Ind. Eng. Chem. Res. 1993,32, 1937-1950
1937
Minimum Utility Cost of Mass Exchange Networks with Variable Single Component Supplies and Targets Ashish Gupta a n d Vasilios Manousiouthakis' Chemical Engineering Department, University of California,Los Angeles, California 90024 Mass exchange network (MEN) synthesis is considered, with streams whose supply and target compositions are allowed to vary between upper and lower bounds. The design task is to determine the minimum mass separating agent (utility) cost needed for the transfer of a component from the rich to the lean streams. The mathematical formulation of this synthesis problem leads to a mixed integer nonlinear program (MINLP). This program is shown to possess certain properties which, in turn, are used to develop a solution procedure. This procedure is guaranteed to converge to the g2obal optimum. Two examples, illustrating that utility cost savings can be achieved over the fixed composition MEN synthesis problem, are discussed. 1. Introduction
Separations involving mass exchange are an integral part of the chemical process industry. They include the unit operations of absorption, adsorption, liquid-liquid extraction, desorption, and stripping. Separation processes also play an important role in hazardous waste minimization, wherein waste components are removed from chemical plant effluents prior to their disposal. Inplant reuse of toxic substances not only reduces the total harmful effluents of a plant but also generates recyclable streams, which can reduce the operating cost of process plants (Hunt and Schechter (1989)). King (1981) classifies separation processes into those using either energy (such as distillation) or mass (such as stripping, adsorption) to achieve component transfer. Therefore, the operating utilities for the latter category are mass separation agents (MSA) such as adsorbents, ion-exchange resins, or solvents,just as steam and cooling water are some of the operating utilities for heat exchange. The design of these separation systems involves choosing the appropriate MSAs and the determination of their optimal flowrates to achieve the specified recovery of the key component. However, systematic methodologies to design such systems have been generally limited to the development of heuristic and evolutionary rules for MSA selection and subsequent cost evaluation. A sampling of papers and teams that have developed algorithms for these approaches include Rudd et al. (19731, Lu and Motard (1985), and Liu (1987). These procedures are unable to rigorously account for thermodynamic or economic constraints at the screening stage. To address these shortcomings, and in analogy to the concept of heat-exchange networks, the notion of massexchange network (MEN) synthesis has been recently introduced by El-Halwagi and Manousiouthakis (1989, 1990a,b). This framework rigorously incorporates thermodynamic constraints and can be used for the selection of the mass separation utilities and the calculation of their minimum cost. Once the minimum utility cost is determined, without any prior commitment to the network structure, a network realizing this operating cost is synthesized. The MEN synthesis concept, therefore, avoids heuristics and evolutionary design for massexchange operations. It has also been shown that in tandem with other design tools (State Space Framework, Bagajewicz and Manousiouthakis (1991)), it provides a rigorous technique for the evaluation of energy based or
* Author to whom correspondence should be addressed.
hybrid energy and mass based separations such as distillation. The classical MEN synthesis minimum utility cost problem can be stated as follows: Given a set of rich streams, R (R& = 1,...,NR), with known flow rates and inlet and outlet compositions for a single component and a set of lean streams and utilities, S {Sjl= 1, ...,Nsj,with known costs and inlet and desired target compositions for the same component, determine the minimum utility cost for the separation. Finding the minimum utility consumption for this task is formulated as a linear program for standalone MENS and as a mixed integer nonlinear program for mass exchange networks that include regeneration by ElHalwagi and Manousiouthakis (1990a,b). In the above probelm statement, it has been assumed that there are no temperature changes within the MEN. Eventually, the synthesis of separation systems must be done with the total annualized cost as the primary criterion. However, knowing the minimum utility consumptionfor a given separation requirement is a significant achievement,as the operating costa are often a dominant component of the total annualized cost of a separation system. Furthermore, the minimum utility cost problem can serve as part of a two-stage synthesis procedure that can be used to automatically synthesize mass exchange networks with minimum total annualized cost. In this paper, we generalize the separation task to reflect real situations where supply and target compositions of the rich and lean streams may not always be fixed but may be subject to upper and lower bounds. First, the advantages of not having fixed stream compositions at the design stage are discussed, and then the characteristics of the problem are explained. From the properties developed in that section, an optimization program is formulated. We then prove some properties of the problem and detail a solution procedure for it. Finally, illustrative examples are solved. 1.1. Variable Stream Compositions. In many engineering problems that require MENS,the inlet and outlet compositions of the lean streams, and sometimes those of the rich streams, are specified not as exact values but only as maximum or minimum allowable targets. To elaborate, if the lean streams are process streams, then other process requirements may impose upper (lower) bounds on the outlet (inlet) compositions of these lean streams. The specification often is that the exit (inlet) composition of a lean stream should be less (greater) than or equal to, rather than exactly equal to, a given target (supply) composition. For such practical situations, the problem
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0888-5885/93/2632-1937$04.00/0 0 1993 American Chemical Society
1938 Ind. Eng. Chem. Res., Vol. 32,No. 9,1993
formulation should incorporate the bounds on the stream compositions as constraints, letting the actual inlet or outlet composition be variable. An example is phenol recovery from lube oil refinery wastewaters in Lewis and Martin (1967). Here, lube oil from the process is used to extract phenol from wastewater. Phenol improves the storage characteristics of the oil, so it is a desirable in the oil, whereas in the wastewater it is a toxic pollutant. A realistic problem for this separation is that the phenol content in the lean stream is greater than a lower bound a t the supply and cannot exceed an upper bound at the target. Similarly, specifications on rich streams, for example in hazardous waste removal or in recycle and reuse tasks, state that at least a certain level of key component removal must be achieved. Removal of a toxic component from a rich stream beyond the upper limit on its target composition, with no change in the utility cost, is an advantage for pollution prevention. However, the resultant reduction in the mass transfer driving force can increase the capital cost of the mass exchange equipment. This trade-off should be investigated through methods that simultaneously minimize operating and capital costs. In this case, the outlet composition of the rich stream, yit, must be less than a maximum acceptable target composition, yiu,which serves as an upper bound on yit.
