Synthesis and Retrofit Design of Operable Heat Exchanger Networks

Ind. Eng. Chem. Res. 1994,33, 1718-1737

1718

Synthesis and Retrofit Design of Operable Heat Exchanger Networks. 1. Flexibility and Structural Controllability Aspects Katerina P. Papalexandri and Efstratios N. Pistikopoulos’ Centre for Process Systems Engineering, Department of Chemical Engineering & Chemical Technology, Imperial College of Science, Technology and Medicine, London SW7 2BY, United Kingdom

A systematic framework for the synthesis and/or retrofit of flexible and structurally controllable heat exchanger networks is presented in this paper. Based on a multiperiod hyperstructure network representation, explicit structural controllability (total disturbance rejection) criteria are developed and included within a synthesis/retrofit mixed-integer nonlinear programming model, where a total annualized (operating and investment) cost is minimized. The proposed framework results in a network structure which features minimum annualized cost, while being (i) flexible to operate within a specified range (or discrete set) of uncertain stream flow rates, inlet temperatures, and/or heat transfer coefficients and (ii) structurally controllable for a given set of disturbances and control variables, 1. Introduction

The design of heat recovery systems,where the integration of hot and cold process streams reduces the utility consumption for the plant cooling and heating requirements, and, thus, operating cost, has been one of the most mature research fields, with the main objective being maximum energy recovery at low investment cost. Grossmann et al. (1987), Gundersen and Naess (19881, Gundersen (1989),and Gundersen et al. (1991) give thorough reviews of the synthesis and retrofit attempts with economic (steady-state) objectives, through both the sequential and the simultaneous synthesis approaches. The uncertain environment of a chemical plant, however, introduces operability considerations in the economic evaluation of a process and, consequently, ita corresponding heat recovery system. Interactions between process design and process control have been proven to have a great impact on the economic optimality of a process design. The traditional practice of sequential design (optimization of the design based on steady-state economics, operability analysis of the economically optimal process and its heat recovery system, design of control system and retrofit remedies to overcome operability bottlenecks that the control system cannot compensate for) may lead to designs that are quite far from optimality. Consequently, operability considerations and requirements (such as flexibility, controllability, and reliability) become a second direction in comprehending the economical optimality of a design and have to be taken simultaneously into account. Flexibility aspects in the synthesis and retrofit design of heat exchanger networks (HEN) have been addressed as an independent problem or as part of the general problem of flexible chemical plants. Quantifying the ability of a design to accommodate uncertainty in the deterministic case, Swaney and Grossmann (1985) introduced a flexibility index, which defines the maximum parameter range that can be achieved for feasible operation. Grossman and Floudas (1987) presented an active set strategy for the efficient computation of the flexibility index of Swaney and Grossmann. The same authors, incorporating the flexibility index concept, developed a systematic two-stage iterative procedure for automatically synthesizing heat exchanger networks flex-

* Author to whom correspondence should be addressed.

ible to operate over a specified range of uncertain stream flow rates and inlet temperatures. A systematic, more straightforwardapproach to the problem of optimal retrofit of existing plants in order to improve their flexibility, though not directly applicable to the HEN retrofit problem, has been proposed by Pistikopoulos and Grossmann (1988,1989). Recently, Papalexandri and Pistikopoulos (1993) have integrated the two-stage iterative scheme of Floudas and Grossmann into an iterative retrofit scheme to improve the flexibility of heat exchanger networks at the minimum totalannualized cost, extending the synthesis of HEN without decomposition to the multiperiod case. Apart from the strict mathematical programming approach, a number of efforts have been made to incorporate flexibilityrequirements within the design of HEN. Saboo et al. (1985) proposed the resilience index to characterize the largest total uncertainty which a HEN can tolerate while remaining feasible. Colberg and Morari (1988) incorporated the resilience concept within a synthesis approach for flexible HEN, proposing a “flexibilityindex target”. Their approach can guarantee flexibilityonly for the convex case, but its ad hoc basis does not differentiate it from intuitive retrofit. Kotjabasakisand Linnhoff (1986) proposed an evolutionary design approach to exploit the trade-offs between flexibility, capital cost, and operating cost, based on information obtained from “sensitivity tables” and “downstream path” connections. Cerda and co-workers (Cerda and Galli, 1990; Cerda et al., 1990; Galli and Cerda, 1991) proposed a different synthesis methodology for obtaining flexible heat exchanger networks, introducing the concept of “transient” and “permanent” streams in order to describe variations in inlet temperatures and flow rates. A set of dominant pinch points is identified and the optimal heat load distribution is obtained through a mixed integer linear programming (MILP) problem. The network structure is then generated, and proper placement of heaters and coolers guarantees its flexibility. Operability considerations in heat exchanger networks are taken into account in the work by Mathisen et al. (1992). The four-way trade-offs between HEN investment cost, operating cost, flexibility,and controllability are explored through “optimal” bypass placements. A mixture of optimizationformulationsand heuristic rules are proposed to derive “optimal” heat exchanger networks which meet the operability criteria. Their approach, however, is

0888-5885/94/2633-1718$04.50/0 0 1994 American Chemical Society

Ind. Eng. Chem. Res., Vol. 33, No. 7, 1994 1719 limited to fixed matches, thus significantly reducing the space where the above trade-offs are exploited. Controllability considerations at the early design stages were attempted by several researchers, whereas a number of tools and controllability measures have been developed to aid exploring the interactions between process design and control. When little information about the process is available, structural controllability is the easiest to assess. The notion was introduced by Lin (1974) and makes use of structural matrices. Concepts of accessibility and disturbability were introduced focusing on the effectiveness of potential feedback control. Morari and Stephanopoulos (1980) define a system as structurally controllable based on the generic rank of ita structural matrix representation. Johnston and Barton (1985) proposed an algorithm to assess the structural controllability of single input-single output (SISO) systems, based mostly on inspection. Caladranis and Stephanopoulos (1986) proposed an operability analysis for HEN based on structural information, where uncertain network parameters are addressed as disturbances. Their approach is mostly designer-driven, whereas various guidelines are provided to reduce the problem dimensionalityand to handle the nonconvexities. Georgiou and Floudas (1989b,c, 1990) have proposed a mathematical linear programming (LP) formulation for the determination of the generic rank of a matrix and formulated mathematical conditions for feasible control systems from a structural controllability point of view. Thus, structural analysis even for large-scale systems can be carried out in a more systematic way. Daoutidis and Kravaris (1992) address the problem of structural evaluation of control configurations for multivariable nonlinear processes. They propose relative order as an analysistool, as it requires only structural information about the process. General structural evaluation guidelines are proposed based on the properties of relative order. Singular value analysis (SVA) was introduced by Doyle and Stein (1981) to assess the performance of a control scheme with less computational effort than dynamic simulations. Arkun et al. (1984) based their analysis on singular values to evaluate the robustness properties of control schemes. Johnston and Barton (1987) proposed a scaling procedure of the variables of the system model, so as to achieve efficient results through SVA on the control scheme performance and the sensitivity to modeling errors. Perkins and Wong (1985) introduced the condition number as a more general measure to relate errors in the process model to errors in the manipulated variables.Later, Skogestad and Morari (1987) formulated a disturbance condition number to measure the effect of rejecting disturbances. To evaluate alternative seta of manipulated and controlled variables, the relative gain array (RGA) (Bristol, 1966) is utilized and a number of pairing rules have been proposed (Bristol, 1966; McAvoy, 1983) to provide stable control schemes. In an attempt of simultaneously taking into account structural controllability and economic aspects, Georgiou and Floudas (1989a) introduced the variable structural matrix concept, based on a superstructure representation of the heat integrated alternatives. Their work on disturbance propagation in heat recovery systems can be used mostly as a screening tool during the HEN synthesis phase. A knowledge based engineering approach to the structural controllability of HEN is proposed by Huang and Fan (1988). They define a disturbance propagation table, where the relations between disturbances and

