Compurers them. Printed in Great
Engng, Britain.
Vol. All
13. No. 6. pp. rights reserved
A RETROFIT
703715,
Copyright
APPROACH FOR NETWORKS A.
Department (Received
1I
December
1989
of Chemical 1987;Jinal
R.
CIRIC and C. A.
Engineering
Princeton
revision received
HEAT
0098I 354189 $3.00 + 0.00 1989 Pergamon Press plc
EXCHANGER
FLOUDAS~
University,
23 May
0
1988;
Princeton,
received
for
NJ 08544,
publicnrion
U.S.A. 1 September
1988)
AbstractThis paper presents a twostage procedure for the optimal redesign problem of existing heat exchanger networks. In the first stage, a mixedinteger linear programming (MILP) model is proposed for the retrofit at the level of matches that is based upon a classification of the possible structural modifications. The objective function of this optimization model seeks to minimize: (a) the cost of purchasing new heat exchangers; (b) the cost of additional area; and (c) the piping cost, and is subject to a set of constraints that describe: (a) the heat few model; (b) the area estimation; (c) the calculation of additional area; and (d) the matchexchanger assignments. The solution of the retrofit model at the level of matches provides information about the process stream matches and their heat loads, the placement/reassignment of new and existing heat exchangers, estimates of the required area of each match and the required increase or decrease of area in each heat exchanger, and estimates of the repiping cost associated with introducing new matches, installing new heat exchangers, moving existing exchangers and repiping streams. In the second stage, the information generated in the first stage is used to postulate a superstructure containing all possible network configurations. The solution of a nonlinear programming problem (NLP) based upon this superstructure gives a retrofitted heat exchanger network. The proposed procedure is demonstrated with three example problems.
INTRODUCTION A great deal of effort has been expended during the past two decades in the design of grassroot heat exchanger networks, to improve the heat recovery in chemical processes (e.g. Cerda and Westerberg, 1983; Linnhoff and Hindmarsh, 1983; Floudas er al., 1986). This research effort has resulted in over 200 publications, and a very good review is presented by Gundersen and Naess (1987). It is interesting to note that, despite the fact that a lot of research has been focused on the synthesis problem of grassroot heat exchanger networks, there has been significantly less systematic study of the optimal redesign of existing heat recovery networks. Recently however, restricted profit margins and the high cost of building new chemical plants provide strong motivation to develop techniques for the optimal redesign 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. 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 their 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. iAuthor
to whom all correspondence
should be addressed.
Tjoe and Linnhoff (1986) proposed a retrofit design philosophy for the retrofit of heat exchanger networks which consists of: (a) the identification of the crosspinch exchangers; (b) the elimination of the crosspinch exchangers from the existing network; (c) the positioning of new exchangers and, where possible, reuse exchangers removed in step (b); and (d) the evolution of improvements by considering heatload loops between streams, process and utility heat exchangers. They stated, however, that such an approach can provide only bounds within which it is expected to find a good retrofit, and that the best retrofit design is difficult to identify due to the complication of process constraints and plant layout. Saboo et of. (I 986) proposed an evolutionary strategy which 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) imposed three targets: (a) maximum utilization of existing exchangers; (b) assignment of existing units to new required matches with minimum piping changes; and (c) minimum number of new stream matches that require the purchase and installation of new units. They developed a MILP assignmenttransshipment model to determine the fewest modifications, at the level of matches. This model does not consider all the potential modification combinations, and does not explicitly take into account the use of existing area and its 703
704
A. R. CIRK and C. A. FLOIJDAS
potential increase or decrease in the different heat exchangers. In an example problem, they demonstrated that the removal of crosspinch heat exchangers, as suggested by Tjoe and Linnhoff (1986) is not always neccessary. In this paper, a systematic twostage approach is proposed for the optimal redesign of heat exchanger networks. In the first stage, a MILP formulation is presented, at the level of matches, that incorporates explicitly the cost associated with each potential match of streams, and involves all possible options for modifications. The solution of this formulation provides information on which exchangers should be reassigned or newly installed, and whether there is a need to increase or decrease the area of the existing exchangers. The second stage takes advantage of this information, and a superstructure is postulated and formulated as a nonlinear programming (NLP) problem. The solution of the NLP provides the actual retrofitted network that minimizes the total modification cost.
PROBLEM
The problem as follows:
STATEMENT
to be addressed
in this paper is stated
Given is an existing heat exchanger network, containing K exchangers of known area; a set H of hot process streams and a set C of cold process streams with fixed flowrate heat capacities, inlet and outlet temperatures; and a set of hot and cold utilities HU and CU, respectively. Associated with each exchanger k is a set of matches M(k) containing matches (ij) that can be assigned to exchanger k. This set of matches may contain only the match originally housed in k, or matches with a particular type of duty (e.g. condensing duties), or it may contain all possible matches. Associated with each existing and potential match (ii) is a set Z(ij) containing all exchangers that are allowed to be assigned to house it. The objective of this problem is to determine an optimally redesigned heat exchanger network that features the minimum total modification cost. This paper proposes a twostage approach for solving this problem. In the first stage, process stream matches and matchexchanger assignments are made
exchanger is equal to the cost per unit area in an existing exchanger. The first two assumptions are requirements of the model: a single minimum temperature approach and constant flowrate heat capacities are required for modelling heat flow using the transshipment model. The assumption of fixed heat transfer requirements is required for estimating the heat transfer area. This assumption can be relaxed in the second stage of the procedure, when detailed information about the flowrates is available and can be used to explicitly calculate variations in the heat transfer coefficients.