rich phase. In this case, a rich phase mass load available at composition x can be transferred to any lean phase with composition y I f ( x ) , where y = f ( x ) continues to denote the equilibrium relation. Employing the y-rich, x-lean notation, the above feasibility condition can be represented on a composition interval diagram (CID) by establishing an equivalence between the rich and lean composition scales through the equilibrium relation y = f ( x ) . These scales are created by mapping the rich phase compositions, y, to lean phase compositions, x , based on the relation x = f l ( y ) - t (where t denotes the minimum allowable mass transfer driving force in x composition units). Then, second law feasibility simply states that any point on the rich scale can transfer mass to lean stream compositions below it. Let us now consider a countercurrent mass exchanger that transfers the key component from a rich stream, with inlet composition yi, outlet composition y" < yi, and flow rate G, to a lean stream, with inlet composition xi, outlet composition xo > x i , and flow rate L. Let also x and y represent the lean and rich key component compositions at an arbitrary cross section of the exchanger. If the two streams have constant flow rates within the exchanger, then a mass balance yields
-
Y - . x' - x o
= y'x' - yoxo x i - xo
(1)
2. Conceptual Problem Formulation
In this section, we focus our attention on variable target compositions. Then, the mass-exchange synthesis problem started in section 1 can be altered as follows: Given a set of rich streams, R 0 (Rili = 1, ..., NR),with known flow rates, inlet compositions, and upper and lower bounds on target compositions for a single component and a set of lean streams and utilities, S (Sjb = 1, ..., Ns), with their costs, inlet compositions, and upper bounds on target compositions for the same component, determine the minimum utility cost for the separation. The element Ri of the set R consists of the flow rate, Gi, the inlet concentration, yis, the outlet concentration, yit, and the upper bound on yit, yi'. Similarly, Sj comprises the cost for the lean stream or utility, ci, the flow (to be determined), Lj, the compositions, x:, X j t , and the upper bound on xi", Xj'. Additional properties such as upper bounds on flow rates of streams can be easily included. The minimum utility cost for any separation task can be found by formulating and solving an optimization program. Such a formulation must include constraints imposed due to thermodynamics as discussed below. 2.1. Thermodynamic Feasibility. In this section, we briefly review the thermodynamic concepts used in massexchange network synthesis. Any mass transfer operation is considered to be feasible if it satisfies the f i s t and second laws of thermodynamics. The first law states that the mass gained by the lean stream must equal the mass lost by the rich stream. This law should hold for both the total mass and the mass of every species in the streams. To understand the second law requirements, consider an arbitrary monotonic equilibrium relation, y = f ( x ) , where y is the key component composition in a rich phase that is in equilibrium with a key component composition x in a lean phase. The second law states that a rich phase mass load available at composition y can be transferred to any lean phase with composition x 5 f l ( y ) , where monotonicity guaranteesthe existence off1(.). The above notation is often reversed in the literature, withy denoting the key component composition in the lean phase that is in equilibrium with a key component composition x in the
=
Equation 1represenb the operating line, on ay-x diagram, which has a slope equal to the ratio of the lean to the rich flow rates, according to eq 2. This operating line should lie above they = f ( x ) equilibrium curve so that the second law requirement, y 1: f ( x ) , is satisfied throughout the exchanger. If f ( x ) is convex with respect to x , i.e. a f ( x i ) + (1 - cy)f(x0) 1 ~ ( C Y X+' (1- C Y ) X O ) , for all CY E [0,11,then the straight operating line remains above the equilibrium line if and only if yi 1f ( x o ) and yo 1f ( x i ) ,i.e. the end-point conditions of the exchanger satisfy thermodynamic feasibility. The above discussion has important ramifications for the construction and use of composition interval diagrams (CID). In a CID, the supplies (and possibly the target compositions if known) of rich and lean streams (xs, y8 and xt, yt) are employed to establish intervals in the diagram. It is desirable that when a rich and lean stream are matched within an interval, second law thermodynamic feasibility be automatically satisfied. For any rich-lean match, with inlet compositions a t the interval edges (upper edge for the rich, lower edge for the lean), the second law thermodynamic requirement is automatically satisfied for the end point conditions of this match, since the outlet compositions of both the rich and the lean stream are within the interval. Therefore, if the equilibrium relation is convex, then the second law is satisfied throughout the match and only first law requirements need be considered. An example of a convex equilibrium relation is the dilute solution linear equation that is employed in the examples presented in this paper y = mj(x + tj) + bj V j = 1, ...,Ns (3) At this point we should remind the reader, if the aforementioned reverse notation is employed (y lean, x rich), then the operating line on the y-x diagram must be below the y = f ( x ) equilibrium curve (y I f ( x ) ) . Thus, if feasibility is to be guaranteed throughout the exchanger by ensuring feasibility only at the end points, the equi-
Ind. Eng. Chem. Res., Vol. 32,No. 9,1993 1939 y-scale
0.4
R, CD-scale
BC-scale
A B x a l e q.
A
0.35 0.1 0.2s B
0.2 0.15 0.1 0.05 n "
0
0.2
0.1
0.3
0.4
0.5
0.6
Figure 1. An adsorption isotherm with convex and nonconvex sections can be approximated with three convex overestimatorsto yield a CID with three lean scales for the adsorbent MSA. y
R
x
S
Figure 2. A CID diagram showing that it is impossible for lean stream S to achieve a concentration greater than 0.01 in the key component.
librium relationship is required to be concave. Such notation is typically encountered in distillation, where the y = f ( x ) equilibrium curve is given in terms of the most volatile component, thus, making the vapor phase lean with its composition indicated by y (Bagajewicz and Manousiouthakis, 1992). In this case, a typical concave equilibrium relation would be the binary vapor-liquid equilibrium with constant relative volatility, aj CYjXj
+
= 1 (CYj- l ) X j Equilibrium relations that describe the transfer of the key component from the rich to the lean stream, and are nonconvex, can also be accommodated in this framework by building piecewise convex overestimators of the f ( 0 ) function. For instance, an adsorption isotherm shown in Figure 1, expressed as Y = X - 1.5X2+ 1.5X3with Y the solute mass ratio in the liquid stream and X the solute mass ratio in the adsorbent (MSA), can be decomposed into a convex section, AB, and two linear overestimators for the concave section, BC and CD,where B is a point of inflection of the isotherm (more overestimators could also be used to reduce approximation error). Then a lean scale is created for each convex section and the lean stream, depending on its inlet and outlet composition, may be represented on each of these scales. This procedure will capture (albeit approximately) any pinch points that may arise due to the nonconvex nature of the equilibrium curve. Again, if the reverse notation is used, then a relation that is nonconcave can be tackled similarly by constructing piecewise concave underestimators. 2.2. Advantages of VariableTargets. With the above thermodynamic background, the advantages offered by consideringvariable targets for stream outlet compositions can be discussed. Consider the lean stream in Figure 2. If the outlet concentration of this stream is fixed at 0.02, there is no feasible mass exchange for this system. Based on second law thermodynamic constraints, rich streams can transfer mass loads to only those lean streams that exist in the same or lower intervals. For the topmost
interval, there is no rich load which can drive the lean composition from 0.01 to 0.02. However, if the outlet composition of the lean stream was variable, it could drop to 0.01 and mass exchange would become feasible. If there are multiple lean streams in the MEN, each with a different cost, and if the cheapest lean stream has a fixed target composition which makes the network infeasible, then the minimum possible cost may not be achieved since this stream is not used. However, a lower outlet compositionfor this least expensive stream will make it a candidate for the separation. Then, the duty on the more expensive lean stream(@will be decreased, lowering their flow rate. As a result, although the mass-exchange capacity per unit flow of the cheapest lean stream may decline, a lower total utility cost is possible. This counterintuitivepossibility,that lowering the target composition of a lean stream away from its bound drops the minimum utility cost for some networks with multiple lean streams, is one possible advantage of specifying variable targets. Besides the situation explained above, savings in the utility cost can also be obtained when the cheapest lean stream is feasible, but the rich mass-load in ita upper interval is less than the rich load in its lower interval. Hence, if the target composition of Sj is fiied, then its flow rate cannot be more than that required to exchange the smallest of the mass loads. Thus, even though the stream can remove a significant part of the rich load from the bottom of the CID, its flow can not exceed the bound. With a variable target, this restriction is no longer there and the stream can remove a larger rich load. Therefore, the flow rate of the more expensive streams can be decreased yielding significant savings. The first example presented in the paper shows this phenomenon. Another advantage of variable targeting is that in some instances, the rich streams can be cleaned more than a given target specification at no additional cost. Consider a rich stream whose exist composition is above the exit composition of the lean stream mentioned above. If the target composition of the rich stream is lowered, the rich load in the upper intervals increases. This rise in the mass load is taken care of by raising the outlet composition of the lean stream, withoutincreasingita flow rate. In effect, the key component recovery is enhanced without any change in the utility cost of the network. In this manner, savings may be obtained due to reduction in the downstream processing of the rich stream. 2.3. Pinch Concept. When the flow rates and inlet and outlet compositions of the rich and lean streams are known, the mass load available (or needed) at a given
1940 Ind. Eng. Chem. Res., Vol. 32, No. 9, 1993
I
Load (Mass Exchanged),kgis
Figure 3. A composition-load diagram with composite c w e . A composite curve shows the compositions of all the rich or all the lean streams in a network as a function of the mass exchanged by the streams. The vertical arrows, GI and G2, only represent the composition range spanned by each stream. 4
Load (Mass Exchanged),kg/s
Figure 4. Properties of the composite curve in the minimum utility cost solution at the pinch point. At the pinch point, the rich load equals the lean load.
