&EPA EPA 600 7 78 1 72a August 1 978 Premixed One-dimensional Flame (PROF) Code User's Manual Interagency Energy/Environment R&D Program Report ------- RESEARCH REPORTING SERIES Research reports of the Office of Research and Development. U.S. Environmental Protection Agency, have been grouped into nine series. These nine broad cate- gories were established to facilitate further development and application of en- vironmental technology. Elimination of traditional grouping was consciously planned to foster technology transfer and a maximum interface in related fields. The nine series are: 1. Environmental Health Effects Research 2. Environmental Protection Technology 3. Ecological Research 4. Environmental Monitoring 5. Socioeconomic Environmental Studies 6. Scientific and Technical Assessment Reports (STAR) 7. Interagency Energy-Environment Research and Development 8. "Special" Reports 9. Miscellaneous Reports This report has been assigned to the INTERAGENCY ENERGY-ENVIRONMENT RESEARCH AND DEVELOPMENT series. Reports in this series result from the effort funded under the 17-agency Federal Energy/Environment Research and Development Program. These studies relate to EPA's mission to protect the public health and welfare from adverse effects of pollutants associated with energy sys- tems. The goal of the Program is to assure the rapid development of domestic energy supplies in an environmentally-compatible manner by providing the nec- essary environmental data and control technology. Investigations include analy- ses of the transport of energy-related pollutants and their health and ecological effects; assessments of, and development of, control technologies for energy systems; and integrated assessments of a wide-range of energy-related environ- mental issues. EPA REVIEW NOTICE This report has been reviewed by the participating Federal Agencies, and approved for publication. Approval does not signify that the contents necessarily reflect the views and policies of the Government, nor does mention of trade names or commercial products constitute endorsement or recommendation for use. This document is available to the public through the National Technical Informa- tion Service. Springfield, Virginia 22161. ------- EPA-600/7-78-172a August 1978 Premixed One-dimensional Flame (PROF) Code User's Manual by Robert M. Kendall and John T. Kelly Acurex Corporation/Energy and Environmental Division 485 Clyde Avenue Mountain View, California 94042 Contract No. 68-02-2611 Task No. ,7 Program Element No. EHE624A EPA Project Officer: W. Steven Lanier Industrial Environmental Research Laboratory Office of Research and Development Research Triangle Park, NC 27711 Prepared for U.S. ENVIRONMENTAL PROTECTION AGENCY Office of Research and Development Washington, DC 20460 ------- ABSTRACT The Prenrixed One-dimensional Flame (PROF) code numerically models com- plex chemistry and diffusion processes in premixed laminar flames. Since this code Includes diffusion, it gives realistic solutions of coupled combustion and pollutant formation processes in the flame zone, as well as downstream in the post-flame region. Experience has shown that the code can be a valu- able aid when interpreting experimental flame data. In addition to flames, the PROF code can also treat problems involving well-stirred reactors, plug-flow reactors, and time-evolution chemical kine- tics. A wide variety of conventional and experimental combustion systems can be treated by the code's flame and reactor options. This PROF code user's manual describes the problems that can be treated by the code. It also describes the mathematical models and solution procedures applied to these problems. Complete input instructions and a description of output are given. Several sample problem input and output listings are pre- sented to demonstrate code options. A program listing and code Fortran vari- able definitions are included in the manual. ii ------- TABLE OF CONTENTS Section Page 1 INTRODUCTION 1-1 2 PROBLEM DESCRIPTION 2-1 3 MATHEMATICAL MODELS 3-1 3.1 Flame Model 3-1 3.1.1 Species Axial Diffusional Flux 3-2 3.1.2 Species Production Terms 3-5 3.1.3 Species Flux at the Wall 3-6 3.1.4 Bulk Gas Volumetric Heat Loss 3-7 3.1.5 Axial Diffusional Heat Flux 3-11 3.1.6 Wall Heat Flux 3-13 3.1.7 Specialized Boundary Conditions 3-14 3.2 Plug-Flow Reactor Model 3-16 3.3 Time-Evolution Chemistry Problem 3-17 3.4 Well-Stirred Reactor Model 3-17 4 SOLUTION PROCEDURES 4.1 Flame Solution Procedure 4-1 4.1.1 Finite Difference Form of the Flame Conservation Equations 4-1 4.1.2 Chemistry Solution Procedure 4-6 4.1.3 Linearized Predictor - Corrector Procedure 4-9 4.2 Plug-Flow Reactor Solution Procedure 4-12 4.3 Well-Stirred Reactor Solution Procedure 4-13 5 CODE INPUT INSTRUCTIONS 5-1 5.1 Card Input Deck 5-1 5.2 Thermochemical Input Data Format 5-12 5.3 PROF Thermochemical Data Update Program (TCUP) 5-15 5.3.1 List Option 5-15 5.3.2 Update Option 5-15 6 OUTPUT 6-1 6.1 Integral and Nonintegral Input Parameters 6-1 6.2 Species Names and Concentrations 6-1 6.3 Thermochemical Data 6-2 6.4 Kinetic Reaction Data 6-2 6.5 First Guess Alphas 6-2 iii ------- Section Page 6.6 Chemistry Solutions at Grid Points 6-2 6.7 Condensed Output for Each Iteration 6-3 6.8 Mole Flux at Grid Points 6-3 6.9 Grid Information Summary Table 6-4 6.10 Kinetic Reaction Summary Information 6-5 7 SAMPLE CASES 7.1 Sample Case 1 - Free Methane/Air Flame 7-1 7.2 Sample Case 2 - Methane/Air Flame Attached to a Flameholder 7-23 7.3 Sample Case 3 - Methane/Air Chemical Evolution in an Internal Combustion Engine 7-36 7.4 Sample Case 4 - Methane/Air Well-Stirred Reactor .... 