Ind. Eng. Chem. Res. 1990,29, 239-251
239
A Mixed Integer Nonlinear Programming Model for Retrofitting Heat-Exchanger Networks Amy R. Ciric and Christodoulos A. Floudas* Department of Chemical Engineering, Princeton Uniuerstiy, Princeton, New Jersey 08544
This paper addresses the problem of determining the optimal retrofit of an existing heat-exchanger network, considering the placement/reassignment of existing exchangers to different process stream matches, their need for additional area, the creation of new process stream matches, the cost of stream repiping, and other issues. A mixed integer nonlinear programming (MINLP) model is proposed that encorporates all possible process stream matches, network configurations, and existing exchanger reassignments in a single mathematical formulation. In addition, it is shown how heat-exchanger rating equations, repiping costs, pressure drop considerations, and varying heat-transfer coefficients can be included in this model. The proposed model simultaneously evaluates the required area of each existing and potential process stream match, the required additional area of each potential match-exchanger assignment, and the estimated repiping cost, while optimizing the selection of process stream matches, the network configuration, and the match-exchanger assignments. The resulting solution of this mathematical model gives the optimal selection of process stream matches, their heat loads, their total required area, the optimal network configuration, and the optimal assignment of existing heat exchangers to new or existing process stream matches, based upon the cost of moving an exchanger and the actual required additional area. It is shown that this mixed integer nonlinear programming formulation can be solved efficiently by applying the Generalized Benders Decomposition technique. The properties of the decomposition are discussed, and the method is illustrated with three example problems. The rising cost of energy in the 1970s sparked an interest in developing new technologies for improving heat recovery in chemical processes. A great deal of research has focused on the design of heat-exchanger networks, an important heat-recovery system. This research effort has resulted in over 200 publications, and a good review is provided by Gundersen and Naess (1988). I t is interesting to note that most of the research has been in the area of grass-roots design of new heat-exchanger networks. Only recently has attention been shifting to developing systematic techniques for the optimal retrofit of existing heat-exchanger networks. The major objectives of these techniques are the reduction of utility use in the existing network, the full utilization of the existing heat exchangers, and the identification of the required structural modifications. In general, structural modifications can be grouped into four broad categories. New exchangers can be purchased, the area of existing exchangers can be increased or decreased, streams can be repiped, and existing exchangers can be reassigned from one match to another. It should be noted that introducing new process stream matches is not a single modification but involves several modifications, including repiping streams, purchasing a new exchanger, and/or reassigning an existing exchanger. Early retrofitting techniques did not address the issue of reassigning existing exchangers but rather were concerned with obtaining retrofitted networks requiring little repiping. Jones et al. (1985, 1986) presented a strategy for the retrofit of heat-exchanger networks that is based upon the generation of a number of alternative designs and the evaluation using simulation runs. They selected the best design based merely upon the full utilization of the existing equipment and the addition of area in some of the heat exchangers. Tjoe and Linnhoff (1987) proposed a retrofit design philosophy in keeping with their "Pinch Technology" approach. It is a two-stage method, using targeting proce-
* Author to whom correspondence should be addressed. om"i/
dures to select the project scope (Tjoe and Linnhoff, 1986) and an evolutionary procedure based upon a remaining problem analysis for retrofit design of heat-exchanger networks. Saboo et al. (1986) proposed an evolutionary strategy that is based upon nonlinear optimization, constrained MILP synthesis, and feasibility evaluation capabilities of RESHEX. Their procedure generates a number of successive retrofit design alternatives without the explicit consideration of economic data. Yee and Grossmann (1987) were the first to address the issue of reassigning exchangers. They developed a MILP model that allowed for exchangers to be reassigned in the case where (a) a match was to be removed and (b) one process stream in the new match also took place in the original match. Their model obtained the process stream matches and exchanger assignments that gave the fewest number of new matches and reassigned exchangers. This model did not consider all the potential combinations and did not explicitly take into account the use of the existing area and its potential increase or decrease in the different heat exchangers. Ciric and Floudas (1989) presented a two-stage approach to the retrofit problem. In the first stage, a MILP formulation at the level of matches selects process stream matches and match-exchanger assignments by minimizing an investment cost that is based upon estimates of the required heat-transfer area. In the second stage, a NLP is formulated on the basis of a postulated network superstructure. The solution of the NLP provides the actual retrofitted network that minimizes the total modification cost. This paper presents a MINLP formulation that selects process stream matches and match-exchanger assignments while simultaneously optimizing the network structure. This approach allows for match-exchanger assignments to be made on the basis of actual area requirements, as opposed to estimates of area and/or piping costs. In addition, the formulation presented in this paper can be expanded to (a) calculate piping costs from the actual network con-
9012629-0239$02.50/0 0 1990 American Chemical Society
240 Ind. Eng. Chem. Res., Vol. 29, No. 2, 1990
figuration, (b) include heat-exchanger rating equations, (c) account for different types of exchangers, (d) handle variable heat-transfer coefficients, and (e) include pressure drop considerations for existing heat exchangers. The MINLP formulation is solved efficiently by application of the Generalized Benders Decomposition algorithm (Geoffrion, 1972). The original formulation is decomposed into two subproblems. The master subproblem is a MILP that selects the process stream matches and heat loads. The primal subproblem is a relaxed MINLP that derives an optimal network configuration and selects the match-exchanger assignments. Theoretical properties and practical aspects of the decomposition of the original problem are discussed, and the method is illustrated with three example problems.
Problem Statement The problem to be addressed in this paper is stated as follows. Given is an existing heat-exchanger network, containing N exchangers of known area; a set H of hot process streams and a set C of cold process streams with fixed flow rate heat capacities F and inlet and outlet temperatures T and To;a set of hot and cold utilities HU and CU respectively; and a minimum temperature approach ATmir The objective is to determine the optimally redesigned heat-exchanger network that features the minimum total modification cost. It should be noted that the minimum temperature approach, AT-, refers to what Jones et al. (1985,1986) term the EMAT-the minimum allowable temperature difference between a hot and a cold stream exchanging heat. Note that AT,, is, in general, different from the targeted HRAT, which is the minimum temperature approach used to calculate the minimum utility consumption. Unless otherwise noted, ATmi,will be taken as equal to HRAT throughout this paper. It is assumed in this paper that the HRAT and the targeted utility levels are given. This assumption can be relaxed by using an outer optimization scheme, such as that employed by Ciric and Floudas (1989). Overview of the Retrofit Problem The tasks of a retrofit problem include (i) selecting the process stream matches that will participate in the retrofitted network; (ii) determining the heat loads of the selected matches; (iii) deriving whether a match will be housed in its original exchanger, a different exchanger, or in a new exchanger; (iv) calculating the heat-transfer area of any new exchangers; and (v) obtaining the amount of area that must be added to an existing exchanger. Three important points must be noted about these five retrofit tasks. (1)The retrofit tasks are interrelated. One cannot derive a network prior to selecting process stream matches or determine the best match-exchanger assignments without knowing the heat-transfer-area requirements of each match. (2) There are many solutions to tasks (9-(iii). For any given retrofit problem, there are several potential combinations of process stream matches. Each of these combinations has several potential network configurations, and for each network configuration, there are many ways to assign the existing exchangers. (3) The modification cost of the retrofitted network is determined by the streams that must be repiped, the number and size of any new exchangers, the amount of additional area that must be added to existing exchangers, and the costs associated with moving any existing ex-
changers. Thus, the modification cost is determined by the results of tasks (iii)-(v). Tasks (i) and (ii) contribute indirectly to the cost of the retrofitted network, as the network configuration and heat-transfer areas will vary with the selected combination of matches and their heat loads. These three points suggest that the best way to approach the retrofit problem is to model all five tasks simultaneously. In this way, the interconnection between each task can be accounted for appropriately. The MINLP formulation presented in this paper is composed of an objective function that is based upon the modification cost that is subject to a set of constraints. The set of constraints is composed of subsets that model each particular task. The transshipment model constraint subset (Papoulias and Grossmann, 1983) models the selection of process stream matches and heat loads. The constraint subset modeling the network configuration task is based upon a generalized match-network hyperstructure (Floudas and Ciric, 1989). An assignment problem is used to determine the placement of existing exchangers and whether to purchase new exchangers. Lastly, a set of rating equations is used to calculate the additional required area of each match. The resulting mathematical formulation simultaneously minimizes the total modification cost while performing the five retrofitting tasks.