composition can be plotted as a function of that composition so as to form composite curves for the rich and lean streams within a composition-load diagram (Figure 3). On this diagram, for the first law of thermodynamics to hold, the abscissa spanned by the rich curve must exactly equal that spanned by the lean curve. The composition axis is a common scale for all the rich streams and several lean streams scales, plotted through eq 3. If the rich composite curve lies above the lean composite curve, the second law of thermodynamics is satisfied. Referring to Figure 4, three well-known properties of the curves, stated in El-Halwagi and Manousiouthakis (1990b), can be deduced. Property 1. At any point along the composite curve on the composition-load diagram, these inequalities must hold Mass lost by the rich streams above any point on the composition scale
Mass gained by the lean
(4)
and Mass lost by the rich streams below any on the composition
Mass gained by the lean below the same
SO
(5)
The above inequalities represent an algebraic statement
of the geometric requirement that the rich composite curve be above the lean composite curve in a composition-load diagram. Property 2. At the minimum utility cost solution, there is no mass transferred across the pinchpoint, if one exits. Remark: A pinch point exists if at the minimum utility cost solution of an MEN problem, the rich and lean composite curves touch each other. Then, from property 1, the mass lost by the rich streams below (or above) the pinch point is equal to the mass gained by the lean streams below (or above) it. Also, the total mass gained by the lean streams must equal the total mass lost by the rich streams. Hence, there is no mass transfer across the pinch point. The slope of a composite curve is inversely proportional to the flow rate(s) of the stream(@present in that section of the curve (Figure 3). As a result, the composite curve is piecewise linear with corner points wherever a stream enters or exists the network. Based on the trapezoidal rule of geometry, the smallest distance between the two composite curves (the pinch point) occurs at the end point of one of these linear segments. Furthermore, these line segments can touch only in regions where their projections onto the abscissa or ordinate overlap. Since the composite curves are free to move along the abscissa, the pinch can occur only at the corner points in the common composition range of the composite curves. From this observation in heat exchange networks by Grimes (1980) and Cerda et al. (1983) and in mass exchange networks by El-Halwagi and Manousiouthakis (1989,1990b1,the following property determines which, from all the entry and exit points of the streams, can be pinch points: Property 3. Only the inlets of the rich or lean streams in the common composition range of t h e y - and x-scales, are pinch point candidates. Remark Let p be a point on the composition scale of the composition-load diagram. The inverse of the slope of a composite curve, at p , is equal to the sum of all the individual stream flow rates at that point, i.e., Gcc = 2iEIGi and LCC = 2jEjLj, where I and J are sets of indices for the rich and lean streams respectively, that exist atp. Let the compositions above p be denoted by p+ and those below, by p-. Then, for p to be the pinch point, at least one of the following relations must be true: 1. Rich Stream:
1 1 +cc >cc G,
2. Lean Stream:
G,
1
-
iff a new rich stream appears at p 1 xjt*, each L, is less than Lj*. By definition, also satisfies eq 6. Hence the new objective value is + ZjEJJ,cjLj*, where, the set J1 is made UP of all j such that xit* < X j u , and the set J comprises all lean stream indices. But this is a solution to the LP and, hence, must have all the pinch inequalities inactive. Therefore, the objectiveso obtained equals CLP*. Thus a new, feasible Lj
Z j l EJlcjlLjl
solution to ( P l )is obtained with an objective lower than C*. Hence, C* must not be the global optimum. For the solution of ( P l ) ,only one of the above three cases is possible. Since of these three, only case I is not contradictory, it is established that if the solution to the linear version of ( P l ) is unpinched, then the solution to ( P l ) is unpinched as well and, in fact, is the same as that of the linear version. If the problem has no active pinch inequality, finding the minimum utility cost, due to the above theorem, becomes equivalent to solvinga linear program. However, if some pinch points exist and the problem is thermodynamically constrained, the linear property of the problem is lost. Nevertheless, restrictions can still be placed on the outlet compositions due to the next theorem: Theorem 3. At the global optimum of problem ( P l ) , the outlet composition of all the lean streams will be at or above the minimum of either the nearest active pinch point composition above this stream’s supply composition or the upper bound of its target composition. Mathematically, let xjt* be the optimal outlet compositions of the lean streams. A set P” is defined which includes only those p E P that are active at the optimal solution. Also construct the set Pj @IXjP 2 X? and p E P“),and define xjP* minpEp,ixjP. Then, xjt* 1 min(xjP*,
=
x j‘)
Proof Let C* be the global minimum utility cost for ( P l ) and let Lj* be the corresponding flow rate of the stream Sj at any such minimum. Now, assume that xjt* < min(xjP*,xju), where X j u is the upper bound on the outlet composition of the stream Sj. Consider now a concentration X: that is infinitesimally larger than xf* and select Lj such that
If Lj*,xjt* is replaced by Lj, X i t in the global solution vector, then eqs 6.9-14 are trivially satisified and the objective function is strictly decreased. Therefore, if Lj, xi” are such that eq 7 is also satisfield, a contradiction is reached. Study first the pinch inequalities strictly above xi”*. For these inequalities, qj,p8 = 1, qjipt = 1 and Lj*(qj,;(xy - xi”) - ?Jj,pt(XjP - Xjt*)j = Lj*(Xit* - xi”) = Lj(Xi”- xis)
Therefore, replacing Lj*, Xit* with Lj, xjt leaves these inequalities unaltered. Now let us examine the pinch inequalities that are at or below xjt* and above x:. For these inequalities, qj,ps = 0,qj,pt = 1, and Lj*(qj,;(X?
