Computers them. Engng, Vol. 13, No. IO, pp. 11331152, 1989 Printed in Great Britain.
0098l354/89 $3.00+ 0.00 Copyright 0 1989Pergamon Press plc
All rights reserved
STRATEGIES FOR OVERCOMING UNCERTAINTIES HEAT EXCHANGER NETWORK SYNTHESIS
IN
C. A. FLOUDAS~ and A. R. CIRIC Department
of Chemical
Engineering,
Princeton
University,
Princeton,
NJ 085445263,
U.S.A.
(Received 15 January 1989; final revision received 10 May 1989; received for publication 8 June 1989)
AbstractCurrent grassroot heat exchanger network design techniques involve a threetask procedure: (a) calculation of the minimum utility consumption; (b) selection of a set of process stream matches that satisfy the minimum utility consumption and minimum units criteria; and (c) derivation of a minimum investment cost heat exchanger network configuration. Section 3 of this paper will present an approach for overcoming the uncertainty associated with the network optimization task that arises from the nonconvexities in the network optimization problem. A global optimum search approach is proposed that decomposes the nonconvex network optimization problem into a set of convex subproblems that represent upper and lower bounds and whose solution can lead to the network configuration with the globally minimum investment cost. Uncertainty arises in the selection of matches task when there are many combinations of matches that satisfy the targetting criteria. When there are many such combinations, exhaustive enumeration may be required to determine the optimal combination. Section 4 presents a decomposition methodology circumventing this uncertainty, based upon a proposed hyperstrucmre that contains all possible network configurations and process stream matches. This hyperstructure is used to derive a mixed integer nonlinear programming (MINLP) formulation that models both the selection of process stream matches and the derivation of a heat exchanger network configuration. A decomposition approach is proposed that efficiently derives the optimal combination of process stream matches and the optimal heat exchanger network configuration.
1. INTRODUCTION Heat exchanger network synthesis has been the subject of an intense research effort: a recent review by Gundersen and Naess (1988) cited over 200 publications on the topic. In general, these published approaches have been based upon a decomposition into three main tasks: (1) targetting; (2) selection of matches; and (3) optimization of the network configuration. In the targetting task, design objectives such as the minimum utility consumption, minimum number of units and minimum heat transfer area are obtained. In the selection of matches task, a set of minimum process stream matches and their heat loads are identified. Lastly, in the optimization task, the minimum investment cost network configuration is obtained. Efficient procedures based on heuristic, evolutionary, thermodynamic and mathematical programming approaches have been developed for completing each task, and these procedures have been incorporated into several approaches to the heat exchanger network synthesis problem (e.g. Linnhoff and Flower, 1978a,b; Floudas et al., 1986). Despite the intense research effort, however, none of the existing grassroots approaches can provide the globally optimal network. The current solution procedures of each task contribute an element of uncertainty. tAuthor
to whom all correspondence
should be addressed.
Uncertainty lThe targetting task identifies the minimum utility consumption and minimum number of units criteria, as well as any pinch points. Uncertainty arises in systems where heat flow across the pinch can provide significant investment savings that can compensate for the increase in utility costs. This particular issue will not be explored in this paper. Uncertainty tThe selection of matches task contributes an element of uncertainty when there are several combinations of matches that satisfy the targetting criteria (minimum utility cost and fewest number of units), but a minimum cost network configuration is derived for only one combination of matches that satisfies the targetting criteria. Currently, this uncertainty is overcome by exhaustive enumeration of all combinations of matches that satisfy the targetting criteria. A minimum cost network is derived for each combination, and the set of matches providing the lowest cost network conliguration is chosen as optimal. For large problems, this procedure can be quite cumbersome. This paper presents an alternative approach to this problem in Section 4. Uncertainty SThe network optimization task also contributes an element of uncertainty. This problem can be formulated as a nonlinear programming (NLP) problem (Floudas et al., 1986). Uncertainty arises because this nonlinear problem is nonconvex, and thus may have several local optima. When conventional solution techniques are employed
1114
C.
A.
FLOUVAS
to solve this nonconvex problem, the final solution depends upon the starting point. This paper addresses the global optimum search in this nonconvex nonlinear (NLP) problem in Section 3. This paper presents systematic approaches for overcoming the inherent Uncertainties 2 and 3 that are associated with the selection of matches and network optimization tasks. Together these tasks can provide the solution to the general problem of sclecting the optimal process stream matches and heat exchanger network satisfying the minimum utility cost criterion.
2.
PROBLEM
STATEMENT
The problem of determining simultaneously the best combination of process stream matches and network configuration can be stated as follows: Given are: (a) a set of hot process streams and hot itff. their inlet and outlet temperatures T’, utilities To,‘, and heat capacity flowrates F’; (b) a set of cold process streams and cold utilities jK. their inlet and outlet temperatures Ti, TOJ, and heat capacity fiowrates Fi; (c) a minimum temperature approach .4Tmi,; (d) the overall heat transfer coefficients U, of each potential match C(j); (e) the minimum utility consumption and the location ofpinch points; and (f) a maximum number of units N,,,. The objective is to obtain simultaneously the set of process stream matches, their heat loads and the minimum cost heat exchanger network configuration. It will be shown in Section 4 that this general problem can be formulated as an MINLP problem. If the matches and heat loads are given (e.g. via the minimum number of matches calculation), then this general problem is reduced to a nonconce.~, nonlinear (NLP) optimization problem. This paper is composed of two sections. Section 3 proposes a new decomposition approach that can be applied to the noncon~.~ nonlinear (NLP) network optimization problem so as to induce special structure and result in a series of convex subproblems that can lead to the globally optimal network configuration. Section 4 presents a decomposition approach that can be applied to the problem of simultaneously optimizing the combinations of matches and the network configuration. It is shown that this simultaneous approach can significantly reduce the number of network evaluations required to obtain the optimal combination of matches and the optimal network configuration.
3. GLOBAL
OPTIMUM
SEARCH
IN
NETWORK
OPTIMIZATIOX
In this section,
a decomposition
approach
is pro
posed that can be applied to the global optimum search for the network optimization problem. A review sented.
of the network optimization accompanied by an analysis
problem is preof the mathemat
and
A.
R.
CIRIC
ical properties of its formulation. The global optimum search approach to the network optimization problem is presented and demonstrated with three examples.
Floudas rl crl. (1986) proposed an approach for the automatic generation of heat exchanger networks that feature minimum investment cost subject to the criteria of minimum utility cost and fewest number of units. Central to this approach is a superstructure that has all possible network configurations embedded within it. This superstructure can be used to formulate an NLP problem. The solution of this NLP is a heat exchanger network with minimum investment cost. A detailed explanation of the conventional network optimization is given in Floudas et (I/. (1986). Prior to presenting the superstructure and its mathematical formulation. it is convenient ti) define the following sets: let ~EHCT be the mdcxcd set of all process streams and utilities. defined as: EICT = (H)u(C). Let the indexed
sets k’~/?~ and ~“ES~,~ be defined
RI = [h ‘I if keH,
k E C‘. k’elf
then
k’~(’ and
and (k ‘, k)~.bfA
by:
(k, k’)eMA
j.
S,,, = :k”lk’ERI.k”~R,.k’#k”j. 3. I. I. Netnork .superstruc twe. A typical superstructure proposed by Floudas Pt d. (I 9X6) is shown in Fig. I. This superstructure is for hot stream k. It contains two matches. with hot stream k exchanging heat uith cold sircams k’ and k”. In this superstructurc. splitters arc located at the inlet of the stream to the network and at the exits of each exchanger. Mixers are located at the inlets of each exchanger, and at the exit of the beat exchanger network. Postulated streams fiow from (a) the splitter at the stream origin to the mixers at the inlet of each exchanger; (b) from the mixer at the rnlet of an exchanger, through that exchanger. and to the splitter following it: (c) from the splitter following an exchanger to a mixer at the inlet of another exchanger; and (d) from the splitter follouing an exchanger to the heat exchanger network exit. This combination of splitters. mixers and streams contatns all possible flow alternatlves within it. For example. if all the bypass streams running from the splitters at Ihe outlets of exchangers, to the mixers preceding other exchangers are set to zero. the resulting network has a parallel structure. In contrast, a series structure is obtained when each splitter has only one outlet stream. All other structures between these two cxtrcmcs can also be derived from this superstructure. Similar superstructures are postulated for each hot and cold process streams in (he heat
Strategies for overcoming uncertainties
Fig.
