Solution of Split Boundary—Value Problems

Even with this simplification, we still must solve a split boundary value problem. A particularly convenient method of solution appears to be a "shooting" technique in which the... 

This set of hyperbolic partial differential equations for the gasifier dynamic model represents an open or split boundary-value problem. Starting with the initial conditions within the reactor, we can use some type of marching procedure to solve the equations directly and to move the solution forward in time based on the specified boundary conditions for the inlet gas and inlet solids streams. 

Outside the limited case of a first-order reaction, a numerical solution of the equation is required, and because this is a split-boundary-value problem, an iterative technique is required. 

The superposition principle states that for linear differential equations, a split boundary-value problem can be solved as a linear combination of initial value solutions. 

Therefore, from the principle of superposition, equation (7.3.15), the solution to the linear split boundary-value problem is known. 

It is seen that the calculation of the global rates involves only the solution of Eq. 4.125, from which Ac, Be, Pa(L), and Pb(L) can be directly calculated with the aid of Eqs. 4.131 and 4.132. The original split boundary value problem has been transformed into an initial value problem. 

Some situations, however result in the form of second-order diffferential equations, which often give rise to problems of the split boundary type. In order to solve this type of problem, an iterative method of solution is required, in which an unknown condition at the starting point is guessed, the differential equation integrated twice and the resulting solution compared with a known boundary condition, obtained at the end point of the calculation. Any error between the known value and the calculated value can then be used to revise the initial starting guess for the next iteration. This procedure is then repeated until... 

The dimensionless model equations are used in the program. Since only two boundary conditions are known, i.e., S at X = l and dS /dX at X = 0, the problem is of a split-boundary type and therefore requires a trial and error method of solution. Since the gradients are symmetrical, as shown in Fig. 1, only one-half of the slab must be considered. Integration begins at the center, where X = 0 and dS /dX = 0, and proceeds to the outside, where X = l and S = 1. This value should be reached at the end of the integration by adjusting the value of Sguess at X=0 with a slider. 

Equations (6.45-6.47) together with the mass balance equations (6.26, 6.28) must be solved to obtain the optimal temperature profile. Equations (6.28, 6.29, 6.45, 6.46) represent a set of two-point boundary value differential equations. The boundary conditions are split, those for equations (6.28, 6.29) defined at z = 0.0 and those for equations (6.45, 6.46) defined at z = 1.0. This set of two-point boundary value differential equations can be solved for any temperature profile y(z ) however from this family of solutions, that which satisfies the condition (6.47) at every z is the optimal solution. There are many algorithms in the literature for solving this problem... 

So far, only a single reaction has been considered. While the reactor point effectiveness cannot be expressed explicitly for a reversible reaction, the internal effectiveness factor can readily be obtained analytically using the generalized modulus (see Problem 4.23). For complex multiple reactions, however, it is not possible to obtain analytical expressions for the global rates and one has to solve the conservation equations numerically. The numerical solution of nonlinear, coupled diffusion equations with split boundary conditions is by no means trivial and often presents convergence difficulties. In this section, the same approach is taken as was used for the reactor point effectiveness. This enables the global rates to be obtained in a straightforward manner and the diffusion equations to be solved as an initial value problem (Akella 1983). 

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