Multicomponent Batch Distillations - American Chemical Society

product cuts and by use of a variable reflux ratio. The effects of these operating parameters, on the recovery of each product cut and of the producti...
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Ind. Eng. Chem. Res. 1999, 38, 4759-4768

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Multicomponent Batch Distillations: Study of Operating Parameters L. Bonny De´ partement de Ge´ nie ChimiquesIUT, 18 Chemin de la Loge, BP 4065, 31029 Toulouse Cedex 4, France

The problem of the optimization of only one multicomponent batch distillation will be formulated by taking into account a new operating parameter corresponding to the start of the collection of product cuts and by use of a variable reflux ratio. The effects of these operating parameters, on the recovery of each product cut and of the productivity, will be explored using digital simulation. Discretization using tiny steps for the reflux ratio can produce significant gains when compared with a constant reflux ratio. Beginning recovery of the main cut other than the first main cut when the mole fraction at the head of the column is equal to the specification of this main cut is not the optimal strategy for a constant reflux ratio. The new formulation of the optimization problem of a batch distillation campaign with the possibility of recycling secondary cuts by use of a variable reflux ratio and by calculation of the optimal moment for the start of recovery of the main cuts will be presented and illustrated by two examples. 1. Introduction Batch distillation is a process commonly used in chemical engineering, particularly in the field of specialization. In batch distillation (Figure 1), the mixture to be separated is initially charged into the reboiler and is vaporized. The overhead vapors are condensed, one part flows back into the column and the other part is collected in different cuts. During the first period, the lightest component (main cut 1) is collected into a product tank until the composition drops to a specified purity level. Then, an intermediate distillate fraction is usually collected (offcut or slop cut or secondary cut). Then, when the concentration of the intermediate component in the distillate reaches a specified level, not necessarily its specified purity level, the second main cut is collected. The procedure is repeated until the heaviest component reaches its specified purity level in the column reboiler. Many authors have been interested in the optimization problem of batch distillation. Sundaram et al.1 proposed a superstructure for a batch distillation system which is represented by a flow sheet of separation steps where the feed is separated into an overhead and a bottom fraction. The reflux ratio of the different steps and the cut location of the steps, defined as the fraction of the feed at the step which is withdrawn as overhead, are optimization variables (the cut location has a different definition in this paper). The overhead and the bottom fraction form the feed to a subsequent separation step or are sent to form product or waste. At the end, all output streams which are suitable are mixed to yield the required product streams and an assigned variable corresponds to the fraction of the bottom stream at one step which is assigned to a main cut or to a slop cut. The authors say the model allows the use of multiple parallel columns at any separation, but the model does not allow for the mixing of waste cuts with fresh feed for subsequent processing. For Farad et al.2 different periods of production of main cuts and offcuts correspond to a batch distillation. The initial time tj-1 of a period j and the final time tj of a period j, the reflux policies for predefined reflux policies (linear, constant,

Figure 1. Batch distillation column for a ternary mixture.

or exponential reflux ratios) are the optimization variables. Farhat et al.2 showed how to produce maximum quantities of main cuts that are produced in a given, finite time (or how to produce minimum quantities of secondary cuts). For Sundaram and Evans1 the objective function is maximization of profit under the horizon time constraint. The approximation of a smooth reflux ratio profile is done by discretizing each cut into a number of subcuts, each subcut having a constant reflux ratio, but allowing the reflux ratio to vary from subcut to subcut. Luyben3 studies ternary systems with slop recycle. He used constant distillate flow rate and flow rates of the form D ) Dmin + Kc(XD1 - XSPECD1) with Dmin given the minimum distillate flow rate and Kc a given constant and XDi and XSPECDi the mole fraction and specified purity of light component i. He supposes that the second main cut begins to be recovered when the mole fraction of the distillate in component 2 is equal to the specification. The curb showing the variation with the time t of a defined capacity factor defined as ∑31Pi/(t + tc), with tc being the time required to empty and

10.1021/ie990196f CCC: $18.00 © 1999 American Chemical Society Published on Web 11/16/1999

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product, etc.) is fixed in the problem and in the examples. The decision variables chosen correspond to (i) the varying reflux ratio policy and the policy of switching product tanks for only one batch distillation and (ii) these preceding variables and the proportions of each slop cut introduced into the reboiler at each batch distillation in a production campaign. 2. Process Description for One Batch and Nomenclature Figure 2. Variation of the three mole fractions at the condenser X1, X2, and X3 with time t (h).

Figure 3. Inputs and outputs of the module for a batch.

recharge the still pot is shown on a graph: the curves show the existence of an optimal reflux ratio for which the capacity factor reaches a maximum point. Farhat2 and Sundaram1 do not show how their objective function varies when a single optimization variable varies. Mujtaba et al.4 presented the simultaneous optimization of design and operation distillation for single- and multiple-separation duties. For the example dealt with by Mujtaba4 for a ternary batch distillation, the recovery of the first component in the first slop cut (amount of component 1 in the slop cut 1/amount of component 1 remaining after the first main cut) is specified, which thus imposes the end of the recovery of the first slop cut and the beginning of the second main cut. The optimization of a campaign of multicomponent batch distillation with reversible reaction was studied by Watje et al.5 by using the reduced sequential quadratic programming code developed by Schmid and Biegler.6 Logsdon and Biegler7 used a relaxed reduced space SQP strategy on a binary batch distillation. Bonny et al.8 presented a methodology for determining the optimal strategy for slop cut recycling in multicomponent batch distillation by arbitrarily fixing the start of the recovery of the main cuts other than the first main cuts and for particular reflux ratio policies: constant reflux ratio during the recovery of one cut or during the time for the whole distillation. Bonny presented superstructures for determining the optimal strategies for handling mixtures with the same NC components but different compositions, with no recycling9 or with recycling10 of slop cuts. The necessity of introducing an optimization variable linked to the beginning of the recovery of a main cut is shown; the problem of the optimization of only one batch distillation is formulated and solved for an example. This paper proposes a formulation of the multicomponent batch distillation campaign optimization problem, with or without recycling. No terminal constraint for the end of the collection of a slop cut (recovery of a