1.
Stream superstructure.
exchanger network. The resulting network superstructure contains all possible network configurations. The heat capacity flowrates of the streams in the superstructure are labelled with the following conventions: .f$F =
Stream
1135
in superstructure
of process stream at the beginning of the network to the mixer preceding the match between k and k’. f9” = Stream in superstructure of process stream k flowing through the exchanger used in the match between k and k’. f$= Stream in superstructure of process stream k flowing from the splitter following match between k and k” to the mixer preceding the exchanger used in the match between k and k’. f 9” = Stream in superstructure of process stream k flowing from the splitter following the exchanger used in the match between k and k’ to the mixer at the end of the network. k flowing from the splitter
Also included in the superstructure are temperature variables, represented by f. Each temperature variable has two superscripts: the first denoting the position with respect to an exchanger (i.e. I = inlet temperature, 0 = outlet temperature) and the second denoting the position of the stream superstructure in the indexed set HCT. Temperature variables also have one subscript, denoting the stream paired with k in the relevant match. For example t$! denotes the temperature of stream k at the inlet to the match between k and k’. In addition to the variables representing the flowrates and temperatures, the superstructure contains several parameters. The heat load of match (ij) is represented by the parameter Qij. The inlet temperature of a process stream or utility is denoted as Tk, and its total flowrate is denoted as Fk. Note that the constant parameters are written in boldface; this
convention will also be used throughout the paper to designate variables that are held constant. 3. f.2. Nonlinear programming formulation. These variables and parameters of the superstructure are used to write the constraints of the NLP problem that minimizes the total investment cost. The constraints include mass balances at the splitting and mixing points, energy balances at the mixing points and the exchangers, and equations to calculate the temperature approaches and the log mean temperature differences. Constraints are also added to maintain the temperature approaches above the minimum temperature approach. 3.2. Objective
function
The objective function minimizes the total investment cost that equals the sum of the investment costs associated with each exchanger, and has the following form:
OBJ = midA
&&Di)R.
(‘1
The quantity inside the brackets equals the heat transfer area of match (ij) and the parameters t(, /I are cost parameters. The function LMTD,, is the log mean temperature difference of match (ii), evaluated with the standard formula: ,T+,fTD
v
=_DT1,DT2, DT1,’
In or with Paterson’s LMTD,=
DT2,
(1984) approximation:
213 x (DTl,,
x DT2,i)“z x [(DTl,
3.3.
Mass
and energy
+ l/3 + DT2,,)/2].
balances
1. Mass balances
at the splitters of the superstructure: &, f $$ = FL
located at the inlet
kcHCT.
(2)
C. A.
1136 2. Mass
balances exchangers:
at mixing points
3. Mass balances
at splitting
FLOUDAS
at the inlets of
points at the outlets of
exchangers: f$” +
c
=0
f&$f::”
k’ER*,
k”E&* k E HCT. 4. Energy
balances exchangers:
(4)
at mixing points at the inlets of
Tkffik + 2 f&t,";;" .fF;;*t$k = 0 k"PSIr k’E RL, 5. Energy
3.4.
balances
k E HCT.
(5)
in heat exchangers:
Qil .f,k,‘(z;,’  tp,‘) = 0
(ij)eMA,
(6)
Qij f:J(tPJ
(ij)~.44A.
(7)
Temperature
1. Temperature
2. Minimum
 rf,‘) = 0
[email protected]
constraints
approaches:
DTl,,
= t;’  tp’
(ij)eMA,
(8)
DT2,
= ty’  t:’
(@MA.
(9)
temperature
approach:
DTl,,
> AT,,,;
(
DT2,
> AT,,,,,;
([email protected]
Equations (I11) form a nonconvex, lem, denoted as (P). 3.5. Nature
(10) (11) NLP prob
of NLP formulation
To solve problem
(P), Floudas et al. (1986) used conventional nonlinear optimization methods and solvers (MINOS 5, Murtagh and Saunders, 1986). A weak point of this approach was that the quality of the solution depended upon the starting point of the optimization: a starting point close to the global optimum gave a good solution, whereas a starting point far from the optimum may converge to a local optimum with a higher cost. If the starting point were particularly bad, the conventional NLP solvers could fail to provide a feasible solution. These difficulties in determining the solution of the NLP arise from the nature of the energy balances in the formulation: these energy balances form constraints that are bilinear with different signs in the flowrates and temperatures, and are thus nonconvex. A nonconvex constraint renders the entire problem nonconvex, meaning that there may be more than one local optimum. Consequently, the solution of the NLP for the grassroots network optimization using conventional solution techniques cannot be guaranteed, and if a solution is reached, it can only be considered as a local optimum.
and A. R. CIRIC 3.6.
Global
optinwm
search
The decomposition approach was developed to solve mathematical optimization problems with no special structure (Floudas ef a/., 1989). The key idea of this approach is to induce special structure in subproblems that arise from a decomposition of the original problem, by an appropriate projection of the variable set and an appropriate partitioning of the constraint set. This approach can be used to decompose nonlinear and nonconveY problems (such a& the network optimization problem) into a series of convex subproblems, whose solution can lead to the overall globally optimal cost network. Nonconvex. NLP problems of this class can be solved using a general threestep method: Step lIdentify the sources of nonconvexities that arise in the particular subproblem. These sources can be identified as the nonconvcx terms of the constraints and objective function. Examples are sums of bilinearterms Gth different signs. ‘These nonconvex terms can appear in an objective function, equality constraints or in inequality constraints. step 2 Partition rhe set of continuous variables into two subsets: (a,! complicating, and (I?) noncomplicating. so that when tither subset oflariables is held fixed, the remainmg problem is easier to soive and its solution is its global solution. Such a partition also decomposes the ccmstramts inro two sets: the first set of constraint> contams equations in\tal!ing only comphcatmg variahles. while the second set contains constraints that may irlvohe both complicating and noncomplicntlng variables. From this partition of the variable and constraint sets. two subproblems are formed. The first subproblem. termed the primal subproblem. consisis of the objective function and the second set of consrraints. containing both complicating and noncomplicating variables. In this subproblem, the complicating variables arc held fixed at values gencratrd either by the second subproblem or by nn initial point. The solution of tbis problem provides an upper bound on the original objective function. The second subproblem, denoted as the master problem. consists of the first set of constraints contaming only complicating variables, and a set of inequalities involving the Lagrangian funcrion. derived from previous solutions of the primal problem. In this subproblem, the complicating variables are calculated. Its solution provides a lower bound on the original objective
.
Strategies for overcoming uncertainties Step &Apply the Generalized Benders Decomposition algorithm (Appendix A) to generate two subproblems and solve these subproblems iteratively. The Generalized Benders Decomposition algorithm consists of selecting a set of values for the complicating variables, solving the primal subproblem, generating the Lagrangian function based on the results of the primal subproblem, and solving the master subproblem to generate new values of the complicating variables. The algorithm terminates when the objective value of the master problem, a lower bound on the final objective cost, meets or exceeds the lowest value of the objective function generated during the solution of primal problems.
remaining constraints that define the temperature approaches and heat transfer area do not add any degrees of freedom. Consequently, if flowrates are fixed and the resulting problem is feasible, there is only one feasible solution to the optimization problem. This is an incentive for selecting the flowrate variables to be the complicating variables in the proposed approach. 3.6.3. Formulation of the primal problem. When the flowrate variables are chosen to be the set of complicating variables, the following primal problem (Pl) is obtained:
subject to T’Y:+ +
Judicious selection of the set of complicating variables can have a profound effect on the solution of a mathematical formulation. In this paper, it will be shown how the appropriate selection of complicating variables changes the character of the heat exchanger network optimization problem from one nonconvex problem to a number of convex subproblems that provide upper and lower bounds and can lead to the globally optimal solution. 3.6.1. Sources of nonconvexities. Problem (P) is a nonconvex NLP problem. Although the objective function (1), for fixed heal loads, is a convex function (see Appendix B), the energy balances, equations (4) and (5), contain sums of terms that are bilinear in the flowrate and temperature variables, and have different signs. Thus, the source of the nonconvexities in problem (P) are the bilinear terms of the form (ft) in the energy balances. 3.6.2. Selection of complicating variables. The appropriate selection of the set of complicating variables for the problem of network optimization becomes apparent when one considers the form of the energy balances in the NLP formulation. These equations are bilinear in the temperature and flowrate variables. Thus, if the flowrate variables are set to fixed values, the energy balances become linear in the temperature variables. Conversely, if the temperature variables are set to fixed values, the energy balances become linear in the flowrate variables. Furthermore, if one set of variables is fixed, the resulting set of linear equations forms a square system, and consequently the other set of variables is also fixed. In other words, fixing the flowrates fixes the temperatures and vice versa. This property of the superstructure equations can be used advantageously when applying the proposed global optimum search approach. By selecting the flowrates as complicating variables, the energy balance constraints in the primal subproblem become a square system of linear equations in terms of the temperature variables. Effectively, this fixes all degrees of freedom in the primal problem, as the
1137
z f$,t$V !dq,,,
 fE+ti+= 0 k’ER,,
Q,,  fi”(ty  ty) = 0
(ij)eMA,
Qu  fF(tpj
(i&VA,
ty) = 0
DT] (r= t!.’ I  t?j 1
(ij)EMA,
DT2, = t?,‘ /  t!j J
(ij)&fA,
DT1,>
AT,,,i”;
DT2, > AT,,,,,;
kEHCT,
(ij)eMA, (ij)44A.