H1

n2

G

T f

Figure 1. Existing heat exchanger network.

controlled variables are heuristically quantified] and propose a structural controllability index, which then they incorporate in the HEN synthesis problem. Simultaneously consideringcontrollabilityand economic aspecta, Luyben and Floudas (1992) proposed a multiobjective optimization approach to include process controllability considerations,based on open-loop indicators, within the processsynthesismodel. In an attempt to assess the interactions between process design and process control, Narraway et al. (1991) combined conventional nonlinear steady-state optimization with linearized dynamics, steady-state economics, and functional controllability analysis to study the impact of disturbances on plant economics. The major difficulty of considering both process economics and controllability aspects during the design phase, when the structure and operating conditions of the process system have not been determined, is that the greater part of the developed control theory refers to closed-loop controllability (for fixed plant design and layout), where process dynamics and stability can be rigorously assessed, whereas even the open-loop indicators are based on linear models. An effort is made in this paper to incorporate both flexibility and controllability considerations in the synthesis and retrofit design of heat exchanger networks. Within an iterative design scheme for flexibility explicit controllability requirements are introduced. Thus, the trade-offs between operability] investment cost, and operating cost are properly exploited, whereas operability is realized through several aspects. In part 1in this series only structural controllability will be explored simultaneously with flexibility. The hyperstructure network representation will be exploited to express the inherent structural controllability characteristics of a network as a function of structural decision variables. Dynamics of the heat exchanger network will also be explored on a hyperstructure basis, and this is the subject of an accompanying paper (part 2). In the next section, the problem of introducing operability requirements at the design stage of HEN will be presented by means of an example. 2. A Motivating Example

To motivate the problem addressed in this paper, consider a HEN retrofit example (Georgiou and Floudas, 1989a). The existing network is shown in Figure 1. It involves two hot and one cold process streams integrated in two process exchangers, whereas a heater provides the additional required heat for the outlet temperature of C1 to reach its target. Stream data and exchanger data are given in Tables 1 and 2 (nominal operation). Process interactions result in the hot stream flow rates varying in the following range of uncertainty: 2 kW/OC IFH1I 4 kW/"C

1720 Ind. Eng. Chem. Res., Vol. 33, No. 7, 1994

Table 1. Stream Data for Motivating Example St-

HI H2 C1

Fcp (kW/OC)

P (OC)

3 4 6.8

250 360 65

'PM (OC) 115 200 310

Table 2. Excbanger Data for Motivating Example exchanger

area (m?

U(kW/(mZoC))

1

49.37 60.97

0.1 0.1

2

3 kW/"C 5 Fm 5 5 kWPC

Also, due to process specificationsthe outlet temperature of H2 must be maintained at its target value and, thus, definesacontrolledvariable.whereas the inlettemperature of H1 may vary within a smallrangeand at high frequency, constituting a 'disturbance" input. Inherent operability characteristics of the existing network are of interest, Le., if it is able to operate feasibly for the whole range of uncertainty of Fm and Fm and to reject the disturbance effect of on the specified control objective q i. Feasible operation of the network is defined by a minimum temperature approach ATrn, = 10 "C and heat exchanger areaadequacy. Taking intoaccount the energy balances around the heat exchangers and expressing TCl and TC2 (see Figure 1) as a function of the uncertain parameters, the following set of constraints result: fl = FH1 - 8.8148 5 0 f2 = FHI- 6.2963 5 0

f3 = FH1 + 1.1852Fm - 14.3536 5 0 f4 = FH1- 3.02774 5 0 f5=

+ 160(1- 0.1471Fm)) 5 0 - 6.097(-19.7059FH1 160(1- 0.1471Fm)

160FH2

ldl

+

135 -0.1471FH1

)

fl-f3 correspond to AT,, = 10 OC constraints, whereas f4 and f5 refer to area adequacy in exchangers 1 and 2, respeetively. f4 is a simplified expression of the initial constraint where the natural logarithm terms have been approximated. Aflexibilityanalysisofthenetworkreveals that itisnot ~ more than 0.022%to able to operate when F H fluctuates the positivedirection (FI=0.00022) duetoareainadequacy at exchanger 1 (see Figure 2, where the feasible region of the network is illustrated). The operating conditions of greater violation are defined by

FH1= 4 kW/K

3 kW/K 5 Fm 5 5 kW/K

We select the vertex of the feasible region to represent the points of greater violation:

FHl= 4 kW/K

F , = 5 kWlK

Furthermore, in the existing network, the outlet temperatureof H2 is 'disturbable" from the disturbance input This is obvious from the 'downstream path" (Kot(see Figure jabasakisand Linnhoff, 1986) from GIto

e,.

-

fI r O

mi 203

400

600

Figure 2. Uncertainty region and feasible operation. Table 3. Cost Data for Retrofit

annualizsd coat area

1200Aast/yr

piping

exchanger reassignment

loo00 $/yT 1OOOOO s/yT

stem

11.05 $/(kW yr)

1). This can also be proved more systematically through structural analysis of the network (Georgiou and Floudas, 1989c). The question arises on how to redesign the network so BS to be flexible over the specified uncertainty range and structurally controllable, as far as disturbance rejection is concerned and at the minimum retrofit cost, exploiting the trade-offs between operating cost and retrofit investment cost (in terms of area overdesign, bypass placement, exchanger reassignment,etc.). Cost dataaregiven inTable 3. In the following the problem of synthesizing and redesigning HENS will he addressed through an iterative scheme, based on the discretization of the uncertainty space. 3. Problem Statement

The problem to be addressed in this paper cnn be stated as follows: Given are 1. Stream integration data (i) In the case of synthesis, these involve a set E of hot process streams, a set C of cold process streams, and a set of hot and cold utilities HU and CU, respectively. (ii) In the case of retrofit design, the existing network isalsogiven,intermsofheatexchangers(asetXofexisting exchangers with known area EA) and network topology. The heat capacity flow rates F, inlet temperatures T' of process streams, and/or the heat transfer coefficients Uarenotfixed. Theyeithervarywithinarangeofvalues, IFLo,Fup), {TL, TLJ,{Urn,U q , respectively, or are described by discrete sets of values TI(?'), U ( n definingmultiperiod operation (in Tperiods). The outlet stream temperatures Toare fixed (either for each period of operation or uniquely) or constrained through some specification. 2. A specified range for the uncertain parameters, where flexibility of the networks is desired (flexibility target). 3. A set of controlled uoriables (outlet or intermediate stream temperatures). 4. A set of disturbance inputs (usually inlet stream temperatures).

F(n,

7,1994 1721

Figure 3. Hyperstructure network representation through binary variables.

