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Ind. Eng. Chem. Res. 1998, 37, 1910-1916
An Optimal Campaign Structure for Multicomponent Batch Distillation with Reversible Reaction R. M. Wajge† and G. V. Reklaitis* School of Chemical Engineering, Purdue University, West Lafayette, Indiana 47906
When resource utilization and/or minimization of waste is comparable in importance to the production rate, the design of the operation policy should encompass an entire campaign of batches rather than a single batch. This notion of campaign optimization is particularly relevant to batch distillation with reversible reaction which produces significant amounts of off-cuts. Reprocessing these off-cuts merely based on consideration of a single batch may lead not only to inefficient production rates but also to the inefficient utilization of reactants. A general strategy for deciding the campaign structure for such an operation is presented. The concept of distillation characteristic is introduced, and its exploitation is shown to result in a simple but effective reprocessing policy for off-cuts. The economic benefits of such a campaign structure are demonstrated with the help of a case study. 1. Introduction With a resurgence of interest in multipurpose batch processing has come a renewed focus on improving the effectiveness of batch distillation from both a design and an operational perspective. A batch still can be very flexible in accommodating different multicomponent batch charges and in allowing the collection over time of a series of product cuts of different compositions. However, in order to achieve reasonably sharp separations between successive cuts and to minimize the production of off-cuts which occur in the transition between desired product cuts, dynamic operating policies must be used. Such policies are normally implemented by adjusting the column reflux rate continuously over time and are determined by solving a dynamic optimization problem. The computational issues associated with the solution of this problem have received considerable attention in recent literature (Logsdon and Biegler, 1993). Typically, the reflux profile is assumed to undergo step-wise changes, where both the timing of these step changes and the reflux level between changes are taken as optimization variables. Using suitable numerical approximations of the ODE (staged column) or PDE (packed column) model (Wajge et al., 1997), the resulting problem can be converted and solved as a large, structured nonlinear programming problem. As a further operational improvement, as shown by Macchietto and Mujtaba (1996), the compositions and amounts of the off-cuts should also be treated as optimization variables and, thus, the problem posed as a two-level optimization in which the cut compositions and amounts are treated as the decision variables of an upper level optimization problem, while the reflux profile optimization problem is treated as a lower level decision problem. This line of development leads naturally to the pursuit of strategies for recycling of off-cuts * Author to whom correspondence should be addressed. Phone: (765) 494-4050. Fax: (765) 494-0805. E-mail:
[email protected]. † Current address: BOC Group Technical Center, Murray Hill, NJ 07974.
for redistillation as well as the collection of off-cuts from successive batches for periodic redistillation. Since the redistillation steps incur utility costs and utilize column time that otherwise could be used for processing fresh batches, such operating strategies which encompass entire campaigns of batches require careful economic balancing of operational costs versus the benefits of better materials recovery and reduced generation of waste materials. The purpose of this paper is to demonstrate that the above notion of campaign optimization can be particularly relevant in the reactive batch distillation case. Batch distillation with reaction occurring in the liquid phase is often used in applications in which the reactions are reversible, and thus, the removal of one or more of the products of reaction can allow equilibrium limitations to be mitigated. However, although the conversion may well be enhanced, it nonetheless often remains incomplete. Therefore, the distillation will result in cuts that contain mixtures of products and reactants in different proportions. The opportunity, therefore, again arises for returning such cuts for redistillation, in this case with the additional degree of freedom that the composition of the recycled cut can be adjusted through the addition of make-up reactants. Since material balance considerations will, in general, result in cuts of different amounts, the further opportunity exists, in principle, for coordinating and accumulating cuts from successive batches so as to achieve efficient batch processing campaigns. The result is an integrated batch campaign in which reactants are judiciously added to intermediate cuts to result in the reduction of both the residuesswhich must be accepted as process wastessand the overall separation load since the generation of high-purity intermediate cuts is avoided. Here, we develop a general optimal campaign structure for reactive distillation batches which offers a simple but efficient reprocessing strategy for off-cuts. The economic benefits offered by such a strategy are analyzed with the help of an ethanol esterification case study.