Elements of the Mathematical Formulation (a) Modification Costs. The objective function represents the total modification cost arising from four sources: (a) the cost of purchasing new exchangers; (b) the cost of purchasing additional area for existing exchangers; (c) the cost of reassigning existing exchangers at different matches; and (d) the cost of repiping streams. Purchasing New Exchangers. The cost of new heat exchangers is modeled with equations of the form dB + CN where A is the total heat-transfer area, CN is a fixed charge cost, cy is a constant cost parameter, and ,t? is an exponent. The existence of the match ij is represented by an integer variable Yij, and the decision to purchase a new exchanger for match zj is represented by an integer variable mij. The cost of purchasing a new exchanger is appropriately modeled by (aA$ + ~N)mijYij
(A)
If (a) match ij is selected for the retrofit network (Le., Yij = 1) and (b) a new exchanger will be purchased for match ij (Le,, mi, = l),then the investment cost of this term will be equal to the cost of purchasing a new heat exchanger. If, however, the match is not selected to participate in the retrofit network or an existing exchanger is used to house match ij, this term of the objective function will be zero. Purchasing New Area. The cost of additional heatexchanger area to an existing exchanger is modeled by cy’(Xg)s;ay.. 1J ZJ (B) Here, a’and /3’ are cost coefficients, and 28 is an integer variable that is equal to one if exchanger n is assigned to match ij in the retrofit network and is zero otherwise. X: represents the additional area that must be purchased if match ij is to be housed in exchanger n. Note that if (a) no additional area is required (X 9 = 0), (b) match ij is not assigned to exchanger n (2: = 0), or (c) match ij is not selected for the retrofitted network (Yi, = 0 ) ,then the cost contribution of this term will be zero.
Ind. Eng. Chem. Res., Vol. 29, No. 2, 1990 241 Exchanger Reassignment Cost. The cost of reassigning an existing exchanger to a different match is given by CP.ZP.Y.. 81 11 IJ (C) The cost parameter c t may take the same value for all possible assignments of existing exchangers to different matches, or the value may vary between different assignments so as to reflect expected repiping costs. Repiping Cost. There are several methods, ranging in complexity from simple to detailed, for incorporating repiping costs within the p r o w e d mathematical formulation for retrofitting heat-exchanger networks. Four cases are presented below. The selection of the appropriate case should be based upon exchanger portability and the cost of stream piping relative to moving and purchasing exchangers. Case A. Existing exchangers are portable, and both stream repiping and exchanger movement are inexpensive relative to equipment cost. In this case, a detailed model of stream repiping is not warranted. Stream repiping costs can be modeled as a lumped cost factor on new matches and exchanger reassignments, via eq C. Case B. Exchangers are portable, and moving them is inexpensive. Stream repiping may be a significant but not dominant cost. In this case, stream repiping costs are independent of exchanger reassignments, as exchangers can be moved to a site convenient for the match. Stream repiping costs can be estimated from the flow rates of the new stream paths. The repiping cost of each new stream is estimated from the formula C,(fIF), where C is the cost of purchasing a single length of pipe that wifl carry the maximum flow rate F and f is the flow rate of a particular stream in the process stream superstructure. The total piping cost is obtained from
is dependent upon the exchanger assignments of the matches. If match ij is assigned to exchanger n, and match ik is assigned to exchanger n’, then the cost per unit flow equals c!$!,,. This parameter rate of repiping stream represents the cost per unit heat capacity-flow rate of piping hot stream i from the outlet of exchanger n‘ to the inlet of exchanger n. This cost is zero if hot stream i already has a pipe linking n and n! Case D. Exchangers are portable, but the cost of moving t h e m is significant, and stream repiping is a significant cost. This is the most general case, requiring that the cost of moving exchangers be traded off against the cost of repiping streams. For systems with portable exchangers, this model will provide the most detailed method of establishing exchanger locations and repiped streams in the retrofitted network. This approach to calculating repiping costs requires the introduction of the concept of a “site”. A site is the physical location of an exchanger and process stream match. By identifying the sites within an existing heatexchanger network, one can accurately model existing piping structures as links from one site to another. This approach can be used to provide detailed estimates of repiping costs. The mathematical formulation of the site model for repiping can be stated as follows: Suppose that there are n’ = 1, ...,N’existing exchanger sites in the plant. Let the integer variables sijntrepresent the assignment of match ij to exchanger site n’and Sijbe an integer variable that equals 1 when a new site must be created for match ij. With this notation, the exchanger reassignment cost, based upon the site of the match, can be calculated from
fy
Here, cnnt represents the cost of assigning exchanger n to site n’, and CnOis the cost of moving exchanger n to a new site. With this scheme, stream repiping costs can be calculated from
The flow rate variables, f , have an upper case superscript denoting the stream position in the superstructure, with superscript I denoting an inlet stream, B denoting a bypass stream, and 0 denoting an outlet stream. The subscripts and superscripts of the flow rate variables are defined in detail in Appendix A. Case C. Some or all exchangers are not portable, the cost of moving any portable units is not significant, and stream repiping is a significant but not domininant cost. In this case, piping cost is a function of the heat-exchanger assignments and is based upon the cost of linking one exchanger to another. In this case, piping cost can be modeled by
The cost coefficients in this formula can be chosen to reflect the cost of piping that particular stream from one site to another. For example, c’,! is the cost per unit flow cCv)i[Ccf;’zn Cfi’mij] fiJ[Ccf;jzg Cbjmij] rate of i in hot stream i from its inlet to site n’. Simi J n n ilarly, c ~ J ,is~ ,the , cost per unit flow rate of piping cold CfSfb[C(Cc:;:izg + c:;imik)z$ + ccct$z,”d + stream j from site .”to site n’. k n n‘ n’ The mapping of process stream matches to sites within c$jmik)mi,] + F,j[C(Cc!;,$z$ + n n‘ the plant requires an assignment model that assures that c!bjmkj)zg + (Cc&iq$ + coB,dmkj)mij]+ each selected match is assigned to a site and that each site n’ is assigned to at most one match. These requirements can fp[Cc?z,g Cf*imij] fP9j[Cc;&g+ c@imij])yij be modeled with the equations n n (D2)