5 . A minimum temperature approach AT^,, to determine feasible heat exchange in each heat exchanger (not utility consumption as HRAT). The objective is, then, the following: (i) in the synthesis case, to synthesize a heat exchanger network; (ii) in the retrofit case,to determine a retrofit design (a) at minimum total (synthesis or retrofit) annualized cost (operating and capital investment cost), (b) able to operate feasibly under the specified uncertainty, and (c) featuring total disturbance rejection with respect to the specified controlled variables and disturbances (structural controllability target). In this work feasible operation of heat exchanger networks under a specified range of uncertain parameters is assessed via the flexibility index evaluation (Swaney and Grossmann, 1985; Papalexandri and Pistikopoulos, 1993). The desired range of operation of the network defines a flexibility index target, F.Total disturbance rejection is defined by the existence of a break point in the downstream path from the disturbance inputs to the specified controlled variables, either physical or due to a manipulating variable. Disturbance propagation is explored through the structural matrix of the network (Georgiou and Floudas, 1989a). Dynamics of the network control scheme assessment will be examined in part 2, where a thorough controllability target will be defined. The following assumptions are made: 1. The heat exchangers are of the countercurrent type. 2. No phase changes are allowed. 3. Constant heat capacities are considered. 4. No hot-hot or cold-cold matches are allowed. 5. Area cost does not depend on the number of shells. This assumption could be relaxed through the approach of Trivedi et al. (1988). In the next sections, the synthesis and retrofit design problem with operability requirements will be discussed. Based on a hyperstructure network representation, an iterative framework to improve the flexibility of HENS (Papalexandri and Pistikopoulos, 1993)will form the basis for the design strategy. Explicit disturbance rejection requirements will be introduced as constraints within a mixed integer nonlinear programming (MINLP) model (for synthesis and the retrofit case). Example problems will be presented, to explore the trade-offs between flexibility, structural controllability, and operating and investment costs. 4. Network Structure Representation

The building block for the proposed synthesis and retrofit approach is the multiperiod hyperstructure net-

work representation, where all structural and operational alternatives are taken into consideration (Floudas and Ciric, 1989; Papalexandri and Pistikopoulos, 1993). The basic features of this representation are (see Figure 3) the following: 1. Each potential stream match corresponds to a potential exchanger unit, which, in the retrofit case, may be an already existing exchanger in the network or a new unit (to be purchased). 2. Each stream entering the network is split toward each potential heat exchange, the stream may contribute. 3. The possibility of multiple heat exchange between two streams is taken into account, i.e., multiple potential exchangers for a stream match are considered, whenever AT,i, constraints and temperature specifications suggest that the two streams might exchange heat more than once. This is not similar to an a-priori network partition into subnetworks, since the final number of subnetworks is not determined by a prespecified pinch point, but the economical evaluation of the multiple heat exchange compared to another heat load distribution. 4. An overall bypass flow is considered for each process stream. 5. Prior to each potential exchanger a mixer is considered for each stream, where the flow from the initial splitter and the bypass flows from all the other potential exchangers of the stream are merged into a flow toward the exchanger. 6. The stream flow toward the exchanger is further split before entering the exchanger, so that potential bypasses in each exchanger are taken into account. 7. After each heat exchange unit a splitter is also considered for each process stream, from which a potential flow is driven toward the final mixer of the stream at the network outlet, and bypass flows are considered toward the mixers prior to the other potential exchangers of the stream. Structural alternatives are represented through binary variables which correspond to the existence of a stream match, stream flow-piping segment, and exchanger and area assignment for the retrofit case (see Appendix A). The network hyperstructure is described by a set of mass and energy balances, at the splitters, mixers and exchangers of the network representation, feasibility constraints and design equations. The synthesis and retrofit models are described in detail in Appendix A.

1722 Ind. Eng. Chem. Res., Vol. 33, No. 7, 1994

6. Synthesis and Retrofit Design with Operability Considerations

The heat exchanger network hyperstructure can be described, as stated above, by a set of equations and inequality constraints, corresponding to mass and energy balances and AT- and area constraints. Let us denote by Ve,the set of the uncertain parameters of the network; U(Ve),the specified range of uncertainty where flexibility is desired; vd,the set of its design variables, Le., the heat exchanger areas and the variables defining the topology of the network (matches, piping layout, etc.); V I , state variables that describe the network operation; V,, control variables correspondingto degrees of freedom at the design stage; V,, control objectives (usu. stream outlet temperatures), a subset of V,; and Vi, disturbance inputs, where Vi (VI U Vel. Also, let h( v@,v,,V,,vd) be the set of equations describing the network hyperstructure and f(Ve,V,,V,,Vd) the set of inequality constraints. The synthesis and retrofit problems with flexibility and disturbance rejection requirements can be formulated as an optimization problem, of the following form: synthesis of HEN

subject to

retrofit of HEN min (total annualized retrofit cost) (RP) Avd*vmva

subject to

h(Ve,Vz,V,,Vd)= 0

V , undisturbable from Vi As has been shown (Halemane and Grossmann, 1983) the flexibility requirements (V Ve E V(Ve))in problems (SP)and (RP) are equivalent to an inequality constraint that involves a max-min-max problem:

subject to

h(V,V,,V,,V,) = 0

Disturbability of a controlled output x from a disturbance input d has been proven (Georgiou and Floudas, 1989a) to be equivalent to the condition that the generic rank of the structural matrix of the system be equal to the number of its active rows (see Appendix B). Thus, for the controlled variables V, to be undisturbable from the disturbance inputs Vi it suffices that GR(M(Vc,V,,Vz,Vi)] I AR(M(Vc,VI,Vz,Vi)}- 1,where M(Ve,VI,Vz,Vi)denotes the structural matrix of the network, G R its generic rank, and AR the number of its active rows. Consequently, problems (SP)and (RP) can equivalently be written as

min

subject to

(total annualized cost (synthesis or retrofit))

h(V,,V,,V,,V,) = 0

Note that problem (P)is an optimization problem which includes as constraints two other optimization problems, in the flexibility and disturbance rejection requirements. To overcome the direct solution of the first (inner) optimization problem in (P)for the flexibilityrequirement, an iterative scheme has been proposed in a previous work (Papalexandri and Pistikopoulos, 1993a). In this approach, starting from a network structure (existing one or, for the synthesis case, the cost-optimal network for nominal operation), critical operating conditions are identified for the current network structure (through a flexibility analysis step) and the network is resynthesized or retrofitted, taking into consideration all critical operating conditions through a synthesis or retrofit step. This synthesis and/or retrofit step is based on a multiperiod consideration of the network operation, where the following occur: (i) “Periods” of operation correspond to the critical operating conditions. (ii) All operational alternatives (heat loads, stream flows, temperatures) are defined for each period of operation. (iii)All structural alternatives are represented through the hyperstructure of Figure 3. (iv)Utility consumption is not prespecified. Design and operation decisions are driven by the minimization of a total (operating and investment) annualized cost. The second (inner) optimization subproblem in (P) for the structural controllability requirement is addressed as follows: First, the hyperstructure representation of Figure 3 enables the direct consideration of all possible control alternatives. Then, the multiperiod synthesis/retrofit MINLP modelisenlargedwithastructural controllability block, where an explicit disturbance rejection criterion is formulated and included. The next section presents in detail the structural controllability modeling aspects. 6. Structural Controllability-Total Disturbance Rejection

Consider the network of the small motivating example. When the retrofit strategy for flexibility only is applied, the network of Figure 4 is obtained, where the exchanger areas are increased with a cost of $25092.8 yrl and a flexibilityindexof 1. However, since the network structure did not alter, structural controllability did not improve is still disturbable from PA. and G! The disturbability of a controlled variable from a disturbance input can be assessed through structural analysis of the network, where only state variables and disturbancesare considered (Georgiou and Floudas, 1989~). In particular, the disturbance propagation can be analyzed on the basis of the generic rank of the structural matrix of the network (see Appendix B). In such an analysis, a controlled variable x is not disturbable from the disturbance input d if thegeneric rank of the network structural matrix, when the x column is not considered, is strictly less than the number of the active rows of the matrix. Based on this, a sufficient total disturbance rejection

Ind. Eng. Chem. Res., Vol. 33, No. 7,1994 1723 H1

H2

+ qt;”’

- yijsM I0 i E H P , j E Ri,s E IS (3)

+ qtLJJ+ qt!/’j

- yijsM I0 j E CP, i E Rj, s E IS (4)

qti?;o‘i + qt?? qQoj

T

Tt

Figure 4. Retrofittednetwork (flexibilityconsidered)for motivating example.