(W
Note that in this’ formulation, the flowrate variables are fixed. As a result, the primal problem (Pl) satisfies three important properties: Property
Property
Property
IFor a given set of matches and for fixed heat loads, the objective function is convex. 2The equality constraints form a square system of linear equations, and if the problem is feasible, there is only one feasible solution to the optimization problem. >The solution of the NLP problem (Pl) is the global optimum of (Pl).
These properties are proven in Appendix B. Note that the primal problem may be infeasible for one of two reasons: (a) the linear equality constraints form a singular system (due to a separated flow system), and hence have no solution; and (b) the temperature approach constraints are violated. The possibility of singular equality constraints can be eliminated by adding bounds to the master subproblem (see Appendix D), while the case of violated temperature approach constraints can be treated as an infeasible primal subproblem (see Appendix C). 3.6.4. Formulation of the relaxed master problem. The relaxed master problem for the network optimization is given below: min
PBD
C. A. FLOUVAS and A. R. CIRIC
1138
estimating the temperatures of the cold bypass streams fM, as the minimum outlet temperature of match Qk,,k. Note that the righthandside of this equation is an upper bound on the enthalpy of,j’F.’ at the inlet of exchanger (k’k), as the estimated inlet is an upper temperature T“’  QVL/FL’  AT,,,,,, bound on rk! that is based upon the maximum outlet temperature of hot stream k’. Upper and lower bounds on the enthalpy of cold stream .f$’ at the outlet of exchanger (k’k) can be used to create another constraint:
< (T”  AT,,,,)I’:.”
n = 1
N.
(Dl)
Note that the parameters IpkgO, l!‘EX.‘+ and d~sX~.” are the Lagrange multipliers of thk energy balances from the 12th primal problem. Similarly, the parameters At. tk5” and tgc*” are the values of the corresponding heat transfer area and temperature variables from the solution to the nth primal problem. The master problem is a linear programming problem, and therefore, its solution is a global solution. It should be noted that both the primal subproblem and the master subproblem are convex. This implies that the Generalized Benders Decomposition applied to the network optimization problem consists of solving a series consisting of nonlinear hut convex primal subproblems and linear and convex master subproblems. Every subproblem has only one solution. From this it is inferred that the global optimum search scheme presented in this paper can lead to the globally optimal solution of the network generation problem. It should be noted, however, that there is no theoretical proof that this scheme determines the global optimum from all starting points.
3.7.
Improvement
qf lower bounds
The efficiency of the global optimum search approach can be improved by adding a set of constraints to the master problem. These constraints arc linearized energy balances, and act as bounds on the flowrate variables. An approximate energy balance for cold process stream k taken at the inlet of exchanger (k’k) provides the constraint process streams: these constraints are: Tkf;” +
1 (Tk + Q,,,, ‘Fk)f;:: ii ES,.A
< (T”  Qkl/Fk’ The lefthandside on the enthalpy exchanger fk’k).
 AT“I,”)f:”
keC;
k’ER,
of this equation is a lower bound of cold stream f:.” at the inlet of This lower bound is obtained by
k&i
O’ER,.
The lefthandside of this equation is a lower bound on the enthalpy of cold stream.f‘:.” at the Inlet of exchanger (k’kj. This lower bound is obtained by estimating the temperatures of the cold bypass streams _f’Ff as the minimum outlet temperature of match Q,,. Note that the righthandside of this equatton is an upper bound on the enthalpy of,/‘:,” at the inlet of exchanger (h‘h) as the estimated Inlet temperature T”  QVk ;F’  A T,,, is an upper bound on ti” that is based upon the maximum outlet temperature of her stream X ‘. Similar constraints may be written for hot process streams:
> (T” + Qkk’ ,F” + A7;,,,,, ,i‘: ’ XEH; Tyf :” +
c
k'ER, ,
(T’  Qtk !Fk)f’,B$  Qkk
cFSki 3 (TV + A T,,, ,,f’:”
k EN
; k ‘ER,
The first equatton is generaied from bounds on the enthalpy of hot streaml:,” at the inlet of exchanger (kk’), while the second equation is based upon bounds of the cnthalpy of hot ctream,f’:’ at the outlet of exchanger (kk’). These constraints have two main effects upon the global optrmization approach: (a) they reduce the number of infeasible flow patterns generated by the master subproblem: and (b) they improve the lower bound provided by the master subproblem. The net effect is to reduce the total number of iterations required by the global optimization approach. 3. 7.1. Global uprimur~l search nigorirhm. The aigorithm that searches for the globally optimum heat exchanger network configuration from the nonlinear, nonconvcs formulation can be summarized as follows: I. Select a set of initial values of the stream flowrates that corresponds to a feasible network structure. Such a structure can be generated by solving a NLP problem that minimizes the infeasibil
Strategies
for overcoming
ity of the network structure. This set comprises the initial point in the optimization. Set the counter Nfeas and the counter Ninceasto zero. 2. Solve the primal problem. If the primal problem is feasible, store the temperature variables and Lagrange multiplier values, and update the value of the upper bound and the counter Nreas. Otherwise, increment the counter Ninfeaa, solve the infeasibility subproblem (Appendix C) and store the temperature variables and Lagrange multiplier values appropriately. 3. Solve the master problem. If the master problem is infeasible or the objective function of the master problem equals or exceeds the upper bound set by the primal problem, stop. Otherwise, store the values of the flowrate variables and return to Step 2. This algorithm was performed automatically using the APROS techniques (Paules and Floudas, 1989) and the following three examples were solved on a VAX II/GPX workstation computer. Example
I
In this example, the global optimal network is sought for a system of two hot streams and one cold stream. The data for these streams are given in Table I, and information about the matches and cost data appears in Table 2. The network chosen as a starting point is shown in Fig. 2. This network has a parallel configuration, and features a net total cost of $81,439. The optimal solution was obtained in six iterations of the global optimum search, consuming a total 39.7 CPU s and averaging 6.6 s per iteration. The optimal network is shown in Fig. 3. This network features a total cost $57,371. It should be noted that the final network has a series configuration, in comparison to the starting network that features a parallel configuration.
1139
uncertainties Table 1. Stream data for Example I T out (K) FC, (kW K‘) T in (K) 500 250 4 350 200 4 150 310 IO
stream Hl HZ Cl AT,“,,=lOK.
Table 2. Match data for Example 1 II (kWm‘K‘)
Q NW)
Match HI Cl H2 Cl
1000 600
Cost of heat exchangers = $13OOA”6.
Example
2
This example involves deriving a network structure for a system of three hot streams and one cold stream. The data for these streams are given in Table 3. There is no utility requirement for this system, and the minimum number of required matches is three. The data for these matches is given in Table 4. This problem was originally solved using MAGNETS, a software package embodying the grassroots design technique of Floudas et al. (1986). MAGNETS used the NLP solver MINOS to derive
H
HZ
1
I 500
K
I 350 K
K
1 200
Cl
1
250
Fig. 3. Optimal
network
in Example
490 K
2.942 A
10.0 150
*
K
1
200 K
Fig. 2. Starting
A (m*) 228.8 124.1
0.05 0.05
point in Example
1.