The solution of modeling equations leads to the module found in Figure 3. For one batch, B moles of the mixture of NC components are introduced into the reboiler. The set of initial concentrations is X0 ) (X0,1, X0,2, ..., X0,NC) for the first, second, ..., NCth component. The feed components are numbered in such a way that component 1 is the lightest component and component NC the heaviest in the mixture. XSPEC is the set of specified purity levels: XSPEC ) (XSPEC1, XSPEC2, ..., XSPECNC). The pressure P, the number of plates (NP), and the vapor flow rate V (mol h-1) are given. At the exit of the batch, (i) for the chosen reflux ratio policy, (ii) for the chosen policy of switching product tanks, and (iii) for a set of specified purity levels XSPEC, the model gives the following information: (i) the number of moles for each main cut Pi, with i ) 1, ..., NC and for each slop cut Sj, with j ) 1, 2, ..., (NC - 1), (ii) their respective concentrations, and (iii) the total time t of the batch. Different policies, concerning the reflux ratio and the commutation of receivers of main cuts and slop cuts, can be used to perform the separation. 2a. Reflux Ratio Policy. Two reflux policies are explored: (i) the reflux ratio R is held constant during the batch; (ii) the most general case of a smooth varying reflux ratio is solved by approximating the reflux ratio as a piecewise constant function. The reflux ratio is constant during the distillation of a fixed percentage, of the initial load B0 defined as (∆B/B0) × 100% with ∆B (mol) a fixed step. The smaller the fixed step, ∆B, the better the approximation of the smooth profile of the reflux ratio. The percentages used in this paper are 100% (constant reflux ratio), 5%, and 1%. 2b. The Policy of Switching Product Tanks. For a multicomponent batch distillation, it is necessary to chose when the recovery of the second, third, ..., (NC 1)th main cut starts. Figure 2 shows the distillate composition profiles for a batch distillation of a ternary mixture. Let us consider the most general case, where the slop cuts exist, and for a fixed ratio policy, where R is constant for example. The time t1 of the end of the recovery of the main cut 1 is imposed by the specification constraint XSPEC1. But the time T(2) for starting the collection of the second product cut must be specified: P2 mol could be produced during the period [T(2), t3], or P′2 moles during the period [t′2, t′3], with the same specification XSPEC2. Numerical examples illustrate this point later on (Table 2). For mixtures of four (or more) components, an identical observation may be made for the collection of the third (or more) main cut. Figure 2 shows that the mole fraction at the condenser of component 2 that we call cut location for component 2 (XCUT2 or XCUT for a ternary mixture) corresponds to a switching time, T(2), for the start of the recovery of the second main cut P2. In Figure 2, the value of XCUT corresponds to two points A and B: the cut location corresponds always to the first point, A, in this

Ind. Eng. Chem. Res., Vol. 38, No. 12, 1999 4761 Table 1. Example 1: Common Input Data and Base Case Conditons vapor flow rate: V ) 91.77 mol h-1 capacity of reboiler: B0 ) 100 mol NP′ ) 10 plates P ) 1 atm set of initial mole fractions: X0 ) (0.300; 0.500; 0.200) specified composition for i component XSPECi: XSPEC1 ) 0.970; XSPEC2 ) 0.900; XSPEC3 ) 0.900

Table 2. Examples 1 and 2: Total Recovered Amount, Q, for Different Cut Locations, Time Interval of the Collection of the Second Main Cut, and Comparison with the Optimal Cut Location (T(2), Switching Time; t′, Time of the End of the Collection of P2) XCUT

[T(2), t′] (h)

total amount Q (mol)

gain ) (Qoptimal - Q)/ Qoptimal × 100%

0.760 0.795 0.900

Example 1 (First Mixture) [3.84; 7.34] 63.0 [4.01; 7.64] 64.4 [4.79; 7.89] 59.1

2.2 optimal 8.2

0.620 0.695 0.900

Example 2 (Second Mixture) [0.836; 1.894] 78.0 [0.892; 1.992] 79.7 [1.129; 2.073] 73.7

7.5 optimal 7.5

Figure 4. Variations of the total amount Q and the main cuts P1, P2, and P3 with the cut location XCUT.

paper: beginning recovery of P2 after the mole fraction 2 at the condenser has reached its maximum value does not allow one to recover a quantity without later gain. As soon as the mole fraction at the condenser of component i equals the cut location XCUTi, the recovery of main cut Pi starts (i ) 2, ..., NC - 1). XCUT is the set of specified cut locations: XCUT ) (XCUT2, ..., XCUT(NC-1)). From the moment when the time equals the switching time, T(i), the recovery of main cut Pi starts. T is the set of specified switching times: T ) (T(2), ..., T(NC - 1)). For a mixture of NC components there are (NC - 2) cut locations and (NC - 2) switching times. For the binary batch distillation, this problem of switching product tanks does not arise: the distillation stops when the specification at the reboiler XSPEC2 is reached. 3. Modeling Equations Model equations are the same as those used previously. The assumptions used by Domenech et al.11 have been retained (theoretical plates, negligible liquid holdup, negligible pressure drop, total condenser, constant vapor and liquid flows throughout the column, no feed stream, and ideal equilibrium with partial pressures predicted by Antoine’s correlation). 4. Effect of the Cut Location for a Batch Distillation 4a. Example 1. A cyclohexane-n-heptane-toluene mixture having 100 mol is treated in one batch. The data are listed in Table 1. The set of specifications is XSPEC ) (0.970; 0.900; 0.900). The reflux ratio R is fixed to 8.25, so the time t (t ) 8.422 h) of the duration of the batch distillation is also fixed and does not depend on the cut location or the first or second specifications. So the maximum of the production rate, defined as the total amount of products per unit of time η ) ∑31Pi/t, corresponds to the maximum recovered amount of main