- xi”) - qj,pt(xjP - Xjt*)) = Lj*(XjP- xi”) > Lj(XjP - x;)
Sincep*was chosen as the active pinch point that is closest to and above XIS, and since xit* < min(xjP*, X i u ) < x j p * , these inequalities are inactive. Therefore, the infinitesimal change in xi”* and Lj* cannot possibly activate them. They remain satisfied. Finally consider the pinch inequalities at or below x:. For these inequalities, q j , p s = 0, qj,pt = 0, and Lj*(qj,;(XiP- xis)
- q j , p t ( X j P - xit*))
=0
Replacing Lj*, xf* with Lj, x j t leaves these inequalities unchanged. In summary, we have demonstrated that an infinitesimal change in Lj*, xjt* with Lj, X j t can lead to a reduction to
Ind. Eng. Chem. Res., Vol. 32, No. 9, 1993 1943 a utility cost if it is assumed that xjt* < min(xjP*, Xj"). This is a contradiction. rn Corollary 1. At the solution to the minimum utility problem, for any lean stream Sj, either 7j,/ = 1 i f xjP* > x j u or ~ j , p t *= 0. The statement follows from Theorem 3 and the definition of 7j,pt. rn Corollary 1immediately suggests a branching solution procedure for (Pl),where each of the P pinch inequalities can be assumed active, one a t a time. This fixesthe integers associated with all the intervals below the pinch point candidate in consideration. Each of the resulting subproblems has a reduced level of difficulty and can be solved globally. The subproblem that has the lowest objective value is the global solution.
< 0. 1 = nk + 1,...,n
A,;
-
AC = diag(~O)~~l,Ihk,l-jft~h+l)
Then Ak can be expanded to A, = w&kWkT = wk(Ak+ + Ak-)WkT e wkAk'WkT + w h l i k - w k T = Ah+ + Ah-
where, Ah+ is p.s.d. and Ah- is n.s.d. Now, there exist some indices k for which Ah- has at least one strictly negative eigenvalue. Define a set of indices
K
= (klk = 1,...,m s.t. Ahhas at least one strictly negative eigenvalue)
4. Solution Procedure 4.1. Mathematical Preliminaries. Problem (P2) has a linear objective function and constraints with bilinear terms (products of outlet concentrations and stream flow rates). It belongs to the class of polynomially constrained polynomial programming (PCPP) problems. These optimization programs are, in general, nonconvex and may be solved through several algorithms (e.g.,Floudas et al., 1989; Floudas and Visweswaran, 1990). In this work we employ algorithms presented by Manousiouthakis and Sourlas (1992) for the global optimization of PCPP problems. These algorithms are guaranteed to converge to the global optimum of PCPP problems and are utilized to identify the global optimum of the MEN synthesis problem considered. Problem (P2)is of the general form:
min c, Tz
(P3)
E
subject to ZT&
+ bkTZ + ck I0,
k = 1,...,m
and let K be the cardinality of the set K. Now, K nonnegative variables {k E R, k E K, are introduced, and the term ZTAkZ, k E K, is expanded to zT
zT Ak+z
+ ZT A,z
= ZT Ak+Z - { k ,
k EK
where {h -zTAk-z, k E K. The introduction of {k in (P3) adds the following constraints zT(-AL)z - {k I O ,
k EK
=
Consider now the symmetric matrix B Zk,=KAk-and let ita eigendecomposition be B = WBABW B ~B . is a symmetric matrix, hence AB is diagonal n.s.d. matrix and W B - = ~ W B ~ If . B has n, strictly negative eigenvalues, then there exists a diagonal, n.d., n, X n, matrix AB and an n X n, real matrix WB,whose columns are eigenvectors associated with the strictly negative eigenvalues in B, such that
Following the method in Manousiouthakis and Sourlas (1992),this problem is converted to one whose constraints are all convex except one that is quadratic, concave, and where WOis an n X (n - n,) matrix. separable. By successivelyunderestimating this concave Equation 24 transforms to constraint, and solvingthe resulting convex subproblems, a nondecreasing sequence of lower bounds to the real objective value is created. This sequence is guaranteed to converge to the global optimum. In order to transform (P3)to a program with only one concave constraint, an eigendecomposition of the matrix of coefficients of the quadratic terms is performed. Such a decomposition leads to two submatrices, one positive semidefinite(p.s.d.) and another which is negative semidefinite (n.s.d.). The n.s.d. terms are then substituted by new linear variables. The quadratic equality that defines these new variables is expressed as two inequalities. One of the inequalities is convex, while the other inequality is concave. The summation of all the concave inequalities results in a single inequality that contains the only nonconvexity in the problem. A mathematical treatment of the method follows. Let the eigendecomposition of Ak be defined as Ak = WkAkWkT. Since Ah, the matrix of quadratic Coefficients, is a symmetric matrix, Ak is real and diagonal and wk is orthogonal (Wk-' = WkT). Notice that, for k = 1, ...,m, Ak can be brought into the form Ah = diag((Ak,l+)&, ~Ak,l-)~*nh+l~) where 2 0. I = 1, ..., nk = diag((Ak,,+j~~,,(0)f=n~+l)All constraints but the last in the above problem are
-
1944 Ind. Eng. Chem. Res., Vol. 32, No. 9, 1993
quadratic and convex. Since i i g is a diagonal, n.d. matrix, the last constraint is quadratic, reverse convex, and separable. Various algorithms can be applied on such problems to determine the global solution with €-convergencein a finite number of steps. One such algorithm has been proposed by Soland (1971). Since the concavity in the problem is due to the T variables, assume a rectangular space C P ( T fRngll IT IL ) where 1 and L are lower and upper bounds of T , respectively. At the first iteration, a linear underestimation of is done and the objective is minimized over the intersection of the convex set defined by all but the last constraint and the convex envelope of the last constraint. The set Cis then partitioned at the minimum obtained and the minimization is done again over each of the resulting sets. Thus the algorithm generates a sequence of lower bounds to (P4) through the solution of a series of convex programs. Problem (P4’+’) is obtained from problem (P49 by refining C into smaller rectangles (branching) and minimizing the objective over the convex Constraintsand the resulting rectangles (bounding). Other algorithms can also be used, e.g., a particular implementation of the generalized Benders decomposition that is guaranteed to converge to the global optimum (Manousiouthakis and Sourlas, 1992). 4.2. Program Properties. In the preliminaries, a technique to obtain global solutions to problem (P2)is presented. For large problems of industrial relevance, (P2) can have a large number of quadratic terms, and hence a methodology that exploits problem properties is needed. The theorems presented in section 3 allow the development of such an algorithm. First, a series of subproblems of (P2)are defined. Let these be indicated as (P2,)where p is an active pinch point candidate, for all p E P. The remaining pinch inequalities may or may not be active and are determined by the solution. Let (P2u)indicate the subproblem in which none of the pinch point candidates is assumed active. Then, from theorem 1,(P2u)is an LP. According to theorem 1, if the solution to this LP is pinched, then (P2)is pinched. And from theorem 2 if the LP is unpinched, then its solution is the global solution. Based on this discussion, the optimum of (P2)is C* = min(P2u, min(P2,)) P€P
where each of the (P2,)subproblems are solved using a nonlinear programming algorithm such as the one described above. The optimum obtained thus is the global optimum, if the solution to each (P2,)is global. We thus have a procedure to solve (P2).First, (P2u)is solved. If the solution is unpinched, it is the global (theorem 2) and no other subproblem needs to be solved. However, if the solution is pinched, then all (P2,)problems should be considered. Each of these subproblems is still an MINLP, though of smaller size than the original one. Moreover,the solvabilityof these subproblemsis enhanced based on theorems 1through 3. Further, if a lower bound of a subproblem exceeds the solution of a previously solved subproblem, then its solution need not be pursued further. Finally, tight bounds on problem variables also accelerate solution times. To derive bounds on the lean stream flow rates, the following properties are proved. Property 4. I f CLP*is the objective value when (P2) is solved with all xf = Xj‘ and i f cj is the cost coefficient for stream Sj, then CLp*/Cj is an upper bound on Lj. Proof: Let problem (P2)be represented in the form of (P3).Let C* be the optimal solution for (P2).Then it is clear that
f
NI
In other words, CLP*IC*. The maximum flow rate of any Sj at the optimum is Lj‘ = C*/Cj. Hence, (CLp*/cj)1 (C*/Cj)and is a valid upper bound on Lj. Property 5. If Sj is a feasible leanstream for the entire network, Le., x,” I{(min(ylu,...,y ~ , ”-) bj)/mj - €1, then the Lj* that solves (P2)when all other Lk are set to zero is a valid upper bound on Lj. Proof: Proceeding with arguments similar to those in the proof above, let CL~*be the optimal solution for (P2) if the constraints
+
k = 1, ...,j - 1,j 1, ...,N , Lk = 0, are added to the problem. Then it follows that CL~*1 C*, where C* is the optimum of (P2), since C* is the solution obtained when some constraints of CL~*are relaxed. Hence, the Lj* corresponding to CL~*is an upper bound on Lj. In the following section we focus on the variable inlet composition problem.