criterion may be developed, exploiting the hyperstructure network representation. To account for all structures embedded within the hyperstructure, the concept of the variable structural matrix, introduced by Georgiou and Floudas (1989a), is extended to the case of the hyperstructure network representation. In particular, a structural matrix is developed for the network hyperstructure model, where the rows and columns refer to the equations and variables of the model. The existence of these rows and variables in the structural matrix is connected directly to the network structural variables (that refer to the existence of stream matches, exchangers, bypass pipes, etc.). Three additional sets of binary variables are introduced: (i) binary variables to denote the existence of an equation row to the set of the active rows of the system when the generic rank is determined, p; (ii) binary variables to denote the existence of a variable column to the set of the active columns of the system when the generic rank is determined, q; (iii) binary variables to denote that a variable j is an output variable of a row i, r. These variables are defined to represent the arbitrary entries in the network structural matrix. For the case of heat exchanger network hyperstructure, where the disturbance and controlled variables are temperatures, the equations that describe the system are the energy balances for the hot and cold process streams. The variable structural matrix is illustrated in Table 4. The presence of manipulating inputs in the networks will be associated with the corresponding affected state temperature variable, which will then constitute a controlled variable with a nonactive column. Since the disturbance rejection criterion involves only controlled variables, disturbances, and state variables, manipulating variables such as stream flow rates and utility loads do not participate in the variable structural matrix. The existence of each “row” and each “column” in the matrix depends on the existence of the corresponding stream match in the network, whereas the presence of each variable in each equation depends also on the existence of certain piping segments in the network. For a certain vector of the structural binary variables, the generic rank of the corresponding network matrix can be evaluated through a simple maximization LP problem (see Appendix B). The explicit formulation of this LP is included within the synthesis/retrofit MINLP model. The structural matrix that corresponds to each vector of the network structural variables results from the following constraints: 1. Constraints (1)-(4) ensure that if a stream match does not exist in the network, the corresponding energy balances and temperature variables do not participate in the network’s structural matrix. pi?

+ p;? + pEtpe- yijsM I0 i E HP, j E Ri, s E IS (1)

where HP and CP are the sets of hot and cold process streams, Ri are R j the seta of possible matches for the hot and cold streams and IS is the set that includes the possibilitiesof multiple heat exchange between two streams (see Appendix A). 2. Similarly, constraints (5)and (6) state that if a bypass stream exists in a exchanger, the final outlet temperature of the stream is a manipulated variable and must not be considered in the disturbance rejection criterion. qtizoPi- (1- a$ I0

i E HP, j E Ri, s E IS (5)

qt,czoJ - (1 - aL) I0

j E CP,i E Rj,s E IS (6)

The outlet temperatures of utility exchangers are also controlled variables: @js

- (1 - yijs)I0

qtZoJ- (1 -yijs) I O

i E HP,j E CU, s E IS (7) j

E CP, i E HU, s E IS (8)

Also, when an input temperature is considered constant, its corresponding q variable is set to zero. When the structure is represented by the vector of the structural binary variables (y, w ,z, a, b) (see Figure 3) the number of the active rows of the structural matrix AR is equal to

where lHPl and lCPl are the cardinalities of HP and CP sets, respectively. Thus, for the structure to meet the disturbance rejection criterion the following condition must be true:

GR = max GR IA R - 1 p m

and the synthesis/retrofit problem for flexibility and disturbance rejection becomes min (total annualized cost (synthesis or retrofit)) Avd,vz,vz

subject to subject to

(PI max min max fj(V,,V,,Vz,V,) I0

V ~ U c v d V, i€4

h(V,,V,,V,,V,) = 0

GR = max GR IAR- 1 par

subject to

(1)-(9), (B.5)-(B.25)

To overcome the maximization problem involved in the disturbance rejection criterion, one should solve parametrically (with respect to the structural binary variables) the optimization problem for the determination of the

- energy balances hot-side ex cold-Bide ex hot-mixer b.e. cold mixer b.e. hot mixer 8.8. cold mixer a.e. hot final mix cold final mix

-

,.hoijr

r;y

thU IC p j r

,.hOij# ece

poijr

p j r

ti”;”j #he

ece

r”@j’ ahpa

,.cOijr ecpr

tF3J

thEOj

IC

IC

t;mJ

,.Ejr

q

T:

rf

Y j r *hpa

clij8 r*C*

qttf

Pijr

,.HIij

pij8 rcae

ehae

clWi’jd recaeij p W r ehpr

fZij8 =pr

p qtja hEOj

qt;JJ

qtfWJ

#Zij ecae

Pijr Pija Pija

rcFQijr ecpr cFQijr rce

qtpJ

Tp

ehe

ahae

qt$oj 0

t p

#A

oi rk rce Oj

qTf

qT7

qTf

#A

Pi”’

q q

b.e., before exchanger; a.e., after exchanger.

generic rank. Instead, we introduce a scalar variable u, such that min (total annualized cost (synthesis or retrofit) Avd,vmv*

GJ

M’u) subject to

subject to

Fluibility Analysis

max min max f,(V,,V,,V,,V,> I0 (P’) v ~ v ( v d Vr i€4

h(V,,V,,V,,V,) = 0 GRlu

(10)

GRIAR-1

(11)

Explicit DiaanbMaRejktion

crivrion

(1)-(9), (B.5)-(B.25)

The introduction of u in the objective function via a penalty parameter M’ (large positive number) penalizes any evaluation of GR less than the maximum. The will lead to minimization of the objective function of (P) the maximization of u and subsequently, due to constraint (8) to the maximization of GR,Le., a consistent evaluation of the generic rank. The penalty parameter M’can be evaluated on the basis of the optimality conditions of P and P’,so that the two problems are equivalent for each network structure (see Appendix C). On the other hand, due to (9), the obtained structure will feature structural total disturbance rejection. Replacing the max-min-max problem in (P’) by the iterative discretization scheme (Papalexandri and Pistikopoulos, 1993) results in a synthesis/retrofit design strategy for cost optimality-flexibility-disturbance rejection, which involves the following steps: 0. (i) Define a range for uncertain parameters where flexibility is desired (flexibility target). (ii) Define a set of controlled and disturbance variables. (iii) The operability target is d e f i e d by both the flexibility target and total disturbance rejection. (iv) Set the existing network as the current heat exchanger network structure (for the retrofit case). 1. Operability analysis. For the current network structure, solve the flexibility index evaluation problem (Papalexandri and Pistikopoulos, 1993). If the current structure is the existing one (in the retrofit case) or the cost-optimal network (in the synthesis case), evaluate the disturbance propagation, as described in Appendix B. (i) If the current structure meets the operability target, stop. Else, go to (ii). (ii) Identify critical operating conditions (periods).

-

Multipsiod

0 Rstroat synmcoip

(“New” srmfm)

2. Multiperiod synthesis/retrofit model with the explicit disturbance rejection criterion. (i) Generate multiperiod heat exchanger network hyperstructure and develop synthesis or retrofit model (PI). (ii) Solve (P’) to obtain “new” heat exchanger network structure. Go to step 1.