Figure 5. Variations of the total amount Q with XCUT for different XSPEC2.

Figure 6. Variations of the optimal cut location XCUT with the specification XSPEC2 and corresponding total amount Q.

cuts Q ) ∑31Pi. Figure 4 shows the variation with XCUT of the quantities of the recovered main cuts P1, P2, and P3 and of the total quantity recovered. The second main cut is recovered only if the cut location XCUT is chosen in the interval [0.752; 0.940] (the maximum value of the mole fraction for component 2 is 0.940). The total amount Q and the second main cut reach a maximum, determined by the golden section search, for XCUT ) 0.795. Table 2 shows the main results for three cut locations and the importance of the choice of XCUT: the gain achieved by starting the recovery of the second main cut P2 at the optimal cut location rather than at the specification of the second main cut is equal to 8.2%. Figure 5 shows the variation of the total recovered amount Q of main cuts with the cut location, for different specifications of the second main cut XSPEC2 (XSPEC1 ) 0.970; XSPEC3 ) 0.900). A discontinuity appears when the second main cut is recovered. When the specification XSPEC2 increases, the value of this discontinuity increases also. Each curve has a maximum. Of course, the total recovered amount corresponding to this optimum decreases when the specifi-

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Table 3. Example 2: Data for the Second Mixture (log10 P ) a - [b/(T + c)]; P (Torr), T (°C)) index of boiling component point (°C) 1 2 3

184.4 245.0 284.0

thermodynamic coefficients a b c 7.63846 7.96718 7.97685

1976.3 2502.2 2654.8

231 247 237

molecular weight 93.1 215.9 286.0

Figure 7. Variations of the total amount Q of main cuts with the cut location XCUT for different specification XSPEC2.

Figure 8. Variations of the optimal cut location XCUT with the specifications XSPEC2 and corresponding total amount Q (second mixture, R ) 1.4).

cation XSPEC2 increases. The optimal cut location XCUToptimal, determined by the golden section search, increases when XSPEC2 increases. Figure 6 shows that the curve of the optimal cut location XCUToptimal, as a function of the specification XSPEC2, is a straight line. The variation of the corresponding total amount is also drawn: for a specification XSPEC2 superior to 0.940, the curve Q ) f(XSPEC2) is horizontal and corresponds to the total amount of main cuts 1 and 2. 4b. Example 2. Let us consider a second mixture. The thermodynamic coefficients are listed in Table 3; the other data correspond to Table 1. For a fixed reflux ratio, R equals 1.4, and for four sets of specifications, (0.97; 0.80; 0.90), (0.97; 0.85; 0.90), (0.97; 0.90; 0.90), and (0.97; 0.95; 0.90), where only XSPEC2 varies, the curves of the total amount Q as a function of the cut location XCUT are drawn (Figure 7). The duration of the batch has the same value in the four cases: t ) 2.18 h. For an increasing XCUT, there is a discontinuity when the recovery of the second main cut P2 begins, and then the curve goes through a maximum and decreases. The second main cut is not collected if XCUT is greater than 0.983 (which is the maximum value of the mole fraction at the condenser for component 2). Similar to the previous example, Figure 8 shows that the variation of the optimal cut location with the specification XSPEC2 is linear and, as expected, the value of the total amount decreases for an increasing specification, XSPEC2. Table 2 shows the main results for three cut locations with XSPEC ) (0.97;0.90;0.90). The gain achieved by starting the recovery of the second main cut P2 at the optimal