7-47 7.5 Sample Problem 5 - Catalytic Combustor Fuel Conversion Efficiency 7-55 8 DEBUG OUTPUT AND PROBLEMS AND PITFALLS 8-1 8.1 Debug Output Description 8-1 8.2 Flame Solution, Problems and Pitfalls 8-5 8.3 Well-Stirred, Plug-Flow and Time-Evolution Chemical Kinetic Solution Problems and Pitfalls 8-7 9 PROGRAM FORTRAN VARIABLES LIST AND DEFINITIONS 9-1 10 PROGRAM AND SUBROUTINES 10-1 10.1 ACEF Main Program 10-1 10.2 READIN Subroutine 10-2 10.3 OBTAIN Subroutine 10-2 10.4 GETDAT Subroutine 10-2 10.5 KINKIN Subroutine 10-2 10.6 EPROP Subroutine 10-2 10.7 FLAME Subroutine 10-3 10.8 RDFLX Subroutine 10-4 10.9 RERAY Subroutine 10-4 11 PROGRAM SOURCE DECK LISTING AND SYSTEM REQUIREMENTS 11-1 12 REFERENCES 12-1 Appendix A Brief Development and Demonstration of Bifur- cation Approximations to Diffusion- Coefficients . . A-l iv ------- LIST OF FIGURES Figure Page 1 Schematic of Unconfined Flame 2-2 2 Schematic of Confined Flame 2-3 3 Schematic of Well-Stirred and Plug-Flow Reactor Problems . . 2-5 4 Planck Mean Absorption Coefficients at One Atmosphere Total Pressure 3-10 5 Schematic of Flameholder Processes 3-15 ------- LIST OF SYMBOLS A cross sectional area B , monochromatic radiation intensity C. J factor constant coefficient \J C specific heat C weighted specific heat defined by Equation (25) C circumference of bounding tube W CV denotes control volume D-. binary diffusion coefficient ' J H diffusion constant defined by Equation (6) E activation energy for kinetic reaction F.. diffusion factor of species i h enthalpy of bulk gas i denotes species when used as subcript J J factor in wall transport expression J. flux of species i in axial direction ------- LIST OF SYMBOLS (Continued) Jw flux of species i at bounding tube wall wi k thermal conductivity of bulk gas spectral absorption coefficient K Planck mean absorption coefficient K equilibrium constant for reaction m pm L characteristic dimension of flame for radiation properties m mass rate of gas M molecular weight p pressure q axial heat transport q.. heat transport at the bounding tube wall Inf Q volumetric heat loss qn radiative flux r, radius of bounding tube W R gas constant S distance along flame axis S_ Schmidt number \f vii ------- LIST OF SYMBOLS (Continued) 3" normalized distance defined by Equation (33) T temperature V volume of well-stirred reactor or radiation integration volume W.. chemical production rate of species i X.j mole fraction of species i Y.. mass fraction of species i o. species concentrations in moles per gram 6 flame zone length scale e emissivity of gas »7 vj third body efficiency of species i in reaction m m ij integral expression which depends upon the particular inter- molecular potential function which is utilized p density a Stefan-Boltzmann constant aref collision cross section for reference species ------- LIST OF SYMBOLS (Concluded) Superscripts P reaction products R reaction reactants IX ------- SECTION 1 INTRODUCTION This User's Manual fully describes and provides instructions for using the Premixed One-dimensional Flame (PROF) computer code * The PROF code can be used to predict the detailed chemical kinetic combustion and/or pollutant formation events which occur in a wide variety of experimental and practical combustion devices. Both steady, free and confined premixed flames, where gaseous diffusion is important, can be treated by the code. Also, well-stirred reactor, plug-flow reactor, and fixed mass time-evolution chemical kinetic problems, where diffusion is not explicitly treated, can be modeled by the code. References 1 and 2, as well as Section 7 of this report, present some examples of problems which can be treated by the PROF code. The key program element in the PROF code is a stable and reliable kinetic chemistry routine. This routine can be applied to any chemical system for which kinetic reaction data is available. To model flame and reactor-type problems, appropriate driver routines are linked to the general chemistry routine. The flame model includes axial gas phase diffusion and is, mathematically, a multivariable boundary value problem. This problem requires a coupled grid solution procedure for all variables. This grid problem is solved in PROF by using a