4. Calculate liquid densities, molar tray and condenser-reflux drum holdups, and liquor and vapor enthalpies. Determine holdup and enthalpy derivatives with respect to time by forward difference approximations.
5. From Eqs. (13-152) to (13-154) compute a new set of values of liquid and vapor molar flow rates.
6. Compute the reboiler molar holdup from Eq. (13-158).
7. Repeat steps 2 through 6 with a corrector step for the same time increment. Repeat again for any further predictor and/or predictor-corrector steps that may be advisable. Distefano (ibid.) discusses and compares a number of suitable explicit methods.
8. Compute condenser and reboiler heat-transfer rates from Eqs. (13-159) and (13-160).
9. Repeat steps 2 through 8 for subsequent time increments until the desired amount of distillate has been withdrawn.
More flexible and efficient methods that can cope with stiffness in batch-distillation calculations utilize stable implicit integration procedures such as the method of Gear (op. cit.). Boston et al. (op. cit.) discuss such a method that also utilizes a two-tier equation-solving technique, referred to as the "inside-out" algorithm, that can handle both wide-boiling and narrow-boiling charges even when very nonideal mixtures are formed. In addition to the features of the Distefano model, the Boston et al. model permits multiple feeds, side-stream withdrawals, tray heat transfer, and vapor distillate and divides the batch-distillation process into a sequence of operation steps. At the beginning of each step, the reboiler may receive an additional charge, distillate or sidestream receivers may be dumped, and a feed reservoir may be refilled. Specifications for an operation step include feed, sidestream withdrawal, and tray heat-transfer rates. In addition, any two of the following five variables must be specified: reflux ratio, distillate rate, boil-up rate, condenser duty, and reboiler duty. An operation step is terminated when a specified criterion, selected from the following list, is reached: a time duration; a component purity in the reboiler, distillate, or distillate accumulator; an amount of material in the reboiler or distillate accumulator; or a reboiler or condenser temperature. The purity is specified to be met as the purity is increasing or decreasing. Finally, column configuration and operating conditions (number ofstages, holdups, tray pressures, and feed, sidestream, and tray heat-transfer rates) can be changed at the beginning of each operation step. In addition, physical properties may be computed from a wide variety of correlations, including equation-of-state and activity-coefficient models.
Example 10: Calculation of Multicomponent Batch Distillation A charge of 45.4 kg ■ mol (100 lb^mol) of 25 mole percent benzene, 50 mole percent monochlorobenzene (MCB), and 25 mole percent orthodichloro-benzene (DCB) is to be distilled in a batch still consisting of a reboiler, a column containing 10 theoretical stages, a total condenser, a reflux drum, and a distillate accumulator. Condenser-reflux drum and tray holdups are 0.0056 and
0.00056 m3 (0.2 and 0.02 ft3) respectively. Pressures are 101.3, 107.6, 117.2, and 120.7 kPa (14.696, 15.6, 17, and 17.5 psia) at the condenser outlet, top stage, bottom stage, and reboiler respectively. Initially, the still is to be brought to total-reflux conditions at a boil-up rate of 45.4 (kg'mol)/h [100 (lb'mol)/h] leaving the reboiler. Then, at t = 0, the boil-up rate is to be increased to 90.8 (kgmol)/h [200 (lb'mol)/h], and the reflux ratio is to be set at 3. The batch is then distilled in three steps. The first step, which is designed to obtain a benzene-rich product, is terminated when the mole fraction of benzene in the distillate being sent to the accumulator has dropped to 0.100 or when 2 h have elapsed. The purpose of the second operation step is to recover an MCB-rich product until the mole fraction of MCB in the distillate drops to 0.400 or 2 h have elapsed since the start of this step. The third step is to be terminated when the mole fraction of DCB in the reboiler reaches 0.98 or 2 h have elapsed since the start of this step. Ideal solutions and an ideal gas are assumed such that Raoult's law can be used to obtain K values. Calculations are made by the method of Boston et al. (op. cit.).
First, the total-reflux condition is computed by making several sets of stage-to-stage calculations from the reboiler to the condenser. For the first set, the reboiler composition is assumed to be that of the initial charge. This composition is adjusted by material balance to initiate each subsequent set of calculations until convergence is achieved. Results are shown in Table 13-29. From these data, the initial-stiffness ratio is approximately [2(99.74)/0.01218] = 16,400. Thus the equations are quite stiff, and an implicit integration method is preferred. Detailed calculated conditions at the end of the first operation step are given in Table 13-30. A short summary of computed conditions for each of the three operation steps is given in Table 13-31.
From Table 13-31, a total of 394 time increments were necessary to distill all but 22.08 lb^mol of the initial charge of 99.74 lb^mol following the establishment of total-reflux conditions. If this problem had to be solved by an explicit integrator, approximately 25,000 time increments would have been necessary.
Instantaneous distillate (or reflux) composition as a function of total accumulated distillate for all three operation steps is plotted in Fig. 13-106. From these results, an alternative schedule of operation steps can be derived to obtain three relatively rich cuts and two intermediate cuts for recycle to the next batch. One example is as follows:
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