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Show correct final temperature. This technique, usually employed to examine chemical kinetics processes in time-varying systems [3J, integrates differential equations for the concentration of each chemical species as they change in time. The rate of each reaction is a function of the temperature, total density, and concentrations of other species that participate in reactions. This, however, is not a competitive computational technique for computing equilibrium states, since it requires considerably more computer time and more complex numerical methods for its implementation than the free energy minimization methods. However, at equilibrium the rates of the forward and reverse reactions are equal, so it is possible to replace the differential equations for the time rate of change of each species concentration with a set of simple algebraic equations for those species concentrations. This makes the equations much easier to solve, and by using equilibrium reactions the formal conservation of each type of atom is made automatic. This approach has been used to calculate equilibria in large multidimensional combustion programs [4J. The primary thermochemical data used by these programs are the heats of formation and the specific heats of each of the chemical species involved. The specific enthalpy HO for a given species is generally expressed as a function of temperature through the specific heat Cp T W(T) W(Tref) + SCpdT I,-.z.\- where Tref is conventionally 298K or room temperature. For virtually every chemical species of interest to combustion, the specific heat is a complicated function of temperature itself. A common procedure, particularly when only a rough estimate of the final state is needed, is to assume that the specific heat is constant over the temperature range of interest, so the above integral can be replaced by the ex~ession S Cp dT Cp (T - Tref). T .. :a.~ Again, this approach can result in errors of the order of lOOK in the product temperatures. The ' most useful compilations of heats of formation and specific heats as functions of temperature are the JANAF tables [5J, which have been assembled and revised over the past 25 years. Other sources of thermochemical data useful for combustion computations have been reviewed recently by Burcat [6J. For use in computer codes, these data are generally fit to polynomial expressions in temperature, so that the repeated integrations of specific heat over temperature can be carried out very efficiently. Errors introduced into the product temperatures by this fitting procedure are negligibly small in most cases. 145 Of course, the entire approach is only as good as the basic thermodynamic data being used. For the small molecules treated by the JANAF tables, the most recent tables are quite reliable, although earlier JANAF versions for some species have changed considerably as more experimental analysis has been done. For larger molecules and for many radical species, specific heats and heats of formation are not as well known and uncertainties in equilibrium product distributions and temperatures are certainly possible. For organizations and individuals who wish to carry out the types of calculations already described (and illustrated below), many of the best computer programs for equilbrium conditions are available to the general public. The original program of Gordon and McBride [lJ from NASA and the program developed by Reynolds [2J at Stanford can be purchased and can even be run on personal computers, while an adaptation of Reynolds' program has been integrated into the widely used CHEMKIN [7J family of combustion modeling programs developed by Sandia Laboratories. Those interested in obtaining such programs should contact the authors of these codes for further information. To summarize this section of the topic, computations of chemical equilibrium require two major elements. The first of these is a numerical model which can solve the relevant equilibrium equations. Such a code can solve time-dependent kinetics equations, algebraic equilibrium relations, or use a free energy minimization technique. In principle, any method should arrive at the same solution, although the computer time and cost requirements of each technique can be widely different, and the most commonly used technique is that of free energy minimization. The second major element is the basic thermodynamic data set which is used by the computer model, and all common models use basically the same thermodynamic data set. It is quite important to include in the equilibrium computation all chemical species which may be present at equilibrium above a certain threshold level, since omission of species such as CO and H2 can result in significant errors in computed equilibrium temperatures. SOME EXAMPLES Let us consider first a reference or baseline set of initial conditions to illustrate the types of information which can be obtained by thermodynamic equilibrium models. This is a mixture of natural gas and air, in which the fuel is defined to consist of 90% CH4, 5% C2H6~ with a 5% impurity level of N2. The air is a mixture of oxygen and nitrogen, with N2/02 = 3.78. All of the gases are assumed to be initially at atmospheric pressure and 60°F. Using Reynolds' |