cut location rather than at the specification of the second main cut is equal to 7.5%. These examples show the necessity of inserting the cut location into the research for the optimal batch distillation. 5. Influence of the Reflux Ratio in Multicomponent Batch Distillation 5a. Variation of the Production Rate with the Reflux Ratio, the Other Operating Parameters Being Fixed. The same mixture of cyclohexanen-heptane-toluene as in example 1 (Table 1), having 100 mol, is treated in one batch with XSPEC ) (0.97;0.90;0.90). The recovery of the second main cut starts when XCUT is equal to 0.80. Figure 9 shows the variation with the reflux ratio of (i) the quotients of each main cut by the total duration t of the batch distillation, noted η1 ) P1/t, η2 ) P2/t, and η3 ) P3/t; (ii) the production rate η ) ∑31Pi/t (so η1 + η2 + η3 ) η); (iii) the total duration t of the batch. The production rate, η ) f(R), has a local maximum which corresponds to the maximum of η3 ) f(R), and a global maximum. The curves η2 and η have a discontinuity for R equal to 7.34 when the recovery of the second main cut P2 starts. The optimal values of the reflux ratio R for η1, η2, η3, and η are reported in Table 4. As expected, the duration t increases with R. For the same mixture and the same set of specifications, Figure 10 shows the variations, with the reflux ratio, of η1, η2, η3, and η, the duration t of the batch for XCUT ) 0.5. The curves of η1, η3 and t are the same as those for the previous example. If η2 has the same general run as that in the previous example, its maximum, for R equal to 14.74, has a smaller value. When the reflux ratio increases, the curve of the production rate η goes first through a local maximum for R equal to 4.80, which corresponds to the maximum of η3, and then reaches a second local maximum for R equal to 9.76, which corresponds to the maximum of η1 + η3 ) (P1 + P2)/t, then decreases, has a discontinuity for R equal to 14.30, and has a global maximum for R equal to 14.54. Table 4 sums up the different maxima of η1, η2, η3, η, and (η1 + η3). Generally, for a fixed cut location, the curve η ) f(R) has different maxima linked to the maxima of the three quotients η1, η2, η3, and their sums. Figure 11 shows the variation, with the cut location XCUT, of the optimal reflux ratio calculated by the golden section search. The curve Roptimal ) f(XCUT) has a minimum [Roptimal ) 9.03; XCUT ) 0.825] and the value of the production rate is 7.71 mol h-1. The production rates which correspond to the pair [XCUT, Roptimal] are also shown: the corresponding curve reaches a maximum; the problem of the optimal pair is shown later on. This optimal pair is [XCUT ) 0.761; R ) 9.252] and the optimal production rate is 7.83 mol h-1. 5b. Influence of the Reflux Ratio on the Optimal Value of the Cut Location XCUT. Let us consider the batch distillation of 100 mol of the mixture of cyclohexane-n-heptane-toluene (Table 1) with the following set of specifications: XSPEC ) (0.97; 0.85; 0.90). For a given value of the reflux ratio R, the optimal value of the start of the recovery of the second main cut XCUToptimal is searched for with the golden section search, the criterion being the production rate (for a fixed reflux ratio R, the total amount Q, and the production rate have the same optimum). Figure 12

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Figure 9. Variations of the production rate, P1/t, P2/t, and P3/t, and of the duration t of the batch with the reflux ratio R (XCUT ) 0.8; XSPEC2 ) 0.9). Table 4. Optimal Reflux Ratios and Corresponding Values of the Production Rate η, of the Quotients P1/t, P2/t, and P3/t, and of (P1 + P3/t) for Two Fixed Cut Locations (P3/t)max ) ((P1 + P3)/t)max ) ηsecond local max (P1/t)max (P2/t)max ηfirst local mix ηglobal max Roptimal value (mol h-1) Roptimal value (mol h-1)

11.40 1.79

Case 1, XCUT ) 0.8 8.60 4.80 9.03 4.30 2.21 7.78

11.40 1.79

Case 2, XCUT ) 0.5 14.74 4.80 14.54 3.39 2.21 6.42

no no

9.76 3.52

shows the variation of the optimal cut location with the reflux ratio R: when the given reflux ratio R increases, the optimal cut location XCUT decreases. The curve of the optimal cut location, XCUToptimal, as a function of the reflux ratio, is linear. The curve of the production rate corresponding to this pair [R, XCUToptimal] has a maximum equal to 9.14 mol h-1 for [R ) 8.21; XCUT ) 0.590], this optimization problem being shown later on. As expected, the total amount Q of main cuts increases with the reflux ratio. 6. Optimization of Only One Batch Distillation 6a. Formulation of the Optimization Problem. Optimization Variables. These variables correspond to the reflux ratio policy and the switching product tanks policy. For a constant reflux ratio policy, the chosen variable linked to the switching product tanks policy is the cut location, but for a varying reflux ratio policy, the chosen variable is the switching time T(i) for removing the degeneration. For a mixture of NC components there are (NC - 2) cut locations (or switching times). In other words, N is the degree of discretization of the reflux ratio: a new reflux ratio is allowed after distilling each other ∆B moles, so N corresponds to the amount distilled divided by the step ∆B. In the last step, the reflux ratio is equal to RN and the batch distillation finishes when the specification XSPECNC into the reboiler is reached. The step ∆B is chosen. So if the degree of discretization is equal to N, there are N optimization variables corresponding to the set R of reflux ratios: R ) (R1, R2, ..., RN).

The number of optimization variables is Nopt ) N + (NC - 2). Objective Function. The chosen objective function is the production rate. The problem is then NC

P(1) Max η )

∑1 Pi/t

R, XCUT (or T) This criterion is nonlinear with respect to the considered variables. Other criteria may be used. In this paper such problems are solved using available software, IMSL Math/Library.12 6b. Problem Example. A mixture of cyclohexanen-heptane-toluene (100 mol) are to be treated during only one batch distillation; the data are in Table 1. The set of specifications is (0.97; 0.90; 0.90). Three different reflux policies are considered: Case 1, the reflux ratio is kept constant for the treatment of the whole batch; Case 2, the step is ∆B ) 5 mol, for a degree of discretization N equal to 17; Case 3, the step is ∆B ) 1 mol (N ) 83; 17.52 mol remain in the reboiler at the end of the optimal distillation). Table 5 presents the optimal solutions and the main results; Figure 13, parts a-c, shows the variations of the three mole fractions in the tanks receiving the different cuts; Figure 14, parts a-c, shows the variations of the three mole fractions at the condenser. Compared with the constant reflux policy, the varying reflux ratio policy allows gains equal to 12.3% (case 2) or 14.3% (case 3). These optimal discrete reflux ratio policies allow a better amount of the three recovered main cuts P1, P2, and P3, and the durations of the batch, for cases 2 and 3, are shorter than the duration with a constant reflux ratio policy. When the distillate is diverted into another tank, a discontinuity appears on the curves of the three mole fractions as a function of the time t. For the cases 2 and 3, the optimal reflux ratio increases during the recovery of the main cuts 1 and 2 or during the recovery of the two offcuts and takes a smaller value when the recovery of another cut starts. The greater the increase in the degree of discretization, the smaller the variations of the mole fractions in a tank: the curves of the three mole fractions in Figure 13, parts a-c, are flattened out when the degree of