5. Variable Inlet Compositions Variable targets on the lean and rich streams have advantages as shown earlier. Similar benefits can also be obtained by having variable inlet compositions as well. Within the context of problem (Pl),the inlet composition, yis or xi”, is allowed to vary between an upper bound, y:“ or and a lower bound, yi”’or x,”’. It is expected that the operating cost for the network would be minimum if the rich and the lean streams are available to the MEN as clean as possible (i.e., y: = y:’ and x,” = x,”’). If the rich stream inlet composition is variable and is decreased, a reduction in the rich load on the MEN is implied. Similarly,lowering the inlet composition of the lean stream enhances feasibility and increases the mass removal capacity per unit of the lean stream. These observations are formalized by the following theorem. Theorem 4. Let the inlet compositions of all the streams in the problem (Pl)be allowed to vary, i.e., let yi”,xi” not be exactly known but rather satisfy the following constraints: i = 1, ...,NR
y? Iy ; 1 y;‘ 1 y;,
,
~ . S l I ~ . S I ~ S I U I X PJ’ 1
1
1
= 1,...,N s
(25) (26)
We denote this variation of (Pl) as (P5). Then, there exists aglobal, minimum utility cost solution of (P5)such that all the inlet compositions are at their lower bounds, i.e. y ; = y:’, xjs =
q’,
i = 1, ...,NR j = 1, ...,Ns
(27)
(28) Proof Let C* be the global optimum value of (P5) and assume that there exists no global solution to (P5) such that all the inlet compositions are at their lower bounds, i.e. yis = yisl,V i ,and x,” = xfl, V j . Therefore, at any global optimum (denoted by the superscript *), at least one stream inlet composition is strictly greater than its lower bound, that is there exists at least one j = 1, ...,N s such that xp* > x;l, or there exists at least one i = 1, ...,NR such that
Ind. Eng. Chem. Res., Vol. 32, No. 9,1993 1946 yis* > y:’. It will be shown that this assumption leads to a contradiction. For clarity, the proof is split into two cases, one for the lean stream inlet compositions and the other for the rich stream inlet compositions. I: Lean stream inlet compositons. Consider a stream Sj for which x,”* > and define
Lj =
Lj*(Xi”*- xjs*) - xjsl)
(xp
Then XI”’ < xi”* imples Lj < Lj*. Let Lj* be replaced by and xJ”* by x f . Then based on the definition of Lj the total mass balance, eq 6, remains satisfied. Let us now consider if an arbitrary (pth) pinch inequality remains satisfied. The following cases need be considered:
Lj
1. xjPIx;.:
Then, ~ j , ~ s=* 1, qj,pt* = 1,and q j g S 1 = 1. When (XI”*, Lj*) is replaced by (xis1, Lj), the left-hand side of the pth inequality is affected only in the term - Lj*(xjt* - x/”*) which changes to the term - Lj(X:* - x p ) . These two terms are equal based on the definition above and, therefore, the inequality is still satisfied. 2. xi”*
Unpinched Solution: Consider a supply composition, yis,that is infinitesimally smaller than y*: and substitute, in the optimum solution, y?* with yp. As a result, the total rich load in the MEN is lowered infinitesimally. To maintain the mass balance, the lean load is correspondingly decreased by defining a new flow rate for any one of the lean streams, Sj, as follows
> xi” > xis*:
Then, qj,pS* = 1,qj,pt* = 0, and qj,psl = 1. Again, in thepth pinch inequality, the only affected term, - L j * ( X j P - XI”:), changes to -Lj(xjP XI”’). For the inequality to remain satisfied, it must hold that
-
-Lj*(xjp - xis*)
+ Lj(xp- xjsl) 2 0
Since all the pinch inequalities were inactivein the original solution, y? can be chosen close enough to y:*, so as to maintain all these inequalities inactive. All flows and compositions continue to be within their bounds. Since the total mass balance also remains satisfied, a new, feasible, lower cost solution to (P5)has been found and a contradiction is reached. Pinched Solution: A set Pi is defined which includes only those p E P that are active at the optimal solution and are below the inlet composition, y:*, of the ith rich stream Ri at the global optimum. Now, consider a supply composition, y,: that is infinitesimally smaller than y*: and substitute, in the optimum solution,y*: withy:. Since the total rich load in the MEN is lowered infinitesimally, the lean load should be correspondingly decreased so as to maintain the mass balance. This can be accomplished by lowering either the flow rate or the target composition of a lean stream. 1. If there existsa lean stream Sj whose inlet composition is above the highest active pinch point below yp, i.e., xp* 1 maxpEp,xjP, then define
Based on the definition of Lj the above becomes
Q
Lj*[-(xip - x;*)(xi”* - ,;I)
+ (Xp
Q
- xis*)(xip - X j q l 2 0
Lj*(XJ.P - xjt*)(xjsl- x;*) I0
which is true. Hence the desired relationship holds and the inequality can only be relaxed by the substitution. 3. x J! l 5
x.p J
5 x?*: J
Then, qjiSe* = 0, qjSist*= 0, and qj,psl= 1. As above, it must hold that
Thus, replacing @is*, Lj*) with (yf, Lj) leaves the mass balance, eq 6, unaffected. The pth pinch inequality (eq 7) is affected as follows: (a) yp Iy:’ i.
x.p J
2 x!*: J
In this case, the left-hand side of the pth pinch inequality undergoes a change in two terms Gi*(y/* - y?) - L j * ( x p- x;*) which become
-Lj*(o - 0 ) I-L J.(xJ.p - xjsl) This obviously holds and the inequality is therefore relaxed. 4.
xp I xjsl:
Then ~ j , ~ s=* 0, ygt* = 0, and q j l j g S 1 = 0. In the left-hand side of the pth pinch inequality, the only altered term is -Lj*(O - 0) which changes in -Lj(O - 0). Clearly, the inequality remains unaffected. Hence, all pinch point inequalities remain feasible by lowering the flow rate of Sj from Lj* to Lj and its supply composition from xf* to x f . Since all the other constraints are also maintained feasible, a new feasible solution with a lower optimal cost has been found. This is a contradiction. 11: Rich stream inlet compositions. Consider a rich stream Ri with an inlet composition y?* > y’: at the global optimum. The associated global solution is either unpinched or features active pinch inequalities.
