Semicontinuous, Pressure-Swing Distillation - American Chemical

ibility. Compared with batch pressure-swing processes, downtime is sharply reduced as liquid holdups are maintained throughout the campaign; dumping a...
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Ind. Eng. Chem. Res. 2000, 39, 122-130

SEPARATIONS Semicontinuous, Pressure-Swing Distillation James R. Phimister and Warren D. Seider* Department of Chemical Engineering, University of Pennsylvania, Philadelphia, Pennsylvania 19104/6393

The merits of semicontinuous, pressure-swing, azeotropic distillation are examined. In continuous operation, two columns at different pressures are fed streams having compositions on opposite sides of the azeotropic compositions. The distillates, which approach the azeotropic compositions at high and low pressure, are cycled between the columns. In contrast, semicontinuous operation involves only a single distillation column, which has lower investment costs and, when the mixture to be separated is changed, shorter downtimes. An optimal-control algorithm is employed to determine desirable campaigns, and to schedule pressure switch-over policies. Simulation results for the dehydration of tetrahydrofuran, involving a pressure-sensitive azeotrope, indicate that switch-over between steady states occurs quickly with on-spec product removed during 93% of the campaign. The column achieves production rates near 89% of the maximum throughput of a single column in the continuous process and shows superior performance when compared to reverse-batch operation. Introduction With rigorous dynamic simulations possible in a fraction of the time required a decade ago, exploration of less common distillation configurations and campaigns is now being pursued more actively. For example, there has been renewed interest in the middle-vessel column; a batch column with two separating sections connected by a large middle vessel.1-3 This paper explores the merits of semicontinuous, pressure-swing distillation to separate minimum-boiling azeotropes. To illustrate this novel process, a typical system, involving the dehydration of tetrahydrofuran (THF), is examined. In continuous operation, the separation is performed using two columns maintained at different pressures. At 1 bar, the azeotrope is at 33 mol % water, and, at 10 bar, it shifts to 19 mol % water. Both THF and water are concentrated in the bottoms products. The distillates, near their respective azeotropic compositions, are cycled between the columns, as shown in Figure 1. In this manner, the azeotropic barrier is overcome, with both water and THF removed continuously. Compared with continuous processing, semicontinuous operation has several advantages, including lower investment costs and greater plant flexibility. Compared with batch pressure-swing processes, downtime is sharply reduced as liquid holdups are maintained throughout the campaign; dumping and recharging of the column are not required. This is because the compositions near the top of the column approach the azeotropic compositions in the distillate, and consequently, do not vary greatly as the tower shifts * Correspondence should be addressed to Warren D. Seider, Department of Chemical Engineering, 220 S. 33rd St., University of Pennsylvania, Philadelphia, PA 19104/6393. Email: [email protected]. Fax: (215) 573-2093. Tel.: (215) 898-7953.

Figure 1. Continuous process for dehydration of THF.

between low- and high-pressure operation. Rather than dump the column, as in batch operation, only to reattain similar tray compositions near the top of the column at the adjusted pressure, the upper section is maintained near the azeotropic compositions. Although some periodic processes are utilized within the chemical industry (e.g., pressure-swing adsorption to separate air), and periodic reactors are being developed,4,5 cyclic distillation has received limited attention. In one example, Sφrensen and Skogestad6 optimized a batch distillation process in which the reflux drum is charged and discharged cyclically to reduce the time to achieve high-purity product. The feasibility of cyclic operations was demonstrated experimentally by Sφrensen and Prenzler.7 In a related study, Watson and coworkers8 enhanced the recovery of ethanol from a

10.1021/ie9904302 CCC: $19.00 © 2000 American Chemical Society Published on Web 12/04/1999