+
+
+
+
%y
4f
+
+
This equation expresses the cost of repiping a stream in a superstructure as a function of the match-exchan er assignments. For example, the cost of repiping stream f Bi ji , the hot bypass stream flowing from match ik to match ij,
&jn’ ij
I1
-j-s..,+s..=y.. $in v v n‘ These constraints are totally unimodular, and as a con-
242 Ind. Eng. Chem. Res., Vol. 29, No. 2, 1990
sequence, the variables sijn, and Si, can be treated as continuous and will still take integer values at the optimal solution. It should be noted that, in cases B-D, piping costs can only be approximated, and a degree of uncertainty is encountered in case B, when a new match is introduced, in case C, when a new exchanger is purchased, and in case D, when a new site is created. In each case, the location of the new element within the plant layout is unknown; consequently,the costs of piping streams to these elements cannot be calculated accurately prior to design. However, the expressions presented above can be used to estimate repiping costs in order to evaluate the trade-off between repiping streams and purchasing additional heat-exchanger area. Equation D1 will be used throughout the discussion of the retrofit model to represent the piping cost calculation. It should be noted, however, that the analysis would also be valid if either D2 or D3 were used to calculate the piping costs. Objective Function. Combining eqs A, B, C, and D1 provides the following objective function for minimizing the modification cost:
OBJ = min C C [ ( c u A t + cN)mij+ C(cu’(XG)fl+ i
l
n
+ (Cp/Fi)(ff”+ Cff$ + k’ f p ) + (Cp/Fj)(fF’ + Cftbj + f?*’)]Yij (1) k’
c$)z;
(b) Heat Flow Model. The heat flow model is represented by the transshipment model of Papoulias and Grossmann (1983), where the temperature range of the problem is partitioned into T temperature intervals by using the inlet temperature of each stream. Heat can flow in one of two ways: between hot stream i and cold stream j in temperature interval t , denoted by qi.r,or from one interval to another via a hot stream residual heat flow, denoted by Ri,. A hot stream can either release heat to a cold stream in the same interval or it may release the heat into a lower temperature interval through a heat residual. A cold stream may absorb heat from any hot stream in the same temperature interval or from the residual heat of a hot stream in a higher temperature interval. Within each interval, hot stream i must release Q 3 heat, and cold stream j must absorb Qj”, heat. The transshipment model consists of (a) energy balances for each stream over each interval; (b) equations calculating the total heat loads, Qij, of each match; (c) constraints connecting the integer variables denoting the existence of a match, Yii, with the total heat load, Qij; and (d) an integer cut restricting the number of matches to be less than or equal to some maximum number Nma:
( c ) Generalized Match-Network Hyperstructure. The network configuration task is modeled with the aid of a generalized match-network hyperstructure (Floudas and Ciric, 1989). This hyperstructure is constructed from smaller superstructures associated with each process stream. Each stream superstructure contains all potential matches and network configurations embedded within it. These stream superstructures contain the Same elements as the superstructure proposed by Floudas et al. (1986). Unlike the superstructure of Floudas et al. (1986),in which only previously selected process stream matches (obtained from a minimum units calculation) appear, the generalized match-network hyperstructure must contain all potential matches, as matches have not been previously selected but are determined simultaneously with the network configuration. A review of the generalized match-network hyperstructure is given in Appendix A. The mathematical model of the network configuration problem utilizes the following sets. Let k E HCT be the indexed set of all process streams and utilities, defined as HCT = (H)U (C) Let the indexed sets k‘ E Rk and k” E Skk‘ be defined by Rk = {k’l if k E H , then k’ E C and (k,k’)E M A or k E C, k’ E H , and (k”,k)E M A ) S k k , = {k”lk’ E R k , k” E Rk, k’ # k”] The mathematical model of the generalized matchnetwork hyperstructure is as follows:
C fkb = Fk
k’€Rh
five
f#@&,,fflk = 0
k “E Shy
fflk +
f@kr- fflk = 0
k EHCT
(7)
k ’ E Rk, k f HCT (8)
k’ E R k , k E HCT
k”ESky
(9) Tkfi v b
+ k “EC
ff,j,@b - fflktfb = 0
k ’ E Rk, k f HCT (10) Qij - fp(tIpi 1 - t 1O i ) = 0 i E H , j E Ri (11) Qij - f$sj(tPsj- t i 4 = 0 (DTl), = tfvi (DT2)ij =
t?tJ
ty - tFj
i E HI j f Ri (12) i E HI j E Ri
(13)
i E H, j E Ri
(14)
”
L
Cgijt - Rit-1 + Ri, = Q f
i = 1, ..., H, t = 1, ..., T
j=l
(2) H
Cqijt= Qjlc,
j = 1, ..., C, t = 1, ..., T
T
Cqilt=Qij
i E H , j E Ri
i = l , ...,H , j = 1 ,
..., C
i = 1, ..., H, j = 1, ..., C
(17)
(18)
(4)
i E H , j E Ri fFpi - Qi,/ATij,,,, I 0 i E H , j E Ri
(5)
f?’
Qij/ATij,max2 0 i E H , j E Ri i E H , j E Ri (DT1)ij 2 AT,,
(20) (21)
fHj - FjY, I0
t=l
Q, - UY, I0
fy,i - FiYij I0
(3)
i=l
-
(DT2)ijI AT,, Here, U represents the minimium of the total heat contents of hot stream i and cold stream j.
i E H , j E Ri
(19)
(22)
Equations 7-9 are mass balances taken at the splitting and mixing points of the generalized match-network hy-
Ind. Eng. Chem. Res., Vol. 29, No. 2, 1990 243 perstructure. Equations 10-12 are energy balances over the mixing points and heat-exchanger units. Individual temperature differences are calculated in eqs 13 and 14, and the log-mean temperature difference is calculated in eq 15. It should be noted that Patersons' approximation (Paterson, 1984) can be used in place of the standard log-mean temperature difference. A minimum temperature approach is specified in eqs 21 and 22. The total required heat-transfer area of each match, Ai., is calculated in eq 16. Note also that Uijis the overah heat-transfer coefficient of match ij. Uij may be treated as a constant or it may be a function of the individual heat-transfer coefficients of each stream, which in turn are functions of the stream flow rates: 1 1 1 -=i = l , ..., H, j = 1 , ..., C (23) Uij hi hj
DT'
T,
I,
Figure 1. Placement of new shell and tube exchanger in series with existing 12 exchangers.
