0 ) Pt (2 0 0 ) Pt (2 2 0 ) Pt (3 1 1 )

Ni (2 2 0 ) g

f Intensity / a.u. e

d c b a

40 50 60 70 80 2 Figure 3. XRD spectra of the Pt electrodeposited Ni disk electrodes at different conditions: a) Teflon holder, b) plain Ni, c) 50 ºC for 20 minutes, d) 50 ºC for 40 minutes, e) 50 ºC for 80 minutes, f) 60 ºC for 80 minutes, and g) 70 ºC for 80 minutes.

The patterns confirmed the presence of polycrystalline Pt deposited on the Ni disk.

On the other hand, a qualitative differentiation of the time and temperature effects on the Pt deposited surface can be observed clearly from the SEM images (Figure 4).

Figure 4a demonstrates the roughness of the Ni disks before the deposition of Pt. Figure

4b, 2c and 2d show a nucleation growth in accordance with the results presented by Yao and Chen [48].At temperatures higher than 50 ºC, e.g., 60 ºC, Pt tends to agglomerate 48 into large particles (Figure 4e and Figure 4f), and at 70 ºC Pt clusters are formed (Figure

4f); these agglomerates and clusters of Pt result in lesser surface porosity. The AFM images of the disk electrodes (inserts in Figure 4), support the results observed from the

SEM images with a 3D view of the surface. The effective surface area of the Pt deposited electrodes was calculated by analyzing the AFM images using the software WSxM 5.0.

[60]. In Table 3, the effective surface area of deposited Pt based on analysis of AFM images is listed. It is observed that the effective surface area increased with the deposition time for the electrodes deposited at 50 °C. But among the electrodes prepared, the highest surface area was found to be the electrode deposited at 60 ºC for 80 minutes

(insert in Figure 4e). The effective surface area for the electrode deposited at 70 °C

(0.314 cm2) was less than the one prepared at 60 °C (0.341 cm2). This loss in surface area is due to the formation of Pt clusters at 70 °C.

Table 3. Surface characteristics of the Pt electrodeposited over Ni disk electrodes

Surface characteristics of the Pt electrodeposited over Ni disk electrodes Deposition Deposition wt wt% wt% Effective 2 2 EASA / cm EASA / cm Temp / Time / % Pt Ni O Surface HUPD CO stripping °C min Area / + + 2 ±SD ± SD cm 50 20 61.99 35.9 2.08 0.226 2.33 ± 0.43 3.06 ± 0.16 50 40 89.46 9.87 0.67 0.248 5.39 ± 0.63 5.48 ± 0.13 50 80 94.54 5.31 0.15 0.305 8.70 ±1.15 9.16 ± 0.66 60 80 96.3 3.55 0.14 0.341 9.79 ± 0.86 9.57 ± 0.37 70 80 93.27 2.37 4.35 0.314 7.86 ± 0.81 6.57 ± 0.02 Plain Ni - 92.83 7.16 0.217 - - +Standard deviation

a) b)

c) d)

e) f)

a) b) 50

Table 3 also lists the chemical composition of the electrode in weight percentage from the EDX analysis, which indicates that Pt distribution over the Ni surface increases with the deposition time. These EDX results are in accordance with the XRD spectra analysis from the earlier discussion.

3.3.2. Electrochemical characterization

The electrochemical active surface area (EASA) of the Pt deposited disk electrodes are also presented in Table 3, where areas calculated based on the CO stripping charge and the HUPD charge are listed. Three measurements were performed for each electrode and the results show that the electrochemical active surface area calculated using CO stripping method has lower standard deviation than the area calculated from the charge associated with HUPD region. These results indicate that the CO stripping is a most reliable technique for the calculation of the EASA in alkaline media, which is in accordance with the literature [58].

The cyclic voltammetry for estimating the electrochemical active surface area by the CO stripping method is shown in Figure 5 for the disk electrode deposited with Pt at

50 °C for 80 minutes. All the other electrodes also exhibited similar cyclic voltammetry curves in 0.1 M KOH solution between -0.8 and 0 V vs. Hg/HgO potential window. The first cycle of the voltammogram for the CO adsorbed electrode surface displayed oxidation features between -0.45 and -0.05 V vs. Hg/HgO. These oxidation features did not appear in the subsequent cycles and as an example the ninth cycle of the voltammetry is shown in Figure 5. Based on these observations, the oxidation features between -0.45 51 and -0.05 V vs. Hg/HgO are attributed to the oxidation of surface adsorbed CO into soluble carbon dioxide (CO2) or otherwise known as CO stripping. The occurrence of multiple peaks for the CO stripping process has been associated to the lower mobility of

CO in alkaline media than in acid media [58, 61-63].

1.E1-03 1st cycle

5.E0.5-04 Q'CO 3 - th 0.E+000 Double layer capacitance 9 cycle

/ A A x10/ Q'HUPD i -5.E-0.5-04

-1.E-1-03

-2.E-2-03 -0.8 -0.6 -0.4 -0.2 0 E / V (vs. Hg/HgO)

The catalytic activity of the Pt deposited disk electrodes was tested by cyclic voltammetry in ammonia solution (0.01 M NH4Cl + 0.1 M KOH). Ammonia oxidation peaks, for these electrodes, were observed from ca. -0.17 to -0.14 V vs. Hg/HgO for the scan rate of 10 mV s-1, and from ca. -0.14 to -0.10 V vs. Hg/HgO for 50 mV s-1. Figure 6 52 shows the fourth oxidative scans for each Pt deposited electrodes in a 0.01 M NH4Cl +

0.1 M KOH solution at a scan rate of 10 mV s-1.

4.E-404

3.E-304 50 °C, 80 min 60 °C, 80 min 50 °C, 40 min 4 - 2.E-204 50 °C, 20 min /Ax10

i 1.E-104

0.E+000 70 °C, 80 min

-1.E-1-04 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 E / V (vs.Hg/HgO)

Figure 6. LSV scans for the Pt deposited disk electrodes in a 0.01 M NH4Cl + 0.1 M

KOH solution at a scan rate of 10 mV s-1. The fourth oxidation scan is displayed. The ammonia oxidation peak is observed from ca. -0.17 to -0.14 V vs. Hg/HgO, while the peak observed ca. 0.63 V vs. Hg/HgO is attributed to the reversible adsorption on ammonia [44]. The highest catalytic activity was presented by the electrode deposited at

50 °C for 80 minutes.

The cyclic voltammograms for the electrodes at the scan rate of 50 mV s-1 were similar to the ones shown in Figure 6. The electrode deposited at 70 ºC showed the lowest catalytic activity towards ammonia oxidation, whilst the electrode deposited at 50 ºC for the same period of time (80 minutes) displayed the highest activity for the 53 electrochemical oxidation of ammonia (Figure 7). The oxidation peak current density (jp), in Figure 7, was calculated based on the effective surface area (column 6 in Table 3) of the electrodes. Although the electrode deposited at 60 ºC for 80 minutes has the highest

EASA, the availability of active sites for ammonia electro-oxidation was substantially greater in the electrode deposited at 50 °C for 80 minutes to generate the highest oxidation current. Therefore, the electrode deposited at 50 ºC for 80 minutes was used for the RDE tests of the electro-oxidation of ammonia.

50 °C 50 °C 50 °C 60 °C 70 °C 20 min 40 min 80 min 80 min 80 min

Figure 7. Histogram of ammonia oxidation peak current and peak current densities for the Pt deposited electrodes in a 0.01 M NH4Cl + 0.1 M KOH solution. The electrode deposited at 50 ºC for 80 minutes had the highest catalytic activity towards ammonia electro-oxidation.

3.3.3. RDE experiments

The ammonia electro-oxidation reaction listed in Equation (2.1) can be simplified for a single ammonia molecule as:

- 1  NH3(aq)  3OH 2 N2  3H 2 O  3e (3.2)

The RDE experiments for the electrode, deposited with Pt at 50 °C for 80 minutes, were performed in 0.1 M KOH and 0.01 M NH4Cl solution at various rotational speeds and scan rates. Figure 8 shows the oxidative scan of the fourth linear sweep voltammetry cycle of the electrode in ammonia (0.1 M KOH and 0.01 M NH4Cl) solution at 50 mV s-1 for varying rotational speeds. The insert of the Figure 8 displays the behavior of the electrode in 0.1 M KOH (blank) solution under the same conditions. In the blank solution, with the increase in rotational speed of the electrode, shift in the current for the oxidation profile to lesser value than static condition (0 rpm) was observed. When the ammonia solution was used, the current profile for different rotational speeds until -0.3 V vs. Hg/HgO was visibly similar to the behavior observed in the blank solution. LSVs were performed in the electrolyte solution in absence of ammonia after deaeration through Ar bubbling for 10 minutes. Under this conditions no shift in the LSV profiles were observed, which points the presence of dissolved oxygen as the cause for the profile’s shift.

7.E7-04 4.E4-04

2.E2-04 4 - 0 rpm

5 0.E+00 0

5.E-04 x10 A / i

-2.E-2-04 4 - 3 2000 rpm 3.E-04 -4.E-4-04 -1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 E / V (vs. Hg/HgO) / Ax10/

i 1.E-104 0 rpm i p 500 rpm -1 -1.E-04 0 1000 rpm 2000 rpm -3.E-3-04 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 E / V (vs. Hg/HgO)

Figure 8. Oxidative LSV scan for the Pt deposited electrode at 50 °C for 80 minutes in a

-1 0.01 M NH4Cl + 0.1 M KOH solution at a scan rate of 50 mV s for different rotational speeds (0, 500, 1000, and 2000 rpm). Insert: LSV profiles of the electrode in 0.1 M KOH

-1 (blank) solution for different rotational speeds at 50 mV s . The peak current (ip) for ammonia oxidation was calculated from the base line to the top of the oxidation peak.

The oxidation of ammonia was observed between -0.3 and 0 V vs. Hg/HgO. The current difference between peak top and the beginning of the ammonia oxidation peak is defined as the peak current (ip), which is shown in Figure 8. On increasing the rotational speed of the electrode, the peak current for ammonia oxidation also increased. The observed effect of rotational speed in the ammonia oxidation region was completely different than the results reported by Endo and his co-workers [38]. Hence, changes in the ammonia oxidation peak current with the rotational rate could be related to mass transfer effects as showed in the Levich equation (Equation (3.1)). 56

Tafel slope analysis for the electro-oxidation of ammonia using the Pt deposited

Ni disk electrodes is shown in Figure 9a. According to Equation (3.3), for total irreversible reactions such as ammonia electro-oxidation (Equation (3.2)), a linear relationship should be observed between the peak potential (Ep) and the logarithm of the scan rate [log (υ)] [57, 64].

RT EP =2.303 log( )+constant (3.3) 2αnα F

where α is the transfer coefficient and nα is the number of electrons transferred in the rate determining step.

Although, previous works on Pt deposited Ni electrodes have reported that the peak potential is independent of the scan rate for ammonia electro-oxidation [47], the results presented in Figure 9a (0 rpm) show that three different linear correlations could be obtained (similar performance was observed for higher rotational speeds as well). At low scan rates (1 to 10 mV s-1), a Tafel slope of 60 mV decade-1 suggests that at least two electrochemical reaction steps are involved (Equation (3.4), and Equation (3.5)), where the second reaction step (Equation (3.5)) will be the rate determining step [33, 57]. Rosca and Koper [41] used single oriented Pt(111) crystal electrodes to oxidize ammonia. They observed similar Tafel slopes, and suggested that the second ammonia dehydrogenation reaction step (Equation (3.5)), dehydrogenation of adsorbed amine (NH2) to adsorbed imine (NH), was the rate determining step. In contrast, at high scan rates (50 to 90 mV s-

1), a Tafel slope of 170 mV decade-1 shows the first electron transfer reaction (Equation 57

(3.4)), dehydrogenation of ammonia to form adsorbed amine-, as the rate-determining step.

 Pt- NH OH  Pt- NH  H O  e (3.4) 3 2 2

 Pt- NH22 OH  Pt- NH  H O  e (3.5)

Figure 9b presents the ammonia oxidation peak currents versus the square root of the scan rate (υ1/2). The linear relationship observed for the different rotational speeds describes a completely diffusion controlled process for the electrochemical oxidation of ammonia. Based on this observation, the changes on Tafel slope displayed in Figure 9a could be explained as mass transfer limitations of the fresh reactant to reach the electrode surface.

The Tafel slope for the intermediate region of scan rate , which ranges from 5 to

50 mV s-1, is 94 mV decade-1 as shown on the insert in Figure 9a. This value was used to calculate αnα using Equation (3.3). The value of αnα was found to be 0.63. Using this value and the number of electrons transferred in Equation (3.2) (n=3), the diffusion coefficient for ammonia in the electrolyte was determined using Equation (3.6), which correlates the peak current of irreversible reactions with the square root of the scan rate

[23].

i 2.99x105 n ( n ) 1/2 1/2 ACb D1/2 p  NH33 NH (3.6)

58 where the units for the constant corresponds to C mol-1 V-1/2 [65].

-0.06 -0.09

-0.1 Ep = 0.047 log(ʋ) - 0.044 Tafel slope 94 mV decade-1 Ep = 0.085log(ʋ) + 0.005 -0.08 -0.11 R² = 0.997 Tafel slope 170 mV decade-1 -0.12 / V Hg/HgO) / (vs

p R² = 0.995 -0.1 E -0.13 -0.14

-0.15 / V(vs /V(vs Hg/HgO) p -0.12 potential Peak -0.16 E -2.4 -2.2 -2 -1.8 -1.6 -1.4 -1.2 log (ʋ / V s-1)

-0.14 Ep = 0.030log(ʋ) - 0.079 Tafel slope 60 mV decade-1 -0.16 R² = 0.990 Peak potentialPeak -0.18 -3.5 -3 -2.5 -2 -1.5 -1 a) log (ʋ / V s-1)

2000 rpm 4 - 10 1.E-03 1000 rpm 500 rpm / Ax10/ p i 8.E8-04 0 rpm

4.E4-04 ip = 2.2742x10-3 ʋ1/2+ 6.3607x10-5

Peak current current Peak R² = 0.998 0.E+000 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 b) ʋ 1/2/ (V s-1)1/2

Figure 9. a) Plot of the peak potential (Ep) for the ammonia electro-oxidation versus the logarithm of the scan rate [log(υ)] in a 0.1 M KOH and 0.01 M NH4Cl solution at 0 rpm.

1/2 Using the slope of the ip vs. υ plot at 0 rpm (Figure 9b) and Equation (3.6), the

-9 2 -1 diffusion coefficient for ammonia was found to be DNH3 = 1.217 x 10 cm s in the electrolyte media at room temperature. This value for ammonia’s diffusion coefficient is almost four orders of magnitude lower than the one determined using the Wilke & Chang

-5 2 -1 correlation [66] for NH3 in water at 25 ºC (DNH3 = 2.4 x 10 cm s ). The low calculated diffusion coefficient value for ammonia is attributed to the presence of other ions in the

+ alkaline electrolyte. Furthermore, the conversion of ammonia molecule to NH4 ion in the electrolyte and the change in the local pH of the diffusion layer will affect the mass transfer profile of ammonia. Therefore, the diffusion coefficient calculated using Eq. (9) is defined as an effective diffusivity of ammonia, which will be used for future calculations.

In Figure 10, a calculated Levich plot (Equation (3.1)), using the effective diffusion coefficient of ammonia and the EASA presented in Table 3, is compared with the experimental currents versus the square root of the rotational speed (ω). It was noticed that the regression curves for different scan rates are in parallel to the Levich curve obtained from Equation (3.1). The slope of the different experimental currents at various scan rates shows that the experimental data is in agreement with the Levich equation. Therefore, the definition of an effective diffusion coefficient calculated from the experimental data at 0 rpm is a suitable representation of the experimental peak currents at the different rotational speeds used in this study. 60

1.E10-03

8.E 8-04 50 mV s-1 4 - 6.E -604 30 mV s-1

/ A x10 A / 4.E -404 p i 10 mV s-1 -1 2.E -204 5 mV s Levich equation 0.E+00 0 6 8 10 12 14 ω(1/2) / (rad s-1)(1/2)

3.3.4. Mathematical approach to the RDE experimental results

The RDE experiments have been widely used to investigate the intrinsic kinetics of electrochemical reactions based on the suppression of the mass transfer control. Then taking into consideration the Levich equation (Equation (3.1)), the kinetic parameters associated neatly with the surface reaction could be calculated by extrapolating ω1/2 or using the Kouteky-Levich plot [57, 67]. However, this approach is valid only for steady state conditions where changes in the potentials do not affect a fast stabilization of the concentration profile in the diffusion layer and a constant current value is rapidly achieved for each potential [67]. 61

The application of a LSV to the RDE system for irreversible reactions has been analyzed by Andricacos and Cheh [68, 69]. In such a system, the potential is continuously changing as a function of time, while the rotation of the RDE guarantees a hydrodynamic control of the mass transfer in the system. This can be described from the following single dimension equation (Equation (3.7)), where the migration effect has been neglected.

2CCC   D NH3  NH3 NH3 (3.7) NH3 y2 y  y  t

-1 where ϑy is the axial velocity towards the electrode surface (in cm s ). The following dimensionless variables, presented in Equations (3.8) to (3.10), were introduced by

Andricacos and Cheh [70].

y   (3.8) 

DtNH   3 (3.9)  2

C ,  c ,1   NH3 (3.10) NH3 Cb NH3

c   In here NH3 is the dimensionless concentration of ammonia, and and are the dimensionless distance and time, respectively. The variable δ is defined as the thickness 62 of the diffusion layer (in cm), which is a function of the rotational speed as shown in

Equation (3.11).

1.61D1/3  1/6  1/2 (3.11) NH3

where ν is the kinematic viscosity (cm2 s-1) and the units for the constant corresponds to rad1/2.

Defining the initial value c ,0 1 and the boundary condition c (1, ) 0 , NH3   NH3 the dimensionless concentration profile at the electrode surface can be represented as

Equation (3.12) [70].

cNH 0,  i() 3  (3.12)  i L, NH3

Andricaicos and Cheh [68], observed relations among the normalized current iP/iL, and the square root of the dimensionless sweep rate (σ1/2), which is defined in Equation

(3.13).

nF 2   (3.13) RTD NH3

At σ1/2 values, approximately lower than three, steady state was observed and the peak current can be represented with Equation (3.1), showing a constant value even at

1/2 high potentials (plateau behavior). o wever, at higher σ values, the ip/iL ratio increased in a transition region and attained a linear relationship with σ1/2 whose slope depends on the reaction mechanism [67-69].

The plot between the experimental normalized current versus the dimensionless scan rate is displayed in Figure 11. The shape of the curve did not exhibit a stable plateau, which strongly suggests that the ammonia electro-oxidation currents observed for the tested settings are not achieving the steady state. In these circumstances the reaction kinetics is limited by the diffusion of the reactant (ammonia). Therefore, a transient state approach could be more suitable for the further development of a mathematical model to describe the ammonia electro-oxidation reaction using the LSV perturbation method coupled with RDE experiments.

3.4. Conclusions

The electrodeposition of Pt over Ni disk electrodes was studied under different deposition temperatures and times. It was concluded after the successful deposition of polycrystalline platinum over Ni, that the morphology of the deposited platinum affects the catalytic activity of the electrode towards ammonia electro-oxidation. Higher Pt loadings did not necessarily provide higher catalytic activity because the formation of Pt clusters reduced the availability of active sites for the electrochemical reaction.

2.5

2 L

i 1.5 / p i 1

0.5

0 0 100 200 300 400 500 600 700 (1/2) σ

Based on the Tafel slope and the correlation of the peak current with the scan rate, the electro-oxidation of ammonia can be concluded as an irreversible reaction, which was controlled by the mass transfer of the reactants. The diffusion coefficient for ammonia was calculated using the results from the LSV experiments, and was lower than its standard value in water. Therefore, the calculated diffusion coefficient value was assumed to be an effective diffusivity which allowed for the agreement among the theoretical and the experimental results. The diffusion of ammonia molecules or ammonium ions towards the electrode surface can be affected by local pH, meaning the pH of the electrolyte on the electrode surface could determine the concentration of

+ ammonia. This could cause the conversion of some of the NH3 molecules into NH4 ions 65 at the electrode surface or diffusion layer. Additionally, the results from the RDE-LSV experiments also suggest that the ammonia electro-oxidation kinetics is controlled by the diffusion of ammonia, even at high rotational speeds and low scan rates. Under these conditions the ammonia electro-oxidation could proceed fast enough to avoid the stabilization of the concentration profiles in the diffusion layer, and achieve a steady state current. The results obtained in this investigation support the future development of a mathematical model for ammonia electro-oxidation as a transient problem rather than as a steady state system.

CHAPTER 4: MATHEMATICAL MODELING OF AMMONIA ELECTRO-

OXIDATION KINETICS IN A PLATINUM DEPOSITED NICKEL ROTATING DISK

ELECTRODE (RDE) SYSTEM

Parts of this chapter have been submitted for publication in Journal of Catalysis by: L.A. Diaz, and G.G. Botte.

3.2. Introduction

The ammonia electrolysis process has proved to be a feasible way for in-situ generation of hydrogen, which can be integrated with a fuel cell as an alternative clean power generation source [25, 56]. Among several hydrogen sources ammonia is identified as one of the most suitable energy carriers since it has a high hydrogen density, it is safer to handle than hydrogen, and there is an available an efficient distribution chain

[18]. Moreover, fuel grade hydrogen can be produced from ammonia electrolysis with a

Faraday’s efficiency above 99.9% [12], whilst the ammonia electrolysis potential to treat low ammonia concentration feedstock [27] leads to its application in wastewater treatment along with hydrogen co-generation [4].

In the ammonia electrolysis process, ammonia is oxidized in the anode side of an electrochemical cell to produce N2 and H2O while H2 is evolved from water in the cathode side. This process requires only five percent of the energy used in the water electrolysis for hydrogen production [8, 9, 25]. Although the ammonia electro-oxidation reaction has been studied for more than five decades, the mechanism and kinetics 67 involved have not yet been fully established. Therefore, a kinetic model that can be reliably applied in the design and scale up of ammonia electrolyzers needs to be developed.

To date, the most accepted mechanism for ammonia electro-oxidation was proposed by Gerisher and Maurer [52]. While most of the experimental evidence supports this mechanism [34, 37, 44], it has been difficult to identify the reaction intermediates. This limitation in understanding the nature and role of the intermediates in the reaction mechanism, along with their interactions with the media and catalysts has not yet permitted full development of the ammonia electro-oxidation process [30].

Within this context, attempts to understand the ammonia electro-oxidation mechanism by means of experimental analysis with different catalysts and support materials have been performed by several researchers [11, 17, 32, 36, 37, 39-41, 44, 50].

However, a comprehensive understanding of the reactions (with postulation of a kinetic rate expression considering the intermediates and the transport and surface phenomena) has not been performed.

In this section, the microkinetic model approach was implemented to develop a kinetic rate expression that describes the behavior of ammonia electro-oxidation on electrodeposited polycrystalline platinum electrode [71]. A kinetic expression was developed from the mathematical manipulation of the most relevant elementary reactions selected from the literature and the analysis of electrochemical experiments performed in a 0.1 M KOH + 0.01 M NH4Cl solution. Hence, the model parameters were adjusted against the experimental results and evaluated through the mathematical prediction of the 68 current density in a RDE system subject to LSV perturbations. The mathematical model considered two possible mechanistic approaches: a combined surface-diffusion controlled process (SDCP) and a surface confined process (SCP).

The combined surface-diffusion controlled (SDCP) mechanism was proposed based on the experimental results presented in Chapter 3, which indicated that the reaction kinetics is strongly affected by the mass transfer of ammonia from the bulk to the electrode surface. Therefore, the SDCP accounts for the effect of the mass transfer through an effective diffusivity, while assuming a rate-determining step with the other intermediate reaction steps and adsorbed intermediates considered in quasi-equilibrium.

On the other hand, the surface confined process (SCP) accounts for the different surface reaction rates independently.

Both mechanisms were evaluated through modeling discrimination techniques, and a mathematical model that describes the intrinsic kinetics of the ammonia electro-oxidation mechanism on polycrystalline platinum electrode was developed.

4.2. Ammonia Electro-Oxidation.

Ammonia in aqueous media can be found either as ammonia (NH3(aq) in alkaline

+ media) or ammonium (NH4 in acid media). However, the oxidation in acid media has shown to be slow and almost non active [31-35]. Consequently, ammonia is recognized as the active molecule toward electro-oxidation, and the mechanism and kinetics of the ammonia electro-oxidation in liquid phase was initially approached from its analogue reaction in the gas phase [33, 36]. 69

An overview of the current background knowledge in the reaction mechanism and kinetics of the ammonia electro oxidation was presented in sections 2.21 to 2.24 in

Chapter 2. A summary of the most relevant information for the development of a kinetic model is presented below.

To date the most accepted ammonia electro-oxidation reaction mechanism on Pt was proposed by Gerisher and Mauerer [52]. In this mechanism the dimerization of partially dehydrogenated adsorbed intermediates NHx (x=1,2), followed by the electro- oxidation of the dimerized species, was presented as the most feasible way to get N2 as main product from the ammonia electro-oxidation [34, 37]. The authors utilized ex-situ thermo programmed desorption (TPD) analysis after reaction at potentials higher than 0.6

V vs. reversible hydrogen electrode (RHE) and concluded that the adsorbed atomic nitrogen N(ads) cannot associate to form molecular nitrogen, and therefore becomes a poison for the catalyst. Within this mechanism it was also considered that adsorbed OH-

(ads) may play an important role in the ammonia electro-oxidation mechanism [34].

Equations (4.1) to (4.7) describe the Gerisher and Mauerer’s mechanism, where the second electron transfer reaction (Eq. 8), was identified as the r.d.s. [44, 52].

OH Pt OHPt e (4.1)

NH33 Pt NH Pt (4.2)

 NH3Pt  OHPt NH2 Pt H 2 O  Pt  e (4.3)

- NH22Pt  OHPt NHPt H O  Pt  e (4.4) 70

NHx Pt NHy Pt N2 H (x y ) Pt (4.5)

- NH2 (xy ) Pt() xyOHPt  N2 ( xyHO  )2  ( xy   1) Pt  ( xye  ) (4.6)

- NHPt OHPt NPt H2 O  Pt  e (4.7)

Although, there are no indisputable proofs for the presence of the intermediates proposed by Gerisher and Mauerer’s mechanism, most of the experimental evidence supports its validity [30, 34, 36-41, 44, 46, 53].

The nature of the reaction intermediates has been analyzed using cyclic voltammetry (CV) in poly crystalline Pt electrodes and differential electrochemical mass spectroscopy (DEMS) [37, 44, 46]. These analyses have confirmed the presence of NHx species as the main reaction intermediates and the effect of monomolecular adsorbed nitrogen (Nads) as a catalyst poison [36, 44]. Moreover, when the CVs of Pt electrodes in presence of ammonia were compared with those obtained in ammonia free solutions, some important conclusions were obtained:

1) Notwithstanding the CV profiles for ammonia and ammonia free solution in the

hydrogen under potential deposition region (ca. -0.75 to -0.4 V vs. Hg/HgO in Figure

12) are different, the calculated charge under both profiles showed the same value

[37]. De Vooys et al. [37] indicated that in the presence of ammonia; this molecule is

more preferable adsorbed than water as observed from the reversible ammonia

adsorption peaks A1 and C1 shown in Figure 12 [44]. Therefore, it can be considered

that one atom of ammonia is adsorbed per active site. 71

2) CVs of the Pt electrodes in ammonia solutions (Figure 12) indicate that ammonia

electro-oxidation starts in the double layer region, prior to the region at which the

partial oxidation of Pt surface takes place [41, 44, 46]. Additionally, potential-

dependent coulometric analysis performed by DeVoys et al. [37] concluded that

ammonia-kind adsorbates might block the adsorption of OH-. Therefore, although the

presence of OH- on the surface at the onset potential of ammonia oxidation is not

- clear [53], it can be assumed that the adsorption of OH may not play a relevant role

in the ammonia-electro oxidation mechanism to N2.

3) In reductive CV scans of ammonia electro oxidation, no noticeable reduction peaks

can be observed pointing towards the irreversible nature of ammonia electro-

oxidation to N2 (which could be caused by catalyst inefficiency and/or blockage of

active sites). Just a small cathodic shoulder (peak C2 shown in Figure 12) attributed

to the reduction of adsorbed intermediates back to NH3 is observed [44, 46].

4) Strongly adsorbed nitrogen species can be reduced by operating the electrodes at

potentials in or close to the hydrogen adsorption region [35].

Studies in preferentially oriented crystal electrodes, mainly Pt(100) and Pt(111), also showed the ammonia electro-oxidation as a surface confined process [40-42]. A

DEMS study performed by Vidal-Iglesias et al. [53] on single Pt basal planes demonstrated that the oxidation of ammonia towards N2 is almost exclusive to Pt(100), while the activity of Pt(111) toward gaseous products is low. Rosca and Koper [34, 41] furthered the kinetic analysis by calculating the Tafel slope in single oriented crystal catalysts and concluded that in Pt(111) the rate determining step (r.d.s.) is the second 72 electron transfer reaction Equation (4.4), while in Pt(100) could be the dimerization of

NH2 to form N2H4 Equation (4.9) [40, 42]. Additionally, density functional theory calculations showed that the adsorption energies of different ammonia oxidation intermediates on Pt clusters from lower to higher are -85, -93, -186, -186, -325, and -380

-1 kJ mol for NH3 < N2H4 < NH2 < NH < N respectively [14].

0.0006 A2

0.0004 d A1 c 0.0002 e f g b

/ A a i -1E-18

-0.0002 a -0.4 V vs Hg/HgO C2 b -0.25 V vs Hg/HgO C1 c -0.2 V vs Hg/HgO d -0.15 V vs Hg/HgO -0.0004 C3 e 0 V vs Hg/HgO f 0.1 V vs Hg/HgO g 0.2 V vs Hg/HgO -0.0006 -0.95 -0.75 -0.55 -0.35 -0.15 0.05 E / V vs. Hg/HgO Figure 12. Open window cyclic voltammetry for ammonia electro oxidation in 0.1 M

The former conclusions along with the adsorption energies values presented for each one of the feasible ammonia electro oxidation intermediates support the assumption that the pathway to obtain N2 requires the formation of N2H4 and its subsequent electro- oxidation, which is recognized to be facile at the potential range where the ammonia electro-oxidation takes place [34]. Within this context, it is proposed that the anodic peak attributed to ammonia electro oxidation over polycrystalline Pt in the CV (peak A2 shown in Figure 12) is the result of a series of consecutive and parallel reactions, which are described in Equations (4.8) to (4.13). It must be taken into account that Equation (4.10) is not an elementary step, but the sum of all the steps in quasi-equilibrium occurring after

NH2 association.

 3(Pt  NH3  OH Pt  NH2,ads  H 2 O  e ) (4.8)

2Pt NH2,ads Pt N2 H 4,ads  Pt (4.9)

 PtNH2 4,ads  4 OH  PtN 2 4 HO 2  4 e (4.10)

 Pt NH2,ads  OH Pt  NHads  H2 O  e (4.11)

 Pt NHads  OH Pt  Nads  H2 O  e (4.12)

 3NH3  9 OH Pt  N22 9HO Pt Nads 9 e (4.13)

In the previous chapter, an experimental analysis of the ammonia electro- oxidation with RDE coupled with linear sweep voltammetry (LSV) on Pt deposited on Ni 74 electrode in a 0.1 M KOH + 0.01 M NH4Cl solution was performed. The results showed a linear relationships of the peak potential and the peak current, with the logarithm of the scan rate ( ) and the square root of the scan rate, respectively (Figure 9). This behavior, which was also observed at high rotational speeds led to the conclusion that under the analyzed conditions the mass transfer of NH3 from the solution bulk to the surface may also play an important role in the reaction rate. An NH3 effective diffusivity, smaller than the theoretical NH3 diffusion coefficient, was calculated allowing agreement among experimental data and the theoretical Levich equation Figure 10. The low value of the effective diffusivity was attributed to possible changes in the local pH within the diffusion layer which could cause ammonia dissociation. However, the validity of this conclusion can be discussed taking into account that this reaction has been previously defined as a surface confined process [41]. Moreover the presence of adsorbed electroactive species in the LSV experiments can mislead to incorrect interpretations as it is mentioned in the literature [72].

Nevertheless, an important feature of the ammonia electro-oxidation mechanism can be obtained plotting the scan rate normalized current density versus scan rate from the results presented in Figure 9b. Figure 13 shows that regardless the rotational speed

1/2 the shape of the current function (ip ) indicates a catalytic reaction following a charge transfer reaction [73].

Hereinafter, Equations (4.8) to (4.12) were used as the base for the development of the mathematical model of ammonia electro-oxidation. Experimental data and electrode characterization were collected as presented in Chapter 3 section 3.2. 75

0.010 0.009 (1/2)

- 0.008 ) 1 - 0.007 0.006 /A (V S (V /A 0.005 2000 rpm (1/2) -

ʋ 0.004 p 1000 rpm i 0.003 0 rpm 0.002 0 0.02 0.04 0.06 0.08 ʋ / V s-1

Figure 13. Variaton of the current function represented by the scan rate normalized peak

1/2 current ip with scan rate. Current function shapes suggest that the ammonia electro- oxidation mechanism proceeds as a catalytic reaction following an electrochemical reaction.

4.3. Mathematical Model of Ammonia Electro-Oxidation

A mathematical model for ammonia electro oxidation in a RDE system under LSV perturbation was developed taking into consideration the following assumptions.

1) Kinetics control by rate determining step.

2) Langmuir adsorption isotherm for the adsorbed intermediates.

3) The RDE is an equipotential electrode and is uniformly accessible.

4) Dilute solution theory was used. The aqueous solution is composed by the species

- + - + NH3, OH , K , Cl , N2, and, NH4 . 76

5) Model equations were discretized using the finite difference method.

6) Transient state was applied (it was numerically solved by using the implicit

method).

7) Ammonia electro-oxidation follows the Rideal-Eley mechanism where OH- reacts

from the liquid phase with an adsorbed molecule.

8) Ammonia-ammonium equilibrium is reached.

4.3.1. Combined surface-diffusion controlled process (SDCP) model in the RDE system

A combined surface-diffusion controlled process requires the development of a reaction kinetics expression that accounts for the presence of the adsorbed intermediates as well as a sluggish mass transfer. This mass transfer limitation was added using the experimental effective diffusivity for NH3, as the limiting reactant, obtained from the experimental electrochemical tests presented in Chapter 3. The material balance of species i in the RDE system is defined on Equation (4.14).

dN dC ii R   0 (4.14) dyi dt

where the flux of species i listed in Table 4, are affected by three different mass transfer effects which are present in Equation (4.15): ionic migration, molecular diffusion, and convection [74].