K
1
1140
C. A. FLCILUAS and A. R. CIRIC
This problem was also solved using the global optimum search technique described in this paper, with the MAGNETS solution taken as the starting point. Two iterations were required to derive the optimal network configuration. The optimal network found by the global optimum search procedure is shown in Fig. 5. Note that the netwotk has a series configuration. indicating that the topology trap was overcome. The optimal network features a total investment cost of $46.266, which is 12% less that the solution provided by MINOS.
Table 3. Stream data for Example 2 Stream
T in (K)
HI HZ H3 Cl
210 210 210 100
Tout
(K)
PC,> (kW Km’) 25 10 50 45
130 160 180 200
Table 4. Mstch data for Example 2 Match
Q (kV
Hl Cl HZ Cl H3 Cl
2000 IO00 1500
U(kWm
‘K
‘1
A cm’) 57.9 36.9 64.7
0.5 IO 2.0
Cost of heat exchangers  %1300A””
Example
a minimum investment cost heat exchanger network. This network is shown in Fig. 4. Note that this network has a parallel configuration, and is an example of a “topology trap”, as any small change in the structure would result in a violation of the minimum temperature approach. The cost associated with this network is $52.542.
The third example is the test problem 7SP4, taken from Papoulias and Grossmann (1983). In this problem, a heat exchanger network must be designed for a system of six hot streams, one cold stream, one hot utility and one cold utility. The stream and match data for this problem are given in Tables 5 and 6. The minimum temperature difference is 2O’F. and there is a pinch point between 410 and 430 F.
3
H
Fig. 4. Solution
Fig
obtained
S. Optimal
by MAGNETS
network
1
in Example
in Example
2.
2.
Strategies for overcoming uncertainties Table 5. Stream data for Example stream
T in (‘F)
T out (“F)
HI H2 H3 H4 H5 H6 Cl CW FIleI
675 590 540 430 400 300 60 80 800
I50 450 115 345 100 230 710 140 801
Table 6. Match data for Exam&
3
FC, (B.t.u.’
sY’)
Match
15 II 4.5 60 12 125 47 11.03 8390
OPTIMIZATION
This section considers the problem of optimizing the matches and the network configuration simultaneously. The proposed approach is based upon a
3675 1540 495 8390
0.1 0.07 0.06
HI HI H3 H4 H5 H6
1182.5 3017.5 1417.5 5100.0 3600.0 8750.0
0.1 0.08 0.06 0.07 0.04 0.055
Cl CW Cl Cl CW Cl
682.6 390.9 211.8
4.1. Generalized
matchnetwork
hyperstructure
The optimization technique presented in this paper is based upon a hyperstructure at the level of matches that contains all possible matches and all possible network configurations. This hyperstructure is constructed from smaller superstructures associated with each process stream, containing all possible matches and network flow configurations embedded within it. These stream superstructures contain the same elements as the superstructure proposed by Floudas
H5
network
114.5 295.4 290.4 2414.5 962.8 2034.9
matchnetwork hyperstructure that contains all possible matches and network configurations embedded within it. The hyperstructure can be used to formulate a matchnetwork optimization problem as an MINLP model that can be solved with a proposed decomposition approach.
430F
6. Optimal
A (ft*)
Cost of heat exchangers = 324A”.
Cl
Fig.
3
U (kB.t.u. h’ ft+‘F‘)
Below pinch
The optimum network configuration was derived for subnetworks above and below the pinch. Twentyseven iterations of the global optimization technique were performed to obtain the optimum network above the pinch, requiring 247 CPU s. Each iteration consumed an average of 9.2 CPU s. Nine iterations were required to find the optimum network below the pinch. 132 CPU s were consumed in this calculation, with an average 14.6 CPU s consumed per iteration. The optimal network is shown in Fig. 6. This network involves an investment cost of $147,072. It should be noted that this network is the optimum for the set of matches obtained by Papoulias and Grossmann (1983) that satisfy the minimum utility and minimum units constraints. MATCHNETWORK
Q (kB.t.u. h‘)
Above pinch HI Cl HZ Cl H3 Cl Fuel Cl
AT,,, = 20°F.
4. SIMULTANEOUS
1141
in Example
3
C. A. FLOUDAS and A. R. CIHIC
I142
*
i Ii 2 SUPERSTRUCTURE
t
t,
t
d?_
)
Hl
Fig. 7. An example
et al. (1986). The primary difference is that the superstructure of Floudas et al. (1986) involved only matches that had been previously selected with a method based upon the minimum number of matches criterion. Here, all potential matches must appear in the hyperstructure, as the matches and heat loads will be determined simultaneously with the network structure. An example of a hyperstructure for a system of two hot streams and three cold streams is shown in Fig. 7. Note that all six potential matches and network configurations are included in this hyperstructure, as opposed to a subset of matches that were previously selected by satisfying the fewest number of matches calculation. 4.2.
Mathematical
The
mathematical
formulation formulation
of
the
general
of a hyperstructure
matchnetwork hyperstructure has two main components. The first component is a transshipment model (Papoulias and Grossmann. 1983) for selecting process stream matches and heat loads. The second component is the hypcrstructure model, which is a generalized matchnetwork optimization model that minimizes investment cost while determining the flou structure, stream temperatures and actual heat exchanger area of the optimal network. The transshipment model involves partitioning the temperature range into T temperature intervals using the inlet and outlet temperatures of the process streams and utilittes. In this model. heat flows in one of two ways: between hot stream i and cold stream j in temperature Interval ET. and from one interval to the interval below it via a hot stream residual heat flow. The first type of heat flow is represented by the variables qq,. while the second type of heat flow ts
Strategies for overcoming uncertainties represented by the variables R, . Within each interval, hot stream i must release Qz and cold stream j must release QE heat. The hyperstructure model contains constraints generated from the generalized matchnetwork hyperstructure. These constraints include mass balances at the splitting and mixing points, energy balances over the mixing points and heat exchangers, equations calculating the total required area, and minimum temperature approach constraints that reflect thermodynamic feasibility criteria. When combined, these two components form the following generalized matchnetwork optimization problem.
4.3. Matchnetwork
optimization problem min C C aA$Y,,,
(MN)
bN jSC
subject to
c qu,R, 
R,.,_ , = QF
icH
jeR,,
(12)
Id,
(13) (14) (1% (16) ; f;+ = FL keHCT, fi! + cf$ v
f3”
=0
/>“+;f&f$“=O
(17)
k’, keHCT,
(18)
k’,keHCT,
(19) (20)
Q,, fy(t$’ Q.
f”“(t%
DTl
 ty’) = 0 /

t1.j) = t
0
&H
J’EC,
(21)
isH
jsC,
(22)
= t”‘ tpJ
icH
jeC,
(23)
DT2; = ,a.;  t;”
ieH
jsC,
(24)
LMTD,
=
DT1,  DT2, ,nDn, DT2,,
Q,
’
jd,
f:,’  F’Yg < 0
iEH
jd,
fyj
ieH
j&,
FiY, GO
DT2,>
> AT,,,;
icH
jeC,
(31)
AT,,,,,,; icH
jd’.
(32)
The objective function minimizes the investment cost, which is based upon the actual heat exchanger area of each match A,. The parameter a scales the investment cost, and p is an exponent that reflects economies of scale. Constraints (12 16) compose the transshipment model. Equations (12) and (13) are energy balances over each interval. Equation (14) calculates the total heat load of each match. Equation (15) connects the heat load to the integer variable representing each match. (Note that U is a large fixed constant.) Equation (16) is an integer cut restricting the solution to an upper bound on the total number of units. Equations (1719) are mass balances taken at the splitting and mixing points of the generalized matchnetwork hyperstructure. Equations (2022) represent energy balances over the mixing points and heat exchanger units. Individual temperature differences are calculated in (2324), and the logmean temperature difference and total heat transfer area in (2526). It should be noted that the logmean temperature difference can also be calculated with Paterson’s (1984) approximation. The minimum temperature approach between a hot stream and a cold stream exchanging heat, AT,,, , is specified in (3132). It should be noted that AT,,, may equal the HRAT (Colbert, 1982), which is the temperature approach used to calculate the utility consumption, or AT,,,,,, can be relaxed by specifying it independently to be some value less than the HRAT. Constraints (2728) connect the existence of a match (z’j), designated as the integer variable Y,, with the flowrate of the streams flowing through the (ii) exchanger. Similarly, equations (2930) place lower bounds on the flowrates through the exchanger. (Note that ATij,m.xequals the largest possible temperature drop through the exchanger, equal to To,’ TJ  AT,,.) If the match is selected, then the Rowrate of the hot and cold streams through the exchanger are bounded from above by the total flowrate and below by a minimum flowrate that is based upon the heat load. If, however, the match is not selected, then the flowrates through the exchanger, fy and f yJ are set to zero. Note that iffy and fp are zero, then all other streams associated with match (ij), namely, fy, ffi, fp’, f f”,f Fi and f pJ are zero, as a result of equations (1920). 4.4. Nature of mathematical formulation
ieH
A,= U, LMTD,
(25)
DTl,
1143
fy
 Q,/ATij,,,,.x a 0
isH
f?
