I n d . Eng. Chem. Res. 1993,32, 500-510
500
SEPARATIONS Synthesis of Separations by Batch Distillation S u r e s h S u n d a r a m and Lawrence B. Evans' Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
The batch distillation synthesis problem for separating a single multicomponent feed stream into specified products using available equipment is addressed. A mathematical programming approach for solving the single column synthesis problem is presented. The method consists of proposing a superstructure which has embedded in it all possible combinations of conducting the desired separation, formulating the superstructure as an optimization problem, and solving the resulting nonlineary program using available optimization software. Standard optimization techniques and a shortcut model for batch distillation simulation are used in the proposed methodology. The sequence of overhead fractions during distillation, the cut location, and the reflux ratio policy during each cut are decision variables in the optimization formulation. The effectiveness of the proposed procedure is demonstrated on a series of example problems. Background and Motivation Batch distillation is widely used for separating liquid mixtures in many batch processes. The flexibility of the batch column provides a distinct advantage over the continuous column. For example, a single batch still can be used to process a wide range of feedstocks under a variety of operating conditions. When separating materials with high solids content, or substances that become very viscous on concentration, a batch still can be used because solids or the thick, viscous fluid stays in the pot and can be removed at the end of the distillation. Most importantly, a number of products can be produced in a single batch column whereas a number of interconnected columns would be required in the continuous case. The schematic of a batch distillation column is shown in Figure 1. Operation of a batch column involves charging the still with the material to be separated and carrying out the fractionation until the desired separation is achieved. As distillation proceeds, a number of Ycutsnare made. A cut refers to the switching from one overhead accumulator to another. Some of the cuts will be desired products while others will be intermediate fractions. The intermediate fractions can be discarded as waste or can be recycled to the next batch, or a number of fractions can be collected and processed further at a later stage. During the course of the entire operation, composition profiles and operating conditions can change over a wide range of values. Instantaneous vapor liquid compositions at any tray depend on the pot liquid composition, the number of stages in the distillation column, the reflux ratio policy, and the thermodynamic properties of the components involved in distillation. Compositions in an accumulator depend on the cut location, in addition to the variables mentioned above. Batch distillation is a dynamic process. The equations which describe the operation of batch distillation are of differential-algebraic form; simulation requires numerical integration of these equations which are usually "stiff". Thus, the model for batch distillation is complex. Currently, there exist sophisticated tools for the rigorous simulation of batch columns given feed conditions,column configuration,and operating policy of the column (Boston
~
Stage 1 Stage 2 Receivers Stage k Stage N
Figure 1. Schematic of a batch distillation column.
et al., 1981). However, there is no systematic procedure to determine the optimal separation system for separating a multicomponent feed stream into products using batch distillation. For the most part, engineers decide the operating variables based on past industrial experience or on heuristics. For example, if the first cut is desired product, then distillation is conducted at constant reflux until the overhead purity of the required component falls below a certain threshold. At this time, the reflux ratio is increased to %queeze" out the last portions of the component. This can result in excess waste product and is clearly inefficient. When a single distillation column is to yield multiple products, the choice of the optimal reflux ratio policy and the location of the cuts is even more difficult. This is because the choice of one cannot be made independent of the other. The two sets of variables should be selected while maximizing a suitable objective. The sequence of overhead fractions that must be taken during distillation is another variable in selecting the optimal separation system. For example, if a feed containing three components A, B, and C (in order of decreasing relative volatility) is to be separated into pure products, the separation system can yield the products in
0888-5885/93/2632-0500$04.00/00 1993 American Chemical Society
Ind. Eng. Chem. Res., Vol. 32, NO. 3, 1993 501 a direct sequence of A, B, and C with alternating waste cub, or the separation system could remove products in the reverse order C, B,and A with alternating waste cuts. Finally, if there are multiple columns, the assignment of separation steps to columns becomes another decision variable. The assignment problem is complicated in the batch distillation case because a single column can be used to separate multiple components. Alternatively, multiple columns can be used in parallel to conduct the same separation. The performance factors, viz., the dynamic nature of batch distillation, the possibility of having a time-varying reflux profile, and interactions between reflux ratio policy and cut location, can make evaluation of even a single separation system fairly time-consuming. The structural factors, viz., the possibility of taking multiple products from a single column, using multiple columns in parallel, various sequences of taking overhead fractions, and the issues of waste cut processing, lead to a large number of possible structural alternatives for conducting a desired separation with a specified set of operating variables even for small-sized problems. Optimization Of Batch Distillation: Literature Review The optimal control problem in binary batch distillation has been formulated in two ways. The uminimum time problem" defines