Table 4. Fixed model parameters

Fixed model parameters SYMBOL PARAMETER NAME VALUE nj Number of nodes 100 T Temperature (K) 298.15 R Universal gas constant (J mol-1 K-1) 8.314 F Farady constan (C mol-1) 96482 ω Disk rotation speed (rpm) 1000 Cutoff Cutoff potential (V) -0.1005 iniV Initial potential (V) -0.40058 Nt number of data points in the output (dimensionless) 604 EASA Electrode active surface area (cm2) 2.33135 EGA Electrode geometric area (cm2) 0.19635 2 -1 v + kinematic viscosity (cm s ) 0.007793 +,++ 2 -1 -6 -5 Ds(1) diffusion coefficient of NH3 (cm s ) 2.65x10 , 2.65x10 Ds(2)+ diffusion coefficient of OH- (cm2 s-1) 5.22 x10-5 Ds(3)+ diffusion coefficient of K+ (cm2 s-1) 1.94 x10-5 Ds(4)+ diffusion coefficient of Cl- (cm2 s-1) 2.01 x10-5 + 2 -1 -5 Ds(5) diffusion coefficient of N2 (cm s ) 3.62 x10 + + 2 -1 -5 Ds(6) diffusion coefficient of NH4 (cm s ) 1.94 x10 z(i) charge number of component i (dimensionless) 0,-1,1,-1,0, and 1 -3 -5 cb(1) bulk concentration of NH4Cl (mol cm ) 1.00 x10 cb(2) bulk concentration of KOH (mol cm-3) 1.00 x10-4 N total electrons transferred in ammonia electrolysis 9

n1 electrons transferred reaction 1 (dimensionless) 1

n2 electrons transferred reaction 2 and 3 (dimensionless) 4

n4 electrons transferred reaction 4 (dimensionless) 1

n5 electrons transferred reaction 5(dimensionless) 1 -1  scan rate V s 0.01, 0.03, and 0.05 -3 -8 Keq NH3 dissociation constant (mol cm ) 2.00 x10 2 -9  Surface concentration mol cm 2.18 x10 Transfer coefficient reaction 1, 4, and 5 0.5 1,4,5 + Transport properties, such as kinematic viscosity and diffusion coefficient, were calculated with Aspen properties using the Electrolyte-NRTL activity coefficient method ++ with SRK equation of state. The first value for NH3 diffusion coefficient was the value used for the mass transfer controlled approximation and was obtained from the experimental data on Chapter 3 using the EGA. 78

Ds() i F C d dC N  z() i  ii Ds() i   C (4.15) i R Temp dy dy y i

The convection is governed by the flow velocity towards the electrode surface in the axial direction ϑy. For the RDE system the axial velocity is defined on Equation (4.16)

[57].

3/2 1/2 2 y 0.51 vy (4.16)

where y is the distance from the disk electrode surface in cm.

In order to facilitate the calculations, a dimensionless distance ξ (Equation (3.8)) and a dimensionless time τ (Equation (3.9)), were defined in terms of the diffusion layer thickness which was calculated using Equation (3.11).

Substituting Equations (3.8), (3.9), (3.11), (4.15), and (4.16) into Equation (4.14),

+ - in the absence of chemical reaction, the mass balance for species i = K , Cl , and N2 in the electrolyte media can be described as shown in Equation (4.17).

z()() i F Ds i d2dC d ()C i   R Temp22i d  d  d  (4.17) 2 Ds() i d C dCDsNH dC i  0.513/2v 1/2 2 i 3 i  0 2d  2 d 2 d 

- + For i = NH3, OH , and NH4 the ammonia dissociation reaction must be included.

Therefore, in order to simplify de calculations, it is assumed that the ammonia

dissociation is at equilibrium resulting that RRRNH     . Equation (4.14) for 3 NH4 OH

- + NH3 can be added to Equations (4.14) for OH , and NH4 respectively, which reduces the mass balances for these three species to two equations consisting of the flux terms as presented in Equations (4.18) and (4.19). As a consequence of the reduction of the number of equations, the equilibrium expression for the ammonia dissociation needs to be defined (Equation (4.20))

dN dC  dN dC OH OH NH3 NH3  0 (4.18) dy dt dy dt

dN dC dN dC NHHN44  NH3 NH3  0 (4.19) dy dt dy dt

CCNH OH 4  K (4.20) C eq NH3

(27)

In Equations (4.17) to (4.20) the unknown variables are the concentrations of species i and the potential in the solution versus a reference electrode ϕ. Therefore, the set of governing equations for the i+1 variables is completed with the electro neutrality condition Equation (4.21).

n  z( i ) Ci 0 (4.21) i1 80

To solve the mass balances presented on the previously listed equations, the following boundary conditions are defined:

1) At the bulk, ξ= 1, the boundary conditions are defined by the bulk

concentrations for each one of the species and a set value (i.e. ϕ= . ) for

the solution potential. These values were also used as the initial condition

to solve the system of equations.

2) At the electrode surface ξ= the flux of species is equal to its rate of

consumption or production as expressed in Equation (4.22).

s() i j N 0 (4.22) nF i

A kinetic expression for the current density of ammonia electro-oxidation was developed from the reaction steps described in Equations (4.23) to (4.27). The first electrochemical reaction in the micro kinetic model is the oxidation of NH3 to adsorbed amine (-NH2) Equation (4.23). Assuming that the reaction can be represented with the

Butler-Volmer model, the net reaction rate for this electrochemical reaction is shown in

Equation (4.28) in terms of the fractional coverage of the species involved.

k1 f  Reaction 1 Pt NH3  OH PtNH2 H 2 O  e (4.23) k1b 81

k2 f

Reaction 2 2PtNH2 PtN2 H4 + Pt (4.24) k 2b

k3 f  Reaction 3 PtN2 H 4 4 OH  N2  Pt 4 H2 O  4 e (4.25)

k4 f  Reaction 4 PtNH2  OH PtNH H2 O  e (4.26) k4b

k5 f  Reaction 5 PtNH OH PtN H2 O  e (4.27) k5b

FF   (1   )    () ( ) RTRT R1 k  C  C   e  k   e (4.28) 11f NH32OH b NH

The second reaction Equation (4.24), is a catalytic reaction and is considered in this research as the rate determining step (r.d.s.). The reaction rate expression is presented in Equation (4.29) in terms of the surface coverage of the species involved.

R2 k 22  2  k  (4.29) 22f NH2 b N2 H 4

The next elemental step considered in the microkinetic analysis is the electro- oxidation of the adsorbed hydrazine Equation (4.25). Compared with the electro- oxidation of ammonia, the electro-oxidation of hydrazine takes place at potentials lower than ammonia electro-oxidation in a relatively faster process [41]. This electrochemical reaction was considered as the irreversible step in the ammonia electro-oxidation process and its reaction rate in with the Butler-Volmer form is presented in Equation (4.30) in 82 terms of the fractional coverage of hydrazine and the concentration of OH- in the vicinity of the surface.

F ()22 4 RT R3  k   C  e (4.30) 3 f N24 H OH

In Equations (4.28) and (4.30) η and η2 represent the overpotential for the ammonia electro-oxidation and hydrazine oxidation, respectively. For ammonia electro oxidation the reaction over potential is presented in Equation (4.31).

   Vd Ujref (4.31) Vd iniV * t

where Vd is the applied potential versus the reference electrode (Hg/HgO), Ujref the formal potential for the electrode reaction versus the reference electrode (Hg/HgO), iniV the initial potential of the potential scan, ʋ the scan rate in s -1, and t the potential sweep time.

Along with the steps considered for the ammonia electro-oxidation to N2

(Equations (4.23) to (4.25)), two additional parallel reactions need to be included within the model to evaluate the effect of the Nads in the catalyst deactivation. Such reactions are the successive dehydrogenation of adsorbed NH2 to form Nads. These reactions rates for 83 the elementary steps presented in Equations (4.26) and (4.27) are presented in Equations

(4.32) and (4.33), respectively.

FF  3 (1   )    3 () ( ) RTRT R4  k   C  e  k   e (4.32) 45f NH2 OH b NH

FF   4 (1   )    4 () ( )    RT    RT R5 k55f NH COH  e kb N e (4.33)

Additionally, the total fraction of active sites is defined in Equation (4.34). Hence substituting Equations (4.28), (4.30), (4.31), (4.32), (4.33), and (4.34) into Equation

(4.29) (r.d.s.) generates the kinetic expression (current density) for ammonia electro- oxidation presented in Equation (4.35).

1           NH3 NH2 N 2 H 4 NH N (4.34)

2F ( Vd   Ujref1 ) EASA 2 2 2 RT n F  k21f K C NH C  e 3 OH (4.35) j  EGA F Vd Ujref  2 1 RT 1e KCC  1 NH3 OH  F22 Vd  Ujref14  Ujref  2 RT KKCC14NH  e 3 OH F33 Vd  Ujref  Ujref  Ujref  1 4 5  3 RT KKKCC  e 5 4 1 NH3 OH 

The Ujrefi variables on Equation (4.35) (and also in Equation (4.41)) represent the formal potential for each electrochemical reaction respectively, which can be calculated as shown in reference [75]. However, in this work the formal potentials are 84 treated as adjustable parameters since apart from the electro-oxidation of hydrazine the standard reduction potential for the other electrochemical reactions are unknown.

4.3.2. Surface confined process (SCP) model in the RDE system

For the development of a surface confined process model, the adsorbed intermediates should be added as implicit variables within the model. The addition of new variables, associated to species present exclusively on the surface of the electrodes, requires the modification of the boundary conditions mostly at ξ= along with the addition of the same number of equations and variables in the central nodes and outer boundary ξ=1. The required equations on the electrode surface come from the individual balances of each species obtained from Equations (4.23) to (4.27). In order to reduce the number of parameters to the minimum possible Equation (4.24) was still considered as the r.d.s. for the formation of N2, whereas Equations (4.23), (4.26), and (4.27) were not longer considered to be in equilibrium. The individual mass balances for the adsorbed intermediates are presented in Equations (4.36) to (4.38), where  is the maximum surface concentration of active sites in mol cm-2 calculated from the adsorption charge associated to a monolayer of hydrogen (210 µC cm-2) [59]. Equation (4.39) accounts for the balance of active sites.

FF   1  F     1 1  11 4 4 RT RRT T k CCC e k  e k   e 1 f NH3 OH 1b NH224 f NH OH 1  F  (4.36)  4 4 Γd kk eRT 2  NH2 42b NH f NH2 dt 85

F 1  F  F 44  44 55 RT RT RT kk CCe  e k  e 4 f NH2 OH 4b NH 5 f NH OH 1  F  (4.37)  55Γd k  eRT NH 5bN dt

F 1  F  55  55   RTTR Γ d N kkC  e  e  (4.38) 55f NH OH b N dt

1       NH2 NH N (4.39)

It can be observed in Equations (4.36) to (4.38) that reactions 1, 4, and 5 are not longer considered in equilibrium with reaction 2, the r.d.s. for the formation of N2. On the other hand, the equilibrium condition was still maintained for reaction 3 regarding reaction 2 so that N2 production rate can still be considered as that of reaction 2 (Equation

(4.29)). Therefore, more than one current term needs to be included in the quantification of the current density. A more general expression for the boundary condition in Equation

(4.22) is presented in Equation (4.40). This expression can therefore be applied when there is more than one electrochemical reaction taking place at the same time.

reac j r   s(,) i r Ni 0 (4.40) r1 nFr 

where r is the reaction number, nr is the number of electrodes transferred in each reaction, and s(i,r) is the stoichiometric coefficient of the dissolved species in the reaction r.

The current density expression developed for the surface confined process is presented in Equation (4.41). Notice that each one of the terms inside the outer 86 parenthesis in Equation (4.41) represent the contribution to the current density from each one of the reactions (Equations (4.23) to (4.27))

F  Vd    Ujref  1  F   Vd    Ujref  1 1  1 1 RTRT n k C C eek  1 1 f NH32OH 1b NH   F  Vd    Ujref  1  F   Vd    Ujref  4 4  4 4 EASA 2 RTRT (4.41) j F n k  n k  C  e k  e 2 2 f NH224 4 f NH OH 4b NH EGA   F Vd   Ujref  1  F   Vd    Ujref  5 5  5 5 RTRT n k C  e k  e 55f NH OH 5bN 

The governing equation for the central nodes and outer boundary can be expressed as in Equation (4.42).

d (,,)NH2 NH N  0 (4.42) dy

4.3.3. Solution methodology

The current density expressions on Equations (4.35) and (4.41) were substituted into Equation (4.22) for each one of the cases under analysis. The systems of nonlinear differential equations were discretized applying finite differences for the space dimension and the implicit method for the time dimension. The discretized equations and their jacobians were generated using the software Maple 14 by extending subroutines developed by Botte [76, 77]. Then, the systems of non linear equations were linearized using the Band-J subroutine [74, 78] and solved using the lower upper decomposition 87 method (LUDEC). Both the Band-J and LUDEC subroutines were implemented in

Fortran.

The unknown parameters for each one of the cases under study were adjusted, using the Gauss-Marquardt-Levenberg method in the software Winpest 3.0.0 [77], against the experimental data obtained from the RDE and LSV experiments in a 0.1 M KOH +

0.01 M NH4Cl solution performed as presented in Chapter 3. The values of the fixed parameters and transport properties used for both models are presented in the Table 4 while the values of the adjusted parameters, which minimized the sum of the square root of the residuals providing the best possible fit of the experimental data, are shown in

Table 5. The two models developed in this paper were discriminated using the Wilks and

Williams test coupled with the t-test [79].

4.4. Results

The results for the best parameter fittings obtained from the kinetic models in

Equations (4.35) and (4.41) are shown in Figure 14. Correlation coefficients of R2=0.992 and R2=0.999 were obtained for the SDCP model and the SCP model, respectively, with the parameters presented on Figure 14. A sensitive analysis and the wide uncertainty range displayed for certain variables, shown in Figure 15 with 95% confidence level, allowed to establish that the cathodic current contributions due to reactions 1, 4, and 5 are not affecting the modeling results, therefore no changes in the predicted current density

were observed when the parameters k1b , k4b , and k5b were removed from the model.

1.8

1.6

1.4

1.2

(geometric) 1 2 - 0.8

0.6 j / mA cm j 0.4

0.2

0 -0.4 -0.35 -0.3 -0.25 -0.2 -0.15 -0.1 E / V vs. Hg/HgO

Figure 14. Comparison of the LSV curves obtained by the SDCP model (solid thin line), the SCP model (solid thick line), and the experimental results (dotted line) obtained at 10

-1 mV s and 1000 rpm in a 0.1 M KOH + 0.01 M NH4Cl solution. A better representation of the experimental data was obtained with the surface confined model.

Although low sensitivity was also observed from the anodic contribution of reactions 4 and 5, their reaction parameters were still considered while it was not possible to obtain a good model fitting when they were removed from Equation (4.41). The low dependence of the final model to the removed parameters can be explained from the low reaction rates and the over potentials in which the reactions are taking place rather than the irreversible nature of the ammonia electro oxidation process. These parameters could still be important if a CV model is done since a cathodic peak attributed to the reduction of adsorbed intermediates back to NH3 (peak C2 on shown in Fig. 1), can be observed in the reductive scan [37].

Table 5. Estimated model kinetic parameters

Estimated model kinetic parameters. Combined Kinetic- Surface confined model Units Diffusion controlled model Ujref -0.74 -0.68 V

Ujref4 -0.24 -0.21 V

Ujref5 -0.20 -0.34 V -4 4 -1 -1 k1f - 1.46x10 cm s mol 6 -2 K1 10 - cm mol -10 -10 -2 -1 k2f 1.80x10 4.16x10 mol cm s -7 -1 k4f - 2.87x10 cm s 3 -1 K4 8.91 - cm mol -7 -1 k5f - 2.08x10 cm s -2 3 -1 K5 2.24x10 - cm mol b 0.1105 0.8895 - D.F+ 301 301 - t calculated 54.53 t at  =0.05 1.65 t + Degrees of freedom (n-p+1). n= number of data, p number of equations

The Wilks and Williams test, along with the t-test [79], were used to evaluate the significance of the models and discriminate among the two different regression equations as presented in [77]. The results of the b constants presented in Table 5 show a higher value for the SCP model than the SDCP model. A higher b value stands for a better goodness of fit for the SCP model regarding the experimental data. Moreover, the one

tailed t value higher that t at t =0.05 rejects the null hypothesis that both models have the same prediction level of the independent value (current density). Therefore, the surface confined prediction model is a better representation of the phenomena taking 90 place during the ammonia electro-oxidation, which discard that mass transfer effects could play any role in the reaction kinetics in the RDE.

Once the parameters were fitted, the ability of the models to predict the current profiles at different scan rates was tested. While the SDCP model was unable to predict changes in current profiles at different scan rate, the SCP model can predict increases in the current peak and peak potential as featured in the experimental results presented in

Chapter 3. A satisfactory fit of the experimental values at different scan rates was then obtained changing the values for the reactions constants of the first electron transfer

reaction ( k1 f in reaction 1) and the reaction constant of the r.d.s. towards the production of

N2 ( k2 f in reaction 2) as presented in Figure 16. 91

3.5

3 0.05 V s-1

2.5 0.03 V s-1 (geometric)

2 2 -

1.5 0.01 V s-1 j / mA cm j 1

0.5

0 -0.4 -0.35 -0.3 -0.25 -0.2 -0.15 -0.1 E / V vs. Hg/HgO Figure 16. Results obtained for the SCP model of ammonia electro-oxidation LSV at different scan rates. Experimental values (dotted lines), Predicted values (continuous

-4 4 -1 -1 lines). Parameter values are as reported in Table 2 except for k1 f =1.46x10 cm s mol

-10 -2 -1 -1 -4 4 -1 -1 and k2 f =4.16x10 mol cm s at  =0.01 V s , =1.97x10 cm s mol and

=5.34x10-10 mol cm-2 s-1 at =0.03 V s-1, and =2.30x10-4 cm4 s-1 mol-1 and

=6.00x10-10 mol cm-2 s-1 at =0.05 V s-1. The SCP model can predict the dependence of the LSV profile with the scan rate validating the application of the model.

Dependence of the calculated reaction constants obtained from potential sweep experiments at different scan rates have been reported in other papers [80, 81]. The differences in the calculated reaction constants at different scan rates can be explained from the theory developed by Nicholson and Shain [73], which describes the dependence of both a dimensionless rate parameter ( ), and the shape of the current profiles, with 92 the potential scan rate. In the particular case of a catalytic reaction following an irreversible electron transfer reaction, as it is the case for the mechanism proposed in this work for ammonia electro oxidation, the reaction constants can be calculated applying the

Nicholson method from Equation (4.43) [73, 82], where n are the total number of electrons transferred before the r.d.s.

nF k ()  (4.43) RT

As shown in Equation (4.43) the scan rate independent rate constant can be obtained from the potential sweep experiments if  is known. Nicholson and Shain

[73, 82], have tabulated several values for different scan rates. However, no values for

0.04< are reported, and the values expected from the rate constant values fitted from the model are lower than that. Therefore, () was calculated from the observed

rate constants k1 f and k2 f using Equation (4.43). The scan rate independent rate

0 -4 4 -1 -1 0 -10 constants for reaction 1 ( k1 f = 1.23x10 cm s mol ) and reaction 2 ( k2 f =3.66x10 mol cm-2 s-1) were obtained from the slopes in Figure 17a and b, respectively.

8.00E8x10--404 2.50E2.5x10-09-9 ψ = 6.32x10-6 ʋ-1+ 1x10-4 7.00E7x10--404 ψ = 1.88x10-11 ʋ-1+ 2.62x10-10 R² = 0.999 2.00E2x10-09-9 R² = 0.999 6.00E6x10-4-04 /b /b 1f -4 2

k 5.00E5x10-04 k -9 = 1.50E-09

= 1.5x10 ψ ψ 4.00E4x10--404 1.00E1x10-09-9 3.00E3x10--404 a) b)

2.00E2x10--044 5.00E5x10-10-10 0 20 40 60 80 100 0 20 40 60 80 100 ʋ-1 / (V s-1 )-1 ʋ-1 / (V s-1 )-1 Figure 17. Dimensionless rate parameters  ( ) obtained from the fitted kinetic

parameters a) k1 f and b) k2 f at different scan rates. b n F RT . The linear dependence of the dimensionless rate parameter with the scan rate was used to calculate

0 0 the scan rate independent rate constants k1 f , and k2 f .

The surface confined model also provides a quantitative and qualitative description of the coverage distribution of the adsorbed intermediates at different potentials, along with the contribution of the different reactions to the final current profile observed during the LSV experiments. Figure 18 and Figure 19 present the current- potential profiles for the different reactions considered within the mechanism developed in this paper (Eq. 30 to 34) at different scan rates, and the adsorbed intermediates’ fractional coverage dependence with the potential at different scan rates, respectively.

1.2 2.5 a) b) 1 0.05 V s -1 2 0.05 V s -1 (geometric) 2 (geometric) -

2 -1 - 0.8 0.03 V s 1.5 -1 0.6 0.03 V s 3 / mA3 cm / - 1 0.01 V s -1 0.4

-1 0.01 V s 0.5 0.2 Reaction Reaction 1 /cm mA Reaction Reaction 2 j j 0 0 -0.4 -0.35 -0.3 -0.25 -0.2 -0.15 -0.1 -0.4 -0.35 -0.3 -0.25 -0.2 -0.15 -0.1 E / V vs. Hg/HgO E / V vs. Hg/HgO

0.25 0.25 d) c) 0.05 V s-1 0.2 0.2 (geometric) (geometric) 2

-1 2 - -1 0.03 V s - 0.05 V s 0.15 0.15

0.1 0.1 0.01 V s -1 0.05 0.05 Reaction Reaction 5 /cm mA Reaction Reaction 4 /cm mA j

j -1 0.01 V s 0.03 V s -1 0 0 -0.4 -0.35 -0.3 -0.25 -0.2 -0.15 -0.1 -0.4 -0.35 -0.3 -0.25 -0.2 -0.15 -0.1 E / V vs. Hg/HgO E / V vs. Hg/HgO Figure 18. Current potential profiles for the different reactions taking place in the ammonia electro oxidation mechanism at different scan rates. a) reaction 1, b) reaction 2-

3, c) reaction 4, and d) reaction 5. The highest contribution toward the total corrent density in the potential range from -0.2 to -0.1 V vs. Hg/HgO is due to the formation of

N2. However the current contribution due to the successive dehydrogenation of adsorbed intermediates increases along with the applied potential.

The results indicate that the main reason for the decrease in the current density in the ammonia electro-oxidation current potential profiles is the successive dehydrogenation of ammonia to form adsorbed imine -NH and atomic nitrogen therefore decreasing the number of active sites in the catalyst at applied potentials higher than -0.2

V vs. Hg/HgO in a 0.1 M KOH + 0.01 M NH4Cl. The results obtained with the model agree with the experimental and phenomenological description of the ammonia electro- oxidation process provided by different authors in previous works [30, 36, 41, 44, 46], 95 and contributes to corroborate that the electro-oxidation of ammonia in Pt towards nitrogen is an slow surface kinetics controlled process, which takes place through the catalytic dimerization of partially dehydrogenated NH3 adsorbed intermediates to hydrazine like adsorbates (Equation (4.24)). Therefore, the electro-oxidation of ammonia to nitrogen proceeds in a electrochemical followed by catalytic (EC’) reaction mechanism where the catalytic process was identified as the r.d.s.

1 0.09 a) b) 0.9 0.03 V s-1 0.08 0.8 0.07 0.7 0.06 0.6 0.05 V s-1 0.01 V s-1 0.05 0.5 0.04 0.01 V s-1 0.4 0.03 V s-1 Fractional Fractional coverage Fractional Fractional coverage 0.03 /

0.3 / 2 NH

NH 0.02

0.2 θ θ 0.1 0.01 0.05 V s-1 0 0 -0.4 -0.35 -0.3 -0.25 -0.2 -0.15 -0.1 -0.4 -0.35 -0.3 -0.25 -0.2 -0.15 -0.1 E / V vs. Hg/HgO E / V vs. Hg/HgO

0.4 1 c) 0.9 d) 0.8 0.3 0.05 V s-1 0.7 0.6 0.03 V s-1 0.2 0.5 0.01 V s-1

Fractional Fractional coverage 0.01 V s-1 0.4 / N

θ 0.3 0.1

-1 Fractional vacant active sites 0.2

0.03 V s /

θ 0.1 0.05 V s-1 0 0 -0.4 -0.35 -0.3 -0.25 -0.2 -0.15 -0.1 -0.4 -0.35 -0.3 -0.25 -0.2 -0.15 -0.1 E / V vs. Hg/HgO E / V vs. Hg/HgO Figure 19. Fractional coverage of the predicted adsorbed intermediates at different scan

    rates. a) NH2 , b) NH , c) N , and d) free space . NH2 is the predominant adsorbed intermediate within the potential region studied.

4.5. Conclusions

A mathematical model for the ammonia electro oxidation in a Pt RDE system was developed based on the microkinetic approach. The ammonia electro-oxidation reaction was observed to be a surface confined process where the decrease of current density reported in the literature even at high rotation rates can be attributed to the effect of adsorbed intermediates and the deactivation of the Pt catalyst by the successive dehydrogenation of NH3 adsorbates.

The electro-oxidation mechanism of ammonia towards molecular nitrogen was successfully modeled as an electrochemical reaction followed by a catalytic (EC’) reaction, where the catalytic reaction, which was considered to be the catalytic dimerization of adsorbed amine –NH2 to form hydrazine, was identified as the r.d.s. In general, slow reaction rates were identified over polycrystalline Pt catalyst. A kinetic expression along with the kinetic parameters, such as reaction rates constants for the different steps relevant to the reaction mechanism developed in this chapter, were obtained. This model can be later use for the process simulation, electrolyzer design and scale up.

CHAPTER 5: HYDRODYNAMIC ANALYSIS OF A FLOW CELL AMMONIA

ELECTROLYZER

Parts of this chapter have been submitted for publication in Chemical Engineering

Journal by: L.A. Diaz, and G.G. Botte.

5.1. Introduction

In the previous chapters the intrinsic reaction kinetics associated with the ammonia electro-oxidation was examined to define a reaction mechanism that agrees with the experimental observations of a well defined hydrodynamic system, such as the rotating disk electrode (RDE). Based on this study, heterogeneous reaction kinetic constants were obtained by means of parameter estimation obtaining a correlation coefficient R2=0.999. The mathematical model of ammonia electro-oxidation on an RDE electrode was fitted and verified with experimental data.

However, in real applications, the prediction and simulation of the electrochemical reactor behavior has to account for flow non-uniformity, dead volumes, and inlet effects that affect the mass transfer of the species, and therefore the performance of the reactor (electrolyzer). In this chapter, Pt deposited on Ni substrate as electrodes, in which the kinetic analysis was performed in the previous chapters, were scaled-up to be tested in a flow cell electrolyzer. Electrochemical characterization and electrochemical tests were performed in the scaled-up electrode before it was arranged in a parallel plate flow single electrolyzer. A hydrodynamic analysis of the single flow cell electrolyzer was 98 done by means of residence time distribution (RTD) test in order to establish the time distribution function and get a better understanding of the behavior of the electrolyzer.

The results obtained from the RTD analysis leaded to the successful simulation of the electrolyzer system as a continuum stirred tank reactor (CSTR). The simulation results were validated experimentally through current density and ammonia fractional conversion.

5.2. Experimental Methods

5.2.1. Electrode preparation

A 14.8 x 14.8 cm Ni gauze, Alfa Aesar 40 mesh woven, framed in a 0.25 cm thick border of Ni foil (Alfa Aesar 0.0127 cm thick, 99+ % Ni on metal basis), was used as electrode substrate. Electrode surface pretreatment was performed before Pt electro deposition following the procedure presented in section 3.2.1. Then the Ni foil frame was covered with a 3M 5480 PTFE film tape to avoid deposition of Pt on the Ni frame.

Pt electro deposition on the framed Ni gauze (FNG) electrode was done in a 6 M

-1 . NaOH (Fisher Scientific, 99.5%) plus 2.4 g L H2Cl6Pt 6H2O (Alfa Aesar, 99.9%) solution using a Solartron 1470E potentiostat. Optimum electro deposition conditions at -

0.6 V vs. Hg/HgO were previously defined in section 3.3.2. (50 °C and 80 min) and used in this study. A three electrode cell was set up in a 15.5 cm length, 19.5 cm height, and

3.2 cm width customized chlorinated polyvinyl chloride (CPVC) container. The auxiliary electrodes were two 15.2 x 19 cm Pt of 0.01 cm thickness and 99.999% purity (ESPI metals) foils connected in parallel and placed at each face of the framed Ni gauze 99 electrode. A Hg/HgO electrode (Koslow Scientific, Inc.) immersed in a Luggin capillary filled with 6 M NaOH was used as reference electrode. The electro deposition solution temperature was controlled with an Isotemp 3013HD heated recirculating bath. Ultrapure water (>18 MΩ) was used to prepare all the solutions used in this work. Finally, Pt loading was measured by mean of weigh change after electro-deposition.

5.2.2. Electrochemical measurements

A three electrodes cell configuration was used to test the electrochemical activity of the electrode in 0.1 M KOH (Fisher Scientific®, >85%) solution, where the electrochemical active surface area of the electrode was measured by means of CO adsorption and striping as described in section 3.2.3. A Hg/HgO electrode in a Luggin capillary and two Pt (15.2 x 19 cm) foils were used as reference and auxiliary electrode, respectively. Cyclic voltammetry and potential steps experiments were performed, with a

Solartron 1470E potentiostat, to examine the catalytic activity towards ammonia electro- oxidation in 0.1 M KOH + 0.01 M NH4Cl (Alfa Aesar® 99.5%) solution. Either two Pt

(15.2 x 19 cm) foil or two 14.8 x 14.8 cm expanded Ni plates, connected in parallel and placed at each face of the FNG electrode, were tested as auxiliary electrodes. The cell voltage was measured using the auxiliary channels in the Solartron 1470E potentiostat.

5.2.3. Residence time distribution (RTD)

The RTD analysis was performed in a flow cell parallel electrode assembly described in the insert of Figure 20 [83]. A single step perturbation was applied using the 100 experimental set up presented in Figure 20. 0.1 M KOH + 0.01 M NH4Cl solution was fed into the electrolyzer at a flow rate of 1.8 mL min-1. Once the flow rate was stabilized, a fixed potential, defined from preliminary electrochemical analysis, was applied. After one hour of applied potential, the three way valves on Figure 20 were open to a solution

-1 of 200 µL L of Yellow 5 dye (Tracer) in 0.1 M KOH + 0.01 M NH4Cl. The concentration of Yellow 5 was measured every 30 s at the exit of the reactor using a

Hewlett Packard 80 µL quartz QS 1.000 compact fluid cell in an Agilent 8453 spectrophotometer. Yellow 5 concentrations at the effluent were measured at 400 nm based on the calibration curve presented in Figure 21. Yellow 5 dye was found to have no significant effect in the ammonia electro-oxidation current-potential profile as is presented in Figure 21.

Gases Collection 20 Flow outlet

Displaced 14.8 1.5 Liquid 28 Splitter 15.3 Wiring port

0.25 WE AE RE capillary inlet UV-VIS Flow inlet

WE FI

Ammonia Tracer Electrolyzer Potentiostat solution solution Effluent RE FI Three way valve AE Working Electrode WE Auxiliary Electrode AE Reference Electrode RE Figure 20. Ammonia electrolysis set up used for the RTD experiments. A constant level of fluid was kept in the upper chamber of the gas collection system to maintain constant pressure inside the system. Insert parallel plate electrolyzer parts and configuration.

Dimension units are in cm. 101

250 1 - L L

μ 200 /

TRACER 150

100

0 Yellow 5Yellow concentration, C 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Absorbance, A / AU

Figure 21. Calibration curve of the tracer (Yellow 5) concentration measured in the UV-

Vis spectrophotometer at 400 nm.

5.3. Results and Discussions.

The amount of Pt deposited on the FNG electrode, measured by weight change, was 0.048 g ± 0.001 g of Pt. This loading was obtained with an experimental charge 249

C which accounts for a Faraday efficiency of 38 % for the deposition process. The FNG

2 electrode geometric area (EGA) was calculated as 251.2 cm (following the procedure described by Zhang et al. [84]). Whereas the electrochemical active surface area (EASA), measured by means of CO adsorption and stripping, was 0.56 m2 ± 0.04 m2. Figure 22 shows the sustained periodic CV cycles at 10 mV s-1 of the deposited Pt electrodes, which were reached at the 5th cycle using the Pt foils as auxiliary electrodes. The CVs presented in Figure 22 were performed before and after the addition of 150 µL of Yellow 5 dye in

750 mL of 0.1 M KOH and 0.1 M KOH + 0.01 M NH4Cl solutions. 102

1 (geometric) 2 - 0

-1 j / mA cm j -2

-3

-4 -0.95 -0.75 -0.55 -0.35 -0.15 E / V vs. Hg/HgO Figure 22. Sustained cyclic voltammograms obtained at 10 mV s-1 with the Pt deposited on FNG electrode in 0.1 M KOH (solid thin line) and 0.1 M KOH + 0.01 M NH4Cl (solid thick line). Sustained CVs obtained at 10 mV s-1 after the addition of 150 µL of Yellow 5 dye in the 0.1 M KOH (dashed thin line) and 0.1 M KOH + 0.01 M NH4Cl (dashed thick line), are also presented, which shows no significant effects of the Yellow 5 dye in the current potential profiles for ammonia electro-oxidation. HUPD region is observed from -

0.75 to -0.4 V vs. Hg/HgO while NH3 electro-oxidation can be observed in the ammonia containing solutions ca. -0.1 V vs. Hg/HgO.

The hydrogen under potential deposition region HUPD, characteristic of polycrystalline Pt [55], can be observed in the ammonia free solutions CVs presented on

Figure 22 at the potential region from -0.75 to -0.4 V vs. Hg/HgO. In presence of ammonia a current peak ca. -0.1 V vs. Hg/HgO attributed to the ammonia electro- 103 oxidation is observed. It is also observed in Figure 22 that the presence of the Yellow 5 dye does not significantly affect the current potential profiles of the ammonia electro- oxidation, therefore Yellow 5 dye can be used as tracer for the RTD tests while a potential is applied. It is fundamental for the RTD studies to use an inert tracer that can be applied in-situ, while the ammonia electrolysis proceeds, because the evolution of gasses (N2 and H2) during ammonia electro-oxidation reaction affects the flow profiles inside the electrolyzer. Misleading results could be obtained from ex-situ RTD analysis.

Potential staircase experiments were performed in 0.1 M KOH + 0.01 M NH4Cl solution, from -0.35 V vs. Hg/HgO to -0.025 V vs. Hg/HgO with a potential increment of

0.025 V and 10 min step size. Figure 23 shows the polarization curves obtained from the average current featured during the last five minutes of each potential step. The decrease in the measured current density after -0.2 V vs. Hg/HgO is attributed to the catalyst deactivation in agreement with the reaction mechanism presented in Chapter 4. Figure

23a shows that same performance can be obtained using either Pt foil or expanded Ni as auxiliary electrodes at the same potential of the working electrode versus the reference electrode. However, using the auxiliary channels of the potentiostat was possible to measure that higher cell voltages are required when expanded Ni is used as counter electrode instead of Pt foil (Figure 23b). This difference can be explained from the higher over potentials required for the hydrogen evolution on Ni than Pt [85]. The electro- oxidation of ammonia on the Pt deposited on FNG electrode is the controlling reaction in the ammonia electrolysis process. Therefore, the use of non noble metals, such as Ni as the cathode, (although increases the voltage required for the process) could reduce the 104 capital cost of the ammonia electrolyzer while still operating at voltages much lower than water electrolysis. Hereinafter, expanded Ni was used as auxiliary electrode for long term potential step experiment, and in the fluid cell electrolyzer.

0 1.1

-0.05 1 0.9 -0.1 0.8 -0.15 0.7

-0.2 0.6 Cell Voltage Cell / V

E / V vs. / Hg/HgO E -0.25 0.5 0.4 b) -0.3 0.3 -0.35 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 -2 j / mA cm (Geometric) -2 j / mA cm (Geometric) Figure 23. Polarization curves of Pt deposited on framed Ni gauze electrode in 0.1 M

Based on the current densities displayed in Figures 22 and 23 the best electrode performance should be expected between -0.2 to -0.1 V vs. Hg/HgO. Hence, constant potential tests were performed in 0.1 M KOH + 0.01 M NH4Cl solution, sweeping the potential at 10 mV s-1 from -0.95 V vs. Hg/HgO to -0.2, -0.15, and -0.1 V vs. Hg/HgO, respectively, and holding each potential for one hour. Then, a reversed sweep scan at 10 105 mV s-1 from the applied potential (either -0.2, -0.15, or -0.1 V vs. Hg/HgO) back to -0.95

V vs. Hg/HgO was performed to evaluate the effect of the applied potential in the amount of adsorbed species. This was made measuring the charge required to reduce the adsorbed intermediates back to ammonia. Figure 24a, b, and c show the positive potential scans, the current profile for the constant potential step, and the following reductive scan, respectively. The measured cell voltage profile is also shown on Figure 24d.

