Nonequilibrium Multistages with Nonlinear Equilibrium Relations

2 Nonequilibrium Multistages with Nonlinear Equilibrium Relations 

Next we consider the case where the equilibrium relation is not linear and the stages are not in equilibrium. Then the rate of mass transfer between the phases RMT becomes nonlinear, i.e., 

This model leads to a set of nonlinear equations that can be solved numerically by using the multi-dimensional Newton method of Section 1.2. It is this more realistic and more accurate nonlinear case which we concentrate on in this book. We now present an efficient MATLAB program to solve this nonlinear and more general case. 

We start with a modified version of the gas-phase molar-balance equation (6.99) and a modified version of the liquid-phase molar-balance equation (6.102) with a quadratic equilibrium relation F(Xj) =Yj = aXj + b. This modification affects the term RMT by a nonlinearly, namely 

Here for simplicity we assume a constant mass-transfer coefficient times area of mass transfer Ka for all trays, i.e., Kgjamj = const = Ka. 

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In this chapter we have presented multistage systems with special emphasis on absorption processes. We have studied multitray countercurrent absorption towers with equilibrium trays for both cases when the equilibrium relation is linear and when it is nonlinear. This study was accompanied by MATLAB codes that can solve either of the cases numerically. We have also introduced cases where the trays are not efficient enough to be treated as equilibrium stages. Using the rate of mass transfer RMT in this case, we have shown how the equilibrium case is the limit of the nonequilibrium cases when the rate of mass transfer becomes high. Both the linear and the nonlinear equilibrium relation were used to investigate the nonequi-librium case. We have developed MATLAB programs for the nonequilibrium cases as well. 

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