Multicomponent Batch Distillations Campaign: Control Variables and

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Ind. Eng. Chem. Res. 2006, 45, 8998-9009

Multicomponent Batch Distillations Campaign: Control Variables and Optimal Recycling Policy Luc Bonny* De´ partement HSE, IUT de l’Oise, UniVersite´ de Picardie Jules Verne, 13 alle´ e de la, Faı¨encerie, 60100 Creil, France

To optimize a process, one first must determine the variables acting on the process. This publication will inventory the control variables in various campaigns of multicomponent batch distillations with and without the recycling of offcuts. The problems that are involved in optimizing these multicomponent batch distillations campaigns then will be stated. The chosen optimization variables are the feed time at the reboiler, the recovery time of a main cut, the proportion of a recycled cut at a batch, and the reflux ratio. The results show that, overall, the optimum values of a batch distillations campaign of a given product are not transferable to another different campaign for the same product and the same composition. The optimal strategy corresponds to the recycling of the second offcut obtained at a batch to the following batch distillation when the added cut and the contents at the reboiler have similar compositions, with the first offcuts being put together for a separate batch distillation. 1. Introduction Batch distillations are frequently an important part of batch processing. In a multicomponent batch distillation (see Figure 1), the mixture of NC components (where NC denotes the number of components) is introduced into the reboiler and is vaporized. The overhead vapors are condensed, and one part flows back into the column, while the other part is collected in different cuts. In the latter set of cuts, the first is the main cut 1, which is rich in the lightest component 1; then the first intermediary cut (the offcut, or the slop or secondary cut); then the second main cut, which is rich in component 2; then the second offcut; and so on, until the recovery, in the column, of the last main cut, which is rich in the heaviest component. A campaign of Nbatch multicomponent batch distillations (where Nbatch represents the number of batch distillations) has Nbatch × (NC - 1) offcuts for a mixture of NC components; therefore, the loss of valuable product in the offcuts can be high. Collecting the offcuts and reprocessing them in subsequent batches can increase recovery. It is necessary to optimize this campaign of multicomponent batch distillations. However, the optimization of a process first necessitates determination of all the variables that are acting on the process. This publication will inventory the control variables in a common campaign of multicomponent batch distillations using an existing column. Two problems that optimize several control variables then will be presented. Control variables are the variables that can be controlled. The heat-transfer rates to be provided at the reboiler and at the condenser are two control variables. When a given quantity of a cut of known composition is added during a batch distillation, the temperature and the pressure of the added cut, the feed location, the feed rate, and the time at which feeding begins are control variables. This publication addresses the optimization of a campaign with an existing column: the number of trays and the size of the reboiler are not control variables. The number of control variables and the number of design degrees of freedom for steady-state process have been the subject of various articles in books and reviews, such as those by Luyben1 and by Henley and Seader.2 * To whom correspondence should be addressed. Fax: 03 44 64 46 55.

Figure 1. Batch distillation column of NP plates for a ternary mixture.

However, when batch processing is examined, there are few detailed analyses of the various control variables, of the variables that are given, and of the variables to be optimized. The effects of the variations of the optimization variables surrounding their optimal values are rarely a focal point of interest. Previous Work. Some articles have examined multicomponent batch distillation with recycled offcuts. Luyben3 traced the curve of a capacity factor that was defined as ∑31 Pi/(t + tc), where t the time of the batch, Pi the amount of main cut i, and tc is the time required to empty and recharge the still pot, according to several variables (such as the distillate flow rate for different volatility, the number of trays, and the size of the initial charge to the still pot). All the offcuts are recycled at the beginning of the next batch. The recovery of the second main

10.1021/ie0609057 CCC: $33.50 © 2006 American Chemical Society Published on Web 11/29/2006

Ind. Eng. Chem. Res., Vol. 45, No. 26, 2006 8999

cut starts when the concentration of the intermediate component in the distillate reaches its specified purity level, XSPEC2. The distillate flow rate is constant or has the following form:

D ) Dmin + Kc(XDi - XSPECi) where Kc is a given constant and Dmin is the minimum distillate flow rate. Quintero-Marmol and Luyben4 compared six strategies for slop handling, with a constant reflux ratio during the batch. The “total slop recycling strategy” is the preceding schema that Luyben3 used. Quintero-Marmol and Luyben4 obtained the optimum reflux ratio (bigger capacity factor) for the first batch and use this value for the other seven batch distillations with recycled offcuts. In the “multicomponent-binary component strategy”, the fresh feed is treated separately and the first slop cuts are saved for six batches; a binary component batch then is conducted. The base case has no second slop cut. They optimize the reflux ratios of the first batch and those of the binary component batch. In the “fed-batch distillation strategy”, the slop cuts are fed into the column at the appropriate time and tray during the next batch distillation, using the following heuristic rules: (i) use the optimum reflux ratio of the first batch of the “total slop recycle strategy”, (ii) begin to feed when the liquid composition on any tray in the column matches that of the mixture to be fed (here, the feed tray is where the composition changes are steeper in component feed), then optimize the feed rate. The three other strategies are particular. In the “segregated initial charge strategy”, the first slop cut is used to fill the reflux drum and the second slop cut feeds the column. The results are best when the optimal variables in the “fed-batch strategy” are used. For the base case, there is only one slop cut: the first one. In the “intermittent distillate strategy”, the reflux drum is withdrawn when the distillate meets specification, then a fixed reflux ratio is used while inventory is built back in the reflux drum, and, last, a total reflux is used until the distillate meets specification. The optimal reflux ratio used during the building of inventory is similar to that of the slop recycle case. In the “accumulated product strategy”, the cuts are collected in the reflux drum. The results of these strategies show increases in capacity factor by segregating the slop cuts and performing a binary batch distillation on each slop cut. Mujtaba and Macchietto5 proposed a method for optimizing the reflux ratio and offcut recycle policies for a multicomponent batch column. The method consists of the decomposition of the multiperiod problem to a sequence of pseudo-binary control problems. However, this is only possible when certain assumptions hold (e.g., good separation) and for a particular choice of product specifications and objective function (minimum batch time). Chiotti et al.6 mixed the fresh feed and the first slop cut of the preceding batch, with the second slop cut being added when the recovery of the second main cuts starts. Successive distillations converged and they searched for the constant optimal reflux ratio for a batch with a good separation. Bonny7 optimized the problem of a batch distillation campaign with the possibility of recycling offcuts at the beginning of each batch. The variables taken into account are the variable reflux ratio of each batch, the time for the start of recovery of the second main cut, and the proportion of each offcut added at each batch. Bonny8 used a superstructure for determining the optimal strategies for handling mixtures with the same NC components but different composition, with the recycling of slop cuts. The binary batch distillation with offcut recycle has been the subject of various studies (see the work of Miladi and Mujtaba,9 Sørensen and

Table 1. Number of Control Variables for the Three Recycling Policies and for Different Numbers of Components (NC) and Batch Distillations (Nbatch) Number of Control Variablesa NC

Nbatch

policy 1

policy 2

policy 3

3 3

3 10

52 (46) 591 (501)

42 (38) 231 (213)

40 (36) 159 (141)

a The value given in parentheses represents the number of control variables when the feed rate is not taken into account.