predictor - linearized corrector itera- tive matrix procedure. The reactor type models do not have explicit diffu- sion terms. These models are initial value problems solved by simple time or space marching in the PROF code. * Program is coded in Fortran V. ------- The PROF code was developed to accurately model the detailed combustion and pollutant formation processes occurring in premixed one-dimensional flames. Previous plug-flow models applied to premixed flame combustion and pollutant formation processes did not incorporate axial diffusion in the formulation. Since ignition processes require upstream diffusion, these plug-flow models could not be directly applied to flames without making some gross assumptions as to the upstream ignition zone starting conditions. In addition, the accuracy of these nondiffusive models is very poor in the flame zone, where diffusion is important. Since the PROF code Includes axial diffusion, predictions of combustion and pollutant formation processes can be achieved in the flame zone as well as downstream of this zone. The accuracy of these predictions is dependent only on the adequacy of elementary kinetic reaction and transport data. Thus, PROF predictions, combined with experimental data, can provide valuable insights into the complex chemical events taking place within as well as downstream of the flame zone. Although the PROF code was developed to treat, primarily, the premixed one-dimensional flame problem, the PROF code formulation has been expanded to include reactor-type problems which do not explicitly include axial diffusion, This single code can now be applied to a wide variety of practical and experi- mental combustion problems. A brief description of the problems that can be treated by the PROF code appears in Section 2. Section 3 presents the mathematical models and boundary conditions for the flame and reactor type problems. Appropriate solution procedures for the mathematical models are given in Section 4. A full set of code input instructions are given in Section 5, and the standard output format is described in Section 6. Listings of input data and selected sections of output for several sample problems are given in 1-2 ------- Section 7. Descriptions of debug output and a short discussion of potential problem areas appears in Section 8. Code Fortran variables and definitions are given in Section 9. Summary descriptions of the program and subroutines are presented in Section 10. A complete listing of the program appears in Section 11. 1-3 ------- SECTION 2 PROBLEM DESCRIPTION The PROF code is applied primarily to premixed one-dimensional lami- nar, laterally unconfined, flame problems. Figure 1 illustrates schemati- cally an unconfined flame which is anchored to a flat flame burner by a flameholder. The flameholder provides a sink for the flame's "excess" heat and reactive species. The PROF code models axial diffusion of heat, q, and species, Jn-, as well as chemical kinetics within the flame. The calculations can also include radiative heat loss to the surroundings and heat and reactive species losses to the flameholder. Very general chemical systems can be treated by the code. Molecular heat and mass transport modeling incorporate accurate approximations for unequal species diffusion and nonunity Lewis number effects. The PROF code unconfined flame option can be applied to a number of experimental and practical flame problems. For example, it can be used to analyze coupled combustion and pollutant formation processes, such as fuel- nitrogen conversion, in experimental flat flames. Also, practical problems such as determining premixed flame speeds and flamability limits of coal or oil-derived gaseous fuel mixtures can be solved with this option. To augment the PROF unconfined flame predictive capability, models for the thermal and chemical effect of confining walls on bulk gas properties are included in the code. Figure 2 illustrates schematically the confined flame problem treated by the PROF code. In this case both axial and radial heat and mass transport are considered. Transfer coefficients are used to 2-1 ------- Post flame zone Luminous zone qf Jf1 Preflame zone flameholder Figure 1. Schematic of unconfined flame. 