Simplified steady-state models for separation process

Abstract

A method in which mass and energy balance equations and equilibrium relationship can be simplified for the solution of separation process has been developed. It has been assumed that the variables such as temperatures, liquid mole fraction, etc. are the continuous function of distance variable, Z. Then the reduced-order model can be obtained by orthogonal collocation method. For example, the liquid mole fraction can be expressed as follows: $$X(z) = \sum^{n+2}_{k=1}\; 1_k (z) X_k$$ Where n+2 is the number of collocation points, $X_k$ is the liquid mole fraction evaluated at collocation point $Z_k$, and $1_k (z)$ is the Lagrange polynomial. The above equation can be applied to the other variables in like manner and this is the most important assumption. As shown in the above equation, our basic idea is that we can reduce the order of the equations by determining the number of collocation points regardless of the number of trays in the column with the several examples we test (1) the effect of the location of collocation points on accuracy and convergency (2) the effect of the number of collocation points on accuracy, (3) the effect of the number of trays on accuracy (4) the effect of the number of components on application of the present model. It has been found that three or four-point collocation method gives satisfactory results. Consequently the complicated equations describing separation process can be solved by reduced-order model without sacrificing accuracy. But it should be mentioned that the present simplified model exhibits some numerical problems in extreme case.