Prenzler,10 Mujtaba and Macchietto,11 Christensen and Jørgensen,12 and Mayur et al.13). Among the studies pertaining to the optimization of just one batch distillation, the following optimized the reflux ratio and other variables: Logsdon et al.,14 Mujtaba and Macchietto,15 Bonny,7 and Miladi and Mujtaba.16 Wajge and Reklaitis17 studied a campaign structure for multicomponent batch distillation with reversible reaction. Scope of This Paper. The case-study approach is used to illustrate the determination of the control variables in a campaign of multicomponent batch distillations without and with recycled offcuts. The number of control variables for a campaign is high: from ∼10 to several hundred (see Table 1). Some values of the control variables will be fixed: the recycled cuts are added at the reboiler at the same temperature as the temperature at the reboiler at the moment of the addition. The values of the other control variables will be obtained via the solution of the optimization problems. The optimization variables retained in this paper are: (i) the feed time for the second offcuts added at the reboiler, (ii) the proportions of the first and second offcuts added at a batch, (iii) the time of the beginning of the recovery of the second offcuts, (iv) the reflux ratios. When a solution has been obtained, it is interesting to study the variation of the objective function with each of the optimization variables retained. This study will be examined in example 1. 2. Process Description and Analysis A quantity of B0 moles of a mixture of NC components of a known composition is introduced into the boiler, to separate these NC components. After putting into operation the process without withdrawal, the distillate is recovered at the top of the column. Figure 1 schematizes this batch distillation column. Operating this simple column necessitates specifying, at every moment, (i) the heat-transfer rate Q0 to be provided at the reboiler, (ii) the heat-transfer rate q to be provided at the condenser, (iii) the pressure P, and (iv) the reflux ratio R. Physical/chemical factors lead to the choice of pressure. The total column pressure can be reduced to maintain maximum column loading. The moment when the recovery of a main cut starts (end of the recovery of the preceding offcut) must be specified. However, the time of the end of the recovery of the second main cut which is imposed by the specification constraint XSPEC2 is not a control variable. For a mixture of NC components, (NC - 2) switching times must be specified (Bonny7). Q, q, P, R are time functions, and the switching times are scalar. The number of control variables for one distillation is

Nv,1 batch ) (NC - 2) + 4 ) NC + 2 For a total quantity Qinitial to be distilled without recycling, which is superior to the maximal capacity of the reboiler Bcapacity, the number of distillations (Nbatch) without recycling is known. The number of control variables is Nbatch × (NC + 2).

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Ind. Eng. Chem. Res., Vol. 45, No. 26, 2006

Figure 2. (a) Recycling policy 1 for a ternary mixture; there are (NC - 1) × (p - 1) possible additions at batch p. (b) Recycling policy 2; there are (NC - 1) possible additions at batch p (p ) 2, ..., Nbatch). (c) Recycling policy 3 for a ternary mixture; there are (NC - 1) possible additions at batch p. (d) Recycling policy 2 with mixing fresh feed and first offcuts.

2.1. The Case of Several Distillations with Recycling. The number of distillations Nbatch must be specified when the quantity to treat is greater than the capacity of the reboiler. This Nbatch value can be deduced from the total time of one distillation and from the limited-time campaign, if the latter exists. For each batch distillation i (i ) 2, ..., Nbatch) and for each added cut, the following decision variables must be specified: (i) the time of the addition (tadd), (ii) the temperature (T) and pressure (P) of the added cut, (iii) the location of the addition (at which tray, reboiler, or reflux drum), (iv) the quantity and the composition of the added cut, and (v) the feed rate (Fadd). Other control variables that will not be taken into account in this paper could be used: for example, Albet et al.18 used a model where the heat-transfer rate at the plates can be provided, whereas some strategies of Quintero-Marmol and Luyben4 require other control variables. The quantity and the composition of the added cut are dependent on the stocking and on the offcuts recycling policy. Three recycling policies are described. 2.2. Recycling Policy 1. This recycling policy is depicted in Figure 2a). Each offcut has its own tank. For a mixture of NC components and for Nbatch batch distillations, (NC - 1) × Nbatch tanks are required. The proportion of a given offcut added to a

given batch is a control variable, but the composition of the added cut is known. There are (NC - 1) possible additions to batch 2, 2 × (NC - 1) possible additions to batch 3, and (Nbatch - 1) × (NC - 1) possible additions to batch Nbatch. The number of additions is (NC - 1) × [1 + 2 + ... + (Nbatch - 1)] ) 1/2 × Nbatch × (Nbatch -1) × (NC - 1). There are six control variables per addition (tadd, T, P, location, Fadd, and proportion). The number of control variables due to recycling is

Nrecycling ) 1 + [3 × Nbatch × (Nbatch -1) × (NC - 1)] including the number of batch distillations as a control variable. The total number of control variables is

Nv ) Nbatch × Nv,1 batch + Nrecycling ) Nbatch × (NC + 2) + [3 × Nbatch × (Nbatch - 1) × (NC - 1)] + 1 2.3. Recycling Policy 2. This recycling policy is depicted in Figure 2b. One tank is designated for each offcut. Thus, Nbatch × (NC - 1) tanks are needed. We must choose the proportions am,p of the first offcut (Sm,1) of batch m added at batch p, the proportions bm,p of the second offcut (Sm,2) of batch m added at batch p, and so on. For a distillation, these quantities of the


different first offcuts are mixed and then added at the same time, tadd 1; all the quantities of the different second offcuts are mixed and then added at the same time, tadd 2; and so on. There are only (NC - 1) additions at each batch i (where i ) 2, ..., Nbatch). For the addition of S1,1, which is the first offcut of batch 1 at the second batch, we must know six variables (tadd, T, P, location, Fadd and a1,1, the latter of which represents the proportion of the first offcut of batch 1); for the addition of S1,1 and S2,1 at the third batch, we must know seven variables (tadd, T, P, location, Fadd, and the proportions a1,3 and a2,3); and so on. The number of additional variables needed to specify the added quantity for all the first offcut, Sm,1, is given as

(5 + 1) + (5 + 2) + ... + [5 + (Nbatch - 1)] ) 1 (Nbatch - 1) 5 + Nbatch 2

(

)

The number of additional variables needed to specify the added quantity for all the offcuts is given as (NC - 1) × (Nbatch - 1) × (5 + 1/2Nbatch). The number of control variables due to this recycling policy is given as

Nv,recycling ) 1 + (NC - 1)(Nbatch - 1)(5 + 1/2Nbatch) The total number of control variables for the campaign is given as

Nv ) Nbatch × Nv,1 batch + Nrecycling ) Nbatch(NC + 2) + 1 (NC - 1)(Nbatch - 1) 5 + Nbatch + 1 2

(

)

