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On the theory of flammability limits

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On the theory of flammability limits

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Content ON THE THEORY OP FLAMMABILITY LIMITS by William Blaine Stine A Dissertation Presented to the t FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (Mechanical Engineering) June 1972 INFORMATION TO USERS This dissertation was produced from a microfilm copy of the original document. While the most advanced technological means to photograph and reproduce this document have been used, the quality is heavily dependent upon the quality of the original submitted. The following explanation of techniques is provided to help you understand markings or patterns which may appear on this reproduction. 1. The sign or "target" for pages apparently lacking from the document photographed is "Missing Page(s)”. If it was possible to obtain the missing page(s) or section, they are spliced into the film along with adjacent pages. This may have necessitated cutting thru an image and duplicating adjacent pages to insure you complete continuity. 2. When an image on the film is obliterated with a large round black mark, it is an indication that the photographer suspected that the copy may have moved during exposure and thus cause a blurred image. You will find a good image of the page in the adjacent frame. 3. When a map, drawing or chart, etc., was part of the material being photographed the photographer followed a definite method in "sectioning" the material. It is customary to begin photoing at the upper left hand corner of a large sheet and to continue photoing from left to right in equal sections with a small overlap. If necessary, sectioning is continued again — beginning below the first row and continuing on until complete. 4. The majority of users indicate that the textual content is of greatest value, however, a somewhat higher quality reproduction could be made from "photographs" if essential to the understanding of the dissertation. Silver prints of "photographs" may be ordered at additional charge by writing the Order Department, giving the catalog number, title, author and specific pages you wish reproduced. University Microfilms 300 North ZM b Rood Ann Arbor, Michigan 4*106 A Xarox Education Company 72-26,059 I STINE, William Blaine, 1937- ( ON THE THEORY OF FLAMMABILITY LIMITS. University of Southern California, Ph.D., 1972 Engineering, mechanical University Microfilms, A X E R O K Company, Ann Arbor, Michigan THIS DISSERTATION HAS BEEN MICROFILMED EXACTLY AS RECEIVED. UNIVERSITY O F SOUTHERN CALIFORNIA THR ORADUATK SCHOOL UNIVERSITY PARK LOS ANOKLSB. CALIFORNIA § 0 0 0 7 This dissertation, •written by ........ Wi1liam B1sin• Stins....... under the direction of Dissertation Com mittee, and approved by all its members, has been presented to and accepted by The Gradu ate School, in partial fulfillment of require ments of the degree of D O C T O R OF P H I L O S O P H Y _ JUNE 1972 Date........... DISSERTATION O i t t e : < 7 PLEASE NOTE: Some pages may have indistinct p rin t. Filmed a s re c e iv e d . University Microfilms, A Xerox Education Company To Jenny, Blaine and Sharon in hopes of a better world tomorrow. 11 ACKNOWLEDGMENTS I would like to express my sincere appreciation to Professor Melvin Gerstein for suggesting this problem and his guidance and support throughout the course of my stud ies. In addition, I would like to thank Professors P. R. Choudhury, M. Ogawa, and R. L. Mannes for their sugges tions and guidance in the preparation of this manuscript. Also, I wish to thank Mr. R. L. Hale for his suggestions in the application of the numerical techniques. Acknowledgment is made to the American Gas Associa tion who supported this research, under their Basic Re search Project BR-48-10, and to North American Rockwell, Los Angeles Division, who provided financial support during the early phases of my study. iii TABLE OF CONTENTS Fag* ACKNOWLEDGMENTS........................................... ill LIST OF T A B L E S .......................................vii LIST OF FIGURES...................................... viii NOMENCLATURE........................................... X ABSTRACT................................................. xiii Chapter I. STATEMENT OF THE PROBLEM................. I II. SURVEY OF THE LITERATURE................. 4 Experimental........................ 4 Limit Theories...................... 6 Non-adiabatic Flam* Theory ........... 13 III. ADIABATIC FLAME STUDIES................... 18 Limit Adiabatic Flame Temperatures . . 18 Adiabatic Flame Structure ........... 26 Equations of Change.......... 26 Boundary Conditions ............. 28 Reaction Kinetics ................ 30 Non-dimensional Equations .... 30 Numerical D a t a ............... 33 Results . . . . . . . . . . . . . 3 I S Comparison with Experiment . . . 41 IV. NON-ADIABATIC FLAME STUDIES ................ 46 Analytical Model .................... 46 Equations of Change.......... 46 Reaction Kinetics ................ 47 iv Chapter Pag* H«at Loss Mod*l............... 47 Conduction............... 48 Radiation............... 49 Boundary Conditions .............. 52 Numerical D a t a ............... 55 Analytical Techniques ................ 56 Starting..................... 58 Integration Parameters ......... 60 Converging the Eigenvalue .... 61 Converging X .................. 62 Computer Program Description . . 65 Results of Integration ................ 67 Effects of Heat Loss Node .... 67 Conduction O n l y ........ 67 Conduction and Radiation . . 72 Minimum Tube Diameter........ 73 Pressure Effects ................ 75 Heat Balance Structure ......... 76 Peak Temperatures............. 78 Plame Temperature-Burning Velocity Relations ........... 60 Flame Thickness............... 63 V. FLAMiABILITY LIMIT MODEL............... 86 M o d e l .............................. 86 Experimental Correlation ............. 87 Influence of Apparatus Size . . . 87 Unification with Quenching Data . 88 Reduced Pressure Flammability Limits..................... 91 Burning Velocity at the Limit .... 97 Relation to Adiabatic Flame Characteristics .................... 101 Effects of Radiation L o s s ......... 101 VI. CONCLUSIONS..............................104 v Pag* APPENDIX............................................... 106 LITERATURE CITED...................................... 119 vi LIST OF TABLES Table Bag* 1. Equilibrium Compositions of Methane-air- diluant Systems ............................... 24 2. Predicted and Observed Minimum Diameters for Flame Propagation ......................... 90 A-l. Cross-listing of Important Program Variables 107 vii LIST OF FIGURES Figure Page 1. Flammability Limit Mixtures of Methane-air- diluent Systems............................... 20 2. Adiabatic Equilibrium Flame Temperatures for Limit Mixtures of Methane-air-diluent Systems 21 3. Limit Adiabatic Flame Temperatures for the Light Paraffin Hydrocarbons .................. 23 4. Structure of an Undiluted Adiabatic Flame . . 36 5. Structure of a Highly Diluted Adiabatic Flame 37 6. Effect of Dilution on the Inflection and Flame Temperatures of Adiabatic Flames . . . 38 7. Effect of Dilution on Flame Thickness and Burning Velocity for Adiabatic Flames .... 40 8. Variation of Adiabatic Flame Thickness with Dilution for Different Diluents ............. 43 9. Conductive and Radiative Heat Loss Models for Circular Tubes ............................... 53 10. Behavior of the Solutions to the Equations of C h a n g e ........................................ 63 11. Calling Sequence Diagram for Program DISTILBR 66 12. Effect of Increasing Heat Loss on Burning Velocity; Conduction Loss O n l y ............. 68 13. Effect of Increasing Heat Loss on Burning Velocity; Conduction and Radiation Loss . . . 69 14. Effect of Dilution on Critical Tube Diameter 74 15. Heat Balance Structure for a Near-limit Flame 77 16. Effect of Heat Loss on Peak Temperature in Non-adiabatic F l a m e s ............. 79 viii Figure Page 17. Relation Between Peak Temperature and Burning Velocity ............................... 81 18. Effect of Heat Loss on Flame Thickness for Non-adiabatic Flames ........................... 84 19. Comparison of Critical Diameter Calculations with Quenching and Flammability Limit Ex periments at Atmospheric and Reduced Pressures 89 20. Variation of Critical Diameter with Pressure; Comparison with Experiments .................. 92 21. Effect of Pressure on the Fuel Required for Plame Propagation in a 5 cm Diameter Tube; Closed Tube, Methane-air Data ................ 94 22. Effect of Pressure on the Fuel Required for Flame Propagation; Open Tube, Propane-air Data 95 23. Variation of Overall Reaction Order with Fuel Concentration for Propane-air Mixtures .... 96 24. Reduced Pressure Flammability Limits; Compari son with Open and Closed 5 cm Tube Propane-air Experimental D a t a ............................. 96 25. Burning Velocity Model Showing Variation of Predicted Burning Velocity with Fuel Concentra tion for Various Diameters; Comparison with Experiments.................................... 99 ix NOMENCLATURE A - steric or frequency factor <units depend on reaction order) B - constant in ateric factor (units depend on reaction order) Cp - specific heat at constant pressure (cal/g °K) C - curve fit parameter, equals 1060 or T, whichever is greater (°K) D - tube diameter (cm) - binary diffusion coefficient, species i into j (cm /sec) EA - activation energy (cal/mole) G - mass flux fraction, reactant species G^ - mass flux fraction, species i h - heat transfer coefficient (cal/cm^ sec °K) - enthalpy (absolute base), species i (cal/mole) Kj - net rate of formation of molecules of species i (mole/cmJ sec) L - mean beam length (ft) m - molecular weight of species. A, B, and M (g/mole) m^ - molecular weight, species i (g/mole) M - mass flux (g/cra^ sec) n - total molar density (mole/cm^) - molar density, species i (mole/cm3) Nu_ - Nusselt number based on diameter ni x pressure (atm) partial prassura of spacias i (atm) Paclat numbar volumatric haat axchanga rata with anvironmant (normal to flama travel) (cal/cm^ sac) reactant heating value (cal/g) dimanaionlaaa reaction rata universal gas constant (cal/mole °K) absolute temperature (°K) internal energy (cal/g) mass average velocity (cm/aec) burning velocity (mass average velocity of un burned gas) (cm/sec) diffusion velocity, species i (cm/sac) mole fraction, reactant species mole fraction, species i distance (cm) overall reaction order absorptivity of gas temperature exponent of steric factor Lewis number emisslvity emissivity contribution of species i dimensionless temperature xi X - thermal conductivity (cal/cm sac °K) £ - dimensionlass distanca p - total mass dansity (g/cm ) * p ^ - mass dansity, spacias i (g/cm3) - Stefan-Boltsmann constant (cal/cm^ sac °K*) - fraction of adiabatic raactant daplation subscripts 0 - raactant boundary (upstraam or cold) 1 - starting boundary ® - product boundary (downstream or hot) A - raactant spacias B - product spacias c - conduction mode g - gas i - general spacias i j - general spacias j M - inert diluent species r - radiation mode s - surface Notei A correlation between these symbols, and those used in the computer program, is found in the Appendix. xii ABSTRACT The purpose of this investigation was to develop a method of analytically predicting the flammability limit of a combustible gas. The flammability limit is defined ex perimentally by diluting a flammable mixture of gases such as methane and air with one of its constituents or with other gases until it is no longer flammable. The flamma bility limit is the borderline composition; a slight change in one direction produces a flammable mixture, in the other direction a non-flammable mixture. The one-dimensional steady-state flame equations with heat loss distributed throughout the flame and with single-step Arrhenius type kinetics, are solved by numeri cal integration on a digital computer. Heat loss due to conduction and radiation are included. A double iteration technique is described which permits starting the integra tion at the cold or reactant boundary and integrating to ward the hot boundary with distance as the independent variable. A generally available integration program is used and a copy of the computer code developed for these solutions is included. Two burning velocities are calculated at low heat loss as with previous closed form approximate solutions. As heat loss increases these approach a single velocity xiii beyond which no solution exists. The important experimental results of flammability limit determination are predicted with this model. These include reduced changes in limits due to increases in tube diameter for diameters greater than about 5 cm and the prediction of quenching diameter and flammability limits at atmospheric and reduced pressures. It is concluded that the flammability limit exists only because of heat loss to the environment. Therefore, the flammability limit and quenching are one and the same phenomenon. Radiation loss accounts for limits in large systems; however, the reduction of radiation loss with size predicts a significant broadening of flammability limits from those determined in 5 cm tubes. This study shows that when using flammability limits to define Msafe" conditions, heat loss must be considered along with the concentration of fuel in the atmosphere being analyzed. xiv CHAPTER I STATEMENT OP THE PROBLEM This rsssarch was carried out in order to attain a better analytical understanding of the experimental phenom enon variously called the flammability limit, inflammabili ty limit, extinction limit, explosion limit1, and propaga tion limit. Early experiments were done to determine safe environments for mining. Later, extensive studies were performed to effect safe processes and handling in the chemical and petroleum industry. Today closed environments as in airplanes, spacecraft and undersea craft are becoming commonplace. It is important to be able to define a safe atmosphere within the constraints of other operational re quirements. Also today, the importance of maintaining steady-state combustion processes near the limits of The term “explosion limit" is often misused to Man flammability limit. It is usual for the term "explosion limit" to relate to the thermal self-ignition of a reactant mixture as described by Semnov (1940). This is based on introducing the reactant into a heated chasiber and, depend ing upon the rates of heat generation and heat loss to the walls of the chamber, the overall process will either be- c o m exothermic with time (an explosion) or cool to the temperature of the chamber. The flammability limit problem deals with the steady-state propagation of a flame into cold reactant whereas the thermal self-ignition problem deals with the transient heating of a reactant in a warm vessel. No known relationships between the results of thermal self-Ignition studies and the flammability limit have been developed. 1 2 flammability is becoming apparent as one means of reducing the generation of air pollutants. The flammability limit is almost universally defined in terms of an experimental result. Coward and Jones (1952) in the definitive work on this subject give the fol lowing definitions "A flammable mixture of gases such as methane and air may be diluted with one of its constituents or with other gases until it is no longer flammable. The dilu tion limit of flammability, or simply the limit of flammability is the borderline composition; a slight change in one direc tion produces a flammable mixture, in the other direction a non-flammable mixture." The determination of flammability limits is necessarily made in an apparatus of "suitable size" so as to reduce heat loss and with an ignition source, usually a spark, hot wire or small flame, energetic enough to ignite weak mix tures. It is generally assumed that the flammability limit is independent of the size of the apparatus. Although there is no A.S.T.M. standard to define the test apparatus and procedure for flammability limit determination, the generally accepted standard apparatus is that used at the U. S. Bureau of Nines (see Coward and Jones, 1952, p. 10). This consists of a vertical glass tube, 5 cm in diameter and 150 cm long. The ignition source is at the bottom and the flame must travel the entire length of the tube before the mixture is classified as flammable. 