Higher current density are reached at higher upper limit potentials as shown in potential scans in Figure 24a. However, Figure 24b shows that the higher potential the faster the drop in the current density. These phenomena can be explained based on the reductive potential scans presented in Figure 24c. The charge of the cathodic peak ca. -

0.5 V vs. Hg/HgO, which is associated to the reduction of adsorbed intermediates back to

NH3 [44], is 0.64, 0.70, and 0.77 C after the step potentials of -0.2, -0.15, and -0.1 V vs.

Hg/HgO, respectively. These results reveal a higher coverage of adsorbed intermediates, possibly leading toward the electrode deactivation, at higher applied potentials as discussed in the reaction mechanism presented in Chapter 4. The applied potential of -

0.2 V vs. Hg/HgO shows a higher and more stable current density profile, which agrees with the more stable measured cell voltage profile presented Figure 24d ca. 0.86 V. An applied potential of -0.2 V vs. Hg/HgO was then used for the RTD analysis.

106

4 4 0.8 a) b) 3 3.5 0.7 3 0.6 0.5

2 (geometric) 2 2.5 - 0.4 (geometric) (geometric) 2 2 - - 2 1 mA /cm -0.2 V

j 0.3

1.5 0.2 -0.15V / mA cm / mA cm 0 -0.1V j j 1 0.1 0 1000 2000 3000 -1 0.5 Time / s

0 -2 0 1000 2000 3000 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 E / V vs. Hg/HgO Time / s 1.4 0.4 -0.2 V 0.2 c) -0.15V d) 1.3 0 -0.1V -0.2 1.2 -0.4 (geometric) 2 - -0.6 1.1 -0.8

Cell voltage/ Cell V 1

/ mA/ cm -0.1V

j -1 -0.15V -1.2 0.9 -0.2 V -1.4 0.8 -1.6 0 1000 2000 3000 -0.95 -0.75 -0.55 -0.35 -0.15 Time / s E / V vs. Hg/HgO Figure 24. Results for the scan hold potential experiments performed to the deposited Pt on framed Ni gauze electrode in 0.1 MKOH + 0.01 M NH4Cl using Expanded Ni as auxiliary electrode. Oxidative scan rate a) and reductive scan rate c) were performed at

10 mV s-1. Upper limit potentials - .2 ( ), - .1 ( ), and -0.1 vs. g gO ( ) were held for one hour. Then current and measured cell voltage profiles are shown in b) and d), respectively. Insert in figure b) zoom in the current profiles. Showing a more stable current obtained at -0.2 V vs. Hg/HgO.

5.3.2. RTD results

The cumulative RTD function F(t/ṫ) (Figure 25) was measured in the experimental ammonia electrolyzer set-up shown in Figure 20, for a flow rate of 1.8 mL min-1. Initially, the experimental data was fitted by the axial dispersion model presented in Equation (5.1) [86]. The parameters of the dispersion model, mean residence time (ṫ), 107

and Peclet number ( Pe uLL L D ); where DL and uL are the axial diffusivity and velocity, respectively, and L is the characteristic length of the electrolyzer, were adjusted using the solver function in Microsoft Excel® minimizing the sum of square residual.

Through this method a correlation coefficient R2= 0.999 was obtained with Pe =2.91 and

ṫ =26477.4 s.

t 1 11t F( t ) 1 erf Pe  (5.1) 22t t

1 0.9 0.8 0.7 0.6 0.5

F(t/ṫ) 0.4 0.3 0.2 0.1 0 0 0.5 1 1.5 t/ṫ Figure 25. Residence time distribution in the ammonia electrolysis system obtained at a

The Pe value indicates that although the RTD function of the electrolysis neither describes a continuous stirred tank reactor (CSTR) Pe < 0.05, nor a plug flow reactor

(PFR) Pe > 50 [87], a high amount of dispersion takes place within the system, which represents a good degree of mixing in the electrolyzer. Therefore, a segregated flow model, such as the laminar flow reactor, would not describe properly the real behavior of the electrolyzer.

Based on the former results, the RTD model of a PFR-CSTR in series was tested and fitted using Equation (5.2). An adequate fitting of the experimental data (R2=0.995) was also obtained with the PFR-CSTR as presented in Figure 26. The mean residence time for the PFR ṫP= 4762.9 s, and a mean residence time for the CSTR ṫs=31685.3 s were calculated using the solver function in Microsoft Excel® minimizing the sum of square residuals.

0 ; tt  p  Ft()  ttp (5.2)  ts 1et ; t p

The PFR section of the PFR-CSTR model represents the dead volume in the electrolyzer system in which no reaction is taking place such as the gas splitter and tubing

(fluid lines) in Figure 20. The total PFR volume calculated from the RTD was VPFR =

140.9 mL. On the other hand, the volume calculated for the CSTR VCSTR = 950.6 mL, could represent the flow electrolyzer. Therefore, in order to verify whether the electrolyzer can be represented by a CSTR model, a simulation of the ammonia 109 electrolysis at constant potential was performed calculating the ammonia electrolyzer as a

CSTR with the mean residence time obtained from the PFR-CSTR model. The simulation results were then compared with experimental data.

1 0.9 0.8 0.7 0.6 0.5 F(t//ṫ) 0.4 0.3 0.2 0.1 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 t/ṫ Figure 26. Residence time distribution in the ammonia electrolysis system obtained at a

5.3.3. Electrolyzer simulation

The kinetic expression, necessary to simulate the behavior of the ammonia electrolyzer as a CSTR at an applied constant potential, was developed in Chapter 4

(from the reaction mechanism described in Equations (4.23) to (4.27)), where Equation

(4.24) was identified as the rate determining step for the formation of N2. The kinetic 110 expression for ammonia consumption reaction rate is presented in Equation (4.41). .

Because the CSTR model was implemented (homogeneous concentration throughout the recter), the value of the concentration over potential  was assumed as 0 V. The mass balance in the CSTR model was developed taking ammonia as the key component as

r shown in Equation (5.3) where NH3 is the ammonia consumption rate.

dCNH() out C C  r t  t 3 (5.3) NH3 ()() in NH3 out NH3 s s dt

Based on the first term inside the big parenthesis in Equation (4.41), ammonia consumption rate can be assumed as a first order reaction if the concentration of OH- and the concentration of unoccupied active sites ( ) remain constant. This assumption is reasonable for OH- concentration because the amount of OH- that is consumed in the ammonia electro-oxidation (Equation (2.1)) is generated in the water reduction reaction

(Equation (2.2)) . However, the concentration of available active sites changes with time as a cause of the formation of adsorbed intermediates and the possible catalyst deactivation as shown in Figure 24 and discussed with more detail in Chapter 4.

   Therefore, the transient mass balance of the adsorbed intermediates ( NH2 , NH , N ) and

presented in Equations (4.36) to (4.39) were added to the ammonia mass balance in

Equation (5.3) to complete the number of equations required by the number of variables in Equation (4.41) (M = 5).

111

The values of the kinetic parameters such as heterogeneous reaction constants kif

and formal potentials Ujrefi vs. Hg/HgO, where i is the number of the reaction in

Equations (4.23) to (4.27), were obtained from the mathematical model of ammonia- electro oxidation in a rotating disk electrode (RDE) system during linear sweep voltammetry (LSV) perturbation and reported in Chapter 4. As the value of the reversed

heterogeneous reaction constants kib were found to be no-significant for the RDE-LSV model, the reversed reaction terms in Equations (4.41), (5.3), and (4.36) to (4.39) were initially excluded from the simulation equations. The linearly independent first order ordinary differential Equations (Equations (5.3), and (4.36) to (4.38)) were solved using the explicit Euler method implemented in an Excel® spread sheet [79], with the initial values and time step size t presented in Table 6. The value of was chosen based on the time interval in which the experimental data was collected by the Solartron 1470E potentiostat.

The simulations results obtained neglecting the effect of the reversed reaction show a rapid decrease of the current profile regardless the applied potential. It was then concluded that although the effect of the reverse reactions in Equations (4.23) to (4.27) may be insignificant when the potential is swept in a positive (oxidative) direction, the reactions in equilibrium become important when the applied potential is kept constant.

Therefore, the value of the missing parameters in Equation (4.41) ( k1b , k4b , k5b ) were calculated fitting the simulated current profile with the experimental current profile obtained at -0.2 V vs. Hg/HgO. 112

Table 6. Parameters used for the electrolyzer simulation as a CSTR reactor

Parameters used for the electrolyzer simulation as a CSTR reactor. SYMBOL PARAMETER NAME VALUE T Temperature (K) 298.15 R Universal gas constant (J mol-1 K-1) 8.314 F Farady constan (C mol-1) 96482 Ujref Formal potential reaction 1 (V vs. Hg/HgO) -0.68

Ujref4 Formal potential reaction 4 (V vs. Hg/HgO) -0.21

Ujref5 Formal potential reaction 5 (V vs. Hg/HgO) -0.34 4 -1 -1 -4 k1f Forward reaction constant reaction 1 (cm s mol ) 1.23x10 4 -1 -1 -9 k1b Backward reaction constant reaction 1 (cm s mol ) 6.83x10 -2 -1 -10 k2f Forward reaction constant reaction 2 (mol cm s ) 3.66x10 -1 -7 k4f Forward reaction constant reaction 4 (cm s ) 2.87x10 -2 -1 -10 k4b Backward reaction constant reaction 4 (mol cm s ) 4.80x10 -1 -7 k5f Forward reaction constant reaction 5 (cm s ) 2.08x10 -2 -1 -11 k5b Backward reaction constant reaction 5 (mol cm s ) 2.50x10 -3 -5 C Initial value for ammonia concentration (mol cm ) 1.00 x10 NH3 /0 t ( , , ) Initial concentration of adsorbed species (dimensionless) 0 NH2 NH N /t=0  Initial concentration of vacant active sites (dimensionless) 1 n1 electrons transferred reaction 1 (dimensionless) 1 n2 electrons transferred reaction 2 and 3 (dimensionless) 4 n4 electrons transferred reaction 4 (dimensionless) 1 n5 electrons transferred reaction 5(dimensionless) 1 2 -9  Surface concentration of active sites mol cm 2.18 x10 Transfer coefficient reaction 1, 4, and 5 0.5 1,4,5 t Time step size Euler method (s) 0.24

113

2.5

2 2 -

1.5

0.5 Cell currentCell / mA Cm

0 0 2000 4000 6000 8000 10000 Time / s

Figure 27. Current profiles of the fluid electrolyzer with a volumetric flow rate of 1.8 mL

The best estimated parameter values that provided a correlation coefficient

R2=0.94 are listed in Table 6. Figure 27 shows the fitted simulation and experimental current profiles of the single fluid cell electrolyzer with the Pt deposited Ni framed electrode as working electrode, expanded Ni electrode as auxiliary electrode, an applied potential of -0.2 V vs. Hg/HgO, and a feed of 1.8 mL min-1 of a 0.1 MKOH + 0.01 M

NH4Cl solution. The noise observed in the experimentally recorded current density can be attributed to the effects of the bubbles growing and leaving the surface of the electrode as the reaction proceeds. 114

In addition to the current profiles, the simulation provides the ammonia conversion profile provided by the ammonia electrolyzer. Figure 28 shows the fractional ammonia conversion profiles compared against experimental measurements. The measurement of the experimental fractional conversion was performed as reported in [4].

0.4

0.35

0.3

0.25

0.2

0.15 fractional fractional conversion

3 0.1 NH 0.05

0 0 2000 4000 6000 8000 10000 Time / s

Figure 28. Ammonia fractional conversion profiles for the fluid cell ammonia electrolyzer. Error bars on the experimental measurement (▲) represent the standard deviation of two samples measurements. Solid continuous line show the results obtained by mean of the electrolyzer simulation as a CSTR reactor.

The results presented in Figure 27 and Figure 28 show adequate agreement among the experimental performance of the ammonia electrolyzer and the electrolyzer simulation as a CSTR reactor, where the maximum error regarding the predicted 115 conversion of the simulation was observed with last experimental fractional conversion measured (ca. 24%). Higher experimental conversions compared with the predicted value could be attributed to the NH3 stripping caused by the evolution of gases (N2 and H2).

The agreement in the experimental and simulated current profiles as well as the close representation of the NH3 conversion in the electrolyzer support the description of the residence time distribution function of the electrolysis system as a PFR-CSTR. Therefore, it confirms that a considerable amount of mixing is taking place within the electrolyzer.

5.4. Conclusions

The hydrodynamic analysis performed to a single fluid cell ammonia electrolyzer established the presence of an extensive degree of mixing within the electrolyzer, which supports that the electrolyzer system can be represented using the residence time distribution function of a PFR-CSTR system. Additionally, the results obtained by means of experimental electrochemical analysis and the simulation of the ammonia electrolyzer as a CSTR explain the effects of the adsorbed intermediates in the ammonia electro- oxidation performance in which the applied potential appears as the most important control variable to reduce the fast poisoning and contamination of the catalyst.

The results obtained through the RTD analysis and simulation support the reaction mechanism and kinetic expression for ammonia electro-oxidation on polycrystalline Pt deposited electrodes developed in Chapter 4.

116

CHAPTER 6: CONCLUSIONS AND FUTURE SCOPE

6.1. Conclusions

The most significant contributions of this work to the field of electrochemical engineering and specifically to the ammonia electrolysis technology are listed below:

1) A mathematical model that accurately represents the kinetics of the ammonia

electro oxidation in a RDE system was developed.

2) The formation of high dehydrogenated ammonia adsorbed intermediates was

identified as the main cause for the catalyst deactivation

3) The Pt deposited on Ni electrode was scaled up and the kinetic parameters

obtained through the mathematical model were tested in a fluid cell reactor,

obtaining adequate agreement simulating the electrolyzer as a CSTR reactor

Although the hydrodynamic analysis presented is this work was developed with the aim to supplement the kinetics of ammonia electro-oxidation in a real reactor, the results are applicable just for electrochemical cells with the same characteristic dimensions of the system used here. However, the validation of a reliable reaction mechanism and a kinetic expression for ammonia electro-oxidation, accomplished through this work, represents a major contribution to the design, scale up, and simulation of the ammonia electrolysis process.

117

6.2. Future Scope and Recommendations

The kinetic parameters obtained in this work provided an accurate description of the ammonia electro-oxidation process on a polycrystalline Pt electrode within the error.

However, an understanding of the contribution of the catalyst surface and morphology can be obtained through multi scale modeling. Using this approach in future work will contribute to the understanding of the interactions of the catalyst and chemical species involved allowing a better design of a less expensive and more active catalyst.

As a continuation of this work it is recommended to follow the experimental procedure described in this work at different temperatures in order to calculate the activation energies of the different reactions and extend the applicability of the model at temperatures apart of 25 °C. Kinetic research coupled with mathematical modeling can be also extended to other catalysts or catalyst mixtures. 118

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131

APPENDIX A: SDCP MODEL CODES FOR FORTRAN AND PEST

Fortran Code c Last modified by Luis Diaz on Oct 8, 2013 c SDCP MODEL IMPLICIT REAL*8(A-H,O-Z) parameter (nne=8,nm=10001,var=6,par=8) c Simbology: The program uses the following nomemclature: c c(1,j): concentration of (mol/cm3). Result c c(2,j): concentration of OH- (mol/cm3). Result c c(3,j): concentration of K+ (mol/cm3). Result c c(4,j): concentration of Cl- (mol/cm3). Result c c(5,j): concentration of N2 (mol/cm3). Result c c(6,j): concentration of NH4+ (mol/cm3). Result c c(7,j): solution potential (V). Result c n: number of equations, 7 c nm: maximum number of nodes to set dimensions, 10001 c nne: maximum number of variables to set dimensions, 7 c var: number of species, 6 c nj: number of nodes in the normal coordinate (max=10001 otherwise change nm in code) c Temp: Temperature (K) c cutoff: Cut off potential (V). Input. This is the value at which the program will stop c nv: number of data points in the output (input value) c omega: disk rotation speed (rpm input, the program transform it into rad/s) c ac: constant for RDE (dimensionless) c vis: kinematic viscosity (cm2/s) c delta: characteristic layer thickness (cm). This is calculated by the program c Ds(i): diffusion coefficient of (i) (cm^2/s) c z(i): charge number of (i) (dimensionless) c cb(1): bulk concentration of NH4Cl (mol/cm3) c cb(2): bulk concentration of KOH (mol/cm3) c s(i): Stoichiometric coefficient (i) in electrochemical reaction (dimensionless) c nn: number of electrons transferred in reaction (dimensionless) c Ujref: Theoretical open circuit potential of reaction (V) c Vd: potential of working electrode (V) c fc: faraday's constant (96485 coul/mol) c R: Universal gas constant (8.314 J/mol K) REAL*8 c,Temp,iapp,omega,ac,vis,delta,Ds,cb,kf,kb,Ke,ioh,iov, #c1ref,c2ref,Ujref,Vd,Ujref2,dx,xr,fc,R,Potgrad,etha,cutoff, #iniV,yref,PN2,as,DT, TIME,DT0,sr,theta,Ujref3,K1,K2,K3,K4,K5,K6, #alpha,Ujref4,TNH3,TNH2,TNH,TN2H4,TN,K7,TOH,K8,TN2,sigma,rho, 132

#Deng,EASA,EGA,CNH3,Keq integer counter, z,betha,s,nn,lr,nv,NUMCOL data fc,R,ac/96487,8.3143,0.51023262/ DIMENSION C(nne,nm), CG(nne,nm), CN(nne,nm), A(nne,nne), #B(nne,nne), D(nne,nne*2+1),X(nne,nne), Y(nne,nne), F(nne), #G(nne), Ds(var), z(var), cb(var), betha(var), s(var), #xr(nm), Potgrad(nm),as(par),ZETA(nm),CNH3(nm,nm)

COMMON/A1/ A,B,C,D COMMON/A2/ G,X,Y,N,NJ 102 FORMAT(A10) 114 FORMAT('0',/,16X,'ANALYTICAL SOLUTION ',//, 1 13X,' ZETA ',10X,'Y1') 108 FORMAT (' ',/,5X,'THIS RUN DID NOT CONVERGE') 111 FORMAT(' ',6X,6f14.5,6f14.5,6f14.5,6f14.5,6f14.5) 109 FORMAT (' ',/,5X,'THIS RUN DID CONVERGE;IE. RESIDUAL L.T. 1E-12') 113 FORMAT(5X,'NUMBER OF ITERATIONS =',I4) 210 FORMAT(//,16X,'NUMBER OF EQUATIONS =',I3,//,16X,'NUMBER OF' 1 ,' NODE POINTS =',I4,//) 211 FORMAT(2X,'DELTA ZETA=',F20.8,//) 212 FORMAT(/,6X,'TIME=',D12.5,//,6X, 'Vd=',D12.5,//,12X,' ZETA ', 1 8X,' C1',8X,'C2',8X,'C3',8X,'C4') 200 FORMAT(/,6X,'INITIAL PROFILE',//,12X,' ZETA ', 1 8X,' C1 ',10X,'C2', 10X,'C3', 10X,'C4', 10X,//) 312 FORMAT(/,6X,'LAST COMPUTED PROFILE',//,12X,'POSITION', 1 8X,' Y1 ',10X,//) 943 FORMAT(' ',5X,'COMPONENT ',I2,' RESIDUAL= ',E14.5,'FOR ', 1I2,' ITERATION' ) 717 FORMAT('1') open(UNIT=5,FILE='DataL.dat',STATUS='old') read(5,*) nj !number of nodes in the normal coordinate read(5,*) Temp !Temperature (K) read(5,*) omega !disk rotation speed (rpm) read(5,*) cutoff !Cutoff overpotential (V) read(5,*) IniV !Initial potential (V) read(5,*) nv !number of data points in the output read(5,*) EASA !electrode active surface area (cm^2) read(5,*) EGA !electrode geometric area (cm^2) read(5,*) vis !kinematic viscosity (cm2/s) read(5,*) Ds(1) !diffusion coefficient of NH3 (cm^2/s) read(5,*) Ds(2) !diffusion coefficient of OH- (cm^2/s) read(5,*) Ds(3) !diffusion coefficient of K+ (cm^2/s) read(5,*) Ds(4) !diffusion coefficient of Cl- (cm^2/s) read(5,*) Ds(5) !diffusion coefficient of Cl- (cm^2/s) 133

read(5,*) Ds(6) !diffusion coefficient of Cl- (cm^2/s) read(5,*) z(1) !charge number of NH3 (dimensionless) read(5,*) z(2) !charge number of OH- (dimensionless) read(5,*) z(3) !charge number of K+ (dimensionless) read(5,*) z(4) !charge number of Cl- (dimensionless) read(5,*) z(5) !charge number of N2 (dimensionless) read(5,*) z(6) !charge number of NH4+ (dimensionless) read(5,*) cb(1) !bulk concentration of NH4Cl (mol/cm3) read(5,*) cb(2) !bulk concentration of KOH (mol/cm3) read(5,*) s(1) !Stoichiometric coefficient in electrochemical reaction NH3 (dimensionless) read(5,*) s(2) !Stoichiometric coefficient in electrochemical reaction OH- (dimensionless) read(5,*) s(3) !Stoichiometric coefficient in electrochemical reaction K+ (dimensionless) read(5,*) s(4) !Stoichiometric coefficient in electrochemical reaction Cl- (dimensionless) read(5,*) s(5) !Stoichiometric coefficient in electrochemical reaction N2 (dimensionless) read(5,*) s(6) !Stoichiometric coefficient in electrochemical reaction NH4+ (dimensionless) read(5,*) nn !number of electrons transferred in reaction (dimensionless) read(5,*) sr !scan rate V/s read(5,*) Keq !Keq dissociation constant read(5,*) sigma !Surface tension dyn/cm read(5,*) Deng !N2 density g/cucm read(5,*) rho !Solution density g/cucm read(5,*) Relec !Electrode radii cm read(5,*) P !SYSTEM PRESSURE read(5,*) Ujref !Theoretical open circuit potential of reaction (V) read(5,*) Ujref3 !anodic transfer coefficient (dimensionless) read(5,*) Ujref4 !anodic transfer coefficient (dimensionless) read(5,*) K1 read(5,*) K2 read(5,*) K4 read(5,*) K5 close (UNIT=5)

open (unit=21,file='short.dat') open (unit=8,file='long.txt',status='unknown') c open (Unit=6,file='tetha.txt',status='unknown') c open (Unit=7,file='CNH3.csv',status='unknown') open (Unit=9,file='time.dat',status='unknown') N=7 134

omega=omega*2*3.1416/60.0 delta=1.61*Ds(1) **(0.1D1 / 0.3D1) *vis **(0.1D1 / 0.6D1)*omega * #* (-0.1D1 / 0.2D1) ccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc ccccc c ----Calculate initial solution concentration based on NH3/NH4+ equilibrium ------ccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc ccccc do j=1,nj c(3,j)=cb(2) c(4,j)=cb(1) c(5,J)=0 c(6,j)=(0.5000000000D0 * (1000*cb(1)) - 0.9976311575D-5 - #0.5000000000D0 * (1000*cb(2)) + 0.2500000000D-13 * #sqrt(0.4000000000D27 * (1000*cb(1)) ** 2 + 0.1 #596209852D23 * (1000*cb(1)) - 0.8000000000D27 * (1000*cb(1)) * #(1000*cb(2)) + 0.1592428 #682D18 + 0.1596209852D23 * (1000*cb(2)) + 0.4000000000D27 * #(1000*cb(2)) ** 2))/1000 c(1,j)=cb(1)-c(6,j) c(2,j)=cb(2)-c(1,j) end do do j=1, nj c(7,j)=0 end do

LLO=nv-1

TIME=(cutoff-IniV)/sr DT=TIME/LLO DT0=DT*Ds(1)/delta**2 c This is the loop to generate voltage-current dv=(cutoff-IniV)/NJ

DZETA=1.D0/(NJ-1) cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc c Calculation of the zeta value and the analytical solution ccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc DO 7 J=1,NJ

ZETA(J)=(J-1)*DZETA 135 c c Analytical solution c In this section you can input the analytical solution if known c for comparison purposes.

7 CONTINUE c Ini Input your guess values in C(1,J) c Input your initial values in CN(1,J) c------DO 3 J=1,NJ CN(1,J)=c(1,j) CN(2,J)=c(2,j) CN(3,J)=c(3,j) CN(4,J)=c(4,j) CN(5,J)=c(5,j) CN(6,J)=c(6,j) 3 CN(7,J)=c(7,j) DO 28 L=0,LLO TIME=L*DT Vd=iniV+sr*TIME c------c This loop assigns to the old value of the dependent variable c The recently calculated C(I,J) c If(L.gt.1) then DO 2 I=1,N DO 2 J=1,NJ 2 CN(I,J)=C(I,J) c endif JCOUNT=0 8 JCOUNT=JCOUNT+1 DO 100 I=1,N DO 100 J=1,NJ 100 CG(I,J)=C(I,J) J=0 DO 9 I=1,N DO 9 K=1,N X(I,K)=0.0 D0 Y(I,K)=0.0 D0 9 CONTINUE 10 J=J+1 DO 11 I=1,N G(I)=0.0 D0 136

DO 11 K=1,N A(I,K)=0.0 D0 B(I,K)=0.0 D0 11 D(I,K)=0.0 D0 IF(J-1) 12,12,14` c THIS IS ZETA=0.0 c Input your equations and jacobians for the left boundary, that is the electrode surface 12 f(1) = s(1)*EASA/EGA*K2*K1**2*C(1,J)**2/(1+K1*C(1,J)*C(2,J)*exp(fc #*(iniV+sr*TIME-C(7,J)-Ujref)/R/Temp)+K1*K4*C(1,J)*C(2,J)**2*exp(fc #*(2*iniV+2*sr*TIME-2*C(7,J)-Ujref-Ujref3)/R/Temp)+K5*K4*K1*C(1,J)* #C(2,J)**3*exp(fc*(3*iniV+3*sr*TIME-3*C(7,J)-Ujref-Ujref3-Ujref4)/R #/Temp))**2*C(2,J)**2*exp(2*fc*(iniV+sr*TIME-C(7,J)-Ujref)/R/Temp)- #(z(1)*Ds(1)*fc*C(1,J)*(-3*C(7,J)+4*C(7,J+1)-C(7,J+2))/R/Temp/dzeta #/2+Ds(1)*(-3*C(1,J)+4*C(1,J+1)-C(1,J+2))/dzeta/2)/delta f(2) = s(2)*EASA/EGA*K2*K1**2*C(1,J)**2/(1+K1*C(1,J)*C(2,J)*exp(fc #*(iniV+sr*TIME-C(7,J)-Ujref)/R/Temp)+K1*K4*C(1,J)*C(2,J)**2*exp(fc #*(2*iniV+2*sr*TIME-2*C(7,J)-Ujref-Ujref3)/R/Temp)+K5*K4*K1*C(1,J)* #C(2,J)**3*exp(fc*(3*iniV+3*sr*TIME-3*C(7,J)-Ujref-Ujref3-Ujref4)/R #/Temp))**2*C(2,J)**2*exp(2*fc*(iniV+sr*TIME-C(7,J)-Ujref)/R/Temp)- #(z(2)*Ds(2)*fc*C(2,J)*(-3*C(7,J)+4*C(7,J+1)-C(7,J+2))/R/Temp/dzeta #/2+Ds(2)*(-3*C(2,J)+4*C(2,J+1)-C(2,J+2))/dzeta/2)/delta f(3) = s(3)*EASA/EGA*K2*K1**2*C(1,J)**2/(1+K1*C(1,J)*C(2,J)*exp(fc #*(iniV+sr*TIME-C(7,J)-Ujref)/R/Temp)+K1*K4*C(1,J)*C(2,J)**2*exp(fc #*(2*iniV+2*sr*TIME-2*C(7,J)-Ujref-Ujref3)/R/Temp)+K5*K4*K1*C(1,J)* #C(2,J)**3*exp(fc*(3*iniV+3*sr*TIME-3*C(7,J)-Ujref-Ujref3-Ujref4)/R #/Temp))**2*C(2,J)**2*exp(2*fc*(iniV+sr*TIME-C(7,J)-Ujref)/R/Temp)- #(z(3)*Ds(3)*fc*C(3,J)*(-3*C(7,J)+4*C(7,J+1)-C(7,J+2))/R/Temp/dzeta #/2+Ds(3)*(-3*C(3,J)+4*C(3,J+1)-C(3,J+2))/dzeta/2)/delta f(4) = s(4)*EASA/EGA*K2*K1**2*C(1,J)**2/(1+K1*C(1,J)*C(2,J)*exp(fc #*(iniV+sr*TIME-C(7,J)-Ujref)/R/Temp)+K1*K4*C(1,J)*C(2,J)**2*exp(fc #*(2*iniV+2*sr*TIME-2*C(7,J)-Ujref-Ujref3)/R/Temp)+K5*K4*K1*C(1,J)* #C(2,J)**3*exp(fc*(3*iniV+3*sr*TIME-3*C(7,J)-Ujref-Ujref3-Ujref4)/R #/Temp))**2*C(2,J)**2*exp(2*fc*(iniV+sr*TIME-C(7,J)-Ujref)/R/Temp)- #(z(4)*Ds(4)*fc*C(4,J)*(-3*C(7,J)+4*C(7,J+1)-C(7,J+2))/R/Temp/dzeta #/2+Ds(4)*(-3*C(4,J)+4*C(4,J+1)-C(4,J+2))/dzeta/2)/delta f(5) = s(5)*EASA/EGA*K2*K1**2*C(1,J)**2/(1+K1*C(1,J)*C(2,J)*exp(fc #*(iniV+sr*TIME-C(7,J)-Ujref)/R/Temp)+K1*K4*C(1,J)*C(2,J)**2*exp(fc #*(2*iniV+2*sr*TIME-2*C(7,J)-Ujref-Ujref3)/R/Temp)+K5*K4*K1*C(1,J)* #C(2,J)**3*exp(fc*(3*iniV+3*sr*TIME-3*C(7,J)-Ujref-Ujref3-Ujref4)/R #/Temp))**2*C(2,J)**2*exp(2*fc*(iniV+sr*TIME-C(7,J)-Ujref)/R/Temp)- #(z(5)*Ds(5)*fc*C(5,J)*(-3*C(7,J)+4*C(7,J+1)-C(7,J+2))/R/Temp/dzeta #/2+Ds(5)*(-3*C(5,J)+4*C(5,J+1)-C(5,J+2))/dzeta/2)/delta f(6) = s(6)*EASA/EGA*K2*K1**2*C(1,J)**2/(1+K1*C(1,J)*C(2,J)*exp(fc #*(iniV+sr*TIME-C(7,J)-Ujref)/R/Temp)+K1*K4*C(1,J)*C(2,J)**2*exp(fc 137