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Figure 10. Variations of the production rate, P1/t, P2/t, and P3/t, and the duration t of the batch with the reflux ratio R (XCUT ) 0.5; XSPEC2 ) 0.9). Table 5. Main Results for the Problem Example case 1 production rate (mol h-1) gain ) (ηCase i - ηCase 1)/ ηCase 1 × 100% t (h) Q ) P1 + P2 + P3 (mol) P1/t (mol h) P2/t P3/t P1 (mol) P2 (mol) P3 (mol) S1 (mol) S2 (mol) optimal T(2) (h) optimal XCUT optimal R

Figure 11. Variations of the optimal reflux ratio R with the cut location XCUT and the corresponding production rate (XSPEC2 ) 0.9).

7.83 9.27 72.5 1.67 4.32 1.84 15.4 40.0 17.0 21.9 5.63 4.21 0.761 (9.25)

case 2

case 3

8.80 12.4

8.95 14.3

9.19 80.9 2.15 4.74 1.91 19.8 43.5 17.5 15.2 3.90 4.64 [0.715; 0.840] Figure 13b

9.02 80.8 2.24 4.76 1.94 20.2 43.0 17.5 14.8 4.40 4.65 [0.683; 0.856] Fiure 13c

The general rule of thumb (Rose13) is to take off the slop cut quickly at a comparatively low and constant reflux ratio. This rule is neither confirmed by the optimal solution obtained for this problem nor by the optimal solution reported in Figure 17, parts a and b. This shows the necessity of a clear formulation of the problem, taking into account the chosen operating parameters and its solution. 7. Optimization of a Campaign of Several Multicomponent Batch Distillations with Potential Recycling of Slop Cuts

Figure 12. Variations of the optimal cut location XCUT with the reflux ratio R, corresponding total amount of main cut Q, and the production rate.

discretization increases. This phenomenon can be found in Figure 14, parts a-c: between two discontinuities, the curves corresponding to the variations of the three mole fractions at the condenser are flattened out when the degree of discretization increases. The optimal switching time T(2) is reported in Table 6. In cases 2 and 3, the beginning of the recovery of the second main cut corresponds to discontinuities of the reflux ratio and of three mole fractions at the condenser.

The superstructure proposed by Bonny et al.8 to solve the problem of the optimization of a campaign of several multicomponent batch distillations with potential recycling of slop cuts is briefly recalled for a ternary mixture. Then, the problem is stated, including the search for the best location of the beginning of the recovery of the main cuts and using a varying reflux ratio policy. Last, the problem is illustrated by two examples. 7a. Operating Strategy for Slop Cut Handling. Qi mol, of a mixture of NC components (NC ) 3), of which the set of initial mole fractions is X0 ) (X0,1 , X0,2, ..., X0,NC), must be treated with only one column with a reboiler of maximal capacity, B0. The number, Nbatch, of batches is fixed and Qi e (NbatchB0). XSPEC is the set of specified purity levels. am,p is the proportion of

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Figure 13. Variations with the time t of the reflux ratio R and the three mole fractions in the tank (Xm,1, Xm,2, and Xm,3): (a) case 1; (b) case 2; (c) case 3.

Figure 14. Variations with the time t of the reflux ratio R and the three mole fractions at the condenser (X1, X2, and X3): (a) case 1; (b) case 2; (c) case 3.

the first offcut, Sm,1, added at batch p and bm,p is the proportion of the second offcut, Sm,2, added at batch p. It is assumed that all of a given slop cut is not necessarily reworked. Figure 15 shows the superstructure for a ternary mixture. Batch 1. B0 mol (composition X0,i; i ) 1-3) are introduced. For the chosen set of reflux ratios R1, for the fixed cut location XCUT1,2 (or the fixed switching time T1(2)) and for the fixed set of specifications XSPEC, the number of moles of the main cuts, P1,1, P1,2, and P1,3, and of the slop cuts, S1,1 and S1,2, their respective compositions, and the total time, t1, of the batch are calculated.

Batch Nbatch: The remaining quantity of the initial mixture and ADDNbatch,

m)(p-1)

Batch p: ADD )

∑

(am,pSm,1 + bm,pSm,2)

m)1

(see Figure 15) ADD moles is the quantity of the previous slop cuts added as well as (B0 - ADD) moles of the initial mixture. The set of reflux ratios Rp, for the fixed cut location XCUTp,2 (or the fixed switching time Tp(2)) and for the fixed set of specifications XSPEC, correspond to Pp,1, Pp,2, and Pp,3 mol of main cuts and Sp,1 and Sp,2 mol of slop cuts, with their compositions and the total time, tp, of the batch.

m)(Nbatch-1)

ADDNbatch )

