Split boundary value problems

Referring to Fig. 4.15, it is seen that the concentration and the concentration gradient are unknown at Z = 0. The above boundary condition relation indicates that if one is known, the other can be calculated. The condition of zero gradient at the outlet (Z=L) does not help to start the integration at Z=0, because, as Fig. 4.17 shows, two initial conditions are necessary. The procedure to solve this split-boundary value problem is therefore as follows ... 

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. 

The continuity equations for mass and energy were used to derive an adiabatic dynamic plug flow simulation model for a moving bed coal gasifier. The resulting set of hyperbolic partial differential equations represented a split boundary-value problem. The inherent numerical stiffness of the coupled gas-solids equations was handled by removing the time derivative from the gas stream equations. This converted the dynamic model to a set of partial differential equations for the solids stream coupled to a set of ordinary differential equations for the gas stream. 

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. 

X = 0, Trx = Trx, inlet) are available at z = 0. This is known classically as a split boundary value problem, and it is characteristic of countercurrent flow heat exchangers. When numerical methods are required to integrate coupled mass and thermal energy balances subjected to split boundary conditions, it is necessary to do the following ... 

Even though it is possible to convince software packages that second-order ODEs can be solved using techniques for first-order ODEs, all numerical methods require that both boundary condition for a second-order ODE must be known at the starting point. In other words, both boundary conditions must be known at the same value of the spatial coordinate. Split boundary value problems do not conform to this requirement. The mass balance for diffusion and chemical reaction is typically classified as a split boundary value problem. 

In the developments just presented, cocurrent plug-flow was assumed in both the tube and shell sides of the reactor. It would be instructive to analyze the effect of countercurrent flow, as well as different combinations of plug and mixed flow on the two sides of the membrane. Countercurrent flow can be achieved merely by changing the direction of sweep gas flow. However, this results in a split boundary value problem because the conditions on the shell side, unlike those on the tube side, are specified at the outlet instead of at the inlet. Substitution of mixed flow for plug flow is straightforward because one has only to use uniform concentrations everywhere in the region. 

The situation in countercurrent case (case 4a in Figure 3.1) design and simulation is shown in Figure 3.6. In both cases we see that boundary conditions are defined at opposite ends of the integration domain. It leads to the split boundary value problem. 

In design this problem can be avoided by using the design parameters for the solid specified at the exit end. Then, by writing input-output balances over the whole dryer, inlet parameters of gas can easily be found (unless local heat losses or other distributed parameter phenomena need also be considered). However, in simulation the split boundary value problem exists and must be solved by a suitable numerical method, e.g., the shooting method. Basically... 

FIGURE 3.6 Schematic of design and simulation in cocurrent case (a) design—split boundary value problem is avoided by calculating from the overall mass balance (b) simulation—split boundary value problem cannot be avoided, broken line shows an unsuccessful iteration, solid line shows a successful iteration—with Y2 assumed the Yprofile converged to Yi. 

The second BC is due to Danckwerts and has been used for chemical reactor models. This leads, of course, to a split boundary value problem, which needs to be solved by an appropriate numerical technique. The resulting longitudinal profiles of solid moisture content and tanperature in a dryer for various Peclet numbers (Pe = u LIE) are presented in Figure 3.10. 

This chapter discusses numerical techniques for solving split boundary-value problems. Split boundary-value problems arise from the description of distributed systems in which part of the boundary information needed to solve a set of differential equations is at one boundary of the system and part at another boundary. An example would be a counter-current heat exchanger where the inlet temperatures are known at either end of the exchanger. 

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. 

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