2.4. Recycling Policy 3. This recycling policy is depicted in Figure 2c. One tank is designated for all the first offcuts, another tank is designated for all the second offcuts, and so on. The recovered offcut is added to the corresponding tank as it is recovered. For the batch distillation i (where i ) 2, ..., Nbatch), we must choose the added proportion ai of the first offcuts from tank 1, the added proportion bi of the second offcuts from tank 2, and so on. For Nbatch batch distillations, we must choose (Nbatch - 1) × (NC -1) proportions. There are six control variables per addition (tadd, T, P, location, Fadd, and proportion); therefore, the number of control variables introduced for this recycling policy is given as

Nv,recycling ) 1 + [6(Nbatch - 1)(NC -1)] The total number of control variables is given as

Nv ) NbatchNv,1 batch + Nv,recycling ) 1 + 6(Nbatch - 1)(NC -1) + Nbatch(NC + 2) ) 7NCNbatch - 4Nbatch - 6NC + 7 2.5. Summary of Recycling Policies. Table 1 gives the number of control variables for these three recycling policies for different mixtures and different numbers of batch distillations. Representation, evaluation, and strategy are three important problems in process synthesis.19 A good representation is sufficiently rich to allow all alternatives and should be capable of providing direct aid, in regard to solving the synthesis problem. The alternatives must be evaluated effectively so that they may be compared. A strategy is needed to locate the best alternative without having to evaluate all of them. Recycling

Table 2. Example 1: Common Input Data and Base-Case Conditions for the Campaign parameter

value

vapor flow rate, V capacity of reboiler, Bcapacity number of plates, NP pressure, P set of initial mole fractions, X0 cyclohexane n-heptane toluene specified composition for component i, XSPECi XSPEC1 XSPEC2 XSPEC3

91.772 mol/h 100 mol 10 1 atm 0.300 0.500 0.200 0.97 0.94 0.97

policy 1 is very general. The offcuts are not mixed and are added separately; the number of decision variables is high, and the manipulations (the addition of cuts, the placement of the added cuts at the correct temperature, etc.) are many. In recycling policy 2, the offcuts of similar composition are mixed, but the offcuts of different composition are not mixed. The number of manipulations for the operators decreases, and the number of decision variables decreases. Recycling policy 3 mixes the offcuts of similar compositions and reduces the number of stocking tanks. The number of decision variables and the number of manipulations decrease even more. 2.6. Initial Charge Policies. In this paper, we assume that we have an initial quantity Qinitial of a product to be distilled. As long as fresh feed remains and the quantity to be recycled is sufficient, the reboiler is filled to the maximum of its capacity (this is the initially full still pot policy). For the given values of the proportions of first offcuts added to a batch, the quantity of fresh feed to be added is deduced to complete the initial charge: the proportion of fresh feed added to a batch is not a control variable. If we consider that the reboiler is not necessarily filled to its maximum capacity, the proportion of the fresh feed added to a batch would be a control variable. 3. Example 1 3.1. System Operation and Objectives. Two hundred fifty moles of a mixture of cyclohexane, n-heptane, and toluene (cyclohexane-n-heptane-toluene) with an initial composition of X0 ) [0.3; 0.5; 0.2] is treated for a column of 10 trays. The reboiler has a capacity of Bcapacity ) 100 mol. The chosen number of distillations is Nbatch ) 3. The chosen recycling policy is the second one presented, with a mixing of the fresh feed and the first offcuts (Figure 2d). For each batch distillation, three of the four control variables of the column are specified: the pressure in the column (P ) 1 atm), the vapor flow rate (V ) 91.772 mol/L), and total condensation. The specifications of main cuts are XSPEC ) [0.97; 0.94; 0.97]. (See Table 2.) For each batch distillation, we must specify when the recovery of the second main cut starts. The cut location (xcutj) is the value of the mole fraction of component 2 at the condenser when the recovery of the second main cut starts at batch j. The add location (xaddj) is the value of the mole fraction for component 3 at the reboiler when the second offcuts are added at batch j. Bcapacity moles of the fresh feed are introduced at the first batch (the initially full still pot policy). The initial mixture introduced at each batch p (for p ) 2, ..., (Nbatch - 1)) then is a mixture of ADD1,p moles of the first offcuts formerly obtained and (Bcapacity - ADD1,p) moles of the initial mixture to be treated m)p-1 with ADD1,p ) Σm)1 am,p × Sm,1. For the last batch, the

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remaining quantity of the fresh feed and ADD1,Nbatch moles of the first offcuts are introduced in the column with

Table 3. Example 1: Optimal Campaign for Three Batch Distillations (Problem P(1)) parameter

m)Nbatch-1 ADD1,Nbatch ) Σm)1 am,NbatchSm,1

The liquid mixture of ADD2,p moles of the second offcuts formerly obtained is introduced at the same temperature and pressure as that of the reboiler at the moment of the addition at m)p-1 batch p (where p ) 2, ..., Nbatch), with ADD2,p ) Σm)1 bm,p × Sm,2. The duration of the introduction is not taken into account, in relation to the duration of the distillation. Therefore, feed rates are not taken into account in those examples. The proposed optimization problem answers the following questions: (i) What must be the reflux ratio Rj, which is kept constant for batch j (for j ) 1-3)? (ii) At what moment must the recovery of the second main cut start? (iii) Into which batch must the offcuts to be recycled be introduced? (iv) What quantities of these offcuts must be introduced? (v) At what time of the distillation must one introduce the second offcuts? There are 38 control variables. Fourteen are the optimization variables: the six proportions; the three cut locations (xcutj); the two add locations (xadd2 and xadd3), corresponding to the two times of the addition for the second offcuts; and the three reflux ratios (Rj) of the three distillations. Twenty four control variables have been fixed: (i) temperature, pressure, time, and location for the addition of the first offcuts (eight variables); (ii) temperature, pressure, and location for the addition of the second offcuts (six variables); (iii) one variable for the number of distillations (Nbatch); and (iv) at each batch, total condensation, a fixed vapor flow rate V, and a constant pressure (nine variables) are considered. The chosen model of batch distillation is the model proposed by Domenech et al.20 with these assumptions: theoretical plates, negligible liquid holdup, negligible pressure drop, total condensation, constant vapor and liquid flows throughout the column, and ideal equilibrium with partial pressures predicted by Antoine’s correlation. 3.2. General Optimization Problem for This Recycling Policy. The chosen objective function is the production rate, which is defined as the ratio of the total amount of main cuts produced relative to the total time t of the Nbatch distillations of the campaign. The general problem for Nbatch distillations (problem P(1)) is then i)Nbatch j)NC

Pi,j ∑ ∑ i)1 j)1

max A,B,R,xcut,xadd

i ) Nbatch

∑ i)1

ti

The constraints are as follows: (i) Conservation of the mass: Σp am,p e 1 and Σp bm,p e 1 with m ∈ {1, 2, ..., (Nbatch -1)} and with p ∈ {(m + 1), ..., Nbatch}. m)p-1 (ii) Limitations of the capacity of the reboiler: Σm)1 am,p m)p-1 × Sm,1 e Bcapacity with p ∈ {3, ..., Nbatch}. Σm)1 bm,p × Sm,2 e (Bcapacity - Bp(tadd)) with p ∈ {2, ,..., Nbatch}. Bp(tadd) is the quantity in the reboiler when the addition starts at distillation p.