OUTLINE
OF RETROFIT
STRATEGY
The proposed strategy for the retrofit of heat exchanger networks is shown in Fig. 1. It consists of the following five steps: In the first step, a heat recovery approach temperature (HRAT) is selected. This selection of the HRAT can be performed either randomly within some reasonable range of values, or using a targetting procedure that provides a good estimate of the optimum heat recovery approach temperature. In the second step, a minimum utility cost calculation is performed which provides information on the hot and cold utilities, the yearly savings in utility cost, and the location of the pinch points that partition the temperature range into subnetworks. In the third step, all potential pairings of matches and exchangers are considered, so that all decisions regarding reassigning heat exchangers, purchasing new exchangers, and repiping streams are incorporated in the proposed retrofit model at the level of matches. The solution of this model corresponds to the minimum modification cost predicted at the level of matches, and provides information on the reasHAAT
Select
m I
Minimum
Utlity
t Consumption
on the basis of estimates of the required heat exchanger area of each match and the total modification cost. In the second stage, a network structure is derived. The first stage required the following assumptions: 1. A heat recovery approach temperature, HRAT (Jones et al., 1986) is specified. 2. The flowrate heat capacities of each of the process streams are constant. 3. The heat transfer coefficients for each match are known and fixed. 4. The cost per unit of heat transfer area in a new
Fig. 1. Outline of retrofit strategy.
Retrofit approach for heat exchanger signment of existing exchangers, stream repiping, the purchase of new exchangers, and additional area for existing exchangers. In the fourth step, a superstructure that has embedded all the alternative network structures is generated and solved as a nontinear programming programming problem, using the approach of Floudas ef af. (1986). The minimum approach temperature (HRAT) is relaxed for each match, and the temperature approach in each match is treated as a variable greater than a specified lower bound. The solution of this NLP problem provides the minimum investment cost heat exchanger network configuration. In the last step, the total profit is calculated. These steps are repeated until the stopping criterion is reached. MATHEMATICAL LEVEL
FORMULATION OF MATCHES
AT THE
The purpose of the mathematical formulation at the level of matches is to select the combination of process matches and matchexchanger assignments that will minimize the cost of structural modifications, such as moving existing exchangers, purchasing new heat exchangers, adding area to existing exchangers and repiping streams. This is accomplished with an objective function that is based upon the cost of reassigning exchangers, purchasing new heat exchangers, increasing the areas of existing exchangers and repiping the appropriate streams, subject to a set of constraints that model the heat flow between streams, calculate the area requirements and new area associated with each match, and assign process matches to existing heat exchangers. To describe mathematically the contributions of each type of modification to the objective function and to the set of constraints, a classification of the structural modifications is required. CATEGORIES
OF STRUCTURAL
MODlFICATlONS
The modifications of reassigning exchangers, purchasing new exchangers and repiping streams arise from the potential pairing of matches and exchangers, and can be divided into six general categories. Each category is based upon characteristics of both the match and exchanger. Category lExisting match @,I is housed in exchanger k in both the original network and the retrofitted network. No structural modifications are required for this case, and consequently the piping costs of this category are zero. Category 2The existing match @,I was housed in exchanger k in the original network, and will be housed in exchanger k’, in the retrofitted network, where k’ is a member of
[email protected] As the piping for this match is already in place, no repiping is required; the only required modification is to move the exchanger k’ to
networks
705
the new match site. The costs of this category, C$ reflect the labor costs of removing exchanger k from the match site, and installing exchanger k’. Category 3The new match (Q) did not exist in the original network and will be housed in exchunger k, a member of the set Z&j), in the retrofitted network; in addition, the original match housed in exchanger k contained either szream i or stream j. This category has a cost Cj that reflects the repiping of one stream. Category 4The match (ij] did not exist in the original network, and wiU be housed in exchanger k, a member of the set
[email protected]], in the retrofitted network; neirher stream i nor stream j participated in the match originaily housed in k. In this case two streams must be repiped. The cost of assignments in this category is Cl. Category 5A new heat exchanger is purchased to house the existing match (ij)_ This can occur when the original exchanger has been reassigned to a new match, whose area requirement is greater than that of match (ij) in the retrofitted network. In this case, no repiping is required, and there are no additional costs associated with this category. All existing matches may enter Category 5. Category 6A new heut exchanger is purchased to house a new match, rhat is, one that did not exisr in the original network. In this case, two streams need to be repiped, and a cost Ci is assigned for the exchanger purchase and installation and stream repiping cost. AH six categories of structural modification can be accounted for in an explicit way by introducing two sets of integer variables. The first set z$ denotes the assignment of a potential match (ij) to an existing exchanger k. The activation of a given z:. variable represents the assignment of the match (ij) to an existing exchanger k. Therefore, the set of zf variables may be divided into four subsets, depending upon the category of the assignment:
* z,,
C’, if the assignment
is in Category
1;
1 C’, if the assignment
is in Category
2;
C3, if the assignment
is in Category
3;
C“, if the assignment
is in Category
4.
The last two categories do not involve reassigning heat exchangers, but rather purchasing new exchangers. To model these categories, a second set of integer variables, denoted as m, is introduced. The variable rn,_ takes the value one if a new exchanger is to be purchased for the match (ij) and is zero otherwise. These variables can be partitioned into two subsets: c?, if match (ij) exists in the original m,,++
network; ( C6, if match (ij) is a new match.