 QlilATi,.m~x z 0
icH
(26) (27) (28) jeC,
(29)
jd,
(30)
Problem (MN) is an MINLP problem, as it contains both integer and continuous variables, a nonlinear objective function and nonlinear constraints. The integer variables Y, appear in the linear constraints (15), (16), (27) and (28). In addition, the integer variables appear in the nonlinear objective function, in terms of the form crA{ Y,,. It should be noted that
1144
C.
A. FL~UDASand A. R. Cnuc
if A, were held constant, the terms aA{ Y, would be linear functions of Yy. The nonlinear objective function and equality constraints also complicate problem (MN). The objective function consists of a sum of concave terms in the area of the heat exchanger. The energy balances (2022) are bilinear in the flowrate and temperature variables, and are thus nonconvex. Equation (25), defining the logmean temperature approach, is a nonlinear equality constraint. Lastly, equation (26), which calculates the heat transfer area, is in general nonconvex, but becomes convex when Q,, is held constant (see Appendix B). It should also bc noted that the heat loads Q,, appear in linear constraints and linearly in nonlinear terms of nonlinear constrain&. Equivalent formulations can bc obtained by substituting equation (25) into equation (26), and/or equation (26) into the objective function. While the first substitution has no consequence other than reducing the constraint set, the substitution of (26) into the objective function provides a formulation in which the Q,, no longer appear linearly. As it will be shown, keeping the formulation linear in Q, allows for a particular type of decomposition in the global optimization approach. 4.5. Nerr decomposition
upproach
The goals of applying a decomposition approach to problem (MN) are twofold: (a) to decompose (MN) in a manner that isolates the integer variables in a pure integer programming (IP) or a mixed integer linear programming (MILP) problem; and (b) to induce special structure in the subproblems thdl can lead to an efficient solution technique. These goals can be accomplished through a careful partitioning of the variable and constraint sets. 4.5. I. Selertion oj’complic~ating ruriables and purtitioning of’constraints. The most obvious choice of complicating variables are the integer variables Y,,. This is the standard choice for solving MINLP problems with either the Generalized Benders Decomposition (Geoffrion, 1972) or the Outer Approximation method (Duran and Grossmann, 1986). Selecting the integer variables as complicating variables partitions the constraint set into two subsets. The first subset, containing the constraints that involve only integer variables, consists of equation (16). The second subset consists of the remaining equations. This partitioning results in a nonconvex, nonlinear prlmal subproblem and a pure integer master subproblem (IP). While this selection provides two solvable subproblems. it has the deficiency that many integer combinations satisfying the minimum units constraint do not satisfy the minimum utility constraint or the transshipment model. Thus. many solutions to the master problem may exist that give infeasible primal subproblems. This can result in an excessively large number of iterations.
This difficulty can be overcome by selecting both the integer variables Y,, and the transshipment variables qr,!, R,, and Q, as complicating variables. In this case, the transshipment model constraints [equations (12 16)]. involve only complicating variables, and thus comprise the first subset of constraints. The network optimization model [equations (17 33)] contains both complicating and noncomplicating variables, and thus comprises the second subset of constraints. The resulting decomposition of problem (MN) will feature an MIL,P master problem containing all the constraints of the transshipment model. and a primal subproblem containing all the constraints of the network optimization model. It is this set of complicating variables that is proposed in this paper. 4.5.2. Primal subproblem. The primal problem is a NLP problem containing the variables and equations associated with network optimization. It should be noted that in this formulation. the complicating variables Y,, and Q, are held constant. To emphasize this, they appear in boldface in the primal subproblem formulation. It is important to note that the unselected matches (those with Y,, = 0) do not affect the optimal solution of the primal subproblem. for three reasons: (a) the investment cost terms for the unselected matches drop out of the objective function: (b) the total heat transfer area of unselected matches is identically zero: (c) the flowrates through the exchangers of unselected matches (f,!’ and .fF’) are identically zero. In addition, the mass balances (18~~19) ensure that all other flowrates associated with these units will also be zero. Therefore, the unselected matches will not appear in or otherwise affect the final fiow pattern; Because the unselected matches do not affect the optimal solution, the constraints and variables associated with these matches can be dropped from the primal subproblem, greatly reducing the size of the primal subproblem. This reduced primal subproblem is the size of a network optimization problem. presented in Section 3 The resulting reduced primal subproblem below: OBJ = min c 2 crAeYii, i&V,tR, subject to
is written
(34) (PI’)
4; f;+ = FL kEHCT. h
(35)
Strategies for overcoming uncertainties Tkfi!
+
1
f$$
t$k f$k$
=
0
kllES~~~
k’ER,, Qi, f/E.i(t:.’  ty) = 0 Qs fy(ty
&H,
 tl”) = 0
DTl.. ‘/ = t!~’ I  tyj
ieH, ieH,
keHCT,
(38)
jcR,,
(39)
jeR,,
(40)
jERi,
(41)
DT2, = to.’ /  1f.j ieH, J’ER,, DT1,DT2, . ICH, jER,, LMTD, = ,,, DTl, DT2, DTl,
a AT,,,in; &H,
DT2,Y 2 ATmi,;
(42) (43)
jeRi,
SH,
jeR,,
A,=
Qij U, LMTD,
ff’
FrYit < 0
ieH;
jeR,,
(47)
f BJ FtY, < 0
iEH;
jcR,,
(48)
fFi  Qii/ATii_
iEH,
20
jER,,
(46)
ieH;
jeRi,
(49)
f 7  Qii/ATij,,,, > 0 ieH;
jER,.
(50)
It should be noted that the above primal formulation corresponds to the nonconvex nonlinear programming problem of Section 3. 4.5.3. Master subproblem. The master problem is constructed from all constraints in (MN) that involve only complicating variables and from the Lagrange functions of previous prima1 problems. The formulation for the master subproblem is present below: min p subject
to
+ af[ff,i,n  F’r;,] + )$.a[fiEj,a  Fj yi,] + ,~b+‘[,:,i,o + J.p[f

F
Q,
AT,,,]
 Q, AT,,,,]
+
xuH’x’“[Q,,
_
fi”“(ty
+
p=[Qii

pytoj.”

Q, U, LMTD,
ty)] [email protected])]
n= 1
N,
(51) (52)
2 q,,, = Q?
_iaC,
t = 1 . . . T,
(53) (54) (55) (56)
1145
Here, N is the total number of iterations, fy, fF”*“, , gx.“, tpJ*n and A; are the values of the ttie ’ toJan I objective function, flowrates and temperature variables, and heat transfer areas obtained in the solution of the nth primal problem, and 2.?, ky, )LpW, k$i’@, )$l”+, k? and kp are the Lagrange multipliers of the upper and lower bounds, energy balances and area calculation equations obtained in the solution of the nth primal problem. Note that the Lagrange function [equation (51)] is linear in both the Y, variable and the Q, variables. This is a consequence of the formulation of (P), as both Y, and Q, appear linearly in the nonlinear terms approach, of (P), when the log mean temperature LMTD, and the A,, are held constant. As a consequence of the linear Lagrange function, the master problem is an MILP. The solution of the master subproblem provides a set of matches and heat loads that satisfy the transshipment model of heat flow, the minimum utility consumption and the Lagrangian constraints (5 1). 4.5.4. Global optimum search of the primal subproblem. The reduced primal subproblem (Pl’) is a nonlinear optimization problem. Because of its nonconvex objective function and constraints, primal subproblem (Pl’) is nonconvex and may have more than one locally optima1 solutions. In Section 3 of this paper, a method was presented for the global optimum search of the network optimization problem. (Pl’) is essentially an equivalent formulation of the network optimization problem of Section 3. The global optimal solution to (Pl’) can be obtained by using the approach presented in Section 3 of this paper for the network optimization associated with the selected matches. Applying the global optimum search approach of Section 3 to the matchnetwork problem will not provide the optimal Lagrange multipliers for the constraints of (Pl’). To obtain these Lagrange multipliers, it is necessary to solve (Pl’) at the globally optimal point. Thus, an algorithm can be established for obtaining the optimal Lagrange multipliers of the globally optimal solution of (Pl’): Step aObtain an initial set of values for the flowrate variables of the selected matches. Step &Solve for the global optimization problem for the selected matches, using the values of the flowrates obtained in Step 1 as a
starting point. Step cUsing the values of the noncomplicating variables obtained in Step 2, solve the prima1 subproblem to obtain the optimal values of the Lagrange multipliers. 4.5.5. Algorithmic strategy. Application of the Generalized Benders Decomposition algorithm to problem (MN) results in the following procedure for determining simultaneously the optimal set of matches and the optimal network structure:
II&l
C.
