Tray-by-Tray Distillation Calculation Method Taking Account of Operating Conditions

Abstract

Numerous tray-by-tray multicomponent distillation calculation methods^3)-5) have been published since the publication of the Lewis-Matheson method¹⁾ and Thiele-Geddes method²⁾. Of these methods, the Tridiagonal-Matrix method announced by Wang and others⁴⁾ has become the principal computerized multicomponent distillation calculation method. Many of the calculation methods based on the Tridiagonal-Matrix method have their respective characteristics in regard to independent variables, discrepancy functions and convergence methods.
The present method is also based on the Tridiagonal-Matrix method and has the two undermentioned characteristics. Meanwhile, Fig. 1 shows a general stage model and the symbols used in the present method.
The first characteristic is that, with the aim of facilitating calculations serving various purposes, two types of operating condition equations are adopted as discrepancy functions in addition to the summation equation and heat balance equation and feed rate F_j, side-cut rates W_j^L and W_j^V and heat exchanger duty Q_j are adopted as variables in addition to the temperature T_j and vapor flow rate V_j of each stage.
E_G=1/S_GG-1=0
E_G=1/S_GUΣδi=γu_i-1=0
Where, S_G: Operating condition.
G: Gross flow rate (V_j, V₁+W₁^L), reflux ratio (W_j^L/L_j, W_j^V/V_j) or heat exchanger duty (Q_j).
U: 1, gross flow rate or reflux ratio.
u_i: l_ij or V_ij.
The second characteristic is that, while obtaining simultaneous solutions of all variables by using the Newton-Raphson method similar to Tomich¹¹⁾, the present method is based on the adoption of analytical differentiation for the purpose of obtaining the differential coefficient while Tomich has adopted numerical differentiation for the same purpose.
Meanwhile, in the case of analytically calculating the differential coefficient, it is also necessary to analytically calculate the differential coefficient by various independent variables of liquid composition (∂l_ij/∂x) at each stage. Thus the present method is based on the adoption of matrix differentiation for analytically obtaining the differential coefficient. That is, by differentiating the matrix A•L=F in terms of the variable x, and by transposing the terms, the following equation can be obtained:
A∂L/∂x=∂F/∂x-L∂A∂x
Here, by applying the foregoing equation to the Tridiagonal-Matrix method, it is possible to obtain ∂l_ij/∂x with ease. An example of calculation by the present method is given in the form of a petroleum refinery debutanizer calculation. Table 1 shows the calculation conditions, and Table 2 and Fig. 2 the calculation results. The required number of iterations has been proved to be five under the convergence conditions whereby each discrepancy function value is 0.0001 or less.
The present method has already been applied to several hundreds of cases over a period of several years and has given satisfactory, stable solutions.