3 Two flammability limits arc normally experienced, tha laan and tha rich, foe any combustibla fual and oxidant gas combination. Thasa corraspond respectively to tha min imum and maximum amount of fual nacasaary for staady-stata flame propagation. Although defined in terms of mixture strength, other important variables are pressure, initial temperature, and possibly tha acceleration due to gravity. Flammability limits are generally determined for mixtures of fuel and oxidant in their vapor state, although some studies have been concerned with "dusts* and "mists." The lean flammability limit is related to the experimen tally determined "flash point" of a pool of liquid fuel. This is the bulk liquid temperature at which the partial pressure of the fuel vapor is just great enough to produce a flammable mixture within the contained vapor space. Gerstein and Stine (1970) have shown how flammability lim it data can be used to describe variations and anomalies in flash point determinations. Modern computing systems have provided the tool for solving many laminar flame problems. However, to date, there is no accepted method of analytically predicting flammability limits. It was the purpose of this research to develop such a method and show its applicability by studying the behavior of flammability limits. CHAPTER IX SURVEY OP THE LITERATURE Exper imental The most dsfinits work, compiling and summarising the body of experimental flammability limit data, was writ ten by Coward and Jones (1952). Subsequently this was up dated by Zabetakis (1965). These authors have compiled data for numerous fuel-oxidant combinations, and in addi tion, present data on the effects of dilution, apparatus sise, direction of propagation, pressure and initial tem perature. A major advance in the experimental search for fun damental flammability limits came with the introduction of the flat flame burner. Results reported by Powling (1949), Egerton and Thabert (1952), Badami and Egerton (1955), and Dixon-Lewis and Isles (1959) show that flat flame lower limit mixtures for hydrocarbons in air are considerably weaker than those determined in tubes. Por example, Badami and Egerton measured limits of 2.53 per cant for ethane compared to approximately 3.0 per cent when measured in a tube. The significant difference between the two methods, other than the relative motion of the flame co ordinates with respect to the laboratory, and direction of 4 5 propagation, baing tha ability to vary tha amount of haat lost (or gainad) at tha hot or cold boundary with tha flat flama burner. Thasa rasults raopanad tha quastion of whathar fundaaantal flammability limits axist and lad Linnatt and Simpson (1957) to concluda that thare is no ex- parimantal avidanca proving that any limits which hava baan obsarvad ara fundamental, and that if ona does axist, thara is probably a considerabla difference between ob sarvad limits and any possible fundamental limit. Kydd and Foss (1964) studied flat flames experimen tally using a 25 cm burnar in an attempt to verify tha up stream/downstream heat loss predictions based on tha as sumptions of Spalding (1957). Burning velocities of below 2 cm/sac ware measured. Their deduced flammability limit criterion was that upstream haat loss should decrease to zero at tha flammability limit. However, instabilities occurred before a valid test of thiB criterion could be made. A recent contribution was mads by Levy (1965) who studied partially propagating flames in tubas optically, using normal and Schliaran photography. Ha found that a 'ball* of hot gas continues to rise in tha tuba after tha flama has gone out for upward propagating flames and that no such phenomenon occurred for downward propagation. Based on thasa observations, ha developed a theory of 6 £lammability limits based on ths buoyancy of ths hot gas and tha matching of burning valocity with tha convactiva valocity. Limit Thaoriaa Tha dilamma of having no adaquata flammability limit thaory is wall dascribad in a survay papar by Bgarton (1953). Attampta at corralating axparimantal rasults with ovarall flama proparties such as flama tamperatura and haat of combustion are shown to have met with limited success. It is pointed out that no adaquata thaory has yat succeeded in predicting tha lower limit concentrations and tha burn ing valocity at the limit and that tha upper limits will probably ba harder to predict due to tha incomplete reac tions which occur. Numerous explanations as to why flammability limits are experienced in tha laboratory, and whether they are fundamental properties have bean proposed. Thasa may ba categorized in three disciplinary areas with soma over lapping; thermal, fluid mechanical, and reaction theories. In addition there is a category of explanations based on empirical criteria. ■arly work by Daniell (1930) resulted in approxi mate solutions to the energy equation including heat loss to the environment, showing that there will be a minimum tube radius (corresponding to a maximum heat loss) below 7 which flama propagation is impossibla. Spalding (1957) aolvad both tha non-adiabatic anargy aquation and tha dif fusion aquation and showad that two burning valocitias ara pradictad for low haat loss and that thara is a maximum haat loss bayond which tha flama aquations cannot ba solvad. Ha ralatad this point of maximum haat loss to tha flammability limit. This tharmal thaory has baan axpandad by Mayar (1957), Barlad and Yang (1960) and subscribad to by Dixon-Lawis and Ialas (1959, p. 481) and othars. Tha non-adiabatic thaoriaa ara discussad in datail in tha fol lowing saction. Tha obsarvad importanca of convaction on flammabili ty limits was pointad out by Linnatt and Simpson (1957). A convaction thaory for upward propagating limit flamas was proposad by Lavy (1965) basad on axparimantal avidanca. Ha obsarvad that a non-luminous "ball** of hot gas continuas to risa aftar a partially propagating limit flama has ax- tinguishad. Bacausa tha valocity of this "ball" is tha sama as that for tha bubbla rising in an amptyimg tuba of liquid, ha postulatad that a flammability limit is raachad whan tha burning valocity is too low comparad to tha spaad of tha bubbla. Basad on this thaory, tha laan flammability limit should incraasa with tha squara root of tha tuba diamatar and tha gravitational accalaration. Ho flammability data 8 at Modified acceleration are known. The increase in lUiits with diameter is contrary to generally accepted experimen- tal results and will be discussed in Chapter V. Lovachev (1970,1972) extended this theory by developing the neces sary analytical formalism. Attempts to predict the existence of flammabillty limits based on fluid mechanical instabilities have been made. Lewis and von Blbe (1951) and Rosen (1954) consider ed the effect of perturbations on the steady-state condi tion. They concluded that for some mixtures perturbations will die while for others they will grow and the flame never return to the steady-state. Layzer (1954), on the other hand, has analyzed the stability of single step, constant enthalpy^ reactions to small disturbances and deduced that such disturbances would die away without lead ing to extinction. Spalding (1957) extends this argument. In their second edition, Lewis and von gibe (1961) aban doned this thesis in favor of one based on the "stretching" of a flame by the flow field in which it is propagating. Extinction is postulated if a critical value of the Karlo- vitz number is exceeded. The Karlovitz number is a measure of the area increase or "stretch” that the cosfoustion wave ^Constant absolute enthalpy of the mixture where the base for each species is obtained by arbitrarily assigning the value of zero to elements in their standard states at a reference temperature of 298°K. 9 surface undergoes in a two dimensional flow fiald. Anothar varaion of tha lnatability arguawnt la baaad on tha preferential diffuaion of ona of tha raacting apa- ciaa through tha mixture. This phenomenon haa baan ra- viawad by Markstein (1953) and ahown to ba aignificant for upward propagation of some mixtures. A third claaaification of flaausability limit thao- xiaa ia baaad on tha raaction kinatics of tha raacting mix- turaa. Thaaa are baaad on tha concept that for certain mixturaa, tha heat of raaction cannot bring tha mixtura to a temperature at which tha chain branching raactiona pre dominate over chain terminating raactiona. Theoriea of this type have baan proposed by van Tiggalan (1947), van Tiggalan and Deckers (1957), van Tiggalan and Bxuger (1964), Weinberg (1955) and Wahnar (1963). A recant sum mary of this type of theory is found in Fenimore (1964). It la fait that limit theories based on the kinetic properties of tha raacting mixtura are still in tha stage of ganaralitias and approximations. This is exemplified by tha theory of van Tiggalan and Brugar (1964) which re quires that tha net heat liberated by chain generation and termination reactions must ba greater than tha heat ab sorbed by chain propagation reactions. In order to pro ceed with tha analytical development, mean values of the important kinetic parameters at soma temperature lying 10 between the initial and final temperature must be defined thereby presupposing an exothermic flame, in addition, it is mentioned that the thermochemical data used may have to be modified to compensate for the effects of heat loss, adding another level of arbitrary assumptions to the theory. Spalding (1957) argues against the kinetic mechanism of extinction from the standpoint that there is normally a radical generation reaction which is independent of radical concentration. This prohibits chain breaking from having the required steeper dependence on radical concentration than the chain branching reaction. Simon, Belles and Spakowski (1953) developed an in terpretation of reduced pressure flammability limits based on the destruction of active particles on the container surfaces. With this theory they related flammability lim its and quenching phenomena. It has since been demonstrat ed that similar results can be obtained by assuming that thermal conductivity rather than the diffusion of active particles is the important mechanism in flame propagation. In addition, there is little experimental evidence to in dicate that different surfaces produce different flammabil- ity limits. It is shown in Chapter V that the results of this study (based on thermal mechanisms only) agree well with the experimental data used by the above authors as 11 avidanca to justify thair activa particla daatructlon mach- anism of flammability limits. Empirical thaorias of flammability limits hava had littla succass in pradicting limits for a wlda ranga of fuals. Ona such traatsiant proposad by Bachart (1949), (1950 a,b) supposas that flamas with valocitias of loss than 1 cm/sac cannot propagata. Obsarvations of a limit* ing valocity at tha flammability limit hava also baan mada by Coward and Hartwall (1926) of 22 and 23 cm/sac, Jost (1946) of 10 to 20 cm/umc and Badami and Egarton (1955) of 3 to 4 cm/sac. If this trand continuas, burning valocitias of around 0.1 cn/iac may soon ba axpactod. In fact, JUcha, Kozak and Zappa (1952) hava maasurad burning valocitias down to zaro; howavar, thair intarprotation of tha thraa- dimansional axparimantal data is quastionad by Spalding (1957) . Ona of tha aarliast amplrical critarion for flasua* ability limits was daducad by hm Chatoliar and Boudouard (1898) who found that tha haat of combustion par unit vol- uma of limit mixturas with air for a numbar of diffarant gasas was approximataly a constant. Khita (1925) axtandad this by postulating that tha flama tamparatura was constant at tha lowar limit. Howavar, Egarton and Powling (1948) found that this critarion is important but no vary rallabia for pradicting limits of diffarant fuals. Egarton (1953) 12 notaa that flat flama burnar limits axhibit a much nora constant flams tamparatura at tha lowar limit for various hydrocarbon fuals than thosa calculatad from tuba propaga tion data. A variation of tha flama taaparatura critarion is tha Burgass-Whaslar aquation which statas that tha hast of combustion of tha fuel timas tha parcantaga of combus- tibla at tha lowar limit, is a constant. This aquation was proposad by Burgass and Whaalar (1911) and had baan modifiad to includa tha affact of diffarant initial tampar aturas by Zabetakls, Lambiris and Scott (1959). This cri tarion is invastigatad in this study and tha rasults pra- santad in Chaptar III. Fann (1951) studiad tha ralationship batwaan tha minimum spark ignition anargy and tha calculatad laan lim it adiabatic flama tamparatura. Using a tharmal thaory of minimum ignition anargy, a ganeral corralation of thasa paramatars was found and pradiction of ona from tha othar was possibla. Howavar, it is fait that tha inaccuracy of tha pradiction is too graat for it to ba usad as an analy tical critarion for flammability limits. This discussion is bast concludad with a quota from Coward and Jonas (1952). "It is apparant that th-s -hvora- tical traatmant of limits of flammability is maagar and mainly qualltatlva; it is bound up with tha unsolvad prob- lam of tha thaory of flama propagation in gsnaral." 13 Mon-adiabatic Fla— Thtorv Tha first known mathematical analyala of a flame with haat loss was by Daniall (1930). Ha solved tha ona* dimensional ataady-stata anargy aquation with haat loss, using thrae diffarant raaction rata approximations but without considaring diffusion. His rasults showad that "cooling introducas a tarm which lowars tha (burning) velocity, and thara is a positiva minimum valocity balow which propagation is impossible." Ha also concluded that "there will ba a minimum radius .... balow which propaga tion is impossible.“ Spalding (1957), using non-Arrhenius kinetics and haat loss only at a boundary beyond tha point of fuel de pletion, solvad tha one-dimensional steady-state anargy and diffusion aquations. A closed form solution was ob tained from which it could ba deduced that a given combus tible has two possible burning velocities. If tha haat loss increases, tha two burning valocitias first becosw coincident and than imaginary. Tha condition of coinci dence Spalding identified as tha flammability limit. The lowar flama spaed is shown to ba unstable. Mayer (1957) followed with an overall anargy balance method again re sulting in two flame speeds which converge at the limit. Tha existence of two flama speeds for a particular mixture was demonstrated experimentally by Spalding and Yumlu 14 (1959). The work of Spalding (1957) has baan axtandad by ■any authors attempting to aliminata ona or mora of his approximations and still obtain aithar a closad form or itarativa solution to tha problam. Chan and Toong (1960) obtainad itarativa solutions basad on truncatad Arrhanius kinatics. Barlad and Yang (1960) solvad tha sama aqua tions by assuming that tha tamparatura distribution was in tha form of a Gaussian arror function. Adlar and Spalding (1961) and Adlar (1963) simplifiad tha formulation by davaloplng a "thin flama” solution tachniqua which was sub- saquantly found by Adlar and Kannarlay (1966) to produca significant arrors unlass soma of tha haat loss was neg- lactad. Tha tarm "thin flame" danotas ona typa of non- Arrhanius kinatics whara tha raaction is assumad to taka placa only ovar soma finita langth of tha total "flama." An axchanga of lattars documenting concern ovar tha validity of tha flama modals was invitad by Barlad (1961 a), who indicatad that haat loss in tha raaction zona dominates tha description of flama extinction limits. This was answered by Spalding (1961) and rebutted by Barlad (1961 b). Tha axchanga ended with a desire for aithar an exact solution or a mathematical investigation of tha rela tive importance of haat loss upstream as opposed to down stream of tha raaction. In this study, haat loss is 15 assumed to be distributed throughout tha flame. Another exchange of letters was concerned with the downstream boundary condition for fuel quantity with ex cess oxidant. All of the aforementioned solutions for the non-adiabatic flame make the assumption that, at an infi nite distance past the flame front, all of the fuel has been depleted. Gray and Yang (1966) pointed out that this seemingly obvious assumption was not necessarily true. They proposed that "heat loss reduces the downstream temper ature too rapidly for the reaction to go to completion, e- ven though the time available is infinite since the time required for completion with a decreasing temperature down stream is even larger." Adler (1967 a) generalized this development and showed that the result is equally true for normal diffusion and a more general heat loss. Zn addi tion, he derived an estimate of the error introduced by the assumption that the reactant concentration is zero at infinity. However, Gray and Yang (1967) showed that these error estimates were invalid. Adler (1967 b) concluded the discussion indicating that the existence of a non-zero fuel concentration at the downstream boundary had not been conclusively shown and that the matter could only be re solved by exact numerical solutions to the flame equations with heat loss. In this study, the possibility of having fuel remaining at the product boundary is investigated. 