S0888-5885(97)00527-7 CCC: $15.00 © 1998 American Chemical Society Published on Web 03/18/1998
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2. Past Work The modeling, simulation, and design aspects of reactive distillation have received increasing attention in recent years from both academia and industry. Agreda et al. (1990) and DeGarmo et al. (1992) present examples of industrially used reactive distillation operations. There are many systems involving reversible reactions (such as Nylon-6,6, MTBE, esterification of ethanol) that are suitable for reactive distillation. A survey of industrially feasible systems and available design techniques is presented by Doherty and Buzad (1992). Mujtaba and Macchietto (1992) discuss different types of batch distillation columns that can be used for reactive distillation (depending on the relative boiling points of the components). The general practice in the domain of design of operation policies seems to be to optimize operation of a single reactive distillation unitsprimary unit (Mujtaba and Macchietto, 1992)sand then aim for the best possible recovery of the reactants/products in subsequent reprocessing units. This approach, however, does not guarantee the most economical operation from the entire campaign viewpoint. Mayur et al. (1970) and Christensen and Jorgensen (1987) observed that a smaller operation time is sufficient for the same amount of product when an off-cut is recycled with an appropriate reflux policy. The two-level approach (or its singlelevel counterpart) presented by Macchietto and Mujtaba (1996) aims at deciding the optimal recycling of the offcuts. The issue of recycling of off-cuts, however, is more complex in the case of distillation with chemical reaction. The notion of campaign optimization for distillation with chemical reactions has not received any particular attention in the literature except for qualitative discussions (Macchietto and Mujtaba, 1996). 3. Development of an Optimal Campaign Structure 3.1. Objectives. The maximization of the production rate or of the purity of the product or, alternatively, the minimization of energy requirements can be selected as the objectives, and the structure of the multiperiod reflux profile, the boil-up rate, and the pressure can be the decision variables (although the pressure is rarely used as a decision variable since excursions from ambient pressure typically ´ınvolve cost penalties in an industrial environment) in the design of the optimal operation policy for a single reactive distillation unit. As mentioned earlier, the presence of a reversible reaction can lead to a large amount of off-cuts if the boiling points of the reactants and products are close. The order of boiling points of the components is also an important factor in the case of reactive distillation because of the presence of reaction. These off-cuts require design of a well-defined reprocessing strategy in order to recover the reactants as well as to reduce waste generation. Thus, maximum utilization of reactants and minimization of waste generation become additional objectives in the overall campaign operation in addition to the production rate. Moreover, the variables associated with the reprocessing strategy (e.g., the quantity of the off-cuts and the stage at which they are to be recycled) add to the decision variable set. If one attempts to determine the optimal operation policy for the combined mixture of various off-cuts, one would obtain batches of off-cuts of still different com-
position which need to be reprocessed. This process could be continued until the final off-cuts are of an insignificant amount. We propose the avoidance of such a proliferation of off-cuts as one of the objectives of the optimal campaign structure since it complicates the design as well as the actual process due to the requirements of a multiplicity of column operating profiles. To accomplish this objective, we make use of the concept of distillation characteristic associated with the chemical equilibrium of a reversible reaction. 3.2. Concept of Distillation Characteristic. Usually, a distillation column is operated at total reflux in the beginning of the operation. During this period, the phase and chemical equilibrium are established inside the column. We refer to the composition profile developed as the distillation characteristic (henceforth referred as DC) of that charge. Clearly, charges with different compositions may have the same DC and hence the same optimal reflux profile due to the presence of reversible reaction. This suggests that, in certain cases, it may be beneficial to leave the reactants in off-cuts (rather than distilling them out) and to utilize them within a suitable reprocessing strategy. To verify if two charges of different composition have the same DC, the composition profiles developed throughout the column need to be compared. However, if we make the assumption of negligible column hold-up (compared to the reboiler initial charge) and stoichiometric feed (common in batch operations), then the condition for having the same DC reduces to maintaining the stoichiometric proportion among the reactants and among the products (stoichiometric constraint). This means that, for reaction 1, mixtures with molar ratios of R1 to R2 in the reactant and R3 to R4 in the products will have the same DC. Since the reactant ratios can easily be adjusted