the optimal operating policy to be that which produces the required quantity of distillate (the first cut) of specified purity in the shortest possible time. The "maximum distillate problem" defines the optimal operating policy to be that which produces the maximum amount of distillate (again only the first cut) of a given purity in a prescribed duration of time. Converse and Gross (1963) and Diwekar et al. (1987, 1989) solved the maximum distillate problem and found that product yield increased by 3-5% over the constant distillate rate and constant overhead composition policies. Coward (1967a,b) solved the minimum time problem for binary systems using both Pontryagin's maximization principle and calculus of variations. Robinson (1970) conducted experiments on an industrially sized still containing a three-component mixture and compared the results for constant reflux, constant overhead composition, and optimal reflux ratio policies. Time savings of a few percent was achieved by using the optimal policy. Kerkhof and Vissers (1978) determined the optimal reflux ratio which maximized a profit function. The profit function took into account startup time and cost of raw materials and products, and included energy, wages, depreciation, and upkeep as operating costs. They developed two dimensionless numbers and gave guidelines on when optimal policies are worthwhile. Mayur et al. (1970) solved the minimum time problem for a binary mixture with a recycled waste cut using a sequential simplex procedure. They showed that reflux policies with the recycled waste cut lead to lower times than corresponding policies without a waste cut. Christensen and Jorgensen (1987) used orthogonal collocation and quadrature to solve the minimum time problem with a recycled waste cut. They introduced a measure of the degree of difficulty of a separation to indicate whether or not a recycled waste cut is advantageous. Luyben (1988) consideredthe operation of ternary batch distillation with recycle of the intermediate waste cuts. The intermediate cuts were located by means of heuristics. He concluded that the fixed reflux ratio policy appeared to be at least as good asthe varible reflux ratio policy. Quintero-Marmol
and Luyben (1990) considered six different handling strategies for the waste cuts when optimizingternary batch distillation. The strategies studied can be classified into two sets: one in which there was some type of mixing of the waste cuts with the next batch and another in which advantage was taken of the separation already done in obtaining the waste cuts. They found that in general mixing of already separated waste cuts has a detrimental effect on the performance of the distillation system. Bernot et al. (1988) developed a simple dynamic model to predict the possible set of overhead fractions that can be obtained for feed of given composition. Their method was applicable to ideal as well as azeotropic systems, and the results for a ternary system were analyzed using the three-component phase plane. Macchietto et al. (1989) showed the benefits of using advanced optimal control policiesfor batch distillation along with supervisorycontrol techniques for conducting batch distillation campaigns. Objectives The present paper addresses the following batch distillation synthesis problem: Given (a) a multicomponent feed stream that is to be separated into specified products. The feed composition and the amount of feed are known. (b) a required set of products. (c) an inventory of available batch distillation units in which the desired separation is to be conducted. Some or all of the units may be utilized. Determine the optimal strategy to achieve the separation of the feed stream into desired products using available equipment. The optimal strategy can be one which meets one of several objectives: maximum profit, minimum cost, maximum production rate, maximum production amount, or minimum time. The choice of the objective function will depend on the specific problem at hand. Various constraints are usually present, and these have to be taken into account when developing the synthesis methodology. First, there are mass and component balance constraints for each separation step. Second,there are composition specifications on some or all of the products. There can also be composition constraints on the wastes. Typically, the products are sold to realize revenue so long as they meet a certain composition which is previously specified. The wastes obtained from the separation can be disposed of or can be recycled for further distillation. In the first case, a waste disposal cost will be incurred. Third, there can be constraints on the availability of equipment and the operating ranges of the available equipment. Finally, there can be a constraint which specifies that the separation has to be completed within a certain time horizon. This paper is a first attempt at solving the batch distillation synthesis problem in chemical engineering research. The paper develops a systematic framework which can be used to provide answers to the questions which describe the synthesis problem: What is the optimal sequence of separation steps? What equipment units should be used at each step? What is the reflux ratio policy at each step? What are the cut locations? Obtaining a solution to the batch distillation synthesis problem is not a simple task. The number of structural alternativesto conduct a desired separation with a specified set of operating variables is large even for small-sized problems. Combined with the dynamic nature of batch
502 Ind. Eng. Chem. Res., Vol. 32, No. 3, 1993
distillation, the solution to the synthesis problem becomes computationally challenging. The number of possible structural alternatives, Nstructures, for separating an ncomponent feed stream into n products using up to M columns, assuming that the waste cuts from any separation step are not reprocessed, can be given by the following formula (Sundaram, 1991):