16 Sokolik (1960) discussed tha offset of haat loaa to tha snvlronmant on flammability limits. Ha developed a charactarization of tha dagraa to which tha walla of a tuba cool a flama which ha calla tha ralativa haat reawval. This is dafinod as tha ratio of tha rata of haat transfar through the lataral surface of tha flama aona (i.a. an arbitrary thicknass which is dafinad to includa prahaat and raaction zonaa) to tha rata of haat libaratad in tha flama. Ha pradicted that tha raducad affact of changing tuba diamatar on limits at larga diamatars is dua to tha transition from laminar to turbulant flow, tharaby in creasing the haat transfer coefficient to tha walls. It should ba noted that axparimantal observations do not con firm this turbulence. A few numerical solutions of tha non-adiabatic flama problem ware obtained by Yang (1961) using an analog computer. Ho programmed tha anargy and diffusion aquations with Arrhenius type kinetics and distributed haat loss and a variable Lewis number. Burning velocity eigenvalues and flama structure information were obtained for adiabatic, quenched and pressure limited flames for a simulated pro pane-air mixtura. Although limited by computer noise (tha starting point had to ba placed at 100°K above cold bound ary temperature, thus reducing the range of validity of tha cold boundary approximations), ha was able to verify 17 numerically tha existence of two burning velocities in general, and the existence of a limiting heat loss beyond which no valid solution could be obtained. Experimental verifications of conduction heat loss causing extinction (i.e. quenching) are numerous as re viewed by Potter (1960). However, tew experimental data are available to show that radiation heat loss causes ex tinction when quenching is no longer significant. Cum mings, Hall and Straker (1962) studied the highly luminous acetylene decomposition flame and concluded that the cri tical pressure limit is determined by loss of heat by radiation. However, Wolfhard (1956) points out that an intrinsic limit due to increased radiation loss at de creasing pressure has never been observed for the less luminous near-stoichiometric flames of hydrocarbons with air or oxygen. CHAPTER III ADIABATIC PLANE STUDIES1 Limit Adiabatic Fla— TMDtiiturti Calculations of tha adiabatic flama tamparatura at tha experimantally obtainad flammability limit with various diluants wara mada for methane-air mixturas. Tha purpoaa of thasa calculations was to datarmina how constant limit flama tamparatura was whan various diluants ara addad to tha fual-air mixtura. Thesa calculations wara mada using a computer pro* gram (McBrida and Gordon, 1968) which calculates product equilibrium composition and properties considering many posslbla atomic combinations including thosa formed from tha diluant (if not inert). Limit methane-air-diluent con centrations ware taken from Coward and Hartwell (1926) for dilutions with argon, helium, nitrogen and carbon dioxide. It is noted in passing that there is some confusion in the re-plotting of these data by Zabatakis (1965) and that the original sources were consulted. Concentration data for dilution with water vapor and carbon tetrachloride were taken from Coward and Jones (1952). These limits are for ^uch of this material appears in Gerstein and Stine (1971). 18 19 upward propagation in larga tubaa at room tamparatura and prassura. A summary of thasa data is praaantad in Figura 1. Nota that tha mixtuxaa throughout this disaartation ara dafinad as parcantaga of total raactant voluma; not as par- cantaga of original oxidant atmoaphara as is somatimas usad. A lina of stoichiomatry is shown for rafaranca. Adiabatic equilibrium flama tamparaturas for math- ana-air-diluant mixtures ara shown in Figura 2. It may ba notad that there is a variation in limit flama tamparatura of ovar 10 par cent. In fact, tha diffaranca between laan and rich limit flama tamparatura is significant even for tha undiluted case. Tha flama tamparatura is found to in crease as tha limit mixtura bacomaa mora rich. A limit critarion baaad on adiabatic flame tamperatura as attempted by Egarton and Powling (1946) will need to at laast include soma compensation for the increasa in tamparatura from laan to rich. In particular nota how tha flama tamparatura of car bon tatrachlorida diluted flames increases, avan beyond tha undiluted rich limit tamparatura. Burdon, Burgoyna and Wainbarg (1955) notad a similar behavior whan methyl bromida was used to diluta naar-limit hydrogan-air and car bon monoxida-air flames. They deduced that this was an indication of chemical intervantion of the reactive diluent and tharafora justifias tha assumption of a flammability DILUENT ADDED - VOLUME % 20 30 STOICHIOMETRIC MIXTURES 40 30 C02 CCI4 METHANE *-V0LUME% Fig. 1. Flammability Limit Mixtures of Methane-air-diluent Systems ADIABATIC FLAME TEMPE RATURE-°K 21 2000 CCL Ht 1800 CO. H20 1600 j-EAN 40 90 30 10 20 0 DILUENT ADDED-VOLUME % Fig. 2. Adiabatic Equilibrium Flama Tamparatura* for Limit Mixturaa of Mathana-air-diluant Syatama 22 limit thaory baaad on tha rata of chain branching baing aqual to tha rata of chain braaking. Howavar, as shown at the and of this chapter, tha efficacy of carbon tetrachlo ride as a diluent can ba attributed largely to thermal con siderations and it is fait that tha increased tamparatura can also be explained if the thermal properties of tha overall raaction ware also modeled. It is notad that an increase in flame temperature from lean to rich is opposite to the behavior noted by Lewis and von Elbe (1961) in their discussion of flanuaabil- ity limits. They note that the adiabatic flame temperature decreases sharply when the fuel-oxygen ratio for limit mix tures changes from lean to rich in a fuel-oxygen-diluent system. As a case in point, they use the propane-air- nitrogen system. In order to clarify this conflict, adia batic flame temperatures were calculated for the first six paraffin hydrocarbons using the limit data of Coward and Jones (1952). Lean and rich limit temperatures along with the temperature at a maximum nitrogen dilution are present ed in Figure 3. It is seen that the methane behavior is an anomaly among the light paraffin hydrocarbons. This is an additional evidence showing that flammability limits can not be easily defined simply by calculating an adiabatic flame temperature. Table 1 shows the product compositions for various ADIABATIC FLAME TEMPERATURE 23 2500 STOICHIOMETRIC s c o I 2000 LEAN LIMIT 1500 MAXIMUM No DILUTION 1000 RICH LIMIT 500 CARBON NUMBER Fig. 3. Limit Adiabatic Flama Tamparatucaa fox tha Light Paraffin Hydrocarbona TABLE 1 EQUILIBRIUM COMPOSITIONS OF METHANE-AIR-DILUENT SYSTB1S Point No. Raactants Adiab. Flam* Tamp. Products CH4 Aix Diluant A CO C02 H h2 h2o Ba 1. .05240 .94760 1533. .00919 .05239 .10473 2. .09480 .90520 2227. .00871 .00896 .08513 .00039 .00364 .18292 3. .14020 .85980 1864. .00758 .08902 .03837 .00018 .09312 .16155 4. .04990 .46650 .48360 1662. .48707 .00252 .04727 .00162 .09796 5. .05830 .57910 .36260 1823. .00562 .00022 .05806 .00011 .11628 .36246 6. .06260 .57830 .35910 1686. .00559 .00493 .05741 .00001 .00314 .12153 7. .06900 .66100 .27000 1789. .00641 .00043 .06852 .00001 .00068 .40684 8. .07120 .70120 .22760 1733. .00680 .00044 .29822 .00006 .14217 9. .08420 .80330 .11250 1836. .00619 .04521 .11115 .00003 Point No. Products (cont. ) n2 NO 0 °2 OH Cl Cl2 HCl 1. .73942 .00100 .09318 .00009 2. .70059 .00197 .00022 .00459 .00287 3. .61015 .00002 4. .36355 .00001 5. .45184 .00053 .00001 .00454 .00033 6. .80738 .00001 7. .51573 .00022 .00084 .00033 8. .54718 .00043 .00449 .00021 9. .49872 .00001 .05201 .01903 .26765 TABLE 1 (cont.) Motes 1. Point description: 1. Lean limit mixture 2. Stoichiometric mixture 3. Rich limit mixture 4. Maximum Argon dilution 5. Maximum Helium dilution 6. Maximum Nitrogen dilution 7. Maximum water dilution 8. Maximum Carbon Dioxide dilution 9. Maximum Carbon Tetrachloride dilution 2. Air was defined asi 76.1% M2' 20*95% °2- 0.95% A. 3. Other product species which were considered but had concentrations of less than 0.5 X 10"5 were: All Points----- For CC14 Diluent c CM c2m ho2 mh3 CC1 Cl CM C(S) cn2 c2n2 h2o (L) mo2 CC12 CIO CH C2 c2o h2o (s ) n2c CC13 cio2 ch2 c2h C3 H *2*4 cci4 C120 ch3 c2h2 HCH MH m2o C0C1 M0C1 ch4 c2h4 HCO mh2 M2°4 C0Cl2 mo2ci Motes (L) and (S) denote liquid and solid phase, respectively. 26 limit flames shown on Figurs 2 along with a stoichiomatric mathana-air flama. It doas not appaar that thara is any unifying composition which could provida tha basis for a limit critarion. It can ba saan from thasa studias that a singla aquilibrium adiabatic flama tamparatura cannot ba usad as an accurata pradictor of tha flammability limit. Any thao ry basad on adiabatic flama tamparatura will hava to in- corporata companaation not only for tha diffarancas ba- twaan laan and rich mixtures, but also for tha type of diluant and tha type of fual. Adiabatic Flasw Structure Aquations of Change In tha axparimantal determination of flammability limits, it is usually assumed that adiabatic conditions hava baan approached. It was therefore assumed that a study of tha behavior of adiabatic burning velocity, thick ness, haat balance, and inflection tamparatura could re veal information about flammability limits and lead to an operational thaory of limits. A mathematical modal is defined basad on tha work of Hirschfalder and Curtiss as summarised in Hirachfalder and Curtiss (1961). Tha complete sat of aquations of change for a ona-dimansional, staady-stata flama at con stant pressure neglecting viscous affects, thermal 27 diffusion, and sxtsrnsl forcss ars as followsi a) Continuity of chsmical apsciss £ [ni(v - Vi>] - X (3.1) b) Mass continuity £ < p v ) . 0 (3.2) c) Conssrvation of snsrgy P v £ - - £ < - x £ ♦ -* ■ £ + < 3-3> d) Diffusion equation (3.4) and where tha mass flux fraction for species i is defined by t . nt«i(v + Vj) (3-5) 1 M Equation (3.1) may be written in terms of the mass flux fraction (3.6) Equation (3.2) implies that M “ constant (3.7) The energy equation (3.3) any be rewritten in the form £ (x&) -■£ (E aiHl)' 4 '" ( 3 ' 8 ) i 28 where q"' is zero for ths adiabatic casa. Tha problem than la tha simultanaous solution of Equations (3.4), (3.6) and (3.8) with tha propar sat of boundary conditions, to obtain tha dasirad information. Boundary Conditions For tha adiabatic casa, at tha hot boundary, z-*•<», tha various quantities approach finite limits corresponding to complete chemical and thermal equilibrium. Accordingly, tha derivatives of all of the quantities approach zero. These ara written as z ■ + ® T - T® (3.9) x t - x iCD ®i “ xi® The cold boundary conditions ara somewhat more dif ficult to specify. If all of the are defined such that they are identically zero at the cold boundary temperature TQ, then the boundary conditions are completely analogous to those at the hot boundary; the various quantities asymp totically approach finite limits. However, the position of the flame is undefined and is sometimes called "unat tached." Much of the work in this area (as in Spalding, 1957) is based on this type of kinetics where the form of is a function of the temperature rise above the cold 29 boundary tamparatura to soma axponant. Anothar typa of approximation obtaining tha aama rasult is to assuma that is zaro for tamparaturas balow soma "ignition" tampara tura. Tha usual form of kinetics used is an Arrhenius typa expression for the tamparatura dependence of Kj, which gives this term a small but finite value at tha cold bound ary tamparatura T0, at least for any T0 above absolute zaro. This than defines what is conuaonly called the "cold boundary difficulty" and has led Hirschfeldar, Curtiss and Bird (1954, p. 762) to define a mathematical artifice which they relate physically to a "flamaholdar." This “flamaholder" which is taken to ba at tha origin of tha co-ordinate system, is a haat sink and a sami-permeable membr ana. Por tha adiabatic flama analysis a change of vari ables permits this difficulty to bo overcome by stating tha cold boundary condition as z ■ - 00 T « Tc (3.10) xi “ xio Gi—► xio ■» T-^T0 and defining tha distance co-ordinate arbitrarily. 30 Raaction Kinatics A simple unimolecular raaction with inart apaciaa 'M' was salactad for examination A + M —» B + M (3.11) whara A, B and M hara symboliza tha reactant, product and inert diluent species respectively. Tha net rata of raac tion may ba defined in tha Arrhenius form for first order concentration dependence as Ka - -nxAAexp (-Ba/RuT) (3.12) whore A a BT$ (3.13) with B being a constant and the exponent is taken as zero for all of these studies (see Williams, 1965, p. 373). Also, in order to simplify the analysis, it was assumed that the diluting species 'M' had a molecular weight equal to that of the reactant 'A1 and the product 'B' which is given the symbol m. Hon-dimensional Equations To simplify the solution, the following disiension- less numbers are definedi x * xA - 1 - Xj| - Xg (3.14) 0 * 0A - 1 - 0B - (3.15) 31 • S Q/Cj>Tod (3.16) R ■ -mXlCA/M2Cpx (3.17) f * M / * (Cp/\)d* (3.18) o © ■ T/T© (3.19) 8 * nmCp,£7£j/ \ (Lewis Number) (3.20) With tha assumption that the Lewis number is equal to 1 and with some rearrangement, Equations (3.4), (3.6) and (3.8) may be written in the form “ ~ Rx (3.21) aO + © - 1 (3.22) dx * x - G (3.23) Equations (3.22) and (3.23) may now be solved giving ax * 1 - © (3.24) which implies a constant enthalpy through the flame and that de * “ a * conBt* (3.25) whereas Equations (3.21), (3.24) and (3.22) are combined to give £2 * E | -----S L r - L _ ] (3.26) d© a I aG + © - 1 | It is noted that the effects of the added diluent 32 'M' cancel in tha abova darivation with tha asaumption that tha molacular waight la tha same aa that of tha xaactant and tha product, and tha boundary conditiona, Equationa (3.9) and (3.10) bacoma Equation (3.26) may now ba solvad by numarical inta- gration aa an eiganvalua problam whara tha dimanaionlaaa raaction rate evaluated at tha hot boundary (Rq, ) forma tha eigenvalue of the problem. Integration ia initiated at tha hot boundary. To initiate tha integration, since Equation (3.26) ia indeterminant at the hot boundary, L‘Hospital*a rule ia applied obtaining An itarativa procedure was used to converge on tha eigenvalue. Tha initial guess waa mada using tha Adams approximation as formulated by Hirschfelder and Curtisa (1961)t T m Tq X - 1 - x M G - 1 - xM T x - 0 (3.27) 0-0 (3.28) 1 I «A (To, - T0) 2 [ »UT®2 I and convarganca was rapid after finding that the selection 33 of subsequent values of Rod can be made fairly accuxataly by assuming that thara is a linaar relationship batwaan tha final valua of G at tha cold boundary, and tha value of Rw usad to obtain this valua. Tha itarativa procedure was terminated whan G - (1.0 + .0001) - xM (3.30) Solutions ware obtained by numerically integrating Equation (3.26) using Hamming's modified predictor-correc- tor method for the solution of general initial valua prob lems (X.B.M. 1966, subroutine DHPCG) on an I.B.N. 370-155 digital computer. numerical Data Values for tha physical properties used in this sec tion ware selected to be representative of typical combus tion problems. Initial calculations were mada using a molecular weight m of 30, a reactant heating valua Q of 2635 cal/g (selected to give an adiabatic flame temperature of 1545°K at a dilution of 84.3 per cent, which is equiva lent to the lean limit for methane), and a steric factor Q ,1 constant B of 0.6 10 sec with no temperature dependence ( » 0). An activation energy EA of 40,000 ca1/mola and a specific heat Cp of 10 cal/mole °K were used except as noted below. Thermal conductivity was calculated using the following relationship! 