Burning-rate eigenvalue

The two-point boundary conditions for equation (42) are e = 0 at T = 0 and = 1 at T = 1. Three constants a, P and A, enter into equation (42). The first two of these constants are determined by the initial thermodynamic properties of the system, the total heat release, and the activation energy, all of which are presumed to be known. In addition to depending on known thermodynamic, kinetic, and transport properties, the third constant A depends on the mass burning velocity m, which, according to the discussion in Section 5.1, is an unknown parameter that is to be determined by the structure of the wave. Since equation (42) is a first-order equation with two boundary conditions, we may hope that a solution will exist only for a particular value of the constant A. Thus A is considered to be an eigenvalue of the nonlinear equation (42) with the boundary conditions stated above A is called the burning-rate eigenvalue. 

FIGURE 5.3. Schematic representation of the solution profiles and of the pseudostationary behavior of the burning-rate eigenvalue as a function of the nondimensional ignition temperature. 

FIGURE 5.6. Dependence of the term y in the formula for the burning-rate eigenvalue, equation (76), on the reaction order n. 

In accordance with the preceding discussion, as n oc, [ ](t) approaches the correct solution from above and Aj ] approaches the correct eigenvalue from below. From the discussion in Section 5.3.3 it is seen that for each [ j(t) a value of A may also be computed from equation (51), and that the resulting sequence of values will approach the correct eigenvalue from above. Thus successively narrower bounds for the burning rate and successively smaller upper bounds for g = dx/d are obtained. 

The physical processes in the gas-phase and subsurface regions must be matched at the interface by requiring continuities of mass and energy fluxes. This procedure eventually determines propellant surface conditions and burning rate as the eigenvalues of the problem. The interfacial boundary conditions are expressed as follows ... 

When the rate of accumulation dpjdt of all quantities in the system is zero, the condition is known as the steady state. All other systems are time dependent. Stationary flames supported on burners are steady-state phenomena, and so for a quasi-one-dimensional stationary flame d AMy)/dy = 0 and AMy = constant. In the hypothetical case of a true one-dimensional adiabatic flame the constant My is the adiabatic mass burning velocity. It is an eigenvalue solution of the physical problem, equal to the product of the density and linear velocity of the gas at any position in the flame. Thus... 

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