June 2001

NASA/TP2001–210770

Effects of Gas-Phase Radiation 

and Detailed Kinetics on the 

Burning and Extinction of a Solid Fuel

Jennifer L. Rhatigan

Lyndon B. Johnson Space Center
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June 2001

NASA/TP2001–210770

Effects of Gas-Phase Radiation 

and Detailed Kinetics on the 

Burning and Extinction of a Solid Fuel

Jennifer L. Rhatigan

Lyndon B. Johnson Space Center

National Aeronautics and

Space Administration

Johnson Space Center

Houston, Texas 77058-3696
Acknowledgments

This document constitutes the dissertation I submitted in partial fulfillment of the requirements for the 

degree of Doctor of Philosophy at Case Western Reserve University. As such, I would like to express my 

gratitude to the following individuals and organizations that have shared in this undertaking, making it 

both possible and enjoyable. While I carried the fire, others sustained it:

Dr. Jackie Sung, who served as thesis advisor. 

Dr. James T'ien, who served as thesis co-advisor

The NASA Glenn Research Center management, who sponsored the fellowship and discretionary funding 

which enabled me to pursue this research; in particular, Dr. Thomas L. Labus, who sponsored, advised, 

and encouraged me throughout this effort.

The Johnson Space Center and International Space Station Program management, who continued to 

support this effort upon my transfer to Houston; in particular, Mr. Rick Nygren and Mr. Bill Stephenson 

provided invaluable support.

My friends and colleagues at NASA, both at the Glenn Research Center and the Johnson Space Center

for all their help and encouragement.

Kathy Weisskopf, for her editorial assistance and support.

My husband, Stephen Frick. This work is for you, in the hope that it might keep you safe when you 

journey into space.

Available from:

NASA Center for AeroSpace Information National Technical Information Service

7121 Standard 5285 Port Royal Road

Hanover, MD 21076-1320 Springfield, VA 22161

This report is also available in electronic form at http://techreports.larc.nasa.gov/cgi-bin/NTRS
iii

Contents

Section Page

Acknowledgments.............................................................................................................................. i

1.0 Introduction............................................................................................................................. 1

1.1 Motivation ................................................................................................................... 1

1.2 Organization of the Report .......................................................................................... 2

2.0 Background ............................................................................................................................. 3

3.0 Formulation for a Computational Study of Solid Trioxane Combustion in 

Stagnation-Point Flows ............................................................................................... 6

3.1 Background.................................................................................................................. 6

3.2 Nomenclature............................................................................................................... 6

3.3 Governing Equations and Boundary Conditions......................................................... 9

3.4 Flame Radiation Model ............................................................................................... 14

4.0 Kinetics and Properties of the Solid Fuel................................................................................ 18

4.1 The Solid Fuel ............................................................................................................. 18

4.2 Kinetics of Trioxane .................................................................................................... 19

4.3 Kinetics Model and Validation.................................................................................... 19

4.4 Thermophysical Properties.......................................................................................... 27

4.5 Radiative Properties..................................................................................................... 28

5.0 Numerical Method .................................................................................................................. 29

6.0 Discussion of Results.............................................................................................................. 33

6.1 The Adiabatic Solution................................................................................................ 33

6.2 Adiabatic Flame Sensitivity and Reaction Path Analysis............................................ 42

6.3 Effects of Surface Radiation........................................................................................ 58

6.4 Effects of Gas-Phase and Surface Radiation ............................................................... 70

6.5 Sensitivity and Reaction Path Analysis for Flame With Gas Phase and Surface Radiation. 89

6.6 Conclusions................................................................................................................. 103

7.0 Surface Radical Interactions in Solid Trioxane Combustion.................................................. 105

7.1 Overview ..................................................................................................................... 105

7.2 Nomenclature............................................................................................................... 105

7.3 Background.................................................................................................................. 105

7.4 Formulation of Surface Radical Interactions in Solid Trioxane Combustion.............. 107

7.5 Numerical Method....................................................................................................... 110

7.6 Discussion of Results................................................................................................... 110

7.6.1 The Surface Modeled as a Radical Sink...................................................................... 110

7.6.2 Surface Radical Interaction Coupled With Surface Emission ..................................... 113

7.7 Conclusions................................................................................................................. 113

8.0 Summary ................................................................................................................................. 114

Appendix 1: CHEMKIN input file................................................................................................... 116

Appendix 2: Transport Property Estimates for Trioxane ................................................................. 118

Appendix 3: Latent Heat of Solid Fuel ............................................................................................ 121

Appendix 4: Surface and Gas-Phase Radiation Effects on the Burning and 

Extinction of a Solid Fuel............................................................................................ 124

A4.1 Background.................................................................................................................. 124
iv

A4.2 Nomenclature............................................................................................................... 125

A4.3 Combustion Model ...................................................................................................... 126

A4.4 Discussion of Results................................................................................................... 128

A4.4.1 Variation of the Flow Parameter (stretch rate) ............................................................ 128

A4.4.2 Parametric Variation of the Surface Radiation Properties........................................... 128

A4.4.3 Radiative Heat Fluxes.................................................................................................. 129

A4.4.4 Flame Standoff Distance and Solid Burning Rate....................................................... 130

A4.4.5 Comparison to Identical Systems With No Gas-Phase Radiation ............................... 131

A4.4.6 Flammability................................................................................................................ 132

A4.5 Conclusions ................................................................................................................. 132

A4.6 Further Studies............................................................................................................. 133

9.0 References............................................................................................................................... 140

List of Tables

Table Page

3-1 Boundary Condition Comparison at x=0 Between the Opposed Jet Formulation and the 

Stagnation-Point Formulation ............................................................................................... 14

3-2 Boundary Condition Comparison at x=l Between the Opposed Jet Formulation and the 

Stagnation-Point Formulation ............................................................................................... 14

4-1 Physical Properties of Trioxane............................................................................................. 18

4-2 Reaction Mechanism for Solid Trioxane Combustion .......................................................... 22

4-3 Thermophysical Properties of Trioxane ................................................................................ 28

6-1 Summary of the Key Features of the Trioxane Extinction Limits......................................... 104

7-1 Example Mole Fractions of Radicals at the Inert Surface for Low, Middle, and 

High Stretch Rates.................................................................................................................. 109

7-2 Boundary Condition Comparison at x=0 Between the Formulation With the Inert Solid 

Surface and With Radical Interaction at the Surface (sink) .................................................. 109

7-3 Effect of the Surface Modeled as Radical Sink for Stretch Rate Near 5 s-1.......................... 111

7-4 Effect of the Surface Modeled as Radical Sink for the Blow-Off Extinction Limit ............. 112

A4-1 Sample Surface Species Mass Fractions (Yw) and Nondimensional 

Temperature (Θ) for Adiabatic Case (Case 6), Over Various Stretch Rates......................... 134

A4-2 Sample Surface Species Mass Fractions (Yw) and Nondimensional 

Temperature (Θ) for the Surface Radiation Case (Case 5), Over Various Stretch Rates...... 134

A4-3 Sample Surface Species Mass Fractions (Yw) and Nondimensional Temperature 

(Θ) for the Gas-Phase and Surface Radiation Case (Case 1), Over Various Stretch Rates... 134
v

List of Figures

Figure Page

Figure 3-1(a & b): Illustration of changes to geometry of problem. ................................................. 13

Figure 3-1(c): Control volume for energy balance on solid surface.................................................. 13

Figure 4-1: Comparison of kinetics mechanisms (Hochgreb vs. GRI 1.2) with experiments for

3560 ppm CH2O, equivalence ratio=0.51, and flow reactor temperature TFR=944 K........... 23

Figure 4-2: Comparison of kinetics mechanisms (Hochgreb vs. GRI 1.2) with Experiments 

for 3480 ppm CH2O, equivalence ratio=0.56, and flow reactor temperature TFR=945 K. .... 23

Figure 4-3: Comparison of kinetics mechanisms (Hochgreb vs. GRI 1.2) with experiments for 

5140 ppm CH2O, equivalence ratio=39.7, and flow reactor temperature TFR=994 K........... 24

Figure 4-4: Comparison of kinetics mechanisms (Hochgreb vs. GRI 1.2) with experiments for 

2080 ppm CH2O, equivalence ratio=0.013, and flow reactor temperature TFR=994 K......... 24

Figure 4-5: Comparison of kinetics mechanisms (Hochgreb vs. GRI 1.2) with experiments for 

2390 ppm CH2O, equivalence ratio=0.70, and flow reactor temperature TFR=994 K........... 25

Figure 4-6: Comparison of kinetics mechanisms (Hochgreb vs. GRI 1.2) with experiments for 

4600 ppm CH2O, equivalence ratio=39.7, and flow reactor temperature TFR=995 K........... 25

Figure 4-7: Comparison of kinetics mechanisms (Hochgreb vs. GRI 1.2) with experiments for 

2340 ppm CH2O, equivalence ratio=0.93, and flow reactor temperature TFR=1095 K......... 26

Figure 4-8: Comparison of kinetics mechanisms (Hochgreb vs. GRI 1.2) with experiments for 

5360 ppm CH2O, equivalence ratio=31.5, and flow reactor temperature TFR=1090 K......... 26

Figure 4-9: Comparison of kinetics mechanisms (Hochgreb vs. GRI 3.0) with experiments for 

5140 ppm CH2O, equivalence ratio=39.7, and flow reactor temperature TFR=994 K........... 27

Figure 6-1(a): Maximum flame temperature vs. stretch rate for the adiabatic flame (20 cm domain).... 35

Figure 6-1(b): Temperature profiles for the adiabatic flame............................................................. 35

Figure 6-2: Nondimensional burning rate for the adiabatic flame. ................................................... 36

Figure 6-3: Axial velocity profile for low stretch adiabatic flame. ................................................... 36

Figure 6-4: Axial velocity profile for mid-stretch rate adiabatic flame............................................. 37

Figure 6-5: Axial velocity profile for high stretch rate turning point adiabatic flame. ..................... 37

Figure 6-6: Major species profiles for low-stretch rate adiabatic flame............................................ 38

Figure 6-7: Major species profiles for middle stretch rate adiabatic flame....................................... 38

Figure 6-8: Major species profiles for high stretch rate (turning point) adiabatic flame................... 39

Figure 6-9: Species profiles of OH, H and O for low stretch rate adiabatic flame............................ 39

Figure 6-10: Species profiles of OH, H and O for middle stretch rate adiabatic flame. ................... 40

Figure 6-11: Species profiles for OH, H and O for high stretch rate (turning point) 

adiabatic flame. ..................................................................................................................... 40

Figure 6-12: Species profiles of HCO, HO2 , and H2O2 for low stretch rate adiabatic flame............ 41

Figure 6-13: Species profiles of HCO, HO2 , and H2O2 for middle stretch rate adiabatic flame. ...... 41

Figure 6-14: Species profiles of HCO, HO2 , and H2O2 for high stretch rate (turning point) 

adiabatic flame. ..................................................................................................................... 42

Figure 6-15: Adiabatic flame, species production rates for low stretch rate. .................................... 47

Figure 6-16: Adiabatic flame, species production rates for mid stretch rate..................................... 47

Figure 6-17: Adiabatic flame, species production rates at the blow-off extinction limit. ................. 48

Figure 6-18(a): Adiabatic flame, CH2O consumption rate of progress by reaction for low stretch rate. . 48

Figure 6-18(b): Adiabatic flame, HCO consumption rate of progress by reaction 

for low stretch rate................................................................................................................. 49

Figure 6-18(c): Adiabatic flame, O2 consumption rate of progress by reaction for low stretch rate........ 49

Figure 6-19(a): Adiabatic flame, CH2O consumption rate of progress by reaction 

for mid stretch rate. ............................................................................................................... 50
vi

Figure 6-19(b): Adiabatic flame, HCO consumption rate of progress by reaction 

for mid stretch rate. ............................................................................................................... 50

Figure 6-19(c): Adiabatic flame, O2 consumption rate of progress by reaction for 

mid stretch rate. ..................................................................................................................... 51

Figure 6-20(a): Adiabatic flame, CH2O consumption rate of progress by reaction at 

the blow-off extinction limit.................................................................................................. 51

Figure 6-20(b): Adiabatic flame, HCO consumption rate of progress by reaction at 

the blow-off extinction limit.................................................................................................. 52

Figure 6-20(c): Adiabatic flame, O2 consumption rate of progress by reaction at 

the blow-off extinction limit.................................................................................................. 52

Figure 6-20(d): Adiabatic flame, H2O2 consumption rate of progress by reaction at 

the blow-off extinction limit.................................................................................................. 53

Figure 6-20(e): Adiabatic flame, HO2 consumption rate of progress by reaction at 

the blow-off extinction limit.................................................................................................. 53

Figure 6-21(a): Adiabatic flame, total heat release for low stretch rate. ........................................... 54

Figure 6-21(b): Adiabatic flame, heat release by reaction for low stretch rate. ................................ 54

Figure 6-22(a): Adiabatic flame, total heat release for mid stretch rate............................................ 55

Figure 6-22(b): Adiabatic flame, heat release by reaction for mid stretch rate................................. 55

Figure 6-23(a): Adiabatic flame, total heat release at the blow-off extinction limit. ........................ 56

Figure 6-23(b): Adiabatic flame, heat release by reaction at the blow-off extinction limit...................... 56

Figure 6-24: Adiabatic flame, normalized Eext sensitivity by reaction at the blow-off extinction limit. .. 57

Figure 6-25: Main reaction path for adiabatic trioxane diffusion flame. .......................................... 57

Figure 6-26(a): Maximum flame temperature vs. stretch rate for flame with surface radiation. ............. 60

Figure 6-26(b): Temperature profiles for flame with surface radiation. ........................................... 61

Figure 6-27(a): Nondimensional burning rate vs. stretch rate for adiabatic flame and 

flame with surface radiation. ................................................................................................. 61

Figure 6-27(b): Dimensional burning rate vs. stretch rate for adiabatic flame and flame 

with surface radiation. ........................................................................................................... 62

Figure 6-28: Species profiles of O2, CO2, and H2O for radiative extinction turning point 

(flame with surface radiation). .............................................................................................. 62

Figure 6-29: Species profiles of C3H6O3, CO, CH2O, and H2 for radiative extinction 

turning point (flame with surface radiation).......................................................................... 63

Figure 6-30: Major species profiles for very low stretch rate (flame with surface radiation)........... 63

Figure 6-31: Major species profiles for low stretch rate (flame with surface radiation)................... 64

Figure 6-32: Major species profiles for mid-stretch rate (flame with surface radiation). ................. 64

Figure 6-33: Major species profiles for blow-off turning point (flame with surface radiation). .............. 65

Figure 6-34: Species profiles of H2O2, H, OH, and HO2 for radiative extinction turning 

point (flame with surface radiation). ..................................................................................... 65

Figure 6-35: Species profiles of H, O, and OH for very low stretch rate (flame with 

surface radiation)................................................................................................................... 66

Figure 6-36: Species profiles of H, O, and OH for low stretch rate (flame with surface radiation)......... 66

Figure 6-37: Species profiles of H, O, and OH for middle stretch rate (flame with surface radiation).... 67

Figure 6-38: Species profiles of H, O, and OH for blow-off turning point (flame with 

surface radiation)................................................................................................................... 67

Figure 6-39: Species profiles of HCO and O for radiative extinction (flame with surface radiation)...... 68

Figure 6-40: Species profiles of HO2, H2O2, and HCO for very low stretch rate (flame 

with surface radiation)........................................................................................................... 68

Figure 6-41: Species profiles of HO2, H2O2, and HCO for low stretch rate (flame with 

surface radiation)................................................................................................................... 69

Figure 6-42: Species profiles of HO2, H2O2, and HCO for middle stretch rate (flame with

surface radiation)................................................................................................................... 69
vii

Figure 6-43: Species profiles of HO2, H2O2, and HCO for blow-off extinction (flame with 

surface radiation)................................................................................................................... 70

Figure 6-44: Maximum flame temperature vs. stretch rate for adiabatic flame, flame 

with surface radiation, and flame with gas phase and surface radiation. .............................. 74

Figure 6-45: Temperature profiles for flame with gas phase and surface radiation.......................... 75

Figure 6-46(a): Nondimensional burning rate vs. stretch rate for adiabatic flame and 

flame with gas phase and surface radiation........................................................................... 75

Figure 6-46(b): Dimensional burning rate vs. stretch rate for adiabatic flame and 

flame with gas phase and surface radiation........................................................................... 76

Figure 6-47: Species profiles of C3H6O3, CO2, O2, and H2O for radiative extinction 

turning point (flame with gas phase and surface radiation)................................................... 76

Figure 6-48: Species profiles of CH2O, CO, and H2 for radiative extinction 

turning point (flame with gas phase and surface radiation)................................................... 77

Figure 6-49: Major species profiles for low stretch rate (flame with gas phase 

and surface radiation). ........................................................................................................... 77

Figure 6-50: Major species profiles for mid-stretch rate (flame with gas phase and surface radiation)... 78

Figure 6-51: Major species profiles at the blow-off extinction limit (flame with gas phase and 

surface radiation)................................................................................................................... 78

Figure 6-52: Species profiles of H2O2, and H for radiative extinction turning point (flame with 

gas phase and surface radiation)............................................................................................ 79

Figure 6-53: Species profiles of HCO, OH, O, and HO2 for radiative extinction turning point 

(flame with gas phase and surface radiation). ....................................................................... 79

Figure 6-54: Species profiles of H, O, and OH for low stretch rate (flame with gas phase and 

surface radiation)................................................................................................................... 80

Figure 6-55: Species profiles of H, O, and OH for mid-stretch rate (flame with gas phase and 

surface radiation)................................................................................................................... 80

Figure 6-56: Species profiles of H, O, and OH at the blow-off turning extinction limit (flame 

with gas phase and surface radiation).................................................................................... 81

Figure 6-57: Species profiles of HO2, H2O2, and HCO for low stretch rate (flame with gas phase 

and surface radiation). ........................................................................................................... 81

Figure 6-58: Species profiles of HO2, H2O2, and HCO for mid-stretch rate (flame with gas phase 

and surface radiation). ........................................................................................................... 82

Figure 6-59: Species profiles of HO2, H2O2, and HCO at the blow-off extinction limit (flame 

with gas phase and surface radiation).................................................................................... 82

Figure 6-60: Radiation flux for radiative extinction point (flame with gas phase and surface radiation). 83

Figure 6-61: dqnet/dx for radiative extinction point (flame with gas phase and surface radiation).... 83

Figure 6-62: Radiation flux for low stretch rate (flame with gas phase and surface radiation). ....... 84

Figure 6-63: dqnet/dx for low stretch rate (flame with gas phase and surface radiation). .................. 84

Figure 6-64: Radiation flux for mid-stretch rate (flame with gas phase and surface radiation)........ 85

Figure 6-65: dqnet/dx for mid-stretch rate (flame with gas phase and surface radiation)................... 85

Figure 6-66: Radiation flux at the blow-off extinction limit (flame with gas phase and surface radiation). 86

Figure 6-67: dqnet/dx at the blow-off extinction limit (flame with gas phase and surface radiation). 86

Figure 6-68: Heat flux breakdown vs. stretch rate for the flame with gas phase and surface radiation. 87

Figure 6-69(a): Ratio of net radiative heat flux to the solid to conductive heat flux to the solid vs. 

stretch rate. ............................................................................................................................ 87

Figure 6-69(b): Surface conduction vs. stretch rate. ......................................................................... 88

Figure 6-69(c): Ratio of net radiative heat loss to the total heat generation vs. stretch rate. ............ 88

Figure 6-70: Flame with gas phase and surface radiation, production rates by reaction for 

radiative extinction................................................................................................................ 92

Figure 6-71: Flame with gas phase and surface radiation, production rates by reaction for low stretch rate. 92
viii

Figure 6-72: Flame with gas phase and surface radiation, production rates by reaction for 

mid stretch rate. ..................................................................................................................... 93

Figure 6-73(a): Flame with gas phase and surface radiation, CH2O consumption rate of 

progress by reaction for radiative extinction. ........................................................................ 93

Figure 6-73(b): Flame with gas phase and surface radiation, HCO consumption rate of 

progress by reaction for radiative extinction. ........................................................................ 94

Figure 6-73(c): Flame with gas phase and surface radiation, O2 consumption rate of progress 

by reaction for radiative extinction........................................................................................ 94

Figure 6-73(d): Flame with gas phase and surface radiation, H2O2 consumption rate of 

progress by reaction for radiative extinction. ........................................................................ 95

Figure 6-73(e): Flame with gas phase and surface radiation, HO2 consumption rate of 

progress by reaction for radiative extinction. ........................................................................ 95

Figure 6-74(a): Flame with gas phase and surface radiation, CH2O consumption rate of

progress by reaction for low stretch rate. .............................................................................. 96

Figure 6-74(b): Flame with gas phase and surface radiation, HCO consumption rate of 

progress by reaction for low stretch rate. .............................................................................. 96

Figure 6-74(c): Flame with gas phase and surface radiation, O2 consumption rate of progress 

by reaction for low stretch rate.............................................................................................. 97

Figure 6-75(a): Flame with gas phase and surface radiation, CH2O consumption rate of progress by 

reaction for mid stretch rate................................................................................................... 97

Figure 6-75(b): Flame with gas phase and surface radiation, HCO consumption rate of progress by 

reaction for mid stretch rate................................................................................................... 98

Figure 6-75(c): Flame with gas phase and surface radiation, O2 consumption rate of progress 

by reaction for mid stretch rate.............................................................................................. 98

Figure 6-76(a): Flame with gas phase and surface radiation, total heat release for radiative extinction. . 99

Figure 6-76(b): Flame with gas phase and surface radiation, heat release by reaction for radiative 

extinction............................................................................................................................... 99

Figure 6-77(a): Flame with gas phase and surface radiation, total heat release for low stretch rate. 100

Figure 6-77(b): Flame with gas phase and surface radiation, heat release by reaction for low stretch rate. 100

Figure 6-78(a): Flame with gas phase and surface radiation, total heat release for mid stretch rate. 101

Figure 6-78(b): Flame with gas phase and surface radiation, heat release by reaction for mid stretch rate. 101

Figure 6-79: Flame with gas phase and surface radiation, normalized Eext sensitivity by reaction 

for radiative extinction. ......................................................................................................... 102

Figure 6-80: Reaction pathways for high-temperature path and chain-branching reaction paths 

for trioxane diffusion flame near low stretch extinction limit............................................... 103

Figure 7-1: Maximum flame temperature vs. stretch rate for the adiabatic (inert surface) trioxane 

flame and the surface sink for formaldehyde. ....................................................................... 112

Figure 7-2: Maximum flame temperature vs. stretch rate for an inert surface including surface 

radiation and a surface that absorbs radicals and radiates..................................................... 113

Figure A3-1: Illustration of enthalpy calculation. ............................................................................. 122

Figure A3-2: Comparison of specific heat data from references....................................................... 123

Figure A4-1: Illustration of geometry and radiation model............................................................... 135

Figure A4-2: Effect of stretch rate (a) on ratio of local radiative absorption length to flame thickness (Dk). 135

Figure A4-3: Nondimensional radiative fluxes vs. stretch rate. ........................................................ 136

Figure A4-4: Nondimensional maximum flame temperature vs. stretch rate.................................... 137

Figure A4-5: Nondimensional flame standoff distance vs. stretch rate............................................. 137

Figure A4-6: Nondimensional fuel burning rate vs. stretch rate. ...................................................... 138

Figure A4-7: Effect of gas-phase radiation on nondimensional maximum flame temperature......... 138

Figure A4-8: Effect of gas-phase radiation on nondimensional burning rate.................................... 139

Figure A4-9: Flammability map: ambient oxygen mass fraction vs. stretch rate.............................. 139
1

Abstract

This is the first attempt to analyze both radiation and detailed kinetics on the burning and extinction of a 

solid fuel in a stagnation-point diffusion flame. We present a detailed and comparatively accurate 

computational model of a solid fuel flame along with a quantitative study of the kinetics mechanism, 

radiation interactions, and the extinction limits of the flame.

A detailed kinetics model for the burning of solid trioxane (a trimer of formaldehyde) is coupled with a

narrowband radiation model, with carbon dioxide, carbon monoxide, and water vapor as the gas-phase 

participating media. The solution of the solid trioxane diffusion flame over the flammable regime is 

presented in some detail, as this is the first solution of a heterogeneous trioxane flame. We identify hightemperature and low-temperature reaction paths for the heterogeneous trioxane flame.

We then compare the adiabatic solution to solutions that include surface radiation only and gas-phase and 

surface radiation using a black surface model. The analysis includes discussion of detailed flame 

chemistry over the flammable regime and, in particular, at the low stretch extinction limit. We emphasize 

the low stretch regime of the radiatively participating flame, since this is the region representative of 

microgravity flames. When only surface radiation is included, two extinction limits exist (the blow-off 

limit, and the low stretch radiative limit), and the burning rate and maximum flame temperatures are 

lower, as expected. With the inclusion of surface and gas-phase radiation, results show that, while flame 

temperatures are lower, the burning rate of the trioxane diffusion flame may actually increase at low 

stretch rate due to radiative feedback from the flame to the surface.

The trioxane diffusion flame model is then extended to include surface absorption of radicals from the gas 

phase. We compare the results to the inert surface case and also include surface radiation both with and 

without surface radical interactions. Results indicate that, for the model studied, surface absorption of 

radicals from the gas phase is unimportant.

1.0 Introduction

This section introduces the scientific and engineering motivations for this study of solid fuel diffusion 

flames. It also provides an orientation to the organization of the text, and gives the reader a ‘roadmap’ to 

the major topics addressed in the study.

1.1 Motivation

This research has been prompted by several considerations. The immediate consideration in this pursuit

is spacecraft fire safety. An understanding of burning and extinction of solid fuel diffusion flames, 

particularly at the low stretch rates characteristic of the microgravity environment endemic in spacecraft, 

is necessary to design and build safer spacecraft. A more fundamentalbut certainly related

consideration is the exploration of the physics that lead to extinction of solid fuel diffusion flames. 

Evidence indicates that thermal radiation within a flame is an important low stretch rate extinction 

mechanism. Moreover, radiative interactions with a solid fuel surface are important as well. A final 

consideration is accuracy of the flame model. By contemporary standards, this includes detailed kinetics 

analyses. This is the first attempt to analyze both radiation and detailed kinetics on the burning and 

extinction of a solid fuel stagnation-point diffusion flame.
2

1.2 Organization of the Report

The focus of this thesis, beginning with Section 2 and concluding with Appendix 3, is a study of the 

effects of radiation and detailed kinetics on the burning and extinction of a solid fuel in a stagnation-point 

diffusion flame. Section 2 provides a background to orient the reader to the current state of the research 

in this area. Section 3 addresses the problem formulation of the solid trioxane stagnation-point diffusion 

flame to include accurate gas-phase radiation and detailed kinetics. The narrowband radiation model of 

Bedir [1998] is used along with surface radiation and emission, coupled with the detailed kinetics model 

of the fuel. Section 4 addresses the physical properties and kinetics of trioxane, and how the kinetics 

mechanism is validated. Section 5 covers the numerical methods used for the solutions.

Section 6 contains the results of the study of the solid trioxane burning in the opposed jet diffusion flame 

configuration with radiation and detailed kinetics. We examine global flame response of trioxane burning 

in air, including flame temperature, fuel burning rate, and flammability, and detailed flame structure for 

cases including surface radiation and a combination of gas-phase and surface radiation. We also examine 

detailed chemistry, reaction pathways, and sensitivity, with emphasis on the low stretch extinction limit.

Section 7 contains a study of the effects of surface radical interactions on the heat balance at the surface, 

since this has potential to affect the kinetics and heat balance of the solid diffusion flame.

Section 8 summarizes our findings from the foregoing sections and suggests potential future research 

directions with this problem.

During our early attempts to understand the interactions between gas-phase radiation and surface 

radiation in the burning of a solid fuel, we developed a relatively simple model and adapted it to facilitate 

parametric comparisons. Appendix 4 contains the parametric study of the effects of both gas-phase and 

surface radiation on the burning and extinction of a solid fuel in a stagnation-point diffusion flame. This 

early study helped establish the understanding of the radiative interactions in the solid flame and sparked 

our interest in pursuing the research that is the focus of the body of this work. Therefore, we refer the 

interested reader to Appendix 4 first, to obtain a fundamental understanding of the interactions between 

gas-phase and surface radiation in a solid fuel. In Appendix 4, we use a two-flux radiation solution with 

carbon dioxide and water vapor as gray participating media. The fuel modeled is PMMA 

(polymethylmethacrylate). The model assumes a one-step finite rate overall gas-phase chemical reaction. 

The results demonstrate that both gas-phase and solid surface radiation become increasingly important as 

flame stretch rate decreases. This early work was followed by a publication [Rhatigan et al. 1998] 

wherein a comparison is made between the results discussed in Appendix 4 and a model that includes a 

non-gray gas analysis.
3

2.0 Background

In this section, we introduce the concepts important to this study, review recent progress in the area, and 

establish relevance to the field of study.

Diffusion Flames: Flames are distinguished by the means with which the fuel and oxidizer come in 

contact for reaction. Diffusion flames are flames in which the course of the reaction is influenced or 

controlled by the flow field of the flame. That is, the reactants and oxidizer come into contact and burn 

by means of fluid transport mechanisms. The flame seeks the stochiometric zone of the mixture brought 

about by the flow field. Burning in a diffusion flame has a rate associated with itthe stretch (or strain) 

rate. Stretch rate is characterized by flow nonuniformity, flame curvature, and flame unsteadiness. 

Burning will cease at either too high a stretch rate (blow-off extinction) caused by insufficient residence 

time for reaction, or too low a stretch rate (radiative extinction), caused by insufficient temperature to 

maintain reaction due to radiative losses. 

Flame configuration: Diffusion flame configurations are distinguished by the orientation of the fuel to 

the oxidizer flow. Counterflow diffusion flames are those in which the fuel and oxidizer flows are 

geometrically opposed to each other and meet at a stagnation-point. This configuration—also known as 

the opposed jet diffusion flameis well studied because the similarity solution for the flow field is 

known, the geometry is simple and quasi-one-dimensional, and comparison between computational 

models and experiments are relatively straightforward. Oxidizer and fuel velocities control flame 

location. Moreover, the flame in this case is typically planar and purely strained.

We used a special case of the counterflow configuration to study burning of solid and liquid surfaces by 

directly impinging the oxidizer stream on a burning surface. Where the oxidizer stream impinges directly 

on a burning surface, it is commonly called the opposed-jet configuration. The flow field is characterized 

by the same counterflow similarity solution, with only a change in boundary condition (on the fuel side). 

The fuel flow is created from solid fuel vaporization from the surface and, in this case, the flow rate of 

fuel from the surface is an eigen-value solution for the given oxidizer velocity. 

Flow Parameter: The counterflow diffusion flame is further distinguished by a constant radial velocity 

gradient (constant stretch rate). For the free stream velocity of the oxidizer jet flowing in direction r

(radial direction parallel to the surface), Ue, the stretch rate, a, is defined by Ue=ar. The stretch rate has 

the units of inverse time, so that it also gives the characteristic time for the problem. It is a convenient 

term in modeling diffusion flames because a number of parametric quantities can be related to the stretch 

rate [Liñán 1974] and thus many modeling studies are posed using the stretch rate, rather than the 

velocity, as the flow parameter. 

Extinction Limits: The flammable regime of the stagnation-point diffusion flame is bounded by two 

extinction limits—a high stretch (blow-off) limit and a low stretch (quench) limit. Much of the research 

emphasis has been on extinction at the blow-off limit, caused by insufficient gas residence time at high 

stretch rates. The emphasis on the blow-off limit is well deserved—it has both fundamental and practical 

significance. It is also easily observed in a laboratory. The low stretch extinction limit has only been 

postulated until recently because the buoyant flows characteristic of earth’s gravity limit the observation 

of very low stretch diffusion flames. 

Importance of Flame and Surface Radiation: T’ien has led an effort to elucidate low stretch extinction 

due to radiative heat loss, beginning with his 1986 paper [T’ien 1986], in which he established the 

possibility that surface radiation can be a significant factor in the physics of low stretch diffusion flames 

and can indeed lead to extinction. He incorporated solid surface emission in the opposed jet diffusion 

flame model, showing that radiative losses at low stretch can be the predominant heat loss from the 

surface and cause flame quenching. 
4

Radiative emission from a hot flame is partly a heat loss and partly a heat feedback mechanism to the 

burning solid. Emission from the burning solid surface can be partially absorbed by the gaseous flame 

and the rest escapes to the ambient as a heat loss. Gas-phase absorption and emission are functions of the 

radiative (optical) thickness of the gas. The optical thickness of the gas varies with stretch rate, typically 

thickening in the low stretch regime due to a thicker flame, and resulting in increased global 

concentrations of the participating media. Radiation can become a predominant heat transfer mechanism 

at low stretch rate and thus become the controlling mechanism for flame extinction (quenching).

Progress in Understanding the Radiative Extinction Limit: T'ien and his co-workers and students 

have contributed to a growing body of work on the radiative extinction of diffusion flames, with emphasis 

on the low stretch regime found in a microgravity environment. The group has established how radiation 

originating from the hot fuel surface, the gaseous flame, or a combination of the two, affects the 

extinction limits of diffusion flames. Starting with inclusion of surface radiation only, Foutch and T’ien 

[1987] studied mixed convection (buoyant and forced) in the stagnation region of a solid cylinder. They 

demonstrated the low stretch extinction limit due to surface radiation loss. Lee and T’ien [1989] studied 

vertical burning of a solid surface PMMA fire, and included surface and gas-phase radiation. They used a 

two-flux model for the gas-phase radiation, and included CO2 and H2O, and soot in the radiative 

treatment. Chao et al. [1990] analytically demonstrated the existence of the low stretch extinction limit in

droplet combustion, considering radiative emission only from the gas phase (i.e., optically-thin 

treatment). Grayson et al. [1994] examined flame spread over a thin solid in a low-speed concurrent flow 

and included solid surface radiation loss. Jiang [1995] also examined flame spread over a thin solid and 

included gas-phase radiation in the model (two-dimensional gray gas model, with a discrete ordinates 

solution). The low stretch extinction mechanism was validated by experiment [Olson 1997] in the lower 

stagnation region of a large radius solid cylinder. Bedir [1998] improved the gas-phase radiation 

treatment by adapting the narrowband radiation model with discrete ordinate solution to the stagnationpoint diffusion flame. The importance of optical thickness in flames was recently demonstrated by Shih 

et al. [1999], who studied a very low pressure opposed jet H2/O2/CO2 diffusion flame using detailed 

kinetics and the narrowband radiation model. They found that the flame is unusually thick due to the 

combined effects of low pressure (slow chemistry), radiation, and low stretch rate. The effect of flame 

radiation was studied directly by varying the amount of CO2, which is a strongly participating gas. Frate 

et al. [2000] examined the opposed jet gaseous diffusion flame of dry CO and oxygen, using both the 

narrowband radiation model and detailed kinetics. The effect of suppressant addition on the flammability 

limits was examined and compared using CO2 and Ar, with the former as a radiatively participating 

species. A reversal in the trend of suppressant effectiveness was found at low stretch rates as compared to 

higher stretch rates.

Building on this body of work, this study takes up the problem of including both accurate radiation and 

flame chemistry models in the burning and extinction of a solid fuel in a stagnation-point diffusion flame. 

The study of the low-stretch solid fuel diffusion flame bears relevance to both the typical materials and 

ventilation-induced flow regime important in spacecraft fire safety.

Detailed Kinetics: In this study, we seek to establish the importance of accuracy in the kinetics portion 

of the model of stagnation-point diffusion flames that is similar to that of the radiation model. The 

solution of accurate, detailed kinetics of even the simplest fuel-oxidizer combinations has been an 

intractable problem until very recently. Comprehensive kinetic mechanisms of fuel oxidation are very 

difficult to obtain. Individual reactions cannot be measured but must be surmised from observation of 

constituent overall reactions. Determining rate constants associated with detailed reaction steps presents 

an additional challenge. The computational power (speed and memory) to perform the quantum theory 

analysis is still quite limited. Detailed mechanisms estimated from experimental data for the most simple 

fuels have only recently become available but are still under debate as to their rigor under extension to 

variation of flame phenomena, flow field, pressure, temperature, etc. In addition, solving the detailed 
5

kinetics problem is computationally complex due to numerical stiffness associated with the varying 

timescales of the flow field and detailed reactions. 