34 X - X 298 T/298 (3.31) where ^298 " X 10-4 except as notad. For tha study of tha affacts of diffarant Inart diluents, tha mixtuxa speci- flc hast and thermal conductivity wara caleuIatad by aver- aging ovar tha molar proportions of each constituent and an average molecular weight varying with dilution was used. Tha maximum integration step size and tha error lim it wara selected by finding values below which no differ ence in tha solutions could be ascertained. Since tha in tegration subroutine will halva tha step size whan tha er ror criterion is not met, tha maximum step size used initi ally is not a critical valua. Tha maximum step size for these studies was selected as one par cant of the total temperature rise. The error critertion, which is defined as the maximum permissible absolute difference between the predicted and corrected values of the dependent variable for a given step was 0.1 per cent. Since the actual flame for these boundary conditions extends over an infinite distance, the flame thickness dis cussed in this chapter is defined as the distance between the point where the flame is 2 per cent above tha cold boundary temperature to the point where the flame is 2 per cent below the adiabatic flame temperature. The cold boundary temperature used was 296°K and was not varied. 35 Results For tha initial studies, integrations wara carriad out to determine tha affacts of various concantrations of inart diluant 'M' on adiabatic flama propartias. Typical rasults of intagrationa are shown in Figures 4 and 5. Tha undiluted flama of Figure 4 is characterized by a short in duction period (to the left of tha inflection temperature) with a comparatively long reaction period. Tha inflection point occurs at 32 par cant of tha adiabatic flame tempera ture, and tha total flama thickness is quite small. On tha other hand, tha differences in structure of a flama con taining 95 par cant inert diluent, as shown in Figure 5, are dramatic. Most of this flama is undergoing induction, and reaction occurs only near tha hot boundary. Also, tha inflection temperature is now 76 per cent of tha adiabatic flama temperature. Tha inflection temperature in tha flama is tha tem perature at which the rate of change of temperature with distance is a minimum and is the dividing line between the region where heat is conducted toward the cold boundary, and where heat is conducted toward the hot boundary. Phys ically, it divides the region of pre-heating sometimes called the induction zone, from the region of heat genera tion called the reaction zone. Figure 6 shows how the in flection temperature varies with dilution. The vary high T A N D G - PE R CENT 36 100 INFLECTION POINT 80 60 40 20 dT/dz ,004 .008 0 DISTANCE - cm Pig. 4. Structure of an Undiluted Adiabatic Flame <XM - 0) dT/dz - K/ cm 37 100 t- z 00 - UJ o 5 60- CL f © 40- o z * 20 INFLECTION POINT dT/dz 0 1000 2000 DISTANCE - cm Fig. 5. Structure of a Highly Diluted Adiabatic Flame (xM - 0.95) TEMPERATURE 38 8 0 0 0 ADIABATIC FLAME ^TEMPERATURE 6 0 0 0 STOICHIOMETRIC AIR * I 4 0 0 0 - INFLECTION TEMPERATURE 200C 0 100 20 8 0 4 0 6 0 TOTAL DILUENT - VOLUME % Fig. 6. Effect of Dilution on the Inflection and Flame Temperatures of Adiabatic Flames 39 values at low di.luti.on ara due to the assumption of con stant specific heat. Since it was the intention of this model to describe near-limit flames, the assumption is valid in that region. It can be seen that the inflection temperature asymptotically approaches the adiabatic flame temperature as dilution increases and never becomes equal to it until there is no nore reactant. For the flame model studied, a flammability limit criterion based on the in flection temperature meeting the flame temperature, was not found. For the overall flame, the total rate of heat gener ation is proportional to the reactant heating value and the burning velocity (or mass flux). Likewise, the total rate of heat loss to the environment from a flame is proportion al to the flame thickness, and the temperature to some power between one and four. Therefore, both the flame thickness and burning velocity are important parameters in determining the overall heat balance when performing a flammability limit experiment. Figure 7 shows these two parameters as a function of the amount of diluent in the flame. The growth of flame thickness and the decrease in burning velocity with increased dilution is seen to be very great for highly diluted flames. The extreme burning ve locity and flame thickness at low dilutions are assumed to be caused by the constant specific heat assumption. It is VELOCITY O R THICKNESS 40 STOICHIOMETRIC AIR BURNING VELOCITY - cm /sec 100- FLAME THICKNESS - cm __— 0.01 100 80 60 40 20 TOTAL DILUENT - VOLUME % Fig. 7. Effect of Dilution on Flame Thickness and Burning Velocity for Adiabatic Flames 4 1 apparent from theaa results that heat loss to the environ ment, i.e. normal to the direction of propagation, must be significant for highly diluted flames. It is based on this large increase in flame thickness, that it was decided that heat loss to the environment must be considered in any valid analysis of the flammability limit problem. ComparIson with Experiment To determine whether experimental evidence confirmed this conclusion, it was postulated that if the experimen tally determined flammability limit was caused by heat loss, then there was probably some maximum flame thickness beyond which flames would not propagate because of the re lation of thickness to the total rate of heat loss from the flame, it was also assumed that adiabatic flame thickness could be correlated with flame thickness for the non-adia- batic case (an assumption shown to be true in Chapter IV). Spalding (1957) showed a similar type of correlation be tween the flammability limit and a property of the adia batic flame. If these assumptions hold, then for a parti cular fuel in a given size tube, the flammability limit should correspond to a particular flame thickness, as de termined by the properties of the reactant/diluent mixture. Six different diluents were studied; their effect on mixture thermal conductivity, mixture specific heat and mixture molecular weight being introduced through molar 4 2 averaging where tha raactant was assumed to be stoichio metric methane-oxygen and the diluent was composed of the nitrogen necessary for the stoichiometric air plus the added diluent. The results of these calculations are pre sented in Figure 8. Coward and Jones (1952) show that the order of effi cacy for these diluents in 5 cm tubes is CC14 > C02 > H20 > N2 > He > A (3.32) This order of efficacy is confirmed in Figure 8 if the flame thickness region above the dotted line 'A1 is taken as the region applicable to a 5 cm tube. For smaller tube diameters, one would expect that the Hallowable flame thickness" would decrease since small er tubes permit a greater rate of total heat loss for a given flame thickness. This is shown to be true in Chapter IV. Data from Coward and Jones (1952) for 2.2 cm diameter tubes gives the order of efficacy as C02 > He > N2 > A (3.33) This order is again confirmed by Figure 8 if the limiting flame thickness for a 2.2 cm tube falls between dotted lines 'A* and 'B1. Note the shift in position of helium. Similarly, for 1.6 cm and 1.7 cm diameter tubes, the order of efficacy is given as \ \ \ \ \ •o ,-'J0 tve 0t ^ %pt . .0-°* t>ofl 44 He > CO2 > N2 > A (3.34) This order is confirmed if the proper flsme thickness fells below the dotted line 'B'. Again note that helium has shifted its relative position. It is shown therefore that the flame thickness (assumed to be related to the total rate of heat loss) varies in such a manner so as to des cribe both diluent effects and tube diameter effects on the flammability limit. Egerton and Powling (1948, p. 190) and Mellish and Linnett (1953) studied the effects of various diluents on flammability limits and came to the conclusion that radical diffusion plays an important part in governing flammability limits. Mellish and Linnett (1953, p. 419) specifically point to the effect of tube diameter on the order of effi cacy as being "indicative of the importance of radical dif fusion." This study does not consider radical diffusion and/or destruction, and it is interesting to note that the aforementioned trends are predicted with arguments based on purely thermal criteria. It is thus important to obtain full solutions to the laminar flame problem with the effect of heat loss from the flame (normal to the direction of propagation) included in the energy equation. Equation (3.3). Since this heat loss to the environment is a function of the temperature of the flame, the order of the energy equation cannot be 4 5 raducad, and tharafora, many of tha othar simplification* mada in adiabatic flama thaory ara not posaibla. CHAPTER IV NON-ADIABATIC FLAKE STUDIES Analytical Modal Equations of Chanqa The diffusion equation and ths conservation of spe cies and energy for a one-dimensional steady-state flame at constant pressure are defined by Equations (3.4), (3.6) and (3.8). For this model, it was decided to include the effects of the diluent species 'M' of Equation (3.11) in the properties of 'A' for simplicity. Equation (3.4) may be simplified for the A —► B re action with equal molecular weights as was done to get Equation (3.23) except without introducing the dimension- similar ly, assuming a constant specific heat and a thermal conductivity described by Equation (3.31), and upon differentiating the conduction term, the energy equa tion, Equation (3.8) becomes less distance and we get for the diffusion equation (4.1) (4.2) The conservation of species equation, Equation (3.6) re- 46 47 mains unchanged sxcapt for dropping tha subscripts as de fined in Equation (3.15) and again m being the molecular weight of either species A or B, giving do . m dz M “ Ka (4.3) Reaction Kinetics For the adiabatic model the reaction order was fixed at unity (Equation 3.12). For the non-adiabatic studies, in order to approximate typical hydrocarbon kinetics for the pressure dependence studies, the reaction order was included as a parameter KA « - (nx)a A exp(- EA/RUT) (4.4) Unless otherwise specified in the text, Cl was equal to one. Heat Loss Model Heat exchange rate per unit volume normal to the In direction of flame propagation, q in Equation (4.2), was considered in two modes; conduction, and conduction plus radiation. Previous analyses of the non-adiabatic flame problem have usually limited the heat loss description to a functional relationship of the difference between gas temperature and surrounding temperature to some arbitrary power between 1 and 5 (see Adler, 1963). In this work, since simple closed form solutions are not pursued, a store 48 realistic and therefore nora complicatad modal vai usad. Conduction,— In ordar to obtain an accuzata modal for conductiva haat loasas to tha anviconmant from a flama In a tube, it would be necessary to re-caat tha flama aqua tions in radial co-ordinanta. Since this would be diffi cult, it is assumed that tha volumetric haat loss rata from a plana flama of thickness Az in a tuba, to tha walla can be written in terms of a haat transfer coefficient a'" _ -h(T - To)('7TdA*> ,a “ (%,rr5» A*) - ‘ ’ and that tha haat transfer coefficient takas the form h - (4.6) D for laminar flow through a tuba, where NuD is tha Nusselt number based on diameter and is a constant of tha ordar unity. Spalding (1957) used a valua of 4 for this, Nbased on solution of tha radial haat conduction aquation." Mayer (1957) references Wusselt's theoretical results of 3.66 for fully developed flow in a tuba with constant wall temperature as reported in Eckert (1950). Barlad and Yang (1960) attribute their value of 4 to the quenching studies of Berlad and Potter (1955), and Williams (1965) obtains a value of 2 by assuming a linear temperature profile from center to wall. In this work a value of 4 is used. 49 Combining Equations (4.5) and (4.6), tha volumetric haat loss rata dua to conduction bacomas Tha haat capacity of tha glass or metal containar in which a flammability limit axparimant is parformad is normally ordars of magnituda graatar than that of tha gas within, and the thermal conductivity of glass or metal is adequate to rapidly distribute this haat. It is therefore assumed that tha walls remain at room temperature, and the entire temperature drop occurs in tha gas. Therefore, tha type of wall and its properties do not enter into this analysis. Radiation.— Tha radiation loss of a reacting high temperature gas to its surroundings can be simplified for engineering purposes by assuming that it emits and absorbs as a gray body. This type of analysis is discussed by Hottel and Sarofim (1967) and results in an equation for qj'', the net volumetric radiation heat loss rate for gas at T to a bounding surface at TQ, where the surface is cylindrical and the gas has a thickness A x 16 X(T - To) T 2 ---- (4.7) C | + 1) (€qT^ - Q.qTp^) (TipAs) I 2 I (%TTD2Ax) 2 (4.8) where and are the emissivities of the bounding sur face and the gas respectively, and Olg the absorptivity of 50 the gas. To simplify tha analysis, it is assumed that € g and 0L g ara aqual which is trua whan T is not too much graatar than T0. Whan T gats largar, the contribution of tha absorption term is negligible and tha approximation therefore unimportant. Tha surface emissivity €, was not treated as a parameter in this work and is taken to be 0.9. With these approximations, Equation (4.8) becomes The gas emissivity €g is a function of composition, temperature, gas layer thickness, gae body shape and total pressure. Previous non-adiabatic flame studies have neg lected most or all of these variables. In this work a gas emissivity model is developed, reflecting the combustion of methane-air, and retaining these functional relation ships. The major radiating constituents for methane-air combustion are water, carbon dioxide and methane. The plotted data for water and carbon dioxide emissivity pre sented in Hottel and Sarofim (1967, pp. 232 and 229) were curve fitted with the resulting equations! (4.9) log €c0 - 0.505 m t C + 4100 (4.10) [6.60 log(Pc02L) + 62 log € h20 “ 0*562 (4.11) 9.18 lO9(»H20L) + SB]2 51 where ^ Cq 2 and € H^0 are the contributions of each of those species to ths total gas emissivity, and tha loga rithms ara to tha basa tan. The parameter C equals 1060 or T, whichever is greater. The mean beam length L for a gas radiating in a cylinder (from Hottal and Sarofim, 1967) may be approximated by 0.9 D1. In order to describe tha variation of partial pressure as the amount of fuel is varied, tha following relation was usadi pco2 " p S §55 (4- 12) where the constant was selected to make Pqq0 *<3ual 0.05 when Q was 415.667 cal/g. Water vapor partial pressure was evaluated as twice the carbon dioxide partial pressure based on the assumption of complete oxidation of hydrogen and carbon, and a hydrogen to carbon ratio of four. To approximate the emissivity of methane, (see Bot- tel and Sarofim, 1967, p. 240), it was noted that the emis sivity of methane behaved as water and carbon dioxide above a temperature of approximately 1060°K, but fell off rapidly below this. To describe this behavior, and the change in composition from reactant to product, the following rela tionship was used to calculate gas emissivityi €g " (€C02 + € H20 * [ 1 " X ( C - tJ] <4’13> ^The units of D must be in feet here. 52 The haat loss models of Equations (4.7) and (4.9) ara shown on Figure 9 at different temperatures for zero reactant concentration. The percentage of the total loss due to radiation is included to show the dominance of con duction loss at small diameters and of radiation loss at large diameters. The percentage of loss due to radiation is approximately the same in the temperature range of 1000°K to 3000°K. It is interesting to note that, for these models in the temperature range noted, and for a 5 cm diameter tube, the temperature dependence of conductive and radiative loss is somewhat similar. This is due to the first power temperature dependence of the thermal conducti vity, and the increasingly negative temperature dependence of the gas emisaivitiea. Boundary Conditions For the non-adiabatic flame with heat loss only to the surroundings, the terms hot and cold boundaries are no longer applicable since the reactants start at T0 and the products end up at T0. Therefore, to be specific, the terms reactant and product boundaries will be used (often in the literature these are designated upstream and down stream, respectively). For an infinite, unattached flame, the reactant boundary condition is still defined by equa tion (3.10) and keeping the unsubscripted nomenclature de fined by Equations (5.14) and (3.15) and with no diluting VOLUMETRIC HEAT LOSS RATE - cal/cm3 sec 53 CONDUCTION xT = 3 0 0 0 °K v>T= 2 0 0 0 ®K v \> r = io o o •! (00 RADIATION = 3 0 0 0 ®K s>r= 2000 ° K >r= iooo® oo 50 i i i .01 . 