Method of Thiele and Geddes

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). 

The stage temperatures are considered the independent variables. The calculations are started by assuming a set of stage temperatures Tj, vapor flows Vj, and liquid flows Lj. The equilibrium coefficients Kj are estimated based on the assumed tray temperatures. Equations 13.1 and 13.5 are combined to give 

a component material balance is written around the top of the column, including any tray / above the feed. The resulting equation is divided by V,  

This equation is written for j reflux ratio, LJVy 

The assumed liquid and vapor rates are used to evaluate A2 , and since the reflux ratio is specified, may be computed. Its value is then substituted in Equation 13.12 written for j = 2, from which L JV is calculated. The procedure is continued through the stage just above the feed. 

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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 ... 

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). 

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. 

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 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. 

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. 

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. 

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. 

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. 

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... 

This method also considers the stage temperatures as the independent variables. The algorithm is applied to a single-feed, two-product column with a partial condenser and reboiler. As in the original Thiele-Geddes method, the problem definition is such that the feed component flow rates, are known and fixed. The column pressure profile is also fixed, as well as its configuration, which defines the number of stages and feed location. In addition, one product rate (the distillate) and one internal flow (such as the reflux rate, Lj) are specified. The solution method, outlined below, is described in detail by Holland (1975). 

A method similar to the Thiele-Geddes method has been proposed by Hummel (Ref. 9). In this method plate-to-plate calculations are made for a few plates at each end and around the feed plate to establish the temperatures. With the values and the known number of theoretical plates the temperature gradient in the tower is drawn. This then gives the temperatures to employ in a Thiele-Geddes type of calculation. For a given number of plates and a given reflux ratio, this method requires an estimation of the distillate, bottoms, and feed-plate compositions. Basically, Hummel s method furnishes a systematic method of successive approximations for the plate temperatures to be used in evaluating the equilibrium constants. 

Together with knowledge of feed condition and reflux ratio, the flow in the column profile can be calculated. Short-cut methods are available for preliminary investigations. The usual ones are the methods of Fenske, Underwood, Erbar-Maddox, and Smith-Brinkley. Two of the earliest rigorous methods were developed by Lewis-Matheson and Thiele-Geddes. 

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