∑

m)1

(am,NbatchSm,1 + bm,NbatchSm,2)

are introduced into the reboiler. The set of reflux ratios RNbatch, for the fixed cut location XCUTNbatch,2 (or the fixed switching time TNbatch(2)) and for the fixed set of specifications XSPEC, correspond to PNbatch,1, PNbatch,2, and PNbatch,3 mol of main cuts and SNbatch,1 and SNbatch,2 mol of slop cuts, with their compositions and the total time tNbatch of the batch. B0 moles are introduced in the reboiler of capacity B0 mol at each batch distillation, except for the last one: different feed policies could be used, but the form of the superstructure would be the same and only minor modifications should be necessary in the constraints statement. 7b. Formulation of the Optimization Problem. Optimization Variables. The optimization variables of concern are as follows: (i) The set of the proportions of the first slop cuts A ) {aij} and second slop cuts B ) {bij}, from the batch i added at the batch j with i ) 1, 2, ..., (Nbatch - 1) and j ) (i + 1), ..., Nbatch. There are 2(1 + 2 + ... + (Nbatch - 1)) ) Nbatch(Nbatch - 1) corre-

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Table 6. Optimal Solution for the Batch Distillation of 250 mol with Recycling batchp

Rp,1

(Sp,1; Sp,2) (mol h-1)

tp (h)

(25.0; 7.30) (25.0; 7.30) (19.1; 3.68)

8.73 8.73 7.99

Example 2: Varying Reflux Ratio (η′′ ) 9.13 mol h-1) [0.715; 0.840] (19.8; 43.5; 17.5) [0.715; 0.840] (19.8; 43.5; 17.5) [0.774; 0.847] a1,3 ) 1, a2,3 )1 (25.0; 30.0; 8.30)

(15.2; 3.90) (15.2; 3.90) (15.0; 2.09)

9.19 9.19 6.30

Tp(2) (h)

XCUTp

am,p and bm,p non-null values

(Pp,1; Pp,2; Pp,3) (mol h-1)

1 2 3

8.62 8.62 6.94

Example 1: Constant Reflux Ratio (η′′ ) 8.35 mol h-1) 4.02 0.774 (13.4; 37.6; 16.7) 4.02 0.774 (13.4; 37.6; 16.7) 4.60 0.745 a1,3 ) 1, a2,3 ) 1 (34.1; 35.5; 7.59)

1 2 3

Figure 13b Figure 13b Figure 17a,b

4.64 4.64 3.72

(i) conservation of the mass:

∑n am,n e 1 and ∑n bm,n e 1 with m ∈ {1, 2, ..., (Nbatch - 1)} and n ∈ {(m + 1), ..., Nbatch}.

(ii) limitation of the capacity of the reboiler for batches 3-Nbatch: m)(n-1)

∑

(am,nSm,1 + bm,nSm,2) e B0

m)1

with n ∈ {3, ..., Nbatch}. Figure 15. Superstructure associated with Nbatch batch distillations for a ternary mixture (am,p, proportion of the first offcut Sm,1 added at batch p; bm,p, proportion of the first offcut Sm,2 added at batch p; i ∈ [1, 2, ..., (m - 1)]; j ∈ [1, 2, ..., (p - 1)]).

sponding variables. (ii) The given degree of discretization is Np at the batch p. The set of reflux ratios for the batch p is Rp ) {Rp,i}, with i ) 1, 2, ..., Np and p ) 1, 2, batch Np corresponding variables. ..., Nbatch. There are ∑N 1 (iii) The cut location XCUTp,2 (or the switching time Tp(2)), for each batch p with p ) 1, 2, ..., Nbatch. There are Nbatch corresponding variables for a ternary mixture. The number, Nopt, of optimization variables for a ternary mixture is Nbatch

Nopt ) Nbatch(Nbatch - 1) + Nbatch +

∑1 Np ) Nbatch

NbatchNbatch +

∑1 Np

Objective Function. The chosen objective function is the production rate defined as the ratio of the total amount of main cuts produced to the total time t of the Nbatch batch distillations of the campaign. R′′ corresponding to the whole sets of reflux ratios Rp, with p ) 1, 2, ..., Nbatch, XCUT the set of cut locations, and T the set of switching times, the problem is then j)Nbatch (i)NC)

P(3)

Max η′′ )

∑ ∑ j)1 i)1

Pi,j

t R′′,A,B,XCUT (or T)

Other types of criteria could be used. Constraints. The constraints are given by the following:

(iii) all the initial mixture must be treated:

{

∑ ∑n (am,nSm,1 + bm,nSm,2) m

}

e (NbatchB0 - Qi)

with m ∈ {1, 2, ..., (Nbatch - 1)} and n ∈ {(m + 1), ..., Nbatch}. The constraints concerning the limitation of the capacity of the reboiler can be eliminated if (Nbatch 1)B0 e Qi. 7c. Example Problems. The problems under consideration are NLP (nonlinear programming) problems. In this paper, these NLP problems are solved using IMSL Math/Library.12 The illustrative examples concern the ternary system: cyclohexane-n-heptane-toluene; 250 mol of this mixture must be treated during a campaign of three batch distillations in a column of which the reboiler has a maximum capacity B0 ) 100 mol. The aim is to obtain the best production rate η′′. The data are listed in Table 1. The set of specifications is XSPEC ) (0.97; 0.90; 0.90). Example 1. The reflux ratio is held constant for each batch. There are 12 optimization variables and 3 constraints. The optimal solution, reported in Table 6 and Figure 16, corresponds to the recycling of the first slop cuts of batch 1 and batch 2 at the last batch and the production rate is 8.35 mol h-1. Without recycling, the optimal production rate is 7.83 mol h-1 with (R ) 9.25; T(2) ) 4.21 h) and with a cut location XCUT ) 0.761 (Table 5, case 1). The recycling allows a gain of the production rate of [(8.35 - 7.83)/7.83] × 100% ) 6.6%. At the last batch, 100 mol (50 mol of the mixture to be treated and 50 mol of the first slop cuts of batches 1 and 2), of which the set of initial mole fractions is (0.458;0.439;0.103), are introduced: the optimal solution for the problem P(1) is (R ) 6.54; T(2) ) 4.52 h) with XCUT ) 0.759 and differs from the optimal values for this last stage of the problem P(3) (R3 ) 6.94; T3(2) ) 4.60) with XCUT3,2 ) 0.745. So for each stage (batch distillation), the optimal solution of the subproblem P(1) is different from the optimal solution of the global problem P(3).