1 I 10 100 I03 I04 10® TUBE D IA M E T E R - cm Fig. 9. Conductive and Radiative Heat Loss Models for Circular Tubes % RADIATION 54 spacias, wa hava z - -09 T - Trt, dT/dz - 0 (4.14) G - 1 x - 1 Thar* has baan considarabla discussion about tha propar product boundary condition as outlinad in Chaptar IX. Tha mannar in which this analysis is formulatad doas not raquira that raactant concantration ba spacifiad at tha product boundary. Tharafora, tha raactant concantration at any point is datarminad analytically and it can ba da- tarminad whathar raactant ramains at tha product boundary. Tha product boundary condition ijt Z “ + 09 T - Tc (4.15) dG “ - 0 dz £ - o Xt was found aftar numarous intagrationa, that tha last two conditions of Equation (4.15) wara only both mat whan G * x * 0. In fact, it is difficult to imagina how anything alsa could ba trua whan singla stap Arrhanius kinatics ara usad. This rasult than carifias tha discus sion of tha product boundary condition notad in Chaptar XX. 55 jtumaricnl Jjata Tha nunarical constants input into tha calculations wars selected to ba raprasantativa of a mathana-alr laan limit flama. A molaculac weight, m of 30 and a spacific heat, Cp of 10 cal/mole °K wara usad as in tha adiabatic integrations. Tha thermal conductivity proportional to tha first power of temperature as defined in Equation (3.31) and ^298 “ x 10~4 was usad throughout. An activa tion energy EA of 40,000 cal/mole was salactad as typical. In addition, tha cold boundary temperature is 298°K for all integrations. Tha important variable representing fuel concentre* tion for lean flames is tha heat of combustion par unit mass of reactant or tha raactant heating valua, Q. Tha baseline raactant heating valua for these studies was sa lactad to give an adiabatic flama temperature of 1545°K, which is tha equilibrium adiabatic flama temperature for mathana-air at tha laan limit (sea Table 1). To arrive at this temperature a raactant heating valua of 415.667 cal/g of raactant was calculated using tha aforementioned spaci fic haat and molecular waight. Egerton (1953) points out that various investigators hava found this Moxygen using capacity to be most important in governing tha limit.M All composition calculations in this study are mada assuming that 5.24 per cant fuel corresponds to a raactant heating 56 value of 415.667 cal/gm raactant. Since it haa been attamptad to maka tha data aa realistic aa poasible within tha acope of global kinatica and conatant apacific heats, a survey of lean limit burning velocity data was mada to establish a valua for tha steric factor conatant B. Looking at tha data of Edmondson, Heap and Pritchard (1970) using a flat flama burner, a burning velocity of around 3 cm/sac could ba projected for a moth* ana-air mixture of 5.24 par cant. Therefore, a valua for g _ l B of 0.6 X 10 sac was usad for all studies dona with 16 3 first ordar kinetics, and a value of 0.197 X 10 cm /mole sac was usad for tha second ordar calculations. These give a calculated burning velocity at tha limit for tha base line raactant heating valua (415.667 cal/g) of 2.24 cm/sac. Mo pre-axponantial temperature dependence was used ( $ * 0). These are the only numerical constraints placed on the flame model; an adiabatic flame temperature of 1545°K at the point where the burning velocity is around 3 cm/sec. There was no further attempt to force the model to match experimental results. The good agreement with experimental results as discussed in Chapter V comes from no more than these two "realistic* constraints. Analytical Techniques Numerical integration of Equations (4.1), (4.2), and (4.3) was done on the University of Southern 57 California, University Computing Cantor*a I.B.M. Syatan/370 Modal 155 digital computar (tha aarly work waa dona on a Syatam/360 Modal 65). All calculationa wara carriad out in tha doubla praciaion moda giving approximataly 16.8 dacimal digit praciaion. Tha numarical tachniqua uaad waa a doubla praciaion modifiad Hamming's pradictor-corractor method as codad in I.B.M. (1966), subroutine DHPCO. This is a modification of Milne*s classical modifiad predictor-corrector method which is more stable. As this is a generally available commercial package, a listing of tha coda is not presented. Hamming's modifiad predictor-corrector method obtains an approximate solution of a general system of first-order ordinary differential equations with given initial values. Because of the first order restrictions, Equation (3.36) and its integral are integrated to obtain the temperature. The calculation procedure gives an estimate for the local truncation error for each step, thereby permitting rapid selection of integration step aise. A Runge-Kutta proce dure followed by one iteration step ia added to the pre dictor-corrector package for calculating the initial four points. Initial values of the independent variables along with their derivatives are needed to start the integra tion . 58 Starting Integration cannot ba initiated at the cold boundary consistent with the boundary conditions of Equation (4.14). It can be seen that Equation (4.3) has a small but finite value at the cold boundary whereas Equation (4.1) is zero. As the distance z starts to increase, the mass flux frac tion G decreases faster than the mole fraction x resulting in the physically impossible situation of reactant dif fusing toward the cold boundary. In addition, it is diffi cult to start numerical integration at an infinite boundary without a change of variable or some other approximation which often hides the important part of the problem. In order to avoid this inconsistency, a technique developed by Yang (1961) for use on an analog computer was applied. With this technique, integration is started at some temperature slightly above the cold boundary tempera ture and the rate of change of fuel mass flux fraction is assumed negligible in this region. In addition, although not necessary, heat loss due to radiation is neglected and the thermal conductivity is assumed constant in this re gion. Using the unity subscript to denote the starting boundary. Equation (4.2) becomes d2T dr 59 which is linear in T and can be solved by standard tech niques getting dz ■> [ « , ffx - * o>\¥f + % <4.17) after apllying the reactant boundary conditions, equation (4.14). How combining Equation (4.17) and Equation (4.16), we have 2 (4.18) Since the starting boundary temperature increment (T^ - T0) is arbitrary, various increments were tried for adiabatic cases and the results compared with the integra tions performed with the adiabatic model described in Chapter ill. For the step sise and error criterion used it was found that a temperature increment of less than .001°K resulted in no detectable difference in temperature and concentration profiles. In addition, the relative magnitudes of the neglected terms were checked in the range of data presented and found to be orders of magnitude less than dominating terms in the summation. With these assumptions, the starting value of the reactant mass flux fraction, 0^, is one and its derivative with respect to distance is calculated by Equation (4.3). The starting value of the reactant mole fraction is an undetermined constant since Equations (4.1) and 60 (4.2) are no longer coupled through G. To eld in the eval uation of this constant, the starting value was formu lated in terms of a new parameter \ which represents the deviation from what the value of x^ would have been for the constant enthalpy adiabatic case. The starting value x^ is then calculated ast where \ has become the undetermined parameter. The deri vative of reactant mole fraction x with respect to distance may now be calculated by Equation (4.1). The distance z is arbitrarily set at zero at this starting boundary. Integration Parameters The Hamming's integration method requires definition of a maximum integration step size along with error cri teria for each equation integrated to determine when the step size must be reduced. The maximum integration step size was set at 0.333 per cent of the integration interval which in turn was empirically defined as 0.015/M. The integration error criterion is defined as the maximum permissible sum of the weighted absolute differ ences between the predicted values and the corrected values of the dependent variable for a given step. The criterion is set in this work such that this difference is limited (4.19) 61 to 0.1 per cent of a typical value of any of the four de pendent variables. If this criterion ia exceeded, the atep aiae ia halved. Converging the Eigenvalue An initial estimate of the eigenvalue M is input into the computer program and integcation proceeds until either G or x pass through a minimum. If it ia 0 which reached a minimum, the value of M must be reduced whereas a minimum x indicates that M was too small. The eigen value is modified and the integration redone until at some minimum value of G the absolute difference between G and x becomes less than 0.001 at which point, no more itera tions are made. The value of this convergence criterion is arbitrary and was selected because it produced a rela tively smooth transition into the continuation phase of the integration where only the temperature changes. Typi cally, the eigenvalue must be correct to six or seven digits to meet this convergence criterion. For the second order kinetic studies at reduced pressure an artifice was necessary to permit G to pass through a minimum, the condition used in the eigenvalue convergence technique to indicate that the eigenvalue used was too largo. For first order studies, exponentiation by 01 was not used which permitted Equation (4.4), and there fore Equation (4.3) to change sign because of x becoming 62 negative. Toe values of & other than one, this is not permitted and to keep the behavior as before, the sign of x is applied to Equation (4.4). This artifice only affects the last point of integration which has no effect on the previous results. Converging Values of M, meeting the requirement that 6 and x must be equal, may be determined for a wide range of values of the parameter X which as discussed previously, relates to the amount of fuel depleted at the starting boundary. However, to meet the complete product boundary condition, Equation (4.15), the temperature must also asymptotically approach T0 at the product boundary. This then forms the criterion for selecting the proper value of X * By observing the behavior of the temperature at the last points of an integration, it was found that the con vergence of X wa* greatly expedited. This behavior is shown on Figure 10. It is found that the temperature ex hibits an asymptotic behavior, after the fuel has been depleted, only when the value of heat loss is less than a certain maximum, and at two values of X for *nY heat loss less than this maximum. Also, as noted on Figure 10, the temperature diverges in the negative direction only when the value of X between the higher and lower burning velocity solution values. All other conditions produce a P E R CENT O F ADIABATIC MASS FLUX 63 100 75 50 REGION OF POSITIVE TEMPERATURE DIVERGENCE (T- 25 0 L I 0 . 0 1 LOCUS OF VALID SOLUTIONS CRITICAL OR MAXIMUM HEAT LOSS POINT REGION OF NEGATIVE TEMPERATURE DIVERGENCE 0 . 1 PARAMETER X Fig. 10. Bahavior of tha Solutions to tha Equations of Changa 64 positive divergence. Since mass flux H varies as burning velocity for the same initial density, it can be seen from Figure 10 that two burning velocities are defined for heat loss up to a certain maximum. Beyond this maximum, valid solutions to the equations of change do not exist. The transition point from negative to positive tem perature divergence becomes very critical when considering the first and second temperature derivatives. In fact, the trend of temperature may be predicted accurately with only a short continuation of the integration past the point where G and x have met their convergence criteria. It is therefore not necessary to carry out the integration for long distances toward the product boundary as may have been thought previously. In the course of this study, it was found that the value of X mt the Unit heat loss point varies over a re latively small range for different values of Q. The varia tion of X approximately linear between a value of 0.135 when Q is 300 cal/g to 0.24 when Q is 700 cal/g. This serves as the initial guess and the limit can be found by alternately trying a range of X values and then a range of diameters. This method of iteration can be seen on Fig ure 10 to result in switches between positive and negative temperature divergence. 65 Computer prpqram Description A computer program called DISTILBR was coded in Fortran IV language to carry out the necessary integra tions, to converge on the eigenvalue M and to print the in tegration results. A listing of the card images i* found in the Appendix along with a cross reference between the symbols used in this dissertation, and those used in the computer code. The program is divided into a main program and four subroutines, one of which (DHPCG) is supplied by the com puting facility library as part of the I.B.N. Scientific Subroutine Package, and therefore does not appear in -the code listing. Figure 11 shows the calling sequence for these subroutines. Input data are read into the main pro gram where all necessary information for an integration are set up. The integration subroutine DHPCG is then called. While performing the integration, DHPCG calls subroutine FCT every time it is necessary to evaluate the derivatives of the dependent variables. When the predic tor -corrector error criteria are met, the accepted values of the dependent variables for that step are passed to subroutine OUTP. Here convergence tests are applied and if the eigenvalue has not been properly selected, the integration is terminated returning control through DHPCG to the main program where another trial value for the MAIN INITIALIZE DATA ITERATE ’FLUX' JHFCC INTEGRATION PACKAGE FCT EVALUATE DERIVATIVES OUTP x or G UVERGEI S CVG. CRITERIA s. MET? v no POINTS STORED UL yes PRINT RESULTS EXIT Fig. 11. Calling Saquanca Diagram for Program DISTILER 67 eigenvalue ia determined. If however, in OUTP, tha con vergence critaria fox N ara mat tha intagration is re-done, but this time storing tha values of tha dependent variables and thair darivativas. Whan tha convergence critaria ara mat this time, tha intagration of Equations (4.1) and (4.3) is terminated and the integration of Equation (4.2) and its integral continues until the temperature diverges. At this point subroutine PRINT is called, terminating tha Integration and printing out the necessary information for selected intagration steps. Program control is than re turned through OUTP and DHPCG to the main program where input data for another case axe read into the program. Tha size of the load module including tha program and all necessary subprograms is about 55,000 hexadecimal bytes. Typical runs require tan seconds of computer tima to converge on the eigenvalue H (about 20 integrations), and with judicious selection, the proper value of X can be determined after about 10 iterations. Therefore, a typical point can be determined in about 100 seconds of computer time using a relatively small amount of the com puters core. Results of integrations Effects of Heat boss Mode Conduction Only.