Solution of the MESH Equations

Another implementation of homotopy-continuation methods is the use of problem-dependent homotopies that exploit some physical aspect of the problem. Vickeiy and Taylor [AIChE J., 32, 547 (1986)] utilized thermodynamic homotopies for K values and enthalpies to gradually move these properties from ideal to ac tual values so as to solve the MESH equations when veiy nonideal hquid solutions were involved. Taylor, Wayburn, and Vickeiy [I. Chem. E. Symp. Sen No. 104, B305 (1987)] used a pseudo-Murphree efficiency homotopy to move the solution of the MESH equations from a low efficiency, where httle separation occurs, to a higher and more reasonable efficiency. 

Solution of the MESH Equations We may identify several classes of methods of solving the equilihrium-stAge model equations ... 

In the category of physical continuation methods are the thermodynamic homotopies of Vickery and Taylor [AIChE J., 32, 547 (1986)] and a related method due to Frantz and Van Brunt (AIChE National Meeting, Miami Beach, 1986). Thermodynamic continuation has also been used to find azeotropes in multicomponent systems by Fid-kowski et al. [Comput. Chem. Engng., 17, 1141 (1993)]. Parametric continuation methods may be considered to be physical continuation methods. The reflux ratio or bottoms flow rate has been used in parametric solutions of the MESH equations [Jelinek et al., Chem. Eng. Sd.,28, 1555(1973)]. 

Thus, there is an equation for each component for each stage. This set of equations may be solved simultaneously by matrix methods, there being a tiidiagonal matrix, Nhy N (N + by N + I including the leboiler or iV + 2 by /V + 2 if there is also a partial condenser), shown in Fig. S.3-13. This approach to the simultaneous (not stagewise) solution of the MESH equations was introduced by Amundson and Pontinen. ... 

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