batch 1

batch 2

batch 3

Rp xcutp txcut,p xaddp tadd,p am,p; bm,p

Optimal Control Variables 10.68 9.90 0.8738 0.8680 5.29 h 5.00 h 0.3450 5.06 h a1,2 ) 0; b1,2 ) 1

Pp,1 Pp,2 Pp,3 Sp,1 Sp,2 tp Xboiler,1 Xboiler,2 Xboiler,3 Xadd,1 Xadd,2 Xadd,3

Main Results production rate ) 6.217 mol/h 18.64 mol 17.11 mol 28.71 mol 27.29 mol 36.65 mol 47.94 mol 13.19 mol 16.55 mol 11.08 mol 22.95 mol 24.98 mol 20.28 mol 17.93 mol 22.64 mol 12.56 mol 11.05 h 12.05 h 11.83 h 0.0111 0.0003 0.6439 0.6327 0.3450 0.3670 0.0000 0.0000 0.6633 0.6334 0.3367 0.3666

8.92 0.8399 5.29 h 0.3670 7.66 h a1,3 ) a2,3 ) b2,3 )1; b1,3 ) 0

(iii) All the mixture must be treated: Σm[Σp am,p × Sm,1] e NbatchBcapacity - Qinitial with m ∈ {1, 2, ..., (Nbatch -1)} and with p ∈ {(m + 1), ..., Nbatch}. The constraints that involve the limitation of the capacity of the reboiler for the first offcuts can be eliminated if (Nbatch 1) × Bcapacity e Qinitial. The problems under consideration are nonlinear programming (NLP) problems. They are solved using IMSL Math/library software.21 3.3. Solution of the General Problem. The optimal values of the 14 control variables, and the main results, are reported in Table 3. The optimal distillation corresponds to the addition of the second offcut of batch j to the following batch (j + 1) when the contents of the reboiler have a composition Xboiler that is similar to the second added cut Xadd. The first two offcuts must be recycled at the last distillation: the contents of the reboiler during the second distillation never have a similar composition to the first offcut that could be added. The cut locations are such that the addition of the second offcuts occurs during the recovery of the second main cut and are not equal to the specification of the second main cut. 3.4. Influence of the Variables on the Addition of a Second Offcut for a Single-Batch Distillation. The last batch of the campaign of example 1 must treat an initial feed of 97.93 mol of composition [0.3979; 0.4962; 0.1059] with the addition of 22.64 mol of composition [0.0000; 0.6334; 0.3666]. The other data are presented in Table 4a. This batch distillation has nine control variables. There are three optimization variables: the reflux ratio (R3), the cut location (xcut3), and the add location (xadd3). The six control variables have been fixed: (i) the temperature, pressure, and location for the addition of the second offcuts (three variables) and (ii) total condensation, a fixed V, and a constant pressure (three variables) are also considered. The solution of the problem, hereafter referenced as problem P(2), j)3

max η3 ) R3,xcut3,xadd3

P3,j ∑ j)1 t3

is shown in Table 4b. The optimal solution of the subproblem

Ind. Eng. Chem. Res., Vol. 45, No. 26, 2006 9003 Table 4. Example 2: (a) Input Data for Problems P(2), P(3), P(4), and P(5) and (b) Optimal Solution for Distillation with the Addition of a Second Offcut (Problem P(2)) parameter

value

(a) Input Data for Problems P(2), P(3), P(4) and P(5) vapor flow-rate, V 91.772 mol/h capacity of reboiler, Bcapacity 100 mol 10 number of plates, NP pressure, P 1 atm initial conditions fresh feed, B0 97.93 mol set of initial mole fractions, X0 cyclohexane 0.3979 n-heptane 0.4962 toluene 0.1059 specified composition for component i, XSPECi XSPEC1 0.97 XSPEC2 0.94 0.97 XSPEC3 added cut added feed, Badd 22.64 mol composition of the added cut, Xadd cyclohexane 0.0000 n-heptane 0.6334 toluene 0.3666

Figure 3. Depiction of study 1. Variations of the optimal pair “reflux ratioadd location” (R3, xadd3)optimal with the cut location xcut3 for problem P(3) and the corresponding production rate η3.

(b) Optimal Solution for Distillation with a Second Offcut (Problem P(2)) Optimal Control Variables R3 xcut3 txcut,3 xadd3 txadd,3

P3,1 P3,2 P3,3 S3,1 S3,2 t3 Xboiler,1 Xboiler,2 Xboiler,3 Xadd,1 Xadd,2 Xadd,3

8.23 0.8613 5.12 h 0.3660 7.13 h Main Results production rate: (P3,1 + P3,2 + P3,3)/t3 ) 7.46 mol/h 26.98 mol 45.03 mol 10.53 mol 23.99 mol 14.04 mol 11.06 h 0.0004 0.6336 0.3660 0.0000 ; 0.6334 ; 0.3666

P(2) [R3 ) 8.23, xcut3 ) 0.8613, xadd3 ) 0.3660] is different from the optimal solution of the global problem P(1) [R3 ) 8.92, xcut3 ) 0.8399, xadd3 ) 0.3670]. The add locations have almost the same value; the added cuts have the same composition, but the times of the addition are different (7.13 and 7.66 h). 3.4.1. Study 1: Influence of the Cut Location. For different values of xcut3, the solution of the problem, hereafter referenced as problem P(3), j)3

max R3,xadd3

P3,j ∑ j)1 t3

is calculated. Figure 3 shows the variations of (R3, xadd3)optimal, relative to xcut3. Whatever the value of xcut3, xadd3 from the (R3, xadd3)optimal couple is a constant value and equal to 0.366: the composition of the added cut does not vary. The curve R3 ) f(xcut3) from the (R3, xadd3)optimal couple has a minimum [xcut3 ) 0.912, (xadd3 ) 0.366; R3 ) 7.91)] and the value of

Figure 4. 1. Depiction of study 2. Variations of the optimal pair “cut location-add location” (xcut3, xadd3)optimal with the reflux ratio R3 for problem P(4) and the corresponding production rate.

the production rate is η3 ) 7.29 mol/h. The production rate curve, which corresponds to [xcut3, (xadd3, R3)optimal], is shown in the figure. As expected, the curve has a maximum, which is the optimal solution of problem P(2): [xcut3 ) 0.8613, (xadd3 ) 0.366; R3 ) 8.23)optimal]. For xcut3 equal to the specification 0.94, the production rate is η ) 7.02. The gain achieved by starting the recovery of the second main cut P3,2 at the optimal cut location (0.861) rather than at the specification of the second main cut (0.94) is [(7.46 - 7.02)/7.02] × 100 ) 6.27%. 3.4.2. Study 2: Influence of the Reflux Ratio (R3). Let us consider again just one batch distillation: that shown in Table 4a. To study the influence of R3, we will choose a value of R3 and then calculate the optimal couple (xcut3, xadd3) of the problem hereafter referenced as problem P(4), j)3

max xcut3,xadd3

P3,j ∑ j)1 t3

with the criterion being the production rate of this sole batch. The variations of the optimal pair “cut location-add location”, (xcut3, xadd3)optimal with the reflux ratio R3, are shown in Figure 4. The optimal value of the add location is constant, again equal to 0.3660, and corresponds to a composition in the reboiler that is similar to the composition of the added cut. The optimal value of the cut location decreases linearly with the reflux ratio R3. As expected, the curve of the production rate that corresponds to [R3, (xcut3, xadd3)optimal] has a maximum for [R3 ) 8.23, (xcut3 ) 0.8613, xadd3 ) 0.3660)optimal], which is the solution of the preceding problem P(2) (see Table 4b).

