Solving 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. 

The complexity of multicomponent distillation calculations can be appreciated by considering a typical problem. The normal procedure is to solve the MESH equations (Section 11.3.1) stage-by-stage, from the top and bottom of the column toward the feed point. For such a calculation to be exact, the compositions obtained from both the bottom-up and top-down calculations must mesh at the feed point and match the feed composition. But the calculated compositions will depend on the compositions assumed for the top and bottom products at the commencement of the calculations. Though it is possible to... 

Rigorous method The mathematical method used to solve the MESH equations. 

In computer-based methods for solving the MESH equations, it is common to replace the energy balance of the condenser (with or without the associated stream splitter) and reboiler with a specification equation. Possible specifications include... 

A great many tearing methods for solving the MESH equations have been proposed. According to Friday and Smith (1964), tearing methods may be analyzed in the following terms ... 

The classic papers by Lewis and Matheson [Ind. Eng. Chem., 24, 496 (1932)] and Thiele and Geddes [Ind. Eng. Chem., 25, 290 (1933)] represent the first attempts at solving the MESH equations for multicomponent systems numerically (the graphical methods for binary systems discussed earlier had already been developed by Pon-chon, by Savarit, and by McCabe and Thiele). At that time the computer had yet to be invented, and since modeling a column could require hundreds, possibly thousands, of equations, it was necessary to divide the MESH equations into smaller subsets if hand calculations were to be feasible. Despite their essential simplicity and appeal, stage-to-stage calculation procedures are not used now as often as they used to be. 

Choosing Return twice places the cursor on the input/Soive command. Hitting enter , the program saves the description and attempts to solve the MESH equations. If all the data has been entered correctly, the next screen appears ... 

The class of simultaneous solution methods in which all of the model equations are solved simultaneously using Newton s method (or a modification thereof) is one class of methods for solving the MESH equations that allow the user to incorporate efficiencies that differ from unity. Simultaneous solution methods have long been used for solving equilibrium stage simulation problems (see, e.g., Whitehouse, 1964 Stainthorp and Whitehouse, 1967 Naphtali, 1965 Goldstein and Stanfield, 1970 Naphtali and Sandholm, 1971). Simultaneous solution methods are discussed at length in the textbook by Henley and Seader (1981) and by Seader (1986). 

By solving the MESH equations the required quantities (compositions in the liquid and vapor phases, temperatures, amount of Hquid and vapor flow) for every theoretical stage can be calculated. 

Modern process simulators (e.g. Aspen-Plus from AspenTech or ChemGad from Chemstations) simultaneously solve the MESH equations using algorithms based on Newton-Raphson methods (Gmehling and Brehm, 1996). However, for highly non-ideal or complex systems, modifications have been developed to enhance convergence behavior. 

Typically, the colunm is completely specified first, including the definition of the number of equilibrium stages, the feed stage, the reflux ratio, and the heat supply. By solving the MESH equations, the internal temperature and concentration profiles and, in turn, the product concentrations are found. If the calculated product qualities do not meet the specifications, then a new column configuration has to be made and the calculation repeated. 

The quality of the design of a distillation column by solving the MESH equations mainly depends on the accuracy of the K-factors (separation factors) [1]. Using one of the -models given in Table 5.6 these values can be calculated for the system to be separated, if the binary parameters are available. However, for the proper design binary parameters have to be used which describe the fC-factors resp. separation factors ctij of the system to be separated over the entire composition and temperature range considered reliably. 

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