The introduction of these integer variables ensures that all possible combinatorial decisions at the level
A. R.
706
CIRIC
and C. A.
FLOUDAS
of matches are taken into account in the mathematical formulation of the optima1 redesign problem of heat exchanger networks. OBJECTIVE
FUNCTION
CONSTRAINTS
The objective function consists of three main components: (a) the cost of purchasing new heat exchangers; (b) the cost of additional heat exchanger area; and (c) the piping cost resulting from structural modifications. The following exponential form is a general expression that takes into account the cost contribution of the first two components of the objective function: CA ‘,
Cl)
where c is a cost constant and b a fixed exponent. In this paper, however, the exponential cost contributions are approximated with a piecewise linear set of fixed charge cost expressions of the area of each exchanger. Therefore, for each potential match (ij) a cost consisting of piecewise linear expressions of the following form is introduced: C, = am,, + PX, ,
(2)
Xii = max(A,
(3)
 A&, 0),
where a is the installation cost of a new heat exchanger, mii represents the integer variable associated with the purchase of a new heat exchanger, /l is the cost per unit of heat exchange area, X,i is the additional area required for the match (ij), if any. Notice that Xii is provided by the maximization problem of A,  A; and zero, as indicated by (3). If a new heat exchanger is purchased for the match (ij), then the existing area for match (ii), AT, equals zero, max(A,  Art 0) equals A,, and the additional area X, is equal to the area of the match. If a heat exchanger with an existing area Ar is reassigned to the match (ij), and the match requires a larger area, then the operator max(A,,  A;“, 0) simply equals the difference between the total required area and the existing area of the heat exchanger. If however, A, is less than or equal to AT the cost contribution is driven to zero, since max(A,  AT, 0) is equal to zero. It should be noted that for the first four categories of matchexchanger assignments the cost contribution comes from the expression: /3X,,= /I max(A,,  A 7,
0),
(4)
as only additional area needs to be purchased. Combining the expressions for the cost of purchasing new heat exchangers, additional area, and the cost of structural modifications results in the following objective function:
The required set of constraints is classified into four main components: (a) the heat flow model; (b) the area estimation at the level of matches; (c) the calculation of any additional required area; and (d) the matchexchanger assignments. (aa) Heat flow
modeI
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 Qij,, or from one interval to another via a hot stream residual heat flow, denoted by R,,. A hot stream can either release heat to a cold stream in the same interval, or it may cascade 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 Qy heat, and cold stream j must absorb Q$ heat. The energy balances of the hot and cold streams can be represented with the set of the following linear equations:
5 Q,+R,,,i,=Q:
i=l..
i1
n,
t = 1.. T,
(6) f,
Qiil = Q$
j = 1 , . C,
t = 1 . . T.
(7) The existence of a match between hot stream i and cold stream j is designated with an integer variable Y,. YU takes the value one if streams i andj exchange heat; it is zero otherwise. If Y, takes the value zero, then the associated continuous variables Q, should take zero values. This is represented by the equation: $_ Q,UY,
i=l..h
j=l._C,
(8)
,I
where U represents the minimum of the maximum heat loads of hot stream i and cold stream j. (a) Area estimation
At the level of matches, where no information is available for the actual structure of the heat exchanger network, it is not possible to incorporate a rigorous expression of the area of each match as a function of the inlet and outlet temperatures of the streams, unless a superstructure at the level of
707
Retrofit approach for heat exchanger networks matches is introduced. Since such a superstructure will increase drastically the size of the problem, an estimation of the area expression of each match (ij) is introduced as follows:
where AT,,. ij, is the logmean temperature difference of match (ij) in interval f, calculated based upon the inlet and outlet temperatures of streams i and j in interval t, hi and h, are the individual heat transfer coefficients of stream i and j, respectively, and Qii, is the heat exchanged between stream i and j in intervai 1. Notice that the log mean temperature difference for each potential match (ij) and for each temperature interval t, ArLM+, is fixed, and can be calculated a priori for each match (ii) for each interval r. Note that if k, and Fz,are specified, then the area of each match A, is a linear expression of Qij, as indicated in equation (9). It should also be noted that the summation of equation (9) over all the matches results in an area target similar to the one proposed by Townsend and Linnhoff (1984). {c) Calculation of additional area The cost expression of additional heat exchange area requires that the new area of each match be calculated. Let the variable S, represent the area of an exchanger assigned to match (ij) and variable XU represent the new area of match (ij). These variables can be calculated using the following two equations:
modelled according 1. 2. 3.
to the following three conditions:
Each exchanger is assigned to only one match. Each match is housed in only one exchanger. No exchanger should be assigned or purchased for a match that will not occur in the retrofit network.
These conditions can be stated with two sets of equations:
c
($3EM(k)
,_I;<1
I~;ii,2;+miiYij=0
mathematically
k=I..x;
(12)
i=l..H.
i=l..C. (13)
The first equation ensures that exchanger k is assigned to at most one match; it also allows for exchanger k to be removed from the network. The second equation requires that if match (i,j) 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. Having presented the objective function (5), as well as the set of constraints (613), the overall mathematical formulation at the level of matches can be stated as follows:
subject to
xi/
A, + S, = 0.