#*(2*iniV+2*sr*TIME-2*C(7,J)-Ujref-Ujref3)/R/Temp)+K5*K4*K1*C(1,J)* #C(2,J)**3*exp(fc*(3*iniV+3*sr*TIME-3*C(7,J)-Ujref-Ujref3-Ujref4)/R #/Temp))**2*C(2,J)**2*exp(2*fc*(iniV+sr*TIME-C(7,J)-Ujref)/R/Temp)- #(z(6)*Ds(6)*fc*C(6,J)*(-3*C(7,J)+4*C(7,J+1)-C(7,J+2))/R/Temp/dzeta #/2+Ds(6)*(-3*C(6,J)+4*C(6,J+1)-C(6,J+2))/dzeta/2)/delta f(7) = z(1)*C(1,J)+z(2)*C(2,J)+z(3)*C(3,J)+z(4)*C(4,J)+z(5)*C(5,J) #+z(6)*C(6,J) s1 = 2*s(1)*EASA/EGA*K2*K1**2*C(1,J)/(1+K1*C(1,J)*C(2,J)*exp(fc*(i #niV+sr*TIME-C(7,J)-Ujref)/R/Temp)+K1*K4*C(1,J)*C(2,J)**2*exp(fc*(2 #*iniV+2*sr*TIME-2*C(7,J)-Ujref-Ujref3)/R/Temp)+K5*K4*K1*C(1,J)*C(2 #,J)**3*exp(fc*(3*iniV+3*sr*TIME-3*C(7,J)-Ujref-Ujref3-Ujref4)/R/Te #mp))**2*C(2,J)**2*exp(2*fc*(iniV+sr*TIME-C(7,J)-Ujref)/R/Temp) s3 = -2*s(1)*EASA/EGA*K2*K1**2*C(1,J)**2/(1+K1*C(1,J)*C(2,J)*exp(f #c*(iniV+sr*TIME-C(7,J)-Ujref)/R/Temp)+K1*K4*C(1,J)*C(2,J)**2*exp(f #c*(2*iniV+2*sr*TIME-2*C(7,J)-Ujref-Ujref3)/R/Temp)+K5*K4*K1*C(1,J) #*C(2,J)**3*exp(fc*(3*iniV+3*sr*TIME-3*C(7,J)-Ujref-Ujref3-Ujref4)/ #R/Temp))**3*C(2,J)**2*exp(2*fc*(iniV+sr*TIME-C(7,J)-Ujref)/R/Temp) #*(K1*C(2,J)*exp(fc*(iniV+sr*TIME-C(7,J)-Ujref)/R/Temp)+K1*K4*C(2,J #)**2*exp(fc*(2*iniV+2*sr*TIME-2*C(7,J)-Ujref-Ujref3)/R/Temp)+K5*K4 #*K1*C(2,J)**3*exp(fc*(3*iniV+3*sr*TIME-3*C(7,J)-Ujref-Ujref3-Ujref #4)/R/Temp)) s4 = -(z(1)*Ds(1)*fc*(-3*C(7,J)+4*C(7,J+1)-C(7,J+2))/R/Temp/dzeta/ #2-3.D0/2.D0*Ds(1)/dzeta)/delta s2 = s3+s4 b(1,1) = s1+s2 s1 = -2*s(1)*EASA/EGA*K2*K1**2*C(1,J)**2/(1+K1*C(1,J)*C(2,J)*exp(f #c*(iniV+sr*TIME-C(7,J)-Ujref)/R/Temp)+K1*K4*C(1,J)*C(2,J)**2*exp(f #c*(2*iniV+2*sr*TIME-2*C(7,J)-Ujref-Ujref3)/R/Temp)+K5*K4*K1*C(1,J) #*C(2,J)**3*exp(fc*(3*iniV+3*sr*TIME-3*C(7,J)-Ujref-Ujref3-Ujref4)/ #R/Temp))**3*C(2,J)**2*exp(2*fc*(iniV+sr*TIME-C(7,J)-Ujref)/R/Temp) #*(K1*C(1,J)*exp(fc*(iniV+sr*TIME-C(7,J)-Ujref)/R/Temp)+2*K1*K4*C(1 #,J)*C(2,J)*exp(fc*(2*iniV+2*sr*TIME-2*C(7,J)-Ujref-Ujref3)/R/Temp) #+3*K5*K4*K1*C(1,J)*C(2,J)**2*exp(fc*(3*iniV+3*sr*TIME-3*C(7,J)-Ujr #ef-Ujref3-Ujref4)/R/Temp)) s2 = 2*s(1)*EASA/EGA*K2*K1**2*C(1,J)**2/(1+K1*C(1,J)*C(2,J)*exp(fc #*(iniV+sr*TIME-C(7,J)-Ujref)/R/Temp)+K1*K4*C(1,J)*C(2,J)**2*exp(fc #*(2*iniV+2*sr*TIME-2*C(7,J)-Ujref-Ujref3)/R/Temp)+K5*K4*K1*C(1,J)* #C(2,J)**3*exp(fc*(3*iniV+3*sr*TIME-3*C(7,J)-Ujref-Ujref3-Ujref4)/R #/Temp))**2*C(2,J)*exp(2*fc*(iniV+sr*TIME-C(7,J)-Ujref)/R/Temp) b(1,2) = s1+s2 b(1,3) = 0 b(1,4) = 0 b(1,5) = 0 b(1,6) = 0 138

s1 = -2*s(1)*EASA/EGA*K2*K1**2*C(1,J)**2/(1+K1*C(1,J)*C(2,J)*exp(f #c*(iniV+sr*TIME-C(7,J)-Ujref)/R/Temp)+K1*K4*C(1,J)*C(2,J)**2*exp(f #c*(2*iniV+2*sr*TIME-2*C(7,J)-Ujref-Ujref3)/R/Temp)+K5*K4*K1*C(1,J) #*C(2,J)**3*exp(fc*(3*iniV+3*sr*TIME-3*C(7,J)-Ujref-Ujref3-Ujref4)/ #R/Temp))**3*C(2,J)**2*exp(2*fc*(iniV+sr*TIME-C(7,J)-Ujref)/R/Temp) #*(-K1*C(1,J)*C(2,J)*fc/R/Temp*exp(fc*(iniV+sr*TIME-C(7,J)-Ujref)/R #/Temp)-2*K1*K4*C(1,J)*C(2,J)**2*fc/R/Temp*exp(fc*(2*iniV+2*sr*TIME #-2*C(7,J)-Ujref-Ujref3)/R/Temp)-3*K5*K4*K1*C(1,J)*C(2,J)**3*fc/R/T #emp*exp(fc*(3*iniV+3*sr*TIME-3*C(7,J)-Ujref-Ujref3-Ujref4)/R/Temp) #) s2 = -2*s(1)*EASA/EGA*K2*K1**2*C(1,J)**2/(1+K1*C(1,J)*C(2,J)*exp(f #c*(iniV+sr*TIME-C(7,J)-Ujref)/R/Temp)+K1*K4*C(1,J)*C(2,J)**2*exp(f #c*(2*iniV+2*sr*TIME-2*C(7,J)-Ujref-Ujref3)/R/Temp)+K5*K4*K1*C(1,J) #*C(2,J)**3*exp(fc*(3*iniV+3*sr*TIME-3*C(7,J)-Ujref-Ujref3-Ujref4)/ #R/Temp))**2*C(2,J)**2*fc/R/Temp*exp(2*fc*(iniV+sr*TIME-C(7,J)-Ujre #f)/R/Temp)+3.D0/2.D0*z(1)*Ds(1)*fc*C(1,J)/R/Temp/dzeta/delta b(1,7) = s1+s2 s1 = 2*s(2)*EASA/EGA*K2*K1**2*C(1,J)/(1+K1*C(1,J)*C(2,J)*exp(fc*(i #niV+sr*TIME-C(7,J)-Ujref)/R/Temp)+K1*K4*C(1,J)*C(2,J)**2*exp(fc*(2 #*iniV+2*sr*TIME-2*C(7,J)-Ujref-Ujref3)/R/Temp)+K5*K4*K1*C(1,J)*C(2 #,J)**3*exp(fc*(3*iniV+3*sr*TIME-3*C(7,J)-Ujref-Ujref3-Ujref4)/R/Te #mp))**2*C(2,J)**2*exp(2*fc*(iniV+sr*TIME-C(7,J)-Ujref)/R/Temp) s2 = -2*s(2)*EASA/EGA*K2*K1**2*C(1,J)**2/(1+K1*C(1,J)*C(2,J)*exp(f #c*(iniV+sr*TIME-C(7,J)-Ujref)/R/Temp)+K1*K4*C(1,J)*C(2,J)**2*exp(f #c*(2*iniV+2*sr*TIME-2*C(7,J)-Ujref-Ujref3)/R/Temp)+K5*K4*K1*C(1,J) #*C(2,J)**3*exp(fc*(3*iniV+3*sr*TIME-3*C(7,J)-Ujref-Ujref3-Ujref4)/ #R/Temp))**3*C(2,J)**2*exp(2*fc*(iniV+sr*TIME-C(7,J)-Ujref)/R/Temp) #*(K1*C(2,J)*exp(fc*(iniV+sr*TIME-C(7,J)-Ujref)/R/Temp)+K1*K4*C(2,J #)**2*exp(fc*(2*iniV+2*sr*TIME-2*C(7,J)-Ujref-Ujref3)/R/Temp)+K5*K4 #*K1*C(2,J)**3*exp(fc*(3*iniV+3*sr*TIME-3*C(7,J)-Ujref-Ujref3-Ujref #4)/R/Temp)) b(2,1) = s1+s2 s1 = -2*s(2)*EASA/EGA*K2*K1**2*C(1,J)**2/(1+K1*C(1,J)*C(2,J)*exp(f #c*(iniV+sr*TIME-C(7,J)-Ujref)/R/Temp)+K1*K4*C(1,J)*C(2,J)**2*exp(f #c*(2*iniV+2*sr*TIME-2*C(7,J)-Ujref-Ujref3)/R/Temp)+K5*K4*K1*C(1,J) #*C(2,J)**3*exp(fc*(3*iniV+3*sr*TIME-3*C(7,J)-Ujref-Ujref3-Ujref4)/ #R/Temp))**3*C(2,J)**2*exp(2*fc*(iniV+sr*TIME-C(7,J)-Ujref)/R/Temp) #*(K1*C(1,J)*exp(fc*(iniV+sr*TIME-C(7,J)-Ujref)/R/Temp)+2*K1*K4*C(1 #,J)*C(2,J)*exp(fc*(2*iniV+2*sr*TIME-2*C(7,J)-Ujref-Ujref3)/R/Temp) #+3*K5*K4*K1*C(1,J)*C(2,J)**2*exp(fc*(3*iniV+3*sr*TIME-3*C(7,J)-Ujr #ef-Ujref3-Ujref4)/R/Temp)) s2 = 2*s(2)*EASA/EGA*K2*K1**2*C(1,J)**2/(1+K1*C(1,J)*C(2,J)*exp(fc #*(iniV+sr*TIME-C(7,J)-Ujref)/R/Temp)+K1*K4*C(1,J)*C(2,J)**2*exp(fc #*(2*iniV+2*sr*TIME-2*C(7,J)-Ujref-Ujref3)/R/Temp)+K5*K4*K1*C(1,J)* 139

#C(2,J)**3*exp(fc*(3*iniV+3*sr*TIME-3*C(7,J)-Ujref-Ujref3-Ujref4)/R #/Temp))**2*C(2,J)*exp(2*fc*(iniV+sr*TIME-C(7,J)-Ujref)/R/Temp)-(z( #2)*Ds(2)*fc*(-3*C(7,J)+4*C(7,J+1)-C(7,J+2))/R/Temp/dzeta/2-3.D0/2. #D0*Ds(2)/dzeta)/delta b(2,2) = s1+s2 b(2,3) = 0 b(2,4) = 0 b(2,5) = 0 b(2,6) = 0 s1 = -2*s(2)*EASA/EGA*K2*K1**2*C(1,J)**2/(1+K1*C(1,J)*C(2,J)*exp(f #c*(iniV+sr*TIME-C(7,J)-Ujref)/R/Temp)+K1*K4*C(1,J)*C(2,J)**2*exp(f #c*(2*iniV+2*sr*TIME-2*C(7,J)-Ujref-Ujref3)/R/Temp)+K5*K4*K1*C(1,J) #*C(2,J)**3*exp(fc*(3*iniV+3*sr*TIME-3*C(7,J)-Ujref-Ujref3-Ujref4)/ #R/Temp))**3*C(2,J)**2*exp(2*fc*(iniV+sr*TIME-C(7,J)-Ujref)/R/Temp) #*(-K1*C(1,J)*C(2,J)*fc/R/Temp*exp(fc*(iniV+sr*TIME-C(7,J)-Ujref)/R #/Temp)-2*K1*K4*C(1,J)*C(2,J)**2*fc/R/Temp*exp(fc*(2*iniV+2*sr*TIME #-2*C(7,J)-Ujref-Ujref3)/R/Temp)-3*K5*K4*K1*C(1,J)*C(2,J)**3*fc/R/T #emp*exp(fc*(3*iniV+3*sr*TIME-3*C(7,J)-Ujref-Ujref3-Ujref4)/R/Temp) #) s2 = -2*s(2)*EASA/EGA*K2*K1**2*C(1,J)**2/(1+K1*C(1,J)*C(2,J)*exp(f #c*(iniV+sr*TIME-C(7,J)-Ujref)/R/Temp)+K1*K4*C(1,J)*C(2,J)**2*exp(f #c*(2*iniV+2*sr*TIME-2*C(7,J)-Ujref-Ujref3)/R/Temp)+K5*K4*K1*C(1,J) #*C(2,J)**3*exp(fc*(3*iniV+3*sr*TIME-3*C(7,J)-Ujref-Ujref3-Ujref4)/ #R/Temp))**2*C(2,J)**2*fc/R/Temp*exp(2*fc*(iniV+sr*TIME-C(7,J)-Ujre #f)/R/Temp)+3.D0/2.D0*z(2)*Ds(2)*fc*C(2,J)/R/Temp/dzeta/delta b(2,7) = s1+s2 s1 = 2*s(3)*EASA/EGA*K2*K1**2*C(1,J)/(1+K1*C(1,J)*C(2,J)*exp(fc*(i #niV+sr*TIME-C(7,J)-Ujref)/R/Temp)+K1*K4*C(1,J)*C(2,J)**2*exp(fc*(2 #*iniV+2*sr*TIME-2*C(7,J)-Ujref-Ujref3)/R/Temp)+K5*K4*K1*C(1,J)*C(2 #,J)**3*exp(fc*(3*iniV+3*sr*TIME-3*C(7,J)-Ujref-Ujref3-Ujref4)/R/Te #mp))**2*C(2,J)**2*exp(2*fc*(iniV+sr*TIME-C(7,J)-Ujref)/R/Temp) s2 = -2*s(3)*EASA/EGA*K2*K1**2*C(1,J)**2/(1+K1*C(1,J)*C(2,J)*exp(f #c*(iniV+sr*TIME-C(7,J)-Ujref)/R/Temp)+K1*K4*C(1,J)*C(2,J)**2*exp(f #c*(2*iniV+2*sr*TIME-2*C(7,J)-Ujref-Ujref3)/R/Temp)+K5*K4*K1*C(1,J) #*C(2,J)**3*exp(fc*(3*iniV+3*sr*TIME-3*C(7,J)-Ujref-Ujref3-Ujref4)/ #R/Temp))**3*C(2,J)**2*exp(2*fc*(iniV+sr*TIME-C(7,J)-Ujref)/R/Temp) #*(K1*C(2,J)*exp(fc*(iniV+sr*TIME-C(7,J)-Ujref)/R/Temp)+K1*K4*C(2,J #)**2*exp(fc*(2*iniV+2*sr*TIME-2*C(7,J)-Ujref-Ujref3)/R/Temp)+K5*K4 #*K1*C(2,J)**3*exp(fc*(3*iniV+3*sr*TIME-3*C(7,J)-Ujref-Ujref3-Ujref #4)/R/Temp)) b(3,1) = s1+s2 s1 = -2*s(3)*EASA/EGA*K2*K1**2*C(1,J)**2/(1+K1*C(1,J)*C(2,J)*exp(f #c*(iniV+sr*TIME-C(7,J)-Ujref)/R/Temp)+K1*K4*C(1,J)*C(2,J)**2*exp(f #c*(2*iniV+2*sr*TIME-2*C(7,J)-Ujref-Ujref3)/R/Temp)+K5*K4*K1*C(1,J) 140

#*C(2,J)**3*exp(fc*(3*iniV+3*sr*TIME-3*C(7,J)-Ujref-Ujref3-Ujref4)/ #R/Temp))**3*C(2,J)**2*exp(2*fc*(iniV+sr*TIME-C(7,J)-Ujref)/R/Temp) #*(K1*C(1,J)*exp(fc*(iniV+sr*TIME-C(7,J)-Ujref)/R/Temp)+2*K1*K4*C(1 #,J)*C(2,J)*exp(fc*(2*iniV+2*sr*TIME-2*C(7,J)-Ujref-Ujref3)/R/Temp) #+3*K5*K4*K1*C(1,J)*C(2,J)**2*exp(fc*(3*iniV+3*sr*TIME-3*C(7,J)-Ujr #ef-Ujref3-Ujref4)/R/Temp)) s2 = 2*s(3)*EASA/EGA*K2*K1**2*C(1,J)**2/(1+K1*C(1,J)*C(2,J)*exp(fc #*(iniV+sr*TIME-C(7,J)-Ujref)/R/Temp)+K1*K4*C(1,J)*C(2,J)**2*exp(fc #*(2*iniV+2*sr*TIME-2*C(7,J)-Ujref-Ujref3)/R/Temp)+K5*K4*K1*C(1,J)* #C(2,J)**3*exp(fc*(3*iniV+3*sr*TIME-3*C(7,J)-Ujref-Ujref3-Ujref4)/R #/Temp))**2*C(2,J)*exp(2*fc*(iniV+sr*TIME-C(7,J)-Ujref)/R/Temp) b(3,2) = s1+s2 b(3,3) = -(z(3)*Ds(3)*fc*(-3*C(7,J)+4*C(7,J+1)-C(7,J+2))/R/Temp/dz #eta/2-3.D0/2.D0*Ds(3)/dzeta)/delta b(3,4) = 0 b(3,5) = 0 b(3,6) = 0 s1 = -2*s(3)*EASA/EGA*K2*K1**2*C(1,J)**2/(1+K1*C(1,J)*C(2,J)*exp(f #c*(iniV+sr*TIME-C(7,J)-Ujref)/R/Temp)+K1*K4*C(1,J)*C(2,J)**2*exp(f #c*(2*iniV+2*sr*TIME-2*C(7,J)-Ujref-Ujref3)/R/Temp)+K5*K4*K1*C(1,J) #*C(2,J)**3*exp(fc*(3*iniV+3*sr*TIME-3*C(7,J)-Ujref-Ujref3-Ujref4)/ #R/Temp))**3*C(2,J)**2*exp(2*fc*(iniV+sr*TIME-C(7,J)-Ujref)/R/Temp) #*(-K1*C(1,J)*C(2,J)*fc/R/Temp*exp(fc*(iniV+sr*TIME-C(7,J)-Ujref)/R #/Temp)-2*K1*K4*C(1,J)*C(2,J)**2*fc/R/Temp*exp(fc*(2*iniV+2*sr*TIME #-2*C(7,J)-Ujref-Ujref3)/R/Temp)-3*K5*K4*K1*C(1,J)*C(2,J)**3*fc/R/T #emp*exp(fc*(3*iniV+3*sr*TIME-3*C(7,J)-Ujref-Ujref3-Ujref4)/R/Temp) #) s2 = -2*s(3)*EASA/EGA*K2*K1**2*C(1,J)**2/(1+K1*C(1,J)*C(2,J)*exp(f #c*(iniV+sr*TIME-C(7,J)-Ujref)/R/Temp)+K1*K4*C(1,J)*C(2,J)**2*exp(f #c*(2*iniV+2*sr*TIME-2*C(7,J)-Ujref-Ujref3)/R/Temp)+K5*K4*K1*C(1,J) #*C(2,J)**3*exp(fc*(3*iniV+3*sr*TIME-3*C(7,J)-Ujref-Ujref3-Ujref4)/ #R/Temp))**2*C(2,J)**2*fc/R/Temp*exp(2*fc*(iniV+sr*TIME-C(7,J)-Ujre #f)/R/Temp)+3.D0/2.D0*z(3)*Ds(3)*fc*C(3,J)/R/Temp/dzeta/delta b(3,7) = s1+s2 s1 = 2*s(4)*EASA/EGA*K2*K1**2*C(1,J)/(1+K1*C(1,J)*C(2,J)*exp(fc*(i #niV+sr*TIME-C(7,J)-Ujref)/R/Temp)+K1*K4*C(1,J)*C(2,J)**2*exp(fc*(2 #*iniV+2*sr*TIME-2*C(7,J)-Ujref-Ujref3)/R/Temp)+K5*K4*K1*C(1,J)*C(2 #,J)**3*exp(fc*(3*iniV+3*sr*TIME-3*C(7,J)-Ujref-Ujref3-Ujref4)/R/Te #mp))**2*C(2,J)**2*exp(2*fc*(iniV+sr*TIME-C(7,J)-Ujref)/R/Temp) s2 = -2*s(4)*EASA/EGA*K2*K1**2*C(1,J)**2/(1+K1*C(1,J)*C(2,J)*exp(f #c*(iniV+sr*TIME-C(7,J)-Ujref)/R/Temp)+K1*K4*C(1,J)*C(2,J)**2*exp(f #c*(2*iniV+2*sr*TIME-2*C(7,J)-Ujref-Ujref3)/R/Temp)+K5*K4*K1*C(1,J) #*C(2,J)**3*exp(fc*(3*iniV+3*sr*TIME-3*C(7,J)-Ujref-Ujref3-Ujref4)/ #R/Temp))**3*C(2,J)**2*exp(2*fc*(iniV+sr*TIME-C(7,J)-Ujref)/R/Temp) 141

#*(K1*C(2,J)*exp(fc*(iniV+sr*TIME-C(7,J)-Ujref)/R/Temp)+K1*K4*C(2,J #)**2*exp(fc*(2*iniV+2*sr*TIME-2*C(7,J)-Ujref-Ujref3)/R/Temp)+K5*K4 #*K1*C(2,J)**3*exp(fc*(3*iniV+3*sr*TIME-3*C(7,J)-Ujref-Ujref3-Ujref #4)/R/Temp)) b(4,1) = s1+s2 s1 = -2*s(4)*EASA/EGA*K2*K1**2*C(1,J)**2/(1+K1*C(1,J)*C(2,J)*exp(f #c*(iniV+sr*TIME-C(7,J)-Ujref)/R/Temp)+K1*K4*C(1,J)*C(2,J)**2*exp(f #c*(2*iniV+2*sr*TIME-2*C(7,J)-Ujref-Ujref3)/R/Temp)+K5*K4*K1*C(1,J) #*C(2,J)**3*exp(fc*(3*iniV+3*sr*TIME-3*C(7,J)-Ujref-Ujref3-Ujref4)/ #R/Temp))**3*C(2,J)**2*exp(2*fc*(iniV+sr*TIME-C(7,J)-Ujref)/R/Temp) #*(K1*C(1,J)*exp(fc*(iniV+sr*TIME-C(7,J)-Ujref)/R/Temp)+2*K1*K4*C(1 #,J)*C(2,J)*exp(fc*(2*iniV+2*sr*TIME-2*C(7,J)-Ujref-Ujref3)/R/Temp) #+3*K5*K4*K1*C(1,J)*C(2,J)**2*exp(fc*(3*iniV+3*sr*TIME-3*C(7,J)-Ujr #ef-Ujref3-Ujref4)/R/Temp)) s2 = 2*s(4)*EASA/EGA*K2*K1**2*C(1,J)**2/(1+K1*C(1,J)*C(2,J)*exp(fc #*(iniV+sr*TIME-C(7,J)-Ujref)/R/Temp)+K1*K4*C(1,J)*C(2,J)**2*exp(fc #*(2*iniV+2*sr*TIME-2*C(7,J)-Ujref-Ujref3)/R/Temp)+K5*K4*K1*C(1,J)* #C(2,J)**3*exp(fc*(3*iniV+3*sr*TIME-3*C(7,J)-Ujref-Ujref3-Ujref4)/R #/Temp))**2*C(2,J)*exp(2*fc*(iniV+sr*TIME-C(7,J)-Ujref)/R/Temp) b(4,2) = s1+s2 b(4,3) = 0 b(4,4) = -(z(4)*Ds(4)*fc*(-3*C(7,J)+4*C(7,J+1)-C(7,J+2))/R/Temp/dz #eta/2-3.D0/2.D0*Ds(4)/dzeta)/delta b(4,5) = 0 b(4,6) = 0 s1 = -2*s(4)*EASA/EGA*K2*K1**2*C(1,J)**2/(1+K1*C(1,J)*C(2,J)*exp(f #c*(iniV+sr*TIME-C(7,J)-Ujref)/R/Temp)+K1*K4*C(1,J)*C(2,J)**2*exp(f #c*(2*iniV+2*sr*TIME-2*C(7,J)-Ujref-Ujref3)/R/Temp)+K5*K4*K1*C(1,J) #*C(2,J)**3*exp(fc*(3*iniV+3*sr*TIME-3*C(7,J)-Ujref-Ujref3-Ujref4)/ #R/Temp))**3*C(2,J)**2*exp(2*fc*(iniV+sr*TIME-C(7,J)-Ujref)/R/Temp) #*(-K1*C(1,J)*C(2,J)*fc/R/Temp*exp(fc*(iniV+sr*TIME-C(7,J)-Ujref)/R #/Temp)-2*K1*K4*C(1,J)*C(2,J)**2*fc/R/Temp*exp(fc*(2*iniV+2*sr*TIME #-2*C(7,J)-Ujref-Ujref3)/R/Temp)-3*K5*K4*K1*C(1,J)*C(2,J)**3*fc/R/T #emp*exp(fc*(3*iniV+3*sr*TIME-3*C(7,J)-Ujref-Ujref3-Ujref4)/R/Temp) #) s2 = -2*s(4)*EASA/EGA*K2*K1**2*C(1,J)**2/(1+K1*C(1,J)*C(2,J)*exp(f #c*(iniV+sr*TIME-C(7,J)-Ujref)/R/Temp)+K1*K4*C(1,J)*C(2,J)**2*exp(f #c*(2*iniV+2*sr*TIME-2*C(7,J)-Ujref-Ujref3)/R/Temp)+K5*K4*K1*C(1,J) #*C(2,J)**3*exp(fc*(3*iniV+3*sr*TIME-3*C(7,J)-Ujref-Ujref3-Ujref4)/ #R/Temp))**2*C(2,J)**2*fc/R/Temp*exp(2*fc*(iniV+sr*TIME-C(7,J)-Ujre #f)/R/Temp)+3.D0/2.D0*z(4)*Ds(4)*fc*C(4,J)/R/Temp/dzeta/delta b(4,7) = s1+s2 s1 = 2*s(5)*EASA/EGA*K2*K1**2*C(1,J)/(1+K1*C(1,J)*C(2,J)*exp(fc*(i #niV+sr*TIME-C(7,J)-Ujref)/R/Temp)+K1*K4*C(1,J)*C(2,J)**2*exp(fc*(2 142

#*iniV+2*sr*TIME-2*C(7,J)-Ujref-Ujref3)/R/Temp)+K5*K4*K1*C(1,J)*C(2 #,J)**3*exp(fc*(3*iniV+3*sr*TIME-3*C(7,J)-Ujref-Ujref3-Ujref4)/R/Te #mp))**2*C(2,J)**2*exp(2*fc*(iniV+sr*TIME-C(7,J)-Ujref)/R/Temp) s2 = -2*s(5)*EASA/EGA*K2*K1**2*C(1,J)**2/(1+K1*C(1,J)*C(2,J)*exp(f #c*(iniV+sr*TIME-C(7,J)-Ujref)/R/Temp)+K1*K4*C(1,J)*C(2,J)**2*exp(f #c*(2*iniV+2*sr*TIME-2*C(7,J)-Ujref-Ujref3)/R/Temp)+K5*K4*K1*C(1,J) #*C(2,J)**3*exp(fc*(3*iniV+3*sr*TIME-3*C(7,J)-Ujref-Ujref3-Ujref4)/ #R/Temp))**3*C(2,J)**2*exp(2*fc*(iniV+sr*TIME-C(7,J)-Ujref)/R/Temp) #*(K1*C(2,J)*exp(fc*(iniV+sr*TIME-C(7,J)-Ujref)/R/Temp)+K1*K4*C(2,J #)**2*exp(fc*(2*iniV+2*sr*TIME-2*C(7,J)-Ujref-Ujref3)/R/Temp)+K5*K4 #*K1*C(2,J)**3*exp(fc*(3*iniV+3*sr*TIME-3*C(7,J)-Ujref-Ujref3-Ujref #4)/R/Temp)) b(5,1) = s1+s2 s1 = -2*s(5)*EASA/EGA*K2*K1**2*C(1,J)**2/(1+K1*C(1,J)*C(2,J)*exp(f #c*(iniV+sr*TIME-C(7,J)-Ujref)/R/Temp)+K1*K4*C(1,J)*C(2,J)**2*exp(f #c*(2*iniV+2*sr*TIME-2*C(7,J)-Ujref-Ujref3)/R/Temp)+K5*K4*K1*C(1,J) #*C(2,J)**3*exp(fc*(3*iniV+3*sr*TIME-3*C(7,J)-Ujref-Ujref3-Ujref4)/ #R/Temp))**3*C(2,J)**2*exp(2*fc*(iniV+sr*TIME-C(7,J)-Ujref)/R/Temp) #*(K1*C(1,J)*exp(fc*(iniV+sr*TIME-C(7,J)-Ujref)/R/Temp)+2*K1*K4*C(1 #,J)*C(2,J)*exp(fc*(2*iniV+2*sr*TIME-2*C(7,J)-Ujref-Ujref3)/R/Temp) #+3*K5*K4*K1*C(1,J)*C(2,J)**2*exp(fc*(3*iniV+3*sr*TIME-3*C(7,J)-Ujr #ef-Ujref3-Ujref4)/R/Temp)) s2 = 2*s(5)*EASA/EGA*K2*K1**2*C(1,J)**2/(1+K1*C(1,J)*C(2,J)*exp(fc #*(iniV+sr*TIME-C(7,J)-Ujref)/R/Temp)+K1*K4*C(1,J)*C(2,J)**2*exp(fc #*(2*iniV+2*sr*TIME-2*C(7,J)-Ujref-Ujref3)/R/Temp)+K5*K4*K1*C(1,J)* #C(2,J)**3*exp(fc*(3*iniV+3*sr*TIME-3*C(7,J)-Ujref-Ujref3-Ujref4)/R #/Temp))**2*C(2,J)*exp(2*fc*(iniV+sr*TIME-C(7,J)-Ujref)/R/Temp) b(5,2) = s1+s2 b(5,3) = 0 b(5,4) = 0 b(5,5) = -(z(5)*Ds(5)*fc*(-3*C(7,J)+4*C(7,J+1)-C(7,J+2))/R/Temp/dz #eta/2-3.D0/2.D0*Ds(5)/dzeta)/delta b(5,6) = 0 s1 = -2*s(5)*EASA/EGA*K2*K1**2*C(1,J)**2/(1+K1*C(1,J)*C(2,J)*exp(f #c*(iniV+sr*TIME-C(7,J)-Ujref)/R/Temp)+K1*K4*C(1,J)*C(2,J)**2*exp(f #c*(2*iniV+2*sr*TIME-2*C(7,J)-Ujref-Ujref3)/R/Temp)+K5*K4*K1*C(1,J) #*C(2,J)**3*exp(fc*(3*iniV+3*sr*TIME-3*C(7,J)-Ujref-Ujref3-Ujref4)/ #R/Temp))**3*C(2,J)**2*exp(2*fc*(iniV+sr*TIME-C(7,J)-Ujref)/R/Temp) #*(-K1*C(1,J)*C(2,J)*fc/R/Temp*exp(fc*(iniV+sr*TIME-C(7,J)-Ujref)/R #/Temp)-2*K1*K4*C(1,J)*C(2,J)**2*fc/R/Temp*exp(fc*(2*iniV+2*sr*TIME #-2*C(7,J)-Ujref-Ujref3)/R/Temp)-3*K5*K4*K1*C(1,J)*C(2,J)**3*fc/R/T #emp*exp(fc*(3*iniV+3*sr*TIME-3*C(7,J)-Ujref-Ujref3-Ujref4)/R/Temp) #) s2 = -2*s(5)*EASA/EGA*K2*K1**2*C(1,J)**2/(1+K1*C(1,J)*C(2,J)*exp(f 143

#c*(iniV+sr*TIME-C(7,J)-Ujref)/R/Temp)+K1*K4*C(1,J)*C(2,J)**2*exp(f #c*(2*iniV+2*sr*TIME-2*C(7,J)-Ujref-Ujref3)/R/Temp)+K5*K4*K1*C(1,J) #*C(2,J)**3*exp(fc*(3*iniV+3*sr*TIME-3*C(7,J)-Ujref-Ujref3-Ujref4)/ #R/Temp))**2*C(2,J)**2*fc/R/Temp*exp(2*fc*(iniV+sr*TIME-C(7,J)-Ujre #f)/R/Temp)+3.D0/2.D0*z(5)*Ds(5)*fc*C(5,J)/R/Temp/dzeta/delta b(5,7) = s1+s2 s1 = 2*s(6)*EASA/EGA*K2*K1**2*C(1,J)/(1+K1*C(1,J)*C(2,J)*exp(fc*(i #niV+sr*TIME-C(7,J)-Ujref)/R/Temp)+K1*K4*C(1,J)*C(2,J)**2*exp(fc*(2 #*iniV+2*sr*TIME-2*C(7,J)-Ujref-Ujref3)/R/Temp)+K5*K4*K1*C(1,J)*C(2 #,J)**3*exp(fc*(3*iniV+3*sr*TIME-3*C(7,J)-Ujref-Ujref3-Ujref4)/R/Te #mp))**2*C(2,J)**2*exp(2*fc*(iniV+sr*TIME-C(7,J)-Ujref)/R/Temp) s2 = -2*s(6)*EASA/EGA*K2*K1**2*C(1,J)**2/(1+K1*C(1,J)*C(2,J)*exp(f #c*(iniV+sr*TIME-C(7,J)-Ujref)/R/Temp)+K1*K4*C(1,J)*C(2,J)**2*exp(f #c*(2*iniV+2*sr*TIME-2*C(7,J)-Ujref-Ujref3)/R/Temp)+K5*K4*K1*C(1,J) #*C(2,J)**3*exp(fc*(3*iniV+3*sr*TIME-3*C(7,J)-Ujref-Ujref3-Ujref4)/ #R/Temp))**3*C(2,J)**2*exp(2*fc*(iniV+sr*TIME-C(7,J)-Ujref)/R/Temp) #*(K1*C(2,J)*exp(fc*(iniV+sr*TIME-C(7,J)-Ujref)/R/Temp)+K1*K4*C(2,J #)**2*exp(fc*(2*iniV+2*sr*TIME-2*C(7,J)-Ujref-Ujref3)/R/Temp)+K5*K4 #*K1*C(2,J)**3*exp(fc*(3*iniV+3*sr*TIME-3*C(7,J)-Ujref-Ujref3-Ujref #4)/R/Temp)) b(6,1) = s1+s2 s1 = -2*s(6)*EASA/EGA*K2*K1**2*C(1,J)**2/(1+K1*C(1,J)*C(2,J)*exp(f #c*(iniV+sr*TIME-C(7,J)-Ujref)/R/Temp)+K1*K4*C(1,J)*C(2,J)**2*exp(f #c*(2*iniV+2*sr*TIME-2*C(7,J)-Ujref-Ujref3)/R/Temp)+K5*K4*K1*C(1,J) #*C(2,J)**3*exp(fc*(3*iniV+3*sr*TIME-3*C(7,J)-Ujref-Ujref3-Ujref4)/ #R/Temp))**3*C(2,J)**2*exp(2*fc*(iniV+sr*TIME-C(7,J)-Ujref)/R/Temp) #*(K1*C(1,J)*exp(fc*(iniV+sr*TIME-C(7,J)-Ujref)/R/Temp)+2*K1*K4*C(1 #,J)*C(2,J)*exp(fc*(2*iniV+2*sr*TIME-2*C(7,J)-Ujref-Ujref3)/R/Temp) #+3*K5*K4*K1*C(1,J)*C(2,J)**2*exp(fc*(3*iniV+3*sr*TIME-3*C(7,J)-Ujr #ef-Ujref3-Ujref4)/R/Temp)) s2 = 2*s(6)*EASA/EGA*K2*K1**2*C(1,J)**2/(1+K1*C(1,J)*C(2,J)*exp(fc #*(iniV+sr*TIME-C(7,J)-Ujref)/R/Temp)+K1*K4*C(1,J)*C(2,J)**2*exp(fc #*(2*iniV+2*sr*TIME-2*C(7,J)-Ujref-Ujref3)/R/Temp)+K5*K4*K1*C(1,J)* #C(2,J)**3*exp(fc*(3*iniV+3*sr*TIME-3*C(7,J)-Ujref-Ujref3-Ujref4)/R #/Temp))**2*C(2,J)*exp(2*fc*(iniV+sr*TIME-C(7,J)-Ujref)/R/Temp) b(6,2) = s1+s2 b(6,3) = 0 b(6,4) = 0 b(6,5) = 0 b(6,6) = -(z(6)*Ds(6)*fc*(-3*C(7,J)+4*C(7,J+1)-C(7,J+2))/R/Temp/dz #eta/2-3.D0/2.D0*Ds(6)/dzeta)/delta s1 = -2*s(6)*EASA/EGA*K2*K1**2*C(1,J)**2/(1+K1*C(1,J)*C(2,J)*exp(f #c*(iniV+sr*TIME-C(7,J)-Ujref)/R/Temp)+K1*K4*C(1,J)*C(2,J)**2*exp(f #c*(2*iniV+2*sr*TIME-2*C(7,J)-Ujref-Ujref3)/R/Temp)+K5*K4*K1*C(1,J) 144