A:x, if A$ is greater than A:! and is zero otherwise. If X $ has a lower bound of zero, then the relationship of X $ can be expressed as
X : ? A?.11 - A ex n
(27)
Substituting (27) into (26) provides
+-
hi = ri(fFi)'
i = 1, ..., H
(24)
hj = r.(f@s')B 1 1 j = 1, ..., C
(25)
Constraints 17 and 18 connect the existence of match ij, designated with the integer variable Yij,with the flow rate of the streams flowing through the ij exchanger. Similarly, eq 19 and 20 place lower bounds on the flow rates through the exchanger. (Note that ATij," equals the largest possible temperature drop through the exchanger, equal to Tovi - Tj - ATmh.) If the match is selected, then the flow rates of the hot and cold streams through the exchanger are bounded from above by the total flow rate and below by a minimum flow rate that is based upon the heat load. If, however, the match is not selected, then the flow rates through the exchanger, f and f FJ,are set to zero. Note that if and f J are zero, then all other streams associated with match ij, namely, f f f/, f f : I , f fkj, and f are zero, as a result of eq 8 and 9. (d) Calculation of Additional Area. Single-Pass Shell and Tube Exchangers. The amount of additional area, X $, that must be added to an existing shell and tube exchanger n for a particular potential match ij is computed with the aid of a rating equation. This equation calculates a lower bound on the total amount of area required to house match i j in existing exchanger n, based upon a dirt factor R d (Kern, 1950). The dirt factor is the inverse of the maximum contribution of fouling to the overall heat-transfer coefficient. In general, the dirt factor may vary from one exchanger to another and may also depend upon the process streams in a particular match. For the simplicity of presentation, it will be assumed that Rd is constant. The rating equation specifies that the difference between the inverses of the design heat-transfer coefficient, Qij/ (A$(LMTD),j),and the heat-transfer coefficient, Uij,must be greater than the dirt factor: 1 - -1 2 R d Qij/AC(LMTD), Uij
F"
y,
?la,
Solving for the area results in
Here, Qij is the heat load of match ij, A$ is the total area that would be required if exchanger n were assigned to house match ij, and LMTD is the log-mean temperature difference of match ij. The additional required area for match-exchanger assignment ij - n,X $, is equal to the difference between the total required area, A$, and the amount of existing area,
Multipass Exchangers. The additional area required for housing a potential match ij in an existing multipass exchanger n is calculated in a slightly different manner. In this case, the additional area Xg is assumed to be provided by a new shell and tube exchanger located in series with the existing unit (Figure 1). The area of this unit is calculated from the heat load in the existing unit Q pn, the heat-transfer coefficient Vi',and the temperature ddferences (DTl), and (DT2)tn. d'he values of Q and (DT2)& are calculated with the aid of a rating equation. The set of equations for calculating additional required heat-transfer area for an existing 1-2 shell and tube exchanger are given in Appendix C. (e) Match-Exchanger Assignments. An assignment model is required for selecting the appropriate pairs of matches and exchangers. The match-exchanger assignment subproblem can be modeled according to the following three conditions: (1) each exchanger is assigned to only one match; (2) each match is housed in only one exchanger; (3) no exchanger should be assigned to or purchased for a match that will not occur in the retrofit network. These conditions can be stated mathematically with two sets of equations:
E,,
cz; I1 i,j
Ezo,V + m.. - Y..= 0 V n
1J
n = 1, ..., N
i = 1, ..., H, j = 1, ..., C
(29) (30)
The first equation ensures that exchanger n is assigned to at most one match it also allows for exchanger n to be removed from the network. The second equation requires that if match ij occurs in the retrofit network, then either a new exchanger will be purchased for it or an existing exchanger will be assigned to it. If, however, a match does not exist, then all the assignment variables associated with the match take the value zero. These equations have the property that when they are isolated, that is, when 2". and mij only appear in (29) and (30), the variables z t a n l mij can be treated as continuous variables in an optimization problem but will still take integer values at the optimal point. The retrofit formulation consists of eq 1-25,29, and 30. This formulation will be denoted as problem P.
Decomposition of MINLP Formulation The retrofitting optimization problem is a mixed integer nonlinear programming (MINLP) problem. In this paper, the Generalized Benders Decomposition (Geoffrion,1972) technique will be used to solve problem P. The key idea of this approach is to partition the variable set into two subsets: the complicating variable subset and
244 Ind. Eng. Chem. Res., Vol. 29, No. 2, 1990
the noncomplicating variable subset. The variables should be partitioned in such a way that, when either subset is held fixed, the resulting optimization subproblem is much easier to solve. From this partitioning of the variable set, the constraint set can be partitioned into two subsets: the first subset involves constraints featuring only the complicating variables, and the second subset contains constraints involving both the complicating and noncomplicating variables. The master subproblem is formed from the first set of constraints and a set of Lagrange functions. The primal subproblem is formed from the second set of constraints, with the complicating variables treated as parameters. The solution to problem P is obtained in an iterative procedure. In the procedure, a set of values of the complicating variables are selected. In the second step, the primal problem is solved with the complicating variables held constant, giving an upper bound on the final solution. In the third step, a Lagrange function is generated from the values of noncomplicating variables and Lagrange multipliers obtained during the solution of the primal problem. In the fourth step, this Lagrange function is appended to the master subproblem, which is then solved to give new values for the complicating variables and a lower bound on the final solution. This four-step procedure is repeated until the upper and lower bounds meet. (a) Selection of Complicating Variables, The goal of selecting complicating variables is to induce and exploit special structures in the primal and master subproblems. The most obvious choice of complicating variables is the integer variables Yij. Selecting only the Yi.variables as complicating gives a pure integer master prohem containing only the maximum units constraint and the Lagrange functions. The primal subproblem is an MINLP that can be solved as a NLP by treating the assignment variables as continuous. As it is shown in Appendix B, these variables appear only in the assignment constraints and will take integer values at the optimal solution. Thus, both the master and primal subproblems can be solved easily. The drawback of this set of complicating variables is that there may be many combinations of integer variables satisfying the maximum units constraint that do not satisfy the transshipment constraints. Thus, the master problem could generate several combinations of integer variables that result in infeasible primal subproblems. A more efficient approach is to select as complicating not just the integer variables Yi.but also the transshipment model variables qVt,Rit, and The primal subproblem resulting from this selection is a MINLP whose integer decisions can be treated as continuous variables and thus can be solved by using conventional NLP techniques. In addition, the complicating variables appear linearly in the primal subproblem, and therefore, the Lagrange functions will also be linear in the complicating variables. The master subproblem will thus be a mixed integer linear programming problem, which will generate few combinations of complicating variables that render the primal problem infeasible. This choice of complicating variables will be used to decompose problem P. (b) Primal Subproblem. The primal subproblem contains all constraints that involve only noncomplicating or both complicating and noncomplicating variables. In addition, the complicating variables are treated as parameters and are held at a constant value. The formulation of the primal subproblem is given as follows:
dip
min
CC[(aA$+ cN)mij+ C(cu’(X$@’+ i J
n
subject to
(LMTD)ij =
(DT1)i; - (DT2)ij
fT,i - F’Y, I 0 fE,j
i
EH,
j E Ri
- FJY;, I0 i E H , j f Ri
fT,i - Qij/AT,,,
i f H , j f Ri
20
f 5’’ - Qij/ATij,,, I 0
(DTl), 2
i E H , j f Ri ATmin, i E H , j E Ri
(DT2)ijI ATmin,
i E H , j E Ri
X $ I Aij(1 + RdUij) - AEx
cz; I1 i,; Cz$+mij-Yij=O
n = 1, ...,N i=l,
..., H , j = 1 , ..., C
n
The primal problem (Pl) is a mixed integer nonlinear programming problem. Note that Yijand Qij are constant parameters (this has been emphasized by writing them in boldface). Notice that problem (Pl) can be solved as a NLP in which the assignment variables are treated as continuous. This is because the assignment variables only appear in the assignment constraints and the objective function. As a consequence, they can be treated as continuous variables and will still take integer values, as shown in Appendix B. (c) Remarks on the Primal Subproblem, It should be noted that the constraints of the primal subproblem involve not only activated matches, that is, matches with Yij = 1 and Q i . > 0, but also inactive matches, that is, matches with $ij = Qij = 0. No terms from the inactive matches contribute to the objective function, and all flow rate, assignment, and area variables will take the value zero. The only variables associated with the unselected variables that are not necessarily zero are the temperature variables.