R1A + R2B h R3C + R4D
(1)
by adding make-up reactants, only the products in the charge need to satisfy the stoichiometric ratio constraint. 3.3. Campaign Structure Principles. With the above understanding of DC, we strive to meet campaign objectives by developing a campaign structure based on the following principles: (i) Use the stoichiometric constraint on the product ratio as the defining condition for off-cuts. (ii) Accumulate off-cuts with the same DC to allow use of uniform charge size. (iii) Adjust the reactant ratio in the off-cut (before it is reprocessed) by the addition of make-up reactants so as to achieve stoichiometric reactant ratios. This general strategy for the campaign structure is shown in Figure 1a. We apply the stoichiometric constraint to one of the off-cuts (a simple material balance can show us that it cannot be applied to all the off-cuts) so that it could be reprocessed by a distillation step with the same reflux policy as that for the fresh charge, after the addition of make-up reactants. The off-cut chosen for the application of the stoichiometric constraint should be the one in which any product appears first. The identity of the product will depend on the boiling points of the species involved. If at least one of the products has a lower boiling point than all the reactants, then, in order to shift the chemical equilibrium favorably (for higher conversions), the product needs to be withdrawn as the distillate and the stoichiometric constraint should be applied to the distil-
1912 Ind. Eng. Chem. Res., Vol. 37, No. 5, 1998 Table 1. Three-Product System moles of off-cut
D
E
F
1 2
D1 D2
E1 E2
F1 F2
Table 2. Applications of the Campaign Principles
Figure 1. Campaign structure strategies for different numbers of products (symbol * denotes the imposition of the stoichiometric constraint to define the off-cut composition whereas symbol 0 denotes the predefined criterion (not the stoichiometric constraint) to define the off-cut composition).
late. This is the conventional distillation where the initial charge is taken into a reboiler. On the other hand, if at least one of the products has a higher boiling point than the reactants, inverted distillation (the initial charge in the condenser) must be employed and the bottom off-cut would satisfy the stoichiometric constraint. For cases other than the above two (i.e., when the boiling points of the products lie between those of the reactants), reactive distillation would not be the choice of operation as it will adversely affect the chemical equilibrium to remove the reactants (either as distillate or as bottom cut), thus leading to lower conversions. However, from a strictly mathematical standpoint, there will always be an off-cut that can satisfy the stoichiometric constraint and, again, it will the one in which any of the products first appears. Irrespective of the cut that satisfies the stoichiometric constraint, the campaign structure and principles remain same. The other off-cut could be subjected to an additional distillation step (after addition of make-up reactants) with the sole objective of collecting a cut that satisfies the stoichiometric constraint (thus recovering some of the reactants) which could then be utilized in the same manner as above. The other off-cut from this step would, however, be waste as far as the campaign structure is concerned. The required purity of the product and the energy consideration would determine the amount of the product cut per second. This is because higher product purity means smaller product cuts and larger off-cuts, which alternatively means
no. of products
campaign strategy
2 1 3 4 or more
Figure 1a Figure 1b Figure 1c extension of Figure 1c
comment general strategy as discussed here degenerate form of general strategy extension of the general strategy extension of the general strategy
higher reprocessing cost. The amount of the off-cut will obviously be governed by the stoichiometric constraint. Though the above discussion considers a chemical system with four components (two reactants, two products), it can be extended to applications involving an arbitrary number of components. The number of productssunlike the number of reactantssdoes play a significant role in defining the campaign structure. As can be seen from Table 2, for one-product systems, any criterion could be chosen to determine the amount of the off-cut since the stoichiometric constraint for the product is absent (typically, it would either be the percentage of the product in the off-cut or the amount of energy that can be spent in the recovery of the reactants). For three-product systems, it may not be possible to obtain a single off-cut that would satisfy the stoichiometric constraint among all the products. It would then be necessary to collect two off-cuts and mix them in prespecified proportion β (this would be an additional decision variable) to yield the required charge. This means that for a charge with reaction 2 and distillation cuts, as shown in Table 1, the stoichiometric constraint would comprise two constraints as given in eqs 3 and 4.