— The primary results of integra tions are shown in Figures 12 and 13. In Figure 12 the BURNING VELOCITY - cm/sec 68 Fig. 4 Q-4II.II 3 =374.10 2 LOCUS OF CRITICAL DIAMETERS 0.2 0 0 .1 I / D — cm-1 12. Effect of increasing Heat Loss on Burning Velocity; Conduction Loss Only BURNING VELOCITY - cm/sac 69 RADIATION DOMINATED CONDUCTION DOMINATED 4 0=416.667 3 LOCUS OF CRITICAL DIAMETERS 2 0=374.10 CONDUCTION ONLY (REE) 0 0.2 0 , 1 Fig. 13. Effect of Increasing Heat Loss on Burning Velocity; Conduction and Radiation Loss 70 eigenvalue is represented as burning valocity and tha haat loss is dua to conduction alone, as dascribad by Equation (4.7). As can ba saan from this aquation, haat loss in- craasas as diamatar dacreasas. For tha basaline casa of Q ■ 415.667 cal/g (which as mentionad previously corres- ponds to a fual concantration of 5.24 per cant), it can ba saen that as the diameter decreases from an infinite value, the higher burning velocity decreases from the adiabatic value and the lower burning velocity increases from zero. As the heat loss continues to increase, a point is reached where the two burning velocities meet, and beyond this point, no solution is possible. It is this point (4.95 cm diameter for this casa) which represents tha quenching point or tha flammability limit as was pointed out by Spalding (1957), Yang (1961), and others. Note that tha shape of tha curve is almost symmetrical about the flamma- bility limit and that v°U»it * °-54 v°adlabatic (4-20) This result is in general agreement with tha theoretical results of Oaniell (1930) who indicated that "the minimum velocity (with heat loss) for a given mixture may be as great as one-half of the velocity with no cooling." From the results of Spalding (1957) a value of 50.3 per cent can be obtained for his example case for weak hydrocarbon- 71 air mixtures. Aa tha fual concantration ia reduced, aa represent- ad by dacraaaing valuaa of Q, tha ahapaa of tha burning valocity cucvaa remain approximataly tha aama and only tha valuaa changa. Thia similarity raaulta in a atraight lina connacting tha flammability limit or quanching pointa with tha origin. It is just thia similarity in shape which ap peared in the highly simplified closed form solution of Spalding (1957), and permitted him to relate tha easily calculated adiabatic burning valocity to tha non-adiabatic dimensionless Paclet number. Since tha locus of limits is linear, this implies that the ratio of the ordinate to the abscissa is a con stant. This constant in fact is tha variable part of the Paclet number which is defined as tha ratio between tha bulk haat transfer and the conductive haat transfer, and for tha radial case is normally written as From tha data of Figure 12, and evaluating tha thermal conductivity at 298°K (as ia typically dona), tha Paclet number based on tha burning valocity at tha limit, ia 66,7, whereas if tha Paclet number is evaluated using tha adiabatic burning valocity as dona by Spalding (1957), a value of 125 is obtained. These may ba compared with (4.21) 72 experimental value* approximated from the data of Putnam and Jenaen (1949) of 46 and of Cullen (19S3) of 30. Al though the absolute numerical agreement ia not preciae, Figure 12 indicates that some consistent interpretation of the Paclet number could be used to predict quenching dia meters as 3jng as the heat loss is due to conduction alone. Conduction and Radiation.— When the radiation heat loss model of Equation (4.9) is added to the conduction loss model, the variation of burning velocity with dia meter behaves differently as shown in Figure 13. Here the abscissa can ba divided into a region of conduction loss dominance and a region of radiation loss dominance as de scribed by Figure 9. The transition tube diameter be tween the two types of heat loss is about 10 cm for this model; however, the transition is gradual and the diameter beyond which radiation accounts for 90 per cent of the loss is 35 cm, whereas conduction accounts for 90 per cent or more of the total loss at diameters less than 3 cm. The effects of radiative heat loss can be seen at the larger diameters where a small change in reciprocal diameter now has a significant effect on the burning velo city. The locus of limit diameter-velocity combinations is not linear as it was for conductive loss alone. This 73 in turn indicates that the Peclet number increases with diameter once radiation loss becomes dominant. Based on these data, it is surmised that the Peclet number is a good indicator of conductive quenching limits; however, it does not describe limits for which the major heat loss mode is the radiation heat transfer. It is interesting to note, comparing Pigures 12 and 13, that the ratio between the burning velocity at the lim it and the adiabatic burning velocity (Equation 4.20) still seems to hold within reasonable accuracy. Minimum Tube Diameter Figure 14 shows how the minimum diameter for propa gation varies with reactant dilution (as indicated by the reactant heating value Q) for conduction heat loss and com bined conduction and radiation. As dilution increases, the minimum tube diameter for flame propagation increases. This phenomena, normally related to conductive quenching, is shown to be very strong for small diameter tubes. If conduction were the only mode of heat transfer, an increase in diameter would permit significantly more dilution, even at very large diameters as shown by the dotted line. How ever, when radiation loss is also considered, this situa tion changes and in the region where radiation loss is dominant (diameters greater than approximately 10 cm), large increases in diameter permit very little increase in PERCENT METHANE 74 TWO BURNING VELOCITY REGION 0.25 atm 0.5 atm 1.0 otm CONDUCTION ONLY— 1.0 atm NON-PROPAGATION 2 - REGION RADIATION DOMINATED 100 50 0 CRITICAL DIAMETER-cm Fig. 14. Effect of Dilution on Critical Tube Diameter 75 reactant dilution. Baaed on the assumption that Q ■ 415.667 cal/g is equivalent to 5.24 per cent methane, in tegration results indicate that to propagate a flame with 4 per cent methane, a tube 500 cm in diameter woald be required. Pressure Effects To analyze the effects of changing pressure, the overall reaction order CL in Equation (4.4) was changed from one to two. Fristrom and Westenberg (1965, p. 337) indicate that stoichiometric methane-oxygen flame structure data correlate well if an overall reaction order of two is used. The data of Diederichsen and Wolfhard (1956) indi cate a value of 1.66 for stoichiometric methane-air; how ever, a value of two was used to minimize computation time. The results are presented in Figure 14 and show that the reactant heating value must be increased in a given size tube for flame propagation at reduced pressure. Zt is shown in the following chapter how this is in agree ment with experimental data. The limit burning velocity remains at approximately 54 per cent of the adiabatic burning velocity. Since the order is two, there is no variation of either of these velocities with pressure. Similarly, there is no variation in flame temperature, and mass flux and flame thickness vary as pressure and 76 reciprocal pressure, respectively. This is as predicted by Hirschfelder, Curtiss, and Bird (1954, p. 765). It nay also be noted that the transition diameter at which the heat loss changes from conduction dominance (smaller dia meters) to radiation dominance (larger diameters) shifts with changing pressure. Heat Balance Structure It is of interest to look at heat balance of an element of gas passing through a flame at conditions near the maximum heat loss. Figure 15 shows the heat exchange rate for an element of gas at different distances into a non-adiabatic flame. Zero distance is the starting bound ary for the integration as discussed previously (296.001°K). The net heat transfer rate shown by dotted lines is made up, as can be seen in Equation (4.2) of conduction in the direction of flame travel, generation due to chemical reaction (including enthalpy diffusion) and heat loss from the system normal to flame travel. Also included on this figure is an inserted plot showing how the percentage of the normal heat loss due to radia tion varies through the flame. The standard thermodynamic convention of heat transfer into the system, and genera tion, being positive is used. When the net heat transfer is positive, temperature is increasing in the flame. The peak temperature (1419°K) occurs at 1.25 cm and the VOLUMETRIC RATE O F HEAT G A IN - col/cm3 sec 77 40 20 o ENERATED -NET GAIN OR LOSS 1 . 0 LOSS'* DISTANCE - cm -4 CONDUCTION IN DIRECTION OF" FLAME -6 -8 Fig. 15. Heat Balance Structure for a Near-limit Flame 78 inflection temperature (1071°K) occurs at 1.05 cm. It is interesting to note the difference (more than an order of magnitude) between the heat generation rate and the normal heat loss rate. This is why attempts to equate these two without solving the total problem and thereby including conduction parallel to the flame, fail to give valid quenching or limit criteria, unless some arbitrary fraction is introduced. Peak Temperatures Figure 16 shows the effect of heat loss on peak flame temperature. When I/O is equal to zero, there is no heat loss and the temperature therefore represents the adiabatic flame temperature. As two burning velocities are defined for heat loss up to a certain maximum (minimum tuba diameter) similarly two peak flame temperatures are obtained which converge at the point of maximum or limit heat loss. The locus of the peak temperature at maximum heat loss is shown by a dotted line. It is interesting to note that the ratio of peak temperature at maximum heat loss to the adiabatic flame temperature remains constanti Tpeak * °*9 Tadiabatic (4.22) limit flame This could provide the basis for a flammability limit PEAK TEMPERATURE 79 4 57.23 1500 0=374.10 Q = 33253" l tooo 500 0.3 0.2 0 . 1 I / D - cm-* Fig. 16. Effect of Heat Loss on Peak Tempexature in Non-adiabatic Flames 8 0 criteria based on the adiabatic flame temperature if it were known how this percentage varies with variations in the reactant properties. This value agrees well with a range of .90 to .93 obtained from the results of Spalding (1956) for a weak hydrocarbon flame and the range being dependent upon wheth er the flame is dominated by radiation or conduction, re spectively. The fact that the magnitude of this temperature is almost equal to the adiabatic flame temperature has impor tant connotations in experimental studies. Temperature measurement and comparison with calculated adiabatic flame temperature appears to be a poor measure of heat loss when the heat loss is somewhat less than the maximum. Also, it may be seen that small heat losses (compared to the maxi mum heat loss) have little effect on the adiabatic flame temperature. f i Temperature-Burning Velocity Relations The relationship between reciprocal peak temperature and burning velocity is shown on Figure 17. This relation ship was reported by Kaskan (1957) where ha showed it to be linear based on experimental data using a flat flame burn er where the temperature and velocity were varied for a given fuel-air ratio by varying the heat loss at the cold boundary. BURNING VELOCITY* cm/sec 8 1 KASKAN (1997) x - INDICATE S CRITICAL CONDITION 7X1 O'4 8XIO-4 1 / TPEAK “ *K -4 9X10 Fig. 17. Relation Between Peak Temperature and Burning Velocity 8 2 The flam* model for this study differs in its heat loss configuration; however, comparable results are ob tained indicating a generality in this relationship. The adiabatic flame temperature/velocity combination is upper most on the figure for a given reactant heating value. As heat loss from the flame increases, the peak temperature and the burning velocity decrease until the limiting heat loss is reached (i.e. quenching or flammability limit). At this point lower (unstable) burning velocities are predict ed as heat loss is reduced back to zero, and are shown be low the symbol denoting the limit. As can be noted, the absolute value of the relation ship varies with fuel-air ratio while the slope remains approximately the same. No significant difference in the curve was noted for the Q 3 415.667 cal/g mixture whether the heat loss mode was radiation and conduction or just conduction alone. The data of Kaskan (1957) are shown for methane-air with an equivalence ratio of 0.6. In this study that would be equivalent to a reactant heating value, 0 of 455 cal/g. It is suspected that the difference in slope between these data and those of Kaskan is due to inaccuracies in the kinetic parameters selected for this study, and the heat loss model. In a recent experimental study, Pritchard, 83 Edmondson and Heap (1972) concluded that tha relationship between the logarithm of burning velocity and reciprocal peak temperature should be linear in order to obtain ac- curate extrapolations to adiabatic burning velocity. The results of this study indicate that some curvature will be encountered in this relationship, especially if the lower unstable flame speed is attained, which is not due to dia meter effects as concluded by the authors above. However, this curvature is predicted to be slight as substantiated by their temperature measurements. Flame Thickness The effect of heat loss on flame thickness is shown in Figure 18. Here flame thickness is defined as the distance from the integration starting boundary (T|_ - 298.001°K) to the peak temperature. It can be seen that the flame thickness does not increase rapidly with heat loss from that of the adiabatic flame. In fact, the ratio between the flame thickness at maximum heat loss (minimum diameter) and the adiabatic flame thickness is a constant for these cases t ^ z limit " 1,64 ^ zadiabatic (4.23) Again, this relationship may lead to an easily calculated quenching and flammability limit criteria once the varia tion of this constant with other fuel properties is 84 E 0 1 CO CO UJ z o ur z < 0 = 332.53 ^Q=374.I0 .,0=415.667 ^0 = 457.23 0 0.2 0 . 1 0.3 I/D - cm-1 Fig. 18. Effect of Heat Loss on Flame Thickness fox Non-adiabatic Flames 85 defined. Also not* that as th* limit diam*t*x is incr*as*d so also is th* flam* thickness increased. This result combined with Equation (4.23) justifies the use of flame thickness as a flammability limit criterion in Chapter 111 in describing the change in the order of efficacy of helium as a diluent. CHAPTER V FLAMMABILXTY LIMIT MODEL Based on the analyses parformed in thla study, an "absolute1 1 or fundamental flammability limit (as defined by Zabetakis and Richmond (1953) as being a function of only the diluting atmosphere, temperature and pressure) does not exist. In fact, a single non-adiabatic flame model developed herein describes three major characteris tics of flammability limit experimentation to such an ex tent that there remains no evidence of a "fundamental" limit and it is concluded that flammability limits and quenching are one and the same physical phenomenon. Model The flammability limit, as observed experimentally may be predicted analytically by solving the equations of change for the non-adiabatic case where heat loss to the environment occurs in both the conduction and radiation mode. The limit is defined as the lowest reactant heating value which results in a solution meeting the boundary conditions, for a particular heat loss configuration re presenting the experimental apparatus. 8 6 87 Experimental Correlation The three major experimental charactor 1stlea de- scrlbad by this modal aret 1) tha apparant lack of da- pandanca on apparatus siza bayond a cartain minimum, 2) tha transition to small tuba quenching as apparatus siza is de- creasad, and 3) tha raducad pressure flammability limits. These are discussed in detail below. Influence of Apparatus Siza Tha predominant method used to reduce heat loss when performing flammability limit experiments is to in crease the dimensions of the apparatus in the direction normal to the direction of flame propagation. Since flam mability limits are not supposed to be influenced by heat transfer, a nominal tuba diameter of 5 cm has been selected to be "beyond the region of significant heat loss." Coward and Jones (1952) conclude that an increase in size above this value "rarely shows more than a few tenths of one per cent increase in the range of flammability." This characteristic is shown by this model to occur as tha dominant heat loss mode shifts from conduction to radiation. Figure 14 shows this "almost” discontinuity which, combined with tha usual inaccuracies of reactant preparation, will give tha appearance in tha laboratory that an adiabatic apparatus has bean attained and that further increase in siza will have no affect on tha 8 8 results. This conclusion is shown to be false in Figure 19 which contains the data of Figure 14 replotted with the abscissa a logarithm scale. The data for upward propaga tion summarized by Coward and Jones to show the lack of dependence on tube diameter are shown. Bazly data from White (1924) are shown separately as a consistent set of experiments. The reduction of flammability limit with ap paratus dimension continues to occur with the limit of zero as the diameter approaches infinity. The implications of this result in the application of flammability limits determined in 5 cm tubes to explo sion hazards in mines, tunnels, nuclear reactor pressure vessels or aircraft fuel tanks is paramount. This study shows that it is necessary to correlate heat loss in addi tion to the amount of fuel in the atmosphere to define “safe" conditions. Unification with Quenching Data It is necessary for a unifying model to predict both the flammability limit composition in a 5 cm tube and quenching diameter for a stoichiometric composition. For a pressure of one atmosphere, it is seen on Figure 19 that the percentage of fuel necessary for propagation increases until the stoichiometric composition of 9.S per cent (Q * 750 cal/g) is reached. At this concentration, the PE R CENT METHANE 69 06 atm 9 8 6 TWO BURNING VELOCITY NON REGION PROPAGATING REGION 0.25 atm ©-LEWIS ft VON ELBE (1961) □ -COWARD a JONES (1952) A-DI PIAZZA,GERSTEIN a WEAST (1951) WHITE (1924) L 0 . 