Initial attempts to solve the counterflow diffusion flame problem used single step, finite rate, global 

reactions to model reaction mechanisms. As these lacked any information on the formation and 

consumption of intermediate species, they were limited. During the 1980s and early 1990s, pursuit of 

‘reduced’ kinetics mechanisms gained popularity as both computer capabilities and understanding of 

some of the important intermediate reactions increased (e.g. [Dixon-Lewis et al. 1984, Bui-Pham and 

Seshardri 1990, Chen 1990, Maas and Pope 1992]). ‘Reduced’ or simplified mechanisms were developed 

from a set of detailed kinetics, with emphasis on only the most important intermediate species (i.e. those 

that control either the rate of the reaction or the formation of other significant species). The problems are 

solved numerically with the set of ‘reduced’ reactions, using the steady state assumption for those species 

deemed less important.

These studies mentioned above focused on gas-phase fuel and oxidizer combinations. Solid fuels, 

however, are more relevant to practical fire safety, as compared to gaseous and liquid fuels. The problem 

of the burning solid poses some kinetically unique considerations. Most studies have assumed one-step 

kinetics for the burning of a solid for several reasons. First, detailed oxidation mechanisms for only the 

most structurally simple fuels are available (e.g., H2, CO, CH2O, CH4) over a range of temperatures that 

typically does not include the condensed phase. Secondly, the phase change mechanism of the fuel must 

be understood (i.e., does the fuel liquefy and then burn as a liquid, or does it sublimate and burn from the 

gas phase?). Lastly, the kinetics mechanism of the solid burning must be known for an accurate estimate 

of the solid surface temperature.

We have found a fuel uniquely suited to this study. Trioxane is a solid fuel that sublimates readily and 

depolymerizes in the gas phase directly to formaldehyde. Detailed oxidation mechanisms for 

formaldehyde are available. Moreover, it is structurally similar to other polymers used in spaceflight 

hardware, such as Delrin and Celcon. All aspects of this fuel can be modeled in detail, so that 

experimental verification is feasible. Study of trioxane serves as both a first step toward understanding 

other more complex solid fuels, and a detailed validation of all the models associated with the solid 

burning modeling through the comparison with future experimental results. 
6

3.0 Formulation for a Computational Study of Solid Trioxane 

Combustion in Stagnation-Point Flows

3.1 Background

In this section and in Sections 4 through 7, we again examine a solid fuel burning in a stagnation-point 

configuration, but with a somewhat different emphasis. While initial studies in Appendix 4 used a 

parametric variation of radiative properties, here we use the most accurate and comprehensive models 

available for both the radiative heat transfer and detailed kinetics to examine the flammable regime of a 

solid fuel burning in a stagnation-point configuration.

The recent emphasis on modeling of diffusion flames with detailed kinetics is the result of the concurrent 

availability of convenient computational capabilities and detailed kinetics mechanisms for a variety of 

simple fuels. The study of detailed kinetics mechanisms for the combustion of fuels has focused 

primarily on simple gaseous fuels, such as H, CO, and CH4 (e.g. [Dixon-Lewis et al. 1984]), since the 

combustion of these fuels forms the mechanisms for the combustion of more complex fuels. Combustion 

of solids in diffusion flames with detailed kinetics has not received the same amount of attention, 

primarily due to the lack of detailed kinetics data for simple solid fuels.

While it has been shown the solid fuels are potentially complex in a radiative sense, comprehensive 

radiation modeling tools are now available that can model both the gas phase and surface radiative 

interactions. Excellent and contemporary reviews of gas-phase radiation studies in diffusion flames can 

be found in Bedir et al. [1996], T’ien and Bedir [1997], and Bedir [1998].

Opposed jet gaseous diffusion flames have been studied in models that include both radiation and detailed 

kinetics (e.g. [Daguse et al. 1996, Frate et al. 2000]). Previous models of the burning of a solid fuel have 

not included both gas-phase radiation and detailed kinetics. This is the first attempt to include accurate 

modeling of both phenomena in the same analysis for a solid fuel. While both simple and sophisticated 

gas-phase radiation models have been studied for solid combustion, they have relied on simplified, onestep overall reaction kinetics. We are interested in observing the detailed kinetics at the low stretch 

radiative extinction limit and also the effects that a more accurate kinetics model may have on the 

outcome of the radiation predictions. The two phenomena are indirectly coupled in the gas phase because 

changes in concentration of radiatively participating species (e.g. H20, CO2, CO) brought about by the 

kinetics affect the optical thickness of the gas, and thus the gas-phase radiative heat transfer.

Insight into the detailed kinetics serves several purposes. First, it allows us to judge the necessary 

accuracy required for future models. Is a single step or reduced model sufficient, or is a detailed kinetics 

analysis necessary for an accurate estimate of the importance of radiation in this type of study? Second, it 

allows us to study in detail the evolution of each participating species. We can know with accuracy the 

concentration of radiatively participating species throughout the flame, and thus better understand the gasphase radiation interaction with a solid fuel. Third, it allows us to evaluate the potential effects of radical 

interactions with the solid surface. Finally, the inclusion of detailed kinetics in this study establishes a 

benchmark for the study of the combustion of solids (or for that matter, liquids) where radiative 

interactions with a surface are potentially important, such as in a microgravity environment.

3.2 Nomenclature

The nomenclature listed below was developed to remain consistent with as much of the notation as 

possible used in the primary references [Kee et al. 1980, Lutz et al. 1996, and Bedir 1998], while 

maintaining some continuity with the earlier work in Appendix 4. This necessitated the use of more 

subscripting than might be desirable, but ensures the continuity of the development.
7

a radial velocity gradient of stagnation-point flow, U ar e = , 1/s

cp constant pressure specific heat of the gas mixture, J/mol-K

cp,f constant pressure specific heat of the gas phase fuel, J/mol-K

cp,s constant pressure specific heat of the solid fuel, J/mol-K

Dkj diffusion coefficient between species k and j, cm2/s

T Dki thermal diffusion coefficient, g/cm-s

F similarity variable related to mass flux, g/cm

2

-s (equation 3-1b)

G similarity variable related to radial component of velocity, g/cm3-s (eqn 3-1a)

E similarity variable related to radial pressure gradient, J/cm4 or g/cm3-s

2

 (Eqn 3-1c)

h enthalpy, J/mol

I total radiation intensity: radiation energy flux per unit solid angle, per unit area normal to the 

path, J/cm3-s

K total number of species

kfi, , kri forward/reverse rate constant of the ith reaction, units depend on reaction

L total latent heat of the fuel at vaporization temperature, J/g

l length of domain, cm

L

o

latent heat of the fuel at the reference temperature, J/g

m& mass flux (total), g/cm2-s

p pressure, atm

Qi heat release rate of reaction i, J/cm3-s

QT total heat release rate due to reaction, J/cm3-s

q radiation heat flux, J/cm2

-s

qa radiation heat flux absorbed by the surface, J/cm2-s

qe radiation heat flux emitted from the surface, J/cm2-s

qi radiation heat flux incident at the surface, J/cm2-s

qnet net radiation heat flux into the surface, J/cm2-s

qr radiation flux reflected from the solid surface, J/cm2-s

r radial coordinate or radius, cm

ri rate of progress of the ith reaction, mol/cm3-s

S direction of travel of radiation flux

T temperature, degrees Kelvin

Tmax maximum flame temperature, degrees Kelvin

u axial velocity, cm/s or pressure pathlength, atm-cm (depending on context)

Ue axial velocity at edge of boundary layer, cm/s

Vk diffusion velocity of species k, cm/s

v radial velocity, cm/s

Wk molecular weight of species k, g/mol

x axial coordinate, cm

Yk mass fraction of species k

Xk mole fraction of species k
8

Zrot rotational relaxation collision parameter (number of collisions needed to establish rotational 

equilibrium of a molecule)

ap polarizability

a total hemispherical absorptivity of the surface

γ spectral line half-width for narrowband model, cm

dn mean line spacing parameter for narrowband model, cm

e total hemispherical emissivity of the surface

k gray gas absorption coefficient, path length-1

kn spectrally dependent absorption coefficient, path length-1

kp Planck-mean absorption coefficient, (atm-cm)-1

l thermal conductivity, J/cm-s-K

m viscosity, kg/m-s, or direction cosine

n spectral frequency, cm-1

nki’ stochiometric coefficient of the kth reactant species in the ith reaction

nki’’ stochiometric coefficient of the kth product species in the ith reaction

r mass density, g/cm3

s Stefan-Boltzman constant, 5.6696x10-8 W/m2-K

4

t transmittance

W solid angle

w& reaction rate, mol/cm3-s

Subscripts:

b black-body value

e equivalent band parameter, or edge of the flame

F gas-phase mixture at fuel-side of domain (x=0)

g gas-phase of mixture

f gas-phase fuel species (trioxane)

i reaction index

k species index

o oxidizer side of domain (x=l)

s solid surface or solid phase of fuel (trioxane)

w mean wall quantity

n spectral quantity

-∞ reference to a distance within the surface such that the temperature decays exponentially to T-¥

∞ quantity at outer limit or edge of domain 

Superscripts:

O reference point at which a property is evaluated, such as latent heat

+ reference to positive x side of surface

- reference to negative x side of surface
9

3.3 Governing Equations and Boundary Conditions

Governing Equations: The governing equations are taken from Kee et al. [1988] and Lutz et al. [1996] 

for an opposed jet diffusion flame. These are expressed in terms of a fixed jet velocity, fixed jet 

separation distance formulation. The momentum and species governing equations from Kee et al. [1988] 

and Lutz et al. [1996] are repeated here for convenience. Figure 3-1 shows the flow configuration.

Conservation of mass in cylindrical coordinates is given by:

Adapting the von Karman similarity analysis, and recognizing that the similarity variables F(x) and G(x)

are functions of x only, the similarity functions are given by:

dx

dF x

r

G x

( )

( ) = − =

ru (3-1a)

The above equation is the continuity equation expressed by the similarity functions. F is defined by:

2

( )

u

F x

r

= (3-1b)

and the radial momentum equation is satisfied by the eigen-value

.

1

const

r

p

r

E =

∂

∂

= (3-1c)

In the limit of potential flow, the eigen-value is related to the stretch rate by [Kee et al. 1988, DixonLewis 1990, Sung et al. 1996]

2 E a = −rO

The axial momentum equation, expressed in terms of the similarity variables, is:

0

3

2

2

 =



























+ + 















−

r

m

r r

G

dx

d

dx

FG G d

dx

d

E (3-1d)

The species conservation equation for species k =1,2... K is given by:

− ( ) − = 0 k k k

k Y V W

dx

d

dx

dY

ru r w& (3-1e)

Since we are interested in the effects of flame radiation, the energy equation is modified to include the 

radiation from participating media in the flame. Adapting Bedir’s formulation [1998] for the energy 

equation to include gas-phase radiation, the energy equation is:

0

1 1

 + + =











 +











− ∑ ∑ = = dx

dq h

dx

dT Y V c

dx

dT

dx

d

dx

dT

uc net

K

k

k k

K

k

r p l r k k pk w& (3-1f)

( ) ( ) 0

1

=

∂

∂

+

∂

∂

r

r r

u

x

r ru
10

The first and third terms represent the convection; the second term represents the conduction; the fourth 

term is the combustion heat release; and the fifth term is the radiative source term. The radiative source 

term can be found by solving the radiative transfer equation [Bedir 1998], described in detail in 

Section 3.4.

The formulation is modified using the technique of Sung et al. [1996] to allow potential flow on the 

oxidizer side, and plug flow on the fuel side, to be consistent with the flow formulation of Appendix 4. 

As such, the stretch rate ahead of the flame on the oxidizer side is chosen as the single parameter to 

characterize the strained flow.

Boundary Conditions for the Solid Fuel Case: The boundary conditions in Kee et al. [1980] and Lutz 

et al. [1996] are formulated for an opposed jet configuration (Figure 3-1(a)). In this work, the boundary 

conditions are written for a jet impinging on a solid fuel surface, also called a stagnation-point 

configuration (Figure 3-1(b)). The fuel surface is modeled as impermeable to all species except fuel 

vapor. Radical interactions at the surface are neglected here, but are considered in Section 7. 

Heat Balance: Figure 3-1(c) shows the heat balance at the solid surface . The surface is modeled as a 

solid heated to a gas (by the flame), so that liquid formation (melting of the solid) and any resulting liquid 

motion is neglected. Performing the energy balance at the fuel surface (x=0), we use the no slip 

condition, such that v=0 at x=0, and

Conduction from gas to solid +net radiation into solid = Latent heat for phase change of solid to gas + 

Conduction into solid, or

g + x= + + net = + s − x=o−

dx

dT

q mL

dx

dT | | l 0

& l

The total latent heat is given by (see Appendix 3):

∫ ∫

= + −

Ts

T

p s

T

T

p f

o

o

s

o

L L c , dT c , dT

where L

o

is the latent heat at the reference temperature, To, at 298K. The specific heat of the solid is 

assumed approximately constant, and the latent heat, L, is evaluated at the surface temperature. The fuel 

is assumed to be thick enough such that the temperature profile into the solid is an exponential decay from 

Tsto T-¥ (infinite slab assumption). The heat flux at the surface on the fuel side is:

( ) = − = T −T−∞ mc

dx

dT

s x 0 p,s s l | &

The total mass flux at x=0 is expressed by:

m& = r F uF = 2F

where the subscript F refers to the fuel side location of the mass balance. Combining the expressions 

above, we have the following boundary condition on axial velocity at x=0:

( )









 + + −

+

= =

∫ −∞

= +





















Ts

T

o

p f p s

o

net

x

g

F F

o

L c dT c T T

q

dx

dT

F u

, ,

0

2

l

r

(3-2a)

Now the equation is somewhat simplified if the ambient temperature is the reference temperature, such 

that T

o

 = T-¥. 
11











 +

+

= =

∫

= +





















Ts

T

p f

o

net

x

g

F F

o

L c dT

q

dx

dT

F u

,

0

2

l

r

(3-2b)

The term in the denominator can be evaluated once for a particular fuel and surface temperature. These 

are addressed for trioxane in Appendix 3.

The net radiation flux into the surface is the difference between the flux the surface absorbs (incident less 

reflected) and the flux the surface emits:

qnet = qa − qe

The absorbed radiation flux into the surface is expressed as the difference between the incident flux and 

the reflected flux, and related to the surface absorptivity by the following expression:

( ) qa = qi − qr = qi − 1−a qi = aqi

Assuming a gray surface, the emitted flux is given by:

4

qe Ts = es

Then the net radiative flux to the solid surface is expressed as: 

4

net i Ts

q = aq − es (3-2c)

where radiation is incident from the flame. In this case, the incident radiation flux, qi, is composed of the 

flame radiation flux as well as flux from the background radiation. The incident radiative flux at the 

surface from the flame is found by solving the radiation transfer equation (Section 3.4). If the flame 

radiation is excluded and the gas is assumed to be transparent, the net radiative flux to the solid surface is 

given by the expression for radiation transfer between two infinite parallel planes at temperatures Ts and 

T¥, where the background is at T¥: 

4 4

= − T + ∞ T∞qnet e ss se s (3-2d)

This development is general, so that other surface properties can be modeled in the future. For our 

purposes specific to the trioxane flame model, the surface and background are modeled as perfect (blackbody) radiators, with absorptivity and emissivity equal to 1.0. This is not intended to be a physical 

representation of the trioxane fuel surface, but an example case for modeling purposes. As pointed out in 

Appendix 4, this case is the most radiatively complex surface condition. The computational results reflect 

a physically realistic, albeit extreme, contribution of radiative heat transfer to this physical situation and 

therefore give an indication of the importance radiation can have. Appendix 4 focused on the full range 

of possible solid surface radiative conditions. The surface radiative condition here is identical to that of 

Case 1 in Appendix 4; however, since the surface temperature is relatively low in comparison to all cases 

in Appendix 4 (thus reducing emission), behaviors of Case 2 (no emission) from Appendix 4 can be 

evident as well. Hence, comparison of the adiabatic and radiative trioxane flames can illustrate whether 

the trioxane surface exhibits the characteristics of Case 1 or Case 2. This will be demonstrated in 

Section 6. Furthermore, transmission of radiation through the solid surface is neglected.

Mass Balance. Now turning to the mass balance at the surface, we start with the general expression for 

the mass balance:
12

= = +∑

k

m r F uF r f u f rkuk

&

The boundary condition at the surface on the fuel species equation is derived from the fact that at the 

surface (x=0), the uk=0 so that: 

m& = r fu f = r FuF = 2F

Also,

f F Vf

u = u +

r f = Yf r F

where Vf is the diffusion velocity of the fuel species. Combining,

( ) f f F f F f F F m& = r u = r Y u +V = r u

then solving for Yf:

( ) F f

F

f

u V

u

Y

+

= (3-3)

Now for all other species at x=0:

rkuk = 0

since uk = 0 . Then from the velocity definition

uk = uF +Vk = 0

so that the boundary condition on non-fuel species at the surface is: 

uF = −Vk(3-4a)

The diffusion velocity is dependent upon the mass fraction, so the diffusion velocity expression can be 

discretized (using the mixture averaged formulation) and solved implicitly for Yk. 

Alternatively:

YkuF = −YkVk(3-4b)

Equation 3-4(b) is a convenient form to solve Yk in the OPPDIF program (discussed in Section 5), since 

the “diffusion velocity” is calculated as a product YkVk. It is the form of the equation used in this work. 

The formulation of equation 3-4(b) is more general because it can be used with either the mixtureaveraged diffusion velocity calculation, or the multicomponent diffusion velocity calculation.

Table 3.1 contains a comparison between the boundary conditions of the opposed flow formulation 

(OPPDIF) and the solid fuel stagnation-point flow formulation (OPPDIFSOLID) at x=0. Note that the 

stagnation-point boundary conditions can be implemented with or without flame and/or surface radiation 

effects.

At x=l, the oxidizer side, the boundary conditions are unchanged from those of Kee et al. [1988] except 

the potential flow approximation for G. This is summarized in Table 3.2.
13

 

Figure 3-1(a & b): Illustration of changes to geometry of problem.

(a) Two gas jets impinging; (b) gas-jet impinging on solid surface.

Figure 3-1(c): Control volume for energy balance on solid surface.
14

Table 3-1: Boundary Condition Comparison at x=0 Between the 

Opposed Jet Formulation and the Stagnation-Point Formulation

Dependent 

Variable

Opposed Flow Boundary Condition 

at x=0

Stagnation-Point Boundary 

Condition at x=0

F

2

F F r u











 +

+

∫

= +





















Ts

T

p f

o

net

x

g

o

L c dT

q

dx

dT

,

0

2

l

G 0 0

T Ts Ts

Yk ( )−∞

r F uFYk + r FVkYk = r F uFYk

( )

F k

F f

F

f

other u V

u V

u

fuel Y

= −

+

=

:

:

(Note that -∞ in the species equation refers to the upstream value on the fuel jet side)

Table 3-2: Boundary Condition Comparison at x=l Between the 

Opposed Jet Formulation and the Stagnation-Point Formulation

Dependent 

Variable

Opposed Flow boundary condition 

at x=l:

Stagnation-Point boundary 

condition at x=l:

F

2

l l r u

2

l l r u

G 0

− d − rlE

d = 1: potential flow

d = 0 : plug flow

T Ts Ts

Yk Yk,l Yk,l

3.4 Flame Radiation Model

Radiation contributes to the heat balance in two places in the model of the solid stagnation-point flame. 

The solid surface both emits radiation and absorbs incident radiation from the flame (and background). 

This effect is modeled in the surface heat balance boundary condition. Radiation also constitutes a heat 

loss from the flame, reducing flame temperature. This effect is modeled in the energy equation as the 

derivative of the radiation flux term. The topic of this section is the formulation of the radiative heat 

transfer equation for solution of the derivative of the radiation flux term in the flame energy balance.

The physics of radiative participation of gases in a flame are complex in that the radiative properties of 

real gases vary with direction, wavelength, temperature, and pressure. The modeling challenge is to 

retain the important dependencies, while simplifying enough to allow for numerical tractability. 
15

Radiation heat transfer models are primarily distinguished by the rigor with which the spectral 

dependence of the radiative properties is treated. Another distinction is the treatment of the directional 

dependence. The radiation heat transfer equation is typically posed in a form to express most 

conveniently and accurately the spectral dependencies. The following discussion addresses the treatment 

methods used in this study, as well as the methods of Appendix 4, in contrast to refinements used here. 

Bedir et al. [1996] and Bedir [1998] provide thorough discussions and comparisons of these models, as 

well as other types of radiation models as applied to diffusion flames.

Spectral Dependence: The simplest models use the gray gas assumption [Seigel and Howell 1981] 

wherein the radiative properties of the gas are spectrally independent. In these models, the radiation 

intensity, I, is attenuated from the black-body radiation intensity over direction, s, by a spectrally 

independent absorption coefficient, k:

(I I)

ds

dI

= k b −

and the derivative of the net radiative heat flux is given by:

= ∫IdΩ

dx

d

dx

dq

p

m

4

where m is the direction cosine, and Ω is the solid angle.

The selection of the gray gas absorption coefficient k for gases that are inherently spectrally dependent is 

a difficult task. In many flame analyses, Planck mean absorption coefficient, kp, is used. The Planck 

mean absorption coefficient is given as a continuous function of the temperature and partial pressure of 

the gas. In Appendix 4, we use the Planck mean absorption coefficients for CO2 and H2O. Using the 

Planck mean absorption coefficient, solving the radiative transfer equation is relatively simple. But, as 

pointed out earlier, in the cases where flame radiation is important, the flame is normally not optically 

thin, thus using the Planck mean absorption coefficient introduces quantitative inaccuracy. 

In the following work, we have chosen to use the most accurate radiation model that is now 

computationally tractable in a problem of this type. The narrowband model is used to replicate the 

spectral properties of the participating gases. Indeed, the narrowband model is typically used as the 

benchmark with which to compare the accuracy of other radiation transfer equation approximations.

Real gases absorb and emit in discrete intervals and quantities across the radiation spectra. These 

irregular patterns are a result of transitions in energy states of the gas molecules in the presence of 

radiative energy, or photons. Emission represents a release of photons, while absorption represents the 

capture of a photon. Since this process occurs in discrete energy transitions, or quanta, at fixed 

frequencies, the emissive and absorptive properties of a gas appear as lines at discrete intervals across a 

frequency spectra. Several phenomena contribute to the 'broadening' of these lines into 'bands,' so that 

photons are absorbed or emitted over ranges of frequencies. If the bands are sufficiently narrow, the 

black-body radiation intensity, Ibn, can be taken out of the integral, because it does not vary much over the 

bandwidth, Dni:

( ) ( )

∫ ∫

Δ Δ

=

i i

I T d I T d b bi

n

n

n

kn n n k n

where i is the band index, in this instance. If the bandwidth is broad, a number of sufficiently narrow 

bands can be used. The narrowband model is thus a numerically convenient representation of the spectral 

absorption and emission bands of the molecular gas.
16

This model is described in detail in Bedir [1998] and is summarized here. We include collision 

broadening here as the primary broadening effect of significance at atmospheric pressure and the 

temperature range of interest (infrared) [Seigel and Howell 1981]. Collision broadening is the result of 

changes of energy state of the molecules because of increased collisions as the density or pressure of the 

gas increases. Collisions can occur with molecules of the same gas (self-broadening) or with other 

constituent gas molecules. Doppler broadening is also included in the model [Shih et al. 1999], since it 

can also be significant in the infrared region [Ozisik 1973]. Doppler broadening is a result of the thermal 

motion of the radiating gas molecules, and contributes to the broadening of the line at high temperatures 

and/or low pressures.

The statistical narrowband model is used for the assumed band spacing of the spectra. The statistical 

model allows for nonuniform spacing and height of the spectral absorption coefficient bands through 

assumption of probability distributions of line strengths and positions.

Here we choose to use CO2, H2O, and CO as the radiatively participating species. These gases are well 

known to be the dominant radiative contributors in hydrocarbon flames. No narrowband radiation data is 

available on the gas-phase trioxane fuel vapor, or its decomposition product, formaldehyde. We have 

assumed these gases to be negligible absorbers, but this should be evaluated in the future. A number of 

studies have included the radiative effects of fuel vapor, including Kashiwagi [1979] and Bedir [1998]. 

The hydrocarbon fuels examined have not been found to be dominant contributors to radiative loss, but 

they do have an effect.

The narrowband model requires a database composed of the reciprocal mean line spacing parameter 

(1/dn) and the mean absorption coefficient, κν, for each spectral band over the spectrum at different 

temperatures. While line-by-line data are available for the gases of interest here, it is not yet numerically 

tractable in a solution of this type. The narrowband data used here for CO, CO2 and H2O are correlated 

with the line-by-line HITRAN'92 database [Rothman et al. 1992]. The temperature and spectral ranges 

are 300-2900 K and 150-9300 cm-1, with a spectral bandwidth of 25 cm-1. It has been demonstrated 

[Soufiani et al. 1985, Soufiani and Taine 1989] that the statistical narrowband model with an exponential 

tailed inverse line strength distribution used here [Malkmus 1967] is in excellent agreement with the lineby-line calculations of Soufiani and Taine [1997] for CO2 and H2O. 

The model is posed in terms of transmittance, rather than absorptance. This yields the most accurate 

representation for the data because the translation to intensity involves several assumptions that can lead 

to inaccuracies [Bedir 1998]. The transmittance of a gas volume is defined as the fraction of the incident 

energy that passes through the gas volume. In the absence of reflection and scattering, then the 

transmitted energy along a path is the incident energy minus the absorbed energy. The average 

transmittance of an isothermal and homogeneous gas over path length (s'-s) of a molar fraction Xk and 

total pressure p is given as [Bedir 1998]:

( )

































′ → = − + −1

2

,

exp 1

n

n n

n

b

p k

p

b

t

u

s s k

where n n n b = 2pg /d is the mean line width to spacing ratio inside the spectral range (band), Dn, gn is 

the line half-width, and the pressure path length isu X p s s = k

′ → . The Curtis Godson approximation 

[Godson 1953] is used to extend the above to non-isothermal, nonhomogeneous media. This 

approximation assumes the transmittance of a non-isothermal, nonhomogeneous medium is that of an 

equivalent isothermal and homogeneous medium. Here the subscript e refers to the equivalent band 

parameters, and the following quantities are used in the transmittance equation:
17

( )

( )

( )

∫

∫

∫

′

′

′

′ → = ′′

′ → = ′′

′ → = ′′

s

s

k

e

e

s

s

e k

s

s

k

pX ds

u

s s

pX ds

u

s s

u s s pX ds

n n

n

n

n n

k b

k

b

k k

,

,

,

1

1

The transmittance of a mixture of gases is the product of the individual gas transmittances:

∏

k

k

t

Then the radiative transfer equation, written in terms of transmittance over a narrowband, for an 

absorbing, emitting, non-gray medium is [Bedir 1998]:

( ) ( )

( ) ( )

( )

( )

I (s )ds

s

s s

s

s

s s

I s I s

s

s s

ds

I s

b

s

s

w

b w w

s s

w

′ ′

















∂

∂ ′ →

∂

∂

+

∂

∂ →  + Ω











∂ ′

∂ ′ → =∂ Ω

∫

′=

n

n

n

n n

n n

t

t t

,

,

(3-5)

where the subscript w refers to the mean wall quantity, and W refers to the solid angle. The first term on 

the right side represents the local emission. The second and third terms represent the local absorption of 

surface emission and absorption of emission from the participating media along the line of sight. These 

two terms are neglected in the optically thin limit, where there is no appreciable absorption by the gas.

From the spectral intensity in equation (3-5), the derivative of the net radiative heat flux is found by 

integration over the solid angle and frequency:

m n

n p

n Δ











= ∑ ∫ Ω

Δ 4

I d

dx

d

dx

dq

(3-6)

Equation (3-6) is used in equation (3-1(e)) to solve the energy equation. Here q = qnet.

Scattering has been neglected in the formulation development. The only potential source of a scattering 

medium in a problem of this type would be soot formation. Trioxane burns with a clear or blue flame in 

standard air, indicating soot is not generated in significant quantities to affect the radiation heat transfer. 

Moreover, our focus is at the extinction limits, where the temperatures are below the threshold for soot 

formation and soot is not observed [Grayson et al. 1994, Olson 1997]. For these reasons, the effects of 

soot radiation have been neglected in this analysis.

Directional Dependence: The two-flux model used in Appendix 4 is among the simplest of the radiation 

models because the radiation intensity is isotropic (independent of position and direction) in the two flux 

directions. That is, the intensity is isotropic in the positive direction and the negative direction, but not the 

same value. Directional dependence can be improved by using the discrete ordinates method, wherein the 

angular variation of radiation intensity is modeled over discrete intervals, or ordinates. The two-flux model in 

essence uses two ordinates. Increasing the number of ordinates gives increased accuracy; however, selecting 

the appropriate ordinates and applying weighting (by Gaussian distribution or other means) are important 

features of the discrete ordinates method. Since the radiative transfer equation is solved for each ordinate, 

increasing the number of ordinates also increases computational complexity (solution time and storage 

requirements). Specifics of the numerical process for the discrete ordinates method are described in Section 5.
18

4.0 Kinetics and Properties of the Solid Fuel

4.1 The Solid Fuel

Criteria for Selection of Fuel: The criteria for fuel selection included the following: the fuel is solid at 

room temperature; the decomposition kinetics are known; the thermodynamic properties are known; and 

the combustion of the fuel has practical application. For these reasons, we have selected 1,3,5-trioxane as 

the fuel to be modeled.

Characteristics of Trioxane: Trioxane is classified as a cyclic trimer of formaldehyde [Kirk-Othmer 

Encyclopedia, Handbook of Thermoplastics 1997]. It is also known as s-trioxane, 1,3,5-trioxane, 1,3,5-

trioxacyclohexane or trioxymethylene [Handbook of Thermoplastics 1997]. Trioxane is a colorless, 

crystalline, flammable solid that sublimes readily. It decomposes directly and exclusively to formaldehyde 

[Kirk-Othmer Encyclopedia, Handbook of Thermoplastics 1997, Melia et al. 1967, Hochgreb and Dryer1

1992]. The kinetics of trioxane decomposition to formaldehyde are known [Hochgreb and Dryer1 1992, 

Irdam and Kiefer 1990], as are the kinetics of the subsequent formaldehyde oxidation [Hochgreb and Dryer2

1992, Hochgreb et al. 1990, Hochgreb 1991]. The thermodynamic properties of trioxane have been reported 

by Melia et al. [1967], among others. Trioxane is a very stable solid fuel and has an almost infinite shelf 

life. It burns steadily with a hot, smokeless blue flame. The military uses it in bar form as a meal warmer 

and campers and hikers use it as a fuel source, particularly in cold and wet weather. It is advertised for use 

to warm hands and food, and to start a fire in windy conditions or with rain-wet wood. Industrially, it is 

used to form acetyl resins, which are polymers of formaldehyde [Kirk-Othmer Encyclopedia]. The physical 

properties of trioxane are listed in Table 4-1 and the molecular structure is sketched below. 

CH2

CH2 CH2

O

O

O

Since the kinetics of formaldehyde oxidation are fairly well understood, any of the acetyl resins that are 

polymers of formaldehyde and that decompose from the solid, such as Delrin or Celcon, could be 

analyzed in a similar manner, given the Arrhenius law for the decomposition of these solids and their 

thermodynamic properties. 

Table 4-1: Physical Properties of Trioxane

Quantity* Value

Molecular formula C3H6O3

Molecular weight 90.08 g/mole

Melting point 336 K

Boiling point 388 K

Heat of formation -465 kJ/mol

Heat of fusion 15.1 kJ/mol

Heat of sublimation 55.6 kJ/mol (6.17E09 erg/g)

Specific gravity at 65oC (molten) 1.17

Specific heat (solid) @298 K

Specific heat (gas) @298 K

113.1 J/mol-K (1.255E07 erg/g-K)

81.88+/-3.50 J/mol-K

* [Melia et al. 1967, NIST Chemistry Web book]
19

4.2 Kinetics of Trioxane

Trioxane decomposition to formaldehyde is homogeneous and first order. It boils (at 115oC) without 

decomposition. Pure trioxane shows no appreciable depolymerization below 250oC (523 K) in the 

absence of a decomposition catalyst [Kirk-Othmer Encyclopedia]. Formaldehyde is the sole product of 

the decomposition of trioxane and this dissociation is well separated from that of its product [Hochgreb 

and Dryer1 1992, Irdam and Kiefer 1990]. The overall reaction for the decomposition is:

C3H6O3 → 3CH2O (1)

Hochgreb and Dryer1 [1992] give the reaction rate for the unimolecular decomposition of trioxane 

between 700 and 800 K (step 1 of Table 4-2). This reaction rate was validated against data gathered from 

a larger temperature range (545 to 1270 K) and found to be most accurate in the low to mid range of the 

temperature distribution. Here, the model is constructed such that the solid trioxane vaporizes upon 

heating and then decomposes directly to gaseous formaldehyde. Since the dissociation of trioxane is well 

separated from that of formaldehyde, the solutions to follow are not sensitive to reaction 1. The 

formaldehyde burns in air at atmospheric pressure via the decomposition kinetics described below. 

4.3 Kinetics Model and Validation

We evaluated three models for the decomposition of formaldehyde against available data over a range 

from fuel-rich to fuel-lean in order to determine the most appropriate model for use with the diffusion 

flame. Hochgreb and Dryer2 [1992] provide a mechanism in a 1992 publication as well as a set of data on 

formaldehyde oxidation. The GRI database [Smith et al.] contains a more recent mechanism (releases 1.2 

and 3.0) optimized for comprehensive description of natural gas combustion. Both of the GRI 

mechanisms are modified to include trioxane decomposition, and exclude the nitrogen chemistry. 

Nitrogen is modeled as a nonreacting third body. Since experimental data are provided in Hochgreb and 

Dryer2 [1992], the three mechanisms are evaluated against these data. The following section contains a 

description of the validation process and a discussion of the important reactions in the selected model.

Hochgreb et al. [Hochgreb and Dryer2 1992, Hochgreb et al. 1990, Hochgreb 1991] studied the oxidation 

of formaldehyde and published the complete reaction mechanism (steps 2 through 38 in Table 4-2). Their 

mechanism is based on a complete review of previously published experimental data, as well as the 

addition of original experimental data to complete the range of temperatures tested. The experiments 

used trioxane as the source of formaldehyde. Trioxane was melted, vaporized, and decomposed directly 

to formaldehyde and forced through a flow reactor at atmospheric pressure, with N as the diluent. In the 

kinetics analysis, the CO/H2/O2 set was extracted from Yetter and Dryer's comprehensive reaction 

mechanism [1991], and the HCO reactions come from the review by Tsang and Hampson [1986] and 

work by Timonen et al. [Timonen et al.1 1987, Timonen et al.2 1987, Timonen et al. 1988]. The CH2O 

reactions are a combination of previous work [Tsang and Hampson 1986, Roth and Just 1984] and new 

mechanisms supported by the experiments. Hippler et al. [1990] give a reaction mechanism for 

HO2+HO2 based on a high and low temperature range, which is used here. The original paper gives 

uncertainty factors for each reaction, which range from 1.5 to 4. Appendix 1 contains the CHEMKIN 

input file (rate constants) for the complete reaction mechanism, along with complete notations of data 

sources and uncertainty factors.

Model Validation: Hochgreb’s kinetic mechanism is compared to the formaldehyde subset of the 1.2 

release of the GRI oxidation kinetics in a perfectly stirred reactor model (PSR) [Glarborg et al. 1986]. 

PSR is a program that predicts temperature and species composition for a given system residence time in 

a perfectly stirred reactor. It is useful in comparing kinetic mechanisms, because flow field effects are

eliminated by the ‘perfectly stirred’ supposition, where mixing rates are infinitely fast.
20

The GRI mechanism and Hochgreb’s mechanism compare poorly [Sung et al. 1999], especially the 

critical residence time and temperature leading to extinction. We communicated with Ms. Hochgreb 

[Hochgreb 1999], who noted that the original paper contained a typographical error in the crucial 

H+O2→OH+O reaction of three orders of magnitude in the pre-exponential term. The correction appears 

here in Table 4-2 (and Appendix 1). 

We validated the corrected formaldehyde reaction mechanism against the data in Hochgreb [1991] in 

order to demonstrate reproducibility. We used the data from Hochgreb’s thesis as opposed to the data in 

the paper [Hochgreb and Dryer2 1992] because the paper contains numerous typographical errors. We 

then compared these to the modified GRI 1.2 model for the formaldehyde subset of reactions (using the 

constituents of formaldehyde oxidation given by Hochgreb [1991]) with the SENKIN code [Lutz et al. 

1988] to select the most appropriate model for the diffusion flame study. Figures 4-1 through 4-8 show 

the comparison of Hochgreb’s mechanism, the GRI 1.2 mechanism, and the flow reactor data for the 

oxidation of formaldehyde from Hochgreb [1991] over a range of equivalence ratios and temperatures. In 

each of the figures, Hochgreb’s mechanism is shown by dotted lines, the GRI mechanism by solid lines, 

and the data from Hochgreb [1991] as data points. Adapting Hochgreb’s approach, all three plotted data 

sets have been matched via time-offset calculation at 50% formaldehyde consumption. The reader may 

note that this matching appears to work better ‘upstream’ rather than ‘downstream.’ We believe this is 

due to the authors’ emphasis on fuel decomposition, rather than product formation.