The overall iterative approach for flexibility and structural controllability is shown in Figure 5. 7. Computational Aspects

The multiperiod synthesis/retrofit model with the explicit disturbance rejection criterion constitutes a mixed integer nonlinear programming model (MINLP) which features not only a large number of continuous variables but also a large number of binary variables, after the introduction of the structural controllability variables. Generalized Benders decomposition (Geoffrion, 1972; Paules and Floudas, 1988) is employed for the solution of the MINLP synthesis and/or retrofit problem. The set of structural variables (y, w ,n, a, b, z, d , u, m) are selected as complicating variables. The MINLP is decomposed into two subproblems, a primal one (consisting of constraints (i) (1)-(9), (A.l)-(A.22), (A.331, (A.341, (B.5HB.25) for synthesis; (ii) (1)-(9), (A.l)-(A.22), (A.331, (A.36), (A.39), (A.43), (B.5)-(B.25) for retrofit), where the complicating variables are considered fixed, and a master subproblem, where the structuralvariables are determined (consistingof (i) (A.23HA.32) and the Lagrangian function of (1)-(9), (A.14), (A.15), (A.18)-(A.22), (A.33) for synthesis; (ii) (A.23)-(A.32) and the Lagrangian function of

Ind. Eng. Chem. Res., Vol. 33, No. 7,1994 1725

(1)-(9),(A.14),(A.15),(A.lB)-(A.22),(A.33), (A.36),(A.39) for retrofit). When the structural variables are fixed, constraints (1)(9)become simple bounds and the remaining constraints for the p , q, and r variables form a totally unimodular matrix (Georgiou and Floudas, 1989; Garfinkel and Nemhauser, 1972). This implies that these variables will take an integer value at the optimal solution and they can be relaxed to continuous variables. Thus, the primal subproblem can be solved as a NLP and the number of binary variables is greatly reduced. The multiperiod synthesis/retrofit model is a nonconvex MINLP, and thus, global optimality cannot be guaranteed with the traditional decomposition techniques; global optimization techniques can be applied to attain the solution of the primal NLPs (Floudas and Visweswaran, 1990, 1993). Note that in the proposed iterative strategy, which has been implemented through the modeling environment GAMS (Brooke et al., 1988),the multiperiod synthesisretrofit step results in an undisturbable structure, due to the explicit rejection criterion in (91,when GR is properly maximized. The consistent evaluation of the generic rank depends upon the proper selection of the penalty parameter M',in problem (P'). As shown in Appendix C when the structural variables are fixed, there is a critical value of M' for which the primal subproblems of (P)and (P') are equivalent. However, very large values of M' can lead to solutions that are far from economical optimality, which corresponds to the "small" part of the objective function. A proper value of M' can be obtained from the optimality conditions of (P)and (P') (see Appendix C) or by solving the multiperiod model for severalvalues of M' and applying the disturbability test to test the consistency of the solutions. Alternatively,the disturbance rejection requirement can be introduced as a screening tool during the synthesis iterations. As the structure is temporarily fixed, the LP generic rank problem can be solved to determine the generic rank of the current network structure. If the structure satisfies the disturbance rejection criterion, the network is optimized to determine the best economical operating conditions. Otherwise, the structure is rejected and a relaxed Lagrangian cut is derived from the solution of relaxed problem. 8. Examples

Example 1. Consider the network of the small motivating example (Figure 1). The retrofit model for flexibility and structural controllability is developed. Two periods of operation are considered, the nominal and the operating condition of greater violation of the network feasibility constraints:

Gl= 3 kW/"C G2= 4 kW/OC pi1= 4 kW/"C

Pi2= 5 kW/OC

The variable structural matrix is also developed, where is considered as disturbance and P;; as controlled variable. The MINLP retrofit model is solved via generalized Benders decomposition, as applied through the modeling system GAMS. The NLP primal featured 366 rows and 328continuous variables, whereas the master MINLP involved 118 rows and 81 variables, 67 of which were discrete. The solution was obtained in five GBD iterations (CPU time = 14.63 s on a SPARC-10 worksta-

H1

HZ

1

I

T: :T Figure 6. Retrofitted network (for flexibility and disturbance rejection) for example 1.

Ll

:T

I

T';

Figure 7. Retrofitted network (for disturbance rejection only) for example 1. H1

H2

G

Tt

Figure 8. Retrofitted network withno reassignment cost for example 1.

tion). The retrofitted structure is depicted in Figure 6.It features a bypass stream of H1 around exchanger 1 and increased exchanger areas.

A,' = 72.61 m2

A,' = 107 m2

The proposed retrofitted network features a total annualized cost (TAC) of $35092.8yr-l. In the retrofitted network the number of the active rows of the system is equal to 7,whereas the generic rank of the corresponding (M*lc=Tmwt, D) is equal to 6. Thus, the network features total disturbance rejection. When only total disturbance rejection is required no area increase is necessary and the bypass stream at exchanger 1 (with a TAC = $6900 yr-l of retrofit) is the optimal choice for an "undisturbable" network (see Figure 7). However, such a retrofit action would only lead to a flexibility index of 0.437. When reassignment of exchangers is not penalized, the retrofit strategy for flexibilityand structural controllability leads to the network structure of Figure 8,which is similar to the one of Figure 6 except from an exchanger reassignment (exchanger 1 is assigned to the match 23241 and exchanger 2 to the match H1-Cl). Note that in this case, the necessary area increase A{ = 61 m2

A,' = 118.6 m2

yields a less expensive retrofit than when exchanger reassignment is penalized (TAC = 34126 yr-l).

1726 Ind. Eng. Chem. Res., Vol. 33, No. 7, 1994 c2

c2

I

12 t W N n6OC

1 I3 t W N

TdC

I

I

2

z I300C

130%

t

z 155°C Figure 9. Existing network for example 2,

2

Figure 10. Proposed network (I) for example 2. c3

Table 5. Stream Data for Example 2 Z$ (oC) 249 206 38 126 93 30 300

stream no.

H1 H2 c1 c2 c3

cw

steam

F ("C) 121 66 130 182 195 40 300

Fcdy (kW/K) 145F518 14 10 12 13

H1 l : ; o c f c

#"< I21OC

195'C

CI

e2 12 kWK

10 k W K

126OC

/

cw H2

14 kWK c

c 66OC

205 OC

Table 7. Cost Data for Example 2 cost 5.031 $ kW-1 y r l 11.06$ kW-1 yrl 1200 $ yr' lo00 $ yrl loo00 $ y r 1 1200A"6 $ yr'

Example 2. When more than one control objectives and disturbances is involved, the disturbance rejection criterion is developed for each controlled variable. Consider the network illustrated in Figure 9. It involves two hot and three cold process streams. Flow rate, inlet temperature data, and bounds on outlet temperatures are given in Table 5. The existing network has resulted from the optimization of the energy recovery and the number of units at a first stage, and of the area capital cost (Georgiou and Floudas, 1989). Exchanger data are given in Table 8. We consider here a 10% overdesign for the heat exchanger areas. A minimum temperature approach of AT- = 10 "C is considered. The heat capacity flow rate of H1varies within the range: 14 kW/"C I Fcp" I 18 kW/"C The outlet temperatures of H1 and C3 are given control objectives, whereas the inlet temperatures of C1 and C2 are disturbance inputs. As can be seen in Figure 9, is not disturbable. However, P;; is disturbable from and Considering the flexibility of the existing network towards the variations of Fcp", a feasibility test reveals

F. el c2.

I3 kWK

2

Table 6. Heat-Transfer Coefficients. in kW/(m' K) c1 c2 c3 cw 0.17143 0.17 0.2222 H1 0.10909 0.1678 0.1875 0.16 H2 0.1

cooling water steam coat fiied cost of making a new match fixed cost of reassigning an exchanger piping cost area coat

195%

P 182OC

2

13OoC

Figure 11. Proposed network (11)for example 2 (when reassignment is not penalized). Table 8. Heat Exchanger Areas for Example 2 exchanger 1 2 3 4 area (existing) (m2) 137.8 83.2 79.6 194.2 area (flexible only) (m2) 137.8 83.2 79.6 530.35 new area I (m2) 168.43 83.2 79.5 194.2 new area I1 (m2) 137.8 83.2 79.5 237.8

