Steady split boundary problem

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. 

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. 

Some models, however, take the form of second-order differential 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. After comparison with the second boundary condition a new starting point is estimated, followed by re-integration. This procedure is then repeated until convergence is achieved. MADONNA provides such a method. Examples of the steady-state split-boundary type of solution are shown by the simulation examples ROD and ENZSPLIT. 

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... 

Note that the initial steady state problem is a split boundary value 7i problem and superposition technique is used. 

The problem (91) is splitted into a steady state term s [R] and transient term 5 [R, t]. Since both boundary conditions (92 b, 92 c) are of the second kind, the average temperature has to be included into the splitting formula [20] ... 

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