However, these two expressions are equal and the pinch inequality does not change. ii. xjs* 5 xip < x?: In this case, the two terms, Gi*(y?*-Yit*) change to
Since ( x j p - XI”*)
-Lj*(xjP-
< (xf* - x?*), it then holds
Hence, this pinch inequality is relaxed.
x?*),
1946 Ind. Eng. Chem. Res., Vol. 32, No. 9, 1993
iii. xjp < x;*: In the pth inequality, only the term Gi*(y:* - Yit*) is affected and it becomes Gi*Q: - yit*). This decreases the rich load in the inequality and relaxes the inequality. (b) y;
< y p < y;*:
i. x.p 2 x t * : I
(eq 7). The changes above can have the following effects: (a) y p 2 y;*: i. x.P 2~ t I 1
* :
The only terms affected in this inequality are Gi*(y?* yit*) - Lj*(xjt* - XI”*), which change to
1
This is an inactive pinch inequality since X j B * I maxpEpixjP. The two affected terms, Gi*(yP - Yit*) Lj*(xjt* - x,?*), change to
Gi*(y; - y?) - L1.*(x 1t * - xjs*) +
which makes the inequality more positive. However, if Yie is close enough toy:*, the inequality will remain feasible. ii. x I?* 5 x: I.P
The two expressions are equal, hence the inequality is unchanged. ii. xjt < xjp < xp: Two terms in the left-hand side of this inequality, Gi*(y?* - Yit*) - Lj*(XjP - XI”*) become
< x 1t*:
The two affected terms, Gi*(yP- Yit*) - Lj*(Xjp- XI”*), are altered to
Gi*(y;* - y?) - L j * ( x p - x:jP) - LI.*(x 1.P - xi’*) Hence the inequality is relaxed. iii. xip < x:
which may be more positive or more negative than before. Nevertheless, if @is* -y:) is sufficiently small, this change will not violate the inactive inequality. iii. xjp < xjs*: Only the term Gi*(yP - yit*) is affected in the left-hand side of the pth inequality and it reduces to Gi*(y? - Yit*). The inequality is not violated. (c) yp 5 y;: i. xjp 2 xp: The only term affected in this inactive inequality is -Lj*(x;* - XI”*), which increases due to the decrease in Lj*. However, if y? is close enough toy:*, the inequality will remain inactive. ii. xjs* 5 xjP 5 xt*: I Again,-Lj*(xjP -XI”*) increases due to the decrease in Lj*. However, ify? is close enough toy:*, the inequalityremains inactive.
In the left-hand side of this inequality, Gi*(y?* - Yit*) Lj*(xjP - XI”*) changes to Gi*(yP - L.*(XjP I - IC?:), which is a lesser quantity than the origmal. Thisinequahty is not violated.
-zt*)
(b) y; i.
< y p < y;*:
XP 2 I
xt*: 1
This is an inactive pinch inequality. The two affected terms, Gi*(yP - yit*) - Lj*(xf* - XI”*), change to
which makes the inequality more positive. But if y: is close enough to y?*, the inequality will remain feasible. ii. xjt < xjp < xjt*: In this inactive inequality, Gi*(yP - Yit*) - L.*(xjP I - XI”*) becomes
iii. xjp 5 x?: This inequality is not affected at all by the two changes.
A new feasible solution that features a lower cost than C* has been found. 2. If a lean stream that satisfies case 1above does not exist in the optimal solution, then there must exist Sj such that, xi”* > maxpEplxjP > XI”*. (Indeed, if all the lean streams had target compositions less than or equal to maxpEplxjP,then the rich load above this point must be zero. This could only be the case if the problem is unpinched. Contradiction.) Let us now infinitesimally decrease y:* to yi* and xjt* to xi” so that xjt remains above maxpEp,xjP and
This change keeps the mass balance, eq 6 , satisfied and is always possible. Consider now thepth pinch inequality
Gi*(y;* - y?) - L I. * ( x p - xp) - Lj*(xjp- xi’*) Hence, the inequality may either relax or be more constrained. In any case, if the decrease in y:* is infinitesimal, the inequality is not violated. iii. xjp < xj”: The only affected term, Gi*(yP - Yit*) decreases to Gi*(y? - yit*). Therefore, the pth pinch inequality remains feasible.
(c) yp I y> i, x.P 2~ t 1 1
* :
The only term affected in this inactive inequality is -Lj*(x:* -x:*), which changesto-Lj*(xf- x p ) . However, if yi* is close enough to y:*, the inequality will remain inactive.
Ind. Eng. Chem. Res., Vol. 32, No. 9,1993 1947 Table I. Stream Data for the 2
X 2
Example
rich streams
y-scale
lean streams
Gi, Y,: it, .rcP, Xi", Cjv stream kg/s kg/kg kg/kg stream kg/kg kg/kg dollars/kg Ri 0.1 0.045 0.02 si 0.0015 0.075 0.7 Rz 1.5 0.03 0.001 Sz 0.004 0.05 0.03
ii. xj" c xjp
0.045
___
0.042
......
0.030
_--
0.028
.....
2
9.050..
R2
-
iii. xjp c xj": The change in the rich load due to the decrease in the inlet composition of Ri does not affect this inequality. Besides, if the decrease in the target composition of Sj is infinitesimal enough, then x: will remain above the pinch point and, hence, there will be no change in the lean load below the pinch. With the above arguments, a new feasible solution has been found for case 2, withy: < y:*. If yfl C y,: then the above procedure can be repeated, until yis = y:'. Thus, the primary assumption for the rich stream case has been contradicted. To summarize, it is established that if y?* is strictly greater than y:', then possibilities exist to either lower the cost of the obtained global solution by lowering the flow rate of a lean stream if the solution is unpinched or case 1 holds or find another solution of the same cost by lowering a lean stream target composition; thus, yis must equal y:' in the optimum solution. Hence it is established that there exists a global, minimum utility cost solution of (P5) such that all the inlet compositions are at their lower bounds, i.e. y: = y,:' x r = x;'.
i = 1, ....NR j = 1,
....N ,
The above theorem suggests that the solution procedure is directly applicable to (P5). Therefore, developed for (Pl) we proceed to illustrate this procedure with two examples. 6. Example Problems 6.1. Example 1. Our first application of the method is an example which has two rich and two lean streams. The waste minimization task is to recover zinc chloride from the effluent of a metal pickling plant. Details on the process have been published by Parthasaradhy (1989), El-Halwagi and Manousiouthakis (1990a), and Lo et al. (1983). The flow and property data for this case are presented in Table I. The two effluent streams from the plant are the spent pickle liquor, R1, and the rinse wastewater, R2. The lean streams are a strong-base ion exchange resin, SI,and the extraction solvent tributyl phosphate, SZ. Linear equilibrium, dependent only on the solute-solvent system, is considered y
x -scale
....................
c xj"":
In this inactive inequality, -Lj*(xjP - xf*) increases to -L,*(xjt xf*). If @in* - y:) is sufficiently small, the inequality remains inactive.