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Figure 5. Depiction of study 3. Variations of the optimal pair “cut location-reflux ratio” (xcut3, R3)optimal with the add location xadd3 for problem P(5) and the corresponding production rate.

0.8613, R3 ) 8.23). The variation of the mole fractions at the reboiler (Xboiler(2) and Xboiler(3)) are also shown. The discontinuities of the η3 curve correspond to the fact that the addition occurs successively during the recovery of P3,1, then of S3,1, then of P3,2, and, finally, S3,2. The optimum xadd3 value (0.366) occurs during the recovery of P3,2, which begins for xadd3 ) 0.219 and ends at xadd3 ) 0.687. The gain achieved by choosing the optimal add location rather than the values 0.219 or 0.687 is 0.8% or 9.25%, respectively. 3.5. Influence of R1, xcut1, R2, xcut2, and xadd2. Figures 7a and 7b show the influence of reflux ratio R1 and cut location xcut1, respectively, for the first distillation, whereas Figures 7c and 7d show the influence of reflux ratio R2 and cut location xcut2, respectively, for the second distillation. Figure 7e shows the influence of the add location xadd2 for the second distillation; when only one variable varies, the others keep the value of the optimal solution. 3.5.1. Influence of R1. Figure 7a shows the variations of the production rate of the campaign and of P1,1/t1, P1,2/t1, P1,3/t1, and Q1/t1 with the reflux ratio R1. The minimal reflux ratio to recover P1,2 is R1 ) 10.5. Then, when R1 increases, P1,2 increases and the second offcut S1,2, which is rich in component 2, decreases. For R1 g 11.30, the second main cut of batch 2 (P2,2) is not recovered: the addition of component 2 is insufficient for the fixed values of R2, xadd2, and xcut2. The optimal solution of the subproblem j)3

max R1

Figure 6. Variations with the add location xadd3 of (i) the production rate η3, (ii) the mole fraction at the reboiler for component 2 (Xboiler(2)) and component 3 (Xboiler(3)) when the addition is done, (iii) composition of the added cut for component 2 (Xadd(2)) and component 3 (Xadd(3)) for constant reflux ratio and cut location (R3 ) 8.23; xcut3 ) 0.8613).

3.4.3. Study 3: Influence of the Add Location. The add location, xadd3, is fixed and the solution of the problem, hereafter referenced as problem P(5), j)3

max R3,xcut3

P3,j ∑ j)1 t3

is calculated. Figure 5 shows the different solutions obtained when xadd3 varies. The curve of the production rate, according to xadd3, attains a maximum as expected for [xadd3 ) 0.3660, (xcut3 ) 0.8613, R3 ) 8.23)optimal], which is the solution of the previous problem (P(2)). When xadd3 increases, the addition of the cut occurs during the recovery of the first main cut, then during the recovery of the first offcut, and, finally, during the recovery of the second main cut (hence the discontinuities of the curves in Figure 5). The optimal value of xcut3 is constant when the addition occurs during the recovery of P3,2. Figure 6 shows the variation of the production rate η3 at the third batch [(P3,1 + P3,2 + P3,3)/t3] with the add location xadd3, the cut location, and the reflux ratio being constant, (xcut3 )

P1,j ∑ j)1 t1

[R1 ) 12.0]

is different from the optimal solution of the global problem P(1) [R1 ) 10.68]. 3.5.2. Influence of xcut1. Figure 7b shows the variations of the production rate of the campaign and of P1,1/t1, P1,2/t1, P1,3/ t1, and Q1/t1 with the cut location xcut1. P1,2 is recovered for 0.869 e xcut1 e 0.959. The optimum value of the first cut location of the campaign (xcut1 ) 0.8738) is different from the optimum value of the first cut location of batch 1 [xcut1 ) 0.893, criterion (P1,1 + P1,2 + P1,3)/t1)]. When xcut1 increases, the second offcut S1,2, which is rich in component 2, decreases. For xcut1 g 0.892, the second main cut of the second batch (P2,2) is zero: the addition of component 2 is insufficient. 3.5.3. Influence of R2. Figure 7c shows the variations of the production rate of the campaign and of P2,1/t2, P2,2/t2, P2,3/t2, and Q2/t2 with the reflux ratio R2. The optimum value of R2 for the campaign (R2 ) 9.90) is different from the optimum value of batch 2 alone [R2 ) 11.15, criterion (P2,1 + P2,2 + P2,3)/t2)]. The second main cut of the second batch (P2,2) is recovered for 9.48 e R2. 3.5.4. Influence of xcut2. Figure 7d shows the variations of the production rate of the campaign and of P2,1/t2, P2,2/t2, P2,3/ t2, and Q2/t2 with the cut location xcut2. The optimum value of the second cut location of the campaign (problem P(1)) is different from the optimum value of that of batch 2 alone [xcut2 ) 0.902, criterion (P2,1 + P2,2 + P2,3)/t2)]. The second main cut of the second batch (P2,2) is recovered for 0.852 e xcut2 e 0.956 (hence the discontinuities of the curves in Figure 7d). 3.5.5. Influence of xadd2. Figure 7e shows the variations of the production rate of the campaign and of P2,1/t2, P2,2/t2, P2,3/ t2, and Q2/t2 with the add location xadd2. P2,2 is recovered for


Figure 7. (a) Variation with the reflux ratio R1 for the first distillation of the production rate of the three distillations (P1,1/t1, P1,2/t1, P1,3/t1), and the production rate for the first distillation (Q1/t1). (b) Variation with the cut location xcut1 for the first distillation of the production rate of the three distillations (P1,1/t1, P1,2/t1, P1,3/t1), and the production rate for the first distillation (Q1/t1). (c) Variation with the reflux ratio R2 of the second distillation of: the production rate of the three distillations (P2,1/t2, P2,2/t2, P2,3/t2), and the production rate of the second distillation (Q2/t2). (d) Variation with the cut location xcut2 for the second distillation of the production rate of the three distillations (P2,1/t2, P2,2/t2, P2,3/t2), and the production rate of the second distillation (Q2/t2). (e) Variation with the add location xadd2 of the second distillation of the production rate of the three distillations (P2,1/t2, P2,2/t2, P2,3/t2), and the production rate of the second distillation (Q2/t2).