2-2 ------- ro co Post-flame zone Luminous zone Preflame zone ! f! f L VI r m Figure 2. Schematic of confined flame. ------- model heat and mass tranfer to the confining walls in terms of bulk and wall gas conditions so that the efficient one-dimensional calculation scheme can be applied to this problem. The details of radial variation of composi- tion and chemical reaction are lost in this method, but this approach is useful to assess the impact of wall effects on bulk gas properties. For the confined flame problem, the state of the gas at the tube wall is assigned as a boundary condition along with the initial bulk species fluxes and temperatures. The wall gases may be in equilibrium or any arbi- trary kinetic state. Given the boundary conditions, the code determines bulk gas properties as a function of distance along the tube axis. A variety of experimental and practical combustion and pollutant formation problems can be treated by this option. Some examples are tubular reactors, surface combustors, and catalytic combustors. Besides flame problems, where axial gas phase diffusion is important, the PROF code can also treat fully-mixed, well-stirred reactor and one- dimensional nondiffusive plug-flow reactor and time-evolution chemical kinetic problems. Figure 3 illustrates schematically some possible appli- cations of the well-stirred reactor and plug-flow reactor PROF code options. The schematic of the jet-stirred combustor illustrates an application of the well-stirred reactor option. In this problem, a specified mass of reactants and products flows continuously through a fixed-volume chamber. Intense mixing of the reactants occurs within the combustor and chemical reaction takes place uniformly throughout the chamber. Depending on the gas and wall temperatures, heat may or may not be lost through the walls 2-4 ------- m water A-16159 water WATER-COOLED SAMPLING PROBE (Plug-Flow Reactor) m JET-STIRRED REACTOR (Well-Stirred Reactor) Figure 3. Schematic of well-stirred and plug-flow reactor problems. 2-5 ------- of the chamber. In this well-stirred reactor case a steady-state fixed volume of fluid is reacting with a mean residence time defined by the con- stant mass flowrate through the reactor. The schematic of a water-cooled sampling probe in Figure 3 illus- trates an application of the plug-flow reactor option. In this problem, a chemically reacting sample is introduced into a tube whose walls are cooled much below the entering gas temperature. The cooling effect of the wall quenches the gas phase reactions, allowing the sample to pass through the probe unaltered by these reactions. Besides cooling the gases, the wall might also provide sites for heterogeneous gas phase reactions which will alter the sample composition. Only radial diffusive heat and mass transfer are considered in this option. Besides the problems illustrated in Figure 2, the plug and well- stirred reactor options can also be applied to a variety of gas turbine, furnace and catalytic combustion and pollutant formation problems. The PROF code can also treat the reaction of a fixed mass of gas in time as the pressure and temperature change. Chemical evolution inside internal com- bustion engines, combustion bombs and other time-dependent combustion systems can be predicted by this option. Of course the option assumes uniformly mixed and reacting mixtures within the system. Therefore, apply- ing this option to spatially nonuniform systems represents only an approxi- mate modeling of the system. The following section describes the assumptions, equations and solu- tion procedures utilized in the PROF code. The solution procedure has been optimized for predicting premixed one-dimensional flames. The following 2-6 ------- discussion will emphasize the development of the flame equations. Well- stirred and plug-flow reactor equations are subsets of the flame equations and the reduction of flame equations to these cases will be only briefly discussed. 2-7 ------- SECTION 3 MATHEMATICAL MODELS In this section the flame model is discussed first. This is followed by discussions of the plug-flow reactor, time-evolution chemical kinetic and well-stirred reactor models. 3.1 Flame Model The flame option governing equations are developed by integrating the steady two-dimensional species, mass and energy equations across a plane perpendicular to the axis. This results in a set of one-dimensional flame equations in terms of bulk gas properties. These properties vary along the flame axis as a result of fluxes of heat and mass at the edge of the flame and chemical reaction and axial diffusive fluxes within the bulk gases. For a large diameter/height ratio free flame with an initially uniform flow, the edge fluxes are relatively small and the bulk properties are equivalent to local conditions across the flame. For a flame confined in a tube (e.g., catalytic combustor) the edge fluxes can be significant. In this case, the edge fluxes are approximately treated by a transfer coefficient approach where the fluxes are assumed to be directly proportional to bulk and wall gas states. Using transfer coefficients permits one-dimensional solution procedures to be applied to an essentially two-dimensional problem. The quasi-one-dimensional flame conservation equations can be written as: Species dY. d(AJ,) *-ars "1 --TIT-- V (1) 3-1 ------- Energy i where global continuity has been incorporated into the above equations. The momentum equation has been replaced by assigning pressure. Assigning a fixed pressure is a valid approximation for the low-velocity flame cases of primary interest. The species conservation expression, Equation (1), shows that the species mass fraction along the flame axis is altered by species bulk gas phase chemical production, W. , axial diffusive flux, J., and wall diffusive flux, J.. . In the energy conservation expression, W1 Equation (2), the enthalpy along the flame axis is altered by the bulk gas heat loss rate, Q, axial diffusive heat flux,S J-h. + k dT/ds, and wall i 1 ' heat loss, q . Expressions for W^ , J., Jw>, Q, K and q , along with boundary conditions, complete the definition of the flame problem. 