The data in Figures 4-1 through 4-8 show that the GRI 1.2 and Hochgreb’s mechanisms compare 

relatively well to the experimental data for fuel-lean flames. For the fuel-rich condition (Figures 4-3, 4-6, 

and 4-8), Hochgreb’s mechanism is a better predictor of the kinetics. Only for the high-temperature, very 

fuel-lean case (Figure 4-7) could a case be made that the GRI mechanism is slightly more accurate. Since 

the diffusion flame contains both fuel-rich and fuel-lean regions, Hochgreb’s mechanism is the better 

model for our study.

The GRI mechanism has been recently updated with the 3.0 release. We compared the formaldehyde 

portion of the GRI 3.0 release to Hochgreb’s mechanism to see if the predictions in the fuel-rich region 

had been improved over the GRI 1.2 release. Figure 4-9 shows these results. The Hochgreb mechanism 

remains a better predictor than the GRI 3.0 mechanism in the fuel-rich region. For these reasons, 

Hochgreb’s mechanism [Hochgreb and Dryer2 1992, Hochgreb 1991] is used in this study. 

Important Reactions: The original paper [Hochgreb and Dryer2 1992] gives a discussion of the 

important reactions, which is summarized below. The important reactions are described in terms of a 

high temperature path and a low temperature path. There is an additional distinction between fuel-lean 

and fuel-rich mechanisms. 

It is important to note that Hochgreb’s model and experiments were conducted using a homogeneous 

system, evaluated at constant temperature and fuel equivalence ratio. The following summary applies to 

this system. It is compared to the heterogeneous diffusion flame system of interest in this work, in later 

sections.

Below 1100 K, the formaldehyde is attacked primarily by OH and H. While the reaction rate of HO2 with 

the formaldehyde is relatively slow, this reaction indirectly controls the reaction progress because it 

produces HCO, which in turn is responsible for production of H and OH. HO2 is the main chain carrier in 

this temperature range. Thus the most important reactions in the low temperature range are:

CH2O+H (or OH)→HCO+H2 (or H2O) (5) or (7)

HCO+O2→CO +HO2 (10)

CH2O+HO2→HCO+H2O2 (8)

H2O2+M→OH+OH+M (35)
21

Note that the equation numbers are consistent with Table 4-2. As the temperature increases beyond 

1100 K, less HO2 is produced, and the thermal decomposition of formyl, HCO+M→H+CO+M (reaction 

9) becomes faster than that of (10). Thus, the rate of production of H is determined from the competition 

between (9) and (10) in lean flames. At the higher temperatures, the branching route, H+O2→O+OH, 

reaction (20), surpasses reaction (35) as the main branching path. So that the important reactions for the 

high temperature range are:

CH2O+H (or OH)→HCO+H2 (or H2O) (5) or (7)

HCO+M→CO+H+M (9)

H+O2→OH+O (20)

Hochgreb found the rate of formaldehyde consumption below 1100 K for fuel-rich cases to be particularly 

sensitive to the concentration of O2. In the presence of oxygen, the formaldehyde is consumed in a 

branching mechanism, rather than a straight chain, allowing multiple pathways for radical attack of the 

formaldehyde. The controlling reactions governing the straight-chain and branching mechanisms are 

outlined as follows:

Straight-chain:

CH2O+H→HCO+H2 (5)

HCO+M→H+CO+M (9)

Branching:

CH2O+H→HCO+H2 (5)

HCO+M→H+CO+M (9)

HCO+O2→CO+HO2 (10)

CH2O+HO2→HCO+H2O2 (8)

H2O2+M→OH+OH+M (36)

Since we are interested here in a diffusion flame burning in standard air, we expect to observe this effect 

in the fuel-rich, lower-temperature regions of the flame; that is, between the flame and the solid surface. 

The region between the flame and the surface contains less oxygen, and is therefore a fuel-rich region, 

while the outer edge of the flame, while low in temperature, is lean.

In Appendix 4, we demonstrated that the maximum flame temperature has a non-monotonic variation 

with stretch rate, and that the maximum flame temperature near the radiative extinction limit is much 

lower than that of the blow-off limit. Hence, we anticipate that, for the radiative flame here, the dominant 

reactions over the low and high stretch rates follow Hochgreb and Dryer’s2 [1992] low- and hightemperature mechanisms, respectively.

We further note that the sensitivity to O2 concentrations limits the experimental accuracy in Hochgreb and 

Dryer2 [1992] because background concentrations were present from air leaks into the experimental 

apparatus. There is also a large discrepancy between observed and predicted results in the very fuel-rich 

region. The paper does not recommend any mechanism changes to alleviate the discrepancy because of 

the many possible explanations for the differences. 

Other Considerations: For the purposes of this study, we assume trioxane burns in air and include the N 

molecule as a nonreacting third body. In reality, small quantities of intermediate NOx species would be 

formed in the combustion. This study is not focused on the formation of NOx species, nor is the 

formation of these species expected to change the results, especially the global burning rate and extinction 

limits. Therefore, the N chemistry is neglected in the kinetics.
22

Table 4-2: Reaction Mechanism for Solid Trioxane Combustion

Rxn # Reactions Considered; k=ATbexp(-E/RT) A b E

1 C3H6O3=>3CH2O 1.90E+15 0.0 47,500

2 CH2O+M=HCO+H+M 4.21E+23 -1.7 92,170

3 CH2O+M=H2+CO+M 8.25E+15 0.0 69,540

4 CH2O+O2=HCO+HO2 2.05E+13 0.0 38,920

5 CH2O+H=HCO+H2 5.18E+07 1.7 1,834

6 CH2O+O=HCO+OH 1.81E+13 0.0 3,078

7 CH2O+OH=HCO+H2O 3.43E+09 1.2 -447

8 CH2O+HO2=HCO+H2O2 1.47E+13 0.0 15,200

9 HCO+M=H+CO+M (1) 1.85E+17 -1.0 17,000

10 HCO+O2=CO+HO2 4.22E+12 0.0 0

11 HCO+H=CO+H2 7.23E+13 0.0 0

12 HCO+O=CO+OH 3.00E+13 0.0 0

13 HCO+O=CO2+H 3.00E+13 0.0 0

14 HCO+OH=CO+H2O 3.00E+13 0.0 0

15 HCO+HO2=CO2+OH+H 3.00E+13 0.0 0

16 CO+OH=CO2+H 9.29E+03 2.2 -2,625

17 CO+HO2=CO2+OH 6.63E+13 0.0 22,950

18 CO+O+M=CO2+M (2) 4.00E+13 0.0 -4,540

19 CO+O2=CO2+O 2.53E+12 0.0 47,690

20 H+O2=OH+O 3.52E+16 -0.7 17,070

21 H2+O=OH+H 5.08E+04 2.7 6,292

22 OH+OH=H2O+O 2.88E+00 3.7 -4,313

23 OH+H2=H2O+H 2.16E+08 1.5 3,430

24 H2O2+OH=H2O+HO2 7.00E+12 0.0 1,430

25 H2O2+H=H2O+OH 1.00E+13 0.0 3,590

26 HO2+HO2=H2O2+O2 (3) 4.20E+14 0.0 11,980

27 HO2+HO2=H2O2+O2 (3) 1.30E+11 0.0 -1,629

28 H+HO2=OH+OH 1.69E+14 0.0 874

29 H+HO2=H2+O2 6.63E+13 0.0 2,126

30 HO2+OH=H2O+O2 1.45E+16 -1.0 0

31 HO2+O=O2+OH 1.81E+13 0.0 -397

32 H2O2+H=H2+HO2 4.82E+13 0.0 7,948

33 O+O+M=O2+M (2) 6.17E+15 -0.5 0

34 H2+M=H+H+M (2) 4.58E+19 -1.4 104,400

35 H+OH+M=H2O+M (2) 2.25E+22 -2.0 0

36 H2O2+M=OH+OH+M (2) 1.20E+17 0.0 45,500

37 O+H+M=OH+M (2) 4.72E+18 -1.0 0

38 H+O2+M=HO2+M (2) 2.00E+15 0.0 -1,000

Rate constants listed are for the forward reaction. Units: mole, cm3, s, cal.

(1) CO enhanced by 1.9, CO2 enhanced by 1.9, H2O enhanced by 16, and H2 enhanced by 1.89.

(2) CO enhanced by 1.9, CO2 enhanced by 1.9, H2O enhanced by 16, and H2 enhanced by 2.5.

(3) duplicate reaction.

Reaction (1) from [Hochgreb and Dryer1 1992]; Reactions 2-38 from [Hochgreb and Dryer2 1992]
23

CH2O: 3560 ppm, eq.ratio=0.51, T=944 K

0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14

Time (sec)

Mole Fraction

900

920

940

960

980

1000

Temperature (K)

O2

Temp

CH2O

CO

H2

CO2

Solid lines: GRI 1.2 mechanism; dashed lines: Hochgreb's mechansim; points: Hochgreb's data

Figure 4-1: Comparison of kinetics mechanisms (Hochgreb vs. GRI 1.2) with experiments for 

3560 ppm CH2O, equivalence ratio=0.51, and flow reactor temperature TFR=944 K.

CH2O: 3480 ppm, eq. ratio=0.56, 945 K

0

0.001

0.002

0.003

0.004

0 0.02 0.04 0.06 0.08 0.1 0.12

Time (sec)

Mole Fraction

900

920

940

960

980

1000

Temperature (K)

CH2O

O2

Temp

CO

H2

CO2

Solid lines: GRI 1.2 mechanism; dashed lines: Hochgreb's mechansim; points: Hochgreb's data

Figure 4-2: Comparison of kinetics mechanisms (Hochgreb vs. GRI 1.2) with Experiments for 

3480 ppm CH2O, equivalence ratio=0.56, and flow reactor temperature TFR=945 K.
24

CH2O: 5140 ppm, eq. ratio=39.7, 994 K

0

0.001

0.002

0.003

0.004

0.005

0.006

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

Time (sec)

Mole Fraction

900

920

940

960

980

1000

Temperature (K)

CH2O

Temp

H2 (top)

CO (bottom)

O2

Solid lines: GRI 1.2 mechanism; dashed lines: Hochgreb's mechansim; points: Hochgreb's data

H2

CO

CH2O

Figure 4-3: Comparison of kinetics mechanisms (Hochgreb vs. GRI 1.2) with experiments for 

5140 ppm CH2O, equivalence ratio=39.7, and flow reactor temperature TFR=994 K.

CH2O: 2080 ppm, eq. ratio = 0.013, 994 K

0

0.001

0.002

0.003

0 0.01 0.02 0.03 0.04 0.05 0.06

Time (sec)

Mole Fraction

940

960

980

1000

1020

1040

Temperature (K)

CH2O Temp

CO

CO2

Solid lines: GRI 1.2 mechanism; dashed lines: Hochgreb's mechansim; points: Hochgreb's data

Figure 4-4: Comparison of kinetics mechanisms (Hochgreb vs. GRI 1.2) with experiments for 

2080 ppm CH2O, equivalence ratio=0.013, and flow reactor temperature TFR=994 K.
25

CH2O: 2390 ppm, eq. ratio = 0.70, 994K

0

0.001

0.002

0.003

0.004

0 0.01 0.02 0.03 0.04 0.05 0.06

Time (sec)

Mole Fraction

940

960

980

1000

1020

1040

Temperature (K)

Solid lines: GRI 1.2 mechanism; dashed lines: Hochgreb's mechansim; points: Hochgreb's data

O2

CH2O Temp

CO

H2

CO2

Figure 4-5: Comparison of kinetics mechanisms (Hochgreb vs. GRI 1.2) with experiments for 

2390 ppm CH2O, equivalence ratio=0.70, and flow reactor temperature TFR=994 K.

CH2O: 4600 ppm; ep. Ratio = 39.7; 995K

0

0.001

0.002

0.003

0.004

0.005

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

Time (sec)

Mole Fraction

960

980

1000

1020

1040

1060

Temperature (K)

Solid lines: GRI 1.2 mechanism; dashed lines: Hochgreb's mechansim; points: Hochgreb's data

CH2O

CO (top)

H2 (bottom)

Temp

Figure 4-6: Comparison of kinetics mechanisms (Hochgreb vs. GRI 1.2) with experiments for 

4600 ppm CH2O, equivalence ratio=39.7, and flow reactor temperature TFR=995 K.
26

CH2O: 2340 ppm, eq. ratio = 0.93, 1095 K

0

0.001

0.002

0.003

0 0.002 0.004 0.006 0.008 0.01 0.012

Time (sec)

Mole Fraction

1050

1070

1090

1110

1130

1150

Temperature (K)

Solid lines: GRI 1.2 mechanism; dashed lines: Hochgreb's mechansim; points: Hochgreb's data

O2

Temp

CH2O

CO

CO2

Figure 4-7: Comparison of kinetics mechanisms (Hochgreb vs. GRI 1.2) with experiments for 

2340 ppm CH2O, equivalence ratio=0.93, and flow reactor temperature TFR=1095 K.

CH2O: 5360 ppm, eq. ratio = 31.5, 1090 K

0

0.001

0.002

0.003

0.004

0.005

0.006

0 0.01 0.02 0.03 0.04 0.05

Time (sec)

Mole Fraction

1050

1070

1090

1110

1130

1150

Temperature (K)

Solid lines: GRI 1.2 mechanism; dashed lines: Hochgreb's mechansim; points: Hochgreb's data

CH2O

Temp

CO

Figure 4-8: Comparison of kinetics mechanisms (Hochgreb vs. GRI 1.2) with experiments for 

5360 ppm CH2O, equivalence ratio=31.5, and flow reactor temperature TFR=1090 K.
27

CH2O: 5140 ppm; eq. ratio = 39.7, 994K

0

0.001

0.002

0.003

0.004

0.005

0.006

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

Time (sec)

Mole Fraction

900

920

940

960

980

1000

Temperature (K)

Solid lines: GRI 3.0 mechanism; dashed lines: Hochgreb's mechansim; points: Hochgreb's data

CH2O

Temp

CO(bottom)

H2 (top)

O2

H2

CO H2 CO

CH2O CH2O

Figure 4-9: Comparison of kinetics mechanisms (Hochgreb vs. GRI 3.0) with experiments for 

5140 ppm CH2O, equivalence ratio=39.7, and flow reactor temperature TFR=994 K.

4.4 Thermophysical Properties

Thermodynamic Data: CHEMKIN II [Kee et al. 1989] is the source for the thermodynamic data, with 

the following exceptions. Trioxane is not contained in the CHEMKIN II database, so thermodynamic 

data were obtained from Lay et al. [1997] and formatted using the THERM program package [Ritter and 

Bozelli 1987] into the NASA Gordon-McBride format. HO2 thermodynamic data was specified on the 

input file using the data from Hills and Howard [1984] (Appendix 1).

Transport Properties: The OPPDIF [Lutz et al. 1996] package uses TRAN [Kee et al. 1986], which 

evaluates gas-phase transport properties using a polynomial fit to the temperature-dependent part of the 

transport property. The pure species properties are calculated from molecular collision data, including the 

dipole moment, the rotational relaxation collision parameter, the polarizability, the Lennard-Jones 

potential well depth, and the Lennard-Jones collision diameter. We added the known properties for 

trioxane (Table 4-3) to the TRAN [Kee et al. 1986] database for multicomponent evaluation of gas-phase 

viscosities, thermal conductivities, diffusion coefficients, and thermal diffusion coefficients. 

Polarizability and rotational relaxation data were not available for trioxane. We show these estimated 

parameters with the given rationale in Appendix 2. 

The rotational relaxation collision parameter, Zrot, for trioxane is here estimated to be 1.0. The quantity 

Zrot is a measure of the number of collisions needed to establish rotational equilibrium of a molecule. 

Thermal conductivity is a weak function of the rotational relaxation collision parameter.
28

Trioxane concentration is highest near the surface, where it is rapidly decomposing to formaldehyde 

during burning. Thus, the effect of these assumptions being in error would be seen very near the surface 

and at relatively low temperatures (compared to the flame).

We estimate polarizability for trioxane to be 0.237 cubic Angstroms (see Appendix 2), using the method 

described by Hirschfelder et al. [1954]. The polarizability of the polar molecule affects diffusion 

coefficients between polar and nonpolar molecules. Transport properties of all other constituent gases 

were available from the existing database [Kee et al. 1986].

Surface Temperature: The Arrhenius law for the solid decomposition of trioxane is not known. If it 

were, we could estimate the surface temperature from this law, as shown in Appendix 4. Since this datum 

is not available, we estimate the surface temperature of the burning trioxane by using the boiling point of 

trioxane (Table 4-1) and keeping it as a constant.

Table 4-3: Thermophysical Properties of Trioxane

Quantity Value

Lennard-Jones potential well depth, K 500

Lennard-Jones collision diameter, Angstrom 6.093

Dielectric Constant, (molten trioxane)

Dielectric Constant, (solid trioxane)

8

3.2

Dipole moment, Debye (10-18cm3/2 erg1/2) 2.18

Polarizability (estimated), Angstrom30.237

Rotational relaxation collision number (estimated) 1.0

[Hochgreb and Dryer1 1992, Walker and Carlisle 1943, Maryott and Acree 1944] 

See Appendix 3 for details on estimated quantities.

4.5 Radiative Properties

Participating Gases: The narrowband properties of CO, CO2, and H2O are taken from Soufiani and 

Taine [1997]. All other gases involved in the combustion do not contribute as participating media, either 

because they are not produced in sufficient quantity, or because they do not absorb or emit a significant 

amount of thermal radiation in the temperature range (spectra) of interest. As discussed in Section 3.4, 

soot is neglected.
29

5.0 Numerical Method

We used the set of computer codes developed at the Sandia National Laboratory for analyses of various 

problems that rely on solution of detailed kinetics models, and modified them as necessary to solve for the 

solid trioxane flame (OPPDIF, PSR, SENKIN, PREMIX). All of these codes rely on three core code 

packages: a chemical kinetics package (CHEMKIN II), a transport property package (TRAN), and a 

solver package that solves the boundary value problem (TWOPNT). This section contains a brief 

description of the code packages used, how they are modified, and the numerical techniques used to 

obtain the solutions.

The OPPDIF Program: Lutz et al. [1996] developed a computer program, called OPPDIF, to solve the 

system of differential equations posed by opposed jet diffusion flames. The governing equations outlined 

in Section 3 are solved by OPPDIF, modified to accommodate the solid surface boundary conditions 

described in Section 3. OPPDIF is formulated to solve the opposed jet problem using the constant 

velocity formulation described in Lutz et al. [1998], and summarized in Section 3. It solves the governing 

equations beginning with a coarse grid, and adds grid points along steep gradients or steep gradient 

derivatives, yielding a nonuniform mesh. New grid points are added until the solution is resolved with 

sufficient accuracy. The governing equations are discretized by finite difference, adjusting for the 

nonuniform grid. The diffusive terms in the energy and species equations are discretized using a secondorder central difference; the convective terms are discretized using an upwind difference, to avoid 

artificial oscillations in the solution. The solution is first found using a given initial temperature profile 

(without solving the energy equation). This solution is then used as a starting point when the energy 

equation is added. The program creates a ‘restart’ file from the solution obtained from ‘scratch,’ so that 

all subsequent solutions can be started from the restart file. Both phases of solution use the two-point 

boundary value problem solver describe below. 

OPPDIF uses a chemical kinetics package of programs called CHEMKIN II [Kee et al. 1989] and a 

transport property package called TRAN [Kee et al. 1986]. The CHEMKIN input file is listed in 

Appendix 1. The input file contains forward rate constants for the reactions. Backward rate constants are 

calculated from the forward rate data and the equilibrium constants. In the analysis, we considered all 

steps to be forward and backward reactions, with the exception of the decomposition of trioxane. This 

step is assumed a forward reaction only. The transport package input is described in Section 4. 

The program allows a diffusion velocity calculation based on either the mixture-averaged formulation or 

the multicomponent formulation. We use the mixture-averaged formulation throughout this study in 

order to increase computational efficiency.

OPPDIF uses a two-point boundary value solver called TWOPNT [Grcar 1992] to solve the stiff system 

of equations typically posed by inclusion of detailed kinetics. TWOPNT uses hybrid Newton/time 

marching solution procedure. A damped Newton’s method is attempted first, using user-specified 

convergence tolerances for both absolute and relative convergence. If the damped Newton’s method does 

not produce a suitable solution, TWOPNT initiates time marching, where the user specifies step size and 

number of time steps. Time-stepping is used to arrive at a better initial solution; then Newton’s method is 

attempted again; and so forth until a solution is obtained or time marching limits are reached. Since 

Newton’s method is fast but fails to converge if the initial solution is too far away, and time marching 

does not have convergence problems, but is slow; this hybrid solution technique takes advantage of the 

strengths of both methods, while avoiding the weaknesses. 

The time-stepping feature of TWOPNT is suppressed for the purposes of this study, primarily because of 

the possibility for dynamic instability near extinction, as described below. 
30

Flame Controlling Method: Once a solution at a single set of jet velocities is found using OPPDIF, a 

modified version of OPPDIF, CFDF, created to accelerate convergence, is used to find all subsequent 

solutions. CFDF adapts the flame controlling methods described in Nishioka et al. [1996]; here the 

stretch rate (a) and solid surface blowing rate (uF), and oxidizer velocity (u0), are solved uniquely for a 

user-specified fixed temperature at a fixed location. In this method, the temperature at a given location is 

used as an internal “boundary” condition, while the stretch rate a, and solid surface blowing rate uF and 

oxidizer velocity (u0) become the flame response. Then by incrementally changing the fixed temperature, 

we can obtain a continuous mapping of the flame response. This has the advantage of solving the flame 

structure near the extinction turning point with a relatively small amount of computational time. The 

CFDF program can solve the flame response by fixing a temperature on either side of the maximum flame 

temperature and moving in either direction with respect to the surface.

Starting Conditions: Starting conditions are set within a subroutine of OPPDIF, which can start from 

ether an initial ‘guess’ at the solution, or a restart from a previously converged solution. This subroutine 

is modified to accommodate the solid surface boundary conditions as described in Section 3 and only 

used to generate initial solutions (including the energy equation) on coarse grids. For this solution, an 

initial guess at the temperature profile is included in the input file. Once a coarse grid solution is found, 

the program is restarted from these solutions and refined to obtain solutions that are more accurate. The 

more accurate solution is then used as the restart file for the CFDF program.

Boundary Conditions: Fuel side boundary conditions as described in Section 3.0 are modified in the 

FUN subroutine of OPPDIF (and CFDF), which sets up the solution to the governing differential 

equations. A version of the program is generated for the sets of boundary conditions listed in Table 3-1 

and the boundary conditions used in Section 7.0 (Table 7-2). Note however that the flame controlling 

method modifies the boundary conditions. A temperature at a fixed point is used as an internal 

“boundary” condition, and the oxidizer velocity is solved, rather than specified as a boundary condition.

Domain Size: The physical domain under study needs to be large enough to accommodate the relatively 

wide flame structures of the solutions that include the full radiation model as well as the movement of the 

flame relative to the surface as the stretch rate changes. In particular, the flame thickness roughly 

inversely scales with the square root of stretch rate. To take advantage of computational efficiency, we 

determine the domain size by solving the optically thin model, because it gives rapid solutions that can be 

scaled to predict a domain size acceptable for use with the more accurate (narrowband) radiation model. 

Optically thin models are formulated with the assumption that the gas does not attenuate the radiation 

intensity in the optically thin limit; that is, it emits, but does not absorb. (Also note that the Planck mean 

absorption coefficients employed herein were generated based on the HITRAN ’92 database [Rothman et 

al. 1992] in the optically thin limit, which are different from those used in Appendix 4.)

The desired domain is large enough that the solution is independent of the domain (no end effects). The 

results of these solutions are scaled to the expected domain increase required for the narrowband solutions 

from previous studies using both models. Based on scaling of the optically thin solutions, we chose a 

domain size of l=20 cm (from surface to oxidizer jet) for the adiabatic solutions. For these adiabatic 

solutions, the domain easily accommodates both extremes of high and low stretch rates, where the flame 

resides near the surface (0.04 cm at the blow-off stretch rate) or outward from the surface (1.5 cm at low 

stretch rate of 2.5 s-1). This 20-cm domain is also used for the narrowband radiation solutions and is 

verified to be large enough (solution farthest from the surface located at 6 cm).

The surface radiation solutions, however, require a larger domain. We used a domain size criterion to 

ensure that effects from the boundaries would not change the solution. The domain is extended so that 

the domain-to-flame thickness ratio is at least 2.0. The flame thickness is estimated using the ‘full width 

at half maximum temperature’ method, where the distance between the points on either side of the flame 

that equal half the maximum flame temperature is used as the measure of the flame thickness. 
31

Additionally, we examine solutions for an adequate ‘tail’ for the temperature profile at the oxidizer side, 

to ensure that heat loss from the boundary does not affect the solution. For a domain of 25 cm, this 

criterion is met for all solutions. The maximum flame temperature location for all solutions ranges from a 

minimum distance of 0.04 cm from the surface (at blow-off extinction) to a maximum distance of 9.7 cm 

(at a stable solution stretch rate of 0.025 s-1, i.e., the upper part of the solution isola).

For completeness, the domain sizes are noted on the figures and text in the discussion sections.

Radiation: The statistical narrowband solver [Bedir 1998] is coupled with the CFDF program to obtain 

the solutions with gas phase radiation. The non-gray narrowband radiative transfer equation is solved 

with the S8 discrete ordinates method, using a 20-direction Gaussian quadrature set. The surface is 

modeled as a perfect (black-body) radiator, with absorptivity and emissivity equal to 1.0, as described in 

Section 3.3. The adiabatic solution at a=100 s-1 is used as a starting point to obtain the narrowband 

solutions. The radiation solver takes much more computational time, so the radiation solver is not called 

at each Newton computation. The solver is called whenever the maximum change in temperature exceeds 

a user-supplied limit. 

Grid Independence: A study was performed to find a grid distribution that produces a solution 

independent of grid size and point distribution, but is also manageable from a computational standpoint. 

Solving the narrowband radiation cases consumes a relatively large amount of CPU time, which is 

primarily dependent on grid size. The grid is initially generated using OPPDIF for the adiabatic case. 

The grid was refined sequentially for representative stretch rates over the full flammability range using 

CFDF.

The GRAD and CURV criteria are used for evaluation. These are criteria used in the two-point boundary 

value problem solver (TWOPNT). “GRAD” is defined as a bound upon the range of change of a 

component C at grid point J such that:

GRAD = ( C (J) – C (J+1) ) / ( C(maximum) - C(minimum) )

“CURV” is defined as a bound upon the range of change of a component’s derivative D such that:

CURV = ( D (J) – D (J+1) ) / ( D (maximum) – D (minimum) )

For cases where GRAD and CURV are 0.4 or less, there is little to no change in key parameters such as 

stretch rate and maximum flame temperature, indicating that the solution is independent of the grid. This 

grid is then checked against the full range of adiabatic solutions by evaluating GRAD and CURV. The 

grid is manually refined where necessary until an acceptable grid is found using the two gridding 

parameters. Unlike some of the opposed flow solutions, where the flame location is maintained at a fixed 

location between the jets by a balance between the jet velocities (based on density ratios), the peak flame 

temperature moves across the domain based on stretch rate. At high stretch rates, the flame is pushed 

close to the surface, requiring grid refinement close to the surface. As stretch rate decreases, the flame 

moves away from the surface, requiring a fine grid at a distance from the surface as well. The resulting 

grid used for the adiabatic solutions contains 174 points for the computational domain of 20 cm.

The surface radiation solutions necessitate a finer grid because the flame moves much further from the 

surface in the low stretch regime. The technique described above is used with the same GRAD and 

CURV criteria to develop a grid with 256 points. Additionally, the domain is extended from 20 cm to 

25 cm as described previously.

For the narrowband radiation solutions, the adiabatic solution with the174-point grid is used to begin and 

to find the high stretch rate solutions. The grid is refined manually for the low stretch rate solutions to 
32

optimize computational time; as the flame moved away from the surface, points are added in this location 

and removed near the surface, while maintaining acceptable gridding parameters (GRAD and CURV). 

Steady-state analysis: One limitation in using the above method, particularly in the extinction regions, is 

that the analysis is steady state. It is well documented [Sohn et al. 2000, Deitrich et al. 2000] that 

dynamic instabilities may occur near the low-stretch extinction limit. Indeed, these instabilities may 

induce extinction. Under strong radiative heat loss, the reaction rate is lowered exponentially due to 

lower temperature. As a result, reactants are consumed at a lower rate, allowing diffusion to occur at a 

higher rate, so that there is sufficient time to increase reactant concentrations. The increased 

concentrations led to increased reaction rate, increased radiation, and the oscillatory behavior observed by 

several researchers. Sohn et al. studied the nonlinear evolution of diffusion flame oscillations that are 

triggered by radiative heat loss. Deitrich et al. [2000] observed transient oscillations near the (low 

stretch) extinction limit in the candle flame experiments performed in low-gravity environments.

It should be noted therefore that, while extinction values are given in the following sections, these are 

based on steady-state solutions. A transient solution may slightly displace the extinction limit based on 

steady-state analysis.

For the narrowband cases, the exact turning point case at low stretch has not been pinned down. 

Solutions are obtained at a very low stretch rate (a=0.068 s-1), but no lower, because of numerical 

stiffness. On the other hand, such a numerical stiffness may be physical and reflect the inadequacy of the 

steady-state analysis when approaching the radiative extinction limit, as explained above. Nevertheless, 

this lowest obtainable solution is believed to be very close to the quench limit, for the following reasons, 

and is used as the case nearest the quench limit in the discussion of results (Section 6.0). This near limit 

case exhibits a ‘leveling off’ of the flame standoff distance, which is observed to be a precursor to 

extinction. It is also at a temperature lower than that of the surface radiation quench limit.
33

6.0 Discussion of Results

6.1 The Adiabatic Solution

We obtained solutions using the formulation described in Section 3 for the solid trioxane combustion and 

the numerical procedures described in Section 5 for the solid fuel trioxane diffusion flame. The solutions 

that do not contain the radiation terms are called the ‘adiabatic’ solutions. We compared all subsequent 

solutions with this ‘base case.’ All cases are run in standard air and standard atmospheric pressure. The 

surface temperature is set to 388 K, which is the boiling point of trioxane. Since this is the first detailed 

analysis of a solid trioxane flame, some commentary on the flame structure is warranted.

Flammability Domain: Figure 6-1 is a plot of the maximum flame temperature versus stretch rate for 

the entire flammability domain for the adiabatic flame. The figure shows that the flammable domain 

ranges from the blow-off stretch rate at 1952 s-1 at maximum flame temperature of 1417 K to the 

asymptotic low stretch maximum flame temperature at approximately 2200 K. The unstable solutions 

below the turning point at 1952 s-1 are shown for completeness. We observe the expected shape for the 

flammability curve where the solution at low stretch rate approaches a constant, maximum flame 

temperature (near 2200 K), and the blow-off occurs at a lower flame temperature (1417 K).

Flame Temperature Profiles: We calculated the maximum flame temperature for each solution in the 

domain (Figure 6-1(a)). Figure 6-1(b) gives several representative flame temperature profiles. Solutions 

at a low stretch rate (5 s-1), middle stretch rate (99 s-1), and the blow-off limit (turning point at 1952 s-1) 

are shown. The figure shows that the temperature profiles are quite different, due to changes in the flow 

regime. At high stretch rates, the flame is blown against the surface, so the peak flame temperature 

(1417 K) is very near the fuel. The flame temperature is low, indicative of the low residence time at the 

blow-off turning point. As the stretch rate decreases, the flame moves away from the surface and 

broadens; and the temperature increases (1952 K at 99 s-1 and 2175 K at 5 s-1). This is resultant from the 

more complete combustion occurring as the residence time increases. Note that the entire domain (20 

cm) is not shown in order to show more detail of the temperature profiles.

Burning Rate: The nondimensional burning rate of the adiabatic flame is presented in Figure 6-2. Figure 

6-2 presents the burning rate in a nondimensional form that is easily compared to the results of Appendix 4. 

Here, the burning rate is presented nondimensionally in an equivalent form to Equation (A4-13): 

a

F

a

m

o o

x

o o

x

r m r m

0 0

| 2 |

= =

=

&

We see that the nondimensional burning rate for the adiabatic flame is relatively flat (around 0.8), except 

at the blow-off extinction limit, indicating that the dimensional burning rate scales with the square root of 

the stretch rate. The trend is consistent with the adiabatic PMMA solution (case 6) in Figure A4-8. (The 

dimensional form of the burning rate, ru|x=0, which is more useful for engineering purposes, is given in 

Section 6.4 in comparison to the radiative flame results.)

Velocity profiles: Corresponding axial velocity profiles are shown for the low stretch rate (5 s-1), middle 

stretch rate (99 s-1), and the blow-off limit (turning point) cases in Figures 6-3 through 6-5. Note that the 

entire domain is not shown, and that the domain and velocity scales are different for each plot to show 

features of each profile. Each profile has similar characteristics: Near the surface, the axial velocity is in 

the positive x-direction, indicating fuel leaving the surface. The stagnation-point occurs where the 

velocity crosses the abscissa, indicating the change in direction of velocity from positive to negative. The 

local maxima in each curve indicate a local velocity increase due to heating by the flame. The location of 

the maxima (or minima in magnitude) corresponds to the location of the peak flame temperatures. The 
34

velocity then approaches the free stream (potential flow) velocity at the outer portion of the flame. In 

addition, the magnitude of the axial velocity gradient is twice that of the stretch rate, which is defined 

based on the radial velocity gradient.

Comparison of the velocity profiles shows that, at high stretch rates, the velocity profile is pushed nearer 

to the surface, so that the stagnation-point and local maxima (or minima in magnitude) are much closer to 

the surface than at low stretch rates. While the velocity profiles are similar, the changes occur over a 

much smaller portion of the domain for the high stretch rate cases. In addition, the blowing velocity from 

the surface, which is related to the fuel-burning rate, is larger at high stretch rate, even though the 

stagnation point is very close to the surface.

Species profiles: Figures 6-6 through 6-14 show species profiles for the same cases (low stretch rate 

(5 s-1), middle stretch rate (99 s-1), and the blow-off limit (turning point)). The species profiles are 

grouped by order of magnitude for ease of comparison. Note that the domain (x axis) is different for each 

plot so that the full range of species profiles is visible. Generally, the reaction occurs near the peak flame 

location so that, at high stretch rates, the reaction occurs much closer to the surface than at low stretch 

rates. The plots are scaled to locate the peak flame temperature at the center.

Figures 6-6 through 6-8 show profiles for C3H6O3, CH2O, CO, H2, CO2, H2O, O2 and the temperature 

profile for each of the three stretch rates. Trioxane gas is produced in approximately the same mole 

fraction from the surface, with only a slight increase of mole fraction at high stretch rate versus low 

stretch rate. The secondary fuel, formaldehyde, is produced in slightly higher quantities at lower stretch 

rate. As stretch rate increases, the ‘overlap’ region of reactants and products near the peak flame 

temperature expands so that, at very high stretch rate, reactants are not fully consumed near the maximum 

flame temperature due to the low residence time of the species in the reaction zone. The intermediate CO 

is produced in somewhat smaller quantity (peak mole fraction near 0.2 versus 0.25) and has a longer ‘tail’ 

from the reaction zone in the high stretch solution. Production of H2 and CO2 drops by half for the high 

stretch case, in comparison to the mid to low stretch cases. Production of H2O decreases somewhat from 

the low stretch solution (peak mole fraction near 0.2) to the high stretch solution (peak mole fraction less 

than 0.15). 

Figures 6-9 through 6-11 show profiles for the intermediates H, O, and OH and the temperature for each 

of the three stretch rates under consideration. First, we observe the similarity: the species peak in the 

same order (H, OH, O) over the domain. The differences again are due to both the lower residence times 

as stretch rate increases and the lower flame temperature. OH is produced in much smaller quantities at 

high stretch rate (approximate peak mole fraction 0.0025) versus low to mid stretch rate (approximate 

peak mole fractions of 0.0055, 0.007). The opposite is true of O and H: Larger quantities of these two 

are produced as the stretch rate increases. 

Finally, Figures 6-12 through 6-14 show profiles for the intermediates HCO, HO2, and H2O2 and the 

temperature for each of the three stretch rates. We observe first that the mole fraction scale is increased 

by an order of magnitude over each of the three plots. This indicates that these intermediates are 

produced in much greater quantities as stretch rate increases. HCO is produced first at lower stretch rate 

in the region between the surface and the peak flame temperature, but its mole fraction peaks at 

approximately 6X10-6 at the low stretch rate and peaks at approximate mole fraction 3.5X10-4 at the high 

stretch rate. At the low and mid stretch rate solutions, HO2 and H2O2 are both produced primarily on the 

oxidizer side of the flame. Figure 6-14 shows a dramatic increase in mole fraction of HO2 at the high 

stretch rate. Also, note the shift in the peak of these radicals (H2O2 and HO2) to the fuel side of the flame. 