R1A + R2B + R3C h R4D + R5E + R6F
(2)
βD1 + D2 R4 ) βE1 + E2 R5
(3)
βD1 + D2 R4 ) βF1 + F2 R6
(4)
This mixture of cuts then could be reprocessed using the same campaign principles as above. 3.4. Formal Representation of Batch Operation. For representing the campaign structure, we make use of the extended STN (State Task Network) representation suggested by Macchietto and Mujtaba (1996). Using this formalism, a process step is represented by a task rectangle while an aggregated distillation cut is shown as a circle. Thus, a typical two-step distillation with recycle of an intermediate off-cut to the next batch can be represented as shown in Figure 2. 3.5. Optimal Campaign Structure. Using this convention, we represent the general campaign strategy for systems having two off-cuts, as shown in Figure 3. The campaign is initiated with a fresh charge of the reactants which is distilled using distillation step A, generating product cut/cuts and an intermediate residue which is processed further to yield two off-cuts: cut 1 (satisfying the stoichiometric constraint) and cut 2.
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4. Multiperiod Reflux Optimization
Figure 2. STN representation of a distillation operation with offcut recycle.
These cuts can either be distillate or residue. Since cut 1 has the same DC as the fresh charge, several batches of type cut 1 can be enriched by adding reactants (Add1) and reprocessed via a distillation A type operation. The amounts of make-up reactants are selected so that the reactants in the recycled charge meet the stoichiometric constraint and also to make the charge size the same as that of fresh charge (normal batch size ) BS). If the total charge exceeds BS, the excess amount must be stored for possible use in place of fresh charge. It should be noted that, in order for two charges to have the same optimal reflux profiles, not only the DC’s but also the quantities of the charges need to be the same. The use of a storage facility is thus required. A mixture of three batches of the main residue is treated in a similar manner except for the additional step of distillation B, essentially aimed at producing an off-cut that would meet the stoichiometric constraint and a residue which can be directed to the purification steps outside the campaign structure or can be sent to disposal. This campaign structure involves two important features. The first is the recurrent use of the same reflux policy which is the result of using the DC concept. The second is the use of the intermediate storage of offcuts to store the mixture of three off-cuts/residues. These two features help in defining an efficient yet operationally simple reprocessing strategy.
The above campaign structure contains distillation operations which use two types of reflux policiessA and B. These policies or profiles are obtained by multiperiod reflux optimization (Macchietto and Mujtaba, 1996) in which the control vector parametrization (CVP) method is used to discretize the reflux profile over different periods. The piecewise continuous reflux policy is obtained by solving a nonlinear programming problem. We consider a piecewise constant reflux profile and use the reduced sequential quadratic programming code (SQP) developed by Schmid and Biegler (1994) to solve the nonlinear programming problem. The duration of each period (not necessarily cuts) is chosen as the decision variable rather than the absolute time so as to avoid the need for additional inequality constraints such as 0 < t1 < t2 and t1 < t2 < t3. The objective function is a weighted sum of the amount of product and total operation time. Moreover, rather than using some prespecified criterion to define the off-cut (as in the case of the single batch optimization problem), the stoichiometric constraint is used to define the off-cut. The multiperiod reflux optimization problem then becomes 0epen
max rj,θj
∑ i)1
amount of product or reactant in
respective cut - wtf j ) 1, 2, ..., total number of periods (5)
subject to (i) the differential contactor model, (ii) purity constraints on the product cuts, (iii) a stoichiometric constraint on the off-cut, and (iv) bounds on the control variables where rj is the reflux level in period j, n is number of cuts, θj ) tj+1 - tj where tj ) time at which period j ends (t0 ) 0, tf ) total time), and w ) weighting factor (a low value of w yields the maximum distillate problem, whereas a high value leads to the minimum
Figure 3. General optimal campaign structure for distillation with reversible reaction.