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quaternary mixture by discharging small quantities of methanol periodically from the condenser, thereby decreasing the time to achieve high-purity product. In addition to these forced-periodic processes, recently, natural limit cycles were reported in certain operating regimes. For example, Lee and co-workers9 used the AUTO package to locate multiple steady states in homogeneous azeotropic distillation towers, with periodic behavior at unstable steady states. In addition, Sundmacher and Hoffman10 reported sustained oscillations in simulations of reactive and nonreactive processes. More recently, Wang and co-workers11 observed limit cycles experimentally in a pilot-scale, azeotropic distillation column. However, no clear advantages to operating at limit cycles were reported. For the separation of azeotropic mixtures, using continuous processing, an extensive body of literature has evolved.12 When using batch processing, Bernot and co-workers13,14 used residue-curve maps to select extractive agents and the sequence for product removal by estimating the performance of towers with an infinite number of stages and infinite reboil ratios. Methods for estimating batch sizes, operating times, heat duties, and optimal reflux and reboil ratios for nonideal mixtures were presented by Bernot and co-workers.15 Safrit and Westerberg16 analyzed distillation regions to sequence batch distillation columns, generating alternative sequences and pruning infeasible sequences. Ahmad and Barton17 used nonlinear dynamics to predict feasible cuts for an n-component azeotropic feed. Their approach was illustrated by Ahmad and co-workers18 for a five-component system with nine azeotropes, using dynamic simulations to confirm the predictions. Safrit and co-workers19 and Safrit and Westerberg20 illustrated the use of a middle-vessel column to separate a ternary, homogeneous, azeotropic mixture into three high-purity products. Pressure-Swing Separation. Pressure-swing distillation (PSD), also known as pressure-sensitive distillation, is considered to be an underutilized technique for separating homogeneous azeotropic mixtures. Frank21 provided an overview of the technique and suggested binary mixtures as candidates for PSD. In addition, because pressure-insensitive azeotropes become pressure sensitive through the addition of an entrainer, many viable candidates for PSD are being discovered.22 One industrial example of PSD involves the dehydration of THF. Both THF and water are formed in equimolar quantities upon the dehydration of 1,4butanediol:

CH2(OH)CH2CH2CH2(OH) f H2O + C4H8O T-x-y diagrams for THF and water are shown in Figure 2 at three pressures. When the VLE data in Appendix A are used, at 1 bar, the binary azeotrope lies at 19 mol % water. It shifts to 30 mol % water at 5 bar, and 33 mol % at 10 bar. The most common PSD in industry is a two-step sequence with a low-pressure column preceding a high-pressure column, as shown in Figure 1. For this configuration, Abu-Eishah and Luyben23 presented design and control strategies that reduce heat consumption through heat integration. An integrated configuration requiring half of the energy required by a nonintegrated system was designed, with a control system to reject disturbances and reduce column interactions. Chang and Shih24 performed an

Figure 2. T-x-y diagram for THF and water.

Figure 3. Reverse-batch dehydration of THF.

economic evaluation for the dehydration of THF, in which PSD was compared with azeotropic distillation using an entrainer. Although n-pentane is a suitable entrainer, azeotropic distillation was not found to be more favorable than PSD, economically or operationally. A potential reverse-batch PSD for the dehydration of THF is shown in Figure 3. Initially, the column is charged with an equimolar mixture and brought to total reflux at 1 bar. Water is removed from the bottom of the tower, with the vapor overhead gradually approaching the azeotropic composition. When the concentration of THF in the bottoms product becomes unacceptable, the contents of the column are dumped. Then, the column is recharged with the contents of the reflux accumulator and brought to total reflux at 10 bar. To provide high recovery, this cycle is repeated many times. In each cycle, approximately 20-30% of the operation time is spent with the column at total reflux, or being dumped or recharged.25 Furthermore, considerable energy is expended as the vapor overhead approaches the azeotropic composition. Although the azeotrope is sensitive to pressure, the swings in composition are small. Consequently, at the pressure changeovers, it is desirable to retain the liquid holdups near the top of the column, since they experience small changes in composition. Semicontinuous Distillation. In semicontinuous PSD, both continuous and periodic operations are

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Figure 5. Performance restrictions on column trays.