(11)
The slack variable S, equals zero if the match (ii) is not assigned to an existing exchanger k, as the Z; for this match is equal to zero. If a match (ij) is assigned to an existing exchanger k, S, can take any value up to the area of exchanger k. Since the cost of the new area is minimized in the objective function, then X, is minimized, and therefore the slack variable S, will be forced to its largest possible value. If the total required area for match (ij), A,, is less than the area of the assigned exchanger, then the slack variable will be equal to the required area and X, will take the value zero. When the required area is greater than the existing area, the slack variable takes the value of the existing area and X, equals the total area minus the existing area. Notice that as Xii is minimized, these constraints become equivalent to the original definition of X,, namely X, = max(A,  A?, 0). (d_I Match exchanger
assignments
An assignment model is required for selecting the appropriate pairs of matches and exchangers. The match&exchanger assignment subproblem can be
2 Q~J,+&,RirQe::
i=l..
I%,
t = 1 . . T,
j1
/t, Q, = Q;
5 Q,,UY,sO
j = 1 . . C,
t = 1 . . T,
i=l..H,
j=l..C,
i=l..H;
j=l..C,
I1
Z: 4+mijYY,60 keZti(f
i=
x A~xi,,zk,
1 . . n,
1 . . H,
i=l..H,
j=
j=
1..
1 __C,
j=l..C,
c,
A. R. CIRK and C. A.
708 Qiil,R,,A,i,S,,X,20
j=l..C,
i=l..H,
r;,, z;, mlj= W,
11.
(Pl)
This mathematical formulation corresponds to a MILP problem, since it involves continuous variables, integer variables and a linear set of constraints. The solution of this mathematical model represents the minimum modification cost at the level of matches, and provides information on the reassignment of existing heat exchangers, the purchase of new exchangers and additional area and the required structural modifications. REMARKS
(a) Restrictions
ON THE MATHEMATICAL FORMULATION
on matchexchanger
assignments
It should be noted that the summation in equation (12) is over the allowed set of matches (ij) E M(k), as opposed to all combinations of (ij). Similarly, the summations in equations (10) and (13) are over the allowed set of exchangers k E Z(ij), as opposed to all exchangers k. These restricted sets allow for cases where a given exchanger can only be reassigned to a subset of the existing and potential matches. For example, consider a retrofit problem with two hot streams (HI, H2), two cold streams (Cl, C2) and three existing exchangers. Exchanger 1 can only be assigned to house match (HI, Cl), exchanger 2 may only be assigned to house matches involving stream Hl, and exchanger 3 may be assigned to any of the matches. In this case, equations (lo), (12) and (13) become:
FLOUDAS
however, that due to the explicit consideration of all potential matches in the assignment problem, there exists a large number of integer variables in the proposed mathematical formulation. This implies that if a standard branch and bound technique were to be used for the solution of (Pl), this could result in computational limitations because of the size of the retrofit problem. This difficulty can be overcome by using an iterative solution method that isolates the assignment constraints in a separate subproblem, to create a linear programming (LP) assignment problem. Assignment problems have the desirable characteristic that the variables take integer values at the optimum. Thus, the zf; and m, variables can be treated as continuous, greatly reducing the number of integer variables. The MILP formulation (PI) cannot be efficiently decomposed into an assignment subproblem and another subproblem. It can be shown, however, that an equivalent mixed integer nonlinear programming (MINLP) formulation can be developed that is easily decomposed to give a pure assignment subproblem. The proposed MILP formulation (Pl) can then he combined with elements of the equivalent MINLP formulation to provide an efficient iterative solution procedure. Alternative MINLP formdation. An alternative form of the MILP formulation is a MINLP formulation, indicated as (P2) which replaces the existing area variables S, and the new area variables X, with a set of variables Xi representing the amount of new area associated with every possible assignment z$. This equivalent formulation is as follows: min
C
(Vk) 0
/3X$z$+
+
C
{C; + gx$}z:,
(i.i. k)e Cr
E
c
{c;+ /?x:j}z:j
W.k)sO
+ z:,+z:,+z:,+z:~~
z:, + z:, + z:, + m,, 
Y,, = 0,
z:z + z:* + ml2 
Y,2 = 0,
z:I + m,, 
Y2, = 0,
z:* + ?n2* 
Y,, = 0.
Thus, the assignment variables representing undesirable matchexchanger assignments (I iz, z i, , t i2, z:, and z&) do not appear in the formulation. Consequently, the assignments will not be made. (b) Decomposition
of the Iarge combinatorial
C V.J. k)E C’
1,
problem
The proposed mathematical model (PI) for the retrofit at the level of matches can be solved with standard branch and bound techniques, as it is classified as a MILP problem. It should be noted,
{c:+
Sxfjlz;5,
subject to 5
j I
Q*.j.r+Ri,rIRi,r=Q~l.
Retrofit approach for heat exchanger networks
Q,, 19Ri. Iv A,j+ X:j 2 0, yi.p z:j, mi,, = Ilo, ‘1.