5 104.6 104.6 104.6 133.7

that 10% overdesign of the exchanger areaswas not enough to compensate for the uncertainty in FcpH1. The multiperiod retrofit model for flexibility and total disturbance rejection is developed for three periods of operation, defined by the nominal and the two extreme values of Fcp". Retrofit cost data are given in Table 7. The model involves 1712rows and 1511variables, of which 225 are binary. The MINLP synthesis problem was solved on a SPARC-2 workstation, and the solution was obtained in 17 GBD iterations (CPU time 590 8). The obtained network is illustrated in Figure 10. It features a total annualized retrofit cost of TAC = $44703.56/yrand an average utility consumption of 1064kW. The areas of the heat exchangers are shown in Table 8. In the retrofitted network bypass streams are added at exchangers 3 and 4 and from exchanger 2 to 3 so that PGtis undisturbable (in structural terms). The area of exchanger 1 has been increased by 22%. Note that the multiperiod retrofit model for flexibility only results in the network topology

Ind. Eng. Chem. Res., Vol. 33, No. 7,1994 1727 H2

H1

ZOC

b

b

&-@-

c2

171'C

I

t

t

9oc

t

5oc

llST

Figure 13. Proposed retrofitted network for example 3.

0.081 K W / d K c U1 c0.U KW/m K

Figure 12. Existing network for example 3. Table 9. Stream Data for Example 3 stream no. ! I (OC) $ Tout (OC) H1 200 115 H2 155 90 c1 20 175 c2 20 175 Steam 300 300 ~~

~~

FcrJI' (kW/K) 60 60 20 50

~

Table 10. Retrofit Cost Data for Example 3 steam cost additional area cost

t

IISC

cost 11.05 $ kW-I yrl 350AA0.8$

of Figure 9, i.e., with no bypass streams and in an area increase of exchanger 4, as shown in Table 8. When stream repiping and exchanger reassignment is not penalized,the multiperiod MINLP retrofit model leads to the network of Figure 11. It features the same number of exchangers as the existing network, but streams C1 and C2 (disturbances) do not affect at all H1 and C3. Exchangers are reassigned and the areas of exchangers 4 and 5 are increased (stream matches H2-Cl and H2-C2 correspondingly) as shown in Table 8). The capital cost for the area increase is $21109.7/yr, whereas the average water consumption is 1018.33 kW. Example 3. The proposed synthesis/retrofit strategy for operability can be applied to the case of fouling, where heat transfer coefficients are uncertain. Consider the network of Figure 12 (Kotjabasakis and Linnhoff, 1987). It involvestwo hot and two cold streams. Stream data are given in Table 9, whereas the areas of the existing exchangers and the nominal values of heat transfer coefficients are shown in Table 11. The heat transfer coefficient between streams H1 and C1, depending on the temperature of stream C1, varies as shown below: 0.081 kW/(m2K)5 UHl-cl 50.12 kW/(m2K) The outlet temperatures of H1 and C1 define two control objectives, whereas the inlet temperature of stream H2 is a disturbance input. As can be seen in Figure 12, both in the existing control objectives are disturbable from structure. The multiperiod retrofit model with total disturbance rejection requirements is developed for the two periods defined by the extreme values of UHl-Cl. Steam at 300 OC is considered available as hot utility. Operating cost and investment cost data are given in Table 10. A AT,k = 5 "C is specified. A high piping cost of 1000 yr-l is also considered, so as to account for pressure drop considerations and to avoid extensive stream splitting.

c2

Table 11. Exchanger Areas for Example 3 exchanger exchanger U (kW/(m2K)) no. area (m2) 1 305 0.12-0.081 2 600 0.225 3 63 0.15 4 173 0.36 Table 12. Stream Data for Example 4 stream FCPN no. IF" (OF) Tout (OF) (kBtu/(h OF)) H1 320 1200 16.67 H2 480 1280 20 H3 440 1150 28 H4 520 1300 23.8 H5 390 1150 33.6 c1 140 1140 14.45 c2 240 1431 11.53 c3 100 2430 16 c4 180 2350 32.76 c5 200 2400 26.35 steam 456 456 water 100 5180

new area (m2) 305 600 253.4 173 cost ($kBtu-' h yr')

11.05 5.031

The retrofit model features 956 rows, 736 continuous variables, and 165binary variables. The model was solved in 10 GBD iterations (considering fixed matches), consuming 190.5s of CPU time on a SPARC-10 workstation. The retrofitted structure is illustrated in Figure 13. The new areas are given in Table 11. The structure features a total annualized cost of $ 1 7 6 0 0 and ~ ~ an ~ average utility consumption of 763.6 kW. Note that the minimum utility consumption, based on an overall minimum temperature approach (HRAT)of 5 "C, is 1850 kW. Exploiting the network interactions and the operating-capital cost tradeoffs, better energy distribution is achieved. In the retrofitted network the area of exchanger 3 (not the fouling exchanger) is increased by 190.4 m2. Due to the existence of the hot bypass in exchanger 3, TD;;. and Fl are undisturbable in structural terms from T&. Example 4. To illustrate the performance of the proposed MINLP retrofit model for larger scale problems, consider the example problem lOSPl (Pho and Lapidus, 1973; Papalexandri and Pistikopoulos, 1993), where uncertainty has been introduced in stream flow rate and inlet temperature. It involves a system of five hot and five cold streams, with heating utility provided by steam a t 456 O F and cooling water 100-180 O F as cooling utility. Stream data for the nominal operating condition and utility cost data are given in Table 12. The existing network is illustrated in Figure 14. Heat exchanger areas are given in Table 14. A heat transfer coefficient U = 0.15 kBtu/h ft2 O R ) is given. Feasible operation of the network is defined by a ATmk = 20 O F . Due to process variations the heat capacity flow rate of H4 varies by f 2 kBtu/(h OF) around ita nominal value,

+

1728 Ind. Eng. Chem. Res., Vol. 33, No. 7, 1994

H1

c3

I c4

c4

I

I

cw

w

cw

e5

I

Figure 16. Retrofitted network for example 4.

Figure 14. Existing network for example 4. Table 13. Retrofit Cost Data for Example 4 additional area cost reaaaignment cost piping cost

cost 350AAO.e $ yr' 5oo$yr' loo00 $ yr'

Table 14. Exchanger Areas for Example 4 exchanaer no. exchanger area (ft2) 1 165 320 2 1415 3 369 4 260 5 15 6 620 7 472 8 157.2 9 185 10

new area (ft2) 165 320 1415 369 260 15 620 472 157.2 185

whereas the inlet temperature of C5 varies by f10 O F . The inlet temperature of C2 varies slightly at high frequencies defining a disturbance input. The outlet temperatures of C1 and C5 must be maintained at their specified values, despite the variations of As can be seen in Figure 14, there is a downstream path ato and i.e., the control objectives are from p disturbable from pa..Furthermore, a flexibility analysis reveals that the existing network is able to operate in only 56% of the uncertainty region of F H and ~ !I$ due !& to AT- constraints in exchanger 2. The multiperiod MINLP retrofit model is developed for the part of the network that limits flexibility (exchangers 1,2,5-10). Retrofit cost data are given in Table 13. Five periods of operation are considered, defined by the nominal operating point and the vertices of the uncertainty region. The model involves 6275 rows, 4318 continuous variables, and 642 binary variables (fixed matches are considered). The retrofitted network is illustrated in Figure 15. It was obtained in four GBD iterations (2356 s of CPU time on a SPARC-10 workstation). Note that the CPU time is comparable to the CPU time required for the solution of the retrofit problem without disturbance rejection considerations. It features atotal annualized cost of $39945/yr,whereas the exchanger

v:

