Prior to the availability of digital computers, the most widely used manual methods for rigorous calculations of simple distillation were those of Lewis and Matheson (LM) [Ind. Eng. Chem., 24, 496 (1932)] and Thiele and Geddes (TG) [Ind. Eng. Chem., 25, 290 (1933)], in which the equilibrium-stage equations are solved one by one by using tearing techniques. The former is a design method, in which the number of stages is determined for a specified split between two key components. Thus, it is a rigorous analog of the FUG shortcut method. The TG method is a rating method in which distribution of components between distillate and bottoms is predicted for a specified number of stages. Thus, the TG method is a rigorous analog of the SB method.
Both the LM and the TG methods suffer from numerical difficulties that can prevent convergence in certain cases. The stage-to-stage calculation used in the LM method proceeds from the top down and from the bottom up and is subject to large truncation-error buildup if the components differ widely in volatility. The TG method avoids that difficulty, but numerical instabilities arise as soon as the stage-to-stage calculation crosses a feed stage. Then, a difference term appears in the equations, and sometimes this results in a serious loss of significant digits, making the TG method basically unsuited for multiple-feed columns.
All stage-to-stage methods that work from both ends of the column toward the middle suffer from two other disadvantages. First, the top-down and the bottom-up calculations must "mesh" somewhere in the column. Usually the mesh is made at a feed stage, and if more than one feed stage exists, a choice of mesh point must be made for each component. When the components vary widely in volatility, the same mesh point cannot be used for all components if serious numerical difficulties are to be avoided. Second, arbitrary procedures must be set up to handle nondistributed components. (A nondistributed component is one whose concentration in one of the end-product streams is smaller than the smallest number carried by the computer.) In the LM and TG equations, the concentrations for these components do not naturally take on nonzero values at the proper point as the calculations proceed through the column.
Because of all these numerical difficulties, neither the LM nor the TG stage-by-stage method is commonly implemented in modern computer algorithms. Nevertheless, the TG method is very instructive and is developed in the following example. For a single narrow-boiling feed, the TG manual method is quite efficient.
Example 3: Calculation of TG Method The TG method will be demonstrated by using the same example problem that was used above for the approximate methods. The example column was analyzed previously and found to have C + 2N + 9 design variables. The specifications to be used in this example were also listed at that time and included the total number of stages (N = 10), the feed-plate location (M = 5), the reflux temperature (corresponding to saturated liquid), the distillate rate (D = 48.9), and the top vapor rate (V = 175). As before, the pressure is uniform at 827 kPa (120 psia), but a pressure gradient could be easily handled if desired.
A temperature profile plus a vapor-rate profile through the column must be assumed to start the procedure. These variables are referred to as tear variables and must be iterated on until convergence is achieved in which their values no longer change from iteration to iteration and all equations are satisfied to an acceptable degree of tolerance. Each iteration down and then up through the column is referred to as a column iteration. A set of assumed values of the tear variables consistent with the specifications, plus the component K values at the assumed temperatures, is as follows, using assumed end and middle temperatures and K values from Fig. 13-14b:
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