Figure 4. Flowsheet for semicontinuous operation.

utilized. The column section operates continuously. It is not emptied or recharged. Liquid levels are maintained on the trays, steam is fed continuously to the reboiler, and cooling water is fed continuously to the condenser. The column operates in two modes, with tanks T1 and T2 alternating as the feed source, and the distillate and bottoms products are sent to alternate tanks, as shown in Figure 4. Operation begins in mode 1, after start-up, during which on-specification products (nearly pure water as a bottoms product and distillate near the azeotrope at 1 bar) are achieved. T1 is recharged periodically with fresh feed from S1. Note that the column operates continuously, but not at a steady state. The column alternates between the modes defined as follows. Mode 1: S2 feeds the column. The condensed overhead and off-specification bottoms product is fed to tank T2. S1 has a zero flow rate. The operating pressure is 1 bar. Mode 2: S3 feeds the column. The overhead and off-specification bottoms product is fed to tank T1. S1 feeds tank T1. The operating pressure is 10 bar. Product is fed to tank T3 or tank T4 when the bottoms composition exceeds 98 mol % in water or THF, respectively. The process switches between modes when tank T1 or tank T2 is empty. In each mode, the column has a fixed pressure, and reflux and reboil ratios. Modes 1 and 2 operate at low and high pressure, respectively. Optimal campaigns are determined by the integration of a dynamic process model, driven by a stochastic optimal-control algorithm. The algorithm seeks to maximize the accumulation of products in tanks T3 and T4 during each mode of the campaign by manipulating the reflux and boilup ratios. In each mode, the objective function,

Fobj ) -(1 - z)(MT3 + MT4) + 108z - tz

(1)

is minimized, where MT3 and MT4 are the molar holdups in tanks T3 and T4, and z is a binary variable which becomes unity when an “undesirable event” occurs along the time horizon; otherwise z is zero. Undesirable events occur when tray, sump or condenser holdups become zero, tray mole fractions become negative, tray residence times become too small, holdup tanks overflow or empty, trays weep, and flooding occurs. The optimal control algorithm predicts one mode into the future, with the reflux and boilup ratios optimized for the next mode of operation. Once an optimal solution

is determined, the process operates under the optimized parameters. After tank T1 or tank T2 is emptied, the procedure is repeated. All code is written in FORTRAN 90. A banded IMSL solver and the tridiagonal solver from Numerical Recipes26 is utilized. For optimization, a rank-selective genetic algorithm is utilized, as described in Appendix B. It provides flexibility for encoding the objective function which incorporates discrete events along the time horizon. The process model is treated as a “black box”. The discontinuous nature of the objective function provides a two-tiered ranking procedure: (1) when the process performs properly, large amounts of product are favored, and (2) when an undesirable event occurs, solutions in which the event occurs at later times are favored. Distillation Model. The batch distillation model in Separation Process Principles27 (p 695) is extended, as described in Appendix C. Proportional level controllers are used for the sump and the reflux drum. An implicitEuler algorithm is used to integrate the stiff ODEs to determine composition changes on all trays and composition and level changes in the reflux accumulator, sump, and hold vessels. The step size is varied to satisfy an error tolerance. The tri-diagonal Jacobian matrixes are inverted efficiently during each time step of integration. The column has a 1.0 m diameter and contains 10 trays (including the reboiler) with 0.5 m spacing. Weir heights are 50 mm. Given 12% area for each downcomer, the volumetric holdup on each tray is 0.029 m3. Allowing for a backup of 0.2 m in the downcomer (40% of the tray spacing), the downcomer has a holdup of 0.019 m3. These sum to 0.048 m3 (48 L) for each tray. The holdup in the reflux accumulator is controlled at 5 m3. This large volume allows for variation in reflux ratios between modes, while ensuring sufficiently large residence times. To allow for rapid transfer between modes, the holdup in the sump is only 1 m3. Furthermore, to evenly distribute liquid holdups throughout the column, 50% of the feed is vaporized. Tanks T1 and T2 have volumes of 150 m3. A proportional controller is used to manipulate the flow rate of S1 such that the total contents in T1 and T2 are 100 metric tons. The fifth tray of the column is fed continuously 6 m3/h (roughly 5400 kg/h), either from tank T1 or tank T2, depending on the operating mode.