A.
FLOUDAS
lIdentify an initial set of process stream matches Y, and heat loads Q, that are solutions to the transshipment model and satisfy the minimum utility consumption and minimum number of units criteria. One method of selecting a set of matches and heat loads is to solve the minimum number of units problem (Papoulias and Grossmann, 1983). Set an upper bound OBJ, = frx, and a lower bound OBJ =  m Set an iteration counter N = 1. Step 2Solve the infeasibility primal suhprohlem for the selected combination of matches and heat loads, then gcneratc and store the error Lagrangian. solution of (Pl’) is not Step 3a If the global (Pl’) as sought, solve the pnmal problem a nonconvex NLP, using the data generated by the inreasihility subproblem in Step 2 as a starting point. The solution of (Pl’) will provide locally optimal values of the noncomplicating variables and Lagrange multipliers. of (Pl’) is sought. Step 3b If the global solution apply the global optimum search procedure described in Section 3 of this paper to the network formed by the selected matches (i.e. matches with Y,, = 1). Using the globally optimal values of the noncomplicating variables as a starting point, solve (Pl’) to obtain the optimal Lagrange multipliers. Step &If the value of the objective function OBJ is less than the upper bound OBJ, . then set OBJ+ = OBJ. Step 4h From the solution of the primal problem, generate a Lagrange function and append it to the master problem. Solve the master problem, obtain a new set of matches Y, and heat loads Q,, and update the lower bound. meet or Step 5 If the upper and lower hounds cross (OBJ, d OBJ ). STOP. Otherwise, return to Step 2, using the values of Y,, and Q, obtained from the master problem.
and A. R. CIRIC
Step
The above algorithmic strategy with applied to solve the following examples. Excrmple
Step
3a was
Table
7
Stream
data for Example
4
FC,, (km’ K
stream
T in (K)
T OUT (K)
HI H2 H3 Cl
500 450 400 300
350 350 120 480
IO 12 N ‘1
C2 c3 CW
340 340
420 400 120
IO H ‘2
AT,,,
Inn
L IO K,
’K
U = 0.8 kW m
six. This problem has the unusual feature that although there arc only 12 potential matches, there are 25 combinations of matches that are solutions to the transshipment model and satisfy the minimum numher of units criterion. A hyperstructure containing the 12 potential matches and every possible network configuration of these 12 matches was formulated. Using this hyperstructure. the mathematical model was generated for this problem. The optimization algorithm was performed on a VAX IIiGPX workstation computer. utilizing APROS techniques (Paules and Floudas. 1989) and GAMS (Kendrick and Meeraus. 1987) as a modelling environment. The solution was obtained m 13 iterations, consuming a total of 2350 CPU s. and avcraging 180s per Iteration. An
exhaustive
search
over
all
combinations
was
MILPNLP iterative procedure. A combination of matches satisfying the minimum units criterion was identified in the MILP step using Papoulias and Grossmann’s (1983) minimum number of units model. uith integer cuts added to eliminate previous solutions. An optimal network configuration for these matches was obtained in the NLP step using the hyperstructure model of Floudas er ul. (1986). The procedure was repeated for 15 iterations, when no more feasible sixunit combinations could be obtained. This exhaustive search required a total of 4507 CPU s. averaging 180.3 CPU s per iteration. Thus, in this example the optimization algorithm required the same amount of CPU time per iteration, but required half the iterations and CPU than the exhaustive enumeration. Both the optimization search and the exhaustive enumeration selected the same match combination and network configuration as optimal This solution features a total investment cost ot’$49,352. The match data for this solution is given in Table 8, and the optimal network is shown in Fig. 8. also
performed
using
a
4
This example involves obtaining the matches, heat loads and network configuration for a system of three hot streams, three cold streams and one cold utility. Stream data is given in Table 7. The minimum tempcraturc approach (AT,,,,,) is IO’C. For this value of the temperature approach, there is no pinch point and only a cold utility is required. The minimum number of units for this problem is
‘) 
Table
8. Match
Mntch HI HZ H? H2
data fur txample Q(kW,
Cl Cl C2 C3
1500 120 x00 280
H3 C3 H3 CW
200 440
Cost of heat cxchangcn
= Sl300~“’
4 A(&) 68 2 3.1 41 8 I.2 3 I .I 15.9
I
Strategies
for overcoming uncertainties
1147
300K c2
C3 Fig.
Example
8. Optimal
network
5
The second example is the test problem lOSP1 (Pho and Lapidus, 1973). This example involves five hot streams, five cold streams and one cold utility. The data for this example are given in Table 9. For a minimum temperature approach of 20”F, there is no pinch point and 6497.8 kB.t.u. h’ of coolant are required. The minimum number of units for this problem is 10. The lOSP1 hyperstructure involves 30 potential matches, and no matches can be selected or eliminated a priori. Thus, there are 30 integer variables in the mathematical formulation. There are at least 843 combinations (Linnhoff et al., 1979) of these integer variables that satisfy the targetting criteria of minimum utility consumption and fewest number of units. A consequence of the large combinatorial nature of this problem is that the master problem consumes most of the computational time. Seventeen iterations of the optimization technique were performed on this problem. The lowest cost network obtained during these iterations is shown in
Cl in Example
4.
Fig. 9, and the match data is given in Table 10. This network requires a total investment cost of $93,253, which is $898 less than the best solution reported in the literature (Linnhoff and Flower, 1978b).
5. CONCLUSIONS
In this paper, decomposition approaches were presented that can be used to overcome two sources of uncertainty in heat exchanger network design: (a) the uncertainty associated with nonconvexities in the network optimization task; and (b) the uncertainty that can arise when there are several combinations of matches satisfying the targetting criteria. In both cases, the basic idea involved selecting a set of complicating variables that induces special structures in the decomposed subproblems. This selection leads to a partitioning of the original problem into two subproblems, referred to as the primal and master subproblems. The solution is obtained by iterating between these subproblems. Section 3 of this paper showed how this approach
Table 9. Stream data for Example 5
stream HI H2 H3 H4 H5 Cl c2 c3 c4 c5 cw AT,,,=20”F,
Tin
CF)
320 480 440 520 390 I40 240 IO0 180 200 100
T out (“F) 200 280 I50 300 150 320 431 430 350 400 180
U=0.15kB.t.u.h‘ft‘“Fl.
Table IO. Match data for Example 5
FC, (kB.t.u. h ’ “Fl) 16.67 20.00 28.00 23.80 33.60 14.45 II.53
16.00
32.76 400
Match
Q (kB.t.u. h‘)
A (ft*)
HI C3 HZ Cl H2 C5 H3 C4 H3 CW H4 C2 H4 C3 HS C2 H5 C5 H5 CW
2000.4 2601 I399 5569.2 2550.8 1956.4 3219.6 245.8 3871 3947
136.8 153.2 130.9 497.6 307.2 188.9 291.1 12.1 516.5 392.8
Cost of heat exchangers = S350A06.
1148
C.
A.
FLOUDAS
and
A. R. CIRIC
225F H 2
280F
480F
241F
44OF H3
319F 9.7
11
52Oi
300F I
&..I
287F
14.1
b
347F
150F
268F
H 5
b3OF
1”““’
Q.
t2OFc5
CZ
Fig.