0.337 e xadd2 e 0.516. The global problem P(1) and the subproblem

3.6. Comparison of the Different Recycling Policies. The values of two sets of proportions, A and B, are fixed and the solution to the problem hereafter referenced as problem P(6),

j)3

max xadd2

P2,j ∑ j)1 t2

have the same optimum add location (xadd2 ) 0.3450) as expected, with the same cut being added at batch 2.

i)3 j)3

∑ ∑Pi,j i)1 j)1 max R,xcut,xadd

i)3

ti ∑ i)1

9006


Table 5. Comparison of the Different Recycling Policiesa optimal cut locations, (xcut1; xcut2; xcut3)

optimal xadd, (xadd2; xadd3)

optimal reflux ratios, (R1; R2; R3)

composition of the boiler, Xboiler, when second offcut is added

composition of the second added offcut, Xadd

(0.8738; 0.8680; 0.8399)

Optimal Process: η ) 6.217; a12 ) 0, a13 ) 1, a23 ) 1; b12 ) 1, b13 ) 0, b23 ) 1 (0.3450; 0.3670) (10.68; 9.90; 8.92) Batch 2: (0.0000; 0.6633; 0.3367) Batch 2: (0.0111; 0.6439; 0.3450) Batch 3: (0.0000; 0.6334; 0.3666) Batch 3: (0.0003; 0.6327; 0.3670)

(0.8826; 0.8826; 0.8378)

η ) 6.171; a12 ) 0, a13 ) 1, a23 ) 1; b12 ) 0, b13 ) 1, b23 ) 1 (no role; 0.3655) (10.72; 10.72; 8.94) Batch 2: second cut not added Batch 3: (0.0000; 0.6348; 0.3652)

Batch 3: (0.0002; 0.6343; 0.3655)

(0.8738; 0.8594; 0.8700)

η ) 6.156; a12 ) 1, a13 ) 0, a23 ) 1; b12 ) 1, b13 ) 0, b23 ) 1 (0.3434; 0.4024) (10.72; 9.19; 10.54) Batch 2: (0.0000; 0.6597; 0.3402) Batch 3: (0.0000; 0.5957; 0.4043)

Batch 2: (0.0032; 0.6533; 0.3434) Batch 3: (0.0004; 0.5972; 0.4024)

(0.8778; 0.8718; 0.8605)

η ) 6.150; a12 ) 1, a13 ) 0, a23 ) 1; b12 ) 0, b13 ) 1, b23 ) 1 (no role; 0.3090) (10.64; 9,02; 9,80) Batch 2: second cut not added Batch 3: (0.0000; 0.6455; 0.3545)

Batch 3: (0.0053; 0.6857; 0.3090)

(0.8927; 0.8776; 0.8628)

η ) 5.980; a12 ) 1, a13 ) 0, a23 ) 1; b12 ) 0, b13 ) 0, b23 ) 0 (no role; no role) (11.15; 9.72; 10.34) Batch 2: second cut not added Batch 3: second cut not added

(0.8933; 0.8933; 0.8320)


(0.8732; 0.8634; 0.8656)

η ) 5.807; a12 ) 0, a13 ) 0, a23 ) 0; b12 ) 1, b13 ) 0, b23 ) 1 (0.3416; 0.3372) (10.76; 9.98; 10.64) Batch 2 (0.0000; 0.6586; 0.3414) Batch 3 (0.0000; 0.6433; 0.3567)

Batch 2: (0.0119; 0.6465; 0.3416) Batch 3: (0.0116; 0.6512; 0.3372)

(0.8664; 0.8664; 0.8553)

η ) 5.800; a12 ) 0, a13 ) 0, a23 ) 0; b12 ) 0, b13 ) 1, b23 ) 1 (no role; 0.3442) (11.15; 11.15; 10.56) Batch 2: second cut not added Batch 3 (0.0000; 0.6484; 0.3516)

Batch 3: (0.0097; 0.6460; 0.3443)

(0.8830; 0.8830; 0.8830) a


Using the production rate (η) and optimal values of the control variables.

is calculated. The solutions to the nine studied cases are shown in Table 5. Whatever the recycling policy of the first slop cuts, be it recycling at the end of the campaign (A ) [0, 1, 1]), recycling at each batch (A ) [1, 0, 1]), or no recycling of the first offcuts (A ) [0, 0, 0]), the recycling of the second offcuts Si,2 at the following batch (i + 1) leads to a better production rate η than the recycling at the last batch of the campaign:

ηB)[0, 0, 0] e ηB)[0, 1, 1] e ηB)[1, 0, 1] For a given recycling policy of the second offcuts B ) [1, 0, 1] or B ) [0, 1, 1], the recycling of the first offcuts at the last batch leads to an improved productivity:

ηA)[1, 0, 1] e ηA)[0, 1, 1] The addition of the second offcuts must be done when the composition of added cut is similar to the composition of the reboiler. The reboiler contains the three components at the start of the distillation, and then the mole fraction of component 1 decreases and the composition of the reboiler becomes poor in regard to the amount of component 1. The composition of the reboiler is never similar to the composition of the first added offcut, which is rich in components 1 and 2, and the recycling policy of the cut Si,1 of batch i at the next batch (i + 1) is not optimal: an already distilled cut is mixed with the fresh feed, which must be distilled. The optimal cut location is not equal to the specification of the second main cut. If we compare the optimal recycling policy and the optimal policy without recycling, the recycling allows for a gain: [(6.217 - 5.571)/ 5.571] × 100 ) 11.6%. 4. Example 2 4.1. System Operation and Objectives. Eight hundred twenty moles of the same cyclohexane-n-heptane-toluene

mixture as that described for example 1 is treated for the same column (10 trays; capacity of the reboiler Bcapacity ) 100 mol; initial composition X0 ) [0.3; 0.5; 0.2]). The chosen number of batch distillations is Nbatch ) 10. The chosen recycling policy is the second one that was presented previously, with a mixing of the fresh feed and the first offcuts (see Figure 2d). For each batch distillation, the three fixed control variables of the column are the same as those for example 1: P ) 1 atm, V ) 91.772 mol/L, and total condensation. The specifications of main cuts are also the same [0.97; 0.94; 0.97]. For each batch distillation, we must specify when the recovery of the second main cut starts. Bcapacity moles of the mixture to be treated are introduced at the first batch. The initial mixture introduced at each batch p (for p ) 2-10) then is a mixture of ADD1, p moles of the first offcuts formerly obtained and (Bcapacity - ADD1, p) moles of the initial mixture to be treated with m)p-1

ADD1,p )

∑

am,p × Sm,1

m)1

(the initially full still pot policy). Of course, if no fresh feed remains, no fresh feed is added. A liquid mixture of ADD2,p moles of the second offcuts formerly obtained is introduced at the same temperature and same pressure as that of the reboiler at the moment of the addition: m)p-1

ADD2,p )

∑

bm,p × Sm,2

m)1

The addition occurs when the values of the mole fractions of component 3 in the reboiler and in the added cut are equal. The proposed optimization problem answers the following questions:

Ind. Eng. Chem. Res., Vol. 45, No. 26, 2006 9007 Table 6. (a) Optimal Value for the Optimization Variables of the Second Campaign, and (b) Main Results of the Second Campaign parameter

batch 1

batch 2

batch 3

batch 4

batch 5

batch 6

batch 7

batch 8

batch 9

batch 10

7.13 .830 b8,9 ) 1

8.53 .681

32.515 0.9700 0.0300 0.0001 24.348 0.5340 0.4642 0.0018 46.902 0.0237 0.9400 0.0363 7.811 0.0000 0.4615 0.5385 5.871 0.0000 0.0300 0.9700 9.884 h