3.1.1 Species Axial Diffusional Flux Utilizing binary coefficient data, the rigorous expression for species multi component diffusional flux can be obtained by implicitly solving the Stefan-Maxwell relations. This involves a level of effort which, in most practical flame cases, is beyond that consistent with the quality of the transport data input into the code. For the PROF code the simple, yet adequate, bifurcation approximation for the binary diffusion coefficients is use'd with the Stefan-Maxwell relations to develop an explicit expression for species flux. The approximation, which is fully developed in Reference 3 and summarized in Appendix A,assumes that the contributions of species i and j to the diffusion coefficients, V..t can be separated in the following manner: • J 3-2 ------- where I) is a reference self-diffusion coefficient and F. and F. are diffu- sion factors. For binary and ternary systems, Equation (3) is exact. The pressure and temperature dependence of V^ is absorbed into U so that F. and Fj are independent of temperature and pressure. For more complex J systems, equation (3) may be considered as a good correlating function of diffusion data. The primary advantage of this approximation is that it permits an explicit formulation for diffusional mass flux J. in terms of gradients of species i to be developed from the Stefan-Maxwell relations. Neglecting thermal diffusion effects, the flux can be written in the follow- ing form (see Appendix A): rar -\M7\dT **^n (*) where F and where X1 and Yi are the mole and mass fractions, respectively. Computations that consider this formulation are considerably more efficient than those that use arbitrary expressions for the Pi . and apply either the Stefan-Maxwell implicit equations or develop the concentration dependent multi component • diffusion coefficients. 3-3 ------- In Equation (4), D is typically taken as the self-diffusion coeffi- cient of some reference species, e.g., (L. The F^ are taken as constants independent of temperature and pressure. From Reference 4: , T(T/M^y' ? D = 2.628 x io"J —2—ej\ ^ (cmVsec) (6) Pa ref"ij' o with T in °K, P in atmospheres, and collision cross section, a in A. For o $2 as tne reference species, a is equal to 3.467 A. From Appendix A the integral expression for transport properties is approximated by: o(5'1)*«1.07 [T/(E/k)]-°-159 (7) • J where the maximum energy of attraction function, e/k, for Og is 106.7 and thus: D = 0.172 x 10'4 T1'659/? (cm2/sec) (8) = 5 T1.659 3-4 ------- Introducing Equation (8) into Equation (4) gives the axial species diffusional flux expression: J| - - TTTF1""" br + hH(£ - F, ^)) (9) 3.1.2 Species Production Terms Each W. in the species conservation equation represents a summation of contributions from all reactions which include that species. For reac- tions of the type: ZR D BI where y are the stoichiometric coefficients on species B^ . W. can be written as: where Ey? In pi EuJ In p^ln K m flfl • H ._. Pi m and vl denotes third body efficiency of species i. The forward reaction 1m rate coefficient, k^ , has the Arrhenius form: m kf =aT"e-(E/RT' m 3-5 ------- The backward reaction rate coefficient is taken to be equal to the forward reaction rate coefficient divided by the equilibrium constant. As discussed in detail in Reference 5 the solution of these equations is subject to several pitfalls which need to be carefully circumvented. The most discussed problem is associated with k^r becoming very large, forcing m the bracketed term in Equation (11) to zero to maintain a finite net contri- bution of the term to the species conservation equation. Forcing the bracketed term toward zero is equivalent to imposing equilibrium. Because the term appears in several species equations, it is necessary to combine these equations in such a way that this equilibrium relation effectively replaces only one mass balance relation. Even with the rearrangement of equations, the iterative Newton-Raphson procedure used to solve the simultaneous equation set can be defeated if a significant change has occurred in the character of the equations. In the PROF chemical kinetics package a combination of gradual recharacterization of equations and damping has proved highly successful in determining a solution. This is very important if an efficient grid-type coupled diffusion/ kinetics boundary value solution is to be achieved. 