(S1)
x -scale I
Rl
0.376(x1+ e )
+ 0.0001
y = 0.845(x2 + c) (S2) with e = 0.0001. With these data, a composition interval diagram (CID) as shown in Figure 5 is constructed. The example has two pinch point candidates (stream inlets that lie in the common rich and lean composition range), p = 1at yzUand p = 2 at x2S. Hence the example problem can be decomposed into three subproblems, the LP, (P21),and (P~z). For the lowest interval in which a rich stream is present, viz., from y 2 = 0.00346 to y2t =
.- - - - - - - - .
.....
......9.075....
-. ....;(i..........0.033...
I I 0.020
.....
.....
.....
9.053....
-......I.
..........o.a24...
I I I I 0.003
---Q.M)9-. - - - - t - _ _ _0 _.w -
0.001
......9.002...
I 1
- .......
.0007
Figure 5. CID diagram for example 1 (not to scale). The outlet composition of S1 and SZcan be anywhere in the composition range shown by the dashed lines. Table 11. Solutions of the Two Subproblems for Example 1
subproblem
P21 P22
active pinch 1 2
optimal solution
L1
J52
%lt
0.5031 0.5031
0.2396 0.2396
0.075 0.075
rat 0.0416 0.0416
r)Z,lt
0 0
0.001, only SIis feasible. Therefore, any feasible solution to this problem must have LI 1 1.5(0.00346 - 0.001)/ (0.00885 - 0.0015) 0.5031. If the target composition is kept fiied at its upper bound, the optimization is a linear program and has the unique solution L1 = 0.5615 kg/s and L2 = 0.1028 kg/s. Indeed, if stream LZis forced to exit at xzt = 0.05, the maximum feasible flow rate for L2 = 0.1028 kgls. This stream is the cheaper of the two. Since the usage of the cheaper stream is constrained, the utility cost, at 0.396 dollars/s, is not the lowest that can be obtained for this network. This solution has a pinch a t y l . Because a pinch point exists in this network, subproblemsP21 and P 2 2 should be solved. Using Properties 4 and 5 and the objective value for P2v above, upper bounds on LI and L2 are found a t 0.56 kg/s and 13.0 kgls, respectively. Since either pinch candidate 1or pinch candidate 2 must be active, then from theorem 3, x1u 1 x l t L 0.00885. The solutionsto the two subproblems are shown in Table 11. Both subproblems have the same solution since both pinch inequalities are active at the optimum. Each subproblem is convexifiedusing eigendecomposition of the bilinear nonconvexities as explained in section 4. For numerical efficiency, the variables are rescaled to lie in the range [0,11. Each subproblem is solved using the Soland algorithm coupled to a standard nonlinear solver, MINOS (Murtagh and Saunders, 1987). The total time for global solution was approximately 90 s of CPU time on an Apollo 10000. The global optimum of the flexible targets problem is L1 = 0.3593 kg/s and L2 = 0.2396 kgls with a minimum utility cost of 0.359 dollarsls. Whereas the outlet com-
1948 Ind. Eng. Chem. Res., Vol. 32, No. 9, 1993 y-scale
R3
x -scule 1
0 5 L, 50.5579 0 5 L, 5 13.0
x -scale 2
0.050
y; < y; I y;
i = 1,2,3
x i s ~11 x t ~ x uj = l , 2
0.045
(2X3,1t- 1)(0.045- y ; ) 10
0.042
(271,,: - 1)(0.0354- x;) 1 0 (2T1,1 - 1)(0.0088- X ; ) 1 0 0.030
X3,; = 0,l
0.028
12,; = 091
0.020
= 091 The upper bounds on Lj are found as explained in the previous example. The composition variables are redefined so as to be scaled within 0-1 range with the following substitutions 11,;
X;
=
0.003 0.001
I.
.0007 ......__Y ..... 0015.1 ...I______...... Figure 6. CID diagram for example 2 (not to scale). The dashed line below Rs indicates that it can be cleaned from 0.045 to 0.0007.
Table 111. Stream Data for the Example with Rich Target Composition as an Additional Variable rich streams lean streams Gi, ~h Yi", Y?, X' = x;, xju, stream kg/s kglkg kg/kg kg/kg stream kg/kg kglkg do&/kg R1 0.08 0.045 0.02 0.02 Si 0.0015 0.075 0.7 Rz 1.5 0.03 0.001 0.001 Sz 0.004 0.05 0.03 Rs 0.1 0.05 0.045 0.0007
position of SIgoes to its upper bound, Sp uses only 81.9% of its entire composition range. This result suggests that the stream S1 is used only to make the mass exchange from the lowest rich interval feasible. Once this stream is utilized for its entire mass exchange capacity, the rest of the load is taken up by the cheaper solvent extraction stream. 6.2. Example 2. In this example it is demonstrated that rich streams in an MEN can be cleaned over their minimum specification at no additional cost. Hence the advantage of variable rich stream targets is shown. Given are three rich and two lean streams. The flow and property data for this example are similar to the previous case and are given in Table 111. The equilibrium relationships are identical to those in example 1. The composition interval diagram (CID) for this case, shown in Figure 6, is constructed. The mathematical formulation for this problem is (P1) min 0.7L, + 0.03L2 subject to 0.0455 + O.l(O.05 -):y - L,(x: - 0.0015) L,(X; - 0.004) = 0 0.0443 + 0.1(X3.,t(0.03-]):y - L1(xlt - 0.0015) L,(o.O314 4- ~ , , ~ X ; )4- 0 . 0 3 5 ~ , ~5, ,0~ 0.0037 + 0.1(X3,~(0.0035 y : )
- 0.0015
- L1(0.0073 + vl,;x;)
+
0.0088Ll~,,; I O
0.0735 X ; - 0.004 Pzt = 0.046 0.045 - Y ; rgt = 0.044 Also, (Pl)is transformed into the form of (P2)using the definition, qj,ptPjt = Uj,pt. The resulting problem is
(P2)
min 0.7L1 + 0.03L2
subject to 0.046 + 0.0044r; - 0.0735L1p; - 0.046Lzp: = 0
+
0.0443 - O.OOI~X~,; 0.0044X3,,trgt- 0.0735~,p,~ + 0.0314L,(q,,lt - 1)- 0.046L2~,,: I 0 0.0037 - 0.0042X3,;
+ 0.0044X3,:rgt + 0.0073L1(q1,; - 1)0.0735L,u1,~I O 0 5 L, 50.5579 0 5 L, 5 13.0
OIr;51
i=1,2,3
01pt11
j=1,2
ci, P) = {(2,1),(1,2)]
0 5 uj,; 5 1 (1j,pt
0 ' 9 P) = {(2,1),(1,2)1 0 (2X3,: - 1)(0.044r:) 1 0
- 1)5 ujs t -Pi"
5
(271,,: - 1)(0.0314 - 0.046~:) 2 0 (2v1,: - 1)(0.0073- 0.0735~:) 2 0
= 0, 1 12,: = 091
11,: = 0,1 This example has two pinch point candidates, p = 1a t y2* and p = 2 at xz*, and it is partitioned into three subproblems, the LP (all pinch inequalities inactive), (P21) (at least pinch 1active), and (P2d (at least pinch 2 active).
Ind. Eng. Chem. Res., Vol. 32, No. 9,1993 1949 Table IV. Solutions of the Two Subproblems for Example
min 0.7L1+ 0.03L2
(LP)
subject to 0.046
2
+ 0.0044ri - 0.0735L1- 0.046L, = 0 0 IL, 5 0.5579 0 IL, I13.0 OIrfI1
i=1,2,3
All the exit compositions are at their upper bounds, hence this is an LP. No other pinch inequalities are active.