#*C(2,J)**3*exp(fc*(3*iniV+3*sr*TIME-3*C(7,J)-Ujref-Ujref3-Ujref4)/ #R/Temp))**3*C(2,J)**2*exp(2*fc*(iniV+sr*TIME-C(7,J)-Ujref)/R/Temp) #*(-K1*C(1,J)*C(2,J)*fc/R/Temp*exp(fc*(iniV+sr*TIME-C(7,J)-Ujref)/R #/Temp)-2*K1*K4*C(1,J)*C(2,J)**2*fc/R/Temp*exp(fc*(2*iniV+2*sr*TIME #-2*C(7,J)-Ujref-Ujref3)/R/Temp)-3*K5*K4*K1*C(1,J)*C(2,J)**3*fc/R/T #emp*exp(fc*(3*iniV+3*sr*TIME-3*C(7,J)-Ujref-Ujref3-Ujref4)/R/Temp) #) s2 = -2*s(6)*EASA/EGA*K2*K1**2*C(1,J)**2/(1+K1*C(1,J)*C(2,J)*exp(f #c*(iniV+sr*TIME-C(7,J)-Ujref)/R/Temp)+K1*K4*C(1,J)*C(2,J)**2*exp(f #c*(2*iniV+2*sr*TIME-2*C(7,J)-Ujref-Ujref3)/R/Temp)+K5*K4*K1*C(1,J) #*C(2,J)**3*exp(fc*(3*iniV+3*sr*TIME-3*C(7,J)-Ujref-Ujref3-Ujref4)/ #R/Temp))**2*C(2,J)**2*fc/R/Temp*exp(2*fc*(iniV+sr*TIME-C(7,J)-Ujre #f)/R/Temp)+3.D0/2.D0*z(6)*Ds(6)*fc*C(6,J)/R/Temp/dzeta/delta b(6,7) = s1+s2 b(7,1) = z(1) b(7,2) = z(2) b(7,3) = z(3) b(7,4) = z(4) b(7,5) = z(5) b(7,6) = z(6) b(7,7) = 0 d(1,1) = -2*Ds(1)/dzeta/delta d(1,2) = 0 d(1,3) = 0 d(1,4) = 0 d(1,5) = 0 d(1,6) = 0 d(1,7) = -2*z(1)*Ds(1)*fc*C(1,J)/R/Temp/dzeta/delta d(2,1) = 0 d(2,2) = -2*Ds(2)/dzeta/delta d(2,3) = 0 d(2,4) = 0 d(2,5) = 0 d(2,6) = 0 d(2,7) = -2*z(2)*Ds(2)*fc*C(2,J)/R/Temp/dzeta/delta d(3,1) = 0 d(3,2) = 0 d(3,3) = -2*Ds(3)/dzeta/delta d(3,4) = 0 d(3,5) = 0 d(3,6) = 0 d(3,7) = -2*z(3)*Ds(3)*fc*C(3,J)/R/Temp/dzeta/delta d(4,1) = 0 d(4,2) = 0 145 d(4,3) = 0 d(4,4) = -2*Ds(4)/dzeta/delta d(4,5) = 0 d(4,6) = 0 d(4,7) = -2*z(4)*Ds(4)*fc*C(4,J)/R/Temp/dzeta/delta d(5,1) = 0 d(5,2) = 0 d(5,3) = 0 d(5,4) = 0 d(5,5) = -2*Ds(5)/dzeta/delta d(5,6) = 0 d(5,7) = -2*z(5)*Ds(5)*fc*C(5,J)/R/Temp/dzeta/delta d(6,1) = 0 d(6,2) = 0 d(6,3) = 0 d(6,4) = 0 d(6,5) = 0 d(6,6) = -2*Ds(6)/dzeta/delta d(6,7) = -2*z(6)*Ds(6)*fc*C(6,J)/R/Temp/dzeta/delta d(7,1) = 0 d(7,2) = 0 d(7,3) = 0 d(7,4) = 0 d(7,5) = 0 d(7,6) = 0 d(7,7) = 0 x(1,1) = Ds(1)/dzeta/delta/2 x(1,2) = 0 x(1,3) = 0 x(1,4) = 0 x(1,5) = 0 x(1,6) = 0 x(1,7) = z(1)*Ds(1)*fc*C(1,J)/R/Temp/dzeta/delta/2 x(2,1) = 0 x(2,2) = Ds(2)/dzeta/delta/2 x(2,3) = 0 x(2,4) = 0 x(2,5) = 0 x(2,6) = 0 x(2,7) = z(2)*Ds(2)*fc*C(2,J)/R/Temp/dzeta/delta/2 x(3,1) = 0 x(3,2) = 0 x(3,3) = Ds(3)/dzeta/delta/2 x(3,4) = 0 146

x(3,5) = 0 x(3,6) = 0 x(3,7) = z(3)*Ds(3)*fc*C(3,J)/R/Temp/dzeta/delta/2 x(4,1) = 0 x(4,2) = 0 x(4,3) = 0 x(4,4) = Ds(4)/dzeta/delta/2 x(4,5) = 0 x(4,6) = 0 x(4,7) = z(4)*Ds(4)*fc*C(4,J)/R/Temp/dzeta/delta/2 x(5,1) = 0 x(5,2) = 0 x(5,3) = 0 x(5,4) = 0 x(5,5) = Ds(5)/dzeta/delta/2 x(5,6) = 0 x(5,7) = z(5)*Ds(5)*fc*C(5,J)/R/Temp/dzeta/delta/2 x(6,1) = 0 x(6,2) = 0 x(6,3) = 0 x(6,4) = 0 x(6,5) = 0 x(6,6) = Ds(6)/dzeta/delta/2 x(6,7) = z(6)*Ds(6)*fc*C(6,J)/R/Temp/dzeta/delta/2 x(7,1) = 0 x(7,2) = 0 x(7,3) = 0 x(7,4) = 0 x(7,5) = 0 x(7,6) = 0 x(7,7) = 0 DO 30 I=1,N G(I)=-F(I) DO 30 K=1,N 30 G(I)=G(I)+B(I,K)*C(K,J)+D(I,K)*C(K,J+1)+X(I,K)*C(K,J+2) CALL BAND(J) GO TO 10 14 IF (J-NJ) 15,20,20 cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc c Input your equations and jacobians for the central nodes c For the RDE this corresponds to the solition within the boundary layer cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc 15 s1 = Ds(1)/delta**2*(C(1,J+1)-2*C(1,J)+C(1,J-1))/dzeta**2+0.255D0* #sqrt(omega)**3*delta*ZETA(J)**2*(C(1,J+1)-C(1,J-1))/sqrt(vis)/dzet 147

#a+Ds(1)*z(1)*fc*(C(1,J)*(C(7,J+1)-2*C(7,J)+C(7,J-1))/dzeta**2+(C(1 #,J+1)-C(1,J-1))/dzeta**2*(C(7,J+1)-C(7,J-1))/4)/R/Temp/delta**2-Ds #(1)/delta**2*(C(1,J)-CN(1,J))/DT0 f(1) = s1+Ds(6)/delta**2*(C(6,J+1)-2*C(6,J)+C(6,J-1))/dzeta**2+0.2 #55D0*sqrt(omega)**3*delta*ZETA(J)**2*(C(6,J+1)-C(6,J-1))/sqrt(vis) #/dzeta+Ds(6)*z(6)*fc*(C(6,J)*(C(7,J+1)-2*C(7,J)+C(7,J-1))/dzeta**2 #+(C(6,J+1)-C(6,J-1))/dzeta**2*(C(7,J+1)-C(7,J-1))/4)/R/Temp/delta* #*2-Ds(1)/delta**2*(C(6,J)-CN(6,J))/DT0 s1 = Ds(1)/delta**2*(C(1,J+1)-2*C(1,J)+C(1,J-1))/dzeta**2+0.255D0* #sqrt(omega)**3*delta*ZETA(J)**2*(C(1,J+1)-C(1,J-1))/sqrt(vis)/dzet #a+Ds(1)*z(1)*fc*(C(1,J)*(C(7,J+1)-2*C(7,J)+C(7,J-1))/dzeta**2+(C(1 #,J+1)-C(1,J-1))/dzeta**2*(C(7,J+1)-C(7,J-1))/4)/R/Temp/delta**2-Ds #(1)/delta**2*(C(1,J)-CN(1,J))/DT0 f(2) = s1+Ds(2)/delta**2*(C(2,J+1)-2*C(2,J)+C(2,J-1))/dzeta**2+0.2 #55D0*sqrt(omega)**3*delta*ZETA(J)**2*(C(2,J+1)-C(2,J-1))/sqrt(vis) #/dzeta+Ds(2)*z(2)*fc*(C(2,J)*(C(7,J+1)-2*C(7,J)+C(7,J-1))/dzeta**2 #+(C(2,J+1)-C(2,J-1))/dzeta**2*(C(7,J+1)-C(7,J-1))/4)/R/Temp/delta* #*2-Ds(1)/delta**2*(C(2,J)-CN(2,J))/DT0 f(3) = Ds(3)/delta**2*(C(3,J+1)-2*C(3,J)+C(3,J-1))/dzeta**2+0.255D #0*sqrt(omega)**3*delta*ZETA(J)**2*(C(3,J+1)-C(3,J-1))/sqrt(vis)/dz #eta+Ds(3)*z(3)*fc*(C(3,J)*(C(7,J+1)-2*C(7,J)+C(7,J-1))/dzeta**2+(C #(3,J+1)-C(3,J-1))/dzeta**2*(C(7,J+1)-C(7,J-1))/4)/R/Temp/delta**2- #Ds(1)/delta**2*(C(3,J)-CN(3,J))/DT0 f(4) = Ds(4)/delta**2*(C(4,J+1)-2*C(4,J)+C(4,J-1))/dzeta**2+0.255D #0*sqrt(omega)**3*delta*ZETA(J)**2*(C(4,J+1)-C(4,J-1))/sqrt(vis)/dz #eta+Ds(4)*z(4)*fc*(C(4,J)*(C(7,J+1)-2*C(7,J)+C(7,J-1))/dzeta**2+(C #(4,J+1)-C(4,J-1))/dzeta**2*(C(7,J+1)-C(7,J-1))/4)/R/Temp/delta**2- #Ds(1)/delta**2*(C(4,J)-CN(4,J))/DT0 f(5) = Ds(5)/delta**2*(C(5,J+1)-2*C(5,J)+C(5,J-1))/dzeta**2+0.255D #0*sqrt(omega)**3*delta*ZETA(J)**2*(C(5,J+1)-C(5,J-1))/sqrt(vis)/dz #eta+Ds(5)*z(5)*fc*(C(5,J)*(C(7,J+1)-2*C(7,J)+C(7,J-1))/dzeta**2+(C #(5,J+1)-C(5,J-1))/dzeta**2*(C(7,J+1)-C(7,J-1))/4)/R/Temp/delta**2- #Ds(1)/delta**2*(C(5,J)-CN(5,J))/DT0 f(6) = Keq*C(1,J)-1000*C(2,J)*C(6,J) f(7) = z(1)*C(1,J)+z(2)*C(2,J)+z(3)*C(3,J)+z(4)*C(4,J)+z(5)*C(5,J) #+z(6)*C(6,J) b(1,1) = -2*Ds(1)/delta**2/dzeta**2+Ds(1)*z(1)*fc*(C(7,J+1)-2*C(7, #J)+C(7,J-1))/dzeta**2/R/Temp/delta**2-Ds(1)/delta**2/DT0 b(1,2) = 0 b(1,3) = 0 b(1,4) = 0 b(1,5) = 0 b(1,6) = -2*Ds(6)/delta**2/dzeta**2+Ds(6)*z(6)*fc*(C(7,J+1)-2*C(7, #J)+C(7,J-1))/dzeta**2/R/Temp/delta**2-Ds(1)/delta**2/DT0 148

b(1,7) = -2*Ds(1)*z(1)*fc*C(1,J)/dzeta**2/R/Temp/delta**2-2*Ds(6)* #z(6)*fc*C(6,J)/dzeta**2/R/Temp/delta**2 b(2,1) = -2*Ds(1)/delta**2/dzeta**2+Ds(1)*z(1)*fc*(C(7,J+1)-2*C(7, #J)+C(7,J-1))/dzeta**2/R/Temp/delta**2-Ds(1)/delta**2/DT0 b(2,2) = -2*Ds(2)/delta**2/dzeta**2+Ds(2)*z(2)*fc*(C(7,J+1)-2*C(7, #J)+C(7,J-1))/dzeta**2/R/Temp/delta**2-Ds(1)/delta**2/DT0 b(2,3) = 0 b(2,4) = 0 b(2,5) = 0 b(2,6) = 0 b(2,7) = -2*Ds(1)*z(1)*fc*C(1,J)/dzeta**2/R/Temp/delta**2-2*Ds(2)* #z(2)*fc*C(2,J)/dzeta**2/R/Temp/delta**2 b(3,1) = 0 b(3,2) = 0 b(3,3) = -2*Ds(3)/delta**2/dzeta**2+Ds(3)*z(3)*fc*(C(7,J+1)-2*C(7, #J)+C(7,J-1))/dzeta**2/R/Temp/delta**2-Ds(1)/delta**2/DT0 b(3,4) = 0 b(3,5) = 0 b(3,6) = 0 b(3,7) = -2*Ds(3)*z(3)*fc*C(3,J)/dzeta**2/R/Temp/delta**2 b(4,1) = 0 b(4,2) = 0 b(4,3) = 0 b(4,4) = -2*Ds(4)/delta**2/dzeta**2+Ds(4)*z(4)*fc*(C(7,J+1)-2*C(7, #J)+C(7,J-1))/dzeta**2/R/Temp/delta**2-Ds(1)/delta**2/DT0 b(4,5) = 0 b(4,6) = 0 b(4,7) = -2*Ds(4)*z(4)*fc*C(4,J)/dzeta**2/R/Temp/delta**2 b(5,1) = 0 b(5,2) = 0 b(5,3) = 0 b(5,4) = 0 b(5,5) = -2*Ds(5)/delta**2/dzeta**2+Ds(5)*z(5)*fc*(C(7,J+1)-2*C(7, #J)+C(7,J-1))/dzeta**2/R/Temp/delta**2-Ds(1)/delta**2/DT0 b(5,6) = 0 b(5,7) = -2*Ds(5)*z(5)*fc*C(5,J)/dzeta**2/R/Temp/delta**2 b(6,1) = Keq b(6,2) = -1000*C(6,J) b(6,3) = 0 b(6,4) = 0 b(6,5) = 0 b(6,6) = -1000*C(2,J) b(6,7) = 0 b(7,1) = z(1) 149

b(7,2) = z(2) b(7,3) = z(3) b(7,4) = z(4) b(7,5) = z(5) b(7,6) = z(6) b(7,7) = 0 a(1,1) = Ds(1)/delta**2/dzeta**2-0.255D0*sqrt(omega)**3*delta*ZETA #(J)**2/sqrt(vis)/dzeta-Ds(1)*z(1)*fc/dzeta**2*(C(7,J+1)-C(7,J-1))/ #R/Temp/delta**2/4 a(1,2) = 0 a(1,3) = 0 a(1,4) = 0 a(1,5) = 0 a(1,6) = Ds(6)/delta**2/dzeta**2-0.255D0*sqrt(omega)**3*delta*ZETA #(J)**2/sqrt(vis)/dzeta-Ds(6)*z(6)*fc/dzeta**2*(C(7,J+1)-C(7,J-1))/ #R/Temp/delta**2/4 a(1,7) = Ds(1)*z(1)*fc*(C(1,J)/dzeta**2-(C(1,J+1)-C(1,J-1))/dzeta* #*2/4)/R/Temp/delta**2+Ds(6)*z(6)*fc*(C(6,J)/dzeta**2-(C(6,J+1)-C(6 #,J-1))/dzeta**2/4)/R/Temp/delta**2 a(2,1) = Ds(1)/delta**2/dzeta**2-0.255D0*sqrt(omega)**3*delta*ZETA #(J)**2/sqrt(vis)/dzeta-Ds(1)*z(1)*fc/dzeta**2*(C(7,J+1)-C(7,J-1))/ #R/Temp/delta**2/4 a(2,2) = Ds(2)/delta**2/dzeta**2-0.255D0*sqrt(omega)**3*delta*ZETA #(J)**2/sqrt(vis)/dzeta-Ds(2)*z(2)*fc/dzeta**2*(C(7,J+1)-C(7,J-1))/ #R/Temp/delta**2/4 a(2,3) = 0 a(2,4) = 0 a(2,5) = 0 a(2,6) = 0 a(2,7) = Ds(1)*z(1)*fc*(C(1,J)/dzeta**2-(C(1,J+1)-C(1,J-1))/dzeta* #*2/4)/R/Temp/delta**2+Ds(2)*z(2)*fc*(C(2,J)/dzeta**2-(C(2,J+1)-C(2 #,J-1))/dzeta**2/4)/R/Temp/delta**2 a(3,1) = 0 a(3,2) = 0 a(3,3) = Ds(3)/delta**2/dzeta**2-0.255D0*sqrt(omega)**3*delta*ZETA #(J)**2/sqrt(vis)/dzeta-Ds(3)*z(3)*fc/dzeta**2*(C(7,J+1)-C(7,J-1))/ #R/Temp/delta**2/4 a(3,4) = 0 a(3,5) = 0 a(3,6) = 0 a(3,7) = Ds(3)*z(3)*fc*(C(3,J)/dzeta**2-(C(3,J+1)-C(3,J-1))/dzeta* #*2/4)/R/Temp/delta**2 a(4,1) = 0 a(4,2) = 0 150

a(4,3) = 0 a(4,4) = Ds(4)/delta**2/dzeta**2-0.255D0*sqrt(omega)**3*delta*ZETA #(J)**2/sqrt(vis)/dzeta-Ds(4)*z(4)*fc/dzeta**2*(C(7,J+1)-C(7,J-1))/ #R/Temp/delta**2/4 a(4,5) = 0 a(4,6) = 0 a(4,7) = Ds(4)*z(4)*fc*(C(4,J)/dzeta**2-(C(4,J+1)-C(4,J-1))/dzeta* #*2/4)/R/Temp/delta**2 a(5,1) = 0 a(5,2) = 0 a(5,3) = 0 a(5,4) = 0 a(5,5) = Ds(5)/delta**2/dzeta**2-0.255D0*sqrt(omega)**3*delta*ZETA #(J)**2/sqrt(vis)/dzeta-Ds(5)*z(5)*fc/dzeta**2*(C(7,J+1)-C(7,J-1))/ #R/Temp/delta**2/4 a(5,6) = 0 a(5,7) = Ds(5)*z(5)*fc*(C(5,J)/dzeta**2-(C(5,J+1)-C(5,J-1))/dzeta* #*2/4)/R/Temp/delta**2 a(6,1) = 0 a(6,2) = 0 a(6,3) = 0 a(6,4) = 0 a(6,5) = 0 a(6,6) = 0 a(6,7) = 0 a(7,1) = 0 a(7,2) = 0 a(7,3) = 0 a(7,4) = 0 a(7,5) = 0 a(7,6) = 0 a(7,7) = 0 d(1,1) = Ds(1)/delta**2/dzeta**2+0.255D0*sqrt(omega)**3*delta*ZETA #(J)**2/sqrt(vis)/dzeta+Ds(1)*z(1)*fc/dzeta**2*(C(7,J+1)-C(7,J-1))/ #R/Temp/delta**2/4 d(1,2) = 0 d(1,3) = 0 d(1,4) = 0 d(1,5) = 0 d(1,6) = Ds(6)/delta**2/dzeta**2+0.255D0*sqrt(omega)**3*delta*ZETA #(J)**2/sqrt(vis)/dzeta+Ds(6)*z(6)*fc/dzeta**2*(C(7,J+1)-C(7,J-1))/ #R/Temp/delta**2/4 d(1,7) = Ds(1)*z(1)*fc*(C(1,J)/dzeta**2+(C(1,J+1)-C(1,J-1))/dzeta* #*2/4)/R/Temp/delta**2+Ds(6)*z(6)*fc*(C(6,J)/dzeta**2+(C(6,J+1)-C(6 151

#,J-1))/dzeta**2/4)/R/Temp/delta**2 d(2,1) = Ds(1)/delta**2/dzeta**2+0.255D0*sqrt(omega)**3*delta*ZETA #(J)**2/sqrt(vis)/dzeta+Ds(1)*z(1)*fc/dzeta**2*(C(7,J+1)-C(7,J-1))/ #R/Temp/delta**2/4 d(2,2) = Ds(2)/delta**2/dzeta**2+0.255D0*sqrt(omega)**3*delta*ZETA #(J)**2/sqrt(vis)/dzeta+Ds(2)*z(2)*fc/dzeta**2*(C(7,J+1)-C(7,J-1))/ #R/Temp/delta**2/4 d(2,3) = 0 d(2,4) = 0 d(2,5) = 0 d(2,6) = 0 d(2,7) = Ds(1)*z(1)*fc*(C(1,J)/dzeta**2+(C(1,J+1)-C(1,J-1))/dzeta* #*2/4)/R/Temp/delta**2+Ds(2)*z(2)*fc*(C(2,J)/dzeta**2+(C(2,J+1)-C(2 #,J-1))/dzeta**2/4)/R/Temp/delta**2 d(3,1) = 0 d(3,2) = 0 d(3,3) = Ds(3)/delta**2/dzeta**2+0.255D0*sqrt(omega)**3*delta*ZETA #(J)**2/sqrt(vis)/dzeta+Ds(3)*z(3)*fc/dzeta**2*(C(7,J+1)-C(7,J-1))/ #R/Temp/delta**2/4 d(3,4) = 0 d(3,5) = 0 d(3,6) = 0 d(3,7) = Ds(3)*z(3)*fc*(C(3,J)/dzeta**2+(C(3,J+1)-C(3,J-1))/dzeta* #*2/4)/R/Temp/delta**2 d(4,1) = 0 d(4,2) = 0 d(4,3) = 0 d(4,4) = Ds(4)/delta**2/dzeta**2+0.255D0*sqrt(omega)**3*delta*ZETA #(J)**2/sqrt(vis)/dzeta+Ds(4)*z(4)*fc/dzeta**2*(C(7,J+1)-C(7,J-1))/ #R/Temp/delta**2/4 d(4,5) = 0 d(4,6) = 0 d(4,7) = Ds(4)*z(4)*fc*(C(4,J)/dzeta**2+(C(4,J+1)-C(4,J-1))/dzeta* #*2/4)/R/Temp/delta**2 d(5,1) = 0 d(5,2) = 0 d(5,3) = 0 d(5,4) = 0 d(5,5) = Ds(5)/delta**2/dzeta**2+0.255D0*sqrt(omega)**3*delta*ZETA #(J)**2/sqrt(vis)/dzeta+Ds(5)*z(5)*fc/dzeta**2*(C(7,J+1)-C(7,J-1))/ #R/Temp/delta**2/4 d(5,6) = 0 d(5,7) = Ds(5)*z(5)*fc*(C(5,J)/dzeta**2+(C(5,J+1)-C(5,J-1))/dzeta* #*2/4)/R/Temp/delta**2 152

d(6,1) = 0 d(6,2) = 0 d(6,3) = 0 d(6,4) = 0 d(6,5) = 0 d(6,6) = 0 d(6,7) = 0 d(7,1) = 0 d(7,2) = 0 d(7,3) = 0 d(7,4) = 0 d(7,5) = 0 d(7,6) = 0 d(7,7) = 0 DO 31 I=1,N G(I)=-F(I) DO 31 K=1,N 31 G(I)=G(I)+A(I,K)*C(K,J-1)+B(I,K)*C(K,J)+D(I,K)*C(K,J+1) CALL BAND (J) GO TO 10 C THIS IS ZETA=1.0 c Input your equations and jacobians for the right boundary, corresponds to the bulk 20 F(1)=C(1,J)+(0.5000000000D0 * (1000*cb(1)) - 0.9976311575D-5 - #0.5000000000D0 * (1000*cb(2)) + 0.2500000000D-13 * #sqrt(0.4000000000D27 * (1000*cb(1)) ** 2 + 0.1 #596209852D23 * (1000*cb(1)) - 0.8000000000D27 * (1000*cb(1)) * #(1000*cb(2)) + 0.1592428 #682D18 + 0.1596209852D23 * (1000*cb(2)) + 0.4000000000D27 * #(1000*cb(2)) ** 2))/1000 #-cb(1) F(2)=C(2,J)-cb(2)+cb(1)- #(0.5000000000D0 * (1000*cb(1)) - 0.9976311575D-5 - #0.5000000000D0 * (1000*cb(2)) + 0.2500000000D-13 * #sqrt(0.4000000000D27 * (1000*cb(1)) ** 2 + 0.1 #596209852D23 * (1000*cb(1)) - 0.8000000000D27 * (1000*cb(1)) * #(1000*cb(2)) + 0.1592428 #682D18 + 0.1596209852D23 * (1000*cb(2)) + 0.4000000000D27 * #(1000*cb(2)) ** 2))/1000 F(3)=C(3,J)-cb(2) F(4)=C(4,J)-cb(1) F(5)=-Ds(1)/delta**2*(C(5,J)-CN(5,J))/DT0 F(6)=c(6,j)- #(0.5000000000D0 * (1000*cb(1)) - 0.9976311575D-5 - #0.5000000000D0 * (1000*cb(2)) + 0.2500000000D-13 * 153

#sqrt(0.4000000000D27 * (1000*cb(1)) ** 2 + 0.1 #596209852D23 * (1000*cb(1)) - 0.8000000000D27 * (1000*cb(1)) * #(1000*cb(2)) + 0.1592428 #682D18 + 0.1596209852D23 * (1000*cb(2)) + 0.4000000000D27 * #(1000*cb(2)) ** 2))/1000 F(7)=C(7,J) B(1,1)=1 B(2,2)=1 B(3,3)=1 B(4,4)=1 b(6,6) = -1 B(7,7)=1 B(5,5)=-Ds(1)/delta**2/DT0 DO 32 I=1,N G(I)=-F(I) DO 32 K=1,N 32 G(I)=G(I)+Y(I,K)*C(K,J-2)+A(I,K)*C(K,J-1)+B(I,K)*C(K,J) CALL BAND(J) DO 201 I=1,N RES=0.D0 DO 202 J=1,NJ 202 RES=RES+(C(I,J)- CG(I,J))**2 IF(RES.GT.1E-12)GO TO 441 201 CONTINUE GO TO 22 441 IF (JCOUNT-500) 8,8,40 22 PRINT 109 PRINT 113,JCOUNT-1 ccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc c------Calculate current density and surface coverage ------ccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc

iapp=1000* #(0.4D-4*EASA/EGA*sr+EASA/EGA*nn*fc*K2*K1**2*C(1,1)**2/(1+K1*C( #1,1)*C(2,1)*exp(fc*(iniV+sr*TIME-C(7,1)-Ujref)/R/Temp)+K1*K4*C(1,1 #)*C(2,1)**2*exp(fc*(2*iniV+2*sr*TIME-2*C(7,1)-Ujref-Ujref3)/R/Temp #)+K5*K4*K1*C(1,1)*C(2,1)**3*exp(fc*(3*iniV+3*sr*TIME-3*C(7,1)-Ujre #f-Ujref3-Ujref4)/R/Temp))**2*C(2,1)**2*exp(2*fc*(iniV+sr*TIME-C(7, #1)-Ujref)/R/Temp))

theta=1/(1+K1*C(1,1)*C(2,1)*exp(fc*(iniV+sr*TIME-C(7,1)-Ujref)/R 154

#/Te #mp)+K1*K4*C(1,1)*C(2,1)**2*exp(fc*(2*iniV+2*sr*TIME-2*C(7,1)-Ujref #-Ujref3)/R/Temp)+K5*K4*K1*C(1,1)*C(2,1)**3*exp(fc*(3*iniV+3*sr*TIM #E-3*C(7,1)-Ujref-Ujref3-Ujref4)/R/Temp))

TNH2=K1*C(1,1)/(1+K1*C(1,1)*C(2,1)*exp(fc*(iniV+sr*TIME-C(7,1)-Ujr #ef)/R/Temp)+K1*K4*C(1,1)*C(2,1)**2*exp(fc*(2*iniV+2*sr*TIME-2*C(7, #1)-Ujref-Ujref3)/R/Temp)+K5*K4*K1*C(1,1)*C(2,1)**3*exp(fc*(3*iniV+ #3*sr*TIME-3*C(7,1)-Ujref-Ujref3-Ujref4)/R/Temp))*C(2,1)*exp(fc*(in #iV+sr*TIME-C(7,1)-Ujref)/R/Temp)

TNH=1/(1+K1*C(1,1)*C(2,1)*exp(fc*(iniV+sr*TIME-C(7,1)-Ujref)/R/Te #mp)+K1*K4*C(1,1)*C(2,1)**2*exp(fc*(2*iniV+2*sr*TIME-2*C(7,1)-Ujref #-Ujref3)/R/Temp)+K5*K4*K1*C(1,1)*C(2,1)**3*exp(fc*(3*iniV+3*sr*TIM #E-3*C(7,1)-Ujref-Ujref3-Ujref4)/R/Temp))*K4*K1*C(1,1)*C(2,1)**2*ex #p(fc*(2*Vd-2*C(7,1)-Ujref-Ujref3)/R/Temp)

TN=1/(1+K1*C(1,1)*C(2,1)*exp(fc*(iniV+sr*TIME-C(7,1)-Ujref)/R/Te #mp)+K1*K4*C(1,1)*C(2,1)**2*exp(fc*(2*iniV+2*sr*TIME-2*C(7,1)-Ujref #-Ujref3)/R/Temp)+K5*K4*K1*C(1,1)*C(2,1)**3*exp(fc*(3*iniV+3*sr*TIM #E-3*C(7,1)-Ujref-Ujref3-Ujref4)/R/Temp))*K4*K5*K2*K1*C(1,1)*C(2,1) #**3*exp(fc*(3*Vd-3*C(7,1)-Ujref-Ujref3-Ujref4)/R/Temp)

ccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc c Writing variables in the short and long output ccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc c----OUTPUT FILE FOR PARAMETERS OPTIMIZATION------WRITE(21,'(2x,f9.6,11x,f20.14)') Vd,iapp

C-----Surface conversion ------WRITE(9,'(2x,f9.6,4x,f10.7,11x,f20.14,4x,f9.7,4x,f9.7, 14x,f9.7,4x,f9.7)')Vd,TIME,iapp,theta,TNH2,TNH,TN NUMCOL=L+1

155 c--- Concentration file ------

DO J=1,NJ WRITE(8,'(2x,f9.6,9x,E12.6,5x,E12.6,4x,E12.6,4x,E12.6 #,4x,E12.6,5x,E12.6,5x,E12.6)') 1 zeta(J), c(1,J),c(2,j),c(3,j),c(4,j),c(5,j),c(6,j),c(7,j)

END DO CONTINUE

GO TO 28 40 WRITE(16,943) I,RES,JCOUNT-1 PRINT 108

28 CONTINUE

SUBROUTINE BAND(J) parameter (nne=8,nm=10001) IMPLICIT REAL*8(A-H,O-Z) c DIMENSION A(3,3),B(3,3),C(3,1001),D(3,7), c 1 X(3,3),Y(3,3),G(3),E(3,7,1001) DIMENSION A(nne,nne),B(nne,nne),C(nne,nm),D(nne,nne*2+1), #X(nne,nne), Y(nne,nne), G(nne),E(nne,nne*2+1,nm) COMMON/A1/ A,B,C,D COMMON/A2/ G,X,Y,N,NJ 101 FORMAT (15H0DETERM=0 AT J=,I4) IF (J-2) 1,6,8 1 NP1= N+1 DO 2 I=1,N D(I,2*N+1)=G(I) DO 2 L=1,N LPN= L + N 2 D(I,LPN)= X(I,L) CALL LUDEC (N,2*N+1,DETERM) IF (DETERM) 4,3,4 3 PRINT 101, J 4 DO 5 K=1,N E(K,NP1,1)= D(K,2*N+1) DO 5 L=1,N E(K,L,1)= - D(K,L) LPN = L+N 5 X(K,L) = -D(K,LPN) RETURN 6 DO 7 I=1,N DO 7 K=1,N DO 7 L=1,N 7 D(I,K)= D(I,K) + A(I,L)*X(L,K) 8 IF (J-NJ) 11,9,9 9 DO 10 I=1,N DO 10 L=1,N G(I)= G(I) - Y(I,L)*E(L,NP1,J-2) DO 10 M=1,N 10 A(I,L)= A(I,L) + Y(I,M)*E(M,L,J-2) 11 DO 12 I=1,N D(I,NP1)= -G(I) DO 12 L=1,N D(I,NP1)= D(I,NP1) + A(I,L)*E(L,NP1,J-1) DO 12 K=1,N 12 B(I,K)= B(I,K) + A(I,L)*E(L,K,J-1) CALL LUDEC (N,NP1,DETERM) 157

IF (DETERM) 14,13,14 13 PRINT 101, J 14 DO 15 K=1,N DO 15 M=1,NP1 15 E(K,M,J)= - D(K,M) IF (J-NJ) 20,16,16 16 DO 17 K=1,N 17 C(K,J)=E(K,NP1,J) DO 18 JJ=2,NJ M=NJ - JJ + 1 DO 18 K=1,N C(K,M)=E(K,NP1,M) DO 18 L=1,N 18 C(K,M)=C(K,M)+E(K,L,M)*C(L,M+1) DO 19 L=1,N DO 19 K=1,N 19 C(K,1)=C(K,1)+X(K,L)*C(L,3) 20 RETURN END ccccccccccccccccccccccccccccccccccccccccccccccccccccccccc c Ludec(N,M,Determ) subroutine c The dimensions of A, B, C, and D c need to be consistent with the main program c C(i,j): dependent variables c A(i,i): Jacobian c B(i,i): Jacobian c D(i,2*i+1): Jacobian. c X(i,2*i+1): Internal variable used in this subroutine c jcol(i): Internal variable used in this subroutine c Important: Do not change anything here unless c you have modified the dimensions of the variables c described above. ccccccccccccccccccccccccccccccccccccccccccccccccccccccccc SUBROUTINE LUDEC(N,M,DETERM) parameter (nne=8,nm=10001) IMPLICIT REAL*8(A-H,O-Z) c DIMENSION A(3,3),B(3,3),C(3,1001),D(3,7), c 1 X(3,3),Y(3,3),G(3),E(3,7,1001) DIMENSION A(nne,nne),B(nne,nne),C(nne,nm),D(nne,nne*2+1), #X(nne,nne*2+1),JCOL(nne) c IMPLICIT REAL*8(A-H,O-Z) c DIMENSION A(3,3),B(3,3),C(3,1001),D(3,7),X(3,7),JCOL(3) COMMON/A1/ A,B,C,D NM1=N-1 158

DETERM=1.0 DO 1 I=1,N JCOL(I)=I DO 1 K=1,M 1 X(I,K)=D(I,K) DO 6 II=1,NM1 IP1=II+1 BMAX=DABS(B(II,II)) JC=II DO 2 J=IP1,N IF(DABS(B(II,J)).LE.BMAX)GOTO2 JC=J BMAX=DABS(B(II,J)) 2 CONTINUE DETERM=DETERM*B(II,JC) IF(DETERM.EQ.0.0) RETURN IF(JC.EQ.II) GO TO 4 JS=JCOL(JC) JCOL(JC)=JCOL(II) JCOL(II)=JS DO 3 I=1,N SAVE=B(I,JC) B(I,JC)=B(I,II) 3 B(I,II)=SAVE DETERM=-DETERM 4 DO 6 I=IP1,N F=B(I,II)/B(II,II) DO 5 J=IP1,N 5 B(I,J)=B(I,J)-F*B(II,J) DO 6 K=1,M 6 X(I,K)=X(I,K)-F*X(II,K) DETERM=DETERM*B(N,N) IF(DETERM.EQ.0.0) RETURN DO 7 II=2,N IR=N-II+2 IM1=IR-1 JC=JCOL(IR) DO 7 K=1,M F=X(IR,K)/B(IR,IR) D(JC,K)=F DO 7 I=1,IM1 7 X(I,K)=X(I,K)-B(I,IR)*F JC=JCOL(1) DO 8 K=1,M 159