Ind. Eng. Chem. Res., Vol. 38, No. 12, 1999 4767

Figure 16. Optimal solution for the examples 1 and 2 (constant and varying reflux ratios). The first slop cuts of batches 1 and 2 are recycled at the last batch.

Figure 17. Variations with the time t of the reflux ratio and of the three mole fractions. (a) in the receivers and (b) at the condenser for the third distillation for the optimal solution.

Example 2. A varying reflux ratio is used. The chosen step is: ∆B ) 5 mol. So for batches 1 and 2 the chosen degree of discretization is N1 ) N2 ) 17. By using (i) the same strategy as that in the constant reflux ratio (recycling the first slop cuts in the last distillation), (ii) the optimal reflux ratio policy which had been calculated beforehand (Table 5, case 2), so with a degree of discretization which equals 17 for batches 1 and 2, (iii) the optimal values of the constant reflux ratio and cut location for the third batch calculated in example 1 (R3,1 ) 6.94; T3(2) ) 4.60), the computation finds that 80.38 mol are introduced in the third distillation and the third main cut of batch 3, P3,3, is equal to 7.54 mol. The degree of discretization, N3, for the third distillation is chosen to be equal to 15 with ∆B ) 5 mol. There are 58 optimization variables and 3 constraints. The optimal solution corresponds to Table 6 and to Figure 16 with the recycling of the first slop cuts of batch 1 and batch 2 at the last batch and the optimal production rate is 9.13 mol h-1. The optimal values of the reflux ratios (Figures 13b and 14b) and of the cut locations for the first and second distillations correspond to the solution of the preceding solution of problem P(1) (Table 5, case 2) when the goal is to treat 100 mol of a mixture of which the initial set of mole fractions is (0.3; 0.5; 0.2), in only one batch distillation. At the last batch, 83.38 mol (50 mol of the mixture to be treated and 33.38 mol of the first slop cuts of batches 1 and 2), of which the initial set of mole fractions is (0.4157; 0.4582; 0.1261), are introduced. The optimal values of the reflux ratios of the third distillation are shown in Figure 17a,b. The optimal switching time of the third distillation T3(2) is

3.72 h. The optimal value of the cut location for the third distillation is degenerated and corresponds to the interval (0.774; 0.847). These determined values for the third stage of problem P(3) are also the solution of problem P(1) when the goal is to treat 83.38 mol of a mixture of which the initial set of mole fractions is (0.4157; 0.4582; 0.1261), in only one batch distillation with N3 ) 15 and ∆B ) 5 mol. Contrary to example 1, the values of the reflux ratios and of the switching times for this problem P(3) are the optimal values of the corresponding subproblem. On the other hand, the reboiler was filled to its maximum capacity for the constant reflux ratio policy; however, this is not the case for this varying reflux ratio policy. Last, in the two cases, all, the first slop cuts are recycled. The differences observed are to be related (i) to the fact that the optimal recovery of main cuts, for a variable reflux ratio, is closer to a distillation with constant distillation composition than a recovery with a constant reflux ratio and (ii) to the values of secondary cuts set forth in Table 6. For this chosen varying reflux ratio policy, the recycling allows for a gain: [(9.13 - 8.80)/8.80] × 100% ) 3.75%. If we compare the constant reflux ratio policy without recycling and the optimal varying reflux ratio policy with recycling, the gain is [(9.13 - 7.83)/7.83] × 100% ) 16.6%. Figure 17a,b shows the variation of the three mole fractions in the tanks and at the condenser for the third distillation. Conclusion and Future Work The problem of the optimization of only one multicomponent batch distillation has been formulated by taking into account a variable reflux ratio and by calculating the optimal moment for the start of recovery of a main cut other than a first main cut. Other objective functions can be chosen. The chosen operating parameter linked to the start of recovery of the main cut is the corresponding mole fraction at the top of the column or, for avoiding the degeneration for a varying reflux ratio, to the time of the start of recovery; another parameter could be used: recovery of a component in the offcut, the percentage distilled, etc. The effects of these operating parameters have been explored for the multicomponent batch distillation of a ternary mixture. There is an optimal cut location for a constant reflux ratio or an optimal interval for a varying reflux ratio. This optimal cut location depends on the specification and on the reflux ratio. Beginning recovery of the main cut other than the first main cut when the mole fraction at the head of the column is equal to the specification of this main cut is not the optimal strategy for a constant reflux ratio. Each main cut has an optimal constant reflux ratio and an optimal reflux ratio exists for the distillation. The discretization of the reflux ratio, using a small step, approximates a variable smooth reflux ratio and leads to significant gains. During the recovery of the main cuts, a decrease in the discretization step leads to a lessening variation of the concentrations in the receivers (the curves of the mole fractions in the receivers as a function of the time are flatter). The optimal reflux ratio policy does not correspond to a low and constant reflux ratio during the recovery of a slop cut. The formulation of the optimization problem of a batch distillation campaign with the possibility of recycling secondary cuts by using a variable reflux ratio and by calculating the optimal moment for the start of recovery of the main cuts has been presented and illustrated by two examples.