1914 Ind. Eng. Chem. Res., Vol. 37, No. 5, 1998 Table 3. Comparison of Base and Campaign Cases case
maximization of
production rate, kmol of EtOAc/h
base campaign
single batch performance overall reactant utilization overall production rate
3.328 2.410 3.328
utilization ratio AcOH/EtOAc EtOH/EtOAc 2.577 1.183 1.883
2.577 1.394 1.694
kmol/h 10.514 1.334 5.114
waste production kmol/kmol of EtOAc 3.159 0.554 1.537
time problem; an intermediate value capturing the actual process economics would give an overall optimal performance for a given production facility). The value of the integer p depends on the number of products that are of interest. The reflux optimization for distillations A and B differs only in the number of cuts (n) and hence in p. Gradients are evaluated numerically by the finite difference approximation since function evaluation is very expensive computationally (around 50 min of CPU on Sparc 5 (Wajge et al., 1997)). Though such an approximation has the potential for introducing numerical errors, good accuracy can be obtained by a careful selection of the step size for the approximation (Rosen and Luus, 1991). We next demonstrate the economic benefits of the proposed campaign structure with the help of a case study. Figure 4. Optimal composition profiles for distillation A.
5. Case Study Consider the system consisting of ethyl acetate (EtOAc), ethyl alcohol (EtOH), water, and acetic acid (AcOH). The boiling points of these species are 350.2, 351.7, 373.2, and 390.4 K, respectively. The reaction of ethanol with acetic acid to produce ethyl acetate is reversible, with stoichiometry given by eq 6.
EtOH + AcOH h EtOAc + water
(6)
The reaction rate constants of the forward and reverse reactions have been reported by Holland (1981). We assume reactive batch distillation is carried out in a packed column. Since the hold-up in the still pot is much larger than the hold-up in the packing, the column hold-up can be neglected. This not only allows the use of the simple stoichiometric constraint for the off-cuts but also allows the formulation of the distillation model with reaction occurring only in the reboiler. The resulting distributed model has been described by Wajge et al. (1997). 5.1. Base Case. Suppose that the distillation is carried out to generate three cuts: a product cut, an off-cut, and a residue, where the product cut composition is specified. The results of the reflux profile optimization (as outlined in the earlier section) obtained in four iterations are shown in Figure 4. The product cut contains 50.2% ethyl acetate. Assuming that this operation is repeated indefinitely for a series of batches, then the mean production rate, reactant utilization, and waste generation rate will be as shown in Table 3. Due to the close boiling points of one of the reactants and one of the products (EtOH and EtOAc, respectively), the amounts of off-cuts and residue are high (each is almost one-third of the fresh feed) which makes their reprocessing inevitable. 5.2. Campaign Cases. On the basis of the campaign structure outlined in the earlier section, we adopt the campaign strategy for this system, as shown in the Figure 5. The optimal reflux and composition profiles for distillation B (obtained in six optimization iterations) are given in Figure 6. Since the amount of total charge
for reprocessing is almost equal to that of fresh charge, no storage is required. The composition and amounts of the various streams are shown in Table 4. Assuming that this campaign strategy is repeated indefinitely, then in each cycle of the campaign three product cuts are produced and only one residue is cut. As shown in Table 3 for the case labeled maximization of the overall reactant utilization, the mean production rate will be 27.58% lower compared to the base case but the reactant utilization will improve quite significantly. Thus, this more complex campaign strategy has allowed reduction in the acetic acid requirement, ethanol requirement, and waste generation (per unit production rate) by 54.09%, 45.91%, and 82.46%, respectively. As can be seen from Figure 3, the product cut is obtained at three different stages. The net time taken to execute each of these tasks is given in Table 5. Assuming that mixing/storage takes negligible time compared to distillation operation, the main residue yields the product cut of the same amount but requires considerably more time (7.5 h as opposed to 3.5 h of fresh charge or off-cut of main batch). Hence, if production rate is the only criterion corresponding to the case labeled maximization of overall production rate in Table 3, then the main residue should not be reprocessed by distillation and the campaign structure shown in Figure 7 will be the optimal strategy. This strategy allows reduction in the acetic acid and ethanol requirement by 26.93% and 34.27%, respectively, for the same production rate as in the base case. The waste generation is reduced by 51.35%. 6. Discussion We have presented a framework for designing optimal operation policies for a campaign of reactive distillation batches. While no formal proof of optimality can be offered for this strategy, it does effectively balance the objectives of good utilization of reactants and a relatively simple operation recipe. It may seem that the treatment of the off-cut to be equivalent to a fresh charge incurs a loss of the energy required in distilling
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Figure 5. Optimal campaign operation of the ethanol esterification case.