8 D(JC,K)=X(1,K)/B(1,1) RETURN C CLOSE(16) END

PEST Code pcf * control data restart estimation 7 302 2 0 1 1 1 double point 1 0 0 5.0 2.0 0.3 0.03 10 3.0 3.0 0.001 0.1 50 0.000000000000001 3 100 0.00001 3 1 1 1 * parameter groups Ujref relative 0.01 0.001 switch 2.0 parabolic K relative 0.01 0.001 switch 2.0 parabolic * parameter data Ujref none relative -0.7 -0.74 -0.6 Ujref 1.0000 0.0000 1 Ujref4 none relative -0.3 -0.3 -0.2 Ujref 1.0000 0.0000 1 Ujref5 none relative -0.15 -0.25 -0.15 Ujref 1.0000 0.0000 1 K1 log factor 10 1e-1 1E2 K 1.0000 0.0000 1 K2 log factor 1.97e-10 1e-10 1E-9 K 1.0000 0.0000 1 K4 log factor 1.01 1e-3 1E1 K 1.0000 0.0000 1 K5 log factor 1.53e-2 1e-23 1e1 K 1.0000 0.0000 1 * observation groups obsgroup * observation data cd1 0.010084057 1 obsgroup cd2 0.011527062 1 obsgroup cd3 0.012596583 1 obsgroup cd4 0.012622048 1 obsgroup cd5 0.014328189 1 obsgroup cd6 0.015847588 1 obsgroup cd7 0.016467231 1 obsgroup cd8 0.017417917 1 obsgroup cd9 0.01867418 1 obsgroup cd10 0.019879513 1 obsgroup cd11 0.020490668 1 obsgroup cd12 0.021670537 1 obsgroup 160 cd13 0.022621223 1 obsgroup cd14 0.023911439 1 obsgroup cd15 0.025668509 1 obsgroup cd16 0.026525824 1 obsgroup cd17 0.028503589 1 obsgroup cd18 0.02903835 1 obsgroup cd19 0.030438913 1 obsgroup cd20 0.033087252 1 obsgroup cd21 0.032713768 1 obsgroup cd22 0.03387666 1 obsgroup cd23 0.03475944 1 obsgroup cd24 0.037195571 1 obsgroup cd25 0.038460323 1 obsgroup cd26 0.038697994 1 obsgroup cd27 0.039444961 1 obsgroup cd28 0.041787722 1 obsgroup cd29 0.043171309 1 obsgroup cd30 0.045242445 1 obsgroup cd31 0.046116736 1 obsgroup cd32 0.047483347 1 obsgroup cd33 0.048714145 1 obsgroup cd34 0.049435647 1 obsgroup cd35 0.051880267 1 obsgroup cd36 0.053603385 1 obsgroup cd37 0.054494653 1 obsgroup cd38 0.055623592 1 obsgroup cd39 0.05729578 1 obsgroup cd40 0.059799817 1 obsgroup cd41 0.062227461 1 obsgroup cd42 0.062346296 1 obsgroup cd43 0.064111855 1 obsgroup cd44 0.066047179 1 obsgroup cd45 0.067685414 1 obsgroup cd46 0.070596889 1 obsgroup cd47 0.071539086 1 obsgroup cd48 0.073941264 1 obsgroup cd49 0.076513208 1 obsgroup cd50 0.078109002 1 obsgroup cd51 0.080604551 1 obsgroup cd52 0.082548364 1 obsgroup cd53 0.085196702 1 obsgroup cd54 0.087607369 1 obsgroup cd55 0.089729435 1 obsgroup cd56 0.092861604 1 obsgroup 161 cd57 0.095535407 1 obsgroup cd58 0.096570975 1 obsgroup cd59 0.099686168 1 obsgroup cd60 0.102453342 1 obsgroup cd61 0.105381793 1 obsgroup cd62 0.108471521 1 obsgroup cd63 0.110126732 1 obsgroup cd64 0.113776686 1 obsgroup cd65 0.117893494 1 obsgroup cd66 0.121602865 1 obsgroup cd67 0.124972705 1 obsgroup cd68 0.126941983 1 obsgroup cd69 0.130702283 1 obsgroup cd70 0.135769777 1 obsgroup cd71 0.138749157 1 obsgroup cd72 0.142517946 1 obsgroup cd73 0.146099994 1 obsgroup cd74 0.150106454 1 obsgroup cd75 0.155139994 1 obsgroup cd76 0.158594718 1 obsgroup cd77 0.162940709 1 obsgroup cd78 0.169001329 1 obsgroup cd79 0.173635921 1 obsgroup cd80 0.178270513 1 obsgroup cd81 0.182565574 1 obsgroup cd82 0.189093049 1 obsgroup cd83 0.193931359 1 obsgroup cd84 0.198761181 1 obsgroup cd85 0.204473783 1 obsgroup cd86 0.210678703 1 obsgroup cd87 0.217189202 1 obsgroup cd88 0.22198507 1 obsgroup cd89 0.227646742 1 obsgroup cd90 0.234343982 1 obsgroup cd91 0.241261917 1 obsgroup cd92 0.247101843 1 obsgroup cd93 0.253052115 1 obsgroup cd94 0.26119236 1 obsgroup cd95 0.268060214 1 obsgroup cd96 0.274820267 1 obsgroup cd97 0.283216858 1 obsgroup cd98 0.289999829 1 obsgroup cd99 0.297104506 1 obsgroup cd100 0.305972195 1 obsgroup 162 cd101 0.31300557 1 obsgroup cd102 0.321130536 1 obsgroup cd103 0.329417628 1 obsgroup cd104 0.337058763 1 obsgroup cd105 0.345917964 1 obsgroup cd106 0.354892605 1 obsgroup cd107 0.363356252 1 obsgroup cd108 0.372403044 1 obsgroup cd109 0.381553392 1 obsgroup cd110 0.391909074 1 obsgroup cd111 0.399999238 1 obsgroup cd112 0.40905961 1 obsgroup cd113 0.419629196 1 obsgroup cd114 0.428890741 1 obsgroup cd115 0.438022415 1 obsgroup cd116 0.447809383 1 obsgroup cd117 0.459099622 1 obsgroup cd118 0.471281978 1 obsgroup cd119 0.479530873 1 obsgroup cd120 0.490806682 1 obsgroup cd121 0.502501812 1 obsgroup cd122 0.513369336 1 obsgroup cd123 0.525848781 1 obsgroup cd124 0.53227185 1 obsgroup cd125 0.540383235 1 obsgroup cd126 0.556701073 1 obsgroup cd127 0.564985618 1 obsgroup cd128 0.575320079 1 obsgroup cd129 0.588425958 1 obsgroup cd130 0.600324806 1 obsgroup cd131 0.611476687 1 obsgroup cd132 0.622844169 1 obsgroup cd133 0.634731983 1 obsgroup cd134 0.646339683 1 obsgroup cd135 0.657938046 1 obsgroup cd136 0.671082972 1 obsgroup cd137 0.682584569 1 obsgroup cd138 0.694569997 1 obsgroup cd139 0.707780282 1 obsgroup cd140 0.718966116 1 obsgroup cd141 0.731978709 1 obsgroup cd142 0.745873317 1 obsgroup cd143 0.757501135 1 obsgroup cd144 0.769569663 1 obsgroup 163 cd145 0.783019325 1 obsgroup cd146 0.795065945 1 obsgroup cd147 0.807858013 1 obsgroup cd148 0.820852102 1 obsgroup cd149 0.833241147 1 obsgroup cd150 0.84585632 1 obsgroup cd151 0.859362845 1 obsgroup cd152 0.872279436 1 obsgroup cd153 0.885666276 1 obsgroup cd154 0.897737436 1 obsgroup cd155 0.910201602 1 obsgroup cd156 0.924260713 1 obsgroup cd157 0.937588985 1 obsgroup cd158 0.950281485 1 obsgroup cd159 0.962399331 1 obsgroup cd160 0.977328489 1 obsgroup cd161 0.987865819 1 obsgroup cd162 0.997311559 1 obsgroup cd163 1.014826242 1 obsgroup cd164 1.024284714 1 obsgroup cd165 1.037804821 1 obsgroup cd166 1.050385276 1 obsgroup cd167 1.064768639 1 obsgroup cd168 1.076930623 1 obsgroup cd169 1.093570166 1 obsgroup cd170 1.104874836 1 obsgroup cd171 1.119628287 1 obsgroup cd172 1.132604295 1 obsgroup cd173 1.144011673 1 obsgroup cd174 1.156214401 1 obsgroup cd175 1.16974639 1 obsgroup cd176 1.181574786 1 obsgroup cd177 1.193995662 1 obsgroup cd178 1.208297537 1 obsgroup cd179 1.219062353 1 obsgroup cd180 1.231383917 1 obsgroup cd181 1.245224879 1 obsgroup cd182 1.255590747 1 obsgroup cd183 1.268497152 1 obsgroup cd184 1.28059123 1 obsgroup cd185 1.291426498 1 obsgroup cd186 1.303245557 1 obsgroup cd187 1.316504224 1 obsgroup cd188 1.327089089 1 obsgroup 164 cd189 1.338064414 1 obsgroup cd190 1.350830763 1 obsgroup cd191 1.361025167 1 obsgroup cd192 1.371117713 1 obsgroup cd193 1.383714296 1 obsgroup cd194 1.392703367 1 obsgroup cd195 1.403576833 1 obsgroup cd196 1.415782956 1 obsgroup cd197 1.423541229 1 obsgroup cd198 1.434032723 1 obsgroup cd199 1.445067465 1 obsgroup cd200 1.453759447 1 obsgroup cd201 1.463444556 1 obsgroup cd202 1.473825703 1 obsgroup cd203 1.481770717 1 obsgroup cd204 1.491617103 1 obsgroup cd205 1.501853949 1 obsgroup cd206 1.508729443 1 obsgroup cd207 1.517412936 1 obsgroup cd208 1.527284787 1 obsgroup cd209 1.533404825 1 obsgroup cd210 1.541502629 1 obsgroup cd211 1.551255643 1 obsgroup cd212 1.555567681 1 obsgroup cd213 1.563716414 1 obsgroup cd214 1.572510256 1 obsgroup cd215 1.577000547 1 obsgroup cd216 1.584529637 1 obsgroup cd217 1.591863497 1 obsgroup cd218 1.595131478 1 obsgroup cd219 1.602312549 1 obsgroup cd220 1.608967348 1 obsgroup cd221 1.613177527 1 obsgroup cd222 1.618593039 1 obsgroup cd223 1.624543312 1 obsgroup cd224 1.626937002 1 obsgroup cd225 1.631936589 1 obsgroup cd226 1.637988721 1 obsgroup cd227 1.63892243 1 obsgroup cd228 1.643837135 1 obsgroup cd229 1.648149173 1 obsgroup cd230 1.648386844 1 obsgroup cd231 1.652936554 1 obsgroup cd232 1.656900573 1 obsgroup 165 cd233 1.656323371 1 obsgroup cd234 1.659990301 1 obsgroup cd235 1.662052949 1 obsgroup cd236 1.662086902 1 obsgroup cd237 1.664845587 1 obsgroup cd238 1.666356498 1 obsgroup cd239 1.665584066 1 obsgroup cd240 1.666381963 1 obsgroup cd241 1.667638226 1 obsgroup cd242 1.666492311 1 obsgroup cd243 1.667171372 1 obsgroup cd244 1.667595785 1 obsgroup cd245 1.663614789 1 obsgroup cd246 1.66390339 1 obsgroup cd247 1.663810019 1 obsgroup cd248 1.659727164 1 obsgroup cd249 1.659582864 1 obsgroup cd250 1.656985455 1 obsgroup cd251 1.653759915 1 obsgroup cd252 1.653327014 1 obsgroup cd253 1.650407051 1 obsgroup cd254 1.645925248 1 obsgroup cd255 1.644550149 1 obsgroup cd256 1.642012158 1 obsgroup cd257 1.636910712 1 obsgroup cd258 1.634143538 1 obsgroup cd259 1.631070786 1 obsgroup cd260 1.625324232 1 obsgroup cd261 1.622268457 1 obsgroup cd262 1.618431762 1 obsgroup cd263 1.611318597 1 obsgroup cd264 1.608492005 1 obsgroup cd265 1.602957657 1 obsgroup cd266 1.596133093 1 obsgroup cd267 1.593026389 1 obsgroup cd268 1.587237393 1 obsgroup cd269 1.580446782 1 obsgroup cd270 1.57659311 1 obsgroup cd271 1.570320284 1 obsgroup cd272 1.563555137 1 obsgroup cd273 1.559251588 1 obsgroup cd274 1.553708752 1 obsgroup cd275 1.545347812 1 obsgroup cd276 1.540509502 1 obsgroup 166 cd277 1.533684938 1 obsgroup cd278 1.523906458 1 obsgroup cd279 1.520205575 1 obsgroup cd280 1.513465894 1 obsgroup cd281 1.505164372 1 obsgroup cd282 1.499315958 1 obsgroup cd283 1.491837798 1 obsgroup cd284 1.483739995 1 obsgroup cd285 1.478256576 1 obsgroup cd286 1.470472838 1 obsgroup cd287 1.462553289 1 obsgroup cd288 1.456976499 1 obsgroup cd289 1.449702057 1 obsgroup cd290 1.440356479 1 obsgroup cd291 1.434185511 1 obsgroup cd292 1.42719967 1 obsgroup cd293 1.417387238 1 obsgroup cd294 1.411835913 1 obsgroup cd295 1.403907875 1 obsgroup cd296 1.394587762 1 obsgroup cd297 1.389367479 1 obsgroup cd298 1.380582127 1 obsgroup cd299 1.371992004 1 obsgroup cd300 1.366347309 1 obsgroup cd301 1.35783358 1 obsgroup cd302 1.299183074 0 obsgroup * model command line MOD5_SCHEME_1>nul * model input/output DataL.tpl DataL.dat short.ins short.dat * prior information

167

APPENDIX B: SCP MODEL CODES FOR FORTRAN AND PEST

Fortran Code c Last modified by Luis Diaz on Oct 19, 2013 c SCP MODEL 1 IMPLICIT REAL*8(A-H,O-Z) parameter (nne=11,nm=10001,var=6,par=11,reac=4) c Simbology: The program uses the following nomemclature: c Simbology: The program uses the following nomemclature: c c(1,j): concentration of (mol/cm3). Result c c(2,j): concentration of OH- (mol/cm3). Result c c(3,j): concentration of K+ (mol/cm3). Result c c(4,j): concentration of Cl- (mol/cm3). Result c c(5,j): concentration of N2 (mol/cm3). Result c c(6,j): concentration of NH4+ (mol/cm3). Result c c(7,j): solution potential (V). Result c c(8,j): NH2 Surface coverage (dimensionless). Result c c(9,j): NH Surface coverage (dimensionless). Result c c(10,j): N surace coverage (dimensionless). Result c c(11,j): Empti active sites fraction (dimensionless). Result c n: number of equations, 11 c nm: maximum number of nodes to set dimensions, 10001 c nne: maximum number of variables to set dimensions, 7 c var: number of species, 6 c nj: number of nodes in the normal coordinate (max=10001 otherwise change nm in code) c Temp: Temperature (K) c cutoff: Cut off potential (V). Input. This is the value at which the program will stop c nv: number of data points in the output (input value) c omega: disk rotation speed (rpm input, the program transform it into rad/s) c ac: constant for RDE (dimensionless) c vis: kinematic viscosity (cm2/s) c delta: characteristic layer thickness (cm). This is calculated by the program c Ds(i): diffusion coefficient of (i) (cm^2/s) c z(i): charge number of NH3 (dimensionless) c cb(1): bulk concentration of NH4Cl (mol/cm3) c cb(2): bulk concentration of KOH (mol/cm3) c s(i,j): Stoichiometric coefficient of (i) in electrochemical reaction j (dimensionless) c nn: number of electrons transferred in reaction (dimensionless) c Ujref: Theoretical open circuit potential of reaction (V) c Vd: potential of working electrode (V) 168 c fc: faraday's constant (96485 coul/mol) c R: Universal gas constant (8.314 J/mol K)

REAL*8 c,Temp,iapp,omega,ac,vis,delta,Ds,cb,kf,kb,Ke,ioh,iov, #c1ref,c2ref,Ujref,Vd,Ujref2,dx,xr,fc,R,Potgrad,etha,cutoff, #iniV,yref,PN2,as,DT, TIME,DT0,sr,theta,Ujref5,K1,K2,K3,K4,K5,K6, #alpha1,Ujref4,TNH3,TNH2,TNH,TN2H4,TN,K7,TOH,K8,TN2,sigma,rho, #Deng,EASA,EGA,CNH3,Keq,GAMMA,K4f,K4b,K5f,K5b,alpha4,alpha5,ic,ir1, #ir2,ir3,ir4,op1,op2,op3,COH,K1f,K1b data fc,R,ac/96487,8.3143,0.51023262/ DIMENSION C(nne,nm), CG(nne,nm), CN(nne,nm), A(nne,nne), #B(nne,nne), D(nne,nne*2+1),X(nne,nne), Y(nne,nne), F(nne), #G(nne), Ds(var), z(var), cb(var), betha(var), s(var,reac), #xr(nm), Potgrad(nm),as(par),ZETA(nm),CNH3(nm,nm),nn(reac)

read(5,*) EGA !electrode geometric area (cm^2) read(5,*) vis !kinematic viscosity (cm2/s) read(5,*) Ds(1) !diffusion coefficient of NH3 (cm^2/s) read(5,*) Ds(2) !diffusion coefficient of OH- (cm^2/s) read(5,*) Ds(3) !diffusion coefficient of K+ (cm^2/s) read(5,*) Ds(4) !diffusion coefficient of Cl- (cm^2/s) read(5,*) Ds(5) !diffusion coefficient of N2 (cm^2/s) read(5,*) Ds(6) !diffusion coefficient of NH4Cl (cm^2/s) read(5,*) z(1) !charge number of NH3 (dimensionless) read(5,*) z(2) !charge number of OH- (dimensionless) read(5,*) z(3) !charge number of K+ (dimensionless) read(5,*) z(4) !charge number of Cl- (dimensionless) read(5,*) z(5) !charge number of N2 (dimensionless) read(5,*) z(6) !charge number of NH4+ (dimensionless) read(5,*) cb(1) !bulk concentration of NH4Cl (mol/cm3) read(5,*) cb(2) !bulk concentration of KOH (mol/cm3) read(5,*) s(1,1) !toichiometric coefficient in electrochemical reaction 1 NH3 (dimensionless) read(5,*) s(1,2)!Stoichiometric coefficient in electrochemical reaction 2 NH3 (dimensionless) read(5,*) s(1,3)!Stoichiometric coefficient in electrochemical reaction 3 NH3 (dimensionless) read(5,*) s(1,4)!Stoichiometric coefficient in electrochemical reaction 4 NH3 (dimensionless) read(5,*) s(2,1)!Stoichiometric coefficient in electrochemical reaction 1 OH- (dimensionless) read(5,*) s(2,2)!Stoichiometric coefficient in electrochemical reaction 2 OH- (dimensionless) read(5,*) s(2,3)!Stoichiometric coefficient in electrochemical reaction 3 OH- (dimensionless) read(5,*) s(2,4)!Stoichiometric coefficient in electrochemical reaction 4 OH- (dimensionless) read(5,*) s(3,1) !toichiometric coefficient in electrochemical reaction 1 K+ (dimensionless) read(5,*) s(3,2)!Stoichiometric coefficient in electrochemical reaction 2 K+ (dimensionless) read(5,*) s(3,3)!Stoichiometric coefficient in electrochemical reaction 3 K+ (dimensionless) read(5,*) s(3,4)!Stoichiometric coefficient in electrochemical reaction 4 K+ (dimensionless) read(5,*) s(4,1)!Stoichiometric coefficient in electrochemical reaction 1 Cl- (dimensionless) read(5,*) s(4,2)!Stoichiometric coefficient in electrochemical reaction 2 Cl- (dimensionless) 170

read(5,*) s(4,3)!Stoichiometric coefficient in electrochemical reaction 3 Cl- (dimensionless) read(5,*) s(4,4)!Stoichiometric coefficient in electrochemical reaction 4 Cl- (dimensionless) read(5,*) s(5,1) !toichiometric coefficient in electrochemical reaction 1 N2 (dimensionless) read(5,*) s(5,2)!Stoichiometric coefficient in electrochemical reaction 2 N2 (dimensionless) read(5,*) s(5,3)!Stoichiometric coefficient in electrochemical reaction 3 N2 (dimensionless) read(5,*) s(5,4)!Stoichiometric coefficient in electrochemical reaction 4 N2 (dimensionless) read(5,*) s(6,1)!Stoichiometric coefficient in electrochemical reaction 1 NH4+ (dimensionless) read(5,*) s(6,2)!Stoichiometric coefficient in electrochemical reaction 2 NH4+ (dimensionless) read(5,*) s(6,3)!Stoichiometric coefficient in electrochemical reaction 3 NH4+ (dimensionless) read(5,*) s(6,4)!Stoichiometric coefficient in electrochemical reaction 4 NH4+ (dimensionless) read(5,*) nn(1) !number of electrons transferred in reaction 1 (dimensionless) read(5,*) nn(2) !number of electrons transferred in reaction 3 (dimensionless) read(5,*) nn(3) !number of electrons transferred in reaction 4 (dimensionless) read(5,*) nn(4) !number of electrons transferred in reaction 5 (dimensionless) read(5,*) sr !scan rate V/s read(5,*) Keq !Keq dissociation constant read(5,*) sigma !Surface tension dyn/cm read(5,*) GAMMA !Pt Surface concentration mol/cm2 read(5,*) Deng !N2 density g/cucm read(5,*) rho !Solution density g/cucm read(5,*) Relec !Electrode radii cm read(5,*) P !SYSTEM PRESSURE read(5,*) Ujref !Theoretical open circuit potential of reaction (V) read(5,*) Ujref4 !anodic transfer coefficient (dimensionless) read(5,*) Ujref5 !anodic transfer coefficient (dimensionless) read(5,*) K1f read(5,*) K2 read(5,*) K4f read(5,*) K5f read(5,*) alpha1 read(5,*) alpha4 read(5,*) alpha5 close (UNIT=5)

171

open (unit=21,file='short.dat') open (unit=8,file='long.txt',status='unknown') open (Unit=6,file='tetha.txt',status='unknown') c open (Unit=7,file='CNH3.csv',status='unknown') open (Unit=9,file='time.dat',status='unknown') N=11 omega=omega*2*3.1416/60.0 delta=1.61*Ds(1) **(0.1D1 / 0.3D1) *vis **(0.1D1 / 0.6D1)*omega * #* (-0.1D1 / 0.2D1) ccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc ccccc c ----Calculate initial solution concentration based on NH3/NH4+ equilibrium ------ccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc ccccc

do j=1,nj c(3,j)=cb(2) c(4,j)=cb(1) c(5,J)=0 c(6,j)=(0.5000000000D0 * (1000*cb(1)) - 0.9976311575D-5 - #0.5000000000D0 * (1000*cb(2)) + 0.2500000000D-13 * #sqrt(0.4000000000D27 * (1000*cb(1)) ** 2 + 0.1 #596209852D23 * (1000*cb(1)) - 0.8000000000D27 * (1000*cb(1)) * #(1000*cb(2)) + 0.1592428 #682D18 + 0.1596209852D23 * (1000*cb(2)) + 0.4000000000D27 * #(1000*cb(2)) ** 2))/1000 c(1,j)=cb(1)-c(6,j) c(2,j)=cb(2)-c(1,j) end do do j=1, nj c(7,j)=0 c(8,j)=0 c(9,j)=0 c(10,j)=0 c(11,j)=0 end do

LLO=nv-1

TIME=(cutoff-IniV)/sr DT=TIME/LLO DT0=DT*Ds(1)/delta**2

172 c This is the loop to generate voltage-current dv=(cutoff-IniV)/NJ

ZETA(J)=(J-1)*DZETA

7 CONTINUE c Ini Input your guess values in C(1,J) c Input your initial values in CN(1,J) c------DO 3 J=1,NJ CN(1,J)=c(1,j) CN(2,J)=c(2,j) CN(3,J)=c(3,j) CN(4,J)=c(4,j) CN(5,J)=c(5,j) CN(6,J)=c(6,j) CN(7,J)=c(7,j) CN(8,J)=c(8,j) CN(9,J)=c(9,j) CN(10,J)=c(10,j) 3 CN(11,J)=c(11,j) DO 28 L=0,LLO TIME=L*DT Vd=iniV+sr*TIME c------c This loop assigns to the old value of the dependent variable c The recently calculated C(I,J) c If(L.gt.1) then DO 2 I=1,N DO 2 J=1,NJ 2 CN(I,J)=C(I,J) c endif JCOUNT=0 8 JCOUNT=JCOUNT+1 DO 100 I=1,N DO 100 J=1,NJ 100 CG(I,J)=C(I,J) 173

J=0 DO 9 I=1,N DO 9 K=1,N X(I,K)=0.0 D0 Y(I,K)=0.0 D0 9 CONTINUE 10 J=J+1 DO 11 I=1,N G(I)=0.0 D0 DO 11 K=1,N A(I,K)=0.0 D0 B(I,K)=0.0 D0 11 D(I,K)=0.0 D0 IF(J-1) 12,12,14 c THIS IS ZETA=0.0 c Input your equations and jacobians for the left boundary, that is the electrode surface 12 f(1) = s(1,1)*EASA/EGA*K1f*C(1,J)*C(2,J)*C(11,J)*exp(alpha1*fc*(in #iV+sr*TIME-C(7,J)-Ujref)/R/Temp)+s(1,2)*EASA/EGA*K2*C(8,J)**2+s(1, #3)*EASA/EGA*K4f*C(8,J)*C(2,J)*exp(alpha4*fc*(iniV+sr*TIME-C(7,J)-U #jref4)/R/Temp)+s(1,4)*EASA/EGA*K5f*C(9,J)*C(2,J)*exp(alpha5*fc*(in #iV+sr*TIME-C(7,J)-Ujref5)/R/Temp)-(z(1)*Ds(1)*fc*C(1,J)*(-3*C(7,J) #+4*C(7,J+1)-C(7,J+2))/R/Temp/dzeta/2+Ds(1)*(-3*C(1,J)+4*C(1,J+1)-C #(1,J+2))/dzeta/2)/delta f(2) = s(2,1)*EASA/EGA*K1f*C(1,J)*C(2,J)*C(11,J)*exp(alpha1*fc*(in #iV+sr*TIME-C(7,J)-Ujref)/R/Temp)+s(2,2)*EASA/EGA*K2*C(8,J)**2+s(2, #3)*EASA/EGA*K4f*C(8,J)*C(2,J)*exp(alpha4*fc*(iniV+sr*TIME-C(7,J)-U #jref4)/R/Temp)+s(2,4)*EASA/EGA*K5f*C(9,J)*C(2,J)*exp(alpha5*fc*(in #iV+sr*TIME-C(7,J)-Ujref5)/R/Temp)-(z(2)*Ds(2)*fc*C(2,J)*(-3*C(7,J) #+4*C(7,J+1)-C(7,J+2))/R/Temp/dzeta/2+Ds(2)*(-3*C(2,J)+4*C(2,J+1)-C #(2,J+2))/dzeta/2)/delta f(3) = s(3,1)*EASA/EGA*K1f*C(1,J)*C(2,J)*C(11,J)*exp(alpha1*fc*(in #iV+sr*TIME-C(7,J)-Ujref)/R/Temp)+s(3,2)*EASA/EGA*K2*C(8,J)**2+s(3, #3)*EASA/EGA*K4f*C(8,J)*C(2,J)*exp(alpha4*fc*(iniV+sr*TIME-C(7,J)-U #jref4)/R/Temp)+s(3,4)*EASA/EGA*K5f*C(9,J)*C(2,J)*exp(alpha5*fc*(in #iV+sr*TIME-C(7,J)-Ujref5)/R/Temp)-(z(3)*Ds(3)*fc*C(3,J)*(-3*C(7,J) #+4*C(7,J+1)-C(7,J+2))/R/Temp/dzeta/2+Ds(3)*(-3*C(3,J)+4*C(3,J+1)-C #(3,J+2))/dzeta/2)/delta f(4) = s(4,1)*EASA/EGA*K1f*C(1,J)*C(2,J)*C(11,J)*exp(alpha1*fc*(in #iV+sr*TIME-C(7,J)-Ujref)/R/Temp)+s(4,2)*EASA/EGA*K2*C(8,J)**2+s(4, #3)*EASA/EGA*K4f*C(8,J)*C(2,J)*exp(alpha4*fc*(iniV+sr*TIME-C(7,J)-U #jref4)/R/Temp)+s(4,4)*EASA/EGA*K5f*C(9,J)*C(2,J)*exp(alpha5*fc*(in #iV+sr*TIME-C(7,J)-Ujref5)/R/Temp)-(z(4)*Ds(4)*fc*C(4,J)*(-3*C(7,J) #+4*C(7,J+1)-C(7,J+2))/R/Temp/dzeta/2+Ds(4)*(-3*C(4,J)+4*C(4,J+1)-C #(4,J+2))/dzeta/2)/delta 174

f(5) = s(5,1)*EASA/EGA*K1f*C(1,J)*C(2,J)*C(11,J)*exp(alpha1*fc*(in #iV+sr*TIME-C(7,J)-Ujref)/R/Temp)+s(5,2)*EASA/EGA*K2*C(8,J)**2+s(5, #3)*EASA/EGA*K4f*C(8,J)*C(2,J)*exp(alpha4*fc*(iniV+sr*TIME-C(7,J)-U #jref4)/R/Temp)+s(5,4)*EASA/EGA*K5f*C(9,J)*C(2,J)*exp(alpha5*fc*(in #iV+sr*TIME-C(7,J)-Ujref5)/R/Temp)-(z(5)*Ds(5)*fc*C(5,J)*(-3*C(7,J) #+4*C(7,J+1)-C(7,J+2))/R/Temp/dzeta/2+Ds(5)*(-3*C(5,J)+4*C(5,J+1)-C #(5,J+2))/dzeta/2)/delta f(6) = s(6,1)*EASA/EGA*K1f*C(1,J)*C(2,J)*C(11,J)*exp(alpha1*fc*(in #iV+sr*TIME-C(7,J)-Ujref)/R/Temp)+s(6,2)*EASA/EGA*K2*C(8,J)**2+s(6, #3)*EASA/EGA*K4f*C(8,J)*C(2,J)*exp(alpha4*fc*(iniV+sr*TIME-C(7,J)-U #jref4)/R/Temp)+s(6,4)*EASA/EGA*K5f*C(9,J)*C(2,J)*exp(alpha5*fc*(in #iV+sr*TIME-C(7,J)-Ujref5)/R/Temp)-(z(6)*Ds(6)*fc*C(6,J)*(-3*C(7,J) #+4*C(7,J+1)-C(7,J+2))/R/Temp/dzeta/2+Ds(6)*(-3*C(6,J)+4*C(6,J+1)-C #(6,J+2))/dzeta/2)/delta f(7) = z(1)*C(1,J)+z(2)*C(2,J)+z(3)*C(3,J)+z(4)*C(4,J)+z(5)*C(5,J) #+z(6)*C(6,J) f(8) = K1f*C(1,J)*C(2,J)*C(11,J)*exp(alpha1*fc*(iniV+sr*TIME-C(7,J #)-Ujref)/R/Temp)-K4f*C(8,J)*C(2,J)*exp(alpha4*fc*(iniV+sr*TIME-C(7 #,J)-Ujref4)/R/Temp)+K4b*C(9,J)*exp(-(1-alpha4)*fc*(iniV+sr*TIME-C( #7,J)-Ujref4)/R/Temp)-K2*C(8,J)**2-GAMMA*(C(8,J)-CN(8,J))/DT f(9) = K4f*C(8,J)*C(2,J)*exp(alpha4*fc*(iniV+sr*TIME-C(7,J)-Ujref4 #)/R/Temp)-K4b*C(9,J)*exp(-(1-alpha4)*fc*(iniV+sr*TIME-C(7,J)-Ujref #4)/R/Temp)-K5f*C(9,J)*C(2,J)*exp(alpha5*fc*(iniV+sr*TIME-C(7,J)-Uj #ref5)/R/Temp)-GAMMA*(C(9,J)-CN(9,J))/DT f(10) = K5f*C(9,J)*C(2,J)*exp(alpha5*fc*(iniV+sr*TIME-C(7,J)-Ujref #5)/R/Temp)-GAMMA*(C(10,J)-CN(10,J))/DT f(11) = C(11,J)+C(8,J)+C(9,J)+C(10,J)-1 b(1,1) = s(1,1)*EASA/EGA*K1f*C(2,J)*C(11,J)*exp(alpha1*fc*(iniV+sr #*TIME-C(7,J)-Ujref)/R/Temp)-(z(1)*Ds(1)*fc*(-3*C(7,J)+4*C(7,J+1)-C #(7,J+2))/R/Temp/dzeta/2-3.D0/2.D0*Ds(1)/dzeta)/delta b(1,2) = s(1,1)*EASA/EGA*K1f*C(1,J)*C(11,J)*exp(alpha1*fc*(iniV+sr #*TIME-C(7,J)-Ujref)/R/Temp)+s(1,3)*EASA/EGA*K4f*C(8,J)*exp(alpha4* #fc*(iniV+sr*TIME-C(7,J)-Ujref4)/R/Temp)+s(1,4)*EASA/EGA*K5f*C(9,J) #*exp(alpha5*fc*(iniV+sr*TIME-C(7,J)-Ujref5)/R/Temp) b(1,3) = 0 b(1,4) = 0 b(1,5) = 0 b(1,6) = 0 b(1,7) = -s(1,1)*EASA/EGA*K1f*C(1,J)*C(2,J)*C(11,J)*alpha1*fc/R/Te #mp*exp(alpha1*fc*(iniV+sr*TIME-C(7,J)-Ujref)/R/Temp)-s(1,3)*EASA/E #GA*K4f*C(8,J)*C(2,J)*alpha4*fc/R/Temp*exp(alpha4*fc*(iniV+sr*TIME- #C(7,J)-Ujref4)/R/Temp)-s(1,4)*EASA/EGA*K5f*C(9,J)*C(2,J)*alpha5*fc #/R/Temp*exp(alpha5*fc*(iniV+sr*TIME-C(7,J)-Ujref5)/R/Temp)+3.D0/2. #D0*z(1)*Ds(1)*fc*C(1,J)/R/Temp/dzeta/delta 175

b(1,8) = 2*s(1,2)*EASA/EGA*K2*C(8,J)+s(1,3)*EASA/EGA*K4f*C(2,J)*ex #p(alpha4*fc*(iniV+sr*TIME-C(7,J)-Ujref4)/R/Temp) b(1,9) = s(1,4)*EASA/EGA*K5f*C(2,J)*exp(alpha5*fc*(iniV+sr*TIME-C( #7,J)-Ujref5)/R/Temp) b(1,10) = 0 b(1,11) = s(1,1)*EASA/EGA*K1f*C(1,J)*C(2,J)*exp(alpha1*fc*(iniV+sr #*TIME-C(7,J)-Ujref)/R/Temp) b(2,1) = s(2,1)*EASA/EGA*K1f*C(2,J)*C(11,J)*exp(alpha1*fc*(iniV+sr #*TIME-C(7,J)-Ujref)/R/Temp) b(2,2) = s(2,1)*EASA/EGA*K1f*C(1,J)*C(11,J)*exp(alpha1*fc*(iniV+sr #*TIME-C(7,J)-Ujref)/R/Temp)+s(2,3)*EASA/EGA*K4f*C(8,J)*exp(alpha4* #fc*(iniV+sr*TIME-C(7,J)-Ujref4)/R/Temp)+s(2,4)*EASA/EGA*K5f*C(9,J) #*exp(alpha5*fc*(iniV+sr*TIME-C(7,J)-Ujref5)/R/Temp)-(z(2)*Ds(2)*fc #*(-3*C(7,J)+4*C(7,J+1)-C(7,J+2))/R/Temp/dzeta/2-3.D0/2.D0*Ds(2)/dz #eta)/delta b(2,3) = 0 b(2,4) = 0 b(2,5) = 0 b(2,6) = 0 b(2,7) = -s(2,1)*EASA/EGA*K1f*C(1,J)*C(2,J)*C(11,J)*alpha1*fc/R/Te #mp*exp(alpha1*fc*(iniV+sr*TIME-C(7,J)-Ujref)/R/Temp)-s(2,3)*EASA/E #GA*K4f*C(8,J)*C(2,J)*alpha4*fc/R/Temp*exp(alpha4*fc*(iniV+sr*TIME- #C(7,J)-Ujref4)/R/Temp)-s(2,4)*EASA/EGA*K5f*C(9,J)*C(2,J)*alpha5*fc #/R/Temp*exp(alpha5*fc*(iniV+sr*TIME-C(7,J)-Ujref5)/R/Temp)+3.D0/2. #D0*z(2)*Ds(2)*fc*C(2,J)/R/Temp/dzeta/delta b(2,8) = 2*s(2,2)*EASA/EGA*K2*C(8,J)+s(2,3)*EASA/EGA*K4f*C(2,J)*ex #p(alpha4*fc*(iniV+sr*TIME-C(7,J)-Ujref4)/R/Temp) b(2,9) = s(2,4)*EASA/EGA*K5f*C(2,J)*exp(alpha5*fc*(iniV+sr*TIME-C( #7,J)-Ujref5)/R/Temp) b(2,10) = 0 b(2,11) = s(2,1)*EASA/EGA*K1f*C(1,J)*C(2,J)*exp(alpha1*fc*(iniV+sr #*TIME-C(7,J)-Ujref)/R/Temp) b(3,1) = s(3,1)*EASA/EGA*K1f*C(2,J)*C(11,J)*exp(alpha1*fc*(iniV+sr #*TIME-C(7,J)-Ujref)/R/Temp) b(3,2) = s(3,1)*EASA/EGA*K1f*C(1,J)*C(11,J)*exp(alpha1*fc*(iniV+sr #*TIME-C(7,J)-Ujref)/R/Temp)+s(3,3)*EASA/EGA*K4f*C(8,J)*exp(alpha4* #fc*(iniV+sr*TIME-C(7,J)-Ujref4)/R/Temp)+s(3,4)*EASA/EGA*K5f*C(9,J) #*exp(alpha5*fc*(iniV+sr*TIME-C(7,J)-Ujref5)/R/Temp) b(3,3) = -(z(3)*Ds(3)*fc*(-3*C(7,J)+4*C(7,J+1)-C(7,J+2))/R/Temp/dz #eta/2-3.D0/2.D0*Ds(3)/dzeta)/delta b(3,4) = 0 b(3,5) = 0 b(3,6) = 0 b(3,7) = -s(3,1)*EASA/EGA*K1f*C(1,J)*C(2,J)*C(11,J)*alpha1*fc/R/Te 176