WY
The mathematical formulation (P2) corresponds to a MINLP problem since bilinearities of continuous and integer variables appear in its objective function. This equivalent formulation has the advantage that the assignment variables z$ and rnUappear only in the objective function and the assignment constraints. Notice that for fixed values of Qi/,.R,, Y+ A, and Xi, this MINLP reduces to a simple assignment problem:
+
C
IC:
+
to be continuous. This relaxed problem, designated as (RPl), is MILP problem, with integer variables Yi,j representing the potential process stream matches. The solution of this relaxed problem will provide process stream matches, their heat loads and their estimated areas. The cost reported by this relaxed problem is a lower bound on both the final solution and the actual cost of this particular combination of process stream matches, heat loads and estimated areas. The actual cost is evaluated by solving the assignment problem (P3), which also gives the matchexchanger assignments. An iterative solution procedure can be built around (RPl) and (P3). Solving (RPl) gives a lower bound on the final solution, while (P3) provides an upper bound. The procedure is driven forward by adding integer cuts to (RPl) at each iteration that will eliminate all combinations of process stream matches identified in previous iterations. Thus, at each new iteration (RPl) will be forced to identify new combinations of process stream matches. The iterative procedure is terminated when the lower bound provided by (RPl) meets or exceeds the upper bound provided by (P3). Thus, a feasible solution algorithm is: Solve the relaxed MILP (RPl) to obtain process stream matches, heat loads and match areas. Set the lower bound on the optimum equal to the MILP objective value. and the Determine the total cost exchangermatch pairings by solving the assignment problem (P3). Set the upper bound of the optimum equal to the total cost. If the lower bound from step one is greater than or equal to the upper bound from step 2. stop. Otherwise, add an integer cut to the MILP formulation and return to step 1. This algorithm was used to solve the second and third example problems described in this paper via APROS (Paules and Floudas, 1989).
B%flmt.,
zf< c (MEM(k)
709
1,
EXAMPLE 1 Odrfj,mi,jC
1.
(P3)
Algorithmic approach. Any solution technique that implements the assignment problem formulation (P3) above will consist of two steps. The first step selects the matches, and calculates the heat load and area of each match. The second step assigns the matches to existing exchangers or to new exchangers and evaluates the total cost via the assignment problem. The efficiency of the first step in generating good guesses of the matches and their areas largely determines the efficiency of the algorithm. Thus, a method which selects matches and areas that are near the optimum point of the MINLP formulation is desirable. One such method involves relaxing the MILP formulation (Pl), by allowing the I and m variables
The first example, which is taken from Yee and Grossmann (1987) consists of three hot streams, three cold streams, one hot utility and one cold utility. The data for this problem are given in Table 1. The relative cost factors for repiping streams and the cost factors for purchasing area are listed in Table 2. The existing network, which is shown in Fig. 2, involves seven exchangers whose areas are listed in Table 3. This network requires 360 kW of steam and 800 kW of cooling water, at a total cost of $44,800 yr‘. The minimum utility cost of this network is $5500 yr‘, for a heat recovery approach temperature of 10°C. The use of stream is completely eliminated, but 440 kW of cooling water is required. The proposed MILP formulation of this retrofit problem involved 108 integer variables, (12 integer variables denoted the existence of a match, 84 were
710
A.
Table Stream
T in (K)
HI HZ f13 Cl cz c3 SI cw
500 450 400 300 340 340 540 300
U=O.SkW
m
I. Stream
data
Tout(K) 350 350 320 480 420 400 540 320
*K
for
FC,,
Example (kWK
R. Cnuc and C. A.
FLOUDAS
1 ‘)
S KW
’ yr
’
IO 12 8 9 IO 8 80 20
t
’
=o480K
exchangermatch assignment variables, and 12 were new exchanger variables), 64 continuous variables associated with area calculations and the transshipment model, and 82 constraints. The problem was solved on a VaxStation II/GPX workstation computer, using a standard branch and bound technique (ZOOM/XMP). The branch and bound method took 1970 CPU s to solve the full MILP. The solution involved seven process stream matches, with an estimated total modification cost of !630,621. The heat loads and matchexchanger assignments for the seven matches are listed in Table 4. Four of these matches took place in the original network, and all of the existing heat exchangers were utilized. No new exchangers were selected to be purchased, and the area of each exchanger was estimated to increase. The assignment of exchangers to matches with larger required areas and the selection of no new exchangers reflect the maximal use of existing heat transfer area. Figure 3 shows a network configuration derived for these results and based upon a minimum temperature difference of 3”. This network requires an additional 50.9 mz of heat exchanger area. While the amount of additional area is less than the amount predicted by the MILP formulation, the amount of repiping is more than the predicted three streams. This extra piping arises from the two sets of parallel piped streams in the final network, a feature that cannot be predicted by the MILP formulation. Work is currently in progress to incorporate estimated repiping costs at the network superstructure level. and will be reported in a future publication (Ciric and Floudas, 1989). The network derived from the proposed MILP formulation was compared with the network derived by Yee and Grossmann’s (1987) technique that is based upon a minimum units approach. Their solution involved only six units, as opposed to seven in this paper and featured 118.7 m2 of additional heat
Tltble p = c*= cs400 c$ = Cp = Cp: =
171.4 IO 800 3460 4260
2. Cost
factors
cw
Cl
Fig.
2.
network
Existing
in
Example I.
exchange area. The reduction in the additional areas occurs because the proposed method selects process stream matches and matchexchanger assignments based upon an estimate of the optimal use of existing heat exchange area. EXAMPLE
2
This example, which is taken from Tjoe and Linnhoff (1986) consists of three hot streams, two cold streams, one hot and one cold utility. The hot utility is provided by a furnace, and the cold utility is assumed to be cooling water. The data for this problem are listed in Table 5, with relative repiping costs listed in Table 2. Table
3.