Fl,

la

Y

Figure 16. Alternative network structure for example 4.

areas are not increased. A cold bypass is added to exchanger 5, to exploit heat load distribution of C2, and a hot bypass is added to exchanger 9 for totaldisturbance rejection. An alternative network structure (intermediate during GBD iterating) is illustrated in Figure 16; it also features total disturbance rejection, however is has a high repiping cost (TAC' = $60551/yr). 9. Conclusions

The synthesis and retrofit design of heat exchanger networks with flexibility and structural controllability requirements were addressed in this paper. An iterative framework has been proposed, based on a multiperiod synthesislretrofit model, where structural controllability criteria are explicitly included. Disturbance propagation has been explored employing the variable structural matrix concept, with which a tdaldisturbance rejection criterion has been developed. Efficient decomposition techniques have been applied for the solution of the MINLP models, where the problem structure is exploited to reduce the complexity and the size of the resulting subproblems. The simultaneous consideration of operability aspects and overall cost optimality at the synthesis andlor retrofit

Ind. Eng. Chem. Res., Vol. 33, No. 7, 1994 1729 design level accounts properly for the trade-offs between structural controllability, flexibility, operating cost, and investment cost. Total disturbance rejection at the level of structure implies improved controllability; usually, however, as illustrated with the example problems, this is achieved with a large increase of the retrofit cost. A question then arises whether other, less strict, controllability requirementa would compromise an acceptable inherent controllability with a reasonable retrofit (or synthesis) cost. In part 2 this question will be addressed.

here is not strict as in pinch technology. The term is used to denote the possibility of multiple heat exchange between two process streams, so that the multiperiod hyperstructure can then include all the network configuration alternatives. Multiple heat exchange at the final retrofitted network is determined by the solution of the model and is dependent on AT-.) When a fixed HRAT is used, then “subnetworks”have the classical meaning as used by Floudas et al. (1986). For the stream matches define the following seta:

PMt = ((ij)/the pair ( i j )can be present at period of operation t]

Nomenclature Ve = set of the uncertain parameters of the network U(V0) = specified range of uncertainty where flexibility is desired Vd = set of ita design variables V, = state variables V, = control variables V, = controlobjectives (usually stream outlet temperatures), a subset of V , Vi = disturbance inputs p = binary variable to denote the existence of equation rows in structural matrix q = binary variable to denote the existence of equation columns in structural matrix r = binary variable to denote arbitrary entry in structural matrix a = structural (binary) variable to denote the existence of a bypass b = structural (binary)variable to denote the existence of an overall bypass GR = generic rank AR = number of active rows A = exchanger area DT = temperature difference d = binary variable to denote exchanger assignment f = flow M = large positive number m = binary variable to denote exchanger purchase Q = heat load s = subnetwork t = period of operation t = temperature U = heat transfer coefficient u = binary variable to denote area assignment w = structural (binary)variable to denote the existence of an inlet pipe x = structural (binary) variable to denote the existence of a bypass y = structural (binary) variable to denote the existence of a match z = structural (binary)variable to denote the existence of an outlet pipe Appendix A. Multiperiod Hyperstructure Model To formulate the multiperiod hyperstructure model, the following sets are introduced

H T = H ~ H CT=C U C U HCT = (HT)

u

(CT)

T = (t: period of operation) Also, denote by 1st the set of potential subnetworks at each period of operation. (The concept of subnetworks

R, = &’/if k E HP then k’ E C and (k,k’) E PM, or k E CP then k’ E H a n d (k,k’) E PI@ Skkt

{k”/k‘ E Rk,k” E R, and the bypass flow of k from (k,k’) to (k,k”) is possible] The multiperiod hyperstructure model involves the following variables and parameters: Binary-Structural Variables Ykk’s denotes the existence of amatch between the streams

k and k’ in the “subnetwork” s. w:,, denotes the existence of a piping segment between the initial splitter of the process stream k E HCT and the exchanger (k,k’,s),where k’ E Rk. zka denotes the existence of a piping segment between the

outlet of the exchanger (k,k’,s)and the final mixer of the process stream k E HCT,where k’ E Rk. a:,* denotes the existence of a bypass piping segment at the exchanger (k,k’,s) (for the hot or the cold side, akk,s, h a;k,s,respectively). denotes the existence of a piping segment for the bypass flow of process stream k E HCT from exchanger (k,k”,s’) to exchanger (k,k’,s),where k’ E R k and k” E Skk’.

bk denotes the existence of an overall bypass for stream k E HCT. Continuous Variables and Parameters Qijst is the total heat amount that hot stream i exchanges with cold stream j a t subnetwork s E IS,in period t. is the flow of stream k E HCT that goes from the initial splitter of the stream directly to the match (k,k’), where k‘ E Rk in subnetwork s and period t. Ck is the flow of stream k E HCT that goes from the initial splitter of the stream directly to the final mixer of the stream, in period t. is the flow of the stream k E HCT that goes through the exchanger of the match (k,k’), where k’ E Rk, in subnetwork s and period t. is the flow of stream k E HCT that exits the mixer prior to the exchanger of the match (k,k’),in subnetwork s and period t, where k‘ E Rk. is the flow of stream k E HCT that bypasses the exchanger of the match (k,k’),in the subnetwork s and period t, where k‘ E Rk. f;,$f‘ is the bypass flow of stream k E HCT from the splitter after the heat exchanger (k,k”) in subnetwork

f??

f$Fkc

1730 Ind. Eng. Chem. Res., Vol. 33, No. 7,1994 s’ to the mixer prior to exchanger (k,k’)in subnetwork

s, at period t. gp is the flow of stream k E HCT that goes from the splitter after the exchanger of the match (k,k’) in subnetwork s to the final mixer of the stream, in period t. tf;;2“is the temperature of stream k E HCT before the exchanger corresponding to the match W’),in subnetwork s and period t. tcf” is the temperature of stream k E HCT after the exchanger corresponding to the match (k,k’),in subnetwork s and period t. tc?&)lr is the temperature of stream k E HCT after the mixer, which is considered after the exchanger of the match (k,k’)in subnetwork s and period t. DT1ijst is the temperature difference between hot stream i and cold stream j at the inlet of the hot stream to the exchanger of the match (id),in subnetworks and period t. DT2ijat is the temperature difference between hot stream i and cold stream j at the outlet of the hot stream to the in subnetworks and period exchanger of the match (id), I

L.

LMTDijst is the log mean temperature difference between hot stream i and cold stream j at the exchanger of the match (ij)in subnetwork s and period t. Aija is the area required for the heat exchange between the hot stream i and the cold stream j , where i E HT and j E Ri, in subnetwork s, which will be the final area of the corresponding exchanger. Uijt is the overall heat transfer coefficient of match (id) in period t. AV!mzis the largest possible temperature drop through tge exchanger of the match (ij)in subnetwork s and period t, equal to Topi The mathematical formulation of the multiperiod hyperstructure model consists of the following. 1. Mass balances for each stream at the splitters and mixers of the hyperstructure, for each period of operation, are given by

,b t!& - t o i d )

Qijat -et (

=0

1.t

j,t

i E H, j E R i , s E I S , t E T (A.6) Qjjst- tp(t:p- t:,?) = 0 j E C, i E Rj, s E IS, t E T (A.7) 4. The following equations define the log-mean difference between two streams that exchange heat. DTlijat= tz1,t ”’ - to” ht

i E H , j E Ri, s E IS, t E T

DT2ijat= tosi” I,t - trJc ht

i E H , j E Ri, s E IS, t E T

64.8) (A.9)

s

E IS, t E T (A.lO)

To avoid computational awkwardness, the log-mean temperature difference is approximated by the formula proposed by Patterson (1984). 5. Design equations for the heat exchangers are A, 1