#mp*exp(alpha1*fc*(iniV+sr*TIME-C(7,J)-Ujref)/R/Temp)-s(3,3)*EASA/E #GA*K4f*C(8,J)*C(2,J)*alpha4*fc/R/Temp*exp(alpha4*fc*(iniV+sr*TIME- #C(7,J)-Ujref4)/R/Temp)-s(3,4)*EASA/EGA*K5f*C(9,J)*C(2,J)*alpha5*fc #/R/Temp*exp(alpha5*fc*(iniV+sr*TIME-C(7,J)-Ujref5)/R/Temp)+3.D0/2. #D0*z(3)*Ds(3)*fc*C(3,J)/R/Temp/dzeta/delta b(3,8) = 2*s(3,2)*EASA/EGA*K2*C(8,J)+s(3,3)*EASA/EGA*K4f*C(2,J)*ex #p(alpha4*fc*(iniV+sr*TIME-C(7,J)-Ujref4)/R/Temp) b(3,9) = s(3,4)*EASA/EGA*K5f*C(2,J)*exp(alpha5*fc*(iniV+sr*TIME-C( #7,J)-Ujref5)/R/Temp) b(3,10) = 0 b(3,11) = s(3,1)*EASA/EGA*K1f*C(1,J)*C(2,J)*exp(alpha1*fc*(iniV+sr #*TIME-C(7,J)-Ujref)/R/Temp) b(4,1) = s(4,1)*EASA/EGA*K1f*C(2,J)*C(11,J)*exp(alpha1*fc*(iniV+sr #*TIME-C(7,J)-Ujref)/R/Temp) b(4,2) = s(4,1)*EASA/EGA*K1f*C(1,J)*C(11,J)*exp(alpha1*fc*(iniV+sr #*TIME-C(7,J)-Ujref)/R/Temp)+s(4,3)*EASA/EGA*K4f*C(8,J)*exp(alpha4* #fc*(iniV+sr*TIME-C(7,J)-Ujref4)/R/Temp)+s(4,4)*EASA/EGA*K5f*C(9,J) #*exp(alpha5*fc*(iniV+sr*TIME-C(7,J)-Ujref5)/R/Temp) b(4,3) = 0 b(4,4) = -(z(4)*Ds(4)*fc*(-3*C(7,J)+4*C(7,J+1)-C(7,J+2))/R/Temp/dz #eta/2-3.D0/2.D0*Ds(4)/dzeta)/delta b(4,5) = 0 b(4,6) = 0 b(4,7) = -s(4,1)*EASA/EGA*K1f*C(1,J)*C(2,J)*C(11,J)*alpha1*fc/R/Te #mp*exp(alpha1*fc*(iniV+sr*TIME-C(7,J)-Ujref)/R/Temp)-s(4,3)*EASA/E #GA*K4f*C(8,J)*C(2,J)*alpha4*fc/R/Temp*exp(alpha4*fc*(iniV+sr*TIME- #C(7,J)-Ujref4)/R/Temp)-s(4,4)*EASA/EGA*K5f*C(9,J)*C(2,J)*alpha5*fc #/R/Temp*exp(alpha5*fc*(iniV+sr*TIME-C(7,J)-Ujref5)/R/Temp)+3.D0/2. #D0*z(4)*Ds(4)*fc*C(4,J)/R/Temp/dzeta/delta b(4,8) = 2*s(4,2)*EASA/EGA*K2*C(8,J)+s(4,3)*EASA/EGA*K4f*C(2,J)*ex #p(alpha4*fc*(iniV+sr*TIME-C(7,J)-Ujref4)/R/Temp) b(4,9) = s(4,4)*EASA/EGA*K5f*C(2,J)*exp(alpha5*fc*(iniV+sr*TIME-C( #7,J)-Ujref5)/R/Temp) b(4,10) = 0 b(4,11) = s(4,1)*EASA/EGA*K1f*C(1,J)*C(2,J)*exp(alpha1*fc*(iniV+sr #*TIME-C(7,J)-Ujref)/R/Temp) b(5,1) = s(5,1)*EASA/EGA*K1f*C(2,J)*C(11,J)*exp(alpha1*fc*(iniV+sr #*TIME-C(7,J)-Ujref)/R/Temp) b(5,2) = s(5,1)*EASA/EGA*K1f*C(1,J)*C(11,J)*exp(alpha1*fc*(iniV+sr #*TIME-C(7,J)-Ujref)/R/Temp)+s(5,3)*EASA/EGA*K4f*C(8,J)*exp(alpha4* #fc*(iniV+sr*TIME-C(7,J)-Ujref4)/R/Temp)+s(5,4)*EASA/EGA*K5f*C(9,J) #*exp(alpha5*fc*(iniV+sr*TIME-C(7,J)-Ujref5)/R/Temp) b(5,3) = 0 b(5,4) = 0 177

b(5,5) = -(z(5)*Ds(5)*fc*(-3*C(7,J)+4*C(7,J+1)-C(7,J+2))/R/Temp/dz #eta/2-3.D0/2.D0*Ds(5)/dzeta)/delta b(5,6) = 0 b(5,7) = -s(5,1)*EASA/EGA*K1f*C(1,J)*C(2,J)*C(11,J)*alpha1*fc/R/Te #mp*exp(alpha1*fc*(iniV+sr*TIME-C(7,J)-Ujref)/R/Temp)-s(5,3)*EASA/E #GA*K4f*C(8,J)*C(2,J)*alpha4*fc/R/Temp*exp(alpha4*fc*(iniV+sr*TIME- #C(7,J)-Ujref4)/R/Temp)-s(5,4)*EASA/EGA*K5f*C(9,J)*C(2,J)*alpha5*fc #/R/Temp*exp(alpha5*fc*(iniV+sr*TIME-C(7,J)-Ujref5)/R/Temp)+3.D0/2. #D0*z(5)*Ds(5)*fc*C(5,J)/R/Temp/dzeta/delta b(5,8) = 2*s(5,2)*EASA/EGA*K2*C(8,J)+s(5,3)*EASA/EGA*K4f*C(2,J)*ex #p(alpha4*fc*(iniV+sr*TIME-C(7,J)-Ujref4)/R/Temp) b(5,9) = s(5,4)*EASA/EGA*K5f*C(2,J)*exp(alpha5*fc*(iniV+sr*TIME-C( #7,J)-Ujref5)/R/Temp) b(5,10) = 0 b(5,11) = s(5,1)*EASA/EGA*K1f*C(1,J)*C(2,J)*exp(alpha1*fc*(iniV+sr #*TIME-C(7,J)-Ujref)/R/Temp) b(6,1) = s(6,1)*EASA/EGA*K1f*C(2,J)*C(11,J)*exp(alpha1*fc*(iniV+sr #*TIME-C(7,J)-Ujref)/R/Temp) b(6,2) = s(6,1)*EASA/EGA*K1f*C(1,J)*C(11,J)*exp(alpha1*fc*(iniV+sr #*TIME-C(7,J)-Ujref)/R/Temp)+s(6,3)*EASA/EGA*K4f*C(8,J)*exp(alpha4* #fc*(iniV+sr*TIME-C(7,J)-Ujref4)/R/Temp)+s(6,4)*EASA/EGA*K5f*C(9,J) #*exp(alpha5*fc*(iniV+sr*TIME-C(7,J)-Ujref5)/R/Temp) b(6,3) = 0 b(6,4) = 0 b(6,5) = 0 b(6,6) = -(z(6)*Ds(6)*fc*(-3*C(7,J)+4*C(7,J+1)-C(7,J+2))/R/Temp/dz #eta/2-3.D0/2.D0*Ds(6)/dzeta)/delta b(6,7) = -s(6,1)*EASA/EGA*K1f*C(1,J)*C(2,J)*C(11,J)*alpha1*fc/R/Te #mp*exp(alpha1*fc*(iniV+sr*TIME-C(7,J)-Ujref)/R/Temp)-s(6,3)*EASA/E #GA*K4f*C(8,J)*C(2,J)*alpha4*fc/R/Temp*exp(alpha4*fc*(iniV+sr*TIME- #C(7,J)-Ujref4)/R/Temp)-s(6,4)*EASA/EGA*K5f*C(9,J)*C(2,J)*alpha5*fc #/R/Temp*exp(alpha5*fc*(iniV+sr*TIME-C(7,J)-Ujref5)/R/Temp)+3.D0/2. #D0*z(6)*Ds(6)*fc*C(6,J)/R/Temp/dzeta/delta b(6,8) = 2*s(6,2)*EASA/EGA*K2*C(8,J)+s(6,3)*EASA/EGA*K4f*C(2,J)*ex #p(alpha4*fc*(iniV+sr*TIME-C(7,J)-Ujref4)/R/Temp) b(6,9) = s(6,4)*EASA/EGA*K5f*C(2,J)*exp(alpha5*fc*(iniV+sr*TIME-C( #7,J)-Ujref5)/R/Temp) b(6,10) = 0 b(6,11) = s(6,1)*EASA/EGA*K1f*C(1,J)*C(2,J)*exp(alpha1*fc*(iniV+sr #*TIME-C(7,J)-Ujref)/R/Temp) b(7,1) = z(1) b(7,2) = z(2) b(7,3) = z(3) b(7,4) = z(4) 178

b(7,5) = z(5) b(7,6) = z(6) b(7,7) = 0 b(7,8) = 0 b(7,9) = 0 b(7,10) = 0 b(7,11) = 0 b(8,1) = K1f*C(2,J)*C(11,J)*exp(alpha1*fc*(iniV+sr*TIME-C(7,J)-Ujr #ef)/R/Temp) b(8,2) = K1f*C(1,J)*C(11,J)*exp(alpha1*fc*(iniV+sr*TIME-C(7,J)-Ujr #ef)/R/Temp)-K4f*C(8,J)*exp(alpha4*fc*(iniV+sr*TIME-C(7,J)-Ujref4)/ #R/Temp) b(8,3) = 0 b(8,4) = 0 b(8,5) = 0 b(8,6) = 0 b(8,7) = -K1f*C(1,J)*C(2,J)*C(11,J)*alpha1*fc/R/Temp*exp(alpha1*fc #*(iniV+sr*TIME-C(7,J)-Ujref)/R/Temp)+K4f*C(8,J)*C(2,J)*alpha4*fc/R #/Temp*exp(alpha4*fc*(iniV+sr*TIME-C(7,J)-Ujref4)/R/Temp)+K4b*C(9,J #)*(1-alpha4)*fc/R/Temp*exp(-(1-alpha4)*fc*(iniV+sr*TIME-C(7,J)-Ujr #ef4)/R/Temp) b(8,8) = -K4f*C(2,J)*exp(alpha4*fc*(iniV+sr*TIME-C(7,J)-Ujref4)/R/ #Temp)-2*K2*C(8,J)-GAMMA/DT b(8,9) = K4b*exp(-(1-alpha4)*fc*(iniV+sr*TIME-C(7,J)-Ujref4)/R/Tem #p) b(8,10) = 0 b(8,11) = K1f*C(1,J)*C(2,J)*exp(alpha1*fc*(iniV+sr*TIME-C(7,J)-Ujr #ef)/R/Temp) b(9,1) = 0 b(9,2) = K4f*C(8,J)*exp(alpha4*fc*(iniV+sr*TIME-C(7,J)-Ujref4)/R/T #emp)-K5f*C(9,J)*exp(alpha5*fc*(iniV+sr*TIME-C(7,J)-Ujref5)/R/Temp) b(9,3) = 0 b(9,4) = 0 b(9,5) = 0 b(9,6) = 0 b(9,7) = -K4f*C(8,J)*C(2,J)*alpha4*fc/R/Temp*exp(alpha4*fc*(iniV+s #r*TIME-C(7,J)-Ujref4)/R/Temp)-K4b*C(9,J)*(1-alpha4)*fc/R/Temp*exp( #-(1-alpha4)*fc*(iniV+sr*TIME-C(7,J)-Ujref4)/R/Temp)+K5f*C(9,J)*C(2 #,J)*alpha5*fc/R/Temp*exp(alpha5*fc*(iniV+sr*TIME-C(7,J)-Ujref5)/R/ #Temp) b(9,8) = K4f*C(2,J)*exp(alpha4*fc*(iniV+sr*TIME-C(7,J)-Ujref4)/R/T #emp) b(9,9) = -K4b*exp(-(1-alpha4)*fc*(iniV+sr*TIME-C(7,J)-Ujref4)/R/Te #mp)-K5f*C(2,J)*exp(alpha5*fc*(iniV+sr*TIME-C(7,J)-Ujref5)/R/Temp)- 179

#GAMMA/DT b(9,10) = 0 b(9,11) = 0 b(10,1) = 0 b(10,2) = K5f*C(9,J)*exp(alpha5*fc*(iniV+sr*TIME-C(7,J)-Ujref5)/R/ #Temp) b(10,3) = 0 b(10,4) = 0 b(10,5) = 0 b(10,6) = 0 b(10,7) = -K5f*C(9,J)*C(2,J)*alpha5*fc/R/Temp*exp(alpha5*fc*(iniV+ #sr*TIME-C(7,J)-Ujref5)/R/Temp) b(10,8) = 0 b(10,9) = K5f*C(2,J)*exp(alpha5*fc*(iniV+sr*TIME-C(7,J)-Ujref5)/R/ #Temp) b(10,10) = -GAMMA/DT b(10,11) = 0 b(11,1) = 0 b(11,2) = 0 b(11,3) = 0 b(11,4) = 0 b(11,5) = 0 b(11,6) = 0 b(11,7) = 0 b(11,8) = 1 b(11,9) = 1 b(11,10) = 1 b(11,11) = 1 d(1,1) = -2*Ds(1)/dzeta/delta d(1,2) = 0 d(1,3) = 0 d(1,4) = 0 d(1,5) = 0 d(1,6) = 0 d(1,7) = -2*z(1)*Ds(1)*fc*C(1,J)/R/Temp/dzeta/delta d(1,8) = 0 d(1,9) = 0 d(1,10) = 0 d(1,11) = 0 d(2,1) = 0 d(2,2) = -2*Ds(2)/dzeta/delta d(2,3) = 0 d(2,4) = 0 d(2,5) = 0 180 d(2,6) = 0 d(2,7) = -2*z(2)*Ds(2)*fc*C(2,J)/R/Temp/dzeta/delta d(2,8) = 0 d(2,9) = 0 d(2,10) = 0 d(2,11) = 0 d(3,1) = 0 d(3,2) = 0 d(3,3) = -2*Ds(3)/dzeta/delta d(3,4) = 0 d(3,5) = 0 d(3,6) = 0 d(3,7) = -2*z(3)*Ds(3)*fc*C(3,J)/R/Temp/dzeta/delta d(3,8) = 0 d(3,9) = 0 d(3,10) = 0 d(3,11) = 0 d(4,1) = 0 d(4,2) = 0 d(4,3) = 0 d(4,4) = -2*Ds(4)/dzeta/delta d(4,5) = 0 d(4,6) = 0 d(4,7) = -2*z(4)*Ds(4)*fc*C(4,J)/R/Temp/dzeta/delta d(4,8) = 0 d(4,9) = 0 d(4,10) = 0 d(4,11) = 0 d(5,1) = 0 d(5,2) = 0 d(5,3) = 0 d(5,4) = 0 d(5,5) = -2*Ds(5)/dzeta/delta d(5,6) = 0 d(5,7) = -2*z(5)*Ds(5)*fc*C(5,J)/R/Temp/dzeta/delta d(5,8) = 0 d(5,9) = 0 d(5,10) = 0 d(5,11) = 0 d(6,1) = 0 d(6,2) = 0 d(6,3) = 0 d(6,4) = 0 d(6,5) = 0 181 d(6,6) = -2*Ds(6)/dzeta/delta d(6,7) = -2*z(6)*Ds(6)*fc*C(6,J)/R/Temp/dzeta/delta d(6,8) = 0 d(6,9) = 0 d(6,10) = 0 d(6,11) = 0 d(7,1) = 0 d(7,2) = 0 d(7,3) = 0 d(7,4) = 0 d(7,5) = 0 d(7,6) = 0 d(7,7) = 0 d(7,8) = 0 d(7,9) = 0 d(7,10) = 0 d(7,11) = 0 d(8,1) = 0 d(8,2) = 0 d(8,3) = 0 d(8,4) = 0 d(8,5) = 0 d(8,6) = 0 d(8,7) = 0 d(8,8) = 0 d(8,9) = 0 d(8,10) = 0 d(8,11) = 0 d(9,1) = 0 d(9,2) = 0 d(9,3) = 0 d(9,4) = 0 d(9,5) = 0 d(9,6) = 0 d(9,7) = 0 d(9,8) = 0 d(9,9) = 0 d(9,10) = 0 d(9,11) = 0 d(10,1) = 0 d(10,2) = 0 d(10,3) = 0 d(10,4) = 0 d(10,5) = 0 182 d(10,6) = 0 d(10,7) = 0 d(10,8) = 0 d(10,9) = 0 d(10,10) = 0 d(10,11) = 0 d(11,1) = 0 d(11,2) = 0 d(11,3) = 0 d(11,4) = 0 d(11,5) = 0 d(11,6) = 0 d(11,7) = 0 d(11,8) = 0 d(11,9) = 0 d(11,10) = 0 d(11,11) = 0 x(1,1) = Ds(1)/dzeta/delta/2 x(1,2) = 0 x(1,3) = 0 x(1,4) = 0 x(1,5) = 0 x(1,6) = 0 x(1,7) = z(1)*Ds(1)*fc*C(1,J)/R/Temp/dzeta/delta/2 x(1,8) = 0 x(1,9) = 0 x(1,10) = 0 x(1,11) = 0 x(2,1) = 0 x(2,2) = Ds(2)/dzeta/delta/2 x(2,3) = 0 x(2,4) = 0 x(2,5) = 0 x(2,6) = 0 x(2,7) = z(2)*Ds(2)*fc*C(2,J)/R/Temp/dzeta/delta/2 x(2,8) = 0 x(2,9) = 0 x(2,10) = 0 x(2,11) = 0 x(3,1) = 0 x(3,2) = 0 x(3,3) = Ds(3)/dzeta/delta/2 x(3,4) = 0 x(3,5) = 0 183 x(3,6) = 0 x(3,7) = z(3)*Ds(3)*fc*C(3,J)/R/Temp/dzeta/delta/2 x(3,8) = 0 x(3,9) = 0 x(3,10) = 0 x(3,11) = 0 x(4,1) = 0 x(4,2) = 0 x(4,3) = 0 x(4,4) = Ds(4)/dzeta/delta/2 x(4,5) = 0 x(4,6) = 0 x(4,7) = z(4)*Ds(4)*fc*C(4,J)/R/Temp/dzeta/delta/2 x(4,8) = 0 x(4,9) = 0 x(4,10) = 0 x(4,11) = 0 x(5,1) = 0 x(5,2) = 0 x(5,3) = 0 x(5,4) = 0 x(5,5) = Ds(5)/dzeta/delta/2 x(5,6) = 0 x(5,7) = z(5)*Ds(5)*fc*C(5,J)/R/Temp/dzeta/delta/2 x(5,8) = 0 x(5,9) = 0 x(5,10) = 0 x(5,11) = 0 x(6,1) = 0 x(6,2) = 0 x(6,3) = 0 x(6,4) = 0 x(6,5) = 0 x(6,6) = Ds(6)/dzeta/delta/2 x(6,7) = z(6)*Ds(6)*fc*C(6,J)/R/Temp/dzeta/delta/2 x(6,8) = 0 x(6,9) = 0 x(6,10) = 0 x(6,11) = 0 x(7,1) = 0 x(7,2) = 0 x(7,3) = 0 x(7,4) = 0 x(7,5) = 0 184 x(7,6) = 0 x(7,7) = 0 x(7,8) = 0 x(7,9) = 0 x(7,10) = 0 x(7,11) = 0 x(8,1) = 0 x(8,2) = 0 x(8,3) = 0 x(8,4) = 0 x(8,5) = 0 x(8,6) = 0 x(8,7) = 0 x(8,8) = 0 x(8,9) = 0 x(8,10) = 0 x(8,11) = 0 x(9,1) = 0 x(9,2) = 0 x(9,3) = 0 x(9,4) = 0 x(9,5) = 0 x(9,6) = 0 x(9,7) = 0 x(9,8) = 0 x(9,9) = 0 x(9,10) = 0 x(9,11) = 0 x(10,1) = 0 x(10,2) = 0 x(10,3) = 0 x(10,4) = 0 x(10,5) = 0 x(10,6) = 0 x(10,7) = 0 x(10,8) = 0 x(10,9) = 0 x(10,10) = 0 x(10,11) = 0 x(11,1) = 0 x(11,2) = 0 x(11,3) = 0 x(11,4) = 0 x(11,5) = 0 185

x(11,6) = 0 x(11,7) = 0 x(11,8) = 0 x(11,9) = 0 x(11,10) = 0 x(11,11) = 0 DO 30 I=1,N G(I)=-F(I) DO 30 K=1,N 30 G(I)=G(I)+B(I,K)*C(K,J)+D(I,K)*C(K,J+1)+X(I,K)*C(K,J+2) CALL BAND(J) GO TO 10 14 IF (J-NJ) 15,20,20 cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc c Input your equations and jacobians for the central nodes c For the RDE this corresponds to the solition within the boundary layer cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc 15 s1 = Ds(1)/delta**2*(C(1,J+1)-2*C(1,J)+C(1,J-1))/dzeta**2+0.255D0* #sqrt(omega)**3*delta*ZETA(J)**2*(C(1,J+1)-C(1,J-1))/sqrt(vis)/dzet #a+Ds(1)*z(1)*fc*(C(1,J)*(C(7,J+1)-2*C(7,J)+C(7,J-1))/dzeta**2+(C(1 #,J+1)-C(1,J-1))/dzeta**2*(C(7,J+1)-C(7,J-1))/4)/R/Temp/delta**2-Ds #(1)/delta**2*(C(1,J)-CN(1,J))/DT0 f(1) = s1+Ds(6)/delta**2*(C(6,J+1)-2*C(6,J)+C(6,J-1))/dzeta**2+0.2 #55D0*sqrt(omega)**3*delta*ZETA(J)**2*(C(6,J+1)-C(6,J-1))/sqrt(vis) #/dzeta+Ds(6)*z(6)*fc*(C(6,J)*(C(7,J+1)-2*C(7,J)+C(7,J-1))/dzeta**2 #+(C(6,J+1)-C(6,J-1))/dzeta**2*(C(7,J+1)-C(7,J-1))/4)/R/Temp/delta* #*2-Ds(1)/delta**2*(C(6,J)-CN(6,J))/DT0 s1 = Ds(1)/delta**2*(C(1,J+1)-2*C(1,J)+C(1,J-1))/dzeta**2+0.255D0* #sqrt(omega)**3*delta*ZETA(J)**2*(C(1,J+1)-C(1,J-1))/sqrt(vis)/dzet #a+Ds(1)*z(1)*fc*(C(1,J)*(C(7,J+1)-2*C(7,J)+C(7,J-1))/dzeta**2+(C(1 #,J+1)-C(1,J-1))/dzeta**2*(C(7,J+1)-C(7,J-1))/4)/R/Temp/delta**2-Ds #(1)/delta**2*(C(1,J)-CN(1,J))/DT0 f(2) = s1+Ds(2)/delta**2*(C(2,J+1)-2*C(2,J)+C(2,J-1))/dzeta**2+0.2 #55D0*sqrt(omega)**3*delta*ZETA(J)**2*(C(2,J+1)-C(2,J-1))/sqrt(vis) #/dzeta+Ds(2)*z(2)*fc*(C(2,J)*(C(7,J+1)-2*C(7,J)+C(7,J-1))/dzeta**2 #+(C(2,J+1)-C(2,J-1))/dzeta**2*(C(7,J+1)-C(7,J-1))/4)/R/Temp/delta* #*2-Ds(1)/delta**2*(C(2,J)-CN(2,J))/DT0 f(3) = Ds(3)/delta**2*(C(3,J+1)-2*C(3,J)+C(3,J-1))/dzeta**2+0.255D #0*sqrt(omega)**3*delta*ZETA(J)**2*(C(3,J+1)-C(3,J-1))/sqrt(vis)/dz #eta+Ds(3)*z(3)*fc*(C(3,J)*(C(7,J+1)-2*C(7,J)+C(7,J-1))/dzeta**2+(C #(3,J+1)-C(3,J-1))/dzeta**2*(C(7,J+1)-C(7,J-1))/4)/R/Temp/delta**2- #Ds(1)/delta**2*(C(3,J)-CN(3,J))/DT0 f(4) = Ds(4)/delta**2*(C(4,J+1)-2*C(4,J)+C(4,J-1))/dzeta**2+0.255D #0*sqrt(omega)**3*delta*ZETA(J)**2*(C(4,J+1)-C(4,J-1))/sqrt(vis)/dz 186

#eta+Ds(4)*z(4)*fc*(C(4,J)*(C(7,J+1)-2*C(7,J)+C(7,J-1))/dzeta**2+(C #(4,J+1)-C(4,J-1))/dzeta**2*(C(7,J+1)-C(7,J-1))/4)/R/Temp/delta**2- #Ds(1)/delta**2*(C(4,J)-CN(4,J))/DT0 f(5) = Ds(5)/delta**2*(C(5,J+1)-2*C(5,J)+C(5,J-1))/dzeta**2+0.255D #0*sqrt(omega)**3*delta*ZETA(J)**2*(C(5,J+1)-C(5,J-1))/sqrt(vis)/dz #eta+Ds(5)*z(5)*fc*(C(5,J)*(C(7,J+1)-2*C(7,J)+C(7,J-1))/dzeta**2+(C #(5,J+1)-C(5,J-1))/dzeta**2*(C(7,J+1)-C(7,J-1))/4)/R/Temp/delta**2- #Ds(1)/delta**2*(C(5,J)-CN(5,J))/DT0 f(6) = Keq*C(1,J)-1000*C(2,J)*C(6,J) f(7) = z(1)*C(1,J)+z(2)*C(2,J)+z(3)*C(3,J)+z(4)*C(4,J)+z(5)*C(5,J) #+z(6)*C(6,J) f(8) = C(8,J)-C(8,J-1) f(9) = C(9,J)-C(9,J-1) f(10) = C(10,J)-C(10,J-1) f(11) = C(11,J)-C(11,J-1) b(1,1) = -2*Ds(1)/delta**2/dzeta**2+Ds(1)*z(1)*fc*(C(7,J+1)-2*C(7, #J)+C(7,J-1))/dzeta**2/R/Temp/delta**2-Ds(1)/delta**2/DT0 b(1,2) = 0 b(1,3) = 0 b(1,4) = 0 b(1,5) = 0 b(1,6) = -2*Ds(6)/delta**2/dzeta**2+Ds(6)*z(6)*fc*(C(7,J+1)-2*C(7, #J)+C(7,J-1))/dzeta**2/R/Temp/delta**2-Ds(1)/delta**2/DT0 b(1,7) = -2*Ds(1)*z(1)*fc*C(1,J)/dzeta**2/R/Temp/delta**2-2*Ds(6)* #z(6)*fc*C(6,J)/dzeta**2/R/Temp/delta**2 b(1,8) = 0 b(1,9) = 0 b(1,10) = 0 b(1,11) = 0 b(2,1) = -2*Ds(1)/delta**2/dzeta**2+Ds(1)*z(1)*fc*(C(7,J+1)-2*C(7, #J)+C(7,J-1))/dzeta**2/R/Temp/delta**2-Ds(1)/delta**2/DT0 b(2,2) = -2*Ds(2)/delta**2/dzeta**2+Ds(2)*z(2)*fc*(C(7,J+1)-2*C(7, #J)+C(7,J-1))/dzeta**2/R/Temp/delta**2-Ds(1)/delta**2/DT0 b(2,3) = 0 b(2,4) = 0 b(2,5) = 0 b(2,6) = 0 b(2,7) = -2*Ds(1)*z(1)*fc*C(1,J)/dzeta**2/R/Temp/delta**2-2*Ds(2)* #z(2)*fc*C(2,J)/dzeta**2/R/Temp/delta**2 b(2,8) = 0 b(2,9) = 0 b(2,10) = 0 b(2,11) = 0 b(3,1) = 0 187

b(3,2) = 0 b(3,3) = -2*Ds(3)/delta**2/dzeta**2+Ds(3)*z(3)*fc*(C(7,J+1)-2*C(7, #J)+C(7,J-1))/dzeta**2/R/Temp/delta**2-Ds(1)/delta**2/DT0 b(3,4) = 0 b(3,5) = 0 b(3,6) = 0 b(3,7) = -2*Ds(3)*z(3)*fc*C(3,J)/dzeta**2/R/Temp/delta**2 b(3,8) = 0 b(3,9) = 0 b(3,10) = 0 b(3,11) = 0 b(4,1) = 0 b(4,2) = 0 b(4,3) = 0 b(4,4) = -2*Ds(4)/delta**2/dzeta**2+Ds(4)*z(4)*fc*(C(7,J+1)-2*C(7, #J)+C(7,J-1))/dzeta**2/R/Temp/delta**2-Ds(1)/delta**2/DT0 b(4,5) = 0 b(4,6) = 0 b(4,7) = -2*Ds(4)*z(4)*fc*C(4,J)/dzeta**2/R/Temp/delta**2 b(4,8) = 0 b(4,9) = 0 b(4,10) = 0 b(4,11) = 0 b(5,1) = 0 b(5,2) = 0 b(5,3) = 0 b(5,4) = 0 b(5,5) = -2*Ds(5)/delta**2/dzeta**2+Ds(5)*z(5)*fc*(C(7,J+1)-2*C(7, #J)+C(7,J-1))/dzeta**2/R/Temp/delta**2-Ds(1)/delta**2/DT0 b(5,6) = 0 b(5,7) = -2*Ds(5)*z(5)*fc*C(5,J)/dzeta**2/R/Temp/delta**2 b(5,8) = 0 b(5,9) = 0 b(5,10) = 0 b(5,11) = 0 b(6,1) = Keq b(6,2) = -1000*C(6,J) b(6,3) = 0 b(6,4) = 0 b(6,5) = 0 b(6,6) = -1000*C(2,J) b(6,7) = 0 b(6,8) = 0 b(6,9) = 0 188 b(6,10) = 0 b(6,11) = 0 b(7,1) = z(1) b(7,2) = z(2) b(7,3) = z(3) b(7,4) = z(4) b(7,5) = z(5) b(7,6) = z(6) b(7,7) = 0 b(7,8) = 0 b(7,9) = 0 b(7,10) = 0 b(7,11) = 0 b(8,1) = 0 b(8,2) = 0 b(8,3) = 0 b(8,4) = 0 b(8,5) = 0 b(8,6) = 0 b(8,7) = 0 b(8,8) = 1 b(8,9) = 0 b(8,10) = 0 b(8,11) = 0 b(9,1) = 0 b(9,2) = 0 b(9,3) = 0 b(9,4) = 0 b(9,5) = 0 b(9,6) = 0 b(9,7) = 0 b(9,8) = 0 b(9,9) = 1 b(9,10) = 0 b(9,11) = 0 b(10,1) = 0 b(10,2) = 0 b(10,3) = 0 b(10,4) = 0 b(10,5) = 0 b(10,6) = 0 b(10,7) = 0 b(10,8) = 0 b(10,9) = 0 189

b(10,10) = 1 b(10,11) = 0 b(11,1) = 0 b(11,2) = 0 b(11,3) = 0 b(11,4) = 0 b(11,5) = 0 b(11,6) = 0 b(11,7) = 0 b(11,8) = 0 b(11,9) = 0 b(11,10) = 0 b(11,11) = 1 a(1,1) = Ds(1)/delta**2/dzeta**2-0.255D0*sqrt(omega)**3*delta*ZETA #(J)**2/sqrt(vis)/dzeta-Ds(1)*z(1)*fc/dzeta**2*(C(7,J+1)-C(7,J-1))/ #R/Temp/delta**2/4 a(1,2) = 0 a(1,3) = 0 a(1,4) = 0 a(1,5) = 0 a(1,6) = Ds(6)/delta**2/dzeta**2-0.255D0*sqrt(omega)**3*delta*ZETA #(J)**2/sqrt(vis)/dzeta-Ds(6)*z(6)*fc/dzeta**2*(C(7,J+1)-C(7,J-1))/ #R/Temp/delta**2/4 a(1,7) = Ds(1)*z(1)*fc*(C(1,J)/dzeta**2-(C(1,J+1)-C(1,J-1))/dzeta* #*2/4)/R/Temp/delta**2+Ds(6)*z(6)*fc*(C(6,J)/dzeta**2-(C(6,J+1)-C(6 #,J-1))/dzeta**2/4)/R/Temp/delta**2 a(1,8) = 0 a(1,9) = 0 a(1,10) = 0 a(1,11) = 0 a(2,1) = Ds(1)/delta**2/dzeta**2-0.255D0*sqrt(omega)**3*delta*ZETA #(J)**2/sqrt(vis)/dzeta-Ds(1)*z(1)*fc/dzeta**2*(C(7,J+1)-C(7,J-1))/ #R/Temp/delta**2/4 a(2,2) = Ds(2)/delta**2/dzeta**2-0.255D0*sqrt(omega)**3*delta*ZETA #(J)**2/sqrt(vis)/dzeta-Ds(2)*z(2)*fc/dzeta**2*(C(7,J+1)-C(7,J-1))/ #R/Temp/delta**2/4 a(2,3) = 0 a(2,4) = 0 a(2,5) = 0 a(2,6) = 0 a(2,7) = Ds(1)*z(1)*fc*(C(1,J)/dzeta**2-(C(1,J+1)-C(1,J-1))/dzeta* #*2/4)/R/Temp/delta**2+Ds(2)*z(2)*fc*(C(2,J)/dzeta**2-(C(2,J+1)-C(2 #,J-1))/dzeta**2/4)/R/Temp/delta**2 a(2,8) = 0 190

a(2,9) = 0 a(2,10) = 0 a(2,11) = 0 a(3,1) = 0 a(3,2) = 0 a(3,3) = Ds(3)/delta**2/dzeta**2-0.255D0*sqrt(omega)**3*delta*ZETA #(J)**2/sqrt(vis)/dzeta-Ds(3)*z(3)*fc/dzeta**2*(C(7,J+1)-C(7,J-1))/ #R/Temp/delta**2/4 a(3,4) = 0 a(3,5) = 0 a(3,6) = 0 a(3,7) = Ds(3)*z(3)*fc*(C(3,J)/dzeta**2-(C(3,J+1)-C(3,J-1))/dzeta* #*2/4)/R/Temp/delta**2 a(3,8) = 0 a(3,9) = 0 a(3,10) = 0 a(3,11) = 0 a(4,1) = 0 a(4,2) = 0 a(4,3) = 0 a(4,4) = Ds(4)/delta**2/dzeta**2-0.255D0*sqrt(omega)**3*delta*ZETA #(J)**2/sqrt(vis)/dzeta-Ds(4)*z(4)*fc/dzeta**2*(C(7,J+1)-C(7,J-1))/ #R/Temp/delta**2/4 a(4,5) = 0 a(4,6) = 0 a(4,7) = Ds(4)*z(4)*fc*(C(4,J)/dzeta**2-(C(4,J+1)-C(4,J-1))/dzeta* #*2/4)/R/Temp/delta**2 a(4,8) = 0 a(4,9) = 0 a(4,10) = 0 a(4,11) = 0 a(5,1) = 0 a(5,2) = 0 a(5,3) = 0 a(5,4) = 0 a(5,5) = Ds(5)/delta**2/dzeta**2-0.255D0*sqrt(omega)**3*delta*ZETA #(J)**2/sqrt(vis)/dzeta-Ds(5)*z(5)*fc/dzeta**2*(C(7,J+1)-C(7,J-1))/ #R/Temp/delta**2/4 a(5,6) = 0 a(5,7) = Ds(5)*z(5)*fc*(C(5,J)/dzeta**2-(C(5,J+1)-C(5,J-1))/dzeta* #*2/4)/R/Temp/delta**2 a(5,8) = 0 a(5,9) = 0 a(5,10) = 0 191 a(5,11) = 0 a(6,1) = 0 a(6,2) = 0 a(6,3) = 0 a(6,4) = 0 a(6,5) = 0 a(6,6) = 0 a(6,7) = 0 a(6,8) = 0 a(6,9) = 0 a(6,10) = 0 a(6,11) = 0 a(7,1) = 0 a(7,2) = 0 a(7,3) = 0 a(7,4) = 0 a(7,5) = 0 a(7,6) = 0 a(7,7) = 0 a(7,8) = 0 a(7,9) = 0 a(7,10) = 0 a(7,11) = 0 a(8,1) = 0 a(8,2) = 0 a(8,3) = 0 a(8,4) = 0 a(8,5) = 0 a(8,6) = 0 a(8,7) = 0 a(8,8) = -1 a(8,9) = 0 a(8,10) = 0 a(8,11) = 0 a(9,1) = 0 a(9,2) = 0 a(9,3) = 0 a(9,4) = 0 a(9,5) = 0 a(9,6) = 0 a(9,7) = 0 a(9,8) = 0 a(9,9) = -1 a(9,10) = 0 192

a(9,11) = 0 a(10,1) = 0 a(10,2) = 0 a(10,3) = 0 a(10,4) = 0 a(10,5) = 0 a(10,6) = 0 a(10,7) = 0 a(10,8) = 0 a(10,9) = 0 a(10,10) = -1 a(10,11) = 0 a(11,1) = 0 a(11,2) = 0 a(11,3) = 0 a(11,4) = 0 a(11,5) = 0 a(11,6) = 0 a(11,7) = 0 a(11,8) = 0 a(11,9) = 0 a(11,10) = 0 a(11,11) = -1 d(1,1) = Ds(1)/delta**2/dzeta**2+0.255D0*sqrt(omega)**3*delta*ZETA #(J)**2/sqrt(vis)/dzeta+Ds(1)*z(1)*fc/dzeta**2*(C(7,J+1)-C(7,J-1))/ #R/Temp/delta**2/4 d(1,2) = 0 d(1,3) = 0 d(1,4) = 0 d(1,5) = 0 d(1,6) = Ds(6)/delta**2/dzeta**2+0.255D0*sqrt(omega)**3*delta*ZETA #(J)**2/sqrt(vis)/dzeta+Ds(6)*z(6)*fc/dzeta**2*(C(7,J+1)-C(7,J-1))/ #R/Temp/delta**2/4 d(1,7) = Ds(1)*z(1)*fc*(C(1,J)/dzeta**2+(C(1,J+1)-C(1,J-1))/dzeta* #*2/4)/R/Temp/delta**2+Ds(6)*z(6)*fc*(C(6,J)/dzeta**2+(C(6,J+1)-C(6 #,J-1))/dzeta**2/4)/R/Temp/delta**2 d(1,8) = 0 d(1,9) = 0 d(1,10) = 0 d(1,11) = 0 d(2,1) = Ds(1)/delta**2/dzeta**2+0.255D0*sqrt(omega)**3*delta*ZETA #(J)**2/sqrt(vis)/dzeta+Ds(1)*z(1)*fc/dzeta**2*(C(7,J+1)-C(7,J-1))/ #R/Temp/delta**2/4 d(2,2) = Ds(2)/delta**2/dzeta**2+0.255D0*sqrt(omega)**3*delta*ZETA 193