1 -j_ -L _L I 10 CRITICAL DIAMETER-c m 100 Fig. 1 9 . Comparison of Critical Diamatar Calculations with Quanching and Flammability Limit Ex- par imants at Atmospharic and Raducad Prassuras 90 predicted minimum diameter for propagation should agraa with quenching data in the literature. Correlation at these points along with two reduced pressure quenching points are detailed in Table 2 for methane-air experimental data. TABLE 2 PREDICTED AND OBSERVED MINIMUM DIAMETERS FOR FLAME PROPAGATION Fuel Concentration Pressure Predicted D Observed D (per cent) (atm) (cm) (cm) 5.24 (lean limit) 1 6.1 5.0* 9.5 (stoichiometric) 1 0.34 0.34b 9.5 0.5 0.66 0.60b 9.5 l i 0.25 1.4 1. 2C 9.5 H 0.06d 5.7 5.0d aFlammability limit from Coward and Jones (1952). bData from Lewis and von Elbe (1961, Fig. 160) and increased by a factor of 1/0.65 (see Potter, 1960, p. 17 2) to correlate quenching distance with quenching diameter. cData as above but interpolated between points at 0.13 and 0.2 atm. dLow pressure limit of DiPiazsa, Gerstein and Weast (1951). (Corrected to open tube data assuming same relationship between closed and open holds for methane limit pressures as for propane). The close agreement at one atmosphere pressure indi cates the validity of the conclusion that the flammability limit is a quenching phenomena and is therefore described by the same model. The agreement for quenching at reduced 9 1 pressure indicat** that thia model should accurately pre dict reduced pressure flammability limits. The variation of minimum tube diameter for propaga tion (quenching diameter) with pressure is shown in Figure 20 for three fuel-air mixtures. Also, included are the data from Lewis and von Elbe (1961, Fig. 160) for methane- air (increased as in Table 2) along with the quenching dia meter data for propane-air from Simon, Belles and Spa- kowski (1953). It is noted that these later data are the data used in that reference to verify the theory that flammability limits are caused by the destruction of reac tion chain carriers by the vessel surface. It can be seen that this flame model adequately predicts this trend (mi nor modifications to th* numerical data will move the pre dicted line to a better fit with experiment). Reduced Pressure Flammabilitv Limits As pressure in a 5 cm diameter flammability limit apparatus is reduced below one atmosphere, according to Coward and Jones (1952) the lean flammability limit "gen erally raises, often imperceptibly for the first few tenths of an atmosphere, but thereafter increasing until at a suitably low pressure, the limits coincide, below which point no mixture can propagate a flame." Closed tub* experimental data at reduced pressure for methane-air in a 5 cm diameter tube were obtained by OiPiazsa, CRITICAL DIAMETER - cm 92 IO r STOICHIOMETRIC METHANE I . V .6% METHANE 0-LEW IS a VON ELBE (1961), STOICH. METHANE-AIR 0 - SIMON,BELLES 8 SPAKOWSKI (1953), STOICH, PROPANE-AIR O.l*-, I __ 0.01 0 . 1 PRESSURE - otm Fig. 20. Variation of Critical Diameter with Pressure; Comparison with Exper iments 93 Gerstein, and Weast (1951), and Scott Zabetakis and Furno (1952). Simon, Balias and Spakowski (1953) axparimantally datarminad tha affacts of reducing prassura on opan tuba flammability limits for tubes of various sizes using pro pane-air. Similar methane-air data are not available; how ever , a comparison of reduced prassura flammability limits of the light paraffin hydrocarbons on the basis of equiva lence ratio was made by DiPiazza, Gerstain and Weast (1951) which showed that tha lean limits ware similar. Also, Fig ure 20 indicates that for quenching there is a similarity between propane and methane. A comparison of the results of this study with these experimental data is shown in Figures 21 and 22. A reac tion order of two and one are both shown. As discussed previously, the overall reaction order for stoichiometric methane-air has been experimentally determined as 1.66. Gerstein (1956) has noted that the overall reaction order decreases as the fuel-air mixture deviates from stoichio metric as shown in Figure 23, which is for propane-air (similar data are not available for methane-air). Closed tube flammability limits, as shown in Figure 21, are expected to occur at lower initial pressures due to compression heating and the increased pressure after ig nition. This could cause some of the deviations noted Ln Figure 21. The difference between reduced pressure closed PRESSURE-aim 94 1 . 0 .8 .6 .4 .2 0 = 0 cm ± O-DI PIAZZA,GERSTEIN 8 WEAST (1951) - CLOSED TUBE Q-SCOTT,ZABETAKIS 8 FURNO (1 9 5 2 )-CLOSED TUBE 4 6 PER CENT METHANE Fig. 21. Effect of Pressure on the Fuel Required for Flame Propagation In a 5 cm Diameter Tube; Closed Tube, Methane-air Data PRESSURE - atm 95 .0 1.6cm 2.8 cm m 4.7 cm THIS STUDY (5cm) SIMON,BELLES & SPAKOWSKI (1953) 0.01 100 40 60 80 EQUIVALENCE RATIO Fig. 22. Effect of Pressure on the Fuel Required for Flame Propagation; Open Tube, Propane-air Data OVERALL REACTION RATE 96 2.0 8 6 1.6 1 .6 1 . 0 1.4 .8 1.2 EQUIVALENCE RATIO Fig. 23. Variation of Overall Reaction Order with Fuel Concentration for Propane-air Mixtures (From G e rs te in , 1958) 97 and open tuba limit* wai atudiad by DiPiassa, Oaratain, and Weast (1951) and thair data for propane-air ara shown in Figura 24. This modal predicts tha aquivalant of an opan tuba, i.e. constant prassura flammability limit. In ordar to obtain battar agraamant with tha experi- mental data in Figures 21 and 22, other than minor changes in the numerical data used in the model, the need for two modifications to tha flame model is apparent. As dis cussed previously, tha overall reaction order should vary with tha fuel-air ratio. Along with this variation could go a variation in the steric factor A, Equation (3.13), of eithar/or B and ^8. In general, a more complete reaction kinetics modal is needed. Secondly, tha specific heat will vary with tempera ture as tha mixture goes from lean at one atmosphere to stoichiometric at the limit pressure since the temperature increases. This should have the effect of making the low pressure portions of the curve flatter to conform with the experimental data. It is also possible that other factors not considered in this model such as buoyancy effects, wall chain-carrier destruction or preferential diffusion could play a role in shaping the reduced pressure results. Burning Velocity at the Limit Figure 25 describes the variation of burning velo city with fuel concentration. It can be seen that the 96 0 1 Ul <r 3 CO CO I f t J cc CL 0.2 0=6 cm — THIS STUDY ©-OPEN TUBE 0-CLOSED TUBE 0.4 0.6 0i8 EQUIVALENCE RATIO 1.0 Fig. 24. Reduced Pressure Flammability Limits; Comparison with Open and Closed 5 cm Tube Propane-air Experimental Data (From Di Piazza, Gerstein and Weast, 1951) 99 o « 0> s E o U O -J UJ > © f t : 3 m 14 12 10 8 6 4 2 ©-EDMONDSON,HEAP S PRITCHARD (1970) 0-EGERTON 8 THABERT (1952) J L \D*I cm D=IOcm ADIABATIC CRITICAL 2 4 6 6 PER CENT METHANE Fig. 25. Burning Velocity Model Showing Variation of Predicted Burning Velocity with Fuel Concentration for Various Diameters; Comparison with Experiment 100 adiabatic burning valocity approach#a zero gradually as fuel concantration is raducsd. However, for a givan dia- matar and fuel concentration, two burning velocities are possible until tha concantration reaches soma minimum val ue, called the flammability limit. Burning velocities be low this point are unstable as described by Spalding (1957, p. 95). Looking at this curve it is easy to see why findings of burning velocity experiments predict finite flammability limits. Selecting a certain diameter to represent burner heat loss, as fuel concentration is reduced, the burning velocity diverges from the adiabatic solution and finally reaches the flammability limit as a vertical line. Simple extrapolation of this vertical line would indicate that the burning velocity was approaching zero at a finite com position, and that the flame probably went out due to burner instabilities. Agreement with the trend of the experimental burn ing velocity data of Edmondson, Heap and Pritchard (1970) is good as shown on Pigure 25. However, as was noted in Chapter IV, the steric factor constant B was chosen to give a limit burning velocity of 2.24 cm/sec at 5.24 per cent methane concentration so that agreement of the ab solute SMgnitudes is constrained at this point. At reduced pressures, as noted previously, burning 101 velocity for a particular fuel concentration remains con stant with pressure. Therefore, the relationship between these two variables shown on Figure 25 will remain; however the limit diameter curves shown thereon will shift toward higher fuel concentrations as pressure is reduced. Relation to Adiabatic Flame Characteristics Relationships between limit and adiabatic flame properties have been found. For a methane-air mixture, the limit burning velocity will be about 50 per cent of the adiabatic burning velocity. The flame temperature will be about 90 per cent and the flame thickness about 164 per cent of the corresponding adiabatic values (see Equations (4.20), (4.22) and (4.23)). These can be expected to vary with the type of fuel used; however, they are expected to remain essentially the same for most lean hydrocarbon-air flames. The addition of diluents with properties greatly different from the above mixtures are expected to modify these values. For reduced pressures, down to 0.05 atmos pheres, the relations seemed to hold well. Effects of Radiation Loss Radiation loss causes a relatively sharp reduction in the effect of tube diameter increases upon the flamma- bility limit. Indeed, a fairly significant proportion of the heat loss for 5 cm diameter tubes at atmospheric 102 pressure is dus to radiation. However, duo to ths recipro cal diajnatar tarm in Equation (4.9), it can ba saan that as tha siza of tha containing vassal or chamber increases, this term decreases with the result that the flammability limit decreases toward zero. Barlad and Yang (I960, p. 328) failed to include this dependence in their analysis (although they did in their calculation example) leading to the erroneous con clusion that radiation loss remains unchanged as diameter is varied, and therefore, that the “quenching limit" and the “inflammability limit1 ' are different phenomena. Spalding (1957, p. 97) also minimized this functional re lationship when he noted that the appearance of D? in the equation fox conductive loss “shows that conduction can be made insignificant by sufficiently increasing the appara tus size;" however, that the variation in radiative heat losses with diameter "is small with practicable dimensions and has been neglected." Seduced pressure flammability limits have been at tributed to radiation loss by Spalding (1957, p. 100) for reactions of higher order than first. Considering the radiation heat loss model. Equations (4.9) through (4.13), it is seen that, for a given diameter tube, as pressure decreases, so does the gas emissivity and therefore, the contribution of radiation. This trend can be seen on 103 Figure 19. Since conduction loss is unaffected by pres sure, the end result is that conduction plays a more im portant part in the net heat loss as pressure is reduced. For example, for a 5 cm tube at one atmosphere, the limit flame has a radiation component of 27 per cent of the total heat loss, whereas at 0.05 atmospheres this has re duced to 5 per cent. Therefore, it is incorrect to con clude that radiation is more important at low pressures. In summary, this analysis shows that radiation loss is important for large containing vessels, and in fact, causes the apparent lack of dependence on tube diameter in the laboratory. However, radiation loss decreases propor tionally as size is increased resulting in a continual lowering of the flammability limit. At room or tunnel dimensions, the flammability limit will be significantly lower than in a 5 cm diameter tube. In fact, this model shows that the flammability limit reduces from 5.36 per cent to 3.92 per cent if the diameter of the tube is in creased from 5 cm to 5 m; and down to 3.54 per cent in a 50m tube or chamber. CHAPTER VI CONCLUSIONS In this study it has been shown that a non-adiabatic flame modal predicts tha important experimental observa tions of flammability limit determination. This same model, predicts flame quenching experimental results in an accurate manner. A search for analytical causes of flame extinction for adiabatic flames produced no such result. It is concluded that the flammability limit exists only because of heat loss to the environment. Radiation heat loss accounts for flammability limits in large systems; however, the reduction of heat loss with sise predicts significantly lower flammability limits as size increases. The experimental observations, often used to show that heat loss to the apparatus is insignificant, are prediced by this model. This study provides the necessary completeness in flame model and its application to support solidly the above conclusion. In particular, the model uses Arrhenius kinetics, distributed heat loss in both the conduction and radiation mode, and numerous calculations were performed over a wide variation of flame parameters to establish trends of non-adiabatic flame theory. 104 105 Investigation of tha heat loss models indicates that both conductive and radiative loss become zero as the di mension of the flame normal to the direction of propaga tion becomes infinite. As a result, flammability limits are expected to approach zero for very large chambers. This conclusion is important to the application of flamma bility limit data from a 5 cm tube to real systems of larger size. The product of limit burning velocity «md diameter was found to deviate from a constant as the dominant mode of heat loss became radiative. Therefore, the Peclet number cannot be used to predict large diameter flammabil ity limits. However, constant relationships between limit and adiabatic burning velocity, peak temperature and flame thickness indicate that it is possible to relate some adi abatic flame results to the non-adiabatic flame. Finally, a relatively simple method has been devel oped to integrate numerically the equations of change for non-adiabatic flames using available numerical techniques on a digital computer. Minor approximations are necessary to start the integration at a point near the cold boundary. APPENDIX COMPUTER CODE On the following pages is a listing of the card images for program DISTILER. The program is written in Fortran IV language and contains 291 source statements with a program size of 22,514 bytes. Table A-l is a cross ref erence between the Fortran variables used in the program, and the symbols used in this dissertation. The program, as listed, is in the form used for the second order reaction calculations. For first order studies, much time can be saved by removing the exponen tiation by NRO and subsequent sign change of DERY (3) in subroutine FCT. Subroutine DHPCG (I.B.M., 1966) was concatenated from the facility library. Therefore, it does not appear in this listing. 106 107 TABLB A-l CROSS-LISTING OF IMPORTANT PROGRAM VARIABLES Computer Coda Varlabla CLAM CP D OELT DERY(l) DERY(2) DERY(3) DEFY(4) DQL DQLR BA EM IS EX FLUX G Dissertation Symbol ^ 296 CP D s dT/dz d2T/dz2 d(l-G)/dz d(l-x)/dz • 14 I q * III qr ®A X M G Computer Code Variable NRO PT Q STERF TC02 TO VO WMOLE XEPS Y(l) Y(2) Y(3) Y(4) Z Dissertation Symbol a P Q A C T„ m x T dT/dz 1-0 1-x CARD IMAGI LISTIMQ PROGRAM DISTILKR C FORTRAN PROGRAM DISTILER - 1 / 1 7 / 7 2 - W. B. STINE, HECH. ENGR. - U .S .C . c t h i s PROGRAM INTEGRATES th e c o n s e r v a tio n eq u a tio n s f o r a f l a k e k i t h r e s p e c t C TO 0 1 STANCE STARTING AT THF COLD BOUNDARY. AHRRENIUS KINETICS ARE USED. C-NCVG IS POINT WHERE 'FUEL CONVERGENCE CRITERIA WAS MET C-NST IS THF POINT TO START PRINT-OUT C-NOT6: OF SET TO HAKE PIC02I-5*? FOR Q -4 1 5 .6 6 7 FOR EMISIVITV CALC C-HlC RATIO ASSUMtD =4 FOR EMISIVITV CALCULATION C-TO RUN A CASE, NAHEL 1ST IN MUST CONTAIN AT LEAST Q, D, AND FLUMYQUR BEST GUESSI C FOR FULL PRINTOUT, SET MINPRT=.FALSE. C-TO FORCE PRINT-OUT FOR A GIVEN FLUX,SET BYP*.TRUE. C-TO RESTART A CONVERGING CASE,SPECIFY FLJX AND EFLX(THE t DEVIATION! IMPLICIT REAL* 8 I A - H , 0 - Z ) EXTERNAL FCT.QUTP DIMFNSION V I4 ! ,DERY(4»,PRMT( 5 ) , AUXI16,41 LOGICAL BYP,CVG,CVT,KINPRT,STUREO,SM COMMON CLAM,CP,C4,CS,0,EC,6W,FLUX,PT,Q« TO,T04,WMULE, X M,NCVG,NRO,NS,NTP, Y BYP,CVG,CVI,MINPRT,STORED COMMON/FUNCT/Cl,C2,FM,C6,C7,CR COMKON/OUT/AUX COMMON/PRIN/DXFPS,XEPS NAMELIST/IN/BYP,D,DXEPS,FFLX,FLUX,MINPRT,PT, 0 , XEPS NAMELIST/PARAH/CLAM,CP,DELT,EA,NRO,PT,STERF,TO,WHOLE CLAH*C.6D-<r CP*. 