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Figure 6. x-y diagram for THF-water using the Wilson equation at 1 bar. Data of Hayduk and co-workers29 are denoted “O”.

Figure 5 illustrates constraints on tray performance as the vapor and liquid flow rates vary, as modified from Sinnott.28 Note that when one of these constraints is violated during integration of the dynamic model, as indicated above, the objective function (1) is penalized by setting z to unity. To ensure that a seal in the downcomer is maintained, the liquid residence time is constrained such that τdc > 3 s (as recommended by Sinnott28) and the overall stage residence time for the liquid is constrained such that τstage > 8 s. To prevent flooding and weeping, external constraints that bound the liquid-vapor flow factor, FLV, are

0.01 e FLV e 1.0

(2)

where FLV is

FLV )

x

L V

Fv FL

Table 1. Optimal Campaign on-speca (h)

end time (h)

mode

P (bar)

reflux ratio

reboil ratio

MT3 + MT4 (kmol)

0.50 17.8 33.5 52.5 69.2 93.1

16.6 30.6 50.4 68.2 91.3 112.3

1 2 1 2 1 2

1 10 1 10 1 10

4.6 2.3 7.3 2.8 9.0 2.3

4.0 1.8 3.5 1.1 3.5 1.3

786 1290 1826 2299 2884 3561

a Time at which 0.98 mole fraction of water (mode 1) and THF (mode 2) is achieved.

from ASPEN PLUS.33 As illustrated in Figure 6, the vapor-liquid equilibrium (VLE) computations are in excellent agreement with the experimental data of Hayduk and co-workers29 at 1 bar (note that data at 10 bar are not available). Results

(3)

and L and V are the liquid and vapor flow rates in kg/ h. Although a constraint for coning is not applied, it is noted that the liquid flow rates are constrained by the lower bound on the reflux ratio that is specified for the optimizer. The model assumes: (1) instantaneous pressure transfers; (2) tray efficiencies of unity throughout the column; (3) fixed liquid and vapor enthalpies for each species; and (4) fixed liquid densities for each species. Assumptions 1 and 2 are inherent in most equilibriumstage models. Results based upon them are optimistic and should be interpreted with care. Assumptions 3 and 4 can be relaxed, as necessary, with increased computational load. Coefficients for the Antoine equation (for vapor pressures) and interaction coefficients for the Wilson equation (for liquid-phase activity coefficients) are obtained

The results were computed using a 200 MHz PC. On this computer, execution times are approximately 30 min, with each dynamic simulation taking approximately 3 s. Maximum step sizes are 0.05 h over a campaign of three cycles. Initially, the trays, sump, and reflux accumulator are loaded with an equimolar mixture of THF and water. Tank T1 is loaded with 2045 kmol (100 m3 of this mixture). The best results are summarized in Table 1 with profiles of the key variables shown in Figure 7. Beginning in mode 1, at 1 bar, tank T1 feeds the column for 16.6 h, at a uniform flow rate of 6 m3/h, as shown in Figure 7A. The reflux ratio is 4.6, with the condenser duty approaching 3.1 MW, and the reboil ratio is 4.0, with the reboiler duty approaching 2.5 MW. After 0.5 h, water is removed as a product in tank T3, as shown by the solid line in Figure 7B. The overhead tank, T2, fills at a uniform rate, as shown by the dashed line in Figure 7A, and its mole fraction of water approaches

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Figure 7. Optimal campaign A: molar feed holdups in T1 (solid line), T2 (dashed line); B: product holdups in T3 (solid line), T4 (dashed line); C: water mole fractions in T1 (solid), T2 (dashed); D: water mole fractions in reflux accumulator (solid) and sump (dashed). Dotted lines in C and D show azeotropic compositions at 1 bar (0.19 MF water) and 10 bar (0.33 MF water).