It is seen that even for a moderately sized problem of separating a six-component feed using up to four columns, the number of structural alternatives is greater than 17 billion! Exhaustive evaluation of all the alternatives requires extensive computation and thus can be prohibitively expensive. Outline of Synthesis Strategy The synthesis problem is solved using methods similar to those used for solving traditional synthesis problems in chemical engineering (Hendry et al., 1973;Hlavacek, 1978; Nishida et al., 1982; Stephanopoulos, 1980; Westerberg, 1980;Umeda, 1983). The three main problems in process synthesis are representation, evaluation, and strategy (Nishida et al., 1981). A good representation is one which includes all the possible alternatives in the process. It should be able to aid one directly in solving the synthesis problem. Each alternative should then be evaluated wuch that they can be compared. Since the total number of alternatives may be quite large, a good strategy that can locate the better alternatives without evaluating them all is needed. This paper focuses on the representation and strategy parts of batch distillation synthesis. To speed up the evaluation step, we developed and used a shortcut model for batch distillation that is based ont he FenskeUnderwood-Gilliland shortcut equations of continuous distillation design. The model, presented in Sundaram and Evans (19931, has shown excellent agreement with rigorous simulation results when used for distilling multicomponent feed streams under the assumptions of constant molal overflow and zero vapor and liquid holdup. The important point is that the model is perceived as a black box by our solution procedure, and can be easily replaced by a more rigorous model. However, the computation requirement of the method will be greatly increased by the use of a rigorous model. A mathematical programming approach is used to solve the synthesis problem. The proposed algorithmic strategy for solving the batch distillation synthesis problem has three parts: 1. Develop a representation scheme which has embedded in it all possible aternatives of conducting the desired separation. This is commonly referred to as a superstructure in chemical engineering research. 2. Formulate the superstructure as an optimization model with appropriate objective functions and constraints. 3. Solve the optimization problem using optimization software. Representation Scheme for Batch Separation Systems The proposed superstructure is shown in Figure 2. The batch separation system is represented by a flowsheet of separation steps. The feed to any separation step is separated into an overhead and a bottom fraction. Each
of these fractions either forms the feed to a subsequent separation step or is sent to form product or waste. The subsequent separation step is done either in the same column or in a different column. At the end, all output streams are suitably mixed to yield the required product streams. Unwanted streams go to form wastes. To help understand the representation, a term called "number of levels of processing" is defined. When either output stream from a separation step is subsequently separated, the number of levels of processing increases by one. For the fresh feed, the number of levels is 1. If either the overhead or bottom fraction from the first step is subsequently separated,the number of levels of processing increases to 2. The number, therefore, is the maximum number of separations that any output stream from the separation process has undergone. A few variable sets are defined to simplify the discussion of the representation scheme. The index set m denotes the number of levels in the separation system, the index set j the separation steps in any level, the index set k the product and waste streams required from the overall separation system, and the index set i the number of components in the separation system. The actual number of levels in a separation system is denoted by Nl, the number of products plus waste streams by Ns, and the number of components by Ncomp. The number of possible separation steps in any level, m, is 2m-1. Thus, the number of possible separation steps in level one is one, in level two is two, in level three is four, in four is eight, and so on. The total number of possible separation steps in a system with NI levels is given by 2"'-*, which is equal to 2N~- 1. The operating variables for separation step j in level m are the reflux ratio ( R , j ) and the cut location ( X m j ) . The cut location is the fraction of the feed to separation step j in level m which is withdrawn as overhead. The amount of feed entering the separation step is denoted by the variable V m - l j , the top stream by tmj, and the bottom stream by b m j . Component flows are denoted by the variable f. Amount of component i in the top stream of separation step j in level m is denoted by f,j,i. The corresponding component flow in the bottom stream is given by The entering component flow is denoted by f",j,i. Knowing the amount and composition of the stream entering a column, the cut location, the reflux ratio, the number of stages in the column, and the thermodynamic properties of the entering components, the distillation step is simulated. The results of the simulation are the amount of the top and bottom streams leaving the separation step, the composition of the top and bottom steps, and the time required for distillation. If the cut location, X m j , for any step is zero, it means that step is not present and no vapor product is produced. The assignment of the output streams from the separation system to product and waste streams is determined by the assignment values. w, 'denotes the fraction of the top stream t M j that is assignedto product or waste stream k, and %j.denotes the fraction of the bottom stream b N , j that is assigned to product or waste stream k. The representation scheme has several advantages. First, it does not distinguish between product and waste streams until the very end. This allows the reprocessing of streams without regard to whether they are locally a product stream or a waste stream. Second, various alternatives in addition to the direct sequence of separation are embedded in the representation. For example, if only the top stream from any separation step is processed
Ind. Eng. Chem. Res., Vol. 32, No. 3, 1993 503
j=2
j=2
m-
a
Level m
(m=m)
output SbrCalUS
Roduct and waste k=1.2,...N
streams,
s
Figure 2. Representation scheme for batch separation systems.