Ind. Eng. Chem. Res., Vol. 29, No. 2, 1990 245 Table I. Stream Data for Example 1" stream Ti,, K Taut,K FC,, kW/K H1 443 333 30 H2 423 303 15 c1 293 408 20 413 40 c2 353 S 450 450 W 293 313 a
C
$/&W year)
H
i l
+
c$)z$'
j=1,
80 20
T
n
+ (Cp/Fj)(fJiiii+ CfTkr' + f?~'~') + k'
t = l , ..., T
i = 1, ..., H, j = 1, ..., C
Cqijt= Qij t
Qij-UYijIO
+ C(~r'(Xg')8' +
..., C,
i=l
For any given iteration of the Generalized Benders Decomposition algorithm, the total number of potential matches is much greater than the number of activated matches. A primal subproblem containing variables and constraints associated with all potential matches would be much larger than a primal subproblem containing only active constraints. These primal subproblems provide the same optimal network, as no terms from the inactive matches appear in the objective function and all flow rate and assignment variables are zero for these problems. Thus, dropping the constraints and variables associated with the inactive matches gives a significantly smaller primal subproblem that provides the same optimal solution as the full primal subproblem. Strictly speaking, the variables and constraints associated with unselected matches should not be dropped, as the Lagrange multipliers of these constraints are required to generate the Lagrange function. However, in the course of solving the examples in this paper and in the grass-roots design problem (Floudas and Ciric, 1989), it has been observed that including the constraints and variables associated with the unselected matches (Le., those with Yij= 0) leads to premature termination of the Generalized Benders Decomposition algorithm. In the grass-roots simultaneous matches and network design problem (Floudas and Ciric, 1989), it has been observed that, when variables and constraints associated with unselected matches were dropped from the primal problem and the Lagrange multipliers of the dropped constraints were set to zero, the Generalized Benders Decomposition algorithm terminated at the correct solution. It is believed that this is due to a degree of uncertainty associated with the Lagrange multipliers. For this reason, the variables and constraints associated with unselected matches (Yi, = 0) will be dropped from the primal subproblem in this paper, and the Lagrange multipliers of the dropped constraints will be taken to be zero. (d) Master Subproblem. The master subproblem is given below: min p subject to
i = 1, ..., H, t = 1, ..., T
j=1
Cqijt=Q$
U = 0.8 kW/(m2 K).
p ICC[{((~(Aij)fl cN)mij
Cqijt+ Rit-l - Ri, = Q#
i=l,
..., H , j = 1 , ..., C
H C
2 2 Yij IN,,, i=lj=l Here, A$lpl, A?',, A!%', Ab?", Afoi', and A30.' are the Lagrange multlpiiers of constraints (12), (16)-(20), and (30) obtained in the solution of the lth primal subproblem. The values of the area ( A and X),flow rate ( f ) temperature ( t ) ,and assignment variables ( z and m)are held at the values obtained in the lth iteration. The master problem is a mixed integer linear programming problem that contains the transshipment model variables and constraints and the Lagrange functions derived from the previous l = l , ...,L iterations. $Ell,
$?i',
(ld,
Algorithm The technique for solving the retrofit problem using the MINLP formulation can be summarized as follows. (1)Set the iteration counter L to 1. Select an initial set of values for the complicating variables Y1,and Q$. Set the upper bound on the final solution to dBJ+ = and the lower bound to OBJ- = -a. (2) Solve the primal subproblem (Pl), with the complicating variables Yij and Qij held constant at yf;. and €&. If the value of the objective function, OBJ, is less than the upper bound OBJ', i.e., OBJ < OBJ', then update the lower bound by setting OBJ+ = OBJ. (3) With the values of the noncomplicating variables and Lagrange multipliers obtained in step 2, formulate the Lth Lagrange equation and append it to the master subproblem. (4) Update the iteration parameter L = L + 1. Solve the master subproblem to obtain new values of the complicating variables, Y$ and sf;. Update the lower bound by setting OBJ- = p. If the lower bound is greater than or equal to the upper bound, then stop. Otherwise, return to step 2. This algorithm was implemented automatically in the program REHEN (REtrofit of Heat Exchanger Networks) using the APROS decomposition methodology (Paules and Floudas, 1989) that utilizes the modeling system GAMS (Brooke et al., 1988). Example 1. The following example was taken from Yee and Grossmann (1987). It features two hot streams, two cold streams, one hot utility, and one cold utility. The data for these streams are given in Table I. Note that a single average heat-transfer coefficient is assumed for all matches. The existing network, shown in Figure 2, features five process stream matches, whose areas are given in Table 11. The current network requires 1500 kW of steam and 1900 kW of cooling water, at a cost of $158000/year. A minimum utility consumption analysis shows that, for a minimum temperature approach of 10 K, the utility consumption can be reduced to 200 kW of steam and 600 kW of cooling water, with a pinch point at 363-353 K, to give an operating cost of $28 000/year. This problem was solved by using the retrofit optimization technique described in this paper. Three simplifying assumptions were made: (1)the existing exchangers
246 Ind. Eng. Chem. Res., Vol. 29, No. 2, 1990 c2
c1
I353K
H2
423):
363K
303K
CW 375.5K
H I
443K
293K
338K 396.3
333K
H1
S
0.
Figure 3. Retrofit network for example 1.
14I3K
CW
Figure 2. Existing network in example 1. Table 11. Exchanger Data for Example 1 exchanger original match 1 H2-C2 above pinch 2 H2-C1 below pinch 3 H1-Cl across pinch 4 H1-W1 below pinch 5 Sl-C2 above pinch
area, m2 46.74 68.72 38.31 40.23 35.00
Table 111. Cost Data for Example 1 cost of area for an existing exchanger cost of area for a new exchanger cost of a new exchanger cost of moving an existing exchanger
1300X$: 1300A$6 3000 300
-350K
3
7
c 2
c 3
I
I
2
K
320K
I O 4 2 0 K q K 500K H ,
400K
e Y 1480K
S
$: CW
were of the simple countercurrent shell and tube type; (2) the dirt factor, Rd, was taken to be zero; and (3) piping costs were ignored. The cost data in Table I11 have been assumed. Eleven iterations of the Generalized Benders Decomposition algorithm were required to compute the solution. The optimal network structure is shown in Figure 3, and the data on the matches are given in Table IV. This structure has a total cost of $35 784. Note that the network exceeds the minimum total area, exhibited by the split of stream H2 below the pinch. These matches have a total area of 67.6 m2;a series arrangement would have a total area of 56.3 m2. However, putting the exchangers in series would require an increase in area in the existing exchangers assigned to house match H4 - W. Thus, in this case, what may be an optimal structure for a grass-roots network would not be optimal for a retrofitted network. Example 2. The second example is also taken from Yee and Grossmann (1987). This example involves obtaining a retrofit network for a system of three hot process streams and three cold process streams. The data for these streams are given in Table V, and the existing network is shown
e-350K
H2
398K
320K
371K
"3I
H3
300
c 1
Figure 4. Existing network in example 2.
in Figure 4. The data on the existing exchangers are given in Table VI. The existing network requires 360 kW of steam and 800 kW of cooling water, at a total cost to $44800/year. A minimum utility analysis shows that, if the network were
Table IV. Matches in Retrofit Network. E x a m d e 1 match Q,kW exchanger assignment original exchanger Above Pinch Hl-C2 2400 new H2-C1 900 2 2 s-c1 200 5
existing area, m2
retrofit, area, m2
cost, $
0 68.72 35.00
164.71 68.71 5.40
30 796 300 300
51.92 37.5 30.11
4 089
'
Below Pinch H1-C1 H2-C 1 H2- W
900 300 600
1
3 4
3
46.74 38.31 40.23
300
300
Ind. Eng. Chem. Res., Vol. 29, No. 2, 1990 247 Table V. Stream Data for Examole
H2 H3 c1 c2 c3 s1 w1
450 400 300 340 340 540 300
350 320 480 420 400 540 320
c3
CW
2O
12 8 9 10 8
350K
370K
+
c
f
U = 0.8 kW/(m2 K). Table VI. Areas of Existing Exchangers for Examole 2 exchanger area, m2 original match 1 12.50 Hl-C2 2 23.50 Hl-C3 3 5.39 H1-W1 4 33.09 H3-C1 5 11.49 H2-W1 6 45.06 H2-Cl 7 5.75 SI-c1
f320K
T300K
CW
Table VII. Cost Data for Example 2 cost of steam cost of cooling water cost of area for an existing exchanger cost of area for a new exchanger cost of a new exchanger cost of moving an existing exchanger cost of repiping a single stream
500K
390K 420K
I380K
I300
i.OK
c2
$BO/ (kW year) $201 (kW year)
Figure 5. Retrofit network for example 2.