Areas
Exchtintter
of existing Example Area
i
5 6 7
for
Examples
I
and
(m’l
Oriainal
2
repiping
two
streams
for
mlrtch
HZCL H IC2 H3Cl HlC3 Slcl HICW H2CW
45.06 12.50 33.09 23.50 5.75 5.39 11.49
I 2 t
Coot m z of heat transfer arelt Rcltttive cost for moving one sxchttnger Reltttive cost of repiping one stre&tm Reltitive cost of rcpiping two streams Fixed charge cost of a new exchtmger Fixed charge cost of ~1 new exchanger and
exchangers
I
Retrofit
approach
Table
4.
for heat exchanger
Retrofit
network
data
Assignment Heat
Match HICl HIC2 HIC3 H2C HZC3 H3Cl H3CW
I
load
Exchanger
512 800 188 908 292 200 440
Example
711
I
Existing
Estimated
CBtCROOTY
area
area
area
3
33.09 12.5 23.5 45.06 11.49 5.75 5.39
38.82 83.08 23.5 97.96 30.52 25.0 IS.84
28.16 34.72 31.74 47.65 21 x.5 5.75 12.84
3 2 4
I
1 1 3 2 3
I
7 5 6
The existing network, which is shown in Fig. 4, involves six exchangers, whose areas are listed in Table 6. This network requires 17,597 kW of a hot utility and 15,510 kW of water. The solution of the minimum utility consumption can be reduced to 12,410 kW of a hot utility and 10,160 kW of cooling water, for a heat recovery approach temperature of 19”C, an estimate of the optimal value found by Tjoe and Linnhoff (1986) using a targetting approach. A pinch point exists at 159140°C for this level of utility consumption. Retrofit networks were derived for two different scenarios. In the first scenario, it is assumed that all exchangers except the furnace unit can be reassigned to any existing or potential match. In the second scenario, it is assumed that the coolers (exchangers 5 and 6) may only be assigned to cooling duties (i.e. matches Hl CW, H2 CW and H3 CW) and that the remaining exchangers (l4) may only be assigned to processprocess duties. c3 J40K
for
networks
Scenario
Retrofitted
1
The MILP formulation for this problem contained 88 integer variables (11 variables denoting the existence of a match, 66 matchexchanger assignment variables, and 11 variables representing the purchase of new exchangers), 57 continuous variables associated with area calculations and the heat flow model, and 61 constraints. The problem was solved on a VaxStation II/GPX workstation computer, using the relaxation technique, which converged in two iterations consuming 320 CPU s. The solution involved eight process stream matches, and the heat loads, matchexchanger assignments, and areas of the matches in the retrofitted network are listed in Table 7. All of the existing matches remain in the retrofitted network, and all existing exchangers are utilized. Four of the original process stream matches have retained their original exchangers, while a fifth,
C2
1
34OK
1
t
4BOK
363.5K
J21.1K
6
5.46 8
3OOK
Cl cw
Fig. 3.
Retrofit
network
in Example
1.
H3
A. R.
712 Table Stream
T in ( C)
HI HZ H3 Cl c2 CW
159 267 343 26 118 25
Operating
cost
5. Stream T out (“0
data
oil
Example
’>
FC_ (kW/K
b (kW
utility = $95.040
“C mm21 0.4 0.3 0.25 0.15 0.5 0.3
KW’
Scenario
yr’
match H lCl, has been assigned exchanger 5, a larger exchanger. One new match, HIC2 has been selected. The matchexchanger assignment for this match, (exchanger 3) indicates that only one stream needs to be repiped for this new match. Note that a new exchanger is to be purchased for match HICW, while its original exchanger has been reassigned to house H 1C 1. In turn, exchanger 3, which originally houses HICl, has been reassigned to HlC2, a new match. This regrouping gives a considerable savings in heat exchanger area: if HICl and HlCW had retained their original exchangers, an extra 237 m2 of additional heat exchanger area would be required.
t
FLCIUDAS
A network structure was derived for this scenario using the approach of Floudas et al. (1986) in which the HRAT was relaxed for each exchanger. The resulting network is shown in Fig. 5. This network features a total requirement of 1812 mz of new heat exchanger area.
2
228.5 20.4 53.8 93.3 196.1
77 80 90 127 265 30
of hot
for
and C. A.
CIRIC
Zest
H3
H2
2
This scenario differs from Scenario 1 in that the existing coolers (exchangers 5 and 6) are restricted to cooling duties, and the remaining exchangers (14) are restricted to processprocess duties. The MILP formulation for this problem contained 60 integer variables (11 variables denoting the existence of a match, 38 matchexchanger assignment variables, and 11 variables representing the purchase of new exchangers), 57 continuous variables associated with area calculations and the heat flow model, and 61 constraints. It was solved in three iterations with the relaxation method, consuming 436 CPU s. The solution contained the same process stream matches and heat loads as in Scenario 1, but the matchexchanger assignments varied. In this case, all the original matches retained their original exchangers, with the exception of match H2X2. A new exchanger is to be purchased for this match, and its original exchanger is to be assigned to match HlC2. The optimal network structure for this scenario is the same as that of Scenario 1, shown in Fig. 5. The total required additional area for this scenario is 2049 mz, 200 m2 greater than that associated with Scenario 1. The retrofitted network structure presented here involves two split streams, and is thus slightly more complicated than the network derived by Tjoe and Linnhoff (1986) which involves only one split stream. However, the network structure of Tjoe and Linnhoff (1986) requires 2320 m* of additional heat exchanger area, more than the amount of heat exchanger area
Table
6. Areas
of existing
Exchanger
Area
I
Fig.