Figure 6. Optimal composition profiles for distillation B. Table 4. Details of States state
total kilomoles
EtOH
mole fraction AcOH EtOAc
water
fresh charge product cut off-cut main residue Add1 Add2 Add3 Add4 residue
60.000 23.200 18.910 17.900 4.41 16.200 9.450 23.22 19.350
0.500 0.187 0.135 0.043 0 1.000 0.265 0.407 0.009
0.500 0.018 0.057 0.344 1.000 0 0.735 0.5971 0.238
0 0.293 0.407 0.439 0 0 0 0 0.734
0 0.502 0.401 0.175 0 0 0 0 0.020
Figure 7. Optimal campaign operation for maximum production rate. Table 5. Sources of Product Streams
the off-cut out; however, unless the column dynamics (e.g., rate of reaction, mass-transfer rates) is very slow (which necessitates more time to attain equilibrium conditions at total reflux), this loss is more than compensated by an efficient and simple reprocessing policy. The DC principles are valid even if total reflux operation is not allowed to reach final equilibrium, as long as the extent of a reversible chemical reaction is the same for both fresh and recycled off-cut. However, in such a case, to ensure the same extent of reaction, either actual samples may need to be taken or a rigorous model that simulates the tray dynamics is required to
sources fresh charge intermediate cut of main batch main residue (three batches)
steps
total time for distillation
distillation A mixing, distillation A
3.5 3.5
mixing, distillation B, mixing, distillation A
7.5
predict the dynamic composition profiles developed during the total reflux operation. It should be noted that since the duration of the distillation operation is a part of the objective function, the energy requirement would be reduced indirectly within each distillation step. For a rigorous consideration of the energy requirement, the multiperiod reflux optimization problem can be modified so as to incorporate the energy requirement directly into either the objective function or the constraints and the trade-off between the overall energy requirement and the overall reactant utilization can be readily studied. Effectively, duration of distillation B improves the utilization of reactants at the expense of production rate; therefore,
1916 Ind. Eng. Chem. Res., Vol. 37, No. 5, 1998
if a performance intermediate to these two campaign cases is required, the duration of distillation B needs be constrained accordingly. One can also treat the duration of distillation B as an additional control variable and solve the optimization problem with a costbased objective function. Additionally, if batch size is treated as a control variable, the same campaign structure principles discussed here can be embedded within a general MILP formulation that aims at optimal utilization of the existing production facility (distillation units). The campaign structure discussed here would not be applicable to systems involving irreversible reaction since in such a system the DC becomes a function of the composition of the initial charge and hence no offcut (or, for that matter, any charge with different composition) can be reprocessed using the same distillation recipe as that for the fresh charge. In some simple cases, such as when all the products have lower (or higher) boiling points than all the reactants or when the chemical equilibrium constant value is high, the amount of off-cut may be so small that the energy requirement may offset the reactant recovery achieved through the reprocessing steps; only then would the primary distillation step (distillation A) be needed and no reprocessing of the off-cuts would be required. The campaign structure construction is based on a critical assumption of negligible column hold-up which reduces the criterion for having the same DC to the simple stoichiometric constraint on the distillation charge. In the case of significant column hold-up, composition profiles developed within the column cannot be neglected. Hence, two distillation charges with different compositions can have the same optimal reflux policy only if they