0.19 (near the 1 bar azeotrope) within 5 h, as shown by the dashed line in Figure 7C. The mole fractions of water in the reflux accumulator and sump are shown as solid and dashed lines, respectively, in Figure 7D. Once T1 empties, the column switches to mode 2. Tank T2 feeds the column at 10 bar and the reflux and reboil ratios become 2.3 and 1.8, which yield condenser and reboiler duties that approach 1.1 and 0.8 MW, respectively. As shown by the dashed line in Figure 7A, beginning at 16.6 h, tank T2 is emptied. Until 17.8 h, off-specification product is fed to tank T1. Then, THF product is accumulated in tank T4, as shown by the dashed line in Figure 7B. Throughout mode 2, distillate and an equimolar feed stream are fed to tank T1. The feed stream causes tank T1 to fill more rapidly than observed for tank T2 in mode 1. Also, the feed stream causes the mole fraction of water in tank T1 to remain slightly less than 0.5, as shown by the solid line in Figure 7C. Note that the composition of the bottoms product transfers quickly from greater than 98 mol % water (at 17.9 h) to greater than 98 mol % THF (at 18.9 h), as shown by the dashed line in Figure 7D. The column cycles through both modes three times in the campaign, producing 1931 kmol of water and 1630 kmol of THF, a total of 148.9 metric tons. The continuous process in Figure 1, with the same total feed (S2 + F1) of 6 m3/h, produces 1.488 metric ton/h, or 167 metric ton over the same time period. On this basis, the semicontinuous process has an efficiency of approximately 89%. Off-specification product is produced for only 9.5 h; that is, for just 8.5% of the campaign. The small fluctuations in the mole fractions in the feed tanks, T1 and T2, as shown in the curves of Figure 7C, are expected. These occur when the tanks have small holdups, and hence, their compositions are sensitive to changes in their feed compositions.

Process Comparison. The primary motivation for operating semicontinuously is to improve upon batch processing when the product demand is intermittent or seasonal, the upstream operations are batch, and the throughput is small, as often occurs in the production of fine or specialty chemicals. In this section, a quantitative comparison of semicontinuous and batch processing is presented, followed by a comparison of semicontinuous and continuous processing. (i) Comparison of Semicontinuous and Batch Processing. For ideal binary systems, Meski and Morari1 show that production from a batch, middlevessel column (a column with a large internal vessel) is maximized, and operating time minimized, when the distillate and bottoms flow rates are held constant, and the middle-vessel composition does not change over the time horizon. Their results show that the middle-vessel column consistently out-performs a typical batch column. Under these operating conditions, the column performance is equivalent when an external feed tank is used in lieu of a middle vessel, and the feed-tray composition is maintained at the feed-tank composition. Consequently, it is expected that in both modes of operation, a semicontinuous column will out-perform a batch column. In the simulation results below, to examine the batch alternative most favorably, reverse-batch operation is simulated with the liquid holdup maintained during the switch-over between modes. This avoids the downtime that would be incurred were the contents of the column dumped during switch-over. Also, note that the operating parameters of the reverse-batch process, such as the bottoms flow rate and the average boilup rate, are equivalent to those of the semicontinuous process. (a) Batch Model. The model described in Appendix C is modified to accommodate reverse-batch distillation.

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Figure 8. Throughput efficiencies.

The column operates at total reflux until 98 mol % is achieved in either product. The bottoms flow rate is 3 m3/h (the maximum bottoms flow rate of the semicontinuous process) when the sump concentration is on specification; otherwise, it is zero. The liquid reflux rate to the top tray is increased by the bottoms flow rate when the column operates on specification. The vapor boilup rate is maintained at the average boilup rate of the semicontinuous process (1.7 MW). The residencetime constraint is relaxed. The reflux accumulator is sized at 300 m3; that is, the total holdup of tanks T1T4 at the end of the semicontinuous campaign. The reverse-batch process is simulated for 112.3 h, the duration of the campaign for the semicontinuous process. (b) Results. Mass-throughput efficiencies, with respect to the maximum semicontinuous throughput, are shown in Figure 8. The abscissa indicates the approach of the water mole fraction in the distillate to the azeotrope at which pressure changeover occurs. For example, when the column is operating at 1 bar (water mole fraction at the azeotrope is 0.18), and the approach is 0.01, the column is switched to 10 bar when the distillate reaches 0.19 mole fraction of water. Alternatively, when operating at 10 bar (water mole fraction at the azeotrope is 0.33), with an equal approach of 0.01, the column is switched to 1 bar when the distillate reaches 0.32 mole fraction of water. The maximum efficiency of 85% is at equal approaches of 0.045. Because it is desirable to have different approaches at the two operating pressures, an exhaustive search, with a step size of 0.005, is performed. Approaches of 0.06, when operating at 1 bar (with water as the product), and 0.02 when operating at 10 bar (with THF removed as the product), are selected. These approaches give a total efficiency of 95%. Note that a closer approach is desirable to increase the removal of the heavier, more valuable THF product. However, when the minimum residence time is increased, internal flow rates must be

