Approximate Solutions at Operating Refluxes

The method is developed first for conventional distillation columns and then modified as required to describe complex columns. 

The approximate method used consisted of solving the combined component-material balances and equilibrium relationships for the light key (i = l) and the heavy key (i = h) components on the basis of assumed temperature and total vapor rate (or total liquid rate) profiles. As shown in Chap. 2, the equations to be solved may be stated in the following matrix form 

The speed of the approximate method is enhanced by the fact that Eq. (1) is solved for only two components, the light and heavy keys (i = /, i = h). This equation is readily solved for d( and the 1, s by use of the Thomas algorithm as demonstrated in Chap. 2. After vNi has been computed, bt is found by use of the expressions 

This expression is readily developed by use of the equilibrium relationship of the reboiler 

Any number of methods could be used to pick adequate temperature profiles. The following relatively simple procedure was used to select these profiles. The temperatures of all intermediate stages were taken to be linear with stage number between the distillate temperature TD and the bottoms temperature TB. For different numbers of stages, the top and bottom temperatures were estimated as follows. First, the lowest overhead temperature T0 L and the highest bottoms temperature TB v were found by use of a modified form of Fenske s equation which was deduced in the following manner. Since the value of (N)min computed by Eq. (2) on the basis of the specified values of bjdl and bh/dh is less than Nv, a separation better than the specified one can be obtained. Thus, it is desirable to revise Eq. (1) such that each separation is better than the one which was specified. The result may be realized by introducing the factor fi as follows 

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