generate the same composition profiles throughout the column (not just in the reboiler) at the end of total reflux step. This would not necessarily be satisfied by distillation charges satisfying the stoichiometric constraint since the reaction rates would depend upon the absolute composition and these, coupled with the column dynamics (such as boil-up rate), may yield different total reflux conditions. This is not the case when column hold-up is negligible since, given enough time, the reboiler mixture would attain the chemical equilibrium that would be dependent only on the relative proportions of the reactants and products. Therefore, the off-cut composition is no longer defined by the simple stoichiometric constraint but by the requirement that it should yield the same total reflux conditions. Hence, for given total reflux composition profiles (which correspond to those generated by fresh charge), the possible set of feed compositions need to determined. This determination can be viewed as a reverse total reflux calculation which would entail iterative schemes which, in general, pose severe numerical challenges. 7. Conclusion The concept of distillation characteristic is relevant in the design of campaign structures which minimize off-cut fragmentation and reduce operational complexity
by utilizing a minimum number of distinct distillation tasks. In addition to a better reprocessing strategy, it offers insight into the trade-off between the production rate, reactant utilization, and waste generation through the convenient integration of reprocessing steps with the primary distillation step. Based on this understanding and the cost structure associated with the specific chemical system, a suitable campaign strategy can be designed so as to obtain the desired trade-off. Acknowledgment We gratefully acknowledge Professor L. T. Biegler, Department of Chemical Engineering, Carnegie Mellon University, for permission to use his reduced SQP program for the optimization calculations. Nomenclature BS ) normal batch size, kmol DC ) distillation characteristic t ) time tj ) time at which the period j ends tf ) total batch distillation time w ) weighting factor
Literature Cited Agreda, V. H.; Partin, R. L.; Heise, W. H. High-purity Methyl Acetate via Reactive Distillation. Chem. Eng. Prog. 1990, Feb, 40-46. Christensen, F. M.; Jorgensen, S. B. Optimal Control of Binary Batch Distillation with Recycled Waste Cut. Chem. Eng. J. 1987, 34, 57. DeGarmo, J. L.; Parulekar, V. N.; Pinjala, V. Consider Reactive Distillation. Chem. Eng. Prog. 1992, March, 43-50. Doherty, M. F.; Buzad, G. Reactive Distillation by Design. Chem. Eng. Res. Des. 1992, 70A, 448-458. Holland, C. D. Fundamentals of Multicomponent Distillation; McGraw-Hill Book Co.: New York, 1981. Logsdon, J. S.; Biegler L. T. A Relaxed Reduced Space SQP strategy for Dynamic Optimization Problems. Comput. Chem. Eng. 1993, 17 (4), 367-372. Macchietto, S.; Mujtaba, I. M. In Design of Operation Policies for Batch Distillation; Reklaitis, G. V., Ed.; NATO Advanced Study Institute Series F143; Springer-Verlag: Berlin, 1996; pp 174215. Mayur, D. N.; May, R. A.; Jackson, R. The Time-Optimal Problem in Binary Batch Distillation with a Recycled Waste-Cut. Chem. Eng. J. 1970, 1, 15. Mujtaba, I. M.; Macchietto, S. Optimal Operation of Reactive Batch Distillation. Presented at the AIChE Annual Meeting, Miami Beach, FL, Nov 1992; Paper 135g. Rosen, O.; Luus, R. Evaluation of Gradients for Piecewise Constant Optimal Control. Comput. Chem. Eng. 1991, 15 (4), 273-281. Schmid, C.; Biegler, L. T. Quadratic Programming Methods for Reduced Hessian SQP. Comput. Chem. Eng. 1994, 18 (9), 817832. Wajge, R. M.; Wilson, J. M.; Pekny, J.; Reklaitis, G. V. Investigation of Numerical Solution Approaches to Multicomponent Reactive Batch Distillation in Packed Beds. Ind. Eng. Chem. Res. 1997, 36, 1738-1746.
Received for review July 25, 1997 Revised manuscript received February 3, 1998 Accepted February 4, 1998 IE970527T