300Xij,
some of the advantages of the MINLP approach to retrofitting heat-exchanger networks presented in this paper. Most notably, the optimal network derived with the twostage approach required the purchase of an additional 50.4 m2of heat-transfer area. The savings in heat-transfer area is attributed to the fact that the network optimization and the match-exchanger assignments are performed in a single step. Example 3. This example will demonstrate how the retrofit technique presented in this paper models systems with different types of exchangers, with restricted reassignments, and with variable heat-transfer coefficients. This problem is taken from Tjoe and Linnhoff (1987). It involves obtaining a retrofit network for a system of four hot streams and five cold streams, with cooling water used as a cold utility and a furnace used as the hot utility. The data for these streams are given in Table IX. Note that the individual heat-transfer coefficients are not constant but vary with the flow rate. For example, the heat-transfer coefficient of hot stream i in match ij, hf.,is given by
1200A $6 4000 300 50
redesigned with a minimum temperature approach of 10 "C, the use of steam can be eliminated completely and the cooling water consumption reduced to 440 kW, giving an annual operating cost of $8800/year. The task of this example is to obtain a retrofitted network by using these utility levels. The cost factors used in this example are given in Table VII. Note that, in addition to investment costs associated with heat exchangers, an investment cost based upon repiping is also included. All potential streams in the superstructure were assumed to contribute to the piping cost. An optimal retrofit network was obtained in 11iterations of the retrofit approach presented in this paper, with an iteration taking approximately 35 CPU seconds on a MIPS RC 2030 workstation computer. The matches, heat loads, match-exchanger assignments, and the required areas of the optimal solution are given in Table VIII. The optimal network is shown in Figure 5. The optimal network involves seven matches, one more than the minimum number of units. Every existing exchanger is utilized in this retrofitted network. An extra 27.5 m2 of heat-exchanger area is required in this network. The total investment cost of this network is $10 800, with $8250 required to purchase new heat-exchanger area, $1500 required for expenses associated with moving existing exchangers, and $1050 required for repiping. Comparing this network with that derived by using the two-stage approach of Ciric and Floudas (1989) reveals Table VIII. Matches in Retrofit Network, Examole 2 exchanger match Q,kW assignment H 1-C 1 820 6 Hl-C3 480 2 H1-CW 200 3 H2-Cl 800 I H2-C2 400 1 H3-C2 400 4 H3-CW 240 5
H1
hj = ,,i(fjCi)O.S The values of y have been,selected so that the individual heat-transfer coefficient, h;, is equal to the value given by Tjoe and Linnhoff (1987) when the flow rate, f?, equals the total flow rate of the stream, Fi. The existing network is shown in Figure 6. This network contains five heat exchangers and three coolers. The data for this equipment are given in Table X. Note that the smaller exchangers are single-pass (1-1) shell and tube units, while the larger exchangers are have one shell and two tube passes (1-2 exchangers).
original exchanger 2 3 6
existing area, m2 45.06 23.5 5.39 5.75 12.50 33.09 11.49
retrofit area, m2 42.23 25.54 5.00 26.45 15.05 34.66 12.16
additional area, m2 0.0
2.04 0.0
20.70 2.55 1.57 0.67
248 Ind. Eng. Chem. Res., Vol. 29, No. 2, 1990 Table IX. Stream Data for Example 3 stream Th,K Tout,K FC,, kW/K H1 327 30 100 H2 220 160 160 H3 220 60 60 H4 160 45 200 c1 100 300 100 c2 35 164 70 c3 80 125 175 c4 60 170 60 c5 140 300 200 CW 20 40 923.75
c5
h, kW/(K m2) 0.02010f~~~ 0.008623f0.8 0.07560f0,8 0.005771f0.8 0.1256f0.8 0.03341f0.8 0.008027f 0.00756f0.8 0.0 1154f 0.003393f0.8
Table X. Existing Exchangers in Example 3" exchaneer area. m2 orieinal match H1-C1 300 H2-C5 1220 H4-C3 935 H3-C4 726 Hl-C2 830 H1-CW 585 H&CW 98 H4-CW 1191
tvDe 1-1
H4160,--
Y
1-2 1-2 1-2 1-1 1-2
100
164
EO
c3
c4
kW of the cold utility. For this temperature approach, a pinch point exists at 160-140 "C. It should be noted that the minimum temperature approach within the exchangers, or ERAT, was relaxed to 5 "C in the exchangers. The cost data are given in Table XI. Note that, for a minimum temperature approach of 20 "C, the reduction in utility consumption provides a savings of $476 928/year. This problem was solved with the mathematical programming method described above. Five iterations were required to solve the problem, with an iteration consuming approximately 420 CPU seconds on a MIPS RC 2030 workstation computer. The optimal network is shown in Figure 7, and data on the retrofitted matches are given in Table XII. Note that only exchanger 8 required additional heat-transfer area and that exchangers 4 and 7 have been removed from the network. The potential reassignments of these exchangers involved purchasing a large quantity of additional area and, given the cost data in Table XI, were more expensive than simply purchasing new exchangers. The reason for this is that the linear cost function for adding large amounts of area gives a higher cost than the exponential cost function for purchasing new exchangers. The optimal network features a cost of $623 546, with the bulk of the investment going to the purchase of new heat exchangers.
$86.4/(kW year) $O.O/(kW year) 300X,, 1200A 4000 100
c6
The reassignment of existing units was restricted with two rules: (1)an existing process exchanger could only be reassigned to another process duty; (2) an existing cooler could only be reassigned to another cooling duty. The dirt factor, Rd, was taken as 0.5 (m2 "C)/kW for every match-exchanger assignment. Thus, the equation for calculating the required additional area in a 1-1 shell and tube exchanger is = Aij(1
$0
c1
W
Figure 6. Existing network in example 3.
Cost Data for Example 3 furnace fuel cooling water area for an existing exchanger area for a new exchanger a new exchanger moving an existing exchnager
Xg
220
1-2 1-2
" R d = 0.5 (m2 OC)/kW. Table XI. cost of cost of cost of cost of cost of cost of
c2
+ 0.5Uij) - AY
The additional required area associated with reassignments of existing 1-2 shell and tube exchangers was calculated by using the equations derived in Appendix C. The existing network, shown in Figure 6, requires 27 100 kW of the hot utility and 20 500 kW of the cooling utility. A minimum utility analysis shows that, for a minimum temperature approach of 20 "C, the utility consumption can be reduced to 21 580 kW of the hot utility and 17 975 Table XII. Matches in Retrofit Network, Example 3 exchanger match Q,kW assignment H1-C1 Hl-C5 H2-C5 H3-Cl H3-C2 F-C4 F-C5
14080 2620 9600 1920 1680 1800 19780
new
Hl-C2 H1-CW H3-C4 H3-C W H4-Cl H4-C3 H4-C W
7350 5650 4800 1200 4000 7875 11125
new new new
original exchanger Above Pinch 1
2
0.00
300 935
1
3
Below Pinch 5
6
6 4 7
5 new
3
6
0.00
1200
2
new
existing area, m2
0.00 0.00 0.00
585 830 0.00
1191
U, kW/(m2 K)
additional area, m2
0.69 0.40 0.310 0.98 0.52
755.98
0.44 0.23 0.18 0.096 0.37 0.22 0.23
0.00
2619.7 0.00 0.00
496.4 1016.1 1320 0.00 0.00
2006.2 230
Ind. Eng. Chem. Res., Vol. 29, No. 2, 1990 249 c5
4
c2
I
I
140
35
125
1170
300
86.5
H 1 327
--9-0. 140
160
Figure 8. General stream superstructure in hyperstructure.
c1
c3
c4
Figure 7. Retrofit network for example 3.