4.
Existing
network
in
Example
Table Match Above
7. Heat
Retrofit load
603.71 584. I5 1001.34 121.53 1048.28 133.54
2 3 4 5 6
2.
network
data
Exchanger
for
Example Assignment category
2,
exchangers (m*)
Scenario
I
Existing area
Retrofitted area
121.53 603.71
276.35 991.32
1048.28 1001.34 0.00 133.56 548.15
1218.95 946. I8 155.80 123.91 927.22
pinch
H2C2 H3C2 HeaterC2 Below pinch HICl HIC2 HICW H2CW H3Xl
2203.2 9899.2 12.410.1 5711.1 4314.2 871 1.1 161 I.6 3712.2
4
I 5 3 New 6 2
I 1
2 3 5 I
I
for Example
Original H3GC2 H3Xl HlCI H2C2 HlCW H2CW
match
2
Retrofit
t
approach
for heat exchanger
2.
26SC
3.
4.
Fig. 5. Retrofit
network
for Example
2, Scenarios
I and 2.
required by either scenario. The savings in heat transfer area is attributed to: (a) the flexibility in the matchexchanger assignments; and (b) a different network structure. EXAMPLE
3
In this example, the total profit over a 2 yr lifetime is optimized with respect to the heat recovery approach temperature using the golden section search technique. This iterative method can be represented as a loop, shown in Fig. 6. The application of the golden section search begins by selecting a range of temperature approaches. The total profits corresponding to the endpoints of this range are calculated in the following manner: 1.
The minimum utility consumption, yearly savings are calculated.
cost and
713
networks
Process stream matches, heat loads and matchexchanger assignments are calculated using the iterative relaxation technique for retrofit at the level of matches. An optimal heat exchanger network and its investment cost are computed using the technique of Floudas et al. (1986) in which the minimum temperature approach of each match is relaxed. The cost factors in this optimization are based upon the amount of additional area required for each match. The total profit is calculated as the lifetime times the yearly savings, minus the investment cost.
The total profit is calculated repeatedly in a iterative procedure that places increasingly tighter bounds on the optimal payback time and temperature approach. The search stops when either: (a) the range of the temperature approach is sufficiently small; or (b) the convergence of the total profit is within acceptable bounds. This technique was applied to the retrofit problem of Tjoe and Linnhoff (1986) used in Example 2. the cost data in Table 2 was used to calculate the cost of the retrofit network in the second (retrofit assignment problem) step of the algorithm. In the network derivation, a nonlinear expression presented by Tjoe and Linnhoff (1986) was used to cost exchangers: cost of heat exchanger area = $1005 A”.83, fixed charge cost of a new exchanger = $12,900. The costs of moving exchangers and repiping streams were taken as the values in Table 2. Each iteration of the golden section search required three iterations to solve the retrofit assignment problem, and two NLP problems to derive the final networks above and below the pinch. Nine iterations were required to find the optimal heat recovery temperature approach of 19.3”C, yielding a yearly savings in utility costs of $487,465 yr‘. This value of
step
1.
Select a range of minimum
step
2.
Calculate
step
3.
Idelltify two points within the tmprraturr
temperature
approaches.
the total profit for the endpoints
uf lhe
rangr.
approach
range, using thr golden
section
search technique. step 4.
Calculate
the total profit for the interior points.
step 5.
If the convergence
of the total profit is within acceptable
bounds,
then STOP.
Otherwise, step 6.
Drop the section of th rangr betwren thr interior point with the lower total profit and its ncarcst endpoint,
Step i.
If the range of the minimum Otherwise,
forming a new range of temperature
temprraturr
return to Stop 3. Fig. 6.
approaches.
approach is sufficiently small, then
Outline of total profit optimiztion.
STOP.
A. R. CIRIC and C. A. FLOU~AS
714 Table Match Above pinch HZC2 H3C2 HeaterC2 Below pinch HICl HIC2 HICW HZCW H3Cl
8. Retrofit Heat
load
Match Above pinch H2C2 H3C2 HeaterC2 Below pinch HICl HlC2 HICW H2LCW H3Xl
Exchanger
2203.2 9899.2 12.410.1
I

Heat load
2
Existing area
Retrofitted area
5
0.00 603.71
276.35 991.32
I
1001.34 121.53 1048.28 133.56 5x4. I5
1218.95 946.18 755.80 123.91 927.22
1

3 4 5 6 2
9. Retrofit
2, Scenario
Assignment category
New
571 I.1 4314.2 8711.1 161 1.6 3712.2
Table
data for Example
network
network
Exchanger
2203.2 9899.2 12.467.95
4 6 
5711.2 4256.35 8769.55 I61 I.6 3712.2
New 7 2
3
I
3 I I
I
data for Examule
3
Assyqrnent category
I 2 I
3 5 1 I
Existing area
Retrofitted area
121.53 104X.28
249.25 1048.28
1001.3 603.8 0.00 133.6 584. I 5
1200.33 921.119 759.67 123.9 937.48
The data generated during the golden section search can be used to generate plots of the investment cost, utility savings, and total profit as functions of the heat recovery approach temperature. Figure 8 shows the investment cost and utility savings plotted as functions of the heat recovery approach temperature. Both the utility savings and investment cost decrease as the heat recovery approach temperature rises. Figure 9 shows the total profit curve, calculated for an expected project lifetime of 2 yr. The breakeven point occurs at a HRAT of 8°C. At this point, the payback time (defined as the investment cost divided by the annual utility savings) equals the expected project lifetime. Retrofit projects involving a HRAT less than 8°C will not show a profit over 2 yr. The maximum profit occurs at 19.3”C, as calculated during the golden section search.