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Ind. Eng. Chem. Res., Vol. 38, No. 12, 1999

Future studies in multicomponent batch distillation will include models with holdup and recycling of slop cuts at appropriate times. Notation A ) set of proportions ai,j (first slop cut) B ) set of proportions bi,j (second slop cut) R ) set of reflux ratios for only one batch distillation Ri ) set of reflux ratios for the ith batch distillation of the campaign of distillations T ) set of the switching times X0 ) set of initial concentrations XSPEC ) set of specified compositions XSPECi XCUT ) set of specified cut locations ai,j ) proportion of Si,1 added at batch j bi,j ) proportion of Si,2 added at batch j B ) total amount in the reboiler (mol) B0 ) maximal capacity of the reboiler (mol) D ) distillate flow rate (mol h-1) Ki,p ) equilibrium coefficient for component i on plate p L ) liquid flow rate (mol h-1) n ) component index N ) number of the discretization steps (only one batch distillation) Nl ) number of the discretisation steps for the lth batch distillation Nbatch ) number of batch distillations NC ) number of components np ) number of plates in the column (including reboiler and condenser) NP′ ) number of plates in the column (without reboiler and condenser) P ) pressure Q ) total recovered amount of main cuts (mol) Qi ) initial amount of mixture to be treated (mol) Pj ) production cut related to the jth key component (only one distillation) (mol) Pi,j ) production cut related to the jth key component at batch i (mol) R ) reflux ratio (quotient of the liquid flow rate by the vapor flow rate) Ri ) reflux ratio for the ith step (only one distillation) Rp,i ) reflux ratio for batch p during step i S1 and S2 ) first and second slop cut (only one distillation) (mol) Si,1 and Si,2 ) first and second slop cut at batch i (mol) t ) total time (h) ti ) total time for batch i (h) Tp ) temperature on plate p T(i) ) switching time for the main cut Pi: the recovery of Pi starts at time T(i) (only one distillation) (h) Tp(i) ) switching time for the main cut Pp,i: the recovery of Pp,i starts at the time Tp(i) (campaign of p distillations) (h) V ) vapor flow rate (mol h-1) Xi ()Xi,np) ) mole fraction of component i at the condenser Xi,p ) liquid composition of component i on plate p Xm,i ) mole fraction of component i in the receiver X0,i ) initial mole fraction of component i in the mixture

XCUT ) cut location (the recovery of main cut 2 starts as soon as the mole fraction of component 2 at the condenser is equal to the XCUT ternary mixture) XCUTi ) cut location for component i (the recovery of main cut i starts as soon as the mole fraction of component i at the condenser is equal to XCUTi) XCUTi,2 ) cut location at batch i (the recovery of main cut Pi,2 starts as soon as the mole fraction of component 2 at the condenser is equal to XCUTi,2 ternary mixture) XSPECi ) specified composition for component i βj ) weighting factor for product Pj ∆B ) fixed step for the discretization (mol) η′ ) production rate (total amount of products per unit of time with appropriate weighting factors βj on each product (mol h-1) η,η′′ ) production rate (total amount of products per unit of time with βj ) 1) (mol h-1) ηi ) quotient of the main cut i Pi by the total time t of the batch distillation (mol h-1)

Literature Cited (1) Sundaram, S.; Evans, L. B. Synthesis of Separations by Batch Distillation. Ind. Eng. Chem. Res. 1993, 32, 500. (2) Farhat, S.; Czernicki, M.; Pibouleau, L.; Domenech, S. Optimization of Multiple-Fraction Batch Distillation by Nonlinear Programming. AIChE J. 1990, 36, 1349. (3) Luyben, W. L. Multicomponent Batch Distillation: 1. Ternary Systems with Slop Recycle. Ind. Eng. Chem. 1988, 27, 642. (4) Mujtaba, I. M.; Macchietto, S. Simultaneous Optimization of Design and Operation of Multicomponent Batch Distillation ColumnsSingle and Multiple Separation Duties. J. Process Control 1996, 6 (1), 27. (5) Watje, R. M.; Reklaitis, G. V. An Optimal Campaign Structure for Multicomponent Batch Distillation with Reversible Reaction. Ind. Eng. Chem. Res. 1998, 37 (5), 1910. (6) Schmid, C.; Biegler, L. T. Quadratic Programming Methods for Reduced Hessian SQP. Comput. Chem. Eng. 1994, 18 (9), 17. (7) Logsdon, J. S.; Biegler, L. T. A Relaxed Reduced Space SQP Strategy for Dynamic Optimization Problems. Comput. Chem. Eng. 1993, 17 (4), 367. (8) Bonny, L.; Domenech, S.; Floquet, P.; Pibouleau, L. Strategies for Slop Cut Recycling in Multicomponent Batch Distillation. Chem. Eng. Process. 1994, 33, 23. (9) Bonny, L. Strategies for Handling Mixtures in a Campaign of Several Multicomponent Batch Distillations. Chem. Eng. Process. 1995, 34, 401. (10) Bonny L. Strategies for Handling Mixtures in Multicomponent Batch Distillations with Slop Recycle. Chem. Eng. Process. 1995, 34, 391. (11) Domenech, S.; Enjalbert, M. Program for Simulation Batch Rectification as Unit Operation. Comput. Chem. Eng. J. 1981, 5, 181. (12) IMSL Fortran Numerical Librairies V3.0 for Window. Mathematic and statistic functions from Visual Numerics. (13) Rose, L. M. Distillation Design in Practice; Elsevier: New York, 1985; p 95.

Received for review March 18, 1999 Revised manuscript received July 20, 1999 Accepted September 20, 1999 IE990196F