At high stretch, there is much more O2 blown into the fuel side of the flame—while the absolute quantity 

of O2 is small (order 10-3) near the H2O2 and HO2 peaksit is sufficient to produce more of these radicals 

(order 10-4). We explore the reasons for the chemical structures seen here in the following section.
35

Furthermore, the staged combustion of trioxane is quite evident. Its oxidation essentially follows the 

path: C3H6O3→CH2O→HCO→H2/CO→H2O/CO2. Important radicals, such as H, O, and OH, are 

produced in the regime of peak temperature. The controlling pathways for trioxane oxidation will be 

identified in the following section.

0

500

1000

1500

2000

2500

0.1 1 10 100 1000 10000

Stretch Rate (s-1)

Maximum Flame Temperature (K)

Figure 6-1(a): Maximum flame temperature vs. stretch rate for the adiabatic flame 

(20 cm domain).

0

500

1000

1500

2000

2500

0 0.5 1 1.5 2 2.5 3

Distance from Surface (cm)

Temperature (K)

a = 1952 s-1

a = 99 s-1 

a = 5 s-1 

Figure 6-1(b): Temperature profiles for the adiabatic flame.
36

0

0.2

0.4

0.6

0.8

1

1.2

0 1 10 100 1000 10000

Stretch Rate (s-1)

Non-Dimensional Burning Rate

Figure 6-2: Nondimensional burning rate for the adiabatic flame.

Adiabatic Flame, Velocity Profile (Stretch Rate = 5 s-1)

20 cm Domain

-50

-40

-30

-20

-10

0

10

0 1 2 3 4 5 6 7 8 9 10

Distance from Surface (cm)

Velocity (cm/s)

Figure 6-3: Axial velocity profile for low stretch adiabatic flame.
37

Adiabatic Flame, Velocity Profile (Stretch Rate = 99 s-1)

20 cm domain

-200

-150

-100

-50

0

50

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Distance from Surface (cm)

Velocity (cm/s)

Figure 6-4: Axial velocity profile for mid-stretch rate adiabatic flame.

Adiabatic Flame, Velocity Profile (Stretch Rate = 1952 s-1)

20 cm domain

-500

-400

-300

-200

-100

0

100

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

Distance from Surface (cm)

Velocity (cm/s)

Figure 6-5: Axial velocity profile for high stretch rate turning point adiabatic flame.
38

Adiabatic Flame, Species Mole Fraction (a = 5 s-1)

20 cm domain

0.0E+00

1.0E-01

2.0E-01

3.0E-01

4.0E-01

0 0.5 1 1.5 2 2.5

Distance from Surface (cm)

Mole Fraction

0

500

1000

1500

2000

2500

Temperature (K)

CO

Temp

H2O

CO2

CH2O

H2

O2

C3H6O3

Figure 6-6: Major species profiles for low-stretch rate adiabatic flame.

Adiabatic Flame, Species Mole Fraction (a = 99 s-1)

20 cm domain

0.0E+00

1.0E-01

2.0E-01

3.0E-01

4.0E-01

0 0.1 0.2 0.3 0.4 0.5

Distance from Surface (cm)

Mole Fraction

0

500

1000

1500

2000

2500

Temperature (K)

Temp

O2

H2O

C3H6O3

CO

CO2

H2

CH2O

Figure 6-7: Major species profiles for middle stretch rate adiabatic flame.
39

Adiabatic Flame, Species Mole Fraction (a = 1952 s-1)

20 cm domain

0.0E+00

1.0E-01

2.0E-01

3.0E-01

4.0E-01

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

 Distance from Surface (cm)

Mole Fraction

0

500

1000

1500

2000

2500

Temperature (K)

Temp

C3H6O3

CO

CO2

H2

H2O

O2

CH2O

Figure 6-8: Major species profiles for high stretch rate (turning point) adiabatic flame.

Adiabatic Flame, Species Mole Fraction (a = 5 s-1)

20 cm domain

0.0E+00

2.0E-03

4.0E-03

6.0E-03

8.0E-03

0 0.5 1 1.5 2 2.5

Distance from Surface (cm)

Mole Fraction

0

500

1000

1500

2000

2500

Temperature (K)

Temp

OH

H

O

Figure 6-9: Species profiles of OH, H and O for low stretch rate adiabatic flame.
40

Adiabatic Flame, Species Mole Fraction (a = 99 s-1)

20 cm domain

0.0E+00

2.0E-03

4.0E-03

6.0E-03

8.0E-03

0 0.1 0.2 0.3 0.4 0.5

Distance from Surface (cm)

Mole Fraction

0

500

1000

1500

2000

2500

Temperature (K)

Temp

OH

H

O

Figure 6-10: Species profiles of OH, H and O for middle stretch rate adiabatic flame.

Adiabatic Flame, Species Mole Fraction (a = 1952 s-1)

20 cm domain

0.0E+00

2.0E-03

4.0E-03

6.0E-03

8.0E-03

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

Distance from Surface (cm)

Mole Fraction

0

500

1000

1500

2000

2500

Temperature (K)

Temp

H

O

OH

Figure 6-11: Species profiles for OH, H and O for high stretch rate (turning point) adiabatic flame.
41

Adiabatic Flame, Species Mole Fraction (a = 5 s-1)

20 cm domain

0.0E+00

1.0E-06

2.0E-06

3.0E-06

4.0E-06

5.0E-06

6.0E-06

0 0.5 1 1.5 2 2.5

Distance from Surface (cm)

Mole Fraction

0

500

1000

1500

2000

2500

Temperature (K)

HCO

H2O2

HO2

Temp

Figure 6-12: Species profiles of HCO, HO2 , and H2O2 for low stretch rate adiabatic flame.

Adiabatic Flame, Species Mole Fraction (a = 99 s-1)

20 cm domain

0.0E+00

1.0E-05

2.0E-05

3.0E-05

4.0E-05

5.0E-05

6.0E-05

7.0E-05

0 0.1 0.2 0.3 0.4 0.5

Distance from Surface (cm)

Mole Fraction

0

500

1000

1500

2000

2500

Temperature (K)

Temp

HCO

HO2

H2O2

Figure 6-13: Species profiles of HCO, HO2 , and H2O2 for middle stretch rate adiabatic flame.
42

Adiabatic Flame, Species Mole Fraction (a = 1952 s-1)

20 cm domain

0.0E+00

1.0E-04

2.0E-04

3.0E-04

4.0E-04

5.0E-04

6.0E-04

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

Distance from Surface (cm)

Mole Fraction

0

500

1000

1500

2000

2500

Temperature (K)

Temp

HCO

HO2

H2O2

Figure 6-14: Species profiles of HCO, HO2 , and H2O2 for high stretch rate (turning point) 

adiabatic flame.

6.2 Adiabatic Flame Sensitivity and Reaction Path Analysis

We performed sensitivity and detailed analyses to discover the driving kinetic mechanisms responsible 

for the structure of the solid trioxane combustion in the stagnation-point flow. The flame structure is 

quite different from the homogeneous studies that established the kinetic mechanism, so these additional 

analyses are warranted. The adiabatic results are used as the ‘baseline’ with which to compare the 

response of the radiative flame in subsequent sections. The results are also compared where possible with 

the original kinetics study [Hochgreb 1991].

Several parameters are of interest in the sensitivity and detailed kinetics study: species production rates,

reaction rates of progress, reaction heat release, and sensitivity to stretch rate. The following quantities 

are described here, using the definitions in the Nomenclature (Section 3.2).

While the species profiles were given in the previous section, we examined the production rates for the 

species to determine the location and magnitude of the major consumption and production zones. The 

production rate for species k is defined as:

= ∑

i

k ki i v& n r

(6-1)

We examined the rates of progress of the formaldehyde and formyl decomposition reactions to determine 

the dominant path(s) for CH2O, HCO, and O2 consumption. The rate of progress for reaction i is 

expressed as:

∏ ∏

′ ′′

= −

k

ri k

k

i fi k

ki ki r k X k X

n n

[ ] [ ] (6-2)
43

We examined the reaction heat release to determine both the total heat release and the primary reactions 

that contribute to the heat release over the flame domain. The heat release for reaction i is given by:

= ∑

k

i ki ihk Q n r (6-3)

and the total heat release is

= ∑ = ∑

k

k k

i

QT Qi v& h (6-4)

This total heat release term is equivalent to the source term in the energy equation (Equation 3-1(f)).

We examined the variables discussed above for three stretch rates over the adiabatic domain (a=5 s-1, 

a= 99 s-1, and a=1952 s-1). For consistency, these are the same low, medium, and high stretch rates used 

in the foregoing discussion of the flammable domain, flame temperature profiles, velocity profiles, and 

species profiles.

Since the response of the flame at extinction is of particular interest here, we chose the radial pressure 

curvature, E, for sensitivity analysis. Recall from the formulation section (Equation 3-1(c)), that E is the 

eigen-value of the solution and is related to the stretch rate by:

2 E a = −ro

The sensitivity gives a quantitative measure of how a solution to a model depends on parameters in the 

model. The sensitivity is calculated using the residual solution vector, F, and computing its sensitivity to 

the reaction rate constants, ki. The vector Φ satisfies

F(Φ, ki) = 0

where Φ is the vector containing the parameters of the solution, including E. F can be differentiated to give

= 0

∂

∂

+

∂

∂Φ

∂Φ

∂

i i

k

F

k

F

Where the matrix ∂F/∂Φ is the Jacobian of the original system, ∂F/∂ki is the matrix of partial derivatives 

of F with respect to the parameters of the solution, and ∂Φ/∂ki are the sensitivity coefficients. These are 

calculated in a similar manner as that used in the PREMIX [Kee et al. 1998] code, and the sensitivity to a 

particular parameter can be extracted from the sensitivity matrix. We calculated the sensitivity to E at the 

extinction state (Eext) and normalized it to form logarithmic derivatives:

( )

( )i

ext

i

ext

ext

i

k

E

k

E

E

k

ln

ln

∂

∂

=

∂

∂

This expression is then normalized in a post-processor by the maximum value of the sensitivity, 

( )

( )

( )

( )

among all i

i

ext

i

ext

k

E

k

E

ln

ln / max

ln

ln













∂

∂

∂

∂

(6-5)

so that the resulting sensitivity parameters are relative to +/- 1.0. Since E is negative and its magnitude is 

proportional to a

2

 in this analysis, the negative (positive) values of sensitivity coefficient represent the 

promoting (retarding) nature of the corresponding reaction, with respect to the overall reactivity. We can 
44

therefore employ the normalized E sensitivity examined at the extinction turning points to identify the 

dominant reactions controlling the flame extinction.

Species Production Rates and Reaction Rates of Progress: The production rates for each species are 

given in Figures 6-15, 6-16 and 6-17 for the sample stretch rates. In each plot, positive production rates 

indicate species production, and negative production rates indicate species consumption. Following these 

plots are the relevant reaction rates of progress of the sample stretch rates (Figures 6-18(a-c), 6-19(a-c), 

6-20(a-e)). Positive values indicate a forward progressing reaction and negative values indicate a reverse 

progressing reaction.

In general, we see in each production rate plot that trioxane decomposes in the same region that 

formaldehyde is produced, via reaction 1. This occurs in a relative low-temperature region of the flame 

(840 K, 950 K and 1060 K at the location of maximum consumption of fuel for a=5, 99, and 1952 s-1, 

respectively). This is followed by consumption of formaldehyde, with the primary products being CO 

and H2. This occurs at somewhat higher temperatures (1150 K, 1170 K, and 1270K for a=5, 99 and 1952 

s

-1

, respectively). CO and H2 are eventually converted to CO2 and H2O, respectively in the hightemperature region of the flame. 

The rates of progress for the formaldehyde decomposition, reactions 2 through 8, are presented in Figures 

6-18(a), 6-19(a), and 6-20(a) for each of the sample stretch rates. Reactions 5 and 7 consistently 

dominate the major formaldehyde consumption, with reaction 5 having the primary role. Note that the 

consumption zones for H and CH2O coincide to promote reaction 5.

 (5) CH2O+H→HCO+H2

 (7) CH2O+OH→HCO+H2O

Subsequently, formyl radical is consumed to form H2 and CO. The production of H2 and CO is attributed 

to the decomposition of HCO via reactions 9 and 11, rather than formaldehyde decomposition (reaction 3). 

(9) HCO+M→H+CO+M

(11) HCO+H→CO+H2

This is validated by examining the HCO reaction rates of progress, Figures 6-18(b), 6-19(b), and 6-20(b). 

Finally, there is a zone of radical pool formation and then recombination into final products, CO2 and 

H2O.

The primary O2 consumption zone is located downstream of the fuel consumption zone. Figures 6-18(c), 

6-19(c), and 6-20(c) show the O2 reaction rates of progress. Reaction 20 is the dominant reaction, and is 

the primary source of radical pools, producing OH and O. Reaction 20 is highly endothermic (see heat 

release discussion), so the local temperature of the maximum O2 consumption is of interest. The local 

temperatures are 2175 K, 1950 K, and 1410 K for a= 5, 99, and 1952 s-1, respectively, indicating that O2

consumption peaks in the high-temperature region of each sample flame. Also, note that H production 

peaks in opposition to O2 consumption.

Recall that Hochgreb [1991] found the rate of formaldehyde consumption in the fuel-rich zone below 

1100 K to be particularly sensitive to the concentration of O2. She observed that, in the presence of 

oxygen, the formaldehyde is consumed in a branching mechanism, in addition to the straight chain, as 

described in Section 4, allowing multiple pathways for radical attack of the fuel. The region between the 

surface and the flame fits the characterization of a low-temperature, fuel-rich zone. In contrast to this, we 

see a consistent dominance of reactions 5 and 7 in the decomposition of formaldehyde, with HCO 

consumed via 9 and 11, i.e., the straight-chain mechanism. 
45

The production rates increase by an order of magnitude for each sample stretch rate, starting from the low 

stretch case and increasing. The turning point is characterized by overlap of the formaldehyde consumption 

and radical production zones, ascribed to the shortened residence time at the high stretch rate.

Only at the turning point does reaction 6, CH2O+O→HCO+OH, have any importance in the 

decomposition of formaldehyderesulting from the overlap of formaldehyde consumption and the 

radical pool. This is in contrast to the previous premixed homogeneous study [Hochgreb 1991], where 

reaction 5 is the straight-chain decomposition reaction, and reaction 8, CH2O+HO2→HCO+H2O2, is the 

branching path decomposition reaction. Since the primary product of these reactions is HCO, we 

examined the rates of progress of the HCO decomposition (reactions 9 through 15) as well.

The rate of progress for the HCO consumption in Figures 6-18(b), 6-19(b), and 6-20(b) confirms that 

reaction 9 is the dominant path for HCO decomposition over the flame domain. Note that reaction 9 is 

plotted on the right-hand scale for Figures 6-18(b) and 6-19(b). Rate of progress for reaction 11 is an 

order of magnitude less than reaction 9 at the low and middle stretch cases. This confirms the foregoing 

supposition that the presence of CO and H2 suggested HCO decomposition via reactions 9 and 11. Only 

for the high stretch case does reaction 10 (HCO + O2→CO+HO2) appear with any significance. Note that 

reaction 10 also appears as an important O2 rate of progress reaction at the high stretch, particularly near 

the surface. This indicates that reaction 10 is the culprit for the increased HO2 that appears near the 

surface in Figure 6-14.

While we see no evidence of Hochgreb’s low-temperature reaction path in the analysis of the CH2O 

decomposition pathway, the emergence of reaction 10 in the HCO and O2 pathways is a telltale footprint 

of an alternate pathway. This, as well as the increase in production of HO2 and H2O2 on the fuel side of 

the flame, is further explored in Figures 6-20(d) and 6-20(e). Figure 6-20(d) gives the primary rates of 

progress for the HO2 reactions. Since we seek an explanation for the changes in the HO2 concentrations 

on the fuel side of the flame, we note that reaction 10 dominates in this region, and is the primary 

producer of HO2. Indeed, the other dominant reaction here (reaction 28) consumes HO2. Figure 6-20(e) 

presents the rates of progress for the major H2O2 reactions. Again, focusing on the fuel side of the flame, 

the dominant reactions are 8 and 36. While reaction 8 does not appear significant in Figure 6-20(a), it is 

the primary source of H2O2 in this portion of the flame. Reaction 36 consumes the H2O2 in this region.

Thus, we see the emergence of Hochgreb’s low-temperature reaction pathway in a portion of the high 

stretch flame. This is attributed to several factors: While the maximum flame temperature is greater than 

1100 K, a large portion of the flame is not; the region of the flame near the surface is fuel-rich; and 

finally, at the high stretch extinction limit, enough O2 is blown into the fuel-rich region to support the 

low-temperature reaction pathway.

In conclusion, the main reaction paths follow the route illustrated in Figure 6-25. This is essentially 

Hochgreb’s ‘high temperature’ mechanism. A portion of the high stretch extinction limit flame exhibits 

the ‘low temperature’ reaction pathway. This is noteworthy in that the reaction path exists in one region 

of the heterogeneous diffusion flame at the high stretch extinction limit. This pathway is further explored 

with the radiative flame in Section 6.5.

Heat Release: Figures 6-21(a), 6-22(a), and 6-23(a) show the total heat release over the domain for the 

sample stretch rates. Temperature profiles are included for comparison. Figures 6-21(b), 6-22(b), and 

6-23(b) contain the heat release by reaction to understand the primary contributors to the total heat 

release. Positive values indicate exothermicity; negative values indicate endothermicity.

All of the profiles exhibit an endothermic zone nearer to the surface, and an exothermic zone beyond. 

The endothermic zone is comparable in magnitude to the exothermic zone, which is unusual for 

hydrocarbon combustion. As can be observed in the reaction heat release, the endothermic zone 
46

corresponds to the fuel pyrolysis region. In the formaldehyde consumption zone, the exothermicity of 

reaction 5 counterbalances the endothermicity of reaction 9.

Beginning with the low stretch solution, the endothermic region coincides with the decomposition of 

trioxane (reaction 1). The consumption of formaldehyde (reaction 5) is opposed by the thermal 

decomposition of HCO (reaction 9), so that the initial part of the exothermic region resembles a small 

plateau. Reactions 16 and 23 also contribute to this plateau. The recombination zone releases much more 

heat in comparison, with reaction 35 (producing H2O from the recombination of H and OH) accounting 

for more than half of the heat release. Reactions 35, 38, and 23 all lead to H2O formation, indicating that 

the primary heat release is due to formation of H2O, rather than CO2 via reaction 16, which is a major 

source of heat release in most hydrocarbon combustion. The branching reaction 20 represents the 

dominant endothermicity in the heat release zone, while it is also the controlling branching reaction 

generating the radical pool.

The middle stretch rate case is similar to the low stretch rate case, except that reaction 11 contributes to 

the smaller of the two exothermic peaks. This indicates that HCO is consumed via reaction 11 as well as 

reaction 9 (as seen in the reaction progress discussion). The second, larger peak occurs in the 

recombination zone, and again, reaction 35 contributes to more than half of the heat release. Reactions 

35, 38, 23, 16, and 28 contribute to the heat release in this zone, and again, the primary heat release in this 

zone is due to formation of H2O. Reaction 20 accounts for the primary endothermic response in this zone.

At the turning point, the endothermic peak is reduced in relative magnitude, and the exothermic peaks 

reverse in magnitude. The first exothermic peak is larger than the second one. This is due to the overlap 

of the fuel and formaldehyde decomposition with the radical pool zones, as well as the contribution of 

reaction 7 to the decomposition of formaldehyde. The flame is blown against the surface, and the 

residence time is short, so that these zones overlap. Reaction 11 exceeds reaction 9 in heat release in this 

case due to the increased importance of reaction 11 (see rate of progress discussion) as a secondary path 

for HCO decomposition at high stretch. Previously, H is consumed in the radical pool zone, but with the 

overlap (or compression) of these zones at high stretch, H is available for reaction with HCO. In the 

radical pool zone, with the reduced presence of OH (Figure 3-21), reaction 35 is no longer significant, and 

the major heat release comes from reaction 38.

Eigen-Value Sensitivity: The results of the normalized Eext sensitivity are presented in Figure 6-24 for 

the high stretch turning point. At the high stretch turning point (a=1952 s-1), where the temperature is 

relatively low, the residence time is low as well, and the flame is very near the surface, the extinction 

stretch rate is sensitive to a number of reactions. Reaction 5 is the primary path for formaldehyde 

decomposition, and reaction 9 is the primary path for HCO decomposition. Reaction 11 promotes HCO 

decomposition but consumes H, thereby retarding the reaction. Additionally, Eext is sensitive to reactions 

16, 20, 21, 28, 29, and 38. We know from Figure 6-20(c) that the reactions 20 and 38 have a net rate of 

progress in the forward direction, and are competing chain branching and termination reactions. 

Unraveling these reactions reveals a common thread: The presence of the highly reactive, low molecular 

weight radicals, in particular, the H radical, controls the reaction progress. Reactions 11 and 29 consume 

H radical and retard the reaction. Reactions 16 and 20 produce H (reaction 16 is a direct production, 

while reaction 20 produces H via a net cycle with OH and O via reactions 21 and 23). Production of these 

radicals, particularly H radical, is an important control mechanism here because these radicals are highly 

reactive in the short residence time available, and these are the reactants more likely to be forced out of 

the reaction zone due to high stretch rate. Also, we observe from the species profiles in Figure 6-11 that 

the peaks for O, OH, and H are shifted somewhat past the peak temperature location to the lean side of the 

flame with respect to the lower stretch rate solutions in Figures 6-9 and 6-10, indicating that the radical 

pool zone is extended past the peak temperature location. Analyzing production rates also confirms this. 

Note that Eext is not sensitive to reaction 1, the trioxane decomposition, since it is well separated from the 

formaldehyde decomposition.
47

Adiabatic Flame, Production Rates (a = 5 s-1)

-2.0E-04

-1.0E-04

0.0E+00

1.0E-04

2.0E-04

3.0E-04

4.0E-04

0 0.4 0.8 1.2 1.6 2 2.4

Distance from Surface (cm)

Production Rate (mol/cm

3-s)

CH2O

H2

CO

Fuel

H2O

CO2

H

H2

CH2O

O2

Figure 6-15: Adiabatic flame, species production rates for low stretch rate.

Adiabatic Flame, Production Rates (a = 99 s-1)

-3.0E-03

-2.0E-03

-1.0E-03

0.0E+00

1.0E-03

2.0E-03

3.0E-03

4.0E-03

5.0E-03

0.0 0.1 0.2 0.3 0.4 0.5

Distance from Surface (cm)

Production Rate (mol/cm

3-s)

CH2O

Fuel

H2

CO

H2O

H

CO2

O2

CH2O

CO

Figure 6-16: Adiabatic flame, species production rates for mid stretch rate.
48

Adiabatic Flame, Production Rates (a = 1952 s-1)

-0.03

-0.02

-0.01

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0 0.02 0.04 0.06 0.08 0.1

Distance from Surface (cm)

Production Rate (mol/cm

3-s)

CH2O

CO

H2

H2O

Fuel

O2

CO2

H

CH2O

H2

Figure 6-17: Adiabatic flame, species production rates at the blow-off extinction limit.

Adiabatic Flame, CH2O Rate of Progress by Reaction (a = 5 s-1)

-2.0E-05

0.0E+00

2.0E-05

4.0E-05

6.0E-05

8.0E-05

1.0E-04

1.2E-04

1.4E-04

1.6E-04

0 0.5 1 1.5 2 2.5

Distance from Surface (cm)

Rate of Progress (mol/cm

3-s)

5

7

Figure 6-18(a): Adiabatic flame, CH2O consumption rate of progress by reaction for low stretch rate.
49

Adiabatic Flame, Rate of Progress by HCO Reaction (a = 5 s-1)

-2.0E-07

0.0E+00

2.0E-07

4.0E-07

6.0E-07

8.0E-07

1.0E-06

1.2E-06

1.4E-06

1.6E-06

0 0.5 1 1.5 2 2.5

Distance from Surface (cm)

Rate of Progress (mol/cm

3-s)

-2.0E-05

0.0E+00

2.0E-05

4.0E-05

6.0E-05

8.0E-05

1.0E-04

1.2E-04

1.4E-04

9 (Right Hand Axis) 1.6E-04

11

14

Figure 6-18(b): Adiabatic flame, HCO consumption rate of progress by reaction for low stretch rate.

Adiabatic Flame, Rate of Progress by O2 Reaction (a = 5 s-1)

-5.0E-06

0.0E+00

5.0E-06

1.0E-05

1.5E-05

2.0E-05

2.5E-05

3.0E-05

3.5E-05

4.0E-05

4.5E-05

0 0.5 1 1.5 2 2.5

Distance from Surface (cm)

Rate of Progress (mol/cm

3-s)

20

38

29

31

30

Figure 6-18(c): Adiabatic flame, O2 consumption rate of progress by reaction for low stretch rate.
50

Adiabatic Flame, CH2O Rate of Progress by Reaction (a = 99 s-1)

-5.0E-04

0.0E+00

5.0E-04

1.0E-03

1.5E-03

2.0E-03

2.5E-03

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Distance from Surface (cm)

Rate of Progress (mol/cm

3-s)

5

7

Figure 6-19(a): Adiabatic flame, CH2O consumption rate of progress by reaction for mid stretch rate.

Adiabatic Flame, HCO Rate of Progress by reaction (a = 99 s-1)

0.0E+00

5.0E-05

1.0E-04

1.5E-04

2.0E-04

0 0.1 0.2 0.3 0.4 0.5

Distance from Surface (cm)

Rate of Progress (mol/cm

3-s)

0.0E+00

5.0E-04

1.0E-03

1.5E-03

2.0E-03

2.5E-03

Rate of Progress (mol/cm

3-s)

11

9 (Right Hand Axis)

14

Figure 6-19(b): Adiabatic flame, HCO consumption rate of progress by reaction for mid stretch rate.
51

Adiabatic Flame, Rate of Progress by O2 Reaction (a = 99 s-1)

-1.0E-04

0.0E+00

1.0E-04

2.0E-04

3.0E-04

4.0E-04

5.0E-04

0 0.1 0.2 0.3 0.4 0.5

Distance from Surface (cm)

Rate of Progress (mol/cm

3-s)

20

38

29 31

30

Figure 6-19(c): Adiabatic flame, O2 consumption rate of progress by reaction for mid stretch rate.

Adiabatic Flame, CH2O Rate of Progress by Reaction (a = 1952 s-1)

-0.002

0.000

0.002

0.004

0.006

0.008

0.010

0.012

0.014

0.016

0.018

0.00 0.02 0.04 0.06 0.08 0.10

Distance from Surface (cm)

Rate of Progress (mol/cm

3-s)

5

7

6

Figure 6-20(a): Adiabatic flame, CH2O consumption rate of progress by reaction at the blow-off 

extinction limit.
52

Adiabatic Flame, HCO Rate of Progress reaction (a = 1952 s-1)

-2.0E-03

0.0E+00

2.0E-03

4.0E-03

6.0E-03

8.0E-03

1.0E-02

1.2E-02

1.4E-02

1.6E-02

1.8E-02

0.00 0.02 0.04 0.06 0.08 0.10

Distance from Surface (cm)

Rate of Progress (mol/cm

3-s)

9

11

10

15

14

Figure 6-20(b): Adiabatic flame, HCO consumption rate of progress by reaction at the blow-off 

extinction limit.

Adiabatic Flame, Rate of Progress by O2 Reaction (a = 1952 s-1)

-0.001

0

0.001

0.002

0.003

0.004

0.005

0 0.02 0.04 0.06 0.08 0.1

Distance from Surface (cm)

Rate of Progress (mol/cm

3-s)

20

10

38

29

31

30

Figure 6-20(c): Adiabatic flame, O2 consumption rate of progress by reaction at the blow-off 

extinction limit.
53

Adiabatic Flame, H2O2 Rate of Progress by Reaction (a = 1952 s-1)

-6.00E-05

-5.00E-05

-4.00E-05

-3.00E-05

-2.00E-05

-1.00E-05

0.00E+00

1.00E-05

2.00E-05

3.00E-05

0 0.02 0.04 0.06 0.08 0.1

Distance from Surface (cm)

Rate of Progress (mol/cm

3-s)

8

36

36

24

25

25

32

32

Figure 6-20(d): Adiabatic flame, H2O2 consumption rate of progress by reaction at the blow-off 

extinction limit.

Adiabatic Flame, HO2 Rate of Progress by Reaction (a = 1952 s-1)

-5.0E-04

0.0E+00

5.0E-04

1.0E-03

1.5E-03

2.0E-03

2.5E-03

0 0.02 0.04 0.06 0.08 0.1

Distance from Surface (cm)

Rate of Progress (mol/cm

3-s)

38

10

28

29

15

Figure 6-20(e): Adiabatic flame, HO2 consumption rate of progress by reaction at the blow-off 

extinction limit.
54

Adabatic Flame, Total Heat Release (a = 5 s-1)

-15

-10

-5

0

5

10

15

20

0.0 0.3 0.6 0.9 1.2 1.5 1.8 2.1 2.4

Distance from Surface (cm)

Heat Release (J/cm

3-s)

0

500

1000

1500

2000

2500

Temperature (K)

Temp

Total Heat Release

Figure 6-21(a): Adiabatic flame, total heat release for low stretch rate.

Adiabatic Flame, Heat Release by Reaction (a = 5 s-1)

-15

-10

-5

0

5

10

15

20

0 0.5 1 1.5 2 2.5

Distance from Surface (cm)

Heat Release (J/cm

3-s)

Total Heat Release

5

35

1

9

20

23

38

16

Figure 6-21(b): Adiabatic flame, heat release by reaction for low stretch rate.
55

Adiabatic Flame, Total Heat Release (a = 99 s-1)

-250

-200

-150

-100

-50

0

50

100

150

200

250

0 0.1 0.2 0.3 0.4 0.5

Distance from Surface (cm)

Heat Release ( J/cm

3-s)

0

500

1000

1500

2000

2500

Temperature (K)

Temp

Total Heat Release

Figure 6-22(a): Adiabatic flame, total heat release for mid stretch rate.

Adiabatic Flame, Heat Release by Reaction (a = 99 s-1)

-250

-200

-150

-100

-50

0

50

100

150

200

250

0.0 0.1 0.2 0.3 0.4 0.5

Distance from Surface (cm)

Heat Release, (J/cm

3-s)

Total Heat Release

5

1

9

20

11

35

23 38

16

28

Figure 6-22(b): Adiabatic flame, heat release by reaction for mid stretch rate.
56

Adiabatic Flame, Total Heat Release (a = 1952 s-1)

-1500

-1000

-500

0

500

1000

1500

2000

2500

0 0.02 0.04 0.06 0.08 0.1

Distance from Surface (cm)

Heat Release (J/cm

3-s)

0

200

400

600

800

1000

1200

1400

1600

Temperature (K)

Temp

Total Heat Release

Figure 6-23(a): Adiabatic flame, total heat release at the blow-off extinction limit.

Adiabatic Flame, Heat Release by Reaction (a = 1952 s

-1)

-2500

-2000

-1500

-1000

-500

0

500

1000

1500

2000

2500

0 0.02 0.04 0.06 0.08 0.1

Distance from Surface (cm)

Heat Release (J/cm

3-s)

Total Heat Release

11

5

7

1

23 38

28

20

16

9

28

Figure 6-23(b): Adiabatic flame, heat release by reaction at the blow-off extinction limit.
57

Adiabatic Flame, Eext Sensitivity (a = 1952 s-1)

-1.0 -0.5 0.0 0.5 1.0

1

3

5

7

9

1 1

1 3

1 5

1 7

1 9

2 1

2 3

2 5

2 7

2 9

3 1

3 3

3 5

3 7

Reaction Number

Normalized Eext Sensitivity

2

4

6

8

1 0

1 2

1 4

1 6

1 8

2 0

2 2

2 4

2 6

2 8

3 0

3 2

3 4

3 6

3 8

Promote Retard

Figure 6-24: Adiabatic flame, normalized Eext sensitivity by reaction at the blow-off extinction limit.

C3H6O3

CH2O

HCO

 CO

O + OH

H

(rxn 1)

(rxn 5)

(rxn 9)

(rxn 6,7)

+M

 + O2

(rxn 20)

Figure 6-25: Main reaction path for adiabatic trioxane diffusion flame. 
58

6.3 Effects of Surface Radiation

As discussed in Appendix 4, including radiative emission from the solid surface gives a qualitative picture 

of the radiative contribution, and is useful for comparison purposes in order to separate surface radiation 

effects from flame radiation effects. In this section, we compare the trends documented in Appendix 4 for 

the simpler PMMA model to those in the trioxane flame and to the adiabatic solution as well. 

Solutions are obtained using the formulation for an emitting surface described in Section 3, that includes 

black-body emission (ε = 1.0) from both boundaries. The domain is increased from 20 cm for the 

adiabatic solution to 25 cm, as described in Section 5. All cases are run in standard air and at standard 

atmospheric pressure. The surface temperature is set to 388 K, which is the boiling point of trioxane. We 

refer to these solutions in the discussion as the ‘surface radiation’ solutions. 

Flammability Domain: Figure 6-26 is a plot of the maximum flame temperature versus stretch rate for 

the entire flammability domain for the trioxane ‘surface radiation’ flame. In this case, the full solution 

isola, including the unstable solutions (i.e. the lower branch solution in Figure 6-26), is presented. The 

figure shows the expected low stretch extinction limit observed in Appendix 4, in contrast to the adiabatic 

solutions. While the materials and the surface temperatures are quite different, the trend is the same. 

The blow-off extinction limit (1942 s-1 at Tmax=1405 K) is nearly identical to that of the adiabatic solution 

(1952 s-1 at Tmax=1417 K). In this regime, the surface emission has virtually no effect. We contrast this 

with Figure A4-7, where surface radiation effects are small, but noticeable at the high stretch regime. 

Although the materials are different, and therefore the heat transfer characteristics are different as well, 

the primary reason for the larger effect on the PMMA solutions in comparison to the trioxane solutions at 

high stretch is the surface temperature (see Table A4-2 for PMMA with surface radiation). The trioxane 

surface is 388 K. The surface temperature of the PMMA solutions was calculated from the Arrhenius 

equation for the solid and was found to be 675 K at the extinction limit shown in Figure A4-7. Since 

emission is proportional to the fourth power of temperature, the surface emission effect is much stronger 

for the PMMA solutions.

Following the curve from right to left, we observe a peak temperature of 2216 K at a stretch rate of 0.4 s-1. 

As stretch rate decreases, the radiative emission begins to dominate the solution, leading to lower 

temperatures and weaker flames. Radiative quenching occurs at 0.011s-1 and a maximum flame 

temperature of 1013 K. Again, the trioxane flame is sustained at a lower stretch rate than that observed in 

Figure A4-7 for PMMA primarily due to surface temperature differences, resulting in weaker surface 

emission for the trioxane flame.

Flame Temperature Profiles: We calculate the maximum flame temperature for each solution in the 

domain (Figure 6-26a). Figure 6-26b gives several representative flame temperature profiles. We choose 

solutions to compare with the adiabatic temperature profiles in Figure 6-2, at a low stretch rate (5 s-1), 

middle stretch rate (98 s-1), and the blow-off limit (turning point at 1942 s-1). We also include the 

radiative extinction turning point temperature profile (a=0.011 s-1), along with a profile from the region 

on the left side of the isola, above the turning point (a=0.025 s-1). The figure shows that the temperature 

profiles are quite different, due to changes in the flow regime as well as radiative emission from the 

surface. As before, the high stretch solutions are blown against the surface. As stretch rate decreases, the 

flame moves away from the surface, and reaches a maximum temperature. In the region where surface 

radiation effects become important, the flame temperature decreases, while the flame continues to move 

away from the surface. Prior to extinction, the flame can no longer sustain the large standoff distance, 

and ‘falls back’ towards the surface. This drop-off toward the surface before extinction was also 

observed in the PMMA study (Figure A4-5) for all but the absorbing, non-emitting surface. 

Consistent with the results of Appendix 4, including surface radiation in the trioxane flame has introduced 

a flame structure not seen in the adiabatic solution; that is, a low temperature, low stretch rate regime.
59

Burning Rate: The burning rate of the flame is presented in a format that allows comparison with the 

results of Appendix 4. Figure A4-8 gives the nondimensional burning rate for the PMMA fuel. Figure 

6-27(a) presents the burning rate in a nondimensional form that is easily compared to the results of 

Appendix 4. Here, the burning rate is presented nondimensionally in an equivalent form to Equation 

A4-13: 

a

F

a

m

o o

x

o o

x

r m r m

0 0

| 2 |

= =

=

&

It allows us to observe the trend of the burning rate of trioxane in comparison with the PMMA results. 