+
(E,)
min 0.7L1 0.03L,
subject to 0.046
+ 0.0044ri - O.O735L, - 0 . 0 4 6 L g l = 0 0.0443 - 0.0015X3,; + 0.0044X3,,tr,t- 0.0735L10.0314L, = 0
+
0.0037 - 0.0042X3,i 0.0044X3,;r2
- 0.0073L1I0
0 IL, I0.5579 0 IL, I13.0 OIritI1
i=1,2,3
0.6827 Ip; I1 (2X3,; - 1)(0.044r:) 2 0 X3,; = 0 , l
The binary variables associated with the lean streams are zero, hence, a much simpler version of (P2)is obtained.
+
(P2,)
min 0.7L1 0.03L,
subject to 0.046
+ 0.0044ri - 0.0735L1p,t- 0.046Lg; = 0 0.0443 - 0.0015h3,; + 0.0044X3,;r: - 0.0735Llp; + 0.0314L2(72,;- 1) - 0.046L2~2,;I0
+
0.0037 - 0.0042X3,; 0.0044h3,;r,t - 0.0073L1 = 0 0 IL, 50.5579 0 IL, I13.0 OIrfI1
i=1,2,3
0.1 Ip; I1 OIp,tIl
0 Iu,,; I1 (42,; - 1) Iu,,;
- Pl I0
(2X3,; - 1)(0.044r:) 2 0 (277,,: - 1)(0.0314 - 0.046~;) 2 0 X3,; = 0 , l
= 091 These two subproblems are solved using the algorithm described in section 4.1. When the stream target compositions are fixed at their upper bound, the LP solution is L1 = 0.553 kg/s and L2 = 0.1165 kg/s with an objective value of 0.3905 dollars/s. Once again, the cheaper stream is constrained by feasibility. The solution is pinched at p = 1. 772,;
sub- active problem pinch y3t X3,1t P21 1 0.03 0 0.045 0 P2z 2 0.03 0
optimal solutions L1 L2 xlt ~ 2 t qz,+ 0 0.5031 0.2332 0.075 0.0491 0 0 0.5031 0.2332 0.075 0.0427 0 0 0.5031 0.2332 0.075 0.0491 0
The target composition of the rich stream R3, yst, is a problem variable. The solutions to the two subproblems are shown in Table IV. Certain features of the optimal solutions may be noted. Both the pinch inequalities are active giving the same solution to the two subproblems. The global solution was obtained in approximately 160 s of CPU time on an Apollo 10000. Therefore, with variable targets, there is an 8.2% reduction in the utility cost in comparison to the fixed target case. The rich stream is cleaned to 35% of its possible composition range from 0.045 to 0.007, at no additional cost over the case when ~3~ = ~ 3 Since ~ . in the latter case xzt* IxzU, S2 has some mass exchange capacity remaining, which is utilized for the extra clean-up of R3. Any rich stream that has in its lowest interval, a feasible lean stream with variable target composition away from the bound at the optimum, can thus be cleaned more than ita fixed target composition. Since maximizing the removal of the key component from the rich stream is not an explicit objective, the problem has infinite optima, with 0.045 2 y3t* 2 0.03 with the corresponding values of xzt. The two solutions obtained )the limit points are shown. for (R1at 7. Conclusions
We have investigated a new problem in mass exchange network synthesis, that of variable supply and target compositions. Many industrial MEN synthesis problems, such as in waste minimization, have stream supply and target compositions specified within upper and lower bounds. Advantages of variable lean stream target compositions are that reducing the exit composition of certain streams can make them feasible if they were infeasible and that the total mass exchange capacity can be increased, despite a drop in the outlet composition. This may lead to a reduction in the minimum utility cost of the network or may allow greater recovery from the rich streams at no additional cost. A mixed integer nonlinear programming formulation for the minimum utility cost problem is proposed. To help the solution of this otherwise large MINLP, a decomposition of the problem is performed based on rigorous properties established in the paper. In situations where the supply compositions of the rich and lean streams are allowed to vary between upper and lower bounds, it is shown that a property of this problem formulation is that there exists a globally minimum utility cost solution such that all the variable supply compositions are a t their lower bounds. This implies that the minimum utility (MSA) cost problem with variable supply and target compositions is equivalent to a problem with only variable target compositions and all the stream supply compositions fixed at their lower bounds. To solve the variable target problem, a decomposition into subproblems is used and a global algorithm is used to solve each subproblem. The resulting solution procedure is illustrated with example problems. The examples illustrate several intriguing results. Specifically, it is shown that lowering the outlet compositions of lean streams can reduce the operating cost of a
1950 Ind. Eng. Chem. Res., Vol. 32,No. 9, 1993
mass-exchange network. It is also shown that certain rich streams can be cleaned more than their minimum requirement with no increase in the utility consumption. It must also be emphasized that by simple analogy, all the results presented here are valid for determining the minimum utility (steam and cooling water, for example) cost of HENS with variable stream inlet and outlet temperatures. In an HEN, the hot streams correspond to the rich streams and the cold to the lean. There is a simple linear thermal equilibrium relationship, TH= Tc + ATmb. Inlet and outlet temperatures are variable if the HEN is synthesized as part of an entire process integration scheme. The formulation we have presented can be embedded into the flowsheetoptimization procedure for determining what set of inlet and outlet temperatures and stream flow rates minimizes the utility consumption of the process HEN. The implication of these results for standalone HENS is that the utility targets should be considered variables to obtain minimum utility consumption for the heat exchange task.
Acknowledgment We are grateful for financial support from the UCLANSF-ERC under Grant No. CDR 86-22184. Some of the software employed in this study was developed with Grant No. L900926 from The Ralph M. Parsons Foundation. Nomenclature Ak = matrix of coefficients for quadratic terms bj = intercept of equilibrium line for lean stream Sj bk = vector of coefficienb for linear terms C* = optimal solution of problem (P2) cj = unit cost of lean stream Sj, /kg C k = vector of constant terms Gi = mass flow rate of rich stream Ri, kg/s Lj = mass flow rate of lean stream Sj, kg/s Lj = variable to replace q j , p t L j mj = slope of equilibrium line for lean stream Sj n = number of z variables N R = number of rich streams in the network NS = number of lean streams in the network p = index for pinch point candidate P = set of pinch point candidate ' P j = set of active pinch point candidates for stream Sj R = set of rich streams S = set of lean streams Ri = rich stream Sj = lean stream x = mass fraction of key component in the lean stream, kglkg y = mass fraction of key component in the rich stream, kglkg z = problem variable from (P2)in (P3)including x:, y?, Lj Subscripts i = index for rich stream in the network j = index for lean stream in the network k = index for constraint in (P3) p = index for pinch point candidate U = unpinched problem form Superscripts * = optimal solution value 1 = lower bound of a variable p = pinch point candidate composition value
s = supply composition value t = target composition value u = upper bound of a variable Greek Symbols q =
binary variable associated with lean stream
X = binary variable associated with rich stream or eigenvalue A = matrix of eigenvectors K = cardinality of the set with strictly negative eigenvalues
of A 4 = 7%
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Received for review May 11,1993 Accepted May 18, 1993