further, a separation scheme can be achieved whereby components are removed in order of decreasing relative volatility. Third, the number of levels of processing is not fixed by the problem description. As one goes to greater number of levels, the energy cost and equipment usage charge will,in general,increase. But the amount of product obtained may also increase. Hence, the number of levels can be a decision variable during synthesis. Fourth, a single column can be used at various separation steps. Alternatively,different separation steps can be conducted in different columns. Fifth, the representation allows the use of multiple parallel columns at any separation step. Formulation of the Optimization Model A few simplifyingassumptions are made to contain the scope of the problem. These assumptions are listed below: 1. A single column is available to perform the separation system;i.e., parallel columns are not allowed in the process. This is the case that commonly occurs in practice; later we will discuss the case when there is more than one column available. 2. The column is available to us from the plant’s inventory, and we will incur a certain equipment usage
cost depending on the length of time the column is required (Barrera and Evans, 1989). Thus the number of stages and the vapor rate of the column are known. Limits may be imposed on the allowable reflux ratio that can be used in the column SO that flooding and weeping conditions are avoided. 3. The amount of feed is assumed to be much larger than the batch size. The column is always charged with a full batch. The distillation is periodic and end effects, such as column inventory, and changeover costs are neglected. 4. All distillations are conducted under the assumption of constant molal overflow and constant relative volatilities between components. These assumptions are necessary at the synthesis stage so that large number of alternative separation systems can be rapidly evaluated. 5. Unlimited storage is assumed to be available at negligible cost. 6. Filling and cleanout costa are neglected, 7. Schedulingconstraints on the columns and on storage are neglected. The two assumptions on storage and on filling and cleanout costa can be incorporated in the synthesis
504 Ind. Eng. Chem. Res., Vol. 32, No. 3, 1993
methodology. These assumptions will be relaxed later in order to study their effects on the solution procedure. The model in its present form does not allow for the mixing of waste cuts with fresh feed for subsequent processing. Such a procedure would mimic recycling of waste in a continuous process, although in a strict batch process only the forward flow of material is possible. The model effectively assumes that intermediate cuts will be accumulated from several (identical) batch runs until there is enough to make a full batch. To derive the mathematical formulation of the problem, the following additional variables are defined. Let
ck
= dollars per kilomole of product or waste (given), $/kmol C, = energy cost per unit time (given), $/h C, = rental charge for column (given), $/h Cf = cost of feed (specified), $/kmol T H= horizon time for desired separation per kilomole of feed (given), h Y*k,i = target mole fraction of component i in product or waste stream k (specified) ffeed,i= amount of component i in fresh feed (given), kmol Mk = lower limit on amount of stream k produced per batch (given), kmol MY = upper limit on amount of stream k produced per batch (given), kmol RL = lower bound on allowable reflux ratio in column (specified) RU = upper bound on allowable reflux ratio in column (specified) a = relative volatilities of entering components (given) NstaBe = number of stages in column (known) B = boilup rate of column (known), kmol/h T m j = distillation time requirement for stage j in level m, h T,= campaign time requirement for desired separation per kilomole of feed, h
The Objective Function The objective function used in this work is maximization of profit, and is computed per kilomole of fresh feed. The contributions to the objective function are revenue generated by sales of products (+I, cost of disposal of the waste streams (-1, cost of energy (-1, rental cost of the column (-), and cost of the feed (-). Each of the contributions can be calculated in terms of the defined variables. Product Revenue and Waste Disposal. If thenumber of levels in the flowsheet is NL,then the total amount of product or waste stream is given by 24-1
Mk
=
(tNljW',j
+ bNlj@j)
1"1
for k = 1, 2, ...,Ns (2)
The total product and waste contributions are given by Ns
z4-1
for separation. (4)
Equipment Usage Charge. The cost of equipment is computed by assigning to the column an equipment usage charge. The charge can be thought of as a rental charge. The equipment usage cost is given by
Tmj
C,
where C, is the rental charge is dollars per hour for the column. Cost of Feed. The cost of the feed to the separation system is a constant and is given by Cf. The profit objective function reduces to the two limits addressed in chemical engineering literature depending on the values of the cost parameters. If the rental cost of the column is increased to an arbitrarily high value, the optimization algorithm will converge to a solution with the minimum campaign time. If the cost of waste disposal is set very high, the optimization procedure will converge to a solution with minimum wastes or, alternatively, maximum distillate. Constraints The objective function is maximized subject to the following constraints: Mass Balances for Separation Steps. These constraints provide the logicallinking of the various separation steps in the superstructure and have the following form: t m j=
if the stream is a waste. Energy Cost. Energy is expended at each separation step, and its cost can be computed as the product of the energy cost per unit time (Ce) and the total time required
xmjvm-lJ
+ bmj = Vm-lj
tmj
(6)
for m = l , 2 ,...,Nl j = 1 , 2 ,..., P-' Component Balances for Separation Steps. Compositions for the top and bottom streams of any separation step are computed using a batch distillation simulation model. (7) f
. . + p ..=t"mJd. .