Incorporation of Pressure Drop Considerations In some retrofit problems, an aspect of the design may be the pressure drop through an existing exchanger. These aspects can be included in the retrofit model by extending the objective function and appending some constraints. The incorporation of pressure drop considerations requires adding the following term to the objective function: Cpc[(s$
+ s t f ) w i j k + (slfkt+ s$ks)(l - wijk)lzijkyij
Here, w , is an integer variable representing the shell or tube s i l i piping decision. If Wijk = 1, then hot stream i will be piped on the shell side of unit k,while cold stream j will be piped on the tube side. Alternatively, if Wijk = 0, then host stream i will be on the tube side of exchanger k and cold stream j on its shell side. It is important to note that, if Wijk is treated as a continuous variable that is bounded between 0 and 1,it will still take an integer value at the optimal solution. This is because Wijk appears linearly in the objective function and does not appear in the constraint set. As a result, Wijk will be forced to one of its bounds (i.e., 0 or 1) at the optimal point. The s i j k variables represent the excess ressure drops above some value AP,,. For example, S i j f is the excess pressure drop that would occur if hot stream i were piped on the shell side of exchanger k. This excess pressure drop is defined by the equation SIfks = max [O, APtfk"- U,,]
8
The remaining excess pressure drops are similarly defined. In the mathematical formulation, this maximum equation can be approximated by two greater than constraints stfksI APIfns - AP,,
sgt 1 0 A P f f is the pressure drop of hot stream i if it were piped into the shell side of exchanger It. This pressure drop is a complex function of the exchanger geometry, physical properties of stream i, and the flow rate of stream i (Kern, in m e rer,row muaei as a nonlinear ~uncciun01 m e scream
flow rate:
q,, = @&(fy
Similar equations must be written for the pressure drop of the hot stream through the tube side and for the cold stream through both the shell and tube sides. The coefficient cW in the objective function term reflects the relative cost of having pressure drops greater than AP,,. AP,, may be the maximum pressure drop that is allowable before a stream pressure must be pumped up. In this case, cpcrepresents a pumping cost. Alternatively, AP,, could be the maximum allowable pressure drop in an exchanger. In this case, the additional terms in the objective function can be interpreted as a penalty function, and cpc would be assigned a high value.
Conclusions An MINLP model of the retrofit optimization problem is presented that encorporates the major tasks of the retrofit problem, including selection of process stream matches and their heat loads, determination of the appropriate match-exchanger assignments, derivation of a network configuration, and calculation of required areas of new exchangers and additional required area of existing exchangers. The optimization model minimizes the cost of purchasing new heat exchangers, additional area of existing exchangers, and repiping costs. This model can be extended to include (a) complex heat exchangers, (b) variable heat-transfer coefficients, and ( c ) pressure level considerations. The formulation was developed by combining several mathematical models, including (a) the transshipment model of heat flow; (b) a network configuration model that is based upon a generalized match-network hyperstructure; ( c ) an assignment problem for determining match-exchanger assignments; and (d) rating equations for calculating the additional heat-exchanger area required for each potential match-exchanger assignment. The resulting model was a mixed integer nonlinear programming problem. This MINLP was solved with the Generalized Benders Decomposition algorithm (Geoffrion, 1972), with the transshipment variables selected as complicating variables. The original MINLP was decomposed into two subproblems. The first subproblem, the master subproblem, was a mixed integer linear programming problem that selected process stream matches and heat loads. The second subproblem, the primal subproblem, was a MINLP that could be solved as a NLP problem by treating the set of integer variables as continuous. These integer variables would still obtain integer values for the optimal solution. The primal problem determined the optimal network configuration and match-exchanger assignments for a selected combination of process stream matches and heat loads. The optimal solution to the retrofit problem was obtained by iterating over the master and primal subproblems.
250 Ind. Eng. Chem. Res., Vol. 29, No. 2, 1990
streams and two cold streams. Information for this example is given in Table XIII. Note that there are four potential matches, H1-C1, Hl-C2, H2-C1, and H2-C2. The stream superstructure for H1 is shown in Figure 9. Note that all potential matches are included in this superstructure. Similar superstructures can be derived for each of the three other streams.
Figure 9. Stream superstructure for sample hyperstructure. Table XIII. Information for Hyperstructure Examr.de stream stream index stream stream index H1 1 c1 3 H2 2 c2 4
Acknowledgment
Appendix B In this appendix, it will be shown that assignment variables z t and mi, in the primal problem (P1) can be treated as continuous and will still take integer values at the optimal point. Note that the assignment variables appear only in the objective function and the assignment constraints, eq 29 and 30. It will be shown that every solution to these constraints is an integer solution. Let the integer constraint matrix A = aij be defined by the product r
I
We acknowledge financial support from the National Science Foundation under Grant DMC-8617239.
Appendix A: Generalized Match-Network Hyperstructure The optimization technique presented in this paper is based upon a hyperstructure at the level of matches that contains a11 possible network configurations. This hyperstructure is constructed from smaller superstructures associated with each process stream, containing all possible matches and flow structures embedded within it. An example of a stream superstructure is shown in Figure 8. Each stream superstructure in the generalized matchnetwork hyperstructure contains a splitter at the beginning of the network, a mixer at the inlet of each potential match, a splitter at the outlet of each potential match, and a mixer at the end of the network. Streams in the superstructure connect the splitter at the inlet of the network to the mixers preceding each potential match, the splitter at the outlet of each exchanger to the mixers of every other exchanger, and from the splitter at the outlet of each exchanger to the mixer at the end of the network. In addition, the superstructures contain streams for each potential match that pass from the mixer preceding the exchanger, through the exchanger, and to the splitter following it. The following convention is used to label the streams: f i t , stream in the superstructure of process stream k flowing from the splitter at the beginning of the network to the mixer preceding the match between k and k‘; f f f stream in the superstructure of process stream k flowing throu h the exchanger used in the match between k and k’; stream in the superstructure of process stream k flowmg from the splitter following match between k and k“ to the mixer preceding the exchanger used in the match between k and k ’; and f stream in the superstructure of process stream k flowing from the splitter following the exchanger used in the match between k and k’ to the mixer at the end of the network. In addition to the flow rate variables, each superstructure has two temperature variables associated with each potential match within it. The variable ti! represents the temperature of f Etk measured at the outlet of the mixer preceding match between k and k‘and is hereafter referred to as the “inlet temperature”. The variable tPP is the temperature of stream f f t k , measured at the inlet of the splitter following the match between k and k ’, and is referred to as the outlet temperature. An example of a stream hyperstructure is shown in Figure 9. This hyperstructure is for a system of two hot
fs!k,: 8
Note that the vector b is an integer vector whose upper terms are zero if the inequality czg 5 1 ij
is inactive and one if the inequality is active. The lower terms of this vector are equal to the integer variables Yip Theorem 1 in Chapter 3 of Garfinkel and Nemhauser (1972) states that every solution of eq B1 is an integer solution if the matrix A is a totally unimodular matrix. Matrix A is totally mimodular if it satisfies two conditions (theorem 3, Chapter 3, Garfinkel and Nemhauser, 1972): (1)no more than two nonzero elements appear in each column and (2) the rows can be partitioned into two subsets Q1,Q2 such that (a) if a column contains two nonzero elements with the same sign, one element is in each of the subsets, and (b) if a column contains two nonzero elements of opposite sign, both elements are in the same subset. The matrix A in eq B1 satisfies the first condition of total unimodularity, as the assignment variables z$ appear once in constraint (29) and (30) and the assignment variables mij appear once in constraint (30). The matrix A also satisfies the second condition. Let Q1 be constraint (29) and Qz be constraint (30). Each variable z$ and mij have positive signs in constraints (29) and (30) and appear at most once in each of these subsets. Thus, matrix A satisfys condition 2a and is a totally unimodular matrix. From this, it follows that every solution of eq B1 is an integer solution. The assignment variables z$ and mij can thus be treated as continuous and will still take integer values at an optimal point.
