## L y xi K lx

which is the residual curve equation. Note that it is set for a fixed pressure.

In fact, Eq. (21) represents a group of ordinary (constant-coefficient) difference equations, and thus can be solved by such mathematical tools as Gear, Runge-Kutta (L-K) methods, etc. During the calculation, bubble point subroutine, as mentioned in chapter 1 (1.1. the equilibrium ratio), should be called as a sub-function in every dt advance so as to solve the temperature because the equilibrium ratio Kt is the function of temperature and liquid composition.

Map representation of residual curves in composition space for three-component system may be drawn in orthogonal or equilateral cartesian coordinates; or transformations can be made in the coordinate system as long as the linear appearance of straight lines remains invariant. The two types of coordinate systems will be used interchangeably throughout.

As examples, Figs. 7 and 8 respectively show the residual curves for the methanol / ethanol / water system at 101.3 kPa in the homogeneous mixture and for the ethanol / water / benzene at 101.3 kPa in the heterogeneous mixture. It can be seen that arrows may be assigned. The arrows are in the direction of time increasing (or temperature increasing) because the concentration of heavy components in the simple distillation still will become higher and higher as time goes on, which results in temperature increasing.

### 3.1.2. Operating lines

Similar as the x-y diagram for two-component system, if composition space is only used to perform the calculations, it is necessary to assume constant molar overflow. As shown in

Fig. 9, both x„ (the distillate composition) and xB (the bottom composition) are predetermined as a given separation task, and the line DFB represents a limiting overall material balance. The operating lines in Fig. 9 are composed of real lines with arrows representing VLE and dashed lines without arrows representing material balance. Real lines and dashed lines are arranged alternately. The number of arrows is equivalent to the separation stages including reboiler, but excluding condenser. This means that there are six separation stages in Fig. 9.

Fig. 10 illustrates how to draw operating lines in the homogeneous azeotropic distillation column. The operating lines are also identical to those in the heterogeneous azeotropic distillation column, but without decanter at the top.

Fig. 7. The residual curves for the methanol / ethanol / water system at 101.3 kPa in the homogeneous mixture; adapted from the reference [21].