9. Optimal
network
can be applied to the nonconvex, nonlinear network optimization problem. Selecting the flowrate variables as complicating variables partitions the original nonconvex problem into a linear master subproblem and a nonlinear but convex primal subproblem. Through several example problems it was demonstrated that there can be a significant difference between the global optimum and the local optimum found by conventional solution techniques. Section 4 of this paper proposed an approach for obtaining simultaneously the best combination of process stream matches and the network configuration of a minimum investment cost heat exchanger network. This procedure is based upon a generalized matchnetwork hyperstructure that contains all potential process stream matches and network configurations. Based upon this hyperstructure, an MINLP problem is formulated that encorporates all design options at the level of matches. This MINLP was decomposed by selecting as complicating variables the integer variables representing a match, the heat loads of each match and the variables in the transshipment model. This decomposition resulted in an NLP subproblem that derived a heat exchanger network for a single set of matches, and an MILP master problem that identified the sets of matches and their heat loads that satisfy the transshipment model and the minimum utility consumption criterion. It should be noted that the MILP master problem can be computationally inefficient for problems with
in
Example 5.
a large number of binary variables. For these problems, the master problem can be efficiently solved algorithm Relaxation Lagrangian using the (Geoffrion, 1974). This technique involves solving two master problems. In this first prohlem, the original master problem is solved with the binary variables treated as though they were continuous variables. The solution of this problem was used to formulate a second master problem containing all the constraints of the original master problem, except for the Lagrange functions. The objective function of this second master problem is generated by summing the products of the Lagrange functions with their multipliers found from the solution of the first problem. This second master problem is a mixed integer linear programming (MILP) problem that provides a new set of values for the complicating variables and also provides a lower hound on the final solution. The advantage of this technique is that it can significantly reduce the number of constraints in the MILP model. as well as exploit any existing structure. Ackno~ledgentenis~The authors would like to gratefully acknowledge financial support from the National Science Foundation under Grant DMC8617139 REFERENCES
Allgor R.. Automatic synthesis of optimal heat exchanger network configurations. Senior Thesis. Princeton Univcrsity (1988). Avriel M ., Nonlinear Programming: Analysis and Merhod~s PrenticeHall, Englewood Cliffs. New Jersey (1976).
Strategies
for overcoming
Colbert R. W., Industrial heat exchanger networks. I. Chem. E. Annual Research Meeting, Bath, U.K. (1982). Duran M. A. and I. E. Grossmann, An outer approximation for a class of mixed integer nonlinear programs. Math. Program. 36, 3077339 (1986). Floudas C. A., A. Aggatwal and A. R. Ciric, Global optimum search for nonconvex NLP and MINLP problems. Computers them. Engng 13, 11171132 (1989). Floudas C. A., A. R. Ciric and I. E. Grossmann, Automatic synthesis of optimum heat exchanger network configurations. AIChE Jl 32, 276290 (1986). Geoffrion A. M.. Generalized Benders decomposition. J. Optimization Theory Applic. IO, 234260 (1972). Geoffrion A. M., Lagrangian relaxation for integer programming. Mathematical Programming Study 2, pp. 82l 14. NorthHolland, Amsterdam (1974). Gundersen T. and L. Naess, The synthesis of cost optimal heat exchanger networks, an industrial review of the state of the art. Cornput. them. Engng If, 503 (1988). Kendrick D. and A. Meeraus, GAMS: an introduction. U.rer.r Manual for CAMS. Development and Research Dept of the World Bank (1987). Linnhoff B. and .I. R. Flower, Synthesis of heat exchanger networks: I. Systematic generation of energy optimal networks. AIChE JI 24, 633642 (1978a). Linnhoff B. and J. R. Flower, Synthesis of heat exchanger networks: Il. Evolutionary generation of networks with various criteria of optimality. AIChE Jl 24, 642654 (1978b). Linnhoff B., D. R. Mason and I. Wardles, Understanding heat exchanger networks. Comput. them. Engng 3, 295302 (1979). Marsten R., Users Manual for ZOOMIXMP. The Department of Management Information Systems, University of Arizona (1986). Murtagh B. A. and M. A. Saunders, MINOS 5.0 Users GuideAppendix A. MINOS 5.1, Technical Report SOL 8320, Systems Optimization Laboratory, Dept of Operations Research, Stanford University (1986). Paterson W. R., A replacement for the logarithmic mean. Chem Engng Sci. 39, 16351636 (1984). Papoulias S. A. and I. E. Grossmann, A structural optimization approach in process synthesisII. Heat recovery networks. Comput. them. Engng 7, 707731 (1983). Paules G. E. IV and C. A. Floudas, APROS: an algorithmic development methodology for discretecontinuous optimization problems. Opers Res. J. 37, 6 (1989). Pho T. K. and L. Lapidus, Topics in computeraided design: Part Il. Synthesis of optimal heat exchanger networks by tree searching algorithms. AIChE JI 19,1182l 189 (1973).
uncertainties
only complicating variables, while constraint sets g(x, y) and h(x, y) contain both complicating and noncomplicating variables. The Generalized Benders Decomposition algorithm (Geoffrion, 1972) consists of the following three steps: Step ISelect intial values y’ for the set of complicating variables y. Initialize the upper bound of (Cl), Z”, to infinity. Set the scalar counter K = 1. Step 2Solve the Kth primal problem (G2): Z(yK) = min C(x, yK), x subject to h(x,yK)=O, &7(TY9
Generalized
Benders
=s 0,
.xeX,
WI
where yK is the Kth set of selected values for the complicating variables y. The solution of (G2) gives Z(yr), xK, fl Kand AK, where qx are the Lagrange multipliers of the constraint set g(x, yK) c 0, and 1 K are the Lagrange multipliers of the constraint set h(x, yK) = 0. Update the current upper bound: Z” = min[Z”,
Z(yx)].
If Z” = Z(yx), then set y* = yK and x* =xK.
Step 3Formulate
the master
problem
(G3):
subject to /~>L(x*,y,q~,i.~)
k=I...K;
e(y) c 0, f(y)
= 0.
.wY, IIER’,
(G3)
where L(x’,y,
7’ ,A.*)= C(x*, y) + (tJYg(x’,
y) + (1“)‘h(x’,y).
Step &Solve problem (G3). The solution of (G3) gives Z: and y K+ ‘. If Zi equals Z”, then STOP. The solution is Z’, y*, x*. Otherwise, set K = K + 1 and return to Step 2. Infeasible
primal
If a particular (G2) infeasible, 1. Replace
APPENDIX
1149
subproblems
set of values yK renders the primal problem then:
(G2) with:
A Z(y”)
Decomposition
The Generalized Benders Decomposition (Geoffrion, 1972) solves problems of the form:
algorithm
= min a, 0(,1
subject to h(x, YK) = 0,
Z = min C(x, y). ‘.,
gb> Y”) G a. a 30,
subject to
XCX,
e(y) C 0, fcv) = 0, &,y)=O, g(.x,y)~o, xsX={x/x~R~,x~<~ ycY.
(64)
where c is a scalar quantity. The optimal solutiou of (m) will be a set of values (xK, qx,lK) that are minimaJJy infeasible solutions of (GZ). 2. Define a new Lagrangian
function:
L(x”, y, $, i,‘) = ($)rg(x’, (Cl)
Where the set of variables y has been chosen as the set of complicating variables, and x the set of noncomplicating variables. Note that constraint sets e(y) and/‘(y) contain
y) + (~‘Yh(x’,
3. GO to Step 3, and add the constraint: L’(xh, y, nk, I’) < 0, to problem
(G3).
Y).
1150
C. A FLOUDAS and A. R. CIRIC APPENDIX
B
Proof of Properties
l3
(813)
(a) Proof qf Properi~l I The first property of problem (Pl) states that the objective function of (Pl) is convex for a given set of matches with constant heat loads. The objective function is:
(Rl4)
(BIS)
In equation (Bl), w,, is the logmean temperature difference (LMTD), 0 is a constant less than one, and a, $, and U,, arc constants, Since the sum of convex functions is also a convex function, it wll be sufficient to prove that one term of (Bl) is convex. ‘Thus. it will be shown that:
f = M ‘1,
(B16)
(82)
is a convex function.
Properties of u convex multivariable function A multivariable function is convex if the eigenvalues of its Hessian matrix are positwe. This implies that the Hessian matrix is positive definite (Avriel. 1976). To prove that both eigenvalues are positive, it is sufficient to prove that the quantities C, and c‘: are positive. where C‘, = I, + i, = a,, $all, 1_ (‘I=/.,12=~I,,“ 12~a~2.