Figure 6-27(a) shows this quantity versus the stretch rate for both the adiabatic flame and the flame with 

surface radiation. When compared to the PMMA solution in Figure A4-8, we see that the radiative 

trioxane flame behaves very much the same. The flame with surface radiation has a uniformly lower 

burning rate. At very low stretch, the burning rate decreases toward the extinction limit, due to the low 

flame temperatures.

Figure 6-27(b) illustrates the dimensional burning rate, ru|x=0, since this is of more interest for 

engineering purposes. Note this is plotted on a log-log scale, and the effects of the surface radiation flame 

is apparent at the lower stretch rates. 

Species Profiles: The species profiles for the five stretch rates under examination are presented in 

Figures 6-28 through 6-43. They are ordered by increasing stretch rate, and the species are grouped by 

order of magnitude. Note changes in domain so that the full range of species profile is visible. The flame 

temperature profiles are included for reference.

The species profiles at a=5, 98, and 1942 s-1 (Figures 6-28 through 6-43), along with the foregoing 

temperature profiles, establish that the surface radiation solutions are virtually identical to the adiabatic 

solutions for these stretch rates. They are presented here to confirm this fact; the reader is referred to the 

foregoing discussion on the adiabatic solutions for these stretch rates.

We turn to the lower stretch solutions (Figures 6-28, 6-29, 6-30, 6-31, 6-34, 6-35, 6-36, 6-39, 6-40, 6-41) 

to examine the effects of surface radiation on the species distribution in the flame. Scaling necessitates a 

different grouping of species for the low stretch cases. Starting with the fuel, we observe that the initial 

mole fraction of fuel at the surface is greatly reduced at the lower stretch rates, from approximately 0.35 

at a =5 s-1 to 0.09 and 0.03 for a = 0.025 s-1 and 0.011 s-1, respectively. Mole fractions of formaldehyde 

are reduced accordingly, however, the shape of the formaldehyde curve differs at low stretch. Rather than 

peaking away from the surface, the formaldehyde mole fractions are relatively constant close to the 

surface, and then decrease. This indicates that formaldehyde accumulates (it is produced faster than it is 

consumed) in the low temperature region near the surface, and is not rapidly depleted until it reaches the 

higher temperature region of the flame, which is located at a large standoff distance. CO, CO2, H2O, and 

H2 behaviors are similar, for the same reasons.

HCO, because it is the primary product of the formaldehyde decomposition, appears nearer to the surface 

than other intermediate radicals. The reduced mole fractions of fuel and formaldehyde result in reduced 

mole fractions of HCO.

Among the intermediate radicals H, O, and OH, we see the emphasis on the OH radical continued from 

a=5 s-1 to a=0.025 s-1. However, at the low stretch turning point, H dominates, echoing the emphasis on 

H at the high stretch turning point. The profiles of H2O2 and HO2 at the low stretch turning point exhibit 

a different character than the other low stretch cases, indicating that a different reaction path is present 

here. This is explored further in the following sections. The H2O2 profile is similar to the high stretch 

turning point, rather than other low stretch cases. On the other hand, HO2 is present in the high 

temperature region of the low stretch turning point flame, as opposed to the low temperature, fuel-rich 
60

region of the high stretch turning point flame. Note that these two HO2 peaks do occur at the same local 

temperature (around 1000 K).

The primary purpose of the surface radiation solution is to distinguish the effects of surface and gas phase 

radiation, and the surface radiation solution is somewhat artificial. Therefore, we do not pursue the 

reaction path analysis here. Reaction path analysis and turning point sensitivity are discussed for the 

flame with gas phase and surface radiation.

Flame with Surface Radiation, Maximum Flame Temperature vs. Stretch Rate

25 cm domain

0

500

1000

1500

2000

2500

0.001 0.01 0.1 1 10 100 1000 10000

Stretch Rate (s-1)

Maximum Flame Temperature (K)

Figure 6-26(a): Maximum flame temperature vs. stretch rate for flame with surface radiation.
61

Flame with Surface Radiation, Temperature vs. Distance 

25 cm domain

0

500

1000

1500

2000

2500

0 5 10 15 20 25

Distance from Surface (cm)

Temperature (K)

a = 0.011 s -1

a = 0.025 s -1

a = 5 s -1

a = 98 s -1

a = 1942 s -1

Figure 6-26(b): Temperature profiles for flame with surface radiation.

Non-dimensional Burning Rate

0

0.2

0.4

0.6

0.8

1

0.01 0.1 1 10 100 1000 10000

Stretch Rate (s-1)

Non-dimensional Burning Rate

Surface Radiation

Adiabatic

Figure 6-27(a): Nondimensional burning rate vs. stretch rate for adiabatic flame and flame with 

surface radiation.
62

Burning Rate

1.0E-06

1.0E-05

1.0E-04

1.0E-03

1.0E-02

1.0E-01

0.01 0.1 1 10 100 1000 10000

Stretch Rate (s-1)

Burning rate (g/cm

2-s)

Surface Radiation

Adiabatic

Figure 6-27(b): Dimensional burning rate vs. stretch rate for adiabatic flame and flame with 

surface radiation.

Flame with Surface Radiation, Species Mole Fraction (a = 0.011 s-1)

25 cm domain

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0 5 10 15 20 25

Distance from Surface (cm)

Mole Fraction

0

200

400

600

800

1000

1200

Temperature (K)

Temp

CO2

H2O

O2

Figure 6-28: Species profiles of O2, CO2, and H2O for radiative extinction turning point (flame with 

surface radiation).
63

Flame with Surface Radiation, Species Mole Fraction (a = 0.011 s-1)

25 cm domain

0.000

0.005

0.010

0.015

0.020

0.025

0.030

0 2 4 6 8 10

Distance from Surface (cm)

Mole Fraction

0

200

400

600

800

1000

1200

Temperature (K)

Temp

C3H6O3

CO

CH2O

H2

Figure 6-29: Species profiles of C3H6O3, CO, CH2O, and H2 for radiative extinction turning point 

(flame with surface radiation).

Flame with Surface Radiation, Species Mole Fraction (a = 0.025 s-1) 

25 cm domain

0.00

0.05

0.10

0.15

0.20

0.25

0 5 10 15 20 25

Distance from Surface (cm)

Mole Fraction

0

500

1000

1500

2000

2500

Temperature (K)

O2

Temp

H2O

CO2

CO

H2

CH2O

C3H6O3

Figure 6-30: Major species profiles for very low stretch rate (flame with surface radiation).
64

Flame with Surface Radiation, Species Mole Fraction (a = 5 s-1)

25 cm domain

0.0

0.1

0.2

0.3

0.4

0.0 0.5 1.0 1.5 2.0 2.5

Distance from Surface (cm)

Mole Fraction

0

500

1000

1500

2000

2500

Temperature (K)

Temp

C3H6O3

CO

H2O

CO2

H2

CH20

O2

Figure 6-31: Major species profiles for low stretch rate (flame with surface radiation).

Flame with Surface Radiation, Species Mole Fraction (a = 98 s-1)

25 cm domain

0.0

0.1

0.2

0.3

0.4

0.0 0.1 0.2 0.3 0.4 0.5

Distance from Surface (cm)

Mole Fraction

0

500

1000

1500

2000

2500

Temperature (K)

C3H6O3 Temp

CO

H2O

CO2

H2

CH2O

O2

Figure 6-32: Major species profiles for mid-stretch rate (flame with surface radiation).
65

Flame with Surface Radiation, Species Mole Fraction (a = 1942 s

-1)

25 cm domain

0.0

0.1

0.2

0.3

0.4

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

Distance from Surface (cm)

Mole Fraction

0

500

1000

1500

2000

2500

Temperature (K)

Temp

C3H6O3

CO

O2

H2O

CH2O

CO2

H2

Figure 6-33: Major species profiles for blow-off turning point (flame with surface radiation).

Flame with Surface Radiation, Species Mole Fraction (a = 0.011 s-1)

25 cm domain

0.0E+00

2.0E-06

4.0E-06

6.0E-06

8.0E-06

1.0E-05

1.2E-05

1.4E-05

1.6E-05

1.8E-05

2.0E-05

0 2 4 6 8 10

Distance from Surface (cm)

Mole Fraction

0

200

400

600

800

1000

1200

Temperature (K)

Temp

H2O2 H

OH

HO2

Figure 6-34: Species profiles of H2O2, H, OH, and HO2 for radiative extinction turning point (flame 

with surface radiation).
66

Flame with Surface Radiation, Species Mole Fraction (a = 0.025 s-1) 

25 cm domain

0.0E+00

2.0E-04

4.0E-04

6.0E-04

8.0E-04

1.0E-03

1.2E-03

0 5 10 15 20 25

Distance from Surface (cm)

Mole Fraction

0

500

1000

1500

2000

2500

Temperature (K)

Temp

OH

H

O

Figure 6-35: Species profiles of H, O, and OH for very low stretch rate (flame with surface 

radiation).

Flame with Surface Radiation, Species Mole Fraction (a = 5 s-1)

25 cm domain

0.000

0.002

0.004

0.006

0.008

0.0 0.5 1.0 1.5 2.0 2.5

Distance from Surface (cm)

Mole Fraction

0

500

1000

1500

2000

2500

Temperature (K)

Temp

OH

H

O

Figure 6-36: Species profiles of H, O, and OH for low stretch rate (flame with surface radiation).
67

Flame with Surface Radiation, Species Mole Fraction (a = 98 s-1)

25 cm domain

0.000

0.002

0.004

0.006

0.008

0.0 0.1 0.2 0.3 0.4 0.5

Distance from Surface (cm)

Mole Fraction

0

500

1000

1500

2000

2500

Temperature (K)

Temp

OH

H

O

Figure 6-37: Species profiles of H, O, and OH for middle stretch rate (flame with surface radiation).

Flame with Surface Radiation, Species Mole Fraction (a = 1942 s-1)

25 cm domain

0.000

0.002

0.004

0.006

0.008

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

Distance from Surface (cm)

Mole Fraction

0

500

1000

1500

2000

2500

Temperature (K)

Temp

H

O

OH

Figure 6-38: Species profiles of H, O, and OH for blow-off turning point (flame with surface radiation).
68

Flame with Surface Radiation, Species Mole Fraction (a = 0.011 s-1)

25 cm domain

0.0E+00

2.0E-07

4.0E-07

6.0E-07

8.0E-07

1.0E-06

1.2E-06

0 2 4 6 8 10

Distance from Surface (cm)

Mole Fraction

0

200

400

600

800

1000

1200

Temperature (K)

Temp

O

HCO

Figure 6-39: Species profiles of HCO and O for radiative extinction (flame with surface radiation).

Flame with Surface Radiation, Species Mole Fraction (a = 0.025 s-1) 

25 cm domain

0.0E+00

1.0E-07

2.0E-07

3.0E-07

4.0E-07

5.0E-07

0 5 10 15 20 25

Distance from Surface (cm)

Mole Fraction

0

500

1000

1500

2000

2500

Temperature (K)

HO2

Temp

HCO

H2O2

Figure 6-40: Species profiles of HO2, H2O2, and HCO for very low stretch rate (flame with surface 

radiation).
69

Flame with Surface Radiation, Species Mole Fraction (a = 5 s-1)

25 cm domain

0.0E+00

1.0E-06

2.0E-06

3.0E-06

4.0E-06

5.0E-06

6.0E-06

0.0 0.5 1.0 1.5 2.0 2.5

Distance from Surface (cm)

Mole Fraction

0

500

1000

1500

2000

2500

Temperature (K)

Temp

HCO HO2

H2O2

Figure 6-41: Species profiles of HO2, H2O2, and HCO for low stretch rate (flame with surface 

radiation).

Flame with Surface Radiation, Species Mole Fraction (a = 98 s

-1)

25 cm domain

0.0E+00

1.0E-05

2.0E-05

3.0E-05

4.0E-05

5.0E-05

6.0E-05

7.0E-05

0.0 0.1 0.2 0.3 0.4 0.5

Distance from Surface (cm)

Mole Fraction

0

500

1000

1500

2000

2500

Temperature (K)

Temp

HCO

HO2

H2O2

Figure 6-42: Species profiles of HO2, H2O2, and HCO for middle stretch rate (flame with surface 

radiation).
70

Flame with Surface Radiation, Species Mole Fraction (a = 1942 s-1)

25 cm domain

0.0E+00

1.0E-04

2.0E-04

3.0E-04

4.0E-04

5.0E-04

6.0E-04

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

Distance from Surface (cm)

Mole Fraction

0

500

1000

1500

2000

2500

Temperature (K)

HCO Temp

HO2

H2O2

Figure 6-43: Species profiles of HO2, H2O2, and HCO for blow-off extinction (flame with surface 

radiation).

6.4 Effects of Gas-Phase and Surface Radiation

We now turn to the ‘full solution’ that includes both surface and gas-phase radiation in the stagnationpoint diffusion flame. This solution, using both detailed kinetics and the narrowband radiative model, 

yields the most accurate model available for the solid trioxane flame.

We solved, using the narrowband formulation for gas-phase radiative participation by CO2, CO, and H2O 

described in Section 3.4. The model includes black-body emission (ε = 1.0) and absorption (α=1.0) by 

the surface, as well as black-body emission at the oxidizer side boundary. We also use the domain of 

20 cm, used for the adiabatic flame. All cases are run in standard air and at standard atmospheric 

pressure. The surface temperature is set to 388 K, which is the boiling point of trioxane. We refer to 

these solutions in the discussion as the ‘flame and surface radiation’ solutions. 

Flammability Domain: Figure 6-44 is a plot of the maximum flame temperature versus stretch rate for 

the entire flammability domain that compares the three global flame cases: ‘adiabatic,’ ‘surface 

radiation,’ and ‘flame and surface radiation’ for the trioxane flame. The full solution isolas, including the 

unstable solutions, are presented. Note that the low stretch extinction turning point for the flame and 

surface radiation case is not shown, due to numerical difficulties described in Section 5.0 that are possibly 

caused by the transient instability at the low stretch turning point. The figure shows the expected low 

stretch extinction limit observed in Appendix 4, in contrast to the adiabatic solutions. In comparison to 

Figure A4-7, while the materials and the surface temperatures are quite different, the trend is the same. 

We observe that the solutions for the ‘flame and surface radiation’ at low stretch rates are uniformly 

lower in temperature than the ‘surface radiation’ cases, and that extinction occurs at a higher stretch rate.

The blow-off extinction limit (1944 s-1 at Tmax=1415 K) is nearly identical to that of the adiabatic solution 

(1952 s-1 at Tmax=1417 K) as well as the ‘surface radiation’ solution. In this regime, the gas-phase and 

surface losses have virtually no effect (This is further explored in the radiative flux analysis section that 

follows.). We contrast this with Figure A4-7, where surface radiation effects are small, but noticeable in 
71

the high stretch regime. As with the surface radiation solutions, the emission is proportional to the fourth 

power of temperature, and the surface emission effect is much stronger for the higher surface temperature 

PMMA solutions. Furthermore, even though gas-phase radiation is included, the flame is so thin in the 

high stretch region, that the rate of gas radiative loss is negligible in comparison to the rate of combustion 

heat generation.

Following the curve from right to left, we observe a peak temperature of 2001 K at a=22 s-1. As stretch 

rate decreases, the radiative losses begin to dominate the solution, leading to lower temperatures and 

weaker flames. Radiative quenching occurs near a=0.068 s-1 and a maximum flame temperature of 

997 K. This quench limit is higher stretch than that of the ‘surface radiation’ case, due to the additional 

flame radiation heat losses. The differences in the shapes of the low stretch curves are entirely due to 

flame radiation losses. The quench limit and the blow-off limit are not the same temperature, and both 

quench limits are due to reaction rate limitations [Bedir 1998]. The blow-off limit is due to limited 

reaction time at high stretch, while the quench limit results from too low a flame temperature because of 

radiative loss. This is further explored in the global radiative flux discussion that follows.

Flame Temperature Profiles: We calculated the maximum flame temperature for each solution in the 

domain, as shown in Figure 6-44. Several representative flame temperature profiles are given in Figure 

6-45. We then chose solutions to compare with the adiabatic and surface radiation temperature profiles in 

Figure 6-2, at the blow-off limit (turning point at a=1942 s-1) and the low stretch case (a=5.0 s-1). We also 

include the case near the radiative extinction turning point temperature profile (a=0.068 s-1), along with a 

profile from the region on the left side of the isola, above the turning point (a=1.0 s-1). The figure shows 

that the temperature profiles are quite different, due to changes in the flow regime as well as radiation 

from the system. As before, the high stretch solutions are blown against the surface. As stretch rate 

decreases, the flame moves away from the surface, and reaches a maximum temperature. The gas 

radiation effects become apparent where the flame temperature begins to deviate from the adiabatic and 

surface radiation curves (a≈100 s-1). Flame temperature begins to drop where a< 25 s-1. The drop-off in 

the standoff distance toward the surface before extinction that is observed in the PMMA study (Figure 

A4-5) is not observed here, although the solutions near the turning point demonstrate a leveling off of the 

flame standoff distance, indicative of an imminent ‘fall’ toward the surface at extinction. 

Burning Rate: The burning rate of the flame is presented in a format that allows comparison with the 

results of Appendix 4. Figure A4-8 gives the nondimensional burning rate for the PMMA fuel. Figure 

6-46a presents the burning rate in a nondimensional form that is easily compared to the results of 

Appendix 4. Here, the burning rate is presented nondimensionally in an equivalent form to Equation 

(A4-13): 

a

F

a

m

o o

x

o o

x

r m r m

0 0

| 2 |

= =

=

&

Figure 6-46(a) shows this quantity versus the stretch rate for both the adiabatic flame and the flame with 

gas phase and surface radiation. When compared to the PMMA solution in Figure A4-8, we see that the 

radiative trioxane flame behaves somewhat differently. Referring to Figure A4-6, the radiative trioxane 

flame behavior is similar to that of Case 2 (α=1,ε=0). This result is significant in that it confirms the 

findings of Appendix 4. The radiative burning rates (considering both flame and surface radiation) can be 

higher than the adiabatic burning rates, in the low stretch regime. This is the first demonstration of this 

phenomenon with a realistic solid fuel. The increased burning rate is due to the combination of the 

absorption of incident radiation from the flame, and the low surface temperature of the trioxane. In this 

case, the flame radiation absorbed by the surface dominates the surface emission. The resulting emission 

is much lower than that of the PMMA flame, so that the burning rate behavior is more like a non-emitting 

surface. At very low stretch, the burning rate decreases toward the extinction limit due to the low flame 

temperatures.
72

Figure 6-46(b) illustrates the dimensional burning rate, ru|x=0, since this is of more interest for 

engineering purposes. Note this is plotted on a log-log scale, and the effects of the radiative flame are 

apparent at the lower stretch rates. 

Species Profiles: The species profiles for the four stretch rates under examination are presented in 

Figures 6-47 through 6-59. They are ordered by increasing stretch rate, and the species are grouped by 

order of magnitude. Note changes in domain so that the full range of species profile is visible. The flame 

temperature profiles are included for reference.

The species profiles at 1944 s-1, along with the foregoing temperature profiles, establish that the ‘flame 

and surface radiation’ solutions are virtually identical to the adiabatic solutions at high stretch rates 

(greater than a=100 s-1). We present them here to confirm this fact; the reader is referred to the foregoing 

discussion on the adiabatic solutions for this stretch rate. When we compare the ‘flame and surface 

radiation’ case at a=5 s-1 to the adiabatic case at the same stretch, we see that flame radiation has reduced 

the maximum flame temperature (from 2175 K to 1906 K), but that the species profiles are as yet 

unaffected because they demonstrate the same character as the adiabatic flame.

We turn to the lower stretch solutions to examine the effects of flame and surface radiation on the species 

distribution in the flame. Scaling necessitated a different grouping of species for the low stretch cases. 

Starting with the fuel, we observe that in contrast to the ‘surface radiation’ solutions, the initial mole 

fraction of fuel at the surface is increased at the lower stretch rates, from approximately 0.55 at a =5 s-1 to 

0.7 for a = 1.0 s-1 and 0.068 s-1 (due to the increase of the nondimensional burning rate at low stretch). 

Mole fractions of formaldehyde are reduced as stretch decreases, indicating that the conversion of fuel to 

formaldehyde is slowed at decreased stretch rate (in contrast to the formaldehyde accumulation observed 

at low stretch in the surface radiation flame). 

Mole fractions of CO are reduced as stretch rate decreases, although they peak uniformly after the 

formaldehyde mole fraction peak. In addition, at higher stretch (e.g. a=5 s-1), CO is produced in larger 

quantities than formaldehyde. At a=1 s-1, they are produced in approximately equal quantities, and at 

a=0.068 s-1, formaldehyde is produced in larger quantities. This is indicative of an alternate reaction path 

at very low stretch, and is examined in the detailed reaction path analysis that follows. We observe a 

similar trend with products CO2 and H2O. H2O is produced in larger quantities at higher stretch (a=5 s-1), 

but equivalent quantities at very low stretch (a=0.068 s-1), further supporting an alternate reaction path at 

very low stretch. H2 is produced in much smaller quantities at very low stretch, in contrast to the higher 

stretch cases.

HCO, because it is the primary product of the formaldehyde decomposition, peaks closer to the surface 

than other intermediate radicals, except near the quench limit. Near radiative extinction, the HCO peak is 

located closer to the other radical peaks (Figure 6-53). Note also that, near the quenching turning point, 

the characters of the HO2 and H2O2 profiles change dramatically. The combination of changes in these 

profiles (HO2, H2O2, and HCO) lends further support to an alternate (branching) reaction path near the 

quench limit, with HO2 and H2O2 featured with formaldehyde. This is further addressed in the reaction 

pathway analysis that follows.

Among the intermediate radicals H, O, and OH, we see the emphasis on the OH radical continued from a=5 

s

-1

 to a=1 s-1. However, near the low stretch turning point, H dominates, echoing the emphasis on H at the 

high stretch turning point. The H2O2 profile has a similarity to the high stretch turning point, rather than 

other low stretch cases. Also, comparing HO2 for both turning points, both exhibit two peaks, one before 

the peak flame temperature, and one behind the peak flame temperature, although the peaks are reversed in 

relative magnitude. The peak of HO2 on the fuel side at the high stretch limit (Figure 6-59) is a result of low 

temperature and more O2 leakage (order 10-3) into the fuel side, as pointed out in Section 6.1. In comparison, 

the O2 present at the smaller fuel side HO2 peak due to diffusion at low stretch (Figure 6-53) is order 10-4.
73

Radiative Flux Analysis: The radiative fluxes and radiative source term are examined for each of the 

sample stretch rates (Figures 6-60 through 6-67) to shed some light on the effect of radiation on the 

solutions. The fluxes are labeled as follows:

• The q+ is the radiative flux from the surface in the positive x direction. At the surface, this flux is 

representative of the surface emission. Observe that q+ at x=0 is constant, due to the constant 

surface temperature. As temperature increases away from the surface, and the concentrations of 

the participating gases increase, the q+ increases as well. The subsequent decrease in q+ is due to 

absorption. At the flame edge, this flux is representative of the radiative loss from the system. 

Note that this loss at the flame edge is larger at higher temperatures (higher emission).

• The q- is the radiative flux toward the surface. At the edge of the flame, this is representative of 

the background black-body radiation. Moving from the edge to the flame, the q- increases due to 

emission from the flame. At the surface, the q- represents the incident radiation at the surface, 

which is fully absorbed by the black surface (α=1).

• The qnet is the net radiative flux: qnet=q+-q

-

. A negative q indicates a net radiative gain by the 

surface, because more radiation is absorbed than emitted.

• The dqnet/dx is the radiative source term that appears in the energy equation (Equations (3-1(f)) 

and (3-6). A positive dqnet/dx indicates a loss due to emission from the participating species in the 

flame, and a negative dqnet/dx indicates a gain due to absorption by the participating species.

Temperature profiles are provided in the plots as well for convenience. Comparing the radiative fluxes 

over the sample stretch rates, we see that for all but the blow-off turning point, there is a negative net 

radiation between the surface and the flame, indicating that emission from the flame dominates the 

surface emission. At the blow-off limit, radiation is relatively unimportant. This is more apparent in the 

following section, where radiative flux is compared to conductive flux. The qnet is positive beyond the 

flame in the presence of larger amounts of H2O and CO2 and high temperatures. This is where the flame 

emits the most radiation, as indicated by the increase in q+. The almost flat qnet profile at the blow-off 

turning point indicates that the flame is optically thin, with very little absorption. 

The radiative source term, dqnet/dx, increases with increasing temperature, and peaks at the maximum 

temperature location. In the center of the flame, the gas is emitting radiation, indicated by the positive 

source term. For all but the blow-off limit, we see a small negative region on either side of the flame, 

where the slope of qnet is negative. These are the regions where the gas is absorbing radiation.

Global Radiative Fluxes: The radiative fluxes over the entire stretch rate regime are presented in 

Figure 6-68. The radiative fluxes (similar to Section A4.4.3) presented are:

• qnet,e: net radiative flux from the flame edge to the surroundings (loss from the system)

• q

-

w: radiative flux from the flame to the surface (absorbed by black surface)

• q

+

w: radiative flux from the surface to the flame (emission from surface)

• qnet,w: net radiative flux from the surface

Note that, since the surface temperature is held constant, q+

w is constant. The net system radiative loss, qnet,e, 

peaks between a=1 and 10, and is lower at both the high stretch and low stretch limits. The maximum loss 

is not at the location of maximum flame temperature (a=22 s-1), but somewhat lower stretch rate; showing 

that the flame is not optically thin in this region and that the flame has significant absorption. The radiation 

absorbed by the surface, qw, follows the same trend, for the same reasons, since its source is the flame, as 

well. The net radiative flux from the surface, qnet,w, illustrates the feedback from the flame that leads to the 

burning rate increase at low stretch rate. The feedback from the flame to the surface is negative for most of 

the domain, indicating that the flame heat loss is a heat gain for the surface.
74

To fully understand the effect of the radiation, however, we must compare it to both the conduction at the 

surface and the heat generation in the flame. It is the decrease in the conduction to the fuel and flame heat 

generation terms in comparison to the radiation losses that lead to quenching of the flame. That is, the 

conduction to the fuel and the heat generation by the flame each decrease more rapidly than the radiation, so 

that the relative importance of the radiation increases. Figures 6-69(a), 6-69(b) and 6-69(c) illustrate this 

point. 

Figure 6-69(a) is a plot of the net radiative heat flux to fuel/conductive heat flux to fuel

0

,

=













−

x

g

net w

dx

dT

q

l

versus the stretch rate. At high stretch rate, the radiation is negligible, but as stretch decreases, the 

importance of the radiation flux from the surface becomes apparent. Figure 6-69(b) is a plot of the 

surface conduction (denominator of above) versus stretch rate. Note the log-log scale. Conduction falls 

off at low stretch rate due to the increase of flame standoff distance. The ratio in Fig 6-69(a) peaks 

between a=0.1 and 1, as the flame radiation peaks, and the conduction diminishes. At very low stretch, 

the net radiative flux to the surface falls off (Figure 6-68), leading to radiative quenching.

Figure 6-69(b) shows the ratio, net radiative heat loss/heat generation

∫ ∑

=

=

x l

x

k

k k

net e

h dx

q

0

,

w

versus stretch rate. Here, the ratio monotonically increases with decreased stretch rate, illustrating that 

the increase in radiative loss as heat generation decreases leads to flame extinction.

Maximum Flame Temperature vs. Stretch Rate (all cases)

0

500

1000

1500

2000

2500

0.01 0.1 1 10 100 1000 10000

Stretch Rate (s-1)

Maximum Flame Temperature (K)

Adiabatic

Flame and Surface Radiation

Surface Radiation

Figure 6-44: Maximum flame temperature vs. stretch rate for adiabatic flame, flame with surface 

radiation, and flame with gas phase and surface radiation.
75

Flame with Narrow Band Radiation, Temperature vs. Distance

20 cm Domain

0

500

1000

1500

2000

2500

0 3 6 9 12 15

Distance from Surface (cm)

Temperature (K)

a = 0.068

a = 1

a = 5

a = 1944 s-1

Figure 6-45: Temperature profiles for flame with gas phase and surface radiation.

Non-dimensional Burning Rate - Predict Extinction

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0.01 0.1 1 10 100 1000 10000

Stretch Rate (s-1)

Non-dimensional Burning Rate

Flame and Surface Radiation

Adiabatic

Figure 6-46(a): Nondimensional burning rate vs. stretch rate for adiabatic flame and flame with 

gas phase and surface radiation.
76

Dimensional Burning Rate - Unstable Included

0.00001

0.0001

0.001

0.01

0.1

1

0.01 0.1 1 10 100 1000 10000

Stretch Rate (s-1)

Burning Rate (g/cm

2-s)

Flame and Surface Radiation

Adiabatic

Figure 6-46(b): Dimensional burning rate vs. stretch rate for adiabatic flame and flame with gas 

phase and surface radiation.

Flame with Narrow Band Radiation Species Mole Fraction 

(a = 0.068 s-1), 20 cm Domain

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 2 4 6 8 10

Distance from Surface (cm)

Mole Fraction

0

500

1000

1500

2000

2500

Temperature (K)

Temp

C3H6O3

H2O

CO2

O2

Figure 6-47: Species profiles of C3H6O3, CO2, O2, and H2O for radiative extinction turning point 

(flame with gas phase and surface radiation).
77

Flame with Narrow Band Radiation Species Mole Fraction 

(a = 0.068 s-1), 20 cm Domain

0.00

0.02

0.04

0.06

0.08

0.10

0 2 4 6 8 10

Distance from Surface (cm)

Mole Fraction

0

500

1000

1500

2000

2500

Temperature (K)

Temp

CH2O

CO

H2

Figure 6-48: Species profiles of CH2O, CO, and H2 for radiative extinction turning point (flame 

with gas phase and surface radiation).

Flame with Narrow Band Radiation Species Mole Fraction 

(a = 1 s-1), 20 cm Domain

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 1 2 3 4 5

Distance from Surface (cm)

Mole Fraction

0

500

1000

1500

2000

2500

Temperature (K)

Temp

C3H6O3

CO H2O

CH2O

H2

CO2

O2

Figure 6-49: Major species profiles for low stretch rate (flame with gas phase and surface radiation).
78

Flame with Narrow Band Radiation Species Mole Fraction 

(a = 5 s-1), 20 cm Domain

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.5 1 1.5 2 2.5

Distance from Surface (cm)

Mole Fraction

0

500

1000

1500

2000

2500

Temperature (K)

Temp

H2O

C3H6O3

CO

CH2O CO2

H2

O2

Figure 6-50: Major species profiles for mid-stretch rate (flame with gas phase and surface 

radiation).

Flame with Narrow Band Radiation Species Mole Fraction 

(a = 1944 s-1), 20 cm Domain

0.0

0.1

0.2

0.3

0.4

0.5

0 0.02 0.04 0.06 0.08 0.1

Distance from Surface (cm)

Mole Fraction

0

500

1000

1500

2000

2500

Temperature (K)

C3H6O3

Temp

CO

H2O

CH2O

CO2

H2

O2

Figure 6-51: Major species profiles at the blow-off extinction limit (flame with gas phase and 

surface radiation).
79

Flame with Narrow Band Radiation Species Mole Fraction

(a = 0.068 s-1), 20 cm Domain

0.0E+00

4.0E-05

8.0E-05

1.2E-04

1.6E-04

0 2 4 6 8 10

Distance from Surface (cm)

Mole Fraction

0

500

1000

1500

2000

2500

Temperature (K)

Temp

H2O2

H

Figure 6-52: Species profiles of H2O2, and H for radiative extinction turning point (flame with gas 

phase and surface radiation).

Flame with Narrow Band Radiation Species Mole Fraction

(a = 0.068 s-1), 20 cm Domain

0.0E+00

2.0E-06

4.0E-06

6.0E-06

8.0E-06

1.0E-05

1.2E-05

1.4E-05

0 2 4 6 8 10

Distance from Surface (cm)

Mole Fraction

0

500

1000

1500

2000

2500

Temp

Temperature (K)

HO2

OH

HO2

HCO

O

Figure 6-53: Species profiles of HCO, OH, O, and HO2 for radiative extinction turning point (flame 

with gas phase and surface radiation).
80

Flame with Narrow Band Radiation Species Mole Fraction 

(a = 1 s

-1), 20 cm Domain

0.0E+00

4.0E-04

8.0E-04

1.2E-03

1.6E-03

0 1 2 3 4 5

Distance from Surface (cm)

Mole Fraction

0

500

1000

1500

2000

2500

Temperature (K)

Temp

OH

O

H

Figure 6-54: Species profiles of H, O, and OH for low stretch rate (flame with gas phase and 

surface radiation).

Flame with Narrow Band Radiation Species Mole Fraction 

(a = 5 s-1), 20 cm Domain

0.000

0.001

0.002

0.003

0.004

0 0.5 1 1.5 2 2.5

Distance from Surface (cm)

Mole Fraction

0

500

1000

1500

2000

2500

Temperature (K)

Temp

OH

H

O

Figure 6-55: Species profiles of H, O, and OH for mid-stretch rate (flame with gas phase and 

surface radiation).
81

Flame with Narrow Band Radiation Species Mole Fraction 

(a = 1944 s

-1), 20 cm Domain

0.000

0.001

0.002

0.003

0.004

0.005

0.006

0 0.02 0.04 0.06 0.08 0.1

Distance from Surface (cm)

Mole Fraction

0

500

1000

1500

2000

2500

Temperature (K)

Temp

H

O

OH

Figure 6-56: Species profiles of H, O, and OH at the blow-off turning extinction limit (flame with 

gas phase and surface radiation).

Flame with Narrow Band Radiation Species Mole Fraction 

(a = 1 s-1), 20 cm Domain

0.0E+00

2.0E-06

4.0E-06

6.0E-06

8.0E-06

0 1 2 3 4 5

Distance from Surface (cm)

Mole Fraction

0

500

1000

1500

2000

2500

Temperature (K)

Temp

HO2

HCO

H2O2

Figure 6-57: Species profiles of HO2, H2O2, and HCO for low stretch rate (flame with gas phase 

and surface radiation).
82

Flame with Narrow Band Radiation Species Mole Fraction 

(a = 5 s-1), 20 cm Domain

0.00E+00

2.00E-06

4.00E-06

6.00E-06

8.00E-06

0 0.5 1 1.5 2 2.5

Distance from Surface (cm)

Mole Fraction

0

500

1000

1500

2000

2500

Temperature (K)

Temp

HO2

HCO

H2O2

Figure 6-58: Species profiles of HO2, H2O2, and HCO for mid-stretch rate (flame with gas phase 

and surface radiation).

Flame with Narrow Band Radiation Species Mole Fraction 

(a = 1944 s

-1), 20 cm Domain

0.0E+00

1.0E-04

2.0E-04

3.0E-04

4.0E-04

0 0.02 0.04 0.06 0.08 0.1

Distance from Surface (cm)

Mole Fraction

0

500

1000

1500

2000

2500

Temperature (K)

Temp

HCO

HO2

H2O2

Figure 6-59: Species profiles of HO2, H2O2, and HCO at the blow-off extinction limit (flame with 

gas phase and surface radiation).
83

Flame with Narrow Band Radiation (a = 0.068 s-1) 

20 cm Domain

-2000

-1000

0

1000

2000

3000

4000

5000

0 2 4 6 8 10

Distance from Surface (cm)

Radiation Flux (W/m

2

)

-1000

-500

0

500

1000

1500

2000

2500

Temperature (K)

qnet

Temp

q

+

q

-

Figure 6-60: Radiation flux for radiative extinction point (flame with gas phase and surface radiation).

Flame with Narrow Band Radiation (a = 0.068 s-1) 

20 cm Domain

-800

0

800

1600

2400

3200

4000

0 2 4 6 8 10

Distance from Surface (cm)

Radiation Source Term (W/m

3

)

-500

0

500

1000

1500

2000

2500

Temperature (K)

Temp

dqnet/dx

Figure 6-61: dqnet/dx for radiative extinction point (flame with gas phase and surface radiation).
84

Flame with Narrow Band Radiation (a = 1 s

-1) 

20 cm Domain

-6000

-4000

-2000

0

2000

4000

6000

8000

10000

0 1 2 3 4 5

Distance from Surface (cm)

Radiation Flux (W/m

2

)

-1500

-1000

-500

0

500

1000

1500

2000

2500

Temperature (K)

q

+

q

-

qnet

Temp

Figure 6-62: Radiation flux for low stretch rate (flame with gas phase and surface radiation).