mJ4
mJ4
for m = l , 2 ,...,Nl j = l , 2 ,...,2"-'
i*
= 1,2, ...,Nmmp
where F is the batch distillation simulation model. Specification of Initial Mass Flow Rates. Flow rate to the columns in level 1 is obtained from the feed flow rate. The objectivefunction and constraints are computed per kilomole of fresh feed.
vo,l= 1.0 (8) Specification of Initial Component Flow Rates. The initial component flow rates are obtained from the feed amount and composition. fO,l,i
ck is positive if the stream is a product, and it is negative
(5)
m = l ]=1
= ffeed,i
(9)
for i = 1,2, ...,Ncomp Composition Constraints for Products and Wastes. The composition of the product and waste streams obtained by mixing the output streams from the separation system have t o meet the specified compositions.
Ind. Eng. Chem. Res., Vol. 32, No. 3,1993 505 Table I: Input Data for Ternary Feed SeDaration ExamDle feed composition 0.45 mole fraction 0.35 mole fraction 0.20 mole fraction
%A
%E XC
relative volatilities 2.0
aAC
for i = 1, 2, ...,Ncomp k = 1, 2, ...,s Distillation Time Requirement. Time required for distillation is computed from the amount of material distilled,the reflux ratio, and the boilup rate of the column. (11)
for
m = l , 2 ,...,N,
aBC
1.4
acc
1.0
distillation column number of stages vapor rate, B (C,= Cr) waste disposal COSb, ck feed cost, Cf
component A
B (12)
Horizon Time Constraint. The horizon time constraint is given by
T,ITH
(13) Assignment Variable Constraints. Any assignment varible is the fraction of a stream that is assigned to a required product or waste stream. The assignment variables for a particular stream should add up to 1.
cuskj P*J S
= 1.0
(14)
k=l S
= 1.0
=1
j = 1, 2, ...,N , for Bounds on Amount of Product and Waste.
Mk 5 M kIMf for k 1, 2, ...,Ns Bounds on Allowable Reflux Ratio. R~ IR,: IR~
(16)
m = 1, 2, ...,iVl j = 1, 2, ...,2"-1
for The above problem is a nonlinear programming problem (NLP) because of the presence of nonlinearities in the objective function and constraints. The NLP is solved using a generalized-reduced-gradientalgorithm, MINOS. Specified Separation Sequence When the separation sequence is specified, the only optimization variables then are the reflux ratio policy and the location of the cut. In the representation scheme above, the separation sequence can be specified by setting some of the values of the cut location to be identically zero. For example, the 'direct" separation scheme is achieved when the values of the cut locations for all indirect separation steps are set to zero. Fresh feed is charged into a distillation column, and alternate product and waste streams are distilled overhead. This problem is referred to as the optimal operations problem for a single distillation column. The problem of synthesizing the optimal strategy for separating a mixture using the direct separation scheme includesdetermining both the reflux ratio policy and where the cuts should be made such that an objective is
10 kmol/h $l.O/h -$O.l/kmol 0.0
Product Specifications
j = l , 2 ,..., 2"'-'
Computation of Campaign Time. The campaign time, T,is simply given by the sum of all the distillation times.
8
C
Ykj,mole fraction product 1 product 2 product 3
selling price (Ck), $/Ism01
20.90 20.85 20.95
2.5 4.5 6.5
maximized. Currently, there are limited systematic procedures to determine the optimal method of performing a separation in a multicomponent batch distillation column with multiple products. Only Farhat et al. (1990) have addressed the problem of determining simultaneouslythe reflux ratio and cut locations for multifraction batch distillation. However, the reflux ratio policy they used was limited to one of three options: constant, linear, or exponential. Also, the objective function they used was either maximization of a weighted sum of the products or minimization of a weighted sum of wastes. As mentioned earlier, the profit objective function in this work encompasses a much wider range of specific objectives. The optimization model developed above was used to solve the optimal operations problem. Various reflux ratio policiesare possible in the formulation. These range from constant reflux ratio throughout the distillation, to constant reflux ratio during a cut but varying from one cut to another, to continuous reflux ratio profile during each cut. In the last case, the continuous reflux ratio profile during each cut is approximated as a piecewise constant profile. The effectiveness of the solution procedure is demonstrated on an example problem. Ternary Feed Separation Example: Direct Separation The input data for the example are given in Table I. The feed is a three-component stream which is to be separated into essentially pure products. The compasition specifications for the products are given. The available column has eight trays and the vapor rate of the column is 10 kmol/h. The selling price of each product and the disposal cost of any waste from the column is known. Energy and rental costs for the column are also given. The feed cost, Cf,was assumed to be zero. Constant Reflux Ratio during Distillation Solution to the NLP yielded an optimal value for the reflux ratio of 22.71. Product 1 is collected in the accumulator until 44.41 9% of the original pot contents has been distilled. Accumulator composition at the end of product 1cut is 0.9 mole fraction A. A waste cut is then taken until the total amount distilled reaches 57.0 96. Product 2 is then collected and the third cut is made when 70.85% of the initial charge has been distilled. Product
SOB Ind. Eng. Chem. Res., Vol. 32, No. 3, 1993 I .o
+ = 0.6115
$ = 0.7491 Shmol feed
SRmol feed
- A
................ ........ B
40
-
...A.......
