Split-boundary problems

In order to solve the differential equations, it is first necessary to initialise the integration routine. In the case of initial value problems, this is done by specifying the conditions of all the dependent variables, y, at initial time t = 0. If, however, only some of the initial values can be specified and other constant values apply at further values of the independent variable, the problem then becomes one of a split-boundary type. Split-boundary problems are inherently more difficult than the initial value problems, and although most of the examples in the book are of the initial value type, some split-boundary problems also occur. 

With complex kinetics a steady-state split boundary problem of the type of Example ENZSPLIT may not converge satisfactorily. To overcome this, the problem may be reformulated in the more natural dynamic form. Expressed in dynamic terms, the model relations become. 

Split-boundary problems 123, 644 Spouted bed reactor 466 Stability of chemical reactors 361 Stage... 

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... 

In order to overcome the problem of split boundaries, it is sometimes preferable to formulate the model dynamically, and to obtain the steady-state solution, as a consequence of the dynamic solution, leading to the eventual steady state. This procedure is demonstrated in examples ENZPLIT and ENZDYN. 

The problem is thus one of a split boundary type, but one which can be solved by an iterative procedure based on an assumed value for one of the unknown boundary conditions. Assuming a value for dC /dZ at the initial condition Z=0, the equation can be integrated twice to produce values of dCs/dZ and Cs at the terminal condition, Z=L. If the correct value has been taken, the integration will lead to the correct boundary condition that dCs/dZ=0 at Z=L and hence the correct value of Cs- The value of the concentration gradient dCs/dZ is also obtained for all values of Z, throughout the depth of liquid. 

Instead of splitting the problem into concentration intervals and time subintervals, the problem is split into time intervals and concentration subintervals, with water demand plotted on the horizontal axis. The boundaries for time intervals and concentration subintervals are set by the process end-points. However, unlike in a case where time is taken as a primary constraints, the streams that are required or available for reuse in each concentration subinterval are plotted separately. This approach has proven to ease the analysis as will be shown later in this section. 

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 ... 

In this case, the flow rates Lm and Gm, concentrations Yin and Xin, temperatures TGin and TLin, are known and in addition the height of packing Z is also known. It is now, however, required to establish the effective column performance by determining the resulting steady-state concentration values, Yout and X0ut, and also temperature TLout. The problem is now of a split-boundary type... 

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. 

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. 

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