Fig. 7. Retrofit network for Example 3.
the heat recovery approach temperature is in very good agreement with the 19°C reported by Tjoe and Linnhoff (1986). The optimal process stream matches and matchexchanger assignments are given in Table 9, and the optimal network is shown in Fig. 7. This network requires 1757 m2 of additional heat transfer area, and features an investment cost of $647,440 giving a payback time of 1.328 yr and a total profit of $327,490.
0.00.*0
1
0
I
I 10
20
I
30
I
40
HRAT
Fig. 8. Utility savings and investment cost as functions of HRAT.
Retrofit
400,000 200,000
E,
.g
s
9 e”
for heat exchanger
*j\.
networks
AcknowledgementThe authors would like to acknowledge financial support from the National Science Foundation under Grant DMC8617239.
REFERENCES
200.000 400,000 I’
i
600,000 600,000
1,ooo,ooo
ii
’
0
. ,
I
I
20
30
40
I
10
HRAT
Fig. 9. Total
profit as a function
of HRAT.
CONCLUSIONS
paper, a twostage strategy is proposed for the retrofit problem of existing heat exchanger network configurations. Central to this strategy is a mathematical model for retrofit at the ievei of matches. This formulation features three main elements: (a) every possible matchexchanger assignment is considered; (b) the use of existing heat exchanger area is optimized based upon an estimate of the required area of a match; and (c) repiping costs are explicitly modelied by dividing the matchexchanger assignments into six structural modification catrgories. These elements allow the mathematical formulation to identify the process stream matches, heat loads and matchexchanger assignments of a retrofitted network at the level of matches, from the costs of stream repining and additional heat exchange area. As all possible matchexchanger assignments are contained within the model, the use of existing heat exchange area can be optimized by moving an exchanger from one match to another. This was demonstrated in the example problems, where large exchangers were transferred from an existing match with a decreased area requirement to an existing or new match with a larger area requirement. Heat exchanger networks were derived in three examples by using a superstructure methodology that minimized investment cost. Optimization of the HRAT and total profit was demonstrated. The resulting heat exchanger networks featured low amounts of required additional area. In this
715
\*
,/
0
approach
Cerda J. and A. W. Westerberg, Synthesizing heat exnetworks having restricted stream/stream changer matches using transportation problem formulations. Chem. Engng S’ci. 38, 17231740 (1983). Ciric A. R. and C. A, Floudas. A mixedinteger nonlinear programming formulation for the retrofit of heat exchanger networks. Computers &em. Engng (1989). Submitted. Floudas C. A., A. R. Ciric and I. E. Grossmann, Automatic synthesis of optimum heat exchanger network configurations. AICkE Jl32, 276290 (1986). Gundersen T. and L. Naess, The synthesis of cost optimal heat exchanger networks, an industrial review of the state of the art. Proc. Chem. Engng Fundam. CEF’87: The Use Comput. Chem. Engng, Taormina, Italy, pp. 675704, (1987). Jones D. A., A. N. Yilmaz and B. E. Tilton, Practical synthesis techniques for retrofitting heat recovery systems. Paper 35c, AIChE Annl Mtg, Chicago (1985). Jones D. A., A. N. Yilmaz and B. E. T&on, Synthesis techniques for retrofitting heat recovery systems. Chem. Engng Prog. 82, 2833 (1986). Kendrick D. and A. Meeraus, GAMS: an introduction. Users Manual for GAMS. Development and Research Dept of the World Bank (1987). Linnhoff B. and E. Hindmarsh. The pinch design method for heat exchanger networks. C&m. Engng Sci. 38, 745761
r 1983)
Users Manual for ZOOM/XMP. The Department of Management Information Systems, University of Arizona (I 986). Murtagh B. A. and M. A. Saunders, lU1NQ.S 5.0 User’s Appendix A: MINOS 5.1. Technical Report Guide, SOL 8320, Systems Optimization Laboratory, Dept of Operations Research, Stanford University (1986). Papoulias S. A. and I. E. Grossmann, A structural optimization approach in process synthesisII. Heat recovery networks. Comput. &em. Engng 7, 707721 (1983). Paules G. E. IV and C. A. Floudas, APROS: Aigorithmic development methodology for discretecontinuous optimization problems. Opers Res. J. In press (1989). Taboo A. N., M. Morari and R. D. Co&erg, RESHEXan interactive package for the synthesis and anaiysis of resilient heat exchanger networksI. Program description and application. Comput. them. Engng IO, 577589 (1986). Tjoe T. N. and B. Linnhoff, Using pinch technology for process retrofit. Chem. Dzgng 93, 4760 (1986). TownsendD. W. and 8. Linnhoff, Surface area targets for heat exchanger networks. I Ith And Res. Mrg Heat Transfer, University of Bath. U.K. (1984). Yee T. F. and 1. E. Grossmann, Optimization model for structural modifications in the retrofit of heat exchanger networks. Report EDRC062587, Engineering Design Research Center, CarnegieMellon University (1987). Marsten R.,