Model countercurrent backmixing

The balance equations for column reactors that operate in a concurrent mode as well as for semibatch reactors are mathematically described by ordinary differential equations. Basically, it is an initial value problem, which can be solved by, for example, Runge-Kutta, Adams-Moulton, or BD methods (Appendix 2). Countercurrent column reactor models result in boundary value problems, and they can be solved, for example, by orthogonal collocation [3]. The backmixed model consists of an algebraic equation system that is solved by the Newton-Raphson method (Appendix 1). 

Fryer, C. and O.E. Potter. Countercurrent Backmixing Model for Fluidized Bed Catalytic Reactors. Indust. Engng. Chem. Fundam. 11 (1972) 338-344. 

This model is described in Sec. 3.3.1.5 and consists of a countercurrent stagewise extractor cascade with backmixing in both phases. 

Five stage countercurrent extraction cascade with backmixing Equilibrium stage model... 

In a colunm acting as a direct-contact cooler, there is always a certain amount of liquid recirculation caused by the opposing gas flow. The flow pattern in a packed column is tortuous, and the use of a continuous countercurrent model again is only a working approximation. The liquid on the trays tends to be backmixed, and so the liquid flow is not continuously countercurrent but rather proceeds from one more or less well-mixed volume to another. Other calculational techniques, discussed in Section 9.1.3.3, may be more appropriate to these cases. 

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