."
C
-
30
e
a
20
5
IO
s
0 0
20
40
60
80
IW
%distilled
Figure 3. Ternary feed separation: reflux ratio eonstant during a cut.
0
20
an
W
40
IW
%distilled
Figure 4. Ternary feed separation: four-step reflux ratio during each cut. 1.0,
.
.
.
.
.
.
.
.
2 has a composition of 0.85 mole fraction B. A second waste cut is now collected, and at 89.81% distillation, the residue left in the pot is product 3. The composition of thereaidueis0.95molefractioncomponentC. Theoptimal objective function value, is $0.2353/kmol of fresh feed. All the imposed constraints are satisfied.
.
,
*,
Reflux Ratio Constant during a Cut Solution of the NLP resulted in the optimal reflux ratio profile shown in Figure 3. The instantaneous vapor, pot liquid, and accumulator compositions corresponding to the optimal reflux ratio profile are also shown. The cuts aremadeat38.25%,56.07%,73.95% ,and90.15%distilled. The optimal objective function value is 0.6115, which is a 160%improvement over the optimal value for constant reflux policy. Reflux Ratio Follows a Profile during Every Cut
A varying reflux ratio profie for every cut is the most general case of the problem. One way to solve the varying reflux problem is to approximate the smooth reflux ratio profile during the cut as a piecewise constant function. This is done by discretizing each cut into a number of subcuta, each subcut having a constant reflux ratio, but allowing the reflux ratio to vary from subcut to subcut. The NLP is solved by progressively increasing the number of subcuta (alternativelythe degree of discretization),using the solution to the previous problem as the initial guess for the subsequent one with greater subcuts. In the limit of infinite subcuts, the optimization solution will be a smooth reflux ratio profile. The optimal location of the cub and the corresponding reflux ratio policy for a four-step reflux ratio profile are shown in Figure 4. The objective function value corresponding to the optimal reflux ratio profile was $0.74911 b o 1 of fresh feed. There exists a trade-off between the degree of discretization and the corresponding computing effort. More
0.0
0
I
2
3
4
5
Number of sub-culs per cut (degree ol dimtiration)
FwureS. Improvementinobjectivefunctiondecreaseswithincreese in degree of discretization.
subcuta means a better approximation of the smooth profiie, hutitgreatlyincreasesthecomputingtimerequired to solve the problem because of the larger number of variables. Also, when solving the NLP by progressively increasing the number of subcuts, a point may be reached when the increase in the objective function (profit)becomes smaller and smaller (a case of diminishing returns). This is illustrated in Figure 5. A special case of the optimal operations problem occurs when the reflux ratio policy is specified. In this case, the NLP can be reformulated as an assignment problem (Sundaram, 1991), which is a linear program (LP) that can be optimally solved. The assignment problem formulation can be used as an analysis tool when one wants to compare various reflux ratio policies. Unspecified Separation Sequence Themoregeneralcaseofsynthesisiswhentheseparation sequence is unspecified. In addition to the "direct" separation scheme considered above, all possible combi-
Ind. Eng. Chem. Res., Vol. 32, No. 3, 1993 507
9
Total,
I t
= 0.7995 S/kmolfeed
A
0.605 2.300
A
B
IW 17.65
I
C
Figure 6. Ternary feed separation: optimal separation system.
nations of indirect separations are considered. Thus, reprocessing of streams is allowed. The optimization model of the superstructure developed above is solved to obtain the optimal values of all the cuts and the reflux ratio during each cut. The presence or absence of any separation step is determined by the value of the cut location. If the cut location for any separation step is determined to be zero, then that separation step is not present. The solution procedure is demonstrated on the ternary feed separation problem.
