The mathematical problem and boundary conditions

The unknown dependent variables of the problem become p, u, Z, T, and Tfc. Equations (86) and (110) provide two algebraic equations relating these variables the other N + 2 variables are determined by the N + I differential equations given by equations (95) and (105) and by the integro-differential equation given in equation (92). The initial values Pq, Zq, Tq, and 0 arc controlled by the experimenter. Attention will be restricted to lean mixtures, whence Z 1 as x - oo for chemical equilibrium to exist at the hot boundary. Since all the differential equations are of the first order, the additional boundary condition suggests that solutions will exist only for particular values of a parameter, which physically is expected to be the burning velocity Uq.  

The number of differential equations can be decreased if the Lewis number is unity. Equations (95) and (105) are so similar in form that when A/Cp = pj D, it is possible to solve for as a function of T. In a manner similar to that in which equation (5-24) was derived, it may be shown [73] that 

Further simplifications arise through equation (92) for certain size distributions G(,(r) for uniform size distributions, equation (92) becomes an ordinary differential equation. When 

The two remaining differential equations are equations (105) and (113) the other dependent variables are determined by the algebraic expressions given in equations (86), (110), and (111). Since the spatial coordinate X appears in none of these equations except as djdx in equations (105) and (113), X is easily eliminated from the problem to obtain a single differential equation, 

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