The BP and SR tearing methods are generally successful only when applied respectively to the distillation of mixtures having a narrow boiling range and to absorbers and strippers. Furthermore, as shown earlier, specifications for these two tearing methods are very restricted. If one wishes to treat distillation of wide-boiling mixtures and other operations shown in Fig. 13-7 such as rectification, reboiled stripping, reboiled absorption, and refluxed stripping, it is usually necessary to utilize other procedures. One class of such procedures involves the solution of most or all of the MESH equations or their equivalent simultaneously by some iterative technique such as a Newton or a quasi-Newton method. Such simultaneous-correction (SC) methods are also useful for separations involving very nonideal liquid mixtures including extractive and azeotropic distillation or for cases in which considerable flexibility in specifications is desired.

The development of an SC procedure involves a number of important decisions: (1) What variables should be used? (2) What equations should be used? (3) How should variables be ordered? (4) How should equations be ordered? (5) How should flexibility in specifications be provided? (6) Which derivatives of physical properties should be retained? (7) How should equations be linearized? (8) If Newton or quasi-Newton linearization techniques are employed, how should the Jacobian be updated? (9) Should corrections to unknowns that are computed at each iteration be modified to dampen or accelerate the solution or be kept within certain bounds? (10) What convergence criterion should be applied?

Perhaps because of these many decisions, a large number of SC procedures have been published. Two quite different procedures that have achieved a significant degree of utilization in solving practical problems include the methods of Naphtali and Sandholm [Am. Inst. Chem. Eng. J., 17,148 (1971)] and Goldstein and Stanfield [Ind. Eng. Chem. Process Des. Dev., 9, 78 (1970)]. The former procedure is of particular interest because, in principle, it can be applied to all cases. However, for situations involving large numbers of components, relatively small numbers of stages, and liquid solutions that are not too highly nonideal, the latter procedure is more efficient computationally.

Naphtali-Sandholm SC Method This method employs the equilibrium-stage model of Figs. 13-48 and 13-49 but reduces the number of variables by 2N so that only N(2C + 1) equations in a like number of unknowns must be solved. In place of Vj, Lj, x.j, and ytjj, component flow rates are used according to their definitions:

In addition, sidestream flow rates are replaced with sidestream flow ratios by sj = Uj /Lj (13-91)

The MESH equations (13-68) to (13-72) then become the MEH functions:

Mi.j = (tj(1 + sj) + vtj (1 + Sj) _ £,.j _1 - vtj+1 -ftj = 0 (13-93) where f.j = FjZtj

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