42.270 0.9700 0.0300 0.0000 11.999 0.5765 0.4233 0.0002 45.731 0.0468 0.9400 0.0132 0.000

Campaigna,b

Rp xcutp nonnul bm,p nonnul am,p

11.51 .861

Pp,1 X1(Pp,1) X2(Pp,1) X3(Pp,1) Sp,1 X1(Sp,1) X2(Sp,1) X3(Sp,1) Pp,2 X1(Pp,2) X2(Pp,2) X3(Pp,2) Sp,2 X1(Sp,2) X2(Sp,2) X3(Sp,2) Pp,3 X1(Pp,3) X2(Pp,3) X3(Pp,3) tp

19.979 0.9700 0.0298 0.0002 20.078 0.4933 0.4995 0.0071 31.148 0.0223 0.9400 0.0377 15.163 0.0000 0.6386 0.3614 13.631 0.0000 0.0300 0.9700 11.773 h

(a) Optimal Value for the Optimization Variables of the Second 11.13 11.21 11.77 11.33 11.71 11.71 11.71 .857 .861 .856 .860 .859 .860 .860 b1,2 ) 1 b2,3 ) 1 b3,4 ) 1 b4,5 ) 1 b5,6 ) 1 b6,7 ) 1 b7,8 ) 1 Batch 9: a1,9 ) a2,9) a3,9 ) 1; a4,9 ) 0.9475 Batch 10: a4,10 ) 0.0525; a5,10 ) a6,10 ) a7,10 ) a8,10 ) 1; a9,10 ) 0.81407 19.402 0.9700 0.0298 0.0002 20.791 0.5013 0.4916 0.0071 39.531 0.0186 0.9400 0.0414 18.294 0.0000 0.6131 0.3869 17.146 0.0000 0.0300 0.9700 12.955 h

19.531 0.9700 0.0298 0.0002 20.782 0.4973 0.4955 0.0072 40.474 0.0175 0.9400 0.0425 19.174 0.0000 0.6105 0.3895 18.333 0.0000 0.0300 0.9700 13.299 h

(b) Main Results of the Second Campaignb,c 20.349 19.713 20.266 0.9700 0.9700 0.9700 0.0298 0.0298 0.0298 0.0002 0.0002 0.0002 19.246 20.476 19.526 0.4947 0.4957 0.4928 0.4983 0.4971 0.5002 0.0070 0.0072 0.0070 42.975 39.679 43.138 0.0168 0.0178 0.0163 0.9400 0.9400 0.9400 0.0432 0.0422 0.0437 17.600 19.292 17.446 0.0000 0.0000 0.0000 0.5975 0.6151 0.5935 0.4025 0.3848 0.4065 19.004 18.441 18.916 0.0000 0.0000 0.0000 0.0300 0.0300 0.0300 0.9700 0.9700 0.9700 13.938 h 13.322 h 13.900 h

20.266 0.9700 0.0298 0.0002 19.583 0.4917 0.5012 0.0071 41.052 0.0170 0.9400 0.0430 17.829 0.0000 0.6028 0.3972 18.717 0.0000 0.0300 0.9700 13.672 h

20.266 0.9700 0.0298 0.0002 19.583 0.4917 0.5012 0.0071 41.709 0.0167 0.9400 0.0433 17.562 0.0000 0.5997 0.4003 18.709 0.0000 0.0300 0.9700 13.726 h

0.000

5.635 h

Optimal production rate is 6.513 mol/h. 820 mol are treated in 10 batch distillations (Nbatch ) 10). Given in terms of quantities (mol) and composition of the different cuts, as well as time (h) for batch p. a

b

c

(i) What must be the reflux ratio Ri, which is kept constant during batch i? (ii) At what moment must the recovery of the second main cut start? (iii) Into which batch must the offcuts to be recycled be introduced? (iv) What quantities of these offcuts must be introduced? There are 213 control variables. There are 110 optimization variables: 90 proportions, 10 cut locations, and 10 reflux ratios. The 103 control variables have been fixed: (i) the temperature, pressure, time, and location for nine additions of the first offcuts (9 × 4 ) 36 variables); (ii) the temperature, pressure, time, and location for the nine additions of the second offcuts (36 variables); (iii) one variable for the fixed number of distillations; and (iv) total condensation and fixed values of V and P (30 variables) are considered for each batch. The chosen objective function is the production rate. The problem (hereafter referenced as problem P(7)) is i)Nbatch j)NC

Pi,j ∑ ∑ i)1 j)1

max A,B,R,xcut

i)Nbatch

∑ i)1

P(1), which is η ) 6.217 mol/h, are also different. The corresponding values of the quantities and composition for this campaign of 10 distillations are given in Table 6b. The optimal recycling consists of recycling the second offcuts (Si,2) of batch i to the next batch (i + 1), except cut S9,2, which is not recycled. The first offcuts (Si,1) must be recycled at batch 9, with the remaining fresh feed and only the first offcuts at batch 10. It is better not to mix a cut already distilled with a cut to be distilled. The addition of a cut that has already been distilled must be done when the composition of the added cut and the composition in the reboiler are similar. The cut location does not correspond to the specification of the second main cut. The optimal values of the reflux ratios and cut locations (Ri and xcuti, respectively) vary for the first batches and then converge, except, of course, at batches 9 and 10. This convergence with small oscillations also can be observed with the quantities and compositions of the different produced cuts. By extending the results of this example, if a large quantity of fresh feed leads to a value of Nbatch .10, all the first offcuts can be processed after processing the fresh feeds without adding them to any fresh feed, with the second offcuts being added during the batch distillations that involve the fresh feed. After making a judicious choice of Nbatch, the reflux ratios and the switching times remain to be optimized.

ti

The constraints are shown in section 3.2. 4.2. Solution. The optimal values of the decision variables are shown in Table 6a. Although the composition of the two initial cuts in examples 1 and 2 are identical and the column and the basic conditions are the same, the optimal values of the first two reflux ratios (R1 and R2) and the first two cut locations (xcut1 and xcut2) of the first two batches in example 1 (Table 3) and example 2 (Table 6a) are different. The optimal production rate of example 2, problem P(7), which is η ) 6.513 mol/h, and the optimal production rate of example 1, problem

5. Conclusion and Possible Perspectives An inventory of the control variables of batch distillation has been examined for different recycling policies. Other operating parameters could be taken into account in the case of particular strategies. The optimization of campaigns has taken into account the following variables: the reflux ratio, the add location (which determines the time of adding the offcuts), the cut location (which determines the time of the start of the recovery of a main cut), and the proportions of the offcuts recycled at each batch. Batch operation optimization is complex, and much work remains to be done. The considered examples have used a