reduced, which reduces the bottoms flow rate to 2.6 m3/h and the efficiency to 90%. (ii) Comparison of Semicontinuous and Continuous Processing. The most notable advantage of the semicontinuous process (Figure 4), compared with the continuous process (Figure 1), is the reduction in capital cost gained by elimination of one column. To simplify the comparison, the two columns in the continuous process are assumed to have the same number of trays as in the semicontinuous process. Furthermore, it is assumed that the tray volumes vary linearly with the volumetric throughput, with each column receiving half the total throughput. Consequently, the diameters of the towers in the continuous process are reduced by a factor of 21/2. Using Figure 16-28 from Peters and Timmerhaus,30 in which the installation costs of the column per tray are graphed as a function of the diameter, the cost of the semicontinuous column lies between 35 and 55% more than one of the columns in the continuous process. Considering the columns alone, the total depreciable capital investment of the semicontinuous process is reduced by 22.5-32.5%. However, this reduction is offset somewhat by the costs of tanks T1 and T2, potentially larger control costs, and the increase in operating costs due to the inability to achieve heat integration between the condensing and boiling streams at different pressures, as implemented by Abu-Eishah and Luyben23 for the continuous process. Furthermore, the assumption that the two continuous columns are sized identically favors semicontinuous distillation. Because the separation is more difficult at elevated pressures, and the second column has a smaller throughput, the two columns would be sized differently. Potential Design Improvements. The following improvements are anticipated to increase the efficiency of semicontinuous distillation: (1) Smaller holdups on the trays and sump, as well as fewer trays, reduce the overall holdup in the column,

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thereby reducing the time to attain on-specification products when switching between modes. (2) To improve the separation, the column feed should be fed to a tray having a comparable composition. Because the feed composition changes when the modes are switched, a manifold of feed lines, with automatic switching, should be beneficial. (3) To avoid unnecessary mixing, the process feed should be fed semicontinuously directly to the column, rather than to the tank T1. (4) As with the continuous process, increased differences between the operating pressures shift the azeotropic compositions further apart, resulting in less internal circulation of chemicals. Higher pressures, however, require increased capital costs. (5) Structured packing permits smaller liquid residence times for comparable numbers of theoretical stages. This reduces the transfer time between modes. Furthermore, because there is no downcomer in a packed tower, the constraint on the minimum residence time can be relaxed, allowing for larger recirculation and production rates. (6) Two sumps, one for each product, which alternately become active, would reduce the transfer time, although the capital cost would be increased, as well as the complexity and cost of the control system. (7) To increase the “area of satisfactory operation” in Figure 5, and the column productivity, in turn, the tray design can be improved by adjusting the tray separation, hole diameter, hole/cap design, and tray diameter. (8) Larger volumes in tanks T1 and T2 reduce the number of switch-overs, improving the productivity, but at higher investment costs. Potential Problems and Other Considerations. When a semicontinuous design is being selected, the following potential disadvantages should be considered: (1) It is assumed that pressure changes quickly when switching modes. This is justified because column pressures respond rapidly to vapor throttling, changes in feed pressure, and changes in cooling water temperature. For this reason, care is required when implementing pressure controllers. (2) For more difficult separations, or for separations where higher-purity products are required, the addition of trays not only increases the capital cost, but the internal holdup as well, thereby increasing the transfer time. (3) In continuous PSD, a significant reduction in operating costs is achieved by heat integration.23 Designs involving heat integration for the semicontinuous operation are not straightforward. Conclusions Semicontinuous PSD can be competitive with reversebatch and continuous PSD. In this paper, desirable campaigns are determined using a stochastic, optimalcontrol algorithm in which production is maximized. The optimal-control algorithm consistently increases the overall throughput for the dehydration of THF by 10%, compared with a reverse-batch configuration. The semicontinuous distillation operates during large portions of its campaign near steady state, that is, with fixed distillate, bottoms, and feed flow rates, and at a steady operating pressure, thereby simplifying the implementation of automatic controllers.

