Thiele and Geddes method

A wide variety of numerical methods have been proposed for solving the set of equations represented by Eq. (2-1). Two fundamentally different iterative procedures have been proposed for solving these equations namely the Lewis and Matheson method10 and the Thiele and Geddes method.14 In the Lewis and Matheson method, the terminal compositions Xm and xBi are taken to be the independent variables, and in the Thiele and Geddes method, the temperatures (the temperature of each stage) are taken to be the independent variables. Up until about 1963, the Lewis and Matheson choice of independent variables was used almost exclusively, and since then, the Thiele and Geddes choice of the independent variables has become the most popular. 

The Thiele and Geddes method is a stepwise procedure based on using a ratio of the concentration of a component to its terminal concentration. Starting at the top of the column, for any component. 

The section below the feed was calculated on the basis W/F 0.34 compared to the calculated value of 0.334, and it will not be rechecked. This is one of the difficulties with the Thiele and Geddes method, i.e., specifying the reflux ratio and feed condition still leaves trial and error for both the plate temperatures and the ratio of 0/V below the feed plate. 

The Thiele and Geddes method is advantageous when the number of theoretical plates and the reflux ratio are specified and the calculation of the separation is desired. Even in this case, the trial and error involved in obtaining the proper equilibrium constants for each plate is formidable. 

Thiel and Geddes method is used to calculate the number of trays. It involves the simultaneous solution of equilibrium relationships (VLE) and the operating... 

Like the Lewis-Matheson method, the original method of Thiele and Geddes (1933) was developed for manual calculation. It has subsequently been adapted by many workers for computer applications. The variables specified in the basic method, or that must be derived from other specified variables, are ... 

Iteration solutions were first proposed by Thiele and Geddes (Tl) in 1933. In this method, all temperatures and flows must be estimated before the solution can begin. The solution is broken into three parts first, solution of the mass-balance equations under the estimated flows and temperatures second, correction of the temperatures and third, correction of the flows. Assuming values for all temperatures and flows reduces the set of mass-balance equations shown in Table I to a linear set of equations which can be solved for the compositions at each point. Because the starting assumptions are completely arbitrary, the compositions will undoubtedly be wrong (the liquid and vapor mole-fractions will not sum to unity), and better values of temperature and flows must then be obtained for use in the next iteration. 

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. 

Although not widely used anymore, this method is presented because it was the forerunner for many of the more recent and more efficient methods. It is one of the earlier methods for rigorous numerical solution of distillation columns. It was first described by Thiele and Geddes (1933) and later detailed by Lyster et al. (1959). 

For problems where both light and heavy non-keys are present, Lewis and Matheson (1932) and Thiele and Geddes (1933) calculated from both ends of the column and matched conpositions at the feed stage (see Smith, 1963, Chapter 20, for details). Unfortunately, closure can be very difficult. When there are both light and heavy non-keys, and when there is a sandwich conponent, other calculation methods such as the matrix method discussed in Chapter 6 are preferable. 

In addition to the Lewis and Matheson, and Lewis and Cope methods given in Chap. 9, plate-to-plate procedures have been given by Thiele and Geddes (Ref. 14) and Hummel (Ref. 9). 

Edminster (Ref. 4) has presented a modified absorption factor method that determines the molal quantities for each component as a fraction of their values in the distillate and bottoms in a manner somewhat similar to the Thiele and Geddes equations. The geometric mean of the absorption and stripping factors at the ends of the section under consideration is employed, and empirical correction terms are applied to these averages. 

Having obtained and reference can be made to any of the several empirical correlations mentioned earlier [5, 10, 11, 17, 35] for an estimate of the number of trays at reflux ratio R, These can be unreliable, however, particularly if the majority of the trays are in the exhausting section of the tower. A relationship which is exact for binary mixtures and can be applied to multicomponents yields better results for that case [59]. The result of such an estimate can be a reasonable basis for proceeding directly to the method of Thiele and Geddes. 

In some computer applications of the method, where the assumption of constant molar overflow is not made, it is convenient to start the calculations by assuming flow and temperature profiles. The stage component compositions can then be readily determined and used to revise the profiles for the next iteration. With this modification the procedure is similar to the Thiele-Geddes method discussed in the next section. 

The basic procedure used in the Thiele-Geddes method, with examples, is described in books by Smith (1963) and Deshpande (1985). The application of the method to computers is covered in a series of articles by Lyster el al. (1959) and Holland (1963). 

The method starts with an assumption of the column temperature and flow profiles. The stage equations are then solved to determine the stage component compositions and the results used to revise the temperature profiles for subsequent trial calculations. Efficient convergence procedures have been developed for the Thiele-Geddes method. The so-called theta method , described by Lyster et al. (1959) and Holland (1963), is recommended. The Thiele-Geddes method can be used for the solution of complex distillation problems,... 

A modified Thiele-Geddes method, programmed for an IBM 370-155, was used to perform the calculations needed to size each required column. Experimental activity coefficient data were used to allow for nonideal liquid phase behavior while energy balances, using estimated enthalpy data, were used to correct for non-constant molal overflow. The Theta Method was used for convergence, and all plate efficiencies were assumed to be 100%. (See Reference 7 for additional calculational details and a program listing.)... 

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