Non-equilibrium performance

Perhaps one of the clearest expositions on this problem is that given by Westenberg and Favin (8), and their approach is the one followed here. 

The characteristic nozzle flow assumptions are made, i.e. the flow is laminar, steady, one-dimensional and there are no dissipative or external forces of any kind. The reacting gas is considered to be composed of fx chemical species, each of which is present at a concentration xt (moles per unit mass of mixture). The usual flow variable temperature T, density p, pressure P and velocity u then make a total of jx + 4 variables. The cross-sectional area ratio e is generally specified as a function of the axial distance z and the axial distance z along the flow direction becomes the independent variable. A mass flow rate W per unit reference 

There are p + 4 unknowns however, there are 4 flow equations as listed above arid xv element conservation equations. Just as in the solution of the equilibrium flame temperature problem discussed in section II. B. 5., M - a additional equations are required. Except instead of using the equilibrium equations, one must adopt the chemical kinetic rate equations. The form used with the present problem is  

The first four equations are reduced by 2 by eliminating P and u as variables. Westenberg and Favin make use of the fact that for ideal gases the enthalpy HA is a function only of T so that  

Thus for numerical solution, the equations are the (at) equations n. C. 9., the (x-at) equations n. C. 10., n. C. 11. andH. C. 12. for the p + 2 variables T, P, and pX1. With all quantities known at some starting point z = 0, a computing machine can be programmed to calculate the derivatives in equations n. C. 10-12. Various machine integration routines are then available to solve simultaneous, first order differential equations. Such routines should have a variable step-wise feature for automatically doubling or halving the internal to satisfy a chosen precision index. 

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