Flame with Narrow Band Radiation (a = 1 s-1) 

20 cm Domain

-4000

0

4000

8000

12000

16000

20000

0 1 2 3 4 5

Distance from Surface (cm)

Radiation Source Term (W/m

3

)

-500

0

500

1000

1500

2000

2500

Temperature (K)

dqnet/dx

Temp

Figure 6-63: dqnet/dx for low stretch rate (flame with gas phase and surface radiation).
85

Flame with Narrow Band Radiation (a = 5 s-1) 

20 cm Domain

-6000

-4000

-2000

0

2000

4000

6000

8000

10000

0 0.5 1 1.5 2 2.5

Distance from Surface (cm)

Radiation Flux (W/m

2

)

-1500

-1000

-500

0

500

1000

1500

2000

2500

Temperature (K)

q

+

q

-

qnet

Temp

Figure 6-64: Radiation flux for mid-stretch rate (flame with gas phase and surface radiation).

Flame with Narrow Band Radiation (a = 5 s-1) 

20 cm Domain

-5000

0

5000

10000

15000

20000

25000

0 0.5 1 1.5 2 2.5

Distance from Surface (cm)

Radiation Source Term (W/m

3

)

-500

0

500

1000

1500

2000

2500

Temperature (K)

dqnet/dx

Temp

Figure 6-65: dqnet/dx for mid-stretch rate (flame with gas phase and surface radiation).
86

Flame with Narrow Band Radiation (a = 1944 s

-1) 

20 cm Domain

0

400

800

1200

1600

0 0.02 0.04 0.06 0.08 0.1

Distance from Surface (cm)

Radiation Flux (W/m

2

)

0

500

1000

1500

2000

2500

Temperature (K)

Temp

q

+

qnet

q

-

Figure 6-66: Radiation flux at the blow-off extinction limit (flame with gas phase and surface 

radiation).

Flame with Narrow Band Radiation (a = 1944 s

-1) 

20 cm Domain

-4000

0

4000

8000

12000

16000

20000

0 0.02 0.04 0.06 0.08 0.1

Distance from Surface (cm)

Radiation Source Term (W/m3)

-500

0

500

1000

1500

2000

2500

Temperature (K)

Temp

dqnet/dx

Figure 6-67: dqnet/dx at the blow-off extinction limit (flame with gas phase and surface radiation).
87

Radiation Fluxes, Gas Phase and Surface Radiation

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0.01 0.1 1 10 100 1000 10000

Stretch Rate, (s-1)

Radiation Flux (W/cm

2

)

qnet,e

qnet,w

q

+

,w

q

-

,w

Figure 6-68: Heat flux breakdown vs. stretch rate for the flame with gas phase and surface radiation.

(qnet,e=net radiative flux from flame edge to the surroundings; qw= radiative flux from flame to surface; 

q

+

w=radiative flux from surface to flame; qnet,w=net radiative flux from surface).

Ratio of Surface Radiation to Surface Conduction

0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

0.01 0.1 1 10 100 1000 10000

Stretch Rate (s-1)

Surface Radiation/Surface Conduction

Figure 6-69(a): Ratio of net radiative heat flux to the solid to conductive heat flux to the solid vs. 

stretch rate.
88

Surface Conduction

0.1

1

10

100

1000

0.01 0.1 1 10 100 1000 10000

Stretch Rate (s-1)

Conduction at Surface (W/cm

2

)

Figure 6-69(b): Surface conduction vs. stretch rate.

Ratio of Net Radiative Heat Loss to Heat Generation

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.01 0.1 1 10 100 1000 10000

Stretch Rate (s-1)

Net Radiative Heat Loss/Heat Generation

Figure 6-69(c): Ratio of net radiative heat loss to the total heat generation vs. stretch rate.
89

6.5 Sensitivity and Reaction Path Analysis for Flame With Gas Phase and 

Surface Radiation

We performed sensitivity and detailed analyses on the narrowband flame results to discover the differences 

in the driving kinetic mechanisms responsible for the structure of the solid trioxane combustion in the 

stagnation-point flow. We also compared the results, where possible, with the original homogeneous 

kinetics study [Hochgreb 1991].

The parameters of interest here are the same as outlined in Section 6.2; namely: species production rates, 

reaction rates of progress, reaction heat release, and sensitivity to stretch rate. These are examined for 

three stretch rates (a=0.068 s-1, a= 1 s-1, and a=5 s-1). For consistency, these are the same stretch rates 

used in the discussion in Section 6.4. The reader is referred to the adiabatic discussion for the higher 

stretch cases, since the ‘flame and surface radiation’ case and adiabatic case are identical at high stretch 

(greater than a=100 s-1).

Species Production Rates and Reaction Rates of Progress. The production rates for each species are 

given in Figures 6-70, 6-71, and 6-72 for the sample stretch rates. In each plot, positive production rates 

indicate species production, and negative production rates indicate species consumption. Following these 

plots are the relevant reaction rates of progress of the sample stretch rates (Figures 6-73(a-c), 6-74(a-c), 

6-75(a-e)). Positive values indicate a forward progressing reaction and negative values indicate a reverse 

progressing reaction.

As before, we see in each production rate plot that trioxane decomposes in the same region in which 

formaldehyde is produced, via reaction 1. This occurs at reduced rates, as stretch rate decreases. The 

temperatures are low in this region of the flame (715 K, 823 K and 847 K at the location of maximum 

consumption of fuel for a=0.068, 1, and 5 s-1, respectively). This is followed by consumption of 

formaldehyde, with the primary products being CO and H2. This occurs at somewhat higher temperatures 

(895 K, 1035 K, and 1105 K for a=0.068, 1, and 5 s-1, respectively). CO and H2 are eventually converted 

to CO2 and H2O, respectively, in the high-temperature region of the flame. 

The rates of progress for the formaldehyde decomposition, reactions 2 through 8, are presented in Figures 

6-73(a), 6-74(a), and 6-75(a) for each of the sample stretch rates. As with the adiabatic flame, reactions 5 

and 7 consistently dominate the major formaldehyde consumption, with reaction 5 having the primary 

role. Subsequently, formyl radical is consumed to form H2 and CO. At a=1 and 5 s-1, consistent with the 

adiabatic flame, the production of H2 and CO is attributed to the decomposition of HCO via reactions 9 

and 11, rather than formaldehyde decomposition (reaction 3). However, near the quench limit stretch 

rate, while reaction 9 remains important, reaction 10 overtakes reaction 11 as the branching path due to 

the increasing leakage of O2 when approaching the quench limit.

(9) HCO+M→H+CO+M

(10) HCO+O2→CO+HO2

This is seen in the HCO reaction rates of progress (Figures 6-73(b), 6-74(b), and 6-75(b)) and is 

suggestive of Hochgreb’s ‘low-temperature’ reaction path, discussed in Sections 4 and 6.4. (Note that 

reaction 9 is plotted on the right-hand scale for Figures 6-73(b), 6-74(b) and 6-75(b).) Also, near the 

turning point, CO is produced at almost the same rate as CH2O, which is explained by the emphasis here 

on reactions 9 and 10. As before, there is a zone of radical pool formation and then recombination into 

final products, CO2 and H2O.

The primary O2 consumption zone is located after the fuel consumption zone. The O2 reaction rates of 

progress are shown in Figures 6-73(c), 6-74(c), and 6-75(c). For a=5 s-1, reaction 20 is the dominant 

reaction, and is the primary source of radical pools, producing OH and O. At the lower stretch rates, 

reaction 38 asserts itself as the dominant reaction for O2 consumption. At a=1 s-1, reactions 20 and 38 
90

have very similar rates of progress; near the quench limit (a=0.068 s-1), reaction 38 dominates. In contrast 

to reaction 20, reaction 38 is exothermic (see heat release discussion). Another interesting feature is the 

O2 consumption layer on the fuel side of the flame. We see evidence of this by reaction 10. This reaction 

asserts itself in the two low temperature regions of the flame. We see some evidence at the high stretch 

turning point (illustrated and discussed in Sections 6.1, 6.2, and 6.4). At the low temperature quench limit 

(6-73(b) and (c)), it is more apparent. 

Figures 6-73(d) and 6-73(e) illustrate the main reaction pathways for H2O2 and HO2, respectively, for the 

case near the quench limit. There are two distinct reaction zones for H2O2 here. In the radical pool (near 

the peak temperature), reaction 26 is the main producer of H2O2 and reaction 36 is the main consumer. 

Closer to the flame, reaction 8 produces H2O2 in the decomposition of formaldehyde via the branching 

path. We again see reaction 8 in the consumption of HO2, as shown in Figure 6-73(e). Reaction 38 is the 

primary producer of HO2 in the radical pool, while reaction 28 is the primary consumer here. HO2 is also 

active on the fuel side of the flame, with reactions 10 and 8 noteworthy on the fuel side of the radical 

pool.

The production rates increase by an order of magnitude for each sample stretch rate, starting from the low 

stretch case and increasing. The low stretch extinction limit retains the distinct zones of formaldehyde 

consumption and radical production; unlike the high stretch extinction limit, which is characterized by 

overlap of the formaldehyde consumption and radical production zones, ascribed to the shortened 

residence time at the high stretch rate.

In conclusion, the main reaction paths follow the route illustrated in the adiabatic flame discussion. 

However, a branching path appears near the quench limit, illustrated in Figure 6-80. As noted earlier, 

traces of this path are also observable at the high stretch extinction limit in the low-temperature portion of 

the flame. This is Hochgreb’s ‘low-temperature’ mechanism.

Low stretch radiative branching path:

(5) CH2O+H→HCO+H2

(9) HCO+M→H+CO+M

(10) HCO+O2→CO+HO2

(8) CH2O+HO2→HCO+H2O2

(36) H2O2+M→OH+OH+M

Heat Release: The total heat release over the domain for the sample stretch rates is presented in Figures 

6-76(a), 6-77(a), and 6-78(a). Temperature profiles are included for comparison. Figures 6-76(b), 

6-77(b), and 6-78(b) contain the heat release by reaction to understand the primary contributors to the 

total heat release. Positive values indicate exothermicity; negative values indicate endothermicity.

As with the adiabatic solutions, all of the profiles exhibit an endothermic zone nearer to the surface, and 

an exothermic zone beyond. However, as stretch rate decreases, the endothermic zone is greatly reduced, 

so that it is no longer comparable to the exothermic zone. The endothermic zone corresponds to the fuel 

pyrolysis region, and since less fuel is pyrolyzed, the endothermicity is reduced. In the formaldehyde 

consumption zone, the exothermicity of reaction 5 counterbalances the endothermicity of reaction 9, 

although at low stretch, the magnitudes are much reduced in comparison to the adiabatic flame.

The small plateau created by the consumption of formaldehyde (reaction 5) being opposed by the thermal 

decomposition of HCO (reaction 9) at a=5 s-1 essentially disappears near the quench limit. Reactions 16 

and 23, which also contribute to this plateau, are only visible in the primary heat release zone near the 

quench limit. 
91

At the higher stretch cases (a=1, 5 s-1), reaction 35 (producing H2O from the recombination of H and OH) 

accounts for more than half of the heat release, consistent with the adiabatic flame. Also, as before, 

reactions 35, 38, and 23 all lead to H2O formation, indicating that the primary heat release is due to 

formation of H2O, rather than CO2. Reaction 20 accounts for the primary endothermic response in this 

zone.

At the low stretch quenching limit, reaction 38 replaces reaction 35 as the major source of heat release, 

consistent with the low temperature path described above. Reaction 20 no longer plays a part in the heat 

balance, so that there is no longer a large endothermic response to counter the endothermic reactions.

Eigen-Value Sensitivity: The results of the normalized Eext sensitivity are presented in Figure 6-79 for 

the low stretch turning point (quench limit). At the low stretch turning point (a=0.068 s-1), the 

temperature is very low (Tmax =997 K) and the flame is far from the surface, and the molecular interaction 

is inhibited more than anywhere in the flame domain due to low flow and low temperature. In contrast to 

the high stretch turning point sensitivity, the formaldehyde and HCO decompositions are not emphasized 

here. Eext is instead sensitive to reactions 16, 20, 28, 29, and 38. While these reactions also appear in the 

high stretch sensitivity, they receive more emphasis here. We know from Figure 6-73(c), that reactions 

20 and 38 have a net rate of progress in the forward direction and are competing chain branching and 

termination reactions, with reaction 38 playing the stronger role. Reactions 16, 20, and 28 promote the 

reaction, producing radicals H and OH (reaction 20 produces H via the net cycle with OH and O via 

reactions 21 and 23). Reactions 29 and 38 are chain terminators and retard the reactions, consuming H. 

The highly reactive H radical appears to be the key to reaction progress here as well as at the high stretch 

turning point, although the OH radical (via forward reaction 28) has more of a role than at the high stretch 

turning point. Figures 6-52 and 6-53 from Section 6.4 show the species profiles for H, O and OH at the 

quench limit, and H has by far the largest mole fraction in the flame zone among the three radicals, with 

OH next in importance. Unlike the high stretch limit, where the radical pool extends past the peak 

temperature location, the radical pool zone at the low stretch limit is located with the peak temperature 

zone. Also note that here, as with the high stretch extinction limit, no sensitivity to reaction 1 is evident, 

because the trioxane decomposition is well separated from the formaldehyde decomposition.

Another feature of note here is reaction 36. This reaction, while of minor importance in terms of 

sensitivity, has switched directions of progress from the blow-off extinction results. At blow-off, the 

reaction progresses in the reverse direction, producing H2O2, and slightly retarding the overall reaction. 

At the low stretch radiative quench limit, the reaction progresses in the forward direction, producing OH, 

and thereby promotes the overall reaction progress.
92

Flame with Narrow Band Radiation, Production Rates

(a = 0.068 s-1), 20 cm Domain

-5.00E-06

-4.00E-06

-3.00E-06

-2.00E-06

-1.00E-06

0.00E+00

1.00E-06

2.00E-06

3.00E-06

4.00E-06

5.00E-06

0 2 4 6 8 10

Distance from Surface (cm)

Production Rate (mol/cm

3-s)

H2O

CO2

H2

CO

CH2O

Fuel

CH2O

O2

O2

H

H2

CO

Figure 6-70: Flame with gas phase and surface radiation, production rates by reaction for 

radiative extinction.

Flame with Narrow Band Radiation, Production Rates

(a = 1 s-1), 20 cm Domain

-4.0E-05

-3.0E-05

-2.0E-05

-1.0E-05

0.0E+00

1.0E-05

2.0E-05

3.0E-05

4.0E-05

5.0E-05

0 1 2 3 4 5

Distance from Surface (cm)

Production Rate (mol/cm

3-s)

CH2O

H2

CO

Fuel

H2O

CO2

H

H2

CH2O

O

O2

OH

CO

H2,CO2

H2O,CO

Figure 6-71: Flame with gas phase and surface radiation, production rates by reaction for low 

stretch rate.
93

Flame with Narrow Band Radiation, Production Rates

(a = 5 s-1), 20 cm Domain

-2.0E-04

-1.5E-04

-1.0E-04

-5.0E-05

0.0E+00

5.0E-05

1.0E-04

1.5E-04

2.0E-04

2.5E-04

3.0E-04

0 0.5 1 1.5 2 2.5

Distance from Surface (cm)

Production Rate (mol/cm

3-s)

CH2O

H2

CO

Fuel

H2O

CO2 H

H2

CH2O

O

O2

OH

CO

Figure 6-72: Flame with gas phase and surface radiation, production rates by reaction for mid 

stretch rate.

Flame with Narrow Band Radiation, CH2O Rate of Progress by 

Reaction (a = 0.068 s-1), 20 cm Domain

0.0E+00

2.0E-07

4.0E-07

6.0E-07

8.0E-07

1.0E-06

1.2E-06

1.4E-06

0 2 4 6 8 10

Distance from Surface (cm)

Rate of Progress (mol/cm

3-s)

5

7

8

6

Figure 6-73(a): Flame with gas phase and surface radiation, CH2O consumption rate of progress 

by reaction for radiative extinction.
94

Flame with Narrow Band Radiation, HCO Rate of Progress by Reaction 

(a = 0.068 s-1), 20 cm Domain

-1.0E-08

0.0E+00

1.0E-08

2.0E-08

3.0E-08

4.0E-08

5.0E-08

6.0E-08

7.0E-08

8.0E-08

0 2 4 6 8 10

Distance from Surface (cm)

Rate of Progress (mol/cm

3-s)

-2.0E-07

0.0E+00

2.0E-07

4.0E-07

6.0E-07

8.0E-07

1.0E-06

1.2E-06

1.4E-06

1.6E-06

Rate of Progress (mol/cm

3-s)

11

15

9 (Right Hand Axis)

10

Figure 6-73(b): Flame with gas phase and surface radiation, HCO consumption rate of progress by 

reaction for radiative extinction.

Flame with Narrow Band Radiation, Rate of Progress by O2 Reaction

(a = 0.068 s-1), 20 cm Domain

-5.0E-07

0.0E+00

5.0E-07

1.0E-06

1.5E-06

2.0E-06

2.5E-06

3.0E-06

3.5E-06

0 2 4 6 8 10

Distance from Surface (cm)

Rate of Progress (mol/cm

3-s)

38

30

29

31

20

10

Figure 6-73(c): Flame with gas phase and surface radiation, O2 consumption rate of progress by 

reaction for radiative extinction.
95

Flame with Narrow Band Radiation, Rate of Progress by H2O2 Reaction

(a = 0.068 s-1), 20 cm Domain

-5.0E-09

0.0E+00

5.0E-09

1.0E-08

1.5E-08

2.0E-08

2.5E-08

0 2 4 6 8 10

Distance from Surface (cm)

Rate of Progress (mol/cm

3-s)

26

36

8

36

26

25

25

Figure 6-73(d): Flame with gas phase and surface radiation, H2O2 consumption rate of progress by 

reaction for radiative extinction.

Flame with Narrow Band Radiation, Rate of Progress by HO2 Reaction

(a = 0.068 s-1), 20 cm Domain

0.00E+00

1.00E-07

2.00E-07

3.00E-07

4.00E-07

5.00E-07

6.00E-07

0 2 4 6 8 10

Distance from Surface (cm)

Rate of Progress (mol/cm

3-s)

0.0E+00

5.0E-07

1.0E-06

1.5E-06

2.0E-06

2.5E-06

3.0E-06

3.5E-06

Rate of Progress (mol/cm

3-s)

38 (secondary axis)

29

28 (secondary axis)

30

10

8 32

Figure 6-73(e): Flame with gas phase and surface radiation, HO2 consumption rate of progress by 

reaction for radiative extinction.
96

Flame with Narrow Band Radiation, CH2O Rate of Progress by 

Reaction (a = 1 s-1), 20 cm Domain

0.0E+00

5.0E-06

1.0E-05

1.5E-05

2.0E-05

2.5E-05

3.0E-05

3.5E-05

0 1 2 3 4 5

Distance from Surface (cm)

Rate of Progress (mol/cm

3-s)

5

7

Figure 6-74(a): Flame with gas phase and surface radiation, CH2O consumption rate of progress 

by reaction for low stretch rate.

Flame with Narrow Band Radiation, HCO Rate of Progress by Reaction 

(a = 1 s-1), 20 cm Domain

0.0E+00

5.0E-08

1.0E-07

1.5E-07

2.0E-07

2.5E-07

0 1 2 3 4 5

Distance from Surface (cm)

Rate of Progress (mol/cm

3-s)

0.0E+00

5.0E-06

1.0E-05

1.5E-05

2.0E-05

2.5E-05

3.0E-05

3.5E-05

11

9 (Right Hand Axis)

10

14

Figure 6-74(b): Flame with gas phase and surface radiation, HCO consumption rate of progress by 

reaction for low stretch rate.
97

Flame with Narrow Band Radiation, Rate of Progress by O2 Reaction

(a = 1 s-1), 20 cm Domain

-2.0E-06

0.0E+00

2.0E-06

4.0E-06

6.0E-06

8.0E-06

1.0E-05

1.2E-05

1.4E-05

1.6E-05

0 1 2 3 4 5

Distance from Surface (cm)

Rate of Progress (mol/cm

3-s)

20

38

30 29

31

Figure 6-74(c): Flame with gas phase and surface radiation, O2 consumption rate of progress by 

reaction for low stretch rate.

Flame with Narrow Band Radiation, CH2O Rate of Progress by 

Reaction (a = 5 s

-1), 20 cm Domain

0.0E+00

4.0E-05

8.0E-05

1.2E-04

1.6E-04

0.0 0.5 1.0 1.5 2.0 2.5

Distance from Surface (cm)

Rate of Progress (mol/cm

3-s)

5

7

Figure 6-75(a): Flame with Gas Phase and Surface Radiation, CH2O Consumption Rate of 

Progress by Reaction for Mid Stretch Rate.
98

Flame with Narrow Band Radiation, HCO Rate of Progress by Reaction 

(a = 5 s-1), 20 cm Domain

0.0E+00

4.0E-07

8.0E-07

1.2E-06

1.6E-06

2.0E-06

0 0.5 1 1.5 2 2.5

Distance from Surface (cm)

Rate of Progress (mol/cm

3-s)

0.0E+00

4.0E-05

8.0E-05

1.2E-04

1.6E-04

11

9 (Right Hand Axis)

10

14

Figure 6-75(b): Flame with gas phase and surface radiation, HCO consumption rate of progress by 

reaction for mid stretch rate.

Flame with Narrow Band Radiation, Rate of Progress by O2 Reaction

(a = 5 s-1), 20 cm Domain

-1.0E-05

0.0E+00

1.0E-05

2.0E-05

3.0E-05

4.0E-05

5.0E-05

6.0E-05

0 0.5 1 1.5 2 2.5

Distance from Surface (cm)

Rate of Progress (mol/cm

3-s)

20

38

30

29

31

Figure 6-75(c): Flame with gas phase and surface radiation, O2 consumption rate of progress by 

reaction for mid stretch rate.
99

Flame with Narrow Band Radiation, Total Heat Release 

(a = 0.068 s-1), 20 cm Domain

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

0.0 3.0 6.0 9.0 12.0 15.0

Distance from Surface (cm)

Heat Release (J/cm

3-s)

-500

0

500

1000

1500

2000

2500

Temp

Total Heat Release

Figure 6-76(a): Flame with gas phase and surface radiation, total heat release for radiative extinction.

Flame with Narrow Band Radiation, Heat Release by Reaction

(a = 0.068 s-1), 20 cm Domain

-0.5

0.0

0.5

1.0

1.5

2.0

0 2 4 6 8 10

Distance from Surface (cm)

Heat Release (J/cm

3-s)

Total Heat Release

5

35

1 9

23

38

16

20

Figure 6-76(b): Flame with gas phase and surface radiation, heat release by reaction for radiative extinction.
100

Flame with Narrow Band Radiation, Total Heat Release 

(a = 1 s-1), 20 cm Domain

-2

0

2

4

6

8

10

0 1 2 3 4 5

Distance from Surface (cm)

Heat Release (J/cm

3-s)

-500

0

500

1000

1500

2000

2500

Temp

Total Heat Release

Figure 6-77(a): Flame with gas phase and surface radiation, total heat release for low stretch rate.

Flame with Narrow Band Radiation, Heat Release by Reaction

(a = 1 s-1), 20 cm Domain

-4

-2

0

2

4

6

8

10

12

0 1 2 3 4 5

Distance from Surface (cm)

Heat Release (J/cm

3-s)

Total Heat Release

35

1

9

23

38

16 5

20

16

Figure 6-77(b): Flame with gas phase and surface radiation, heat release by reaction for low 

stretch rate.
101

Flame with Narrow Band Radiation, Total Heat Release 

(a = 5 s-1), 20 cm Domain

-18

-12

-6

0

6

12

18

24

30

0.0 0.5 1.0 1.5 2.0 2.5

Distance from Surface (cm)

Heat Release (J/cm

3-s)

-1500

-1000

-500

0

500

1000

1500

2000

2500

Temperature (K)

Temp

Total Heat Release

Figure 6-78(a): Flame with gas phase and surface radiation, total heat release for mid stretch rate.

Flame with Narrow Band Radiation, Heat Release by Reaction

(a = 5 s-1), 20 cm Domain

-15

-10

-5

0

5

10

15

20

25

30

0.0 0.5 1.0 1.5 2.0 2.5

Distance from Surface (cm)

Heat Release (J/cm

3-s)

Total Heat Release

35

1

9

23

38

16

5

20

16

Figure 6-78(b): Flame with gas phase and surface radiation, heat release by reaction for mid 

stretch rate.
102

Flame with Narrow Band Radiation, Eext Sensitivity

(a = 0.068 s-1), 20 cm Domain

-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0

1

3

5

7

9

11

13

15

17

19

21

23

25

27

29

31

33

35

37

Reaction Number

Normalized Eext Sensitivity

2

4

6

8

1 0

1 2

1 4

1 6

1 8

2 0

2 2

2 4

2 6

2 8

3 0

3 2

3 4

3 6

3 8

Figure 6-79: Flame with gas phase and surface radiation, normalized Eext sensitivity by reaction for 

radiative extinction.
103

C3H6O3

CH2O

HCO

 CO

O + OH

H

(rxn 1)

(rxn 5)

(rxn 9)

(rxn 6,7)

+M

 + O2

(rxn 20)

HO2

H2O2

(rxn 8)

(rxn 10)

(rxn 8) OH

(rxn 36)

+O2

High Temperature Path

Low Stretch Radiative 

Branching Path

+M

Figure 6-80: Reaction pathways for high-temperature path and chain-branching reaction paths for 

trioxane diffusion flame near low stretch extinction limit.

6.6 Conclusions

We presented an accurate and comprehensive model of the solid trioxane combustion in a stagnationpoint flow configuration, using the narrowband model for the radiative heat transfer and detailed kinetics 

for the trioxane decomposition. In addition, we presented detailed reaction path analyses and sensitivity 

at the extinction turning points for the trioxane diffusion flame.

The flame with gas phase and surface radiation was compared to the adiabatic solution and a solution 

with surface radiation only. This study confirms and emphasizes previous parametric results (Appendix 

4) showing that radiative effects (both gas phase and surface) are amplified at low stretch rate and can 

lead to extinction (quenching). Gas phase radiation effects are amplified at low stretch due to (1) a 

decrease in flame heat generation, brought about by the large flame standoff distance and (2) a longer 

optical length, also due to the thicker radiating layer. But the increasing loss from the flame lowers the 

flame temperature. Thus, the net amplification of gas radiation at low stretch is somewhat modulated by 

this interactive flame-radiation effect. 

The stagnation-point trioxane diffusion flame in air has two distinct extinction limits. The high stretch 

blow-off turning point was found to occur at a=1952 s-1 at a maximum flame temperature of 1417 K. The 

low stretch turning point (quench limit) is near a=0.068 s-1 at a maximum flame temperature of 997 K.

Reaction path analysis shows that a straight-chain process dominates the trioxane decomposition over the 

entire flammability regime with the exception of near the extinction limits. Here, a branching path asserts 

itself due to low temperatures. Sensitivity analysis at the turning points shows that the extinction flow 
104

parameter (the radial pressure curvature) is sensitive to the fuel, formaldehyde, and formyl radical 

decomposition reactions at the high stretch extinction limit, but not near the low stretch extinction limit. 

The key to the rate of progress at both extinction limits is the H radical, because of its high reactivity. 

Table 6-1 is a summary of the key features of the extinction limits.

Table 6-1: Summary of the Key Features of the Trioxane Extinction Limits

Extinction Characteristics Trioxane Flame Type High Stretch Low Stretch

Adiabatic

Low temperature (1400K)

Low residence time

Close to Surface

Narrow flame

Straight-chain reaction path

No extinction

Surface Radiation

Low temperature (1400K)

Low residence time

Close to surface

Narrow flame

Low temperature (1000K)

High residence time

Very far from surface

Very wide flame

Surface And Gas Phase 

Radiation

Low temperature (1400K)

Low residence time

Close to surface

Narrow flame

Straight-chain reaction path

Low temperature (1000K)

High residence time

Far from surface

Wide flame

Branching ‘low temperature’ 

reaction path

We conclude that, while the parametric studies of Appendix 4 give much insight into the radiative physics 

of the flame, including detailed kinetics with an equally accurate radiation model allows more accurate 

extinction analysis and yields insight into the detailed chemistry and sensitivity at extinction.

While Appendix 4 gave us some indication that it is possible for a radiative flame to have a higher 

burning rate than the adiabatic counterpart, here we have demonstrated this fact with a realistic fuel. 

Moreover, we see that flame radiation must be included to observe this effect—the flame with surface 

radiation only does not exhibit the higher burning rate, since the increase in burning rate at low stretch is 

due to feedback of the flame radiation to the surface. 

The activity of the radical intermediate species, particularly at high stretch, where these species are blown 

against the surface, lead us to question the role these may play in the surface heat balance. In the next 

section, we investigate the possible effects of radical recombination at the solid surface.
105

7.0 Surface Radical Interactions in Solid Trioxane Combustion

7.1 Overview

As observed in Section 6, the flame interaction with the solid surface can be critical in the determination 

of the flame structure and extinction boundaries. In Appendix 4, we parametrically examined the role of 

gas-phase radiation and the extremes of surface radiative properties on the solid surface combustion. In 

Section 6, we examined in detail the role of flame radiation and detailed kinetics in combustion of a solid. 

We now turn to the examination of intermediate radical interaction with the solid surface. Solid surfaces 

can induce chemical reactions via either catalytic behavior or third-body behavior. Here we investigate 

the possible effects of surface-induced radical recombination in the solid trioxane flame. 

Evidence indicates that intermediate radicals produced during combustion may have an effect on the 

flame structure and extinction boundaries because of recombination at the surface. We investigate the 

role of surface quenching of radicals in this section to determine whether this effect plays a significant 

role in the heat balance at the trioxane flame surface. We also consider the combined effects of surface 

radical interactions and surface radiative emission over the flammability regime of solid trioxane burning 

in air. The trioxane flame computational model of Section 3 is adapted to allow for surface radical 

recombination.

7.2 Nomenclature

The nomenclature in this section is adapted from Section 3, with the following additions:

Subscripts:

k non-fuel species (as in Section 3)

I non-fuel species that do not react at the surface (inert)

r radical species that may react at the surface

7.3 Background

Vlachos and coworkers are leading an effort [Bui-Pham and Seshardri 1990, Vlachos et al. 1993, and 

Vlachos et al. 1994, Aghalayam et al. 1998, Fernandes et al. 1999, Gummalla et al.1, Gummalla et al.2, 

Park et al.] to understand the effects of interactions of radicals with a solid surface on flame structure, 

particularly on the ignition and extinction behaviors of various flames. Vlachos et al. examined the role 

of intermediate species interacting with a solid surface in a methane-air flame [1994] and a hydrogen-air 

flame [1993]. The surface is modeled as a sink for various radicals in order to gauge the effect of radical 

absorption by the surface. Each radical species is selectively removed from the surface (its surface 

concentration is set to zero) and the effect on ignition and extinction temperatures is observed in the premixed flame. We assume an infinitely fast radical destroying mechanism at the surface. This model is 

not intended to simulate a physical situation, but to indicate the potential effect of each radical on the 

kinetics when surface recombination can occur. The investigators found that ignition and extinction 

temperatures can be altered significantly by surface chemistry. For the case of extinction in a hydrogenair flame, the H radical had by far the largest effect in increasing extinction temperature, while O and OH 

had lesser effects. For the case of extinction in a methane-air flame, again the H radical plays the largest 

role in increased extinction temperature, particularly under fuel-rich conditions. O and OH also increase 

extinction temperature, but to a lesser extent. Other radicals, particularly CH3, have a strong role in 

increasing ignition temperature.
106

Aghalayam et al. [1998] extended the treatment of the surface reactions to a cycle of absorption-surface 

reaction-desorption (of stable species) and included the heat of reaction. They investigated the role of 

wall quenching in ignition, extinction, and auto-thermal behavior in a premixed hydrogen-air flame 

impinging on a flat surface. They found that, for ignition, radical quenching retards the system due to the 

kinetics of the surface reaction. Extinction is also retarded due to thermal feedback from the 

recombination of radicals at the wall. 

The investigators propose a mechanism that involves removing H, O, OH, HO2, and H2O2 radicals at the 

surface via reaction at vacant sites. ‘Sticking’ coefficients are assumed to be unity (a surface ‘clean’ of 

radicals), to provide the fastest rate of radical removal (the most extreme case). The pre-exponential term 

is taken to be 1013 s-1, which the authors estimate from transition state theory. The surface recombination 

reactions have zero activation energy. These assumptions result in an extreme (maximum) estimate of the 

potential effects of surface radical recombination on the heat balance and flame structure.

All real surfaces behave as sources and sinks for radicals. To determine the potential effect of a 

chemically reactive surface, the investigators [Aghalayam et al. 1998] looked at three cases for the 

premixed flame:

1. A reactive surface. Capable of absorbing out radicals from the gas phase (a sink for radicals). In

this case, surface diffusion assists radical recombination into stable species. Since the formation of 

the stable species is exothermic, heat is released by the surface to the flame.

2. An inert surface. This is the base case used for comparison. Only the gas-phase reactions are used 

and the diffusive flux of each species at the wall is zero. This can be considered an ‘extreme’ case 

in the same sense as a radiatively nonparticipating surface (no absorption or emission).

3. A reactive surface with heat of reaction set to zero, to understanding the effect of the heat of surface 

reaction (as compared to case 1). 

The investigators also examined the recombination of one radical at a time at the wall to isolate the most 

important radicals.

The investigators found that, while the actual mechanism used for the radical recombination may be 

debated, these details are less important to the outcome than the specific radical that is quenched and the 

effect of the heat of recombination. 

Quenching of H radicals: For ignition, the radical recombination case requires higher temperature and 

power to ignite, indicating that the radical quenching of H radicals at the wall tends to make the mixture 

less ignitable (retards ignition). The heat of reaction at the surface shows no effect so that the ignition 

retardation is due entirely to the kinetics of radical quenching at the surface. 

For extinction, heat loss due to radical recombination does have an effect. The flame is sustained at a 

lower power (supplied to the surface) due to heat feedback from the surface reactions. Temperatures 

across the flame are higher (due to the heat of absorption of H). Concentrations of H are lower, and H2

higher near the surface when the radical recombination is considered. The investigators found a trade-off 

effect for extinction: Radical recombination kinetically retards the flame; however, the heat feedback 

from the radical recombination counters this effect and sustains the flame.

The closer a flame is to extinction, the more important the thermal and chemical coupling between the 

gas-phase and the surface becomes. Away from the extinction limit, the coupling is weak.

Quenching of other radicals: Fuel-rich and fuel-lean cases were considered in evaluation of the effect of 

other radicals. For the fuel-lean case, H, O, and OH have important effects (as discussed above) on 

ignition and extinction. The effect of H radical is the most pronounced, with O and OH having 
107

decreasing effects in that order. For the fuel-rich case, only the H radical played any significant role in 

ignition and extinction effects because the concentrations of O and OH are relatively small in a fuel-rich 

situation. The effect of thermal feedback on extinction behavior is only important in cases where the 

radical concentrations are sufficiently high.

The role of strain rate: An increase in strain rate tends to contract the non-extinction zone of the flame 

(making ignition harder), and thus the effect of wall radical recombination has the same qualitative effect, 

although more pronounced. Ignition near a reactive surface requires a higher ignition source at higher 

strain rates, compared to an inert surface at the same conditions.

Effect of surface radiation: The investigators studied surface radiation in the auto-thermal regime (selfsustained flame) using a surface emissivity of 0.3 and a surface temperature of 1000 K. Since they 

considered only emissivity, which represents a heat loss to the system, they found the auto-thermal 

regime greatly reduced and the effect of surface radical recombination is thus less significant. 

Mass balance adjustments: A comment is warranted here on maintaining a mass balance in the model. 

When the mass fraction of a species is set to zero at the solid boundary to simulate a reactive surface 

(sink), the mass balance is not strictly maintained unless the product of the recombined species is adjusted 

as well. Warnatz [1978] noted that, for a species, or combination of species, having significant 

interactions at the surface, the reactants and products need to be accounted for in the mass balance so that 

the mass is adjusted for strict accounting of radicals reacting at the surface. The studies cited here did not 

make use of this adjustment because the mass of the species of interest did not change enough to 

necessitate this adjustment.

7.4 Formulation of Surface Radical Interactions in Solid Trioxane Combustion

Application to solid trioxane combustion: Here we examine the effect of radical recombination on the 

solid trioxane surface by adapting the ‘radical sink’ technique. Nothing is known about the activity of 

surface radicals on the burning of trioxane, so we consider the potential extremes in this situation to 

investigate if further study into this effect is merited. 

Starting with the approach described by Vlachos et al. [1993, 1994], we examine the effect of the radicals 

produced in the trioxane decomposition, namely H, O, OH, HO2, H2O2, and HCO, in order to observe the 

effect on the flame. 