The results from solving the optimization problem are shown in Figure 6. There are eight separation steps in the optimal sequence with the cut locations and reflux ratios as shown in the figure. The contributions of various components to the objective function are also shown. The value of the objective function is $0.7995/kmol of feed. This compares with an optimal profit of $0.6115/kmol on using a direct separation scheme with constant reflux ratio during each cut. Effects of Cleanouts and Intermediate Storage
Ternary Feed Separation: Optimal Separation System The problem was solved with four levels of processing. Thus, there were 15 allowed separation steps and a maximum of 16output streams from the separation system. These 16 streams were to be mixed to obtain the products specified in Table I. The total number of variables was a 110: 15reflux ratios, 15cut locations, and 80 assignment variables (5 assignment variables per output stream-1 each for the 3 products, and 1each for 2 waste streams). The total number of constraints were 31: 15 composition constraints on the 3 product and 2 waste streams, and 16 summation constraints on the assignment variables.
Cleanouts in batch distillation are encountered any time the pot is dumped and refilled with fresh feed. Storage is required for the fresh feed, the product and waste streams, and any intermediate streams that undergo indirect separation. The number of "indirect" separations gives us a measure of the number of cleanouts required during a campaign. Each indirect separation leads to an additional cleanout. In terms of the representation scheme, any time an overhead stream is further distilled, a cleanout is incurred. More indirect separations mean more cleanouts. Figure 7 shows that, for a four-level separation system, there can be a maximum of seven indirect separations.
508 Ind. Eng. Chem. Res., Vol. 32,No. 3, 1993 Product revenue Energy plus r e n d Winte disposal Fresh Feed Clcanwa-
Storage
I
-
2 m a
-1.7421 -0.0273
-:-0 I
.n n
i2lI
= -0.1106
- W
33.60
-
U
+-@g C
Figure 8. Optimal ternary feed separation system with cleanout costa and intermediate storage considered; Celt= Ccli(i) = $O.l/kmol, C,t = $O.O5/gal.
v
Product revenue Energy plus renul Waste dispod Fresh Feed
r
7
J
Clcanwts
Stmaze
2.3804 -1.7403
I
= -0.0338 I.O.0 = -0.1
--ni
7
t
Figure 7. Cleanout and intermediate storage requirements for fourlevel separation system.
These are labeled 1through 7 in the figure, and the flows of the indirect streams are given by the expressionsstor(1) through stor(7). Each intermediate stream has an associated clean out cost per kilomole which can be expressed as C,n(i) dollars per kilomole, where i varies from 1to 7. Additionally, the fresh feed has a cleanout cost of Cclf per kilomole.
W
C
Figure 9. Optimal ternary feed separation system with cleanout costs and intermediate storage considered; Cell= C,li(i) = $O.l/kmol, C,t = $O.O5/gal.
The additional terms for cost of storage are, per kilomole of fresh feed, 7
The effects of including storage in the synthesis problem are many-fold. In batch distillation, storage is required for the fresh feed, the product and waste streams, and any intermediate streams that undergo indirect Separation. Frequently, the intermediate storage available in the plant may be limited, and less than that calculated assuming unlimited storage. This means that storage constraints can change the optimal solution. Also, scheduling of the available storage tanks has to be done in conjunction with scheduling of the various batch distillation operations. Even with a single column in use, the storage scheduling problem is quite difficult. To circumvent the problem of scheduling of storage, it will be assumed that there is unlimited intermediate storage available. However, the storage is not free. Instead, there is a fixed cost for storage expressed in terms of cost per gallon of Storage, Cst$/gal. The additional terms in the objective function for cleanouta are, per kilomole of fresh feed,
c,,+ CC,, stor(i) + C, r=l
Ternary Feed Separation Including Cleanouts and Storage The ternary feed separation example problem considered earlier was solved again taking into account the cose of intermediate storage and cleanouta. Two interesting solutions are presented in Figures 8 and 9. In the first example, the cleanout cost was assumed to be the same for the fresh feed and for the intermediate streams. C,u and C,li(i) were taken to be equal to $0.1/ kmol. The cose of intermediate storage was assumed to be $O.O5/gal. The optimal solution for this example is shown in Figure 8. The corresponding optimal profit is $0.5286/kmol of fresh feed. The number of indirect separations in the optimal system is one, compared to the two indirect separations obtained when cleanout costa and intermediate storage costa were neglected. For comparison, when the optimal profit for the solution shown in Figure 6 is recomputed taking into account cleanout and intermediate costa, the value obtained is $0.4948/kmol fresh feed.