(B3) (B4)
In theses equations, the a,, are defined to be the secondorder derivatives of / with rcspcct to Y, and x,. The
Hessian elements
Let .x, and x2 be the temperature exchanger. The log mea” temperature function of .x, and I?:
differences in a” difference w is a
il = M,(S,.S,). The quantities
mea” As the log mea” is always less than or equal to the arithmetic mea”, this quantity is always less than or equal to zero Thus. t, and cr, are always less than or equal to ~cro. while c’,: is always greater than or equal to zero. It should also be noted that d,, and d,? equal squared quantities. and thus they ale always greater than or eyual to zero. In addition. as the value of ii’is between Y, and v?, the term (rt,  .I )(it ~x2) is less thdn or equal to zero. giving ri,: greater than or equal to zero. As c,, and cIz are less than or equal to zero. and d,, and & are greater than or equal to zero, the quantity Inside the hrackcts in equation (86) is less than or equal to zero. Thus, C, is greater than or cqu,il to xro fbr al? values of .x, and 1: greater than zero. Fquation (013) can be simphlicd by noting that: i,,(‘., <‘’ ;* 0
(BS)
C, and C? can be show” to be equal:
and
giving C. = /,‘(/I
where c;, and d,, are functions
of the partial derivatives
of IV:
The constraints of problem (Pl) mvolvmg variables and fixed fiowratcs are:
Proqf jbr the standard LMTD .function The standard is.
+ I)n, “I’’“[c,,,di, + c>:d,,  2c,ld,2]
form of the log mea” temperature
(B17)
It can be shown that the term inside the brackets is strictly negative. Therefore. C? is also positive. Thus, the eigenvalues of the Hessian matrix are positive and i = M’” is a convex function. A similar proof can be easily performed for Paterson’s approximation (Paterson, 1984) of the !og mea” temperature difference.
temperature
difference
@lo) The parameters (BIO):
c,, and d,, can be evaluated
directly
from
(BI 1)
’
(Bl2)
Q: =/.F’fi!  ty,, = 0 (ij)eMA.
(Bl9)
Q #I it’(iPx’ ii”, .= 0
(920)
(,i)GMrl.
There are 4MA temperature vartables. As the flowrates and heat loads are hxed in (PI). the total number of degrees of freedonl rquals IMA. Constramt (BlY) runs over the set of all streams (IICT). and each streams set of matches R,, giving a total of 2MA cquatwns. Constraints (B19) and (820) provide MA equations each, giving a net total of4MA equations. Thus, the number of constraimng equations equals the “umber of temperature variables. and all degrees of freedom are consumed. The constraints (Bl g)(B2O) form a square system 01 linear equations. If the determinant of thts system is not
Strategies
for overcoming
uncertainties
1151
equal to zero, then this system has one unique solution. The conditions under which the determinant is equal to zero are discussed in Appendix D. (c) Proof
of Property
3
Property 3 states that the solution of the primal problem (Pl) is the global optimum solution of (Pl). Problem (Pl) is a member of a general class of nonlinear optimization problems of the form: minS(x), Feasible values of the complicating variables give values of the error Lagrangian that are less than or equal to zero. Thus, it is appropriate to add the constraint:
subject to hi(X) = 0, g,(x)>0
(PC)
wheref(x) is a convex function, h,(x) are linear equations that form an aSi& set, and gi(x) are concave functions, The solution of all problems of the class (PC) corresponds to a global optimum solution. This is proven in Theorem 4.38 in Avriel (1976).
LE.” .< 0 , to the master
problem. APPENDIX
Conditions APPENDIX Infeasible
Primal
C Problems
subproblem
The mathematical programming problem (P2) shown below minimizes the total infeasibility of the primal problem: min o!
Qfj  f&
(ij)~hfA,
 :;,J) = 0
m1,
 AT,,,
t
cc;
DT2,
AT,,,
B a;
(01) The matrix superstructures
The energy balances three units in it are:
Lo,” =
C )c$L”(DTl,#  AT,,) (~,EYA
ff
Gv
Here, the llowrate variables arc held constant. When a is equal to zero, the solution of the energy balances give values of the temperatures that satisfy the minimum temperature approach constraints, indicating that the primal problem (Pl) is feasible. When a is greater than zero, the primal problem is infeasible. The primaldual solution of (P2) gives values of the Lagrange multipliers J.y.“, ItRn, J.f;mix.k,n,Ayax’.” and AF.CEX./,n, These multipliers can be used to formulate an error Lagrangian:
(El)
determinant
0 A=
0
for
a three unit stream
for a stream
superstructure
&‘I)
Tf;+f,B,,y+f:r$‘ffr:=O,
@‘2)
Tf:+f&ro+f~rpf~r:=O,
(D3)
f:(rl,  t?) = Q,,
(D4
f:(t:  t?) = Qa>
CDS)
f:(r:
(W
 rp) = Qa.
A is given by:
0
0
f::
0
r:,
fBII
f::
0
0
f”u
fR 51
0
fB 31
fF
f?
f::
0
0
0
0
0
ff
ff
0
0
0
0
0
0
r:
r:
The determinant
with
Tf:ff::tP+f::tPfktl=O.
For this system, the matrix
(z&W& (ij)aWA.
Problems
At=b,
ksHCT, (ij)d4.4,
Primal
where A is a square matrix whose elements are determined by the Rowrate variables, t is a composite vector of the temperature variables, and b is a vector with elements of the form  T’fy for mixer energy balances and Q, for exchanger energy balances. If matrix A is not singular, then equation (El) is solvable for the temperature variables. If A is singular, then its determinant, det A is zero, there is no solution to (El), and infeasible. This problems (Pl) and (P2j are inherently appendix will discuss what type of flow patterns create singular matrices, and how they can be avoided.
subject to
Q,,  ff”(t’.’  to,‘) = 0
D
@feasible
The energy balances for the primal subproblem (Pl) form a square system of linear equations. These equations can be written in the form:
During the iterations of the global optimization algorithm, the master problem (Dl) may generate values of the complicating variables for which the primal problem is infeasible. When this occurs, it is necessary to solve an alternative problem (PZ). The values of the noncomplicating variables and Lagrange multipliers obtained from the solution of (P2) can be used to add an error Lagrangian to the master problem (Dl). This Lagrangian acts to restrict the feasible space of (Dl) to a closer approximation of the feasible space of (Pl). The iqfeasibility
for
of this matrix
det (A) = fffv%(fv&f&
0
(D7)
is:
+ f$T&f::
+ fsfif:,f::
0
+ ffi&f&
+ f;,ffzf::
f:f:F:)
(D8)
The matrix A is singular if: (a) f:, ff or f: equal zero; if (b) f&f& = fvf and f$ and ff, are zero; if [c) f$,f$ = ffr: and f& and f:: are zero; or (d) f$f& = fvf and f:: and f& are zero.
1152
C A. FLOUDAS and A. R. Cnw
1
I
Fig. 10. An example
energy balances. It is these “separated” flow systems that give rise to a singular energy balance matrix. It should be noted that these “separated” fIow systems cannot arise for superstructures containing less than three matches (Allgor. 1988). Thus, only those superstructures containing three or more units can give rise to a singular energy balance matrix.
T
of separated
flow systems
Case (a) cannot occur because of the nonzero lower bounds on ff. f: and fy. Cases (M) occur in a phenomenon termed “separation”, where a closed flow loop is established. An example of a separated system is shown in Fig. 10. This flow systems satisfies the mass balances, but not the
in general, an uunit can separate into closed loops wtth 2, 3. n  I units (,41&w, 1988) As these separated flow systems render the primal problem infeasible, it IS highly desirable to add constraints tn the master suhp~oblem that will eliminate separated 110~ system One type of constramt that accomplishes this in un upper bound on the flowrates through the exchanger,. ./ ’ ‘These flowrates must be Less than or equal to the turn of the inlet streams of each potential closed loop. This constraint is formulated hy generatmg for each potential closed loop a set .V\ representing the units inside the potential closed loop, and a set M, representing the units external to Ihe ioop. The appropriate lower bound of fL is: !a F.,‘L
\,
k
c I,,
/
Flow structures with closed loops do not satisfy this equation, and flow structures without closed !cops do.