3.1.3 Species Flux at the Wall For a flame situated in a bounding tube with reactive walls, a net species flux to the wall develops which is rigorously a function of the detailed radial distribution of species concentration at the wall. For efficient computations, the PROF formulation utilizes a transfer coefficient approach to establish the species flux at the wall. In this approach, the species flux is assumed to be directly proportional to the difference between the bulk and wall gas composition. Since it is not necessary to 3-6 ------- know the detailed radial species concentration distribution, this formula- tion allows an efficient axial -direction, one-dimensional computational scheme to be retained. For consistency, the wall flux proportionality factor is developed in a manner similar to the axial species diffusion case. Analogous to the axial diffusion expression, the wall species flux is written as: Wi - pu [0] • se- =-L (v. - y (13) with the J factor being represented by a simple Reynold's number relation, C, • Re"m. The Schmidt number, u/pU, adjusts the J factor to a transfer J coefficient appropriate to the reference species; the y-jF^ corrects it further for the specific species in question. Consistent with the accuracy of typical J-factor correlations (see Reference 6) the molecular weight and ]iy gradients (as in Equation 9) have been neglected in the driving potential formulation. 3.1.4 Bulk Gas Volumetric Heat Loss The term Q in the energy equation is a bulk gas phase volumetric heat source, or sink term. If desired, Q can be assigned as a constant or as a function of distance along the flame axis. However, most often Q, for uncon- fined flames, is the volumetric heat loss due to radiation. The PROF code formulation includes logic to model heat loss due to radiation for nonsooting f 1 ames . The radiation heat loss, is written as: /QA ds = -/($ • q) dV (14) 3-7 ------- where V • q is the divergence of the radiative flux. For conditions where the flame radius is small compared to all absorption coefficients at all frequencies (i.e., /fw 1C dr « 1 for all v), and where radiation back from O the environment can be neglected, Qr becomes (Reference 7): Qr = -4KpaT4 (15) where 1C is the Planck mean absorption coefficient given by: This is the emission-dominated or optically thin limit approximation which is valid for typical nonsooting laboratory flames. To complete the defini- tion of radiative heat loss, K values are now defined. The major radiating species in hydrocarbon flames are C(L, 1^0, and CO. For each of these species, there are roughly one to four important radiation bands scattered through the 1 to 15 micron wavelength range of interest.* Each band typically occupies a relatively narrow region of the spectrum and can be separated from adjacent bands by a region which neither absorbs nor emits thermal radiation. The properties of the band systems are required to properly calculate emission losses. Rigorously, the spectral absorption coefficient, K^, is available only for the individual line spectra and applying this property requires much effort. Therefore, its usefulness is limited in applications requiring Approximately 2 to 20 percent of the energy released in combustion becomes infrared radiation, whereas only 0.4 percent becomes visible or ultraviolet radiation (Reference 8). Therefore, only infrared radiation need be considered. 3-8 ------- just heat loss from combustion products. An alternate (and simpler) approach is to use a wide band absorption model. In the PROF code the wide band model correlation parameters of Edwards and Balakrishnan (Reference 9) are utilized to determine the emissivity of COgj H^O, and CO. The Planck mean absorption coefficient is determined from the gas emissivity by using the optically thin limit expression (Reference 10): - 2 L where e is the gas emissivity and L is the characteristic dimension of the system. Calculated Planck mean absorption coefficients for temperatures between 400°K and 2600°K are given in Figure 4 for the pure gaseous com- ponents COp. 1^0 and CO at a total pressure of 1 atm. For all cases, the path length, L, is taken as 0.001 feet, a number which is small enough to ensure recovery of optically thin data. As shown, COg is a strong radiator at high temperatures compared to 1^0 and CO. The emissivities, from which the K values were obtained, have been compared to available data (Refer- ence 11). For an assumed pressure-pathlength product of 0.01 atm-foot at a total pressure of 1 atm, the calculated results were within 10 percent of the reported data. This agreement lends confidence to the use of the IC's given in Figure 4. To establish 1C for a mixture of gases at any pressure, a simple summation scheme was derived which considers radiatively noninteracting gases. This expression has the form: KP - 3-9 ------- 16 » 14 12 I 4-> C X 10 • at o o o I/) .O IB C 10 Q) O C |