Table 2. Extended Antoine Coefficients for THF and Watera C1 C2 C3 C4 C5 C6 C7 a

H2O

THF

7.36 -7258 0.0 0.0 -7.304 4.16530 × 10-6 2.0

5.490 -5305 0.0 0.0 -4.763 1.42910 × 10-17 6.0 Ci7

ln Psi ) C1i + C2i /(T + C3i ) + C4i T + C5i ln T + C6i T ; Ps, Pascal.

For the THF-water separation, semicontinuous distillation provides production rates near 89% of those achieved in continuous distillation. Installation costs for the column are reduced, but additional costs for holding tanks and more complex control offset this reduction somewhat. Consequently, semicontinuous distillation may not be as favorable as continuous distillation for the processing of large throughputs. Experimental data are needed to confirm the theoretical predictions herein. Design improvements are suggested that should increase the efficiency of a semicontinuous PSD column. Appendix A: Equilibrium Model and Physical Property Data Vapor pressures are computed using the extended Antoine equation, with parameters in Table 2, obtained from the PURECOMP databank in ASPEN PLUS. The liquid-phase activity coefficients are estimated using the Wilson equation:

zjkxj

∑j zijxj) - ∑j ∑k zjkxk

ln γi ) 1 - ln(

(A1)

where

ln zij ) Aij +

Bij T

(A2)

Interaction coefficients, Aij and Bij, obtained from ASPEN PLUS, are shown in Table 3. Liquid densities and enthalpies of the saturated liquid and vapor for THF and water are provided in Table 4. Appendix B: Genetic Algorithm Genetic algorithms (GAs) attempt to mimic a Darwinian evolution in which a population of solutions evolves over generations. Note that a collection of papers on GAs with sample software is provided by Chambers.31 Evolutionary rules, concerning the reproduction, death, and mutation of members of the population, are applied which implement a survival of the fittest methodology on the population. Through these rules, superior solutions remain and evolve, and while not ensured, convergence toward an optimal solution becomes increasingly likely. Intrinsic to GAs, a population representing potential solutions is created and evolves. Each individual, termed a genome (also referred to as a “chromosome”), is made up of genes (in this case variables) and represents a potential solution. The genome undergoes a number of operations; specifically, mutation, crossover (creation of

Ind. Eng. Chem. Res., Vol. 39, No. 1, 2000 129 Table 3. Wilson Interaction Coefficients for THF and Water Ai,j

water

THF

Bi,j

water

THF

water THF

0.0 -2.999

-23.709 0.0

water THF

0.0 -45.07

7500 0.0

Condenser (stage 0): V1 - L0 - S0 -

water THF

950 850

0 0

40 624 27 870

6 192 14 350

42 426 42 127

offspring), and death. Commonly, a GA uses an objective function to evaluate the fitness of each genome for survival. The fittest members are most likely to survive and evolve. The least fit members are most likely to be removed from the population. In this paper, the rankselective GA is the same as the multiobjective, rankselective GA employed by Phimister and co-workers,32 but only one objective function is involved herein. Genome Representation. Each genome is comprised of two continuous variables, x1 and x2. These represent the recycle fractions, ro and rb, such that 0.5 e xi e 0.9, i ) 1,2. Note that they are related to the reflux and reboil ratios by