9008


constant reflux ratio. The optimization of a campaign with a variable reflux ratio necessitates the discretization using tiny steps for the reflux ratio.7 This case will be addressed in another study. A quantitative comparison among the recycling policies will be examined in another study. At the end of a batch distillation, the column holdup either could be treated as an extra offcut (if allowed to drain into a separate vessel) or its effect on the bottom product composition could be taken into account. Thus, another work can check to determine if neglecting the column holdup could be introducing an appreciable error. The addition of offcuts at the reboiler is optimal when the composition of the added offcuts and the composition of the reboiler are similar. This condition is encountered during the distillation for the second offcuts, but it does not occur for the first offcuts. Optimal campaigns recycle each second offcut at the next batch, with the first offcuts being recycled alone or with the remaining fresh feed. If the addition of the first offcuts or of the other offcuts is envisaged into the column and not into the reboiler, the variables to be taken into account are the feed time, the feed location, and the feed rate. This unfavorable effect of the mixture of products of different compositions has already been shown by Quintero-Marmol and Luyben4 in the case of recycling offcuts and by Bonny8 in the case of a batch distillation campaign of several cuts of different compositions. When a solution has been found for the campaign of a given product, it is not certain that the optimal variables of this campaign are also the solution for another different campaign of the same product. The method described allows one to optimize batch distillation campaigns with the recycling of offcuts at the reboiler 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 the batch distillations of the campaign X0 ) set of initial mole fractions of the mixture to be treated Xadd ) set of the mole fractions of the added cut Xboiler ) set of the mole fractions at the reboiler XADD ) set of specified add locations XCUT ) set of specified cut locations XSPEC ) set of specified compositions ai,j ) proportion of Si,1 added at batch j bi,j ) proportion of Si,2 added at batch j Bcapacity ) maximal capacity of the reboiler (mol) Badd ) amount of the added cut (mol) B0 ) total initial amount in the reboiler (mol) Bp(tadd) ) quantity in the reboiler when the second offcuts are added at distillation p (mol) D ) distillate flow rate (mol/h) Fadd ) feed rate of the added cut L ) liquid flow rate (mol/h) Nbatch ) number of batch distillations NC ) number of components NP ) number of plates in the column P ) pressure Qj ) total recovered amount of main cuts at batch j (mol) Qinitial ) initial amount of mixture to be treated (mol) Pi,j ) production cut related to the jth key component at batch i (mol) R ) reflux ratio (quotient of the liquid flow rate to the distillate flow rate) Ri ) reflux ratio for batch i

Si,1 and Si,2 ) first and second slop cuts at batch i (mol) T ) temperature t ) total time (h) tadd ) feed time of the added cut (h) ti ) total time for batch i (h) txadd,j ) time for adding the second offcuts at batch j (h) txcut,j ) the recovery of Pj,2 starts at the time txcut,j (h) V ) vapor flow rate (mol/h) Xi(Pp,j or Sp,k) ) mole fraction of component i of the cut Pp,j or Sp,k Xadd,j ) mole fraction of component j of the added cut Xboiler,i ) mole fraction of component i in the reboiler when the second offcuts are added xaddj ) add location for batch j: value of the mole fraction for component 3 at the reboiler when the second offcuts are added at batch j xcutj ) cut location for batch j: the recovery of the second main cut of batch j starts as soon as the mole fraction of component 2 at the condenser is equal to xcutj XSPECi ) specified composition for component i η ) production rate of the campaign (total amount of products per unit of time) (mol/h) ηi ) quotient of the amount of main cuts at batch i to the total time ti of batch i (mol/h) Literature Cited (1) Luyben, W. L. Design and Control Degrees of Freedom. Ind. Eng. Chem. Res. 1996, 35, 2204. (2) Henley, E. J.; Seader, J. D. Equilibrium Stage Separation Operation in Chemical Engineering; Wiley: New York, 1981. (3) Luyben, W. L. Multicomponent batch distillation. 1. Ternary systems with slop recycle. Ind. Eng. Chem. 1988, 27, 642. (4) Quintero-Marmol, E.; Luyben, W. L. Multicomponent batch distillation. 2. Comparison of Alternative Slop Handling and Operating Strategies. Ind. Eng. Chem. 1990, 29, 1915. (5) Mujtaba, I. M.; Macchietto, S. An Optimal Recycle Policy For Multicomponent Batch Distillation. Comput. Chem. Eng. 1992, 16 (S), S273. (6) Chiotti, O. J.; Salomone, H. E.; Iribarren, O. A. Selection of Multicomponent Batch Distillation Sequences. Chem. Eng. Commun. 1993, 119, 1. (7) Bonny, L. Multicomponent Batch Distillations: Study of Operating Parameters. Ind. Eng. Chem. Res. 1999, 38, 4759. (8) Bonny, L. Strategies for handling mixtures in multicomponent batch distillations with slop recycle. Chem. Eng. Process. 1995, 34, 401. (9) Miladi, M. M.; Mujtaba, I. M. The Effect of Off-cut Recycle On the Optimum Design and Operation of Binary Batch Distillation With Fixed Product Demand. Comput. Chem. Eng. 2005, 29, 1687. (10) Sørensen, E.; Prenzler, M. A Cyclic Operating Policy For Batch DistillationsTheory and Practice. Comput. Chem. Eng. 1997, 21 (S), S1215. (11) Mujtaba, I. M.; Macchietto, S. Optimal Recycle Policies In Batch DistillationsBinary Mixtures. In Re´ cents Progre` s en Ge´ nie des Proce´ de´ s 2(6); Domenech, S., Joulia, X., Koehret, B. Eds.; Lavoisier Technique et Documentation: Paris, 1988. (12) Christensen, F. M.; Jørgensen, S. B. Optimal Control of Binary Distillation With Recycled Waste Cut. Chem. Eng. J. 1987, 34, 57. (13) 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. (14) Logsdon, J. S.; Diwekar, U. M.; Biegler L. T. On the simultaneous Optimal Design and Operation of Batch Distillation Columns. Trans. Inst. Chem. Eng. 1990, 68, 434. (15) 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, 27. (16) Miladi, M. M.; Mujtaba, I. M. Optimisation of Design and Operation Policies of Binary Batch Distillation with Fixed Product Demand. Comput. Chem. Eng. 2004, 28, 2377. (17) Wajge, R. M.; Reklaitis, G. V. An Optimal Campaign Structure for Multicomponent Batch Distillation with Reversible Reaction. Ind. Eng. Chem. Res. 1998, 37, 1910.

Ind. Eng. Chem. Res., Vol. 45, No. 26, 2006 9009 (18) Albet J.; Le Lann J. M.; Joulia X.; Koehret, B. Evolutions et Tendances En Simulation de Colonnes de Rectification Discontinue. Chem. Eng. J. 1994, 54, 95.

(21) IMSL Fortran Numerical Libraries V3.0 for Windows, Mathematic and Statistic Functions; Visual Numerics: Houston, TX.

(19) Nishida, N.; Stephanopoulos, G.; Westerberg, A. W. A Review of Process Synthesis. AIChE J. 1981, 27, 321.

ReceiVed for reView July 13, 2006 ReVised manuscript receiVed October 3, 2006 Accepted October 4, 2006

(20) Domenech, S.; Enjalbert, M. Program for simulation batch rectification as unit operation. Comput. Chem. Eng. 1981, 5, 181.

IE0609057

Multicomponent Batch Distillations Campaign: Control Variables and

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