#(J)**2/sqrt(vis)/dzeta+Ds(2)*z(2)*fc/dzeta**2*(C(7,J+1)-C(7,J-1))/ #R/Temp/delta**2/4 d(2,3) = 0 d(2,4) = 0 d(2,5) = 0 d(2,6) = 0 d(2,7) = Ds(1)*z(1)*fc*(C(1,J)/dzeta**2+(C(1,J+1)-C(1,J-1))/dzeta* #*2/4)/R/Temp/delta**2+Ds(2)*z(2)*fc*(C(2,J)/dzeta**2+(C(2,J+1)-C(2 #,J-1))/dzeta**2/4)/R/Temp/delta**2 d(2,8) = 0 d(2,9) = 0 d(2,10) = 0 d(2,11) = 0 d(3,1) = 0 d(3,2) = 0 d(3,3) = Ds(3)/delta**2/dzeta**2+0.255D0*sqrt(omega)**3*delta*ZETA #(J)**2/sqrt(vis)/dzeta+Ds(3)*z(3)*fc/dzeta**2*(C(7,J+1)-C(7,J-1))/ #R/Temp/delta**2/4 d(3,4) = 0 d(3,5) = 0 d(3,6) = 0 d(3,7) = Ds(3)*z(3)*fc*(C(3,J)/dzeta**2+(C(3,J+1)-C(3,J-1))/dzeta* #*2/4)/R/Temp/delta**2 d(3,8) = 0 d(3,9) = 0 d(3,10) = 0 d(3,11) = 0 d(4,1) = 0 d(4,2) = 0 d(4,3) = 0 d(4,4) = Ds(4)/delta**2/dzeta**2+0.255D0*sqrt(omega)**3*delta*ZETA #(J)**2/sqrt(vis)/dzeta+Ds(4)*z(4)*fc/dzeta**2*(C(7,J+1)-C(7,J-1))/ #R/Temp/delta**2/4 d(4,5) = 0 d(4,6) = 0 d(4,7) = Ds(4)*z(4)*fc*(C(4,J)/dzeta**2+(C(4,J+1)-C(4,J-1))/dzeta* #*2/4)/R/Temp/delta**2 d(4,8) = 0 d(4,9) = 0 d(4,10) = 0 d(4,11) = 0 d(5,1) = 0 d(5,2) = 0 d(5,3) = 0 194

d(5,4) = 0 d(5,5) = Ds(5)/delta**2/dzeta**2+0.255D0*sqrt(omega)**3*delta*ZETA #(J)**2/sqrt(vis)/dzeta+Ds(5)*z(5)*fc/dzeta**2*(C(7,J+1)-C(7,J-1))/ #R/Temp/delta**2/4 d(5,6) = 0 d(5,7) = Ds(5)*z(5)*fc*(C(5,J)/dzeta**2+(C(5,J+1)-C(5,J-1))/dzeta* #*2/4)/R/Temp/delta**2 d(5,8) = 0 d(5,9) = 0 d(5,10) = 0 d(5,11) = 0 d(6,1) = 0 d(6,2) = 0 d(6,3) = 0 d(6,4) = 0 d(6,5) = 0 d(6,6) = 0 d(6,7) = 0 d(6,8) = 0 d(6,9) = 0 d(6,10) = 0 d(6,11) = 0 d(7,1) = 0 d(7,2) = 0 d(7,3) = 0 d(7,4) = 0 d(7,5) = 0 d(7,6) = 0 d(7,7) = 0 d(7,8) = 0 d(7,9) = 0 d(7,10) = 0 d(7,11) = 0 d(8,1) = 0 d(8,2) = 0 d(8,3) = 0 d(8,4) = 0 d(8,5) = 0 d(8,6) = 0 d(8,7) = 0 d(8,8) = 0 d(8,9) = 0 d(8,10) = 0 d(8,11) = 0 195

d(9,1) = 0 d(9,2) = 0 d(9,3) = 0 d(9,4) = 0 d(9,5) = 0 d(9,6) = 0 d(9,7) = 0 d(9,8) = 0 d(9,9) = 0 d(9,10) = 0 d(9,11) = 0 d(10,1) = 0 d(10,2) = 0 d(10,3) = 0 d(10,4) = 0 d(10,5) = 0 d(10,6) = 0 d(10,7) = 0 d(10,8) = 0 d(10,9) = 0 d(10,10) = 0 d(10,11) = 0 d(11,1) = 0 d(11,2) = 0 d(11,3) = 0 d(11,4) = 0 d(11,5) = 0 d(11,6) = 0 d(11,7) = 0 d(11,8) = 0 d(11,9) = 0 d(11,10) = 0 d(11,11) = 0 DO 31 I=1,N G(I)=-F(I) DO 31 K=1,N 31 G(I)=G(I)+A(I,K)*C(K,J-1)+B(I,K)*C(K,J)+D(I,K)*C(K,J+1) CALL BAND (J) GO TO 10 C THIS IS ZETA=1.0 c Input your equations and jacobians for the right boundary, corresponds to the bulk 20 F(1)=C(1,J)+(0.5000000000D0 * (1000*cb(1)) - 0.9976311575D-5 - #0.5000000000D0 * (1000*cb(2)) + 0.2500000000D-13 * #sqrt(0.4000000000D27 * (1000*cb(1)) ** 2 + 0.1 196

#596209852D23 * (1000*cb(1)) - 0.8000000000D27 * (1000*cb(1)) * #(1000*cb(2)) + 0.1592428 #682D18 + 0.1596209852D23 * (1000*cb(2)) + 0.4000000000D27 * #(1000*cb(2)) ** 2))/1000 #-cb(1) F(2)=C(2,J)-cb(2)+cb(1)- #(0.5000000000D0 * (1000*cb(1)) - 0.9976311575D-5 - #0.5000000000D0 * (1000*cb(2)) + 0.2500000000D-13 * #sqrt(0.4000000000D27 * (1000*cb(1)) ** 2 + 0.1 #596209852D23 * (1000*cb(1)) - 0.8000000000D27 * (1000*cb(1)) * #(1000*cb(2)) + 0.1592428 #682D18 + 0.1596209852D23 * (1000*cb(2)) + 0.4000000000D27 * #(1000*cb(2)) ** 2))/1000 F(3)=C(3,J)-cb(2) F(4)=C(4,J)-cb(1) F(5)=-Ds(1)/delta**2*(C(5,J)-CN(5,J))/DT0 F(6)=c(6,j)- #(0.5000000000D0 * (1000*cb(1)) - 0.9976311575D-5 - #0.5000000000D0 * (1000*cb(2)) + 0.2500000000D-13 * #sqrt(0.4000000000D27 * (1000*cb(1)) ** 2 + 0.1 #596209852D23 * (1000*cb(1)) - 0.8000000000D27 * (1000*cb(1)) * #(1000*cb(2)) + 0.1592428 #682D18 + 0.1596209852D23 * (1000*cb(2)) + 0.4000000000D27 * #(1000*cb(2)) ** 2))/1000 F(7)=C(7,J) f(8) = C(8,J)-C(8,J-1) f(9) = C(9,J)-C(9,J-1) f(10) = C(10,J)-C(10,J-1) f(11) = C(11,J)-C(11,J-1) B(1,1)=1 B(2,2)=1 B(3,3)=1 B(4,4)=1 b(6,6) = -1 B(7,7)=1 B(5,5)=-Ds(1)/delta**2/DT0 b(8,1) = 0 b(8,2) = 0 b(8,3) = 0 b(8,4) = 0 b(8,5) = 0 b(8,6) = 0 b(8,7) = 0 b(8,8) = 1 197 b(8,9) = 0 b(8,10) = 0 b(8,11) = 0 b(9,1) = 0 b(9,2) = 0 b(9,3) = 0 b(9,4) = 0 b(9,5) = 0 b(9,6) = 0 b(9,7) = 0 b(9,8) = 0 b(9,9) = 1 b(9,10) = 0 b(9,11) = 0 b(10,1) = 0 b(10,2) = 0 b(10,3) = 0 b(10,4) = 0 b(10,5) = 0 b(10,6) = 0 b(10,7) = 0 b(10,8) = 0 b(10,9) = 0 b(10,10) = 1 b(10,11) = 0 b(11,1) = 0 b(11,2) = 0 b(11,3) = 0 b(11,4) = 0 b(11,5) = 0 b(11,6) = 0 b(11,7) = 0 b(11,8) = 0 b(11,9) = 0 b(11,10) = 0 b(11,11) = 1 a(8,1) = 0 a(8,2) = 0 a(8,3) = 0 a(8,4) = 0 a(8,5) = 0 a(8,6) = 0 a(8,7) = 0 a(8,8) = -1 198

a(8,9) = 0 a(8,10) = 0 a(8,11) = 0 a(9,1) = 0 a(9,2) = 0 a(9,3) = 0 a(9,4) = 0 a(9,5) = 0 a(9,6) = 0 a(9,7) = 0 a(9,8) = 0 a(9,9) = -1 a(9,10) = 0 a(9,11) = 0 a(10,1) = 0 a(10,2) = 0 a(10,3) = 0 a(10,4) = 0 a(10,5) = 0 a(10,6) = 0 a(10,7) = 0 a(10,8) = 0 a(10,9) = 0 a(10,10) = -1 a(10,11) = 0 a(11,1) = 0 a(11,2) = 0 a(11,3) = 0 a(11,4) = 0 a(11,5) = 0 a(11,6) = 0 a(11,7) = 0 a(11,8) = 0 a(11,9) = 0 a(11,10) = 0 a(11,11) = -1 DO 32 I=1,N G(I)=-F(I) DO 32 K=1,N 32 G(I)=G(I)+Y(I,K)*C(K,J-2)+A(I,K)*C(K,J-1)+B(I,K)*C(K,J) CALL BAND(J) DO 201 I=1,N RES=0.D0 DO 202 J=1,NJ 199

202 RES=RES+(C(I,J)- CG(I,J))**2 IF(RES.GT.1E-12)GO TO 441 201 CONTINUE GO TO 22 441 IF (JCOUNT-500) 8,8,40 22 PRINT 109 PRINT 113,JCOUNT-1 ccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc c------Calculate current density ------ccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc

iapp=1000* #(0.4D-4*EASA/EGA*sr+EASA/EGA*nn(1)*fc*K1f*C(1,1)*C(2,1)*C(11,1 #)*exp(alpha1*fc*(iniV+sr*TIME-C(7,1)-Ujref)/R/Temp)+EASA/EGA*nn(2) #*fc*K2*C(8,1)**2+EASA/EGA*nn(3)*fc*K4f*C(8,1)*C(2,1)*exp(alpha4*fc #*(iniV+sr*TIME-C(7,1)-Ujref4)/R/Temp)+EASA/EGA*nn(4)*fc*K5f*C(9,1) #*C(2,1)*exp(alpha5*fc*(iniV+sr*TIME-C(7,1)-Ujref5)/R/Temp))

ic=1000*(0.4D-4*EASA/EGA*sr) ir1=1000*(EASA/EGA*nn(1)*fc*(K1f*C(1,1)*C(2,1)*C(11, #1)*exp(alpha1*fc*(iniV+sr*TIME-C(7,1)-Ujref)/R/Temp))) ir2=1000*(EASA/EGA*n #n(2)*fc*K2*C(8,1)**2) ir3=1000*(EASA/EGA*nn(3)*fc*(K4f*C(8,1)*C(2,1)*exp(alph #a4*fc*(iniV+sr*TIME-C(7,1)-Ujref4)/R/Temp))) ir4=1000*(EASA/EGA*nn(4)*fc*(K5 #f*C(9,1)*C(2,1)*exp(alpha5*fc*(iniV+sr*TIME-C(7,1)-Ujref5)/R/Temp) #))

ccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc c Writing variables in the short and long output ccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc c------c----OUTPUT FILE FOR PARAMETERS OPTIMIZATION------WRITE(21,'(2x,f9.6,11x,f20.14)') Vd,iapp WRITE(6,'(2x,f9.6,4x,f10.7,4x,f10.7,4x,f9.7,4x,f9.7 1 )')Vd,C(11,1),C(8,1),C(9,1),C(10,1)

WRITE(9,'(2x,f9.6,4x,f20.14,4x,f12.10,4x,f12.10,4x,f12.10, #4x,f12.10,4x,f12.10,4x,f12.10,4x,f12.10,4x,f12.10,4x,f12.10, #4x,f12.10,4x,f12.10)') 1Vd,TIME,ic,ir1,ir2,ir3,ir4,C(2,1),C(1,1),C(8,1),C(9,1), 200

1C(10,1) c--- Concentration file ------DO J=1,NJ WRITE(8,'(2x,f9.6,9x,E12.6,5x,E12.6,4x,E12.6,4x,E12.6 #,4x,E12.6,5x,E12.6,5x,E12.6)') 1 zeta(J), c(1,J),c(2,j),c(3,j),c(4,j),c(5,j),c(6,j),c(7,j)

END DO CONTINUE

GO TO 28 40 WRITE(16,943) I,RES,JCOUNT-1 PRINT 108

28 CONTINUE

close (unit=20) PRINT 717 STOP END ccccccccccccccccccccccccccccccccccccccccccccccccccccccccc c Band(J) subroutine c c The dimensions of A, B, C, D, X, Y and G c need to be consistent with the main program c E is not a common variable but its dimensions need c to be consistent with the maximum number of nodes c and equations used in the program. c c C(i,j): dependent variables c A(i,i): Jacobian c B(i,i): Jacobian c D(i,2*i+1): Jacobian. c X(i,i): Jacobian c Y(i,i): Jacobian c F(i): Equation c G(i): Equation used in band(j) c E(i,2*i+1,j) c c Important: Do not change anything here unless c you have modified the dimensions of the variables c described above. 201 ccccccccccccccccccccccccccccccccccccccccccccccccccccccccc SUBROUTINE BAND(J) parameter (nne=11,nm=10001) IMPLICIT REAL*8(A-H,O-Z) c DIMENSION A(3,3),B(3,3),C(3,1001),D(3,7), c 1 X(3,3),Y(3,3),G(3),E(3,7,1001) DIMENSION A(nne,nne),B(nne,nne),C(nne,nm),D(nne,nne*2+1), #X(nne,nne), Y(nne,nne), G(nne),E(nne,nne*2+1,nm) COMMON/A1/ A,B,C,D COMMON/A2/ G,X,Y,N,NJ 101 FORMAT (15H0DETERM=0 AT J=,I4) IF (J-2) 1,6,8 1 NP1= N+1 DO 2 I=1,N D(I,2*N+1)=G(I) DO 2 L=1,N LPN= L + N 2 D(I,LPN)= X(I,L) CALL LUDEC (N,2*N+1,DETERM) IF (DETERM) 4,3,4 3 PRINT 101, J 4 DO 5 K=1,N E(K,NP1,1)= D(K,2*N+1) DO 5 L=1,N E(K,L,1)= - D(K,L) LPN = L+N 5 X(K,L) = -D(K,LPN) RETURN 6 DO 7 I=1,N DO 7 K=1,N DO 7 L=1,N 7 D(I,K)= D(I,K) + A(I,L)*X(L,K) 8 IF (J-NJ) 11,9,9 9 DO 10 I=1,N DO 10 L=1,N G(I)= G(I) - Y(I,L)*E(L,NP1,J-2) DO 10 M=1,N 10 A(I,L)= A(I,L) + Y(I,M)*E(M,L,J-2) 11 DO 12 I=1,N D(I,NP1)= -G(I) DO 12 L=1,N D(I,NP1)= D(I,NP1) + A(I,L)*E(L,NP1,J-1) DO 12 K=1,N 12 B(I,K)= B(I,K) + A(I,L)*E(L,K,J-1) 202

CALL LUDEC (N,NP1,DETERM) IF (DETERM) 14,13,14 13 PRINT 101, J 14 DO 15 K=1,N DO 15 M=1,NP1 15 E(K,M,J)= - D(K,M) IF (J-NJ) 20,16,16 16 DO 17 K=1,N 17 C(K,J)=E(K,NP1,J) DO 18 JJ=2,NJ M=NJ - JJ + 1 DO 18 K=1,N C(K,M)=E(K,NP1,M) DO 18 L=1,N 18 C(K,M)=C(K,M)+E(K,L,M)*C(L,M+1) DO 19 L=1,N DO 19 K=1,N 19 C(K,1)=C(K,1)+X(K,L)*C(L,3) 20 RETURN END ccccccccccccccccccccccccccccccccccccccccccccccccccccccccc c Ludec(N,M,Determ) subroutine c The dimensions of A, B, C, and D c need to be consistent with the main program c C(i,j): dependent variables c A(i,i): Jacobian c B(i,i): Jacobian c D(i,2*i+1): Jacobian. c X(i,2*i+1): Internal variable used in this subroutine c jcol(i): Internal variable used in this subroutine c Important: Do not change anything here unless c you have modified the dimensions of the variables c described above. ccccccccccccccccccccccccccccccccccccccccccccccccccccccccc SUBROUTINE LUDEC(N,M,DETERM) parameter (nne=11,nm=10001) IMPLICIT REAL*8(A-H,O-Z) c DIMENSION A(3,3),B(3,3),C(3,1001),D(3,7), c 1 X(3,3),Y(3,3),G(3),E(3,7,1001) DIMENSION A(nne,nne),B(nne,nne),C(nne,nm),D(nne,nne*2+1), #X(nne,nne*2+1),JCOL(nne) c IMPLICIT REAL*8(A-H,O-Z) c DIMENSION A(3,3),B(3,3),C(3,1001),D(3,7),X(3,7),JCOL(3) COMMON/A1/ A,B,C,D 203

NM1=N-1 DETERM=1.0 DO 1 I=1,N JCOL(I)=I DO 1 K=1,M 1 X(I,K)=D(I,K) DO 6 II=1,NM1 IP1=II+1 BMAX=DABS(B(II,II)) JC=II DO 2 J=IP1,N IF(DABS(B(II,J)).LE.BMAX)GOTO2 JC=J BMAX=DABS(B(II,J)) 2 CONTINUE DETERM=DETERM*B(II,JC) IF(DETERM.EQ.0.0) RETURN IF(JC.EQ.II) GO TO 4 JS=JCOL(JC) JCOL(JC)=JCOL(II) JCOL(II)=JS DO 3 I=1,N SAVE=B(I,JC) B(I,JC)=B(I,II) 3 B(I,II)=SAVE DETERM=-DETERM 4 DO 6 I=IP1,N F=B(I,II)/B(II,II) DO 5 J=IP1,N 5 B(I,J)=B(I,J)-F*B(II,J) DO 6 K=1,M 6 X(I,K)=X(I,K)-F*X(II,K) DETERM=DETERM*B(N,N) IF(DETERM.EQ.0.0) RETURN DO 7 II=2,N IR=N-II+2 IM1=IR-1 JC=JCOL(IR) DO 7 K=1,M F=X(IR,K)/B(IR,IR) D(JC,K)=F DO 7 I=1,IM1 7 X(I,K)=X(I,K)-B(I,IR)*F JC=JCOL(1) 204

DO 8 K=1,M 8 D(JC,K)=X(1,K)/B(1,1) RETURN C CLOSE(16) END

PEST Code pcf * control data restart estimation 7 302 2 0 1 1 1 double point 1 0 0 5.0 2.0 0.3 0.03 10 3.0 3.0 0.001 0.1 50 0.000000000001 3 100 0.00001 3 1 1 1 * parameter groups Ujref relative 0.01 0.001 switch 2.0 parabolic K relative 0.01 1D-4 switch 2.0 parabolic * parameter data Ujref none relative -0.68 -0.74208 -0.630544 Ujref 1.0000 0.0000 1 Ujref4 none relative -0.21 -0.6 -0.15 Ujref 1.0000 0.0000 1 Ujref5 none relative -0.34 -0.6 -0.15 Ujref 1.0000 0.0000 1 K1f log factor 1.46E-4 1e-4 1E-2 K 1.0000 0.0000 1 K2 log factor 4.16E-10 3E-10 5.853398e-10 K 1.0000 0.0000 1 K4f log factor 2.87E-7 1e-7 1E-1 K 1.0000 0.0000 1 K5f log factor 2.077e-7 1E-7 1e-3 K 1.0000 0.0000 1 * observation groups obsgroup * observation data cd1 0.010084057 1 obsgroup cd2 0.011527062 1 obsgroup cd3 0.012596583 1 obsgroup cd4 0.012622048 1 obsgroup cd5 0.014328189 1 obsgroup cd6 0.015847588 1 obsgroup cd7 0.016467231 1 obsgroup cd8 0.017417917 1 obsgroup cd9 0.01867418 1 obsgroup cd10 0.019879513 1 obsgroup 205 cd11 0.020490668 1 obsgroup cd12 0.021670537 1 obsgroup cd13 0.022621223 1 obsgroup cd14 0.023911439 1 obsgroup cd15 0.025668509 1 obsgroup cd16 0.026525824 1 obsgroup cd17 0.028503589 1 obsgroup cd18 0.02903835 1 obsgroup cd19 0.030438913 1 obsgroup cd20 0.033087252 1 obsgroup cd21 0.032713768 1 obsgroup cd22 0.03387666 1 obsgroup cd23 0.03475944 1 obsgroup cd24 0.037195571 1 obsgroup cd25 0.038460323 1 obsgroup cd26 0.038697994 1 obsgroup cd27 0.039444961 1 obsgroup cd28 0.041787722 1 obsgroup cd29 0.043171309 1 obsgroup cd30 0.045242445 1 obsgroup cd31 0.046116736 1 obsgroup cd32 0.047483347 1 obsgroup cd33 0.048714145 1 obsgroup cd34 0.049435647 1 obsgroup cd35 0.051880267 1 obsgroup cd36 0.053603385 1 obsgroup cd37 0.054494653 1 obsgroup cd38 0.055623592 1 obsgroup cd39 0.05729578 1 obsgroup cd40 0.059799817 1 obsgroup cd41 0.062227461 1 obsgroup cd42 0.062346296 1 obsgroup cd43 0.064111855 1 obsgroup cd44 0.066047179 1 obsgroup cd45 0.067685414 1 obsgroup cd46 0.070596889 1 obsgroup cd47 0.071539086 1 obsgroup cd48 0.073941264 1 obsgroup cd49 0.076513208 1 obsgroup cd50 0.078109002 1 obsgroup cd51 0.080604551 1 obsgroup cd52 0.082548364 1 obsgroup cd53 0.085196702 1 obsgroup cd54 0.087607369 1 obsgroup 206 cd55 0.089729435 1 obsgroup cd56 0.092861604 1 obsgroup cd57 0.095535407 1 obsgroup cd58 0.096570975 1 obsgroup cd59 0.099686168 1 obsgroup cd60 0.102453342 1 obsgroup cd61 0.105381793 1 obsgroup cd62 0.108471521 1 obsgroup cd63 0.110126732 1 obsgroup cd64 0.113776686 1 obsgroup cd65 0.117893494 1 obsgroup cd66 0.121602865 1 obsgroup cd67 0.124972705 1 obsgroup cd68 0.126941983 1 obsgroup cd69 0.130702283 1 obsgroup cd70 0.135769777 1 obsgroup cd71 0.138749157 1 obsgroup cd72 0.142517946 1 obsgroup cd73 0.146099994 1 obsgroup cd74 0.150106454 1 obsgroup cd75 0.155139994 1 obsgroup cd76 0.158594718 1 obsgroup cd77 0.162940709 1 obsgroup cd78 0.169001329 1 obsgroup cd79 0.173635921 1 obsgroup cd80 0.178270513 1 obsgroup cd81 0.182565574 1 obsgroup cd82 0.189093049 1 obsgroup cd83 0.193931359 1 obsgroup cd84 0.198761181 1 obsgroup cd85 0.204473783 1 obsgroup cd86 0.210678703 1 obsgroup cd87 0.217189202 1 obsgroup cd88 0.22198507 1 obsgroup cd89 0.227646742 1 obsgroup cd90 0.234343982 1 obsgroup cd91 0.241261917 1 obsgroup cd92 0.247101843 1 obsgroup cd93 0.253052115 1 obsgroup cd94 0.26119236 1 obsgroup cd95 0.268060214 1 obsgroup cd96 0.274820267 1 obsgroup cd97 0.283216858 1 obsgroup cd98 0.289999829 1 obsgroup 207 cd99 0.297104506 1 obsgroup cd100 0.305972195 1 obsgroup cd101 0.31300557 1 obsgroup cd102 0.321130536 1 obsgroup cd103 0.329417628 1 obsgroup cd104 0.337058763 1 obsgroup cd105 0.345917964 1 obsgroup cd106 0.354892605 1 obsgroup cd107 0.363356252 1 obsgroup cd108 0.372403044 1 obsgroup cd109 0.381553392 1 obsgroup cd110 0.391909074 1 obsgroup cd111 0.399999238 1 obsgroup cd112 0.40905961 1 obsgroup cd113 0.419629196 1 obsgroup cd114 0.428890741 1 obsgroup cd115 0.438022415 1 obsgroup cd116 0.447809383 1 obsgroup cd117 0.459099622 1 obsgroup cd118 0.471281978 1 obsgroup cd119 0.479530873 1 obsgroup cd120 0.490806682 1 obsgroup cd121 0.502501812 1 obsgroup cd122 0.513369336 1 obsgroup cd123 0.525848781 1 obsgroup cd124 0.53227185 1 obsgroup cd125 0.540383235 1 obsgroup cd126 0.556701073 1 obsgroup cd127 0.564985618 1 obsgroup cd128 0.575320079 1 obsgroup cd129 0.588425958 1 obsgroup cd130 0.600324806 1 obsgroup cd131 0.611476687 1 obsgroup cd132 0.622844169 1 obsgroup cd133 0.634731983 1 obsgroup cd134 0.646339683 1 obsgroup cd135 0.657938046 1 obsgroup cd136 0.671082972 1 obsgroup cd137 0.682584569 1 obsgroup cd138 0.694569997 1 obsgroup cd139 0.707780282 1 obsgroup cd140 0.718966116 1 obsgroup cd141 0.731978709 1 obsgroup cd142 0.745873317 1 obsgroup 208 cd143 0.757501135 1 obsgroup cd144 0.769569663 1 obsgroup cd145 0.783019325 1 obsgroup cd146 0.795065945 1 obsgroup cd147 0.807858013 1 obsgroup cd148 0.820852102 1 obsgroup cd149 0.833241147 1 obsgroup cd150 0.84585632 1 obsgroup cd151 0.859362845 1 obsgroup cd152 0.872279436 1 obsgroup cd153 0.885666276 1 obsgroup cd154 0.897737436 1 obsgroup cd155 0.910201602 1 obsgroup cd156 0.924260713 1 obsgroup cd157 0.937588985 1 obsgroup cd158 0.950281485 1 obsgroup cd159 0.962399331 1 obsgroup cd160 0.977328489 1 obsgroup cd161 0.987865819 1 obsgroup cd162 0.997311559 1 obsgroup cd163 1.014826242 1 obsgroup cd164 1.024284714 1 obsgroup cd165 1.037804821 1 obsgroup cd166 1.050385276 1 obsgroup cd167 1.064768639 1 obsgroup cd168 1.076930623 1 obsgroup cd169 1.093570166 1 obsgroup cd170 1.104874836 1 obsgroup cd171 1.119628287 1 obsgroup cd172 1.132604295 1 obsgroup cd173 1.144011673 1 obsgroup cd174 1.156214401 1 obsgroup cd175 1.16974639 1 obsgroup cd176 1.181574786 1 obsgroup cd177 1.193995662 1 obsgroup cd178 1.208297537 1 obsgroup cd179 1.219062353 1 obsgroup cd180 1.231383917 1 obsgroup cd181 1.245224879 1 obsgroup cd182 1.255590747 1 obsgroup cd183 1.268497152 1 obsgroup cd184 1.28059123 1 obsgroup cd185 1.291426498 1 obsgroup cd186 1.303245557 1 obsgroup 209 cd187 1.316504224 1 obsgroup cd188 1.327089089 1 obsgroup cd189 1.338064414 1 obsgroup cd190 1.350830763 1 obsgroup cd191 1.361025167 1 obsgroup cd192 1.371117713 1 obsgroup cd193 1.383714296 1 obsgroup cd194 1.392703367 1 obsgroup cd195 1.403576833 1 obsgroup cd196 1.415782956 1 obsgroup cd197 1.423541229 1 obsgroup cd198 1.434032723 1 obsgroup cd199 1.445067465 1 obsgroup cd200 1.453759447 1 obsgroup cd201 1.463444556 1 obsgroup cd202 1.473825703 1 obsgroup cd203 1.481770717 1 obsgroup cd204 1.491617103 1 obsgroup cd205 1.501853949 1 obsgroup cd206 1.508729443 1 obsgroup cd207 1.517412936 1 obsgroup cd208 1.527284787 1 obsgroup cd209 1.533404825 1 obsgroup cd210 1.541502629 1 obsgroup cd211 1.551255643 1 obsgroup cd212 1.555567681 1 obsgroup cd213 1.563716414 1 obsgroup cd214 1.572510256 1 obsgroup cd215 1.577000547 1 obsgroup cd216 1.584529637 1 obsgroup cd217 1.591863497 1 obsgroup cd218 1.595131478 1 obsgroup cd219 1.602312549 1 obsgroup cd220 1.608967348 1 obsgroup cd221 1.613177527 1 obsgroup cd222 1.618593039 1 obsgroup cd223 1.624543312 1 obsgroup cd224 1.626937002 1 obsgroup cd225 1.631936589 1 obsgroup cd226 1.637988721 1 obsgroup cd227 1.63892243 1 obsgroup cd228 1.643837135 1 obsgroup cd229 1.648149173 1 obsgroup cd230 1.648386844 1 obsgroup 210 cd231 1.652936554 1 obsgroup cd232 1.656900573 1 obsgroup cd233 1.656323371 1 obsgroup cd234 1.659990301 1 obsgroup cd235 1.662052949 1 obsgroup cd236 1.662086902 1 obsgroup cd237 1.664845587 1 obsgroup cd238 1.666356498 1 obsgroup cd239 1.665584066 1 obsgroup cd240 1.666381963 1 obsgroup cd241 1.667638226 1 obsgroup cd242 1.666492311 1 obsgroup cd243 1.667171372 1 obsgroup cd244 1.667595785 1 obsgroup cd245 1.663614789 1 obsgroup cd246 1.66390339 1 obsgroup cd247 1.663810019 1 obsgroup cd248 1.659727164 1 obsgroup cd249 1.659582864 1 obsgroup cd250 1.656985455 1 obsgroup cd251 1.653759915 1 obsgroup cd252 1.653327014 1 obsgroup cd253 1.650407051 1 obsgroup cd254 1.645925248 1 obsgroup cd255 1.644550149 1 obsgroup cd256 1.642012158 1 obsgroup cd257 1.636910712 1 obsgroup cd258 1.634143538 1 obsgroup cd259 1.631070786 1 obsgroup cd260 1.625324232 1 obsgroup cd261 1.622268457 1 obsgroup cd262 1.618431762 1 obsgroup cd263 1.611318597 1 obsgroup cd264 1.608492005 1 obsgroup cd265 1.602957657 1 obsgroup cd266 1.596133093 1 obsgroup cd267 1.593026389 1 obsgroup cd268 1.587237393 1 obsgroup cd269 1.580446782 1 obsgroup cd270 1.57659311 1 obsgroup cd271 1.570320284 1 obsgroup cd272 1.563555137 1 obsgroup cd273 1.559251588 1 obsgroup cd274 1.553708752 1 obsgroup 211 cd275 1.545347812 1 obsgroup cd276 1.540509502 1 obsgroup cd277 1.533684938 1 obsgroup cd278 1.523906458 1 obsgroup cd279 1.520205575 1 obsgroup cd280 1.513465894 1 obsgroup cd281 1.505164372 1 obsgroup cd282 1.499315958 1 obsgroup cd283 1.491837798 1 obsgroup cd284 1.483739995 1 obsgroup cd285 1.478256576 1 obsgroup cd286 1.470472838 1 obsgroup cd287 1.462553289 1 obsgroup cd288 1.456976499 1 obsgroup cd289 1.449702057 1 obsgroup cd290 1.440356479 1 obsgroup cd291 1.434185511 1 obsgroup cd292 1.42719967 1 obsgroup cd293 1.417387238 1 obsgroup cd294 1.411835913 1 obsgroup cd295 1.403907875 1 obsgroup cd296 1.394587762 1 obsgroup cd297 1.389367479 1 obsgroup cd298 1.380582127 1 obsgroup cd299 1.371992004 1 obsgroup cd300 1.366347309 1 obsgroup cd301 1.35783358 1 obsgroup cd302 1.299183074 0 obsgroup * model command line MOD5_SCHEME_3>nul * model input/output DataL.tpl DataL.dat short.ins short.dat * prior information

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