We also consider the role of formaldehyde as an intermediate radical. Formaldehyde has a duel role in 

the trioxane flame. Trioxane decomposes directly to formaldehyde, so that formaldehyde can be 

considered a secondary fuel. However, formaldehyde also behaves like an intermediate, in that its 

concentration is initially small at the surface, increases rapidly, and then is entirely consumed. While 

CH2O is more stable than the other radicals examined (as evidenced by the gas chromatography data 

produced by Hochgreb [1991]), we examine this potential effect in order to encompass all possible 

surface radical interactions. 

Table 7-1 is a comparison of surface mole fractions of trioxane, formaldehyde, and intermediate radicals 

for several oxidizer velocities using an inert trioxane surface. Three stretch rates are shown for each 

species. These data are identical to the ‘adiabatic’ cases described in Section 6.1. They represent the 

range of possible stretch rates for this flame. At the high stretch rate, the flame is at the extinction turning 

point (blow-off), where it is driven very close to the surface. The middle stretch rate case represents a 

flame further from the surface and not near the extinction limit. The low stretch rate case is a flame with 

a larger standoff distance whose temperature is approaching the flame sheet limit.
108

The concentration of CH2O is roughly an order of magnitude smaller than that of trioxane at each stretch 

rate, but relatively large at the surface compared to the other radicals (at least two orders of magnitude), 

so the effect of formaldehyde is of particular interest for study. HCO is more concentrated at the surface 

than the other species. In addition, the concentrations of radicals at the surface are generally larger at the 

high stretch rate, which is the blow-off extinction point. This is also of particular interest for study, since 

the temperature is low enough in the flame (1415 K peak) for the ‘low-temperature’ reaction path 

discussed in Section 3.0 to dominate, and produce higher concentrations of radical intermediates that may 

interact at the surface. Another item of note is that the mole fractions at the surface in the solid trioxane 

flame are generally smaller than the surface mole fractions in the cited references. This is primarily due 

to the larger surface temperatures studied in the cited references.

Surface radical sink boundary condition: In order to gauge the importance of surface reaction effects, 

we begin with the mole fraction at the surface set to zero; all other boundary conditions remain as before

(Section 3.0). This approach assumes an infinitely fast radical absorption mechanism at the surface. 

Some commentary on the species boundary conditions is necessary. While the boundary conditions do 

not change from the solid fuel stagnation-point formulation (with the exception of the radical species), the 

rationale for using these boundary conditions does change.

For the reactive surface, we can no longer use the same rationale as in Section 3 in the mass balance. 

Recall that at x=0 we used the following condition for the nonreactive surface:

m& = r fu f = r FuF = 2F

For a reactive surface, the mass flux not only includes fuel, but also radicals involved in the surface 

reactions. (Realistically, it would also include products of the radical recombination, but here we are 

using the ‘radical sink’ approximation and neglect the products). We separate the non-fuel species, k, into 

two types: those that are inert, I, and the radicals that react at the surface, r. Then the full mass balance 

for the reactive surface is expressed by:

= +∑ +∑r

r r

I

m& r f u f rIuI r u

We use the boundary condition from Section 3.0, where the nonreactive, non-fuel species uI=0, but we no 

longer claim that ur=0, so that

= −∑rm r f u f rrur

&

However, we can claim, as a first-order approximation, that 

r f >> rr

so that

f f m& ≈ r u

and the boundary conditions, Equations 3.3 and 3.4, on the fuel species remain the same. We verify the 

validity of this assumption by examining the mole fractions of the radicals at the surface. Table 7-1 lists 

example mole fractions for the fuel and the intermediates of interest in the trioxane flame (trioxane, and 

potential ‘r’ species: formaldehyde, H, HCO, H2O2, HO2, O and OH). Inert, or ‘I,’ species include the 

rest of the species, such as H2, O2, H2O, CO, CO2, and N2. Since both the mole fraction and the molecular 

weight of the fuel are large compared to the intermediates at the surface, we deduce that the fuel density is 

also much larger at the surface, since
109

= >> 1

r

f

r

f

r

f

M

M

X

X

r

r

For the radical species, the Yr are set to zero to simulate a species sink. The boundary conditions for the 

simple surface sink approximation are summarized in Table 7-2.

Table 7-1: Example Mole Fractions of Radicals at the Inert Surface 

for Low, Middle, and High Stretch Rates* 

Mole Fractions for Adiabatic Trioxane

Stretch rate, s-15 99 1952

C3H6O3 3.79E-01 4.05E-01 4.12E-01

CH2O 7.08E-02 5.77E-02 6.12E-02

H 5.24E-11 4.14E-09 2.36E-06

HCO 9.31E-07 1.39E-05 7.49E-06

H2O2 1.40E-09 1.29E-07 1.65E-04

HO2 1.18E-11 1.41E-08 1.64E-04

O 0.00E+00 0.00E+00 1.44E-12

OH 6.36E-16 1.35E-11 1.70E-07

*Adiabatic cases in standard air; surface temperature 388K; 20 cm separation 

distance.

Table 7-2: Boundary Condition Comparison at x=0 Between the Formulation With the Inert Solid 

Surface and With Radical Interaction at the Surface (sink)

Dependent 

Variable

Inert Surface Boundary 

Condition at x=0

With Radical 

Interaction

F











 + ∫

= +





















s

o

T

T

p f

o

x

g

L c dT

dx

dT

,

0

2

l











 + ∫

= +





















s

o

T

T

p f

o

x

g

L c dT

dx

dT

,

0

2

l

G 0 0

T Ts Ts

Y ( )

F k

F f

F

f

other u V

u V

u

fuel Y

= −

+

=

:

:

( )

F k

r

F f

F

f

other u V

radical Y

u V

u

fuel Y

= −

=

+

=

:

: 0

:
110

Surface heat balance: As noted previously, the extreme situation to examine for our flame is the radical 

sink case, since including heats of reaction tends to have a trade-off effect in sustaining the flame near the 

extinction limit. Moreover, the effect of thermal feedback from radical recombination heats of reaction 

on extinction behavior is not significant unless radical concentrations are sufficiently high. Therefore, our 

study attempts first to determine the potentially important radicals via the ‘surface sink’ method described 

above. Heats of reaction from the surface reactions are not included here. Only a case found to have 

significant radical recombination effect using the sink method would warrant examination of the potential

for thermal feedback due to heats of reaction from the radical recombination at the fuel surface.

7.5 Numerical Method

We adapted the CFDF program described in Section 5 to accommodate the radical interaction model for 

the surface boundary conditions. The surface radical sink is simulated by setting the mass fraction of the 

radical of interest to zero at the surface in the subroutine FUN. The solutions are run using the fixed 

temperature method described in Section 4. We used the domain size (20 cm) and number of grid points 

(174) throughout the study for the inert and reactive surface cases. 

We adapted the surface radiation model of Section 3 to include the reactive surface in the same manner. 

For these cases, a domain of 25 cm is used, and 256 grid points are necessary to achieve accuracy over the 

entire domain.

7.6 Discussion of Results

7.6.1 The Surface Modeled as a Radical Sink

We solve over the entire flammability range from a stretch rate of approximately 2 s-1 to the blow-off 

extinction limit. In all cases, solutions are for standard air and atmospheric pressure, with the surface 

temperature set to 388 K. We first solve for the surface simulated as a radical sink, for one radical of 

interest at a time. Then we run the extreme case, called ‘all sink’, where the surface is a sink for all 

radicals (except formaldehyde). We first discuss results for all radicals and the ‘all sink’ case, since these 

are similar. We address results for the intermediate formaldehyde separately.

Individual radical sink: Results indicate that individual radical recombination at the surface has little to 

no effect on the flame compared to the inert surface case. We examined formaldehyde separately because 

of its unique role in the kinetics of the flame. Plots are of no use in examining the results because the 

differences are indistinguishable from the inert case. We therefore present the data in tables for sample 

stretch rates. Table 7-3 and 7-4 illustrate results at the two extremes of the domain, where any differences 

are most apparent. Table 7-3 gives results at the very low stretch rate, where the flame is not near the 

surface and the maximum flame temperature approaches the flame sheet limit. Table 7-4 presents the 

comparison at the blow-off extinction turning point, where the flame is blown against the surface, and the 

flame temperature is low. The tables list the stretch rates, fuel, and oxidizer side velocities, and 

maximum flame temperature for the inert case (adiabatic case of Section 6) and each case of radical 

surface sink. 

The only difference worthy of mention is the turning point of HO2. For the HO2 radical, the turning point 

occurs at a stretch rate slightly lower than the inert surface case (1945 s-1 versus 1952 s-1). Referring back 

to Figure 6-14, we see that for the inert surface, the HO2 radical is produced in relatively large quantities 

near the surface at the blow-off limit due to the low temperature of the flame. This explains why we 

notice a slight decrease in the blow-off limit when the HO2 radical is suppressed at the surface. The effect 

is small because the concentration of the radical is still very small in an absolute sense.
111

The effects of the radical sink in the trioxane flame are very slight compared to previous findings for the 

following reasons. First, the surface temperature of the trioxane is low (388 K) in comparison to that 

studied by previous researchers. Second, the lower temperature leads to smaller mole fractions of radicals 

at the surface. The most important radicals (OH, O and H) in Vlachos et al. [1994] have surface mole 

fractions in the 10-2 to 10-3 range. Aghalayam et al.’s study [1998] include somewhat lower radical mole 

fractions: XOH=O(10-3 to 10-4) and XH=O(10-5). Here, these radicals are far less concentrated at the 

surface, as indicated by the inert surface mole fractions given in Table 7-1.

‘All sink’: Cases are also run for the surface acting as a sink for all of the radicals of interest in 

combination. This case is called ‘all sink’ in Tables 7-3 and 7-4. The 'all sink' case also blows off at a 

stretch rate very slightly lower (1943s-1 versus 1952 s-1), primarily due to the effect of HO2 described 

above. Otherwise, the data are virtually identical. We conclude that the surface behaving as a pure sink 

for radicals is an insignificant effect for the solid trioxane flame, due to the low surface temperature and 

resulting low concentrations of radicals.

Formaldehyde: As discussed in Section 4.3, formaldehyde has a unique role in the kinetics of the solid 

trioxane flame. For the case where the surface is modeled as a sink for formaldehyde, the turning point 

occurs at a stretch rate somewhat lower than the inert surface (1761 s-1 versus 1952 s-1), although the 

turning point temperatures are very close (Table 7-4). At the low stretch end of the regime, the difference 

is smaller; Table 7-3 shows that, at a stretch rate near 5 s-1, the maximum flame temperatures are 2148 K 

for the formaldehyde sink case, and 2175 K for the inert case. The formaldehyde sink case gives 

uniformly lower maximum flame temperatures across the flammability domain, as illustrated in Figure 7-

1. We conclude that the effect of formaldehyde being absorbed by the surface is very small in spite of the 

relatively large concentration at the surface in the inert (adiabatic) flame. Since formaldehyde is 

produced in larger quantities between the surface and the flame, as shown in Figures 6-6 through 6-8, the 

surface absorption effect is minor. In a true cycle of absorption-desorption of formaldehyde on a trioxane 

surface, we would expect the observed effect to be even smaller than the extreme sink model, because 

formaldehyde is more stable than the other radicals involved, and its recombination into higher molecular 

weight species requires higher energy bonds, and is thus less likely to occur.

Aghalayam et al. [1998] suggest that surface chemistry plays a minor role at lower surface temperatures; 

but at high surface temperatures the surface recombination of radicals can be important for accurate 

prediction of wall heat flux. This study demonstrates their commentary to be true for a lower surface 

temperature reactive surface in the solid trioxane diffusion flame.

Table 7-3: Effect of the Surface Modeled as Radical Sink for Stretch Rate Near 5 s-1

a (s-1) uF (cm/s) uO (cm/s) Tmax (K)

Inert 5.1 0.6 -94.0 2175.4

H 5.0 0.6 -92.9 2176.2

HCO 5.1 0.6 -94.2 2175.5

H2O2 5.1 0.6 -93.8 2175.5

HO2 5.1 0.6 -95.1 2175.0

O 5.0 0.6 -92.7 2176.2

OH 5.1 0.6 -94.7 2175.2

CH2O 5.0 0.5 -96.6 2147.9

All sink 5.1 0.6 -94.2 2175.3
112

Table 7-4: Effect of the Surface Modeled as Radical Sink for the Blow-Off Extinction Limit

a (s-1) uF (cm/s) uO (cm/s) Tmax (K)

Inert 1951.9 11.3 -38913.9 1416.6

H 1951.5 11.3 -38907.0 1416.4

HCO 1951.7 11.3 -38910.6 1416.8

H2O2 1949.5 11.3 -38867.8 1416.8

HO2 1945.0 11.2 -38776.3 1416.6

O 1951.9 11.3 -38913.9 1416.6

OH 1951.8 11.3 -38913.7 1415.7

CH2O 1761.0 9.8 -35106.5 1415.3

All sink 1943.0 11.2 -38736.8 1417.0

Maximum Flame Temperature vs. Stretch Rate 

(adiabatic inert surface and surface sink for Formaldehyde) 

0

500

1000

1500

2000

2500

1 10 100 1000 10000

Stretch Rate (s-1)

Maximum Flame Temperature (K)

Adiabatic Case

CH2O Sink

Figure 7-1: Maximum flame temperature vs. stretch rate for the adiabatic (inert surface) trioxane 

flame and the surface sink for formaldehyde.
113

7.6.2 Surface Radical Interaction Coupled With Surface Emission

The ‘all sink’ case described above is coupled with the model for a surface emitting radiation (as 

described in Section 3) in order to gauge the potential effects of radical interactions in the presence of 

surface radiation. Figure 7-2 contains two cases: the full isola of the inert surface that emits as a black 

body, and the surface that both absorbs radicals and emits as a black body. The full isola of the ‘all sink’ 

case is not produced—but the blow-off and radiative extinction turning points are shown. Figure 7-2 

illustrates that there is virtually no difference if surface radical interaction is included with surface 

radiation. Surface radiation is clearly the dominant effect when compared to the radical recombination at 

the surface. This is consistent with Aghalayam et al.’s [1998] observations with partial surface emission. 

Although they studied a higher surface temperature (1000 K), they observed a decreased significance of 

radical recombination in the presence of surface emission. 

Maximum Flame Temperature vs. Stretch Rate

0

500

1000

1500

2000

2500

0.01 0.1 1 10 100 1000 10000

Stretch Rate (s-1)

Maximum Flame Temperature (K)

Inert with Surface Radiation Allsink with Surface Radiation

Figure 7-2: Maximum flame temperature vs. stretch rate for an inert surface including surface 

radiation and a surface that absorbs radicals and radiates.

7.7 Conclusions

We conclude that the effect of radical interactions at the surface in a solid trioxane flame is not 

significant, and that the only observable effect is near the extinction turning point. Even in the extreme 

case, where the surface absorbs all radicals, the effect is very small. In comparison to previous studies 

that found that surface radical interactions could be important, the trioxane flame has a relatively low 

surface temperature, and resulting smaller mole fractions of intermediates. When coupled with surface 

radiation effects, the surface radiation dominates and no surface radical interaction effect is observable. 

We conclude that the inert model of the solid trioxane diffusion flame is adequate and no representation

of radical recombination at the surface is necessary to accurately represent the physics of the flame under 

study. While including heats of reaction as in Aghalayam et al. [1998] and mass compensations, ala 

Warnatz [1978], is acknowledged to be more accurate representations, these are not necessary in this 

study because the most extreme surface sink model yields very minor differences.
114

8.0 Summary

The goal of this work has been to add to the increasing body of knowledge on microgravity combustion 

so that space travel will be both safer and more feasible. Knowledge of how fires burn and extinguish in 

the unique microgravity environment of space is an essential component of spacecraft fire safety. Two 

numerical models were developed for the study of gas-phase radiation effects on the burning and 

extinction of a solid fuel in a stagnation-point diffusion flame configuration. 

Early studies described in Appendix 4 present a relatively simple model formulation using PMMA in a 

stagnation-point configuration with one-step kinetics and Planck mean absorption coefficient. Results 

showed that both the gas-phase and solid surface radiation become more important as flame stretch rate 

decreases. The interaction of gas-phase and surface radiation, flame temperature, solid burning rate, 

extinction limits, and flammability boundaries were examined parametrically. The nondimensional 

analysis shows that the gas radiation parameter is inversely proportional to the stretch rate. Moreover, the 

nondimensional surface radiation parameter is inversely proportional to the square root of stretch rate. 

Even though, at high stretch rates, surface radiative losses can be large (and thus more important), gasphase radiation can be comparable or dominant at low stretch rates.

The body of this thesis presents a solid trioxane stagnation-point diffusion flame model enhanced by 

detailed kinetics coupled with a narrowband radiation model, with carbon dioxide, carbon monoxide, and 

water vapor as the gas-phase radiation participating media. This model validates several of the 

assumptions of the first model, and expands the results to include detailed reaction path analysis and 

sensitivity.

Gas phase radiation effects are amplified at low stretch due to (1) a decrease in flame heat generation,

brought about by the large flame standoff distance and (2) a longer optical length also due to the thicker 

radiating layer. But the increasing loss from the flame lowers the flame temperature. Thus, the net 

amplification of gas radiation at low stretch is somewhat modulated by this interactive flame-radiation 

effect. 

The stagnation-point trioxane diffusion flame in air has two distinct extinction limits. The high stretch 

blow-off turning point is found to occur at a=1952 s-1 at a maximum flame temperature of 1417 K. The 

low stretch turning point (quench limit) is near a=0.068 s-1 at a maximum flame temperature of 997 K.

Reaction path analysis shows that a straight-chain process dominates the trioxane decomposition over the 

entire flammability regime with the exception of near the extinction limits. Here, a branching path asserts 

itself due to low temperatures. Sensitivity analysis at the turning points shows that the extinction flow 

parameter (the radial pressure curvature) is sensitive to the fuel, formaldehyde, and formyl radical 

decomposition reactions at the high stretch extinction limit, but not near the low stretch extinction limit. 

The key to the rate of progress at both extinction limits is the H radical, because of its high reactivity. 

We demonstrated with a realistic fuel that it is possible for a radiative flame to have a higher burning rate 

than the adiabatic counterpart. Moreover, flame radiation must be included to observe this effect—the 

flame with surface radiation only does not exhibit the higher burning rate, since the increase in burning 

rate at low stretch is due to feedback of the flame radiation to the surface. 

The trioxane diffusion flame model is extended to include surface absorption of radicals from the gas 

phase and the results compared to the inert surface case. Surface radiation is also included both with and 

without surface radical interactions. Results indicate that, for the model studied, surface absorption of 

radicals from the gas phase is unimportant.

Some comments on potential improvements that could be undertaken in future work are appropriate. The 

radiation model could be improved by including the fuel and formaldehyde as participating species. 
115

While we believe we have included the primary participating species, including the fuel can have an 

effect on the results [Bedir 1998, Kashiwagi 1979]. Accurate measurements of these species absorbing 

and emitting bands could be included if made available.

The trioxane surface was assumed to absorb and emit as a black-body in this model; measurements of real 

surface absorption and emission could improve the model. The polarizability and the rotational relaxation 

collision parameter for trioxane were estimated for this analysis. While the solutions presented are not 

greatly dependent on either quantity, availability of accurate measurements of these two quantities could 

improve our confidence in the results. Finally, knowledge of the surface temperature relationship to the 

trioxane burning rate would be usefula solid pyrolysis law (Arrhenius relationship) similar to that used 

in Appendix 4 for PMMA could be implemented in the model to include a surface temperature solution, 

rather than an imposed surface temperature.

The computer codes that generated these solutions were adapted for trioxane specifically. These codes 

could be generalized so that any solid (or liquid) fuel could be studied in the future.

Finally, and most importantly, experimental study on trioxane flames should be undertaken in order to 

validate the model predictions.
116

Appendix 1: CHEMKIN input file 

ELEMENTS

C H O N 

END

SPECIES

C3H6O3 CH2O CO CO2 H HCO H2 H2O H2O2 HO2 O OH O2 N2

END

THERM

! All of the thermodynamic data are from the CHEMKIN database except:

! The heat of formation for HO2 (H(f)=3.0kcal/mol), from

! Hills and Howard. J.Chem. Phys. 81, 4458, 1984. The fit below is from NASA 

! Gordon-McBride.

HO2 L 5/89H 1O 2 00 00G 200.000 3500.000 1000.000 1

 4.01721090E+00 2.23982013E-03-6.33658150E-07 1.14246370E-10-1.07908535E-14 2

 1.11856713E+02 3.78510215E+00 4.30179801E+00-4.74912051E-03 2.11582891E-05 3

-2.42763894E-08 9.29225124E-12 2.94808040E+02 3.71666245E+00 1.00021620E+04 4

!

! Trioxane data from Lay et al, J Phys Chem A, 101: 2471-2477, 1997. Curve fit

! to NASA format using THERM (Bozelli)

C3H6O3 9/28/98 THERMC 3H 6O 3 G 300.000 5000.000 1390.000 01

 1.35345410E+01 1.75029470E-02-6.07166082E-06 9.51875850E-10-5.56106710E-14 2

-6.32346609E+04-5.25370825E+01-4.03811034E+00 5.69255009E-02-3.89747992E-05 3

 1.30526128E-08-1.70320983E-12-5.70209398E+04 4.23309293E+01 4

!

REACTIONS

!Trioxane decomposition:

C3H6O3=>3CH2O 1.90E15 0 47500

!Hochgreb and Dryer, J PHYS CHEM, 96: 295-297, 1992.

!

!Oxidation of CH2O:

!Following rxns from Hochgreb and Dryer, Comb. and Flame, 91: 257-284, 1992.

!except as noted. References listed in above are noted with HD prefix.

!All data from HD fitted to 700-2500 K; Uncertainty Factors listed with UF 

!prefix

!Third body efficiency of N2 assumed to be 1.0 in all rxns

!

!Consumption of CH2O:

CH2O+M=HCO+H+M 4.21E23 -1.66 92170 !UF 3

CH2O+M=H2+CO+M 8.25E15 0 69540 !UF 3

!HD: Roth and Just, 20th Symposium, 1984, p. 807.

!HD: Third body efficiencies for above 2 rxns are all 1.0

CH2O+O2=HCO+HO2 2.05E13 0 38920 !UF 3

!HD: Tsang and Hampson, J PHYS CHEM REF DATA 15: 1087-1279, 1986.

CH2O+H=HCO+H2 5.18E07 1.66 1834 !UF 2 

!HD: best fit for 300<T<1700K used, since CH20 conc. highest near surface.

CH2O+O=HCO+OH 1.81E13 0 3078 !UF 1.5 

!HD: Tsang and Hampson, 1986

CH2O+OH=HCO+H2O 3.43E09 1.18 -447 !UF 3

!HD: Tsang and Hampson, 1986

CH2O+HO2=HCO+H2O2 1.47E13 0 15200 !UF 1.5

!HD

!

!Formation and Consumption of HCO:

HCO+M=H+CO+M 1.85E17 -1.00 17000 !UF 2

!HD: Timonen, Ratajczak, Gutman,& Wagner, J PHYS CHEM, 91: 5325-5332,1987.

CO/1.9/ CO2/1.9/ H2O/16/ H2/1.89/

HCO+O2=CO+HO2 4.22E12 0 0 !UF 2

!HD: Timonen, Ratajczak, Gutman, & Wagner, J PHYS CHEM, 92: 651-655, 1988.

HCO+H=CO+H2 7.23E13 0 0 !UF 2

!HD: Timonen, Ratajczak, Gutman, & Wagner, J PHYS CHEM, 91: 692-694, 1987.

HCO+O=CO+OH 3.00E13 0 0 !UF 3
117

!HD: Tsang and Hampson, 1986 

HCO+O=CO2+H 3.00E13 0 0 !UF 2

!HD: Tsang and Hampson, 1986

HCO+OH=CO+H2O 3.00E13 0 0 !UF 3

!HD: Tsang and Hampson, 1986

HCO+HO2=CO2+OH+H 3.00E13 0 0 !UF 3

!HD: Tsang and Hampson, 1986

!

!Oxidation of CO:

CO+OH=CO2+H 9.29E03 2.23 -2625 !UF 1.5

!HD: Yetter, Dryer & Rabitz, Comb Sci Tech, 79: 97-128, 1991.

!HD: fit from the original non-Arrhenius expression

CO+HO2=CO2+OH 6.63E13 0 22950 !UF 2

!HD: Yetter et al, 1991

CO+O+M=CO2+M 4.00E13 0 -4540 !UF 4

!HD: Yetter et al, 1991

CO/1.9/ CO2/1.9/ H2O/16/ H2/2.5/

CO+O2=CO2+O 2.53E12 0 47690 !UF 3

!HD: Yetter et al, 1991

!

!H2-O2 Chain Reactions:

H+O2=OH+O 3.52E16 -.70 17070

!HD: corrected Ref. 26; Masten, Hanson & Bowman (J PHYS CHEM, 94:

!7119-7128, 1990).

H2+O=OH+H 5.08E04 2.67 6292 !UF 2

!HD: Yetter et al, 1991

OH+OH=H2O+O 2.88E00 3.66 -4313 !UF 2.5

!HD: Yetter et al, 1991

OH+H2=H2O+H 2.16E08 1.51 3430 !UF 2

!HD: Yetter et al, 1991

!HD: fit from the original non-Arrhenius expression

!

!Formation and Consumption of H2O2 and HO2:

H2O2+OH=H2O+HO2 7.00E12 0 1430 !UF 2

!HD: Yetter et al, 1991

H2O2+H=H2O+OH 1.00E13 0 3590 !UF 2

!HD: Yetter et al, 1991

HO2+HO2=H2O2+O2 4.20E+14 0.00 11980 !UF 1.5

DUP

HO2+HO2=H2O2+O2 1.30E+11 0.00 -1629

DUP

!HD: Hippler, Troe, & Willner, J.Chem.Phys.93: 1755-1760, 1990.

! Rate constant is the sum of the above two rates.

H+HO2=OH+OH 1.69E14 0 874 !UF 3

!HD: Yetter et al, 1991

H+HO2=H2+O2 6.63E13 0 2126 !UF 3

HO2+OH=H2O+O2 1.45E16 -1.00 0 !UF 2.5

HO2+O=O2+OH 1.81E13 0 -397 !UF 2

H2O2+H=H2+HO2 4.82E13 0 7948 !UF 2

O+O+M=O2+M 6.17E15 -.50 0 !UF 3

CO/1.9/ CO2/1.9/ H2O/16/ H2/2.5/

H2+M=H+H+M 4.58E19 -1.40 104400 !UF 2

CO/1.9/ CO2/1.9/ H2O/16/ H2/2.5/

H+OH+M=H2O+M 2.25E22 -2.00 0 !UF 2

CO/1.9/ CO2/1.9/ H2O/16/ H2/2.5/

H2O2+M=OH+OH+M 1.20E17 0 45500 !UF 2

CO/1.9/ CO2/1.9/ H2O/16/ H2/2.5/

O+H+M=OH+M 4.72E18 -1.00 0 !UF 30

CO/1.9/ CO2/1.9/ H2O/16/ H2/2.5/

H+O2+M=HO2+M 2.00E15 0 -1000 !UF 3

CO/1.9/ CO2/1.9/ H2O/16/ H2/2.5/

END
118

Appendix 2: Transport Property Estimates for Trioxane

This appendix outlines the rationale for the estimations of rotational relaxation collision parameter and 

polarizability for trioxane. These data were not available for trioxane after an extensive literature search, 

nor could an estimate be made by back calculating from other known properties. The effect of these two 

quantities on transport properties is also discussed, along with potential effects of estimation error on the 

results.

Rotational Relaxation, Zrot 

The quantity Zrot is a measure of the number of collisions needed to establish rotational equilibrium of a 

molecule when it has been excited out of equilibrium by addition of energy to the system. This quantity 

was first defined by Parker [1959] and the theory was expanded by Brau and Jonkman [1970]. Their 

analyses are applicable to diatomic gases only; however, certain analogies may be drawn to more 

complex species, such as trioxane. Gas-phase thermal conductivity can be thought of as a measure of the 

collision efficiency of a species in the transmission of thermal energy, and indeed, thermal conductivity is 

a weak function of Zrot [Kee et al. 1986].

The quantity Zrot for a same species collision of a diatomic gas is given by Parker [1959] as:

2

16

1













=

e

d

Z a rot

In this expression, a is a constant characterizing the spatial rate of variation of the collision force between 

two molecules, d is the nuclear separation distance, and e is a parameter based on the geometry of the 

collision. The quantities a and e are typically estimated by fitting experimentally known values of the 

vibrational relaxation collision parameter, Zvib. None of these quantities is available for trioxane.

Zrot is zero for monatomic species and some linear molecules with no rotational component. Since 

trioxane is a nonlinear molecule, we know that its rotational component is non-zero. The vast majority 

(all but one of 100 +) of the nonlinear molecules in the TRAN database have rotational relaxation 

collision parameter greater than or equal to 1.0; a large portion of these are equal to 1.0. That is, it takes 

one or more collisions for the molecule to relax to the equilibrium state. Another molecule, benzene, has 

an analogous molecular weight (benzene m. wt. = 78, trioxane m. wt. = 90) and structure (ring) to 

trioxane. Z rot for benzene (and all other high molecular weight molecules in the TRAN database) is one 

collision. Therefore the quantity Zrot = 1 was chosen for trioxane because it is a high molecular weight, 

nonlinear molecule, and is similar in structure and molecular weight to benzene. 

We evaluated the thermal conductivity of trioxane with the TRAN program using the values of Zrot equal 

to one and zero, to gather further insight into the effect of rotational relaxation on thermal conductivity. 

The value of thermal conductivity at 300 K was increased by 6 percent for Zrot =1. We compared the 

value to that of benzene and found it to be comparable, but somewhat largerwhich is expected, given its 

larger molecular weight.

With the known information outlined above, the selection of Zrot =1 is reasonable and conservative. 

Larger estimates (Zrot > 1), while possible, would be unsupportable by the known data.

The effect of this assumption being in error would be to slightly increase thermal conductivity if Zrot > 1. 

During burning of the solid, the gas-phase trioxane is formed through pyrolysis very near the surface and 

rapidly decomposes to formaldehyde. Thus, an underestimation of thermal conductivity of trioxane 

would have the largest effect at the surface and tend to underestimate the conduction heat transfer from 

the gas to the surface. Any potential error then would be most noticeable at low stretch rates, where heat 

feedback between the flame and the surface is increasingly important, and where the concentration of 
119

trioxane would tend to be larger due to increased residence time at low velocities. Thus, the effect of a 

small underestimation of thermal conductivity would also somewhat underestimate the flammable region 

of trioxane.

Polarizability, ap

Trioxane is a dielectric molecule with a permanent dipole [Walker and Carlisle 1943]. These electrical 

characteristics are important in molecular collisions, because they affect the intermolecular forces present 

between molecules of trioxane and other species when they collide. Transport properties are thus affected 

by molecular electrical characteristics. The polarizability, ap, is used in the calculation of binary 

diffusion coefficients (for polar-nonpolar collisions only) in the transport package, TRAN [Kee et al. 

1986].

An excellent and exhaustive discussion of polarizability and other electrical characteristics of molecules 

is given by Hirschfelder et al. [1954]. The following discussion is a brief summary from Hirschfelder et 

al. [1954] of the pertinent information necessary to understand the effect of polarizability.

If Ε is an electric field strength applied to a molecule, and m

ind is the dipole moment induced by the 

electric field, then

= pΕ

ind m a

and a is the polarizability.

There exists a quantity called the electric susceptibility, X, that relates polarizability, dipole moment, and 

the dielectric constant. X is expressed via the dielectric constant, e, by the Clausius-Mosotti relation:













+

−

=

2

1

4

3

e

e

p Nr

M

X

where M is the molecular weight, N is Avagadro’s number, and r is the density.

X for molten trioxane is calculated from the dielectric constant and found to be 0.237 cubic Angstroms. 

For solid trioxane, the value is 0.143. We can use the electric susceptibility to predict the variation of the 

dielectric constant with density, as shown in the above expression, since the electric susceptibility remains 

constant with respect to these two quantities.

We can find polarizability and dipole moments by measuring the dielectric constant, as a function of 

temperature. If m is the dipole moment, and k is the Boltzman constant, then

kT

X

3

2

m

= a +

When we plot this expression as 1/T, the result is a straight line, with a as the intercept and m

2

/3k as the 

slope. Polarizability is then defined at the intercept, where 1/T → 0. (Or X ≈ a).

Then the intercept gives a, the polarizability, as approximately 0.237 cubic Angstroms.

In the calculation of the binary diffusion coefficient in the TRAN program [Kee et al. 1986], the 

polarizability for a polar molecule appears in the calculation of a collision integral, which is internal to the 

TRAN computational package. A priori insight into the variation of the diffusion coefficient with 

polarizability of a polar molecule is therefore not available. For this reason, we used the TRAN program 

to assess differences in diffusion coefficients with different polarizabilities.
120

The differences in polar-nonpolar diffusion coefficients for trioxane were assessed for polarizabilities of 

zero and 0.237. The value for solid trioxane lies within this range, so that the extremes of the range of 

known values of polarizability for trioxane are included. (The dielectric constant for gas phase trioxane is 

unknown, possibly because trioxane decomposes so rapidly to formaldehyde in the gas phase. It is 

possible that gas phase trioxane has a higher dielectric constant, and thus a higher polarizability, but we 

don’t consider this possibility here because no evidence for this exists.) The results showed no 

differences to the third significant figure in the binary diffusion coefficients. Thus, we conclude that the 

polarizability does not significantly affect the transport characteristics of this problem, and any error in 

the estimate will have little effect. The value a = 0.237 is retained in the database for trioxane.
121

Appendix 3: Latent Heat of Solid Fuel

This appendix contains the detailed latent heat calculation for a solid fuel (generalized), and the specific 

application for trioxane.

Generalized latent heat calculation: We desire the change in latent heat (enthalpy) of the fuel for the 

heat balance at the surface. The surface has temperature Ts, and the enthalpy is calculated from reference 

quantities, designated by the 'o' superscript. Figure A3-1 illustrates the change in enthalpy.

The total enthalpy of the solid, where cp,s is the constant pressure specific heat of the solid fuel, is:

( )

∫

= +

s

o

T

T

p s

o

hs Ts hsc , dT

The total enthalpy of the gas-phase fuel, where cp,f is the constant pressure specific heat of the gas-phase 

fuel, is:

( )

∫

= +

s

o

T

T

p f

o

hf Ts hfc , dT

Here the subscripts refer to the phase of the fuel: s refers to the solid phase, and f refers to the gas phase. 

Then the change in enthalpy for the fuel at the solid-gas interface is:

∫ ∫

Δ = + − −

s

o

s

o

T

T

T

T

p s

o

p f s

o

hfs hfc , dT h c , dT

For simplicity, allow

fs L = Δh

and

o

s

o

f

o

L = h − h

Then the general expression for the enthalpy at the fuel interface is:

∫ ∫

= + −

s

o

s

o

T

T

T

T

p f p s

o

L L c , dT c , dT

Specific application to trioxane: Now for most solids, including trioxane, we approximate cs as a 

constant, particularly when Tsis not large compared to T

o

, so that

( ) ∫

= + − −

s

o

T

T

o

p f p s s

o

L L c , dT c , T T (A3-1)

For trioxane, L

o

=55.6 kJ/mol (6.17x109 erg/g), T

o

=300 K, Ts =388 K (boiling point of trioxane), and the 

integral can be calculated. 

When the integration is carried out numerically using the transport data for trioxane (referenced in 

Section 4) the result is:
122

cP f dT 9.445x10 erg / g

8

388

300

, = ∫

Approximating the integral from the NIST database [National Institute of Standards Chemistry Web 

book], we start with:

Cp,f (trioxane gas, 300 K) = 82.38 J/mol/K (9.145x106 erg/g/K)

and

Cp,f (trioxane gas, 400 K) = 109.75 J/mol/K (1.218x107 erg/g/K)

Assuming a linear profile for specific heat over the temperature range 300 to 400 K, we can interpolate to 

get the midpoint value over the range of integration:

Cp,f (trioxane gas, 344 K) = 94.42 J/mol/K (1.048x107 erg/g/K)

Then approximating the integral:

cP f dT c f midpo T cP f

(344K)(388K 300K) 9.224x10 erg / g

8

, int ,

388

300

, ≈ Δ = − = ∫

This is less than 3 percent different from the value calculated from numerical integration. We conclude 

that the specific heat is a near-linear function of temperature over the range evaluated and that the 

transport data used in Section 4 are consistent with the NIST data for trioxane. The data from references 

[Lay et al. 1997] and [National Institute of Standards Chemistry Web book] used to approximate the 

integral are compared in Figure A3-2. Upon investigation, we found that both references use a single 

source [Dorofeeva 1992], so this result is not surprising. It does confirm that the polynomial 

approximation used to convert the data from Lay et al. [1997] via the THERM program into the NASA 

format is accurate.