Ind. Eng. Chem. Res., Vol. 32, No. 3, 1993 509 In the second example, the cleanout cost for intermediate streams was increased to 10 times that for cleaning fresh feed, Le., Cclf = $O.l/kmol and Ccli(i)= $l.O/lmol. Cost of intermediate storage was the same as above, i.e., C,t = $0.05/kmol. The optimal solution is shown in Figure 9, and the corresponding optimal profit was $0.4064/lmol. There are no indirect separations in the optimal solution, which is almost identical to the optimal direct separation scheme shown in Figure 3. Extension to Multiple Columns The presence of multiple columns introduces discrete variables into the optimization model. The discrete nature of the problem is introduced by assigning to each separation step in the superstructure a set of integer variables. The set of integer variables for separation step j in level m is denoted by, Y m j . Each Y m j has Ncolentries, with Ncol being the total number of columns that can be used at that step. The entries are represented by ymj,i. It takes a value of 1if column 1is used at stage j in level m,and is 0 otherwise. Thus, if Y I J= (O,l,l], then columns 2 and 3 are used in parallel at separation step 1 in level 1. The outlet streams from any column are fed into a splitter. The splitter simply splits the stream into streams which can form part of the feed to the columns in the subsequent level. Each column is preceded by a mixer which mixes the various feed streams to that column from the preceding level. In summary, the superstructure consists of (1)an initial splitter for the feed stream, (2) a splitter at each product stream of each column, (3) a mixer a t the inlet of each column, (4) a splitter for each output stream from the separation system, and (5) a mixer prior to each desired product/waste stream. The optimization model for this superstructure is a mixed-integer nonlinear programming problem or MINLP (Sundaram, 1991). The model is complicated by the presence of nonidentical parallel units which lead to pathdependent batch completion times (Wellons, 1989). This makes the calculation of the horizon time constraint nontrivial. The MINLP belongs to the class of problems for which the outer approximation/equality relaxation (OA/ER) algorithm (Duran and Grossmann, 1986; Kocis and Grossmann, 1987) is suitable. However, solution to the NLP subproblems for the existing and nonexisting units a t each iteration of the OA/ER algorithm is computationally prohibitive. Heuristic or evolutionary methods which exploit the trade-offs on using multiple columns can be used to prune alternatives (Sundaram, 1991). Conclusions In this paper, we addressed the bath distillation synthesis problem for separating a single multicomponent feed stream into products using a single column. A mathematical programming approach was presented which consisted of postulating a superstructure which has embedded in it all possible alternatives of conducting the desired separation, formulating the superstructure as an optimization problem, and solvingthe resulting nonlinear programming problem using available optimization software. The decision variables in the optimization model are the sequence of overhead fractions that should be taken, the location of the cuta, and the reflux ratio policy during each cut. The effects of nonzero cleanout costa and nonzero intermediate storage costa were incorporated and discussed. The proposed synthesis methodology was demonstrated on a series of example problems. Extension of
the optimization model to multiple columns resulta in a MINLP to which available solution techniques are applicable. Nomenclature Variables b,, = amount of top stream from separation step j in level m,kmol B = boilup rate, kmol/h C,u = cleanout cost for fresh feed, $/kmol Cell = cleanout cost for intermediate streams, $/kmol C, = energy cost per unit time, $/h Cf = cost of fresh feed, $/kmol c k = dollar per kilomole of product or waste, $/kmol C, = rental charge for column, $/h Cat = cost of intermediate storage, $/gal = amount of component i in bottom stream from separation step j in level m,kmol fern,, = amount of component i in entering stream to separation step j in level m,kmol FmJ,,= amount of component i iii top stream from separation step j in level m,kmol M, Ncol= number of columns Mk = lower bound on amount of stream k produced per batch, kmol = upper bound on amount of stream k produced per batch, kmol n,Ncomp = number of components N1 = total number of levels in separation scheme Ns = number of product plus waste streams Natage= number of stages in column NatmCtuea = number of possible structural alternatives for conducting specified separation R,, = reflux ratio of separation step j in level m RL = lower bound on reflux ratio RU = upper bound on reflux ratio stor(i) = amount of indirect stream that needs intermediate storage, kmol T,= compaign time requirement for desired separation per kilomole of fresh feed, h TH= horizon time for desired separation per kilomole of feed, h t,, = amount of top stream from separation step j in level m,kmol TmJ= distillation time requirement for separation step j in level m,h V,-I, = amount of feed into separation step j in level m, kmol w"k = fraction of top stream t N l J that is assigned to product or waste stream k @J = fraction of bottom bN1, that is assigned to product or waste stream k X,, = cut location on separation step j in level m y , ,,I= binary variable denoting the presence or absence of column 1 in separation step j in level m Y, = vector of integer variables of length Ncolwhich denotes the presence or absence of columns in separation step j in level m Y*k,,= target mole fraction of component i in product or waste stream k a = relative volatility Subscripts i = component j = separation step k = product or waste stream m = level
e,,,
,
,
510 Ind. Eng. Chem. Res., Vol. 32,No.3, 1993
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Received for review October 8, 1992 Accepted November 17, 1992
