Lewis-Matheson method, multicomponent

The method proposed by Lewis and Matheson (1932) is essentially the application of the Lewis-Sorel method (Section 11.5.1) to the solution of multicomponent problems. Constant molar overflow is assumed and the material balance and equilibrium relationship equations are solved stage by stage starting at the top or bottom of the column, in the manner illustrated in Example 11.9. To define a problem for the Lewis-Matheson method the following variables must be specified, or determined from other specified variables ... 

Multicomponent distillation, 393 absorption factor method, 398 azeotropic, 420-426 bubblepoint (BP) method, 406-409 computer program references. 404 concentration profiles, 394 distribution of non-kevs. 395 Edmister method, 398,399 extractive, 412, 417-422 feed tray location, 397 free variables, number of 395 Lewis-Matheson method 404 MESH eauations. 405-407 molecular, 425-427 nomenclature, 405 number of theoretical trays, 397 packed towers, 433-439 petroleum, 411-415 reflux, minimum, 397 reflux, operating, 397 SC (simultaneous correction) method, 408-411... 

The Lewis and Matheson method appears to be the most satisfactory method of handling the general multicomponent design problem. In most cases, it requires relatively little trial and error and will handle cases of normal and abnormal vapor-liquid equilibria. It is especially well suited to the cases in which the reflux ratio and the separation of the key components are specified and the problem is to determine the number of theoretical plates and the component concentrations in the column. In some specific cases, other methods may have advantages, but unless a number of problems of the same type are to be handled, it is more desirable to have one method that will apply to essentially all cases. 

The general method of stage-by-stage calculation for multicomponent systems was first shown by Lewis and Matheson (LI) and by Underwood (Ul) in 1932. The method of Lewis and Matheson was further improved by Robinson and Gilliland (Rl), but substantially unchanged. In its most basic form, the concept of the method is simple. Consider the example cited above. If the amounts of each component in both of the products could be exactly calculated, it would only be necessary to start at one end and calculate until a stage was reached at which the composition matched that of the other product. 

The two principal tray-by-tray procedures that were performed manually are the Lewis and Matheson and Thiele and Geddes. The former started with estimates of the terminal compositions and worked plate-by-plate towards the feed tray until a match in compositions was obtained. Invariably adjustments of the amounts of the components that appeared in trace or small amounts in the end compositions had to be made until they appeared in the significant amounts of the feed zone. The method of Thiele and Geddes fixed the number of trays above and below the feed, the reflux ratio, and temperature and liquid flow rates at each tray. If the calculated terminal compositions are not satisfactory, further trials with revised conditions are performed. The twisting of temperature and flow profiles is the feature that requires most judgement. The Thiele-Geddes method in some modification or other is the basis of most current computer methods. These two forerunners of current methods of calculating multicomponent phase separations are discussed briefly with calculation flowsketches by Hines and Maddox (1985). 

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. 

In Chap. 9 the Lewis and Matheson procedure for SorePs plate-to-plate method was presented. Many other design methods have been proposed based on alternate methods of analysis or approximations. None of them illustrates the phenomena involved in multicomponent rectification so well as the Lewis and Matheson ihethod. A number of the methods require less effort to obtain certain design factors than the stepwise procedure and are useful in cases where similar systems are to be analyzed repeatedly. When a new type of problem is to be considered, the information obtained by the plate-to-plate method is well worth the effort involved. Actually a detailed analysis by methods of Chap. 9 does not usually require over a few hours, and the confidence in the result and the insight obtained of the operation justify the effort involved. 

One of the more widely used methods for calculating the number of theoretical stages in multicomponent systems was developed by Lewis and Matheson [6]. Again, the molar flow rates of vapor and liquid in each section are assumed to be constant. On each equilibrium stage, the summation of the concentrations of all vapor components must equal unity. The same is true for the summation of the concentrations of all liquid components. Further, the vapor and liquid compositions of each component are related by the K value for that component. 

The general methods of design for multicomponent distillation apply. Since in most cases all the components of the feed streams are found in the bottoms, the method of Lewis and Matheson can be used, starting at the bottom and computing to the top. There will be an optimum solvent-circulation rate small solvent rates require many trays, but the column diameters are small large ratios require fewer trays but larger column diameters and greater solvent-circulation costs. 

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