Figure A3-1: Illustration of enthalpy calculation.
123

0

40

80

120

160

200 250 300 350 400 450 500

Temperature (K)

Specific Heat (J/mol-K)

Cp (J/mol/K) [NIST] Cp(J/mol/K) [Lay et al.]

Figure A3-2: Comparison of specific heat data from references.

 [Lay et al. 1997] and [NIST Chemistry Web book]
124

Appendix 4: Surface and Gas-Phase Radiation Effects on the 

Burning and Extinction of a Solid Fuel

A4.1 Background

Radiative loss from flames has long been recognized as a mechanism for flame extinction. For example, 

in the early theory of flammability limits for premixed flames, gaseous radiative loss was included in the 

one-dimensional analysis [Spalding 1957]. Surface radiative loss was introduced for a solid 

monopropellant to explain the observed low-pressure deflagration limit of ammonia perchlorate 

[Friedman et al. 1959].

In analyses of diffusion flame extinction, research has concentrated on the high stretch regime, which 

most often occurs in higher speed flows. In the absence of convective flow, Bonne [1971] hypothesized 

that radiation can lead to diffusion flame extinction in microgravity. Surface radiative loss has also been 

included in a flat-plate boundary layer combustion analysis of a solid [Silbulkin et al. 1981 and 1982]. 

Unfortunately, the parabolic equations used are unsuitable for flame extinction analysis [Chen and T’ien 

1986], and the extinction condition found appears to be the artificial consequence of the local similarity 

assumption.

T'ien [1986] used flame stretch rate as the flow parameter to investigate combustion and extinction 

characteristics of a diffusion flame anchored to a solid surface in the stagnation-point region. Only 

surface radiative loss was included. He found that the importance of surface radiative loss is inversely 

proportional to the square root of the stretch rate. At low stretch rate, he found a radiatively quenched 

flammability boundary in a map using oxygen percentage and stretch rate as the coordinates.

Heat loss from gas-phase radiation has been included in several studies of diffusion flames. Negrelli et al. 

[1977] used the exponential wide-band gas radiation model and assumed the flame chemistry to be 

infinitely fast. Liu and Rogg [1991] employed detailed chemical kinetics for methane and air using the 

optically thin radiation approximation in the gas-phase. Sohrab and Williams [1981] assumed one-step 

kinetics and used asymptotic techniques. All of these studies concluded that radiation lowers the flame 

temperature, as expected, but did not examine the radiation quench limit.

Chao et al. [1990] studied the spherical droplet combustion problem using an asymptotic expansion, and 

treated gas radiation as a pure loss. They identified a large droplet diameter limit beyond which flame 

quenching occurs. Kinoshita and Pagni [1981] studied a gray radiation diffusion flame above a gray 

pyrolyzing solid fuel in the stagnation-point region using the optically thin model. They recognized flame 

and surface radiation interaction, and that flame radiation effects may cancel surface emission effects. In 

a series of papers [Bhattacharjee1 and Altenkirch 1990, Bhattacharjee2 and Altenkirch 1990, 

Bhattacharjee and Altenkirch 1992], Bhattacharjee and Altenkirch included both gas-phase and solid 

surface radiation in their model of opposed-flow flame spread over a solid. They pointed out that gasphase emission is not a pure heat loss from the flame system because the portion of the gas-phase 

radiation incident on the fuel surface may be absorbed. They assumed in their model that the amount of 

incident radiation is balanced by the surface emission, thus detailed gas-surface radiative interaction 

analysis was not performed.

Chen and Cheng [1994], in their model of downward flame spread over a solid, incorporated a two-flux, 

optically thin model to study the effect of gas-phase radiation from CO2 and H2O. Although the accuracy 

of the two-flux model is somewhat questionable in this two-dimensional configuration, the method 

provides a direct coupling between the gas-phase radiative flux and the solid radiative flux. For surface 

radiative properties, they selected an absorptivity and emissivity equal to 0.92, and presented detailed 

radiative and convective heat fluxes along the solid surface.
125

The objective here is to study the influence of gas-phase radiation from carbon dioxide and water vapor 

on the burning and extinction of solid fuels. In particular, we are interested in the interaction between 

gas-phase radiation and the solid surface. In the analysis, we vary not only flame stretch rate but also the 

surface radiant properties to obtain a more complete understanding of the effect of gas-phase and solid 

surface radiation. This study is unique in that it offers the only parametric variation over the 

comprehensive set of surface radiation parametersincluding the gas-phase flame radiation so that 

surface absorption as well as emission can be evaluated. Also, this section provides insights into the 

calculations for a real fuel with detailed chemistry that is taken up in the following sections.

A4.2 Nomenclature

a Stagnation-point flow radial velocity gradient, Ue = ar, s-1 (r = radius)

cp Specific heat of the gas, J/kg-K

cs Specific heat of the solid, J/kg-K

Dk Ratio of local radiative emission length to flame thickness, Equation A4-2

F Nondimensional parameter of the two-flux radiation model, see Equation (A4-4)

f Modified stream function [T’ien et al. 1978]

-fw Nondimensional burning rate, see Equation (A4-13)

Kp Planck mean absorption coefficient, 1/m

L Latent heat of the condensed fuel scaled to cpTe

m mass burning rate of the condensed fuel, kg/m-s

Pr Prandtl Number, μcp/λ

Q Nondimensional heat loss to solid, see Equation (A4-10)

qf Nondimensional heat of combustion

q

+ Nondimensional gas-phase radiative flux away from the surface, scaled to σTe4

q

- Nondimensional gas-phase radiative flux towards the surface, scaled to σTe4

qnet Nondimensional net heat flux away from the surface, see Equation (A4-8)

S See Equation (A4-3)

T Temperature, degrees Kelvin

Ue Radial velocity at edge of boundary layer

w Nondimensional reaction rate

Y Species mass fraction

y direction normal to surface

a Total hemispherical absorptivity of the surface

e Total hemispherical emissivity of the surface

h Nondimensional distance normal to the fuel surface [T’ien et al. 1978], 

∫

∫

=

y

r

e e e

e

dy

U dr

U

0

0

2

r

r m

h

Θ Nondimensional temperature, Θ = T/Te, where Te = 300K

Θc Nondimensional temperature far inside the condensed fuel

l Thermal conductivity, kJ/m-s-K

m Viscosity, kg/m-s

r Density, kg/m3

s Stefan-Boltzman constant, 5.6696x10-8 W/m2-K

4

t Nondimensional optical thickness, see Equation (A4-12)
126

Subscripts

e Edge of boundary layer

F Fuel

f Flame

o Oxidizer

P Product

w Wall (fuel surface)

Superscripts

' d/dη

A4.3 Combustion Model

An axi-symmetric stagnation-point configuration is adopted for the solid-fuel laminar diffusion flame 

analysis. The flame model, with one-step finite-rate chemical kinetics, is similar to that described in 

references T’ien et al. [1978] and T’ien [1986]; here we add a two-flux gas-phase radiation model (Figure 

A4-1). The modified energy equation, the surface boundary conditions and the two-flux radiation 

equations are described as follows.

The nondimensional energy equation is identical to T’ien et al. [1978] except that the gas-phase radiation 

has been included:

F q w

SD f

f

k Θ + Θ + ( − Θ ) = −

Pr

2

' ' '

Pr

1 4

 (A4-1)

The last term on the left-hand side is the gas-phase radiative contribution, where

= pΘ

e

e

k K

a

D

r

m

2 (A4-2)

and

a

T

S

e

e

e

e

r

m

l

s

3

=

(A4-3)

The nondimensional parameter Dk can be regarded as the ratio of local radiative emission length to flame 

thickness. The nondimensional parameter S represents the rate of black-body radiative heat transfer to 

gas-phase conduction. (The coefficient term S/Pr is similar to that defined by Ozisik [1973] relating the 

Boltzman number and the Reynolds number.) Kp is the local Planck mean absorption coefficient of the 

mixture, adjusted for local partial pressures. In this work, only combustion products H2O and CO2 are 

assumed to participate radiatively. The Planck mean absorption coefficients are taken from data in Seigel 

and Howell [1981] and T’ien and Lee [1982] and modeled as polynomial functions of temperatures and 

summed using the partial pressure of the gases as the weighting factor. Soot and fuel vapor contributions 

are neglected.

The quantity a in Equations (A4-2) and (A4-3) is the flame stretch rate which is the flow parameter in this 

problem. Note that the product SDk is inversely proportional to stretch rate, a. Therefore, the importance 

of the gaseous radiation term in Equation (A4-1) increases at small stretch rate.
127

The gas-phase radiation flux is modeled using the two-flux (Schuster-Schwartzchild) radiation model 

[Ozisik 1973, Siddall 1974, Seigel and Howell 1981, Lee and T’ien 1989], chosen because of its 

simplicity and its capability to model both gas-phase absorption and emission. In addition, the method 

allows for detailed accounting of the radiative fluxes (versus a global approach) and permits study of the 

interaction between gas-phase and surface radiation. The quantity F in Equation (A4-1) is given by

F = 

1

2

(q

+ + q-

) (A4-4)

where q

+

 is the nondimensional gas-phase radiative flux away from the surface (in the positive h

direction) and q

-

 is the nondimensional gas-phase radiative flux toward the surface (in the negative h

direction); see Figure A4-1. These nondimensional fluxes are scaled to the background radiative flux, 

sTe

4

. The two-flux radiation equations are:

( )

4 +

+

= D Θ − q

d

dq

k

h

(A4-5)

and

( )

4

= − Θ

−

−

D q

d

dq

k

h

(A4-6)

with boundary conditions:

: 1

0 : (1 )

4

4

→ ∞ = Θ =

= = − + Θ

−

+ −

e e

w w w

q

q q

h

h a e

(A4-7)

where a and e are the surface absorptivity and emissivity. At the surface, the radiation flux leaving is 

equal to the reflection of incident radiation plus the radiation emitted from the surface. Transmission of 

radiation through the solid surface is negligible. At the outer boundary of the flame, the radiation 

entering the system is the background emission from the surroundings. The net flux of gas-phase 

radiation is the difference between the flux away from the surface and the flux toward the surface:

qnet = q

+ - q-

(A4-8)

The energy balance at the fuel surface, with radiation, is:

w Q Sqnet w

f

w

,

' Θ = − Pr + (A4-9)

where Θw' is the temperature gradient at the surface from the gas phase, fwPrQ is the heat transferred to 

the solid and to pyrolyze it to fuel gas [T’ien 1986, Oh et al. 1994], and Sqnet,w is the net radiation from 

the surface. Also,

L

c

c

Q

p

s

= Θw − + (A4-10)

Note that the radiation parameter S appears in the surface boundary condition (Equation A4- 9) as well as 

the energy equation (Equation A4-1). It is inversely proportional to the square root of the stretch rate, so 
128

the effect of surface radiative transfer is also amplified at small stretch rate. However, this amplification 

rate is smaller than that of the gas-phase radiation (which is proportional to the inverse stretch rate).

A4.4 Discussion of Results

A4.4.1 Variation of the Flow Parameter (stretch rate)

The dependency of Equations (A4-1) and (A4-9) on S and Dk, and thus stretch rate, a, suggests the 

relative importance of radiation to the heat balance. At higher stretch rate, surface radiative effects are 

more significant than gas-phase radiation effects, because the optical thickness of the gaseous flame is 

smaller at higher stretch rate. Note that

SD a

S a

k 1/

1/

∝

∝

(A4-11)

Moreover, gas-phase radiation effects could take precedence due to larger amplification by SDk at low 

stretch rates. This is due to an increase of the optical thickness of the flame as stretch rate decreases. 

Figure A4-2 shows the quantity Dk, as given by Equation (A4-2), over a range of stretch rates, and 

illustrates the dependence of optical thickness on stretch rate. Dk can be interpreted as the ratio of the 

local radiation absorption length to the flame thickness. The nondimensional optical thickness of the 

entire flame,t, is related to Dk by

t Dk dh ∫

∞

=

0

( ) (A4-12)

Figure A4-2, which is a plot of Dk versus nondimensional distance from the surface, h, for a range of 

stretch rates, illustrates the increased importance of gas-phase radiation emission at low stretch rate for a 

typical case that includes full surface absorption and emission. One can see that, as stretch rate decreases, 

the peak values of Dk increase. This means that the optical thickness t (the integrated value of Dk) 

increases as stretch rate decreases as well. While the flame is considered optically thin (t << 1), the 

relative optical thickness increases almost tenfold between a stretch rate of 60 and 1.4 s-1. Thus at the low 

stretch limit, gas-phase radiation (particularly gas-phase emission) becomes increasingly important in the 

heat balance. The significance of this increased emission is shown later.

A4.4.2 Parametric Variation of the Surface Radiation Properties

The model equations are solved numerically by the time-marching scheme described in T’ien et al. 

[1978], using the properties of PMMA from T’ien [1986] (with the exception of surface radiative 

properties, which are varied parametrically) and the two-flux radiation equations are solved by integration 

in η, the nondimensional spatial coordinate. To understand how gas-phase radiation interacts with the 

surface processes (absorption, emission, reflection) and the consequences of this interaction on the flame, 

we chose four cases with the extreme combinations of surface absorptivity and emissivity. We computed 

these cases with variation of absorptivity and emissivity in Equation (A4-7) over the range of stretch rates 

from blow-off to low speed quenching. These cases are:

Case 1: a = 1; e = 1; full absorption and full emission (a ‘black’ surface)

Case 2: a = 1; e = 0; full absorption and no emission (i.e., largest radiation energy gain to surface; a 

‘cold’ surface)

Case 3: a = 0; e = 0; no absorption (fully reflecting) and no emission
129

Case 4: a = 0; e = 1; no absorption (fully reflecting) and full emission (largest radiation energy loss from 

the surface)

These parametric extremes permit us to consider a series of interesting questions: What is the importance 

of surface absorption in both the presence and absence of surface emission? How does a fully reflecting 

surface affect the flame heat balance? How does the flame respond in the presence of large surface 

radiation loss (non-absorbing; fully emitting surface)? Additionally, these cases could represent some 

physically realizable situations. For example, Case 2 mimics a liquid surface behavior, with little 

emission (due to low surface temperature) but full absorption.

A4.4.3 Radiative Heat Fluxes

Radiative heat fluxes, normalized by the background flux sTe

4

, are plotted on a semi-log scale as 

functions of stretch rate in Figure A4-3(a), (b) and (c) and can be described as follows (refer to Figure 

A4-1 for illustration):

q

+

e

 : radiative flux from flame outer boundary to surroundings (loss from system)

q

-

w : radiative flux from the gas-phase to the surface

q

+

w : radiative flux from the surface to the gas-phase (includes both surface emission and reflection of 

incident radiation)

Figure A4-3(a) illustrates the radiative flux at the flame outer boundary toward the ambient for the four 

combinations of surface absorptivity and emissivity. The radiative flux at the flame boundary is the total 

(global) heat loss from the system (including the gas flame and the solid radiation). Case 2 has the 

smallest loss, because of full absorption by the surface and no surface emission. Case 4, that of no 

surface absorption and full emission, has the largest loss to the ambient. This is due to loss of both the 

gas-phase and surface emissions, less a small amount of gas-phase absorption. Case 3, that of no surface 

emission or absorption, represents the situation where the system heat loss is from gas-phase radiation 

only. At high stretch rate, Figure A4-3(a) shows that qe

+

 is small due to the smaller optical thickness and 

consequently smaller gas emission, DkΘ

4

. At low stretch rate, Dk is amplified and qe

+

 correspondingly 

increases until it peaks at a = 5 s-1. The decrease of qe

+

 at extremely low stretch rate is due to decreased 

flame temperature (to be discussed later).

The difference between Cases 2 and 3 is as follows. In Case 2, the gas-phase radiation to the surface is 

absorbed; while in Case 3, it is reflected (and since these flames are optically thin, most of this reflected 

radiation is lost to the ambient environment). Case 1: full emission and full absorption, has high qe

+

 at 

large stretch rate due to large surface emission. Note that surface emission depends on surface 

temperature and is independent of optical thickness. The value of qe

+

 is amplified at lower stretch rate 

due to the increased gas-phase emission (and loss). The curves for Cases 1 and 3 almost coincide for a < 

5 s-1. In this range of stretch rate, the reflected gas-phase radiative flux from the surface in Case 3 is 

almost equal to the emitted surface radiative flux in Case 1. Consequently, the overall losses from the 

flame are approximately equal. This is considered a coincidence rather that a general observation since 

the result can be altered by material properties such as the solid pyrolysis surface temperature.

The nature of the difference between Case 1 and 4 is the same as that between Cases 2 and 3. While the 

incident gas-phase radiation is fully absorbed on the surface in Case 1, this incident flux is fully reflected 

from the surface in Case 4, and most of it is lost to the ambient environment. Therefore, Case 4 has the 

highest possible radiative loss from the system.

The trend of flame radiative loss (qe

+

) versus stretch rate, as shown in Figure A4-3(a), correlates very well 

with the maximum flame temperature curves shown in Figure A4-4. Higher loss results in lower flame 

temperature and vice versa. Since the curves in Figure A4-3 represent the full flammable range from 
130

blow-off to extinction, the flammability ranges can be compared: higher loss results in a narrower 

flammable range (i.e., higher a at the quenching limit and lower a at the blow-off limit).

The radiative fluxes at the solid fuel surface are given in Figures A4-3(b) and A4-3(c). Figure A4-3(b) 

shows the incident gas-phase radiative flux onto the fuel surface. The four curves also correlate well with 

the flame temperature curve shown in Figure A4-4 (i.e., at a given stretch rate, high flame temperature 

produces high qw

-

). Figure A4-3(c) shows the total radiative flux from the surface to the gas phase. It is 

the sum of surface emission (eΘw

4

) and reflection (1-a)qw

-

 (Equation A4-7). For Case 2, qw

+

 is zero and 

for Case 4, qw

+

 is the highest. As mentioned previously, Figure A4-3(c) shows that qw

+

 for Cases 1 and 3 

are approximately equal for a < 5 s-1 for the particular solid fuel properties chosen in the computation.

Using the two-flux radiation model, approximately half of the radiation emitted from the gas phase goes 

toward the flame edge (ambient environment) directly and the other half goes toward the fuel surface. 

Our model includes absorption, but the amount of absorption is small. (The amount of absorption is 

greater when the more accurate spectral radiation models are employed [Bedir et al 1997]. This will be 

discussed in more detail in later sections.) Therefore, for simplicity in this discussion, we will neglect the 

effect of gas-phase absorption. The fate of the incident flux and the amount of surface radiation emitted 

depend on the values of surface absorptivity and emissivity. The total amount of radiative flux from the 

surface, qw

+

, for the four cases is shown in Figure A4-3(c). For each case, the difference between qe

+

 in 

Figure A4-3(a) and qw

+

 in Figure A4-3(c) should be approximately one-half of the radiation emitted in the 

flame (i.e., the portion emitted directly to the ambient) which in turn should be approximately equal to 

qw

-

. A check between Figure A4-3(a), A4-3(b), and A4-3(c) shows this is indeed the case. The 

approximation is more exact at higher stretch rate because the gas-phase absorption is less.

A4.4.4 Flame Standoff Distance and Solid Burning Rate

The distance between the solid fuel surface and the location of maximum temperature (very close to the 

location of maximum consumption rate of fuel vapor) is referred to as the flame standoff distance. Here we 

use the nondimensional flame standoff distance, hf, which is a function of (a)

1/2. This distance is shown in 

Figure A4-5 for the four cases as a function of stretch rate. (Note that the trend in the physical coordinate is 

different than in the nondimensional coordinate.) The curves for Cases 1, 3 and 4 all display a maximum at 

an intermediate stretch rate. Case 2, however, monotonically increases with decreased stretch rate. This is 

related to the increase of nondimensional burning rate, fw, as follows. Here the burning rate is an eigenvalue of the solution and the nondimensional solid burning rate, fw, is related to the mass burning rate by:

a

m

f

e e

w

r m

− = (A4-13)

Note that the nondimensional burning rate varies with the inverse square root of the stretch rate. Figure A4-

6 gives the nondimensional solid burning rate as a function of stretch rate for the four cases studied. For 

Cases 1, 3, and 4, -fw displays a maximum at high stretch rate, then decreases at lower stretch rate until the 

quenching limit is reached. For an adiabatic system (no radiative loss) and infinitely fast chemical kinetics, 

-fw is a constant (approximately 1.2 for the present fuel in air). The nondimensional solid burning rate falls 

at low stretch rate for these three cases due to radiative loss. (Note that the mass burning rate, m, (which is 

dimensional) falls at a more rapid rate, because it is dependent on the product –fw(a)1/2. For Case 2 in the 

dimensional domain, the radiative burning rate can be higher than the adiabatic limit—see discussion below). 

The nondimensional burning rate is determined by the heat transfer rate to the solid surface which, 

according to Equation (A4-9), is the sum of two terms: the conduction flux, Θw', and the net radiative flux 

at the surface, Sqnet,w. To obtain a rough estimate of Θw', we can use Figures A4-4 and A4-6 and Θw'=(Θf

- Θw)/hf and qnet,w can be obtained from Figures A4-3(b) and A4-3(c) and Equation (A4-8).
131

Case 3 is radiatively neutral at the solid surface (qnet,w=0), so -fw is determined by the conduction flux 

only. Radiation, however, has an influence through the flame temperature. Since gas-phase radiation loss 

(emission) lowers the flame temperature (see Figure A4-4 discussion), the conductive flux is smaller, 

yielding a smaller -fw at low stretch rate.

Case 1 is approximately radiatively neutral at the solid surface for a < 5 s-1, as previously discussed. At higher 

stretch rates, qw

+

 is larger while flame temperature is lower than that of Case 3, due to surface emission. Both 

the conductive and radiative fluxes are lower, which produces a lower -fw compared with Case 3.

Case 4 has the highest qw

+

, the lowest qw

-

, and the lowest Θf. Therefore, both the conductive and radiative 

fluxes are the lowest of the four cases, resulting in the lowest nondimensional burning rate.

The curve for Case 2 in Figure A4-6 displays a qualitatively different trend: nondimensional burning rate 

increases monotonically with decreasing stretch rate. Furthermore, the value exceeds that given by the 

adiabatic, fast-kinetic limit (-fw = 1.2). This is because gas-phase radiative feedback becomes dominant at 

low stretch rate. Note that qw

+

=0. Comparing with the other three cases, this case has the smallest 

radiative loss (FigureA4-3a), the highest flame temperature (Figure A4-4), and the largest qw

-

 (Figure A4-

3). Note also that qnet,w is negative (it is positive in all other cases), which means that there is a net 

radiative heat input to the solid. This radiative heat feedback is especially pronounced at low stretch. The 

corresponding standoff distance, hf, is also larger at low stretch rate than the other cases, due to the larger 

nondimensional burning rate (a blowing effect).

This unusual behavior for Case 2 can have implications to the burning of a liquid fuel. For many liquids, 

the vaporization temperature is close to that of ambient, so the surface radiative emission is negligible, but 

the surface can be radiatively absorbing. Case 2 mimics this situation. Therefore, gas-phase radiation can 

overdrive the liquid burning rate above the classical adiabatic value, particularly at low stretch rate. This 

“anomaly” will be further explored in the following sections using a realistic solid fuel.

A4.4.5 Comparison to Identical Systems With No Gas-Phase Radiation

The effects of gas-phase radiation on flame temperature, burning rate, and flammability are discussed in 

comparison with identical systems with no gas-phase radiation (Cases 5 and 6) in order to elucidate the 

fundamental physical processes affected by gas-phase radiation. Here we compare the case with both 

gas-phase and surface radiation (i.e., Case 1: full absorption and full emission) to the case with surface 

emission only (no gas radiation) (Case 5: Kp = 0; e = 1) and the adiabatic case, which neglects all of the 

radiation effects (Case 6: Kp = 0; e = 0).

Figures A4-7 and A4-8 compare the effects of surface radiation only (Case 5) and the combination of 

surface and gas-phase radiation (Case 1) to the case where radiation effects are neglected (Case 6). When 

radiation heat transfer is entirely neglected in the heat balance, no extinction limit at low stretch rate is 

observed. Adding surface emission (Case 5) leads to extinction at low stretch rates [T’ien 1986]. Adding 

gas-phase radiation further narrows the flammable range.

Figure A4-7 is a plot of nondimensional maximum flame temperature versus stretch rate. The addition of 

gas-phase radiative participation yields reduced flame temperatures over the range of stretch rates, due to 

the increased gas-phase radiative heat loss from the system. Compared with the surface radiation case, 

the temperature reduction is small at high stretch rates but becomes increasingly large at low stretch rates.

Figure A4-8 is a plot of the nondimensional burning rate versus stretch rate for the three cases discussed 

above. When radiative effects are entirely neglected (Case 6), the nondimensional burning rate 

approaches a constant level as stretch rate decreases. When surface emission is added (Case 5), the 

nondimensional burning rate is lower than the adiabatic case due to radiative heat loss from the surface. 
132

The addition of gas-phase radiation to the model yields increased burning rate over the case with surface 

emission only. This is a reverse of the trend of the nondimensional flame temperature, Figure A4-7, and 

due to the absorption of radiation by the fuel surface. This burning rate increase is more pronounced at 

lower stretch rates, although it is diminished somewhat at the quenching limit due to the flame 

temperature drop there.

Tables A4-1, A4-2, and A4-3 contain nondimensional maximum flame temperatures, and species mass 

fractions at the surface for various stretch rates, ranging from low to high stretch (Cases 1, 5, and 6, 

respectively). These are included primarily for comparison to the model with detailed chemistry in the 

following chapters. We observe that, for both the adiabatic surface and the radiating surfaces, more 

oxygen reaches the surface when approaching the high stretch extinction limit, which is indicative of 

incomplete combustion. Also note that, for the radiating surface, less fuel is pyrolyzed at the surface in 

the low stretch regime.

A4.4.6 Flammability

Using oxygen mass fraction at the edge of the boundary layer (Yoe), and stretch rate as parameters, Figure 

A4-9 presents the comparison of flammability boundaries for three cases: adiabatic (Case 6), surface 

radiation only (Case 5) and gas-phase and surface radiation (Case 1). As pointed out first in T’ien [1986], 

the flammable boundary for the adiabatic case is monotonicthe solid is flammable in a very low oxygen 

stream as long as the stretch rate is sufficiently low. With surface radiative loss, this is no longer true. A 

low-stretch quenching branch exists and the U-shaped flammability boundary defines a minimum oxygen 

percentage below which the stagnation-point region is not flammable for any stretch rate.

The extinction boundary with gas-phase radiation added (Case 1) has a similar shape to the one with 

surface radiation only. The flammable range is narrower, however, because of the additional system heat 

loss. The difference between these two cases is largest in the quenching branch at low stretch rate. At 

high stretch rates, the blow-off boundaries are close. We note that this is opposite to the findings in Oh et 

al. [1994] which report a large difference at high stretch rate and little difference at low stretch rate. Due 

to a series of complex mathematical transformations and approximations used in Oh et al. [1994], we 

were not able to pin down the source of this discrepancy. The results we obtained are compatible with the 

flame temperature drops and are consistent with the notion (as reflected by the nondimensional 

parameters) that gas-phase radiation takes precedence over surface radiation as stretch rate decreases.

A4.5 Conclusions

We have extended the model of an axi-symmetric stagnation-point diffusion flame for a solid fuel [T’ien 

1986, Oh et al. 1994] to include a gas-phase radiation model in order to investigate the effect of surface 

absorption and emission on flame temperature, burning rate, radiative fluxes, and the extinction limit of 

an optically thin flame. This study has demonstrated analytically and numerically that radiation effects 

(both gas-phase and surface) are amplified at low stretch rate.

The increase of surface radiative influence is due to the increase of flame standoff distance at low stretch 

rate (∝ a

-1/2). The increase of the gas-phase radiative effect is due to an increase of the optical thickness 

of the gas at low stretch rate. The rate of increase of the gas-phase radiative effect is more than that of the 

surface radiative effect (∝ 1/a). Thus, although at high stretch rate, surface radiation is more important 

than that in the gas phase, the latter may become comparable or more dominant at low stretch rates, as 

illustrated in our numerical study.

The radiation emitted from the surface results in a heat loss to the flame. Depending on the nature of the 

interaction with the surface, the influence of gas-phase radiation on the burning of a solid fuel is more 
133

complicated, as pointed out in De Ris [1979] and Bhattacharjee and Altenkirch [1992]. For a quasi-onedimensional, flat flame like the one we have here, approximately half of the radiation emitted from the gasphase is lost to the ambient environment; the other half is incident to the surface. This incident portion can 

be reflected at the surface (and lost to the ambient environment in an optically thin flame), or can be 

absorbed by the surface, or a combination of these, depending on surface absorptivity. In addition, the 

emissivity of the surface will determine how much radiative flux will join the reflected flux as a system loss 

to the ambient environment (less the small amount of flux absorbed in the gas-phase in the optically thin 

flame). In this work, we have studied the four limiting cases of surface radiative properties. In reality, solid 

fuels will have characteristic surface radiative properties bounded by these four extreme cases. 

Nevertheless, these cases help us to identify and understand the physics involved in the radiative exchange.

We have also used the results of Case 1 (gas-phase radiation with surface emission and absorption equal 

to unity) as a 'model' solid to compare with those of Case 5 (surface radiation only) in order to separate 

the effect of gas-phase radiation. With gas-phase radiation, the computed results show a lower flame 

temperature, a narrower flammable domain, but a slightly higher solid fuel burning rate.

A4.6 Further Studies

Using the two-flux radiation model along with the Planck mean absorption coefficient has led to 

fundamental understanding of the effects of flame radiation on the burning and extinction of a solid fuel. 

Computationally this model is very efficient. However, it does have well known limitations. Bedir et al 

[1997] and Rhatigan et al. [1998] have shown that, while the use of Planck mean absorption coefficient 

gives a qualitatively correct trend for many of the combustion problems, it over-predicts the radiative heat 

flux. The amount of over-prediction depends on the stretch rate (related to the optical length) and can be 

as much as 100% near the low-stretch quench limit. In the following sections, the combustion model has 

been extended, using the methods established by Bedir [1998], to include the more accurate narrowband 

radiation model with a discrete ordinate solution. When quantitative results are required, it is 

recommended that, unless numerical simplicity and computational time are major factors, the more 

accurate radiation models (i.e., narrowband and a discrete ordinate solution) be used.

The surface radiative fluxes for Case 1 show that, at low stretch, the incident flux is approximately equal to 

the surface radiative emission, so that the surface is radiatively neutral. This may not be true for 

multidimensional situations. In Jiang [1995], where gas-phase radiation is modeled using the S-N discrete 

ordinate scheme for a concurrent-flow flame spread problem, Jiang found that the incident flame radiation is 

always smaller than that emitted from the surface. We note that the radiatively neutral assumption was used 

in Bhattacharjee and Altenkirch [1992] for their opposed-flow flame spreading analysis.

Although we believe we have captured the essential influence of gas-phase radiation on the burning of a 

solid fuel in this work, quantitatively more accurate results can be predicted by improving the present 

models (e.g., Rhatigan et al. [1998]). The combustion model can be improved by including detailed 

kinetics for the gas and solid phases (as described in Section 3). A more sophisticated radiation model, 

which includes spectral analysis, was used to demonstrate the limitations of the results using the Planck 

mean absorption coefficient model. In this work, soot radiation has not been included. In low oxygen 

and/or low stretch rate situations, soot is not observed [Grayson et al. 1994, Olson 1997]; but in higher 

oxygen streams, soot may be present. To be consistent with the spirit of this work, accounting for the 

soot radiation effect requires that we predict the soot concentration, which is a difficult and contemporary 

research problem.

The other important limitation in this analysis is the use of one-step kinetics. The extension of the model 

to include detailed kinetics is taken up in the body of the thesis. This brings the accuracies of the 

radiation model (using narrowband spectral analyses) and the kinetics model (using detailed kinetics) to 

an equivalent level for study of the stagnation-point diffusion flame.
134

Table A4-1: Sample Surface Species Mass Fractions (Yw) and Nondimensional Temperature (Q) 

for Adiabatic Case (Case 6), Over Various Stretch Rates

stretch rate, s-1 Θmax Θsurface YF,w YO,w YCO2,w YH2O,w

6.25 7.815 2.024 0.5927 4.46E-12 0.09692 3.17E-02

101 6.87 2.291 0.576 1.28E-04 0.1008 3.29E-02

133.7 6.086 2.311 0.5706 1.41E-03 0.1008 3.29E-02

Table A4-2: Sample Surface Species Mass Fractions (Yw) and Nondimensional Temperature (Q) 

for the Surface Radiation Case (Case 5), Over Various Stretch Rates

stretch rate, s-1 Θmax Θsurface YF,w YO,w YCO2, w YH2O,w

0.88 4.923 1.734 0.1062 1.73E-03 2.11E-01 6.89E-02

6.25 7.214 1.993 0.3794 1.86E-08 1.48E-01 4.83E-02

25 7.261 2.128 0.4651 1.69E-06 1.27E-01 4.16E-02

100 6.371 2.245 0.5054 7.25E-04 1.17E-01 3.82E-02

106.3 6.097 2.252 0.5039 1.53E-03 1.17E-01 3.81E-02

Table A4-3: Sample Surface Species Mass Fractions (Yw) and Nondimensional Temperature (Q) 

for the Gas-Phase and Surface Radiation Case (Case 1), Over Various Stretch Rates

stretch rate, s-1 Θmax Θsurface YF,w YO,w YCO2,w YH2O, w

1.33 6.296 2.035 0.2529 2.72E-02 1.75E-01 5.72E-02

6.25 6.381 2.055 0.4678 1.5E-06 1.26E-01 4.14E-02

100 6.139 2.255 0.5088 1.23E-03 1.16E-01 3.78E-02

101.1 6.034 2.256 0.5075 1.59E-03 1.16E-01 3.78E-02
135

Figure A4-1: Illustration of geometry and radiation model.

Figure A4-2: Effect of stretch rate (a) on ratio of local radiative absorption length to flame 

thickness (Dk).
136

Figure A4-3: Nondimensional radiative fluxes vs. stretch rate.

(a): Radiative Flux from the Flame Edge to the ambient (qe

+

), (b): Radiative 

Flux from the Gas-phase to the Surface (qw

-

), (c): Radiative Flux from the 

Surface to the Gas-phase (qw

+

)
137

Figure A4-4: Nondimensional maximum flame temperature vs. stretch rate.

Figure A4-5: Nondimensional flame standoff distance vs. stretch rate.
138

Figure A4-6: Nondimensional fuel burning rate vs. stretch rate.

Figure A4-7: Effect of gas-phase radiation on nondimensional maximum flame temperature.
139

Figure A4-8: Effect of gas-phase radiation on nondimensional burning rate.

Figure A4-9: Flammability map: ambient oxygen mass fraction vs. stretch rate.
140

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Brau, A.; Jonkman, R. M. “Classical Theory of Rotational Relaxation in Diatomic Gases,” The Journal of 

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Effects of Gas-Phase Radiation and Detailed Kinetics on the Burning and Extinction of a 

Solid Fuel

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Jennifer L. Rhatigan, Ph.D.

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Lyndon B. Johnson Space Center

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13. ABSTRACT (Maximum 200 words)

This is the first attempt to analyze both radiation and detailed kinetics on the burning and extinction of a solid fuel in a stagnationpoint diffusion flame. We present a detailed and comparatively accurate computational model of a solid fuel flame along with a 

quantitative study of the kinetics mechanism, radiation interactions, and the extinction limits of the flame.

A detailed kinetics model for the burning of solid trioxane (a trimer of formaldehyde) is coupled with a narrowband radiation model, 

with carbon dioxide, carbon monoxide, and water vapor as the gas-phase participating media. The solution of the solid trioxane 

diffusion flame over the flammable regime is presented in some detail, as this is the first solution of a heterogeneous trioxane flame. 

We identify high-temperature and low-temperature reaction paths for the heterogeneous trioxane flame.

We then compare the adiabatic solution to solutions that include surface radiation only and gas-phase and surface radiation using a 

black surface model. The analysis includes discussion of detailed flame chemistry over the flammable regime and, in particular, at the 

low stretch extinction limit. We emphasize the low stretch regime of the radiatively participating flame, since this is the region 

representative of microgravity flames. When only surface radiation is included, two extinction limits exist (the blow-off limit, and the 

low stretch radiative limit), and the burning rate and maximum flame temperatures are lower, as expected. With the inclusion of 

surface and gas-phase radiation, results show that, while flame temperatures are lower, the burning rate of the trioxane diffusion flame 

may actually increase at low stretch rate due to radiative feedback from the flame to the surface.

The trioxane diffusion flame model is then extended to include surface absorption of radicals from the gas phase. We compare the 14. SUBJECT TERMS 15. NUMBER OF 

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fuel, solid fuel, diffusion flame, extinction, gas-phase radiation, kinetics, surface 

radiation 155

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