3 3333333? 333 333 3DC OELT-l.DO EA=ACrOC.OO HCsA.C NRO*2 0F=832O. STERF*1.97D15 TO=299.0DC TQA*T0**4 WM0LE=3r.CD3 C-START HERE TO READ MORE CASE DATA I CONTINUE CVG*.FALSE. MOLD=C NB*0 N I- 0 NTP=0 STORED*.FALSE. SN*.FALSE. C-OEFAULT VALUES FDR NAMELIST IN BYP*.FALSE. OXEPS*0.0 E F L X *C .0 l MINPRT*.TRUE. P T -l.D C XFPS-C.DO C-READ DATA AT THIS POINT READIS f IN I IF IB Y P I MINPRT-.FALSE. IFISTDREDI EF LX-C.O l o to IFCDKFPS.EQ.C.CI CXEPS-C.50*XEPS P C L * C * 9 0 * D * 0 * P T /( 0 F + 3 C .^ 8 ) 110 x 3 O UJ or 0£ a o o S' * * CM * * O • • eg m •O in - * > ♦ U 3 a. a o o O ' —1* ”1 • o o M O D T < — or u o o o z < X * * r» — a I 0<DC * * _i ,0 ee '0 -O • • • ■ — <C 9 1 I - UJ u . u a H I I M < — ■» t — u x or or or Hi U U 3 3 Z o or UJ a IL X o z <1 • a uj c m U U J J 3 « UJ Z l l a • UJ H H Z Z ^ b a > « u- t _ ) u x a < cm 3 ^ * "s, — C M — > * 0*4 ^ X O I < "v z z z — # • >■ * *G V \ H # —> ♦ a v (v u o * o * * • X •> CD X i / > O U. a eg —* O X o c, ♦ • O 0 ao o • O' • i- i C M (j v I I ( I I I I I *■ 7 ► > > a o m o + x # O « - > uj * 0 0 3 □ it in it * — o g» (T I O O ' * ■ — • • • * - • n cs o w I I I I I I •• H N I" “ > > ► cm a a a o N. O eg in o t O' O O o c* • • • C ) O M II M H I a lA • O I I U U . I - C C O > ►- K a x x uj a a o a a a a in i o PRMT ( < i I *0. P05DC N D IN -4 Cl*?98.*FL'JX*Q /C LA N C2 - 2 9 8 . *FLUX*CP/CLAM C 4 * 1 6 . / D * * 2 C5 * 2 9 8 . 0 0 * 5 . 1 5 0 - 1 2 / I0*CLAM» C 6 -IW M 0 L F *S T F R F /F L U X » *tP T /8 2 .C 5 7 |**N R C C 7 *-E A /1 .9 8 T D G C 8=F LU X *C P*298,00/(D E LT*C L A M I CALL DHPCGIPRMT,Y,DERY,NDIM,IHLF,FCT,GUTP,AUXI I F ( BY*M GO TO 1 I F ( STORED! GO TO L I F ( CVGI GO TO 2 C THIS °ART CONVERGES THE EIGENVALUE I F U M . E Q . O I . A N D . U H L F . L T . i n i GO TO 1 I F I I H L F . G E . I 1 I GO TO 8 9 CONTINUE I F I N B . L F . O I DFLX=EFLX*FLUX NB*NR+1 I F ( CM^MOLOI• F O.C > SW=.TRUF. I F I.N O T * SW» D F IX =D F L X *2 .0 IF ($•)) D FLX*0FLX/2.C IFUFLUX+N*O FLX» . L E . C . O I DFLX*DFLX/A.O FLUX* DABSIFLUY*M*DFLXI mold** FFLX*DFLX/FLUX PRINT 2 0 0 0 , FLUX,OFLX#M,FFLX GO TO 2 C - IF IHLF IS EXCEEOEO* TRY INCREASING FLUX UP TO 9 TIMES 6 PRINT 40CC I F tM .F Q .O ) M=1 I F ID E R Y IA I.G T .C .O C I M * - l NI*N!♦I IF 1 N I . G T . 91 GO TO 1 GO TO 9 ♦ 111 FORMAT ( IH ,'F D R THE NEXT ITERATION; FL UX= • ,G2A* 16 * * DFLX=* « K G I Z . ^ . ' M* • * 1 3 * * F F L X = » ,G I2 . 4 ) t 30DC F0RMAT(1H1» 4CDC FORMAT 11H * * T HIS CASE TERMINATED DUE TO EXCEEDING IHLF L I M I T * I FNO SUBROUTINE < = C T 1 X, Y, DFR V I C -TH IS SUBROUTINE IS CALLED BY DHPCG,AND CALCULATES THE O E R IV IT IV E S . IM P L IC IT R E A L * 8 IA -H .O -Z I LOGICAL BYP.CVG.CVI»MINPRT,STORFD,SM DIMENSION Y (A I,D E R Y (4 » COMMON CLAMf CP*CA*CS*D*EC * EM* FLUX*PTtO#TO*TOA*MMOLE f X Mf NCVG»NRU«NS*NTP, Y BYP,CVG*CVltMINPRT,STORED C0MH0N/FUNCT/Clf C 2 *F M ,C 6 *C T #C8 C C IF (C V I I GO TC 2 DEFY I 3 l*C 6*D F X P IC 7 /Y I 11) * ( I 1 .DC - V ( A I I / Y| I I| « * N R Q DERY( 3 1*O SIG N ID ERVI3 1 , ( 1 , 0 0 - Y ( V ) | ) O E R Y IA I*! V U I - Y I 3 I I * C 8 / Y ( 1) GO TO 3 2 O ERY (3l*C.CD 0 DEPY<M*C.COO 3 OFRY( 1 1»YI2» IF ( X»GT, 0* 01 GO TCi 1 112 DERYI2I-I mi-TDI*FM**2 RETURN C-P0SSI9LE EXIT P O INT» >>»» » » » > » 1 IFIC5.E0.C.0CI GO TO 1C TC02=Y(I1 IFtY<ll.LT.lC6C.PI TC02U06G.D0 EMISM lC.**t.5C5-tTC02*4lCO.I/CCU-10.**<C.562-t YI 11 *2625. I/E * II X *( l.DC-f l.DO-Y(4| 1*1 (TCG2-Y(1I1/ITC02-T0I I I IFtEMIS.GT.l.CI EMISU.C IFIYIll.LE.TOI EMIS=C.D0 1C IFtDERYt31.LE.C.DCI GO TO 11 0ERYI2»-l-Cl*0ERYt3t+C2*0FRYli|-OERY(1»**2*C5*EMIS*(Y<il**4-T04ll X /Y(l)*C4*tYUl-TOI RETURN C-POSS I RLE EXIT P 3 I N T » » » » » » » > » 11 OFRY(.21— <C2*DERYtl l-DERVm**2+C5*EMlS*t Y(ll**4-T04l I X /Yt 1I*C4*< Yt ll-TOI RETURN END SUBROUTINE OUTPt X, Y , DERY,IHLF,NDIM.PRMTI C -TH IS SUBROUTINE IS CALLED BY DHPCG. IT CHECKS FOR CONVERGENCE.AND STORES C- POINTS IF CONVERGENCE HAS BEEN ATTAINED. IM P LIC IT R E A L * 8 (A -H .O -Z I LOGICAL BYP.CVG.CVl.HINPRT * STORED.SW DIMENSION Y I4 I .D E R Y ( 4 | .P R M T t S t DIMENSION X S < 2 C 0 l.vS t2 C 0 .4 l,D E R Y S (2 C C , 4 1 . IH L F S I2 0 C I 113 o n DIMENSION A U X I1 6 ,* ) COMMON C L A M ,C P ,C 4 ,C 5 ,D ,F C ,E N ,F L U X ,P T ,Q ,T G ,T 0 4 ,M M G L E , X M,NCVG»NRO,NS*NTP» V BYP,CVG,CVl,MINPRT,STOREO COMMON/OUT/AUX I F I Y I l l . L T . T O I GO TO IT |F(CVG.DR.BYP> GO TO 10 I F U 0 E R Y I 3 » . G E . C . C I . A N O . < O F R Y I M . G E . C . C n RETURN C-POSSIBLE EXIT PQI NT >>>>>>>>>> > » » » IF ID E R Y I4 1 I 1 , 2 , 2 1 M*1 GO TO 8 C -IM P LlE D THAT OEKYI 3 ) <C 2 M=-l IF I DABS I Y I 3 I - Y I 4 I I . L E . C . C 0 1 I CVG=.T«UE. C-STOP INTEGRATION 8 PRMT151 = 1.C RETURN C-POSSI8LF EXIT P O I N T » » » » » > » » » 10 CONTINUE STORED-.TRUE. IF IN S . E Q .C l GO TO 11 I F U H L F . G E . l l I GO TO 11 IF IX -X O L D - PRMTC3I I 1 2 , 1 1 , 1 1 11 XOLD*X NS*NS+l C - L IM IT SIZE OF ARRAY TO THAT DIMENSIONED , I F I N S . G T . 2 0 0 I NS*20C C PUT RESULTS OF EACH INTEGRATION STEP INTO AN ARRAY NMAX*NS IH L F S IN S I* IH L F X S IN S I*X 00 13 1 = 1 ,4 YSINS. n = v m 13 D E R Y S IN S *I) -D6RYI I » IF IN T P . G T .P I GO TO 12 I F I Y I 2 I . L E . C . 0 I NTP=NS-1 12 CONTINUE I F I C V I I GO TO 15 IP ( B Y P .A N O .< O E R Y m .G E .C .P > ) GO TO 16 I F i n E R Y I T I . G E . C . C I RETURN C-POSS 1 RLE EXIT P J I N T > » » » » > » > » » C-PAST THE POINT OF SIGNIFICANT G £ X C V I -# TRUE• NCVG=NS AU XU 5 , 1 ) * C . 5 A U X I1 5 , 2 1 * 0 . 5 AUX( 1 5 , 3 1 * 0 .C AU XI1 5 , 4 ) * 1 . 0 PRMT|4 I * 0 . 5 0 0 15 CONTINUE IF(M INPR T.AND .(NS.EO .(NCVG +1D 111 GO TO 14 I F I D E R Y I 1 I . G T . C . 0 ) GO TO 14 16 CONTINUE I F I l H L F . G E . i l I GO TO 14 1 E f I X - P R N T I 2 M . G E . 0 . 0 1 GO TO 14 IF ID A B S IX -P R N T I2 11.G E.G .1*0AB S(PR M T( 3 ) 1 1 RETURN C-POSS IBLE EXIT P O I N T > » » > » » » » > » 14 CONTINUE 115 CALL PRINT*XS»YS*DERYS*IHLFS,NMAXI PRMT( 5 ) = L ■ 0 RETURN C-POSS I RLE EXIT P O I N T > » » > > » > > » » » 17 CONTINUE G M .O C -Y * 3» X F *1 ,0C -Y < 4» PRINT lC C C ,X F t G PRMT(5 1 - 1 , C RETURN ICTC FORMAT* IH , 'EXECUTION TERMINATED BECAUSE T<TO BEFORE CVG. CRITERIA X MET: X = ' , G I 2 * 4 * ' G » ' t G l 2 . 4 l END SUBROUT INF PR INT IX S ,Y S .D E R Y S *IH L F S f NMAXI C-THIS SUBROUTINE IS CALLED BY OiJTP.AND PRINTS THE INTEGRATION POINTS* IM P L IC IT R F A L * 8 I A - H , 0 - 7 I LOGICAL BYP,CVG,CVI,M!NPRT,STORED,SW DIMENSION XSC2CCI, Y S I2 0 C , 4 1,DEPYS* 20C, 4 » , IH L F S I2 C O I• MM*QI COMMON CLAM,CP,C4*CSt O .E C ,EW tFL U X,pTt Q ,T 0 f T04,WM0LE* X M,NCVG*NRO*NS«NTP* V BYP,CVG.CVI.MINPRT,STORED COMMON/PRIN/DXEFS*XEPS NAMELIST/OUT/D*fLUX,NCVG,Q*VO,XEPS PRINT 803 VO*82.C57DO*FLUX*TO/<WMOLr *P T ) WRITE ( f»t OUT ) 117 I / O 3 « / * U J at at O UJ u . o Of at I A > at UJ Ct -* — < I I a Z ( / ) >■ o o u> a CT X o H- <J O • Z < • CJ HILO I Of 1 1 X b. h OIL z Of Q. I UJ I X *4 < H X *- — Z if) » HZ — — H- X - ^ Q. 1 (0 — < f < * : * Z z »- x »fl. w D C Z Q .Z M — a . * z — x OZX--I/I * • *- i/» (/i > rj f J T Z > I u j ID • II t o o a • i — a o c i- a . o z Ct o Z U Z u* • I— HZ • • H 7 H- — —■ *4 H — «/) L L U , O II X O f z - » o o in a A A A A A A A A A A A A A A A Z A IX A 3 h- I- Z IU M a O & X — < 4 X * Z C L X 3 _1 ID * o O • 00 CT C M a z ( / ) >- or UJ O — I Of * — a . x Z UJ O O tJ o in c -i At I / ) U _J * - T X Z ------- — ~ a at a a Z 7 C u “ C l l O l / > t / » 0 a. 1 UJ a . — ■ ► z m a H - H » Z i / i * i / ) a •• z > > z i/> i i i « # — > a u & x ia — Z O Z < ► * . — _j # a if Hifl U o u H > It H H O X I I U O X o u L a o o QL*QH-QG-QC DQC*( CLAM/ 2 9 8 ■ PDCI*(Y$(NP * 1I*D E R Y S IN P *2 I+DERYS<N P , 1 1* * 2 1 PQG*FLUX*Q*OERYSINP.TI OOH-FLUX*CP*OERYS|NP,11 ! F (C 5 .E Q .0 .C I G O T O 2 TC02*T l F t T . L T . 1 0 6 f . I TC02*1C 60. E M I S = ( 1 C . * * I . 5 C 5 - ( T C 0 2 * M 0 C . » / E C J M C . * * ! C . 5 6 2 - ( T + 2 6 2 5 . I / E H M X * ( 1 . DC-( EX l * ( (TCQ2-T »/<TC02-T0m I F ( E M IS .G T * 1 . 0 1 E M I S ' l . C D 3 L R = t - 5 . i 5 0 - l 2 / 0 ) * F M I $ * t T * * W O M 0 Q L - D Q L R - 1 6 . M Y S I N P . 1 I - T D >*C L A M *Y S IN P .1 1 / < 2 9 8 . * D * * 2 » P L R -0 .0 IF (O O L .N F .T .C » PLR-1CO.*OOLR/OQI GO TO 3 2 CONTINUE EM IS=0*O DQLR=0.0 PLR*C.C OQL» - 1 6 . * ( Y S ( N P » 1 » - T O ) * C l A M * Y S ( N P .l » / l 2 9 8 . * 0 * * 2 1 3 CONTINUE PRINT 3C 0O>XS(NP),GG .O C.OH,OL.OQG.DOC.DQH,DQL. DQLRfPLR.EMIS 5 CONTINUE return 800 FORMAT!IH1,'RFSULTS QF INTEGRATION:*******PRQGRAM O I S T I L F R * * * * * * * I 900 FORMAT ( 1H .'X tO IS T A N C E I Y I (TEMP ) Y2 I0T /D Z 1 Y3( 1-GI Y M l - X I X O E R Y d l 0E R VI2I 0ERY(3» DERY ( A 1 1, 7 X , *G' # 10X* • X • . 5X* Y ' I H L F M 1CG0 FORMAT ( IH ,5 G 1 1 • A, * G 1 0 . 3 . 2G11 .<► * I 2» 1500 FGRMAT(1H1,'HEAT BALANCE PARAMETERS: GC ,G .H , CL < CAL/GMI 5DQC ETC. (C XAL/CM3*SECM > 2CCG FORMATI1H0.' OISTANCF QG QC OH QL ■• X* DOG OOC OOH DOL OQLR t-RAD EMI YS' > 30CC FORMAT( IH . 1 2 G 1 0 .3 I ENO LITERATURE CITED Adler, J., and Spalding, D. B. 1961. Proc. Roy. Soc. (London) A261, 53. Adler, J. 1963. Combustion and Flame T_, 39. Adler, J., and Kennerley, J. A. 1966. Combustion and Flame 10, 191. Adler, J. 1967 a. Combustion and Flame 11., 85. Adler, J. 1967 b. Combustion and Flame 11, 442. Badami, G. N., and Egerton, A. E. 1955. Proc. Roy. Soc. (London) A228, 297. Bechert, K. 1949. Ann. Physik (Leipzig) 4, 191. Bechert, K. 1950 a. Z. Electrochem. 54, 239. Bechert, K. 1950 b. Ann. Physik (Leipzig) ,5, 349. Berlad, A. L., and Potter, A. E. 1955. Fifth Symposium (International) on Combustion, p. 728, Reinhold. Berlad, A. L., and Yang, C. H. 1960. Combustion and Flame £, 325. Berlad, A. L. 1961 a. Combustion and Flame j>, 301. Berlad, A. L. 1961 b. Combustion and Flame 5., 389. Burdon, M. C., Burgoyne, J. H., and Weinberg, F. J. 1955. Fifth Symposium (International) on Combustion, p. 647, Reinhold. Burgess, N. J., and Wheeler, R. V. 1911. J. Chem. Soc. (London) 99, 2013. Chen, T. N., and Toong, T. Y. 1960. Combustion and Flame 4, 313. Coward, H. F., and Hartwell, F. J. 1926. J. Chem. Soc. (London) 129. 1522. 119 120 Coward, H. F., and Jonas, 0. W. 1952. Limits of Flamma- bllitv of Qasas and Vapors. U. S. Bureau of Minas Bullatin 503. Cullan, R. B. 1953. Trans. Am. Soc. Mach. Bngrs. 7_5* ^3. Cummings, G. A. McD., Hall, A. R., and Straker, R. A. M. 1962. Eighth Symposium (International) on Combustion, p. 503, Williams and Wilkins. Daniell, P. J. 1930. Proc. Roy. Soc. (London) A126. 393. Diederichsen, J., and Wolfhard, H. G. 1956. Trans. Fara day Soc. 52^ 1102. Di Piazza, J. T., Gar stein, M., and Weast, R. C. 1951. Ind. Eng. Cham. ,42, 2721. Dixon-Lewis, G., and Isles, G. L. 1959. Seventh Sympo sium (International) on Combustion, p. 475, Butter- worths. Eckert, E. R. G. 1950. Introduction to the Transfer of Heat and Mass. McGraw-Hill. Edmondson, H., Heap, M. P., and Pritchard, R. 1970. Combustion and Flame 1£, 195. Egarton, A. C., and Powling, J. 1948. Proc. Roy. Soc. (London) A193. 172 and 190. Egarton, A. C., and Thabart, S. K. 1952. Proc. Roy. Soc. (London) A211. 445. Egarton, A. C. 1953. Fourth Symposium (International) on Combustion, p. 4, Rainhold. Fenimore, C. P. 1964. The International Encyclopedia of Physical Chemistry and Chemical Phvales. Topic 19 (A. F. Trotman-Dickenson, Ed.) Volume 5, Chemistry in Premixed Flames. Pergamon. Fenn, J. B. 1951. Ind. Eng. Chem. 43, 2865. Fristrom, R. M., and Westernberg, A. A. 1965. Flame Structure, McGraw-Hill. Garstein, M. 1958. Combustion and Propulsion - Third AGARD Colloquium (M. W. Thring, J. Fabri, O. Lutz, A. H. Lefebvre, Eds.) p. 307, Pergamon. 121 Gerstein, M., and Stine, W. B. 1970. Anomalies In Flaah Polnti of Mlxturaa of Haloaenated Hvdrocarbona and flammable Llauida. Fapar presented at tha Fall 1970 maatlng of the Western States Section, Tha Combustion Institute, WSS/CI 70-20. Gerstein, M., and Stine, W. B. 1971. Proceedings of the 1971 Conference on Natural Gas Research and Technolo gy, 3424 S. State Street, Chicago, Illinois 60616. Gray, B. F., and Yang, C. B. 1966. Combustion and Flame 10, 199. Gray, B. F., and Yang, C. H. 1967. Combustion and Flame 11, 441. Hirschfelder, J. O., Curtiss, C. F., and Bird, R. B. 1954. Molecular Theory of Gases and Liquids. John Wiley and Sons. Hirschfelder, J. O., and Curtiss, C. F. 1961. Advances in Chemical Physics, Vol. Ill (I. Prigogine, Ed.), p. 59, interscience Press. Bottel, H. C., and Sarofim, A. F. 1967. Radiative Trans fer , McGraw-Hill. I.B.M. 1966. System/360 Scientific Subroutine Package (360 A-CM-03X) Version III Programmer 1s Manual, pp. 337-343, I.B.M., White Plains, New York. Jost, W. 1946. Explosion and Combustion Processes in Gases. McGraw-Hill. Kaskan, W. E. 1957. Sixth Symposium (International) on Combustion, p. 134, Reinhold. Kydd, P. H., and Foss, W. I. 1964. Combustion and Flame ft, 267. Layzer, D. 1954. J. Am. Chem. Soc. , 2 2 , , 222, 229. Le Chateller, H., and Boudouard, O. 1B98. Bull. Soc. chim. Paris ,19, 483. Levy, A. 1965. Proc. Roy. Soc. (London) A283, 134. Lewis, B., and von Elbe, G. 1951. Combustion. Flames and Explosions of Gases. Academic Press. 122 Lewis, B., and von Elba, 0. 1961. Combustion, F l a w and Bxploalona of Gaaaa, 2nd Ed., Academic Praaa. Linnett, J. W., and Simpson, C. J. S. M. 1957. Sixth Symposium (Intarnational) on Combustion, p. 20, Rein- hold. Lovachav, L. A. 1970. Doklady Akad. Nauk SSSR 193, 634. Lovachav, L. A. 1972. Combuation and Plama 17., 275. Macha, H., Kozak, W., and Zappa, A. 1952. Machr. Cham. 83, 171. Markatain, G. H. 1953. Fourth Symposium (Intarnational) on Combustion, p. 44, Williams and Wilkins. Mayer, E. 1957. Combustion and Flama 1., 438. McBride, B., and Gordon, S. 1968. 1CPRG One-Dimensional Equilibrium Referanca Program. MASA-Lawis Research Canter, Cleveland. Mellish, C. E., and Linnatt, J. W. 1953. Fourth Symposi um (Intarnational) on Combustion, p. 407, Williams and Wilkins. Potter, A. E. 1960. Progress in Combustion Science and Technology. Volume I, p. 145, Pergamon Press. Powling, J. 1949. Fuel 26., 25. Pritchard, R., Edmondson, H., and Heap, M. P. 1972. Combustion and Flame 18, 13. Putnam, A. A., and Jensen, R. A. 1949. Third Symposium on Combustion, Flame and Explosion Phenomena, p. 89, Williams and Wilkins. Rosen, J. B. 1954. J. Chem. Phys. 22., 733 and 743. Scott, G. S., Zabetakis, M. G., and Furno, A. L. 1952. U. S. Bureau of Mines, Report of Investigation 4639. Semenov, N. N. 1940. Progress of Physical Science (U.S.S.R.) 23, 251 (translated in NACA TM 1024). Simon, D. M., Belles, F. E., and Spakowski, A. E. 1953. Pourth Symposium (International) on Combustion, p. 126, Williams and Wilkins. 123 Sokolik, A. S. 1960. Self-lonition. Flam* and Intonation in Gases (translated in NASA TT F-125). Spalding, D. B. 1957. Proc. Roy. Soc. (London) A240, 83. Spalding. D. B., and Yumlu, V. S. 1959. Combustion and Plane 5, 553. Spalding, D. B. 1961. Combustion and Flame _5, 389. Tiggelen, A. van. 1947. Bull. Soc. Chim. Beiges 55., 202. Tiggelen, A. van, and Deckers, J. 1957. Sixth Symposium (International) on Combustion, p. 61, Reinhold. Tiggelen, A. van, and Burger, J. 1964. Combustion and Flame 8, 343. Wehner, J. F. 1963. Combustion and Flame T_, 309. Weinberg, F. J. 1955. Proc. Roy. Soc. (London) A230. 331. White, A. G. 1924. J. Chem. Soc. (London) 125, 2387. White, A. G. 1925. J. Chem. Soc. (London) 127. 672. Williams, F. A. 1965. Combustion Theory, Addison Wesley. Wolfhard, H. G. 1956. Selected Combustion Problems II. p. 328, Butterworths. Yang, C. H. 1961. Combustion and Flame 5., 163. Zabetakis, M. G., and Richmond, J. K. 1953. Pourth Sympo sium (international) on Combustion, p. 121, Williams and Wilkins. Zabetakis, N. G., Lambiris, S., and Scott, G. S. 1959. Seventh Symposium (International) on Combustion, p. 484, Butterworths. Zabetakis, M. G. 1965. Flamnabilltv Characteristics of romliii«t;iblo Gases and Vapors. U. S. Bureau of Nines Bulletin 627.

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Creator Stine, William Blaine (author)

Core Title On the theory of flammability limits

Contributor Digitized by ProQuest (provenance)

Degree Doctor of Philosophy

Degree Program Mechanical Engineering

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag engineering, mechanical,OAI-PMH Harvest

Language English

Advisor Gerstein, Melvin (committee chair), Choudhury, P. Roy (committee member), Mannes, Robert L. (committee member), Ogawa, Masaru (committee member)

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c18-506123

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Rights Stine, William Blaine

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