R)

ro , 1 - ro

R′ )

rb 1 - rb

Note that the reflux and reboil ratios lie between 1 and 9. Crossover. A new member is introduced to the population by “crossing over” two parent genomes. Randomly selected genes are obtained from the parents, with genes taken from each parent. Mutation. A mutation operation is included to add diversity to the population. This operation adds a new member with a random genetic makeup. Specifications. For crossover, parent genomes are determined by rounding w2Npop and selecting the corresponding member in the ranked population, where w is a random number between zero and one, and Npop is the number of genomes in the current population. Similarly, genomes are selected for removal by rounding (1 - w2)Npop and removing the corresponding member from the population. The GA favors selection of the fittest parent genomes, near the top of the ranked population list, and removal of weaker members of the population. Optimal solutions for each mode are determined by evolving 100 randomly created genomes over 50 generations. In each generation, 20 genomes are created by crossover, 10 are created by mutation, and 30 are removed. The best genome is not allowed to be mutated or removed. Appendix C: Column Model and Process Specifications The MESH equations are integrated with some approximations intended to reduce the computation times. These approximations do not affect significantly the best campaigns selected by the predictive control algorithm. Mass and Energy Balances. The mass and energy balances for the condenser (stage 0), trays (stages 1,..., N - 1), and the sump (stage N) are

(C1)

V1hV,1(y1) - L0hL,0(x0) -

Table 4. Physical Property Data density hL (1 bar) hV (1 bar) hL (10 bar) hV (10 bar) (kg/m3) (kJ/kmol) (kJ/kmol) (kJ/kmol) (kJ/kmol)

dM0 )0 dt

S1hL,1(x1) -

d(M1hL,1(x1)) ) 0 (C2) dt

Intermediate trays (stages 1,..., N - 1): Vi+1 + Li-1 - Li - Vi + Fi -

dMi )0 dt

(C3)

Vi+1hV,i+1(yi+1) + Li-1hL,i-1(xi-1) LihL,i(xi-1) - VihV,i(yi) + FihL,i(zi) -

d(MihL,i) (C4) dt

Sump (stage N): LN-1 - VN - SN -

dMN )0 dt

(C5)

LN-1hL,N-1(xN-1) - VNhV,N(yN) d(MNhL,N(xN)) SNhL,N(xN) ) 0 (C6) dt where Vi, Li, Fi, and Si are the vapor, liquid, feed, and sidedraw molar flow rates to or from stage i; Mi is the liquid molar holdup on stage i; xi, yi, and zi are vectors of the liquid, vapor, and feed mole fractions on stage i; and hL,i and hV,i are the liquid and vapor enthalpies on stage i as a function of the temperature and pressure (not shown) and the liquid and vapor mole fractions, respectively. For the dehydration of THF, the feed is to tray 5; hence, Fi ) 0, i ) 1,..., N; i * 5. These differential equations are integrated with an implicit-Euler approximation and a step-size adjustment algorithm. The following ODEs are integrated to determine the liquid mole fractions at the end of a time step:

Condenser (stage 0):

[

]

dM0 L0 + D + V1K1,j dx0,j dt )x0,j + x1,j dt M0 M0 j ) 1, ..., Nc (C7)

[ ]

Intermediate trays, (i ) 1,..., N - 1):

[ ]

Li-1 dxi,j x ) dt Mi i-1,j

[

Li + Ki,jVi +

]

dMi dt

Mi

Sump (stage N):

[

xi,j +

[

]

Ki+1,jVi+1 xi+1,j Mi j ) 1, ..., Nc (C8)

]

dMN KN,jVN + LN-1 dxN,j dt ) x xN,j dt MN N-1,j MN j ) 1, ..., Nc (C9)

[ ]

where Nc is the number of chemical species. The righthand sides are evaluated using Vi, Li, and Si, i ) 1, ..., N, at the end of the previous time step. Note that So is

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Ind. Eng. Chem. Res., Vol. 39, No. 1, 2000

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Received for review June 14, 1999 Revised manuscript received September 29, 1999 Accepted October 12, 1999 IE9904302