Appendix C: Equations for Calculating the Additional Required Area for 1-2 Exchangers Consider an existing 1-2 shell and tube exchanger n assigned to house match ij. Any additional heat-transfer area, X 8,needed to fulfill the heat-transfer duty of match i j is provided with a new shell and tube exchanger that will be placed in series with the existing unit (Figure 1). Let Aijn be an estimate of the additional area Xij,, computed with the equation A.. = LJn
Qij
- Q Bn
UijLMTD(DTlij, DT2fn)
Ind. Eng. Chem. Res. 1990,29, 251-258
E,,
Here, Qij is the heat duty of match ij,Q is the amount of heat transferred between streams in the 1-2 shell and tube exchanger, and (DT2)& is the temperature difference between the hot and cold streams at the hot end of the 1-2 exchanger. Note that Qij,the heat duty of match ij,may be less than Q the heat transferred between streams in the 1-2 shell and tube exchanger. In this case, A,, is less than zero, indicating that no new exchanger area needs to be purchased. If A,, is greater than zero, then it is equal to the amount of area in the new shell and tube exchanger. Thus, A,, is a lower bound on the total additional area: Xij, 1 Aijn
E,,
The heat load Q $, and temperature difference (DT2)$, can be calculated from a set of equations derived from the rating equation of a 1-2 exchanger to give Qtn
= f P’(ATC)ijn
(DT2)$, = (DT2)ij + (Rij - l)(ATC)ij, Rij = f $ t j / f
,E”
Here, (ATQj, is the temperature rise of the cold stream in the 1-2 shell and tube exchanger, f Fj is the flow rate of the cold stream, and Rij is the ratio of the cold stream flow rate f F J to the hot stream flowrate f y . The temperature rise is obtained from (ATC)ij, =
where
sij,= exp[ -
UijAZx(Ri? + 1)lI2 (1 R&j)f Fj
+
]
Literature Cited Brooke, A.; Kendrick, D.; Meeraus, A. GAMS A Users Guide; The Scientific Press: Redwood City, CA, 1988.
25 1
Ciric, A. R.; Floudas, C. A. A Retrofit Approach for Heat Exchanger Networks. Comput. Chem. Eng. 1989,13(6), 703-715. Floudas, C. A,; Ciric, A. R. Strategies for Overcoming Uncertainties in Heat Exchanger Network Synthesis. Comput. Chem. Enp. 1989, 13 (lo), 1133-1152. Floudas, C. A.; Ciric, A. R.; Grossmann, I. E. Automatic Synthesis AIChE J. of Optimum Heat Exchanger Network Configurations 1986,32, 276-290. Garfinkel, R. S.; Nemhauser, G. L. Integer Programming; John Wiley and Sons Inc.: New York, 1972. Geoffrion, A. M. Generalized Benders Decomposition. J. Optimization Theory Appl. 1972, 10(4), 234-260. Gundersen, T.; Naess, L. The Synthesis of Cost Optimal Heat Exchanger Networks, an industrial review of the state of the art. Comput. Chem. Eng. 1988,12(6), 503-530. Kern, D. Q. Process Heat Transfer; McGraw-Hill Book Co: New York, 1950. Jones, D. A.; Yilmaz, A. N.; Tilton, B. E. Practical Synthesis Techniques for Retrofitting Heat Recovery Systems. AIChE Annual Meeting, Chicago, IL; American Institute of Chemical Engineers: New York, 1985; paper 35c. Jones, D. A.; Yilmaz, A. N.; Tilton, B. E. Synthesis Techniques for Retrofitting Heat Recovery Systems. Chem. Eng. Prog. 1986,82, 28-33. Paterson, W. R. A Replacement for the Logarithmic Mean. Chem. Eng. Sci. 1984,39, 1635-1636. Papoulias, S. A.; Grossmann, I. E. A Structural Optimization Approach in Process Synthesis-11. Heat Recovery Networks. Comput. Chem. Eng. 1983, 7,707-721. Paules, G. E., IV; Floudas, C. A. APROS: An Algorithmic Decomposition Methodology for Solution of Discrete-Continuous Optimization Problems. Oper. Res. J. 1989, 37 (6). Saboo, A. N.; Morari, M.; Colberg, R. D. RESHEX-An Interactive Package for the Synthesis and Analysis of Resilient Heat Exchanger Networks I Program Description and Application. Comput. Chem. Eng. 1986,10,577-589. Tjoe, T. N.; Linnhoff, B. Using Pinch Technology for Process Retrofit. Chem. Eng. 1986, 93, 47-60. Tjoe, T. N.; Linnhoff, B. Achieving the Best Energy Saving Retrofit. AIChE Spring Meeting, Houston, TX; American Institute of Chemical Engineering: New York, 1987; paper 17d. Yee, T. F.; Grossmann, I. E. Optimization Model for Structural Modifications in the Retrofit of Heat Exchanger Networks. Engineering Design Research Center, Report EDRC-06-25-87; Carnegie-Mellon University: Pittsburgh, PA, 1987.
Received for review March 10, 1989 Revised manuscript received September 11, 1989 Accepted October 23, 1989
Simulation of a Dry Fluidized Bed Process for SO2 Removal from Flue Gases Ourania Faltsi-Saravelou* and Iacovos A. Vasalos Chemical Process Engineering Research Institute and Department of Chemical Engineering, Aristotelian University of Thessaloniki, P.O. Box 1951 7, 54006 Thessaloniki, Greece
A fluidized bed reactor model, the large particle fluidized bed model (LPFBM), has been developed in order to predict the performance of a fluidized bed reactor, which operates a t low-to-medium u0/umtratios (uo/ud = 2-5) in the presence of large particles. The LPFB model was applied to simulate a pilot-plant unit of the dry regenerable process of SO2 removal from flue gases, over a wide range of operating conditions. The conversions predicted by the model fit the experimental data reported in the literature satisfactorily. The LPFB model was used to study the effects of the solid sorbent composition and particle size on SO2 conversion. From the simulation, it can be concluded that SO, sorbents containing 5 wt 5% or less of Cu, Na, Fe, Ce, Co, or Ni could be well considered for indistrial applications. Regenerable processes for removing SO2from flue gases are increasingly gaining commercial significance. In a regenerable process, the flue gases are fed to a sorption
* To whom correspondence should be addressed. 0888-5885/90/2629-0251$02.50/0
reactor where SO2reacts with a solid metal oxide sorbent, forming a sulfate that can be in turn regenerated by a reducing gas to form the oxide and a S02/H2Smixture. The feasibility of the regenerable processes depends mainly on the solid sorbent properties and resistance and to a 0 1990 American Chemical Society