Qijst

UijrLMTDijst

i E H , j E R,, s E IS, t E T (A.ll)

6. Energy balances at the mixers of the hyperstructure of each process stream (i) after the exchangers and (ii) at the final mixer, for each period of operation, are given by

e.

k E HCT, k‘ E Rh, s E I S , t E T (A.2)

cy - gy - fy”= 0 k E HCT, k’ E Rk, s E IS, t E T (A.3)

k E HCT, k‘ E R h , s E I S , t E T (A.4) 2. Energy balances at the mixers of the hyperstructure

of each process stream prior to heat exchangers, for each period of operation, are given by

k E HCT, t E T (A.13) 7 . The following constraints give an upper bound to the flow of a stream to and through an exchanger respectively and ensure that such a flow is not present if the corresponding match does not exist &ijs = 0). If the match (ij)is selected, then the flow of the hot and the cold stream through the exchanger is bounded from a minimum based on the heat load and the corresponding maximum temperature drop. This is described by the constraints in (A.14)-(A.16). The constraints in (A.17) account for feasible operation for each period of operation.

fTPis 1tt - FiStyijsI 0

{p - F’”yij8 I0

=0

k E HCT, k‘ E Rk,s E IS, t E T (A.5)

3. Energy balances over the exchangers for the hot and cold streams, for each period of operation, are given by

j E C,i

E Rj, s E IS, t E T (A.14)

fE,‘”-FiptyijaSO J.t i E H , j E R , s E I S , tE T

ffp- FiityijaI0

j E C , i E Rj, s E IS, t

E T (A.15)

fp--Qijst

20

iEH,jERi,sEIS,tET

tp-- Qijst

L O

j E C, i E R j , s E IS, t E T

AT;-

f$”t;ff

i E H, j E R , s E IS, t E T

Aq;-

(A.16)

Ind. Eng. Chem. Res., Vol. 33, No. 7,1994 1731 DTlijSt1 ATmin i E H , j E Ri, s E I S , t E T DT2ij8t1 ATmin j E C , i E Rj,s E I S , t E T

(A.17)

8. The following constraints provide an upper bound to stream flows to and from heat exchangers and bypass flows. When the corresponding piping segments do not k k exist in the retrofitted network (wk0 = 0,Zkrs = 0, 0, X:,ktJs = 0 , or bk = 0 ) ,these flows are forced equal to zero.

- w:,,M

ffSt

I0

k E HCT, k’ E R,, s E I S , t E T (A.18)

e:t- z:,jt4 $5’-

k E HCT, k’ E Rk, s E I S , t E T

I O

(A.19)

k E HCT, k’ E Rk, s E I S , t E T

a:,jt4 I O

then it leaves the network only from that exchanger. k k Zk/s-ykt/st

u

k E HP CP,k’ E Rk n (HPor CP),k” E R, (HU or C V ,

I0

(A.32)

13. Constraint (A.33) defines the existence of a match

unit: Qijst - MyijsI

O

( i j )E PM, t E T

(A.33)

Synthesis of heat exchanger networks is driven by the minimization of the total annualized synthesis cost (operating and capital investment cost). In particular, the terms of the objective function are defined as follows: 1. The operating cost of the hot and cold utility consumption, CHUi, i E HU, and CCUj, j E CU, respectively, is a term of the objective function:

(A.20)

c&tss,t -

Xk,k,,ss,M k

I0

k E HCT, k’ E R,, s, s’ E IS, t E T (A.21) &-b#IO

kEHCT,tET

(A.22)

9. Constraints in (A.23)-(A.27) ensure that if a match does not exist (Ykk’s = 0), the piping segments connected with the corresponding exchanger do not exist either: k

WkJs

- Ykkta I0

k

zkts - Y k k J s IO

- Y k k t 8 I0 k

-

k E HCT, k‘ E Rk, s E Is

(A.23)

k E HCT, k’ E Rk, S E IS

(A.24)

k E HCT, k‘ E Rk, s E Is

(A.25)

I0 k E HCT, k’ E R,, k” E S k k ’ , s, s’ E IS (A.26)

Xk/k/?ss/ Ykkt8

10. Constraint (A.28) prevents cyclic piping between two heat exchangers of the same process stream, while constraint (A.29) prevents the presence of multiple bypasses when the process stream exchanges heat only once.

2. The cost of purchasing an exchanger includes the cost of the required area plus a standard cost NE of installation. Let the cost of purchasing area A be given by a function aAb. Then the relative term in the objective function will be

3. Piping cost is assumed to correspond to a standard cost for each piping segment. Pressure drop considerations are not explicitly taken into account. However, relatively high piping costs drive the network toward simple piping layouts. Therefore, the total annualized synthesis cost to be minimized will be

fi.im\

k

(A.34)

k E HCT (A.29) 11. Constraints in (A.30) and (A.31) ensure that if a process stream contributes to a heat exchange (k,k’,s)then the stream enters and exits the exchanger.

k E HCT, k‘ E R,, s E IS (A.30)

k E HCT, k‘ E R,, s E IS (A.31) 12. Constraint (A.32) ensures that if a hot or cold stream exchanges heat with a cold or hot utility, respectively,

Constraints (A.l)-(A.33) along with the objective function in (A.34) define the mixed integer nonlinear programming (MINLP) model for the multiperiod synthesis of heat exchanger networks without decomposition. It should be noted that restrictions on the network structure (e.g., no stream splitting requirements) and/or on undesired (forbidden) matches (due to layout restrictions) can be easily included in the model. In this respect, the user has the flexibility of interacting with the model and simplifying the solution procedure as well as the network structure. Stetrofit of heat exchanger networks is driven by the minimization of a total annualized retrofit cost. That is, only structural modifications are penalized, when investment cost is concerned. In the retrofit problem, however, the existing equipment can be utilized. Existing heat exchanger units can be assigned to existing or new matches; the area of existing exchangers can be assigned to the

1732 Ind. Eng. Chem. Res., Vol. 33, No. 7, 1994

location of different exchangers. Thus, the retrofit problem requires the introduction of the following variables: Binary Variables dCs is the decision variable to denote the assignment of existing exchanger n to the match ( i j , s ) . VI,,is the decision variable to denote the assignment of the area of existing exchanger n to the location of existing exchanger 1. nijs is the decision variable to denote the purchase of a new exchanger unit for the match ( i j , s ) . Continuous Variables AAn is the additional area to be purchased for existing exchanger n. AEA, is the area (from existing exchangers) allocated to exchanger unit n. NAijs is the area of the new exchanger purchased for the match (i4,s). The match-exchanger assignments in the retrofit problem are modeled as follows: 1. (A.35) ensures that for a match in a subnetwork s either an existing unit will be assigned or a new one will be purchased. When a match ( i j )is not selected @ijs = 0), then neither an existing unit is assigned to it (d;s = 0) nor a new one is purchased for it (nijs = 0). dCs + nijs- yij8= 0

(A.35)

2. (A.36) defines the total existing area assigned to the location of an exchanger n. Constraints (A.37) and (A.38)

P(EA,~,,)x X

AEA, =

nE

(A.36)

5. (A.42) gives an upper bound to the matches present in each period of operation with a maximum number of unitsMAXU, equal to the number of process streams and utilities plus one.

The notation of the binaries d and n over subnetwork cover for the case when a match is present in several subnetworks and more than one exchanger must be assigned to it (existing or new). The objective function of the retrofit problem involves the following: 1. The operating cost of the hot and cold utility consumption, CHUi, i E HU, and CCUj, j E CU, respectively, is given by

2. The cost of purchasing a new exchanger, which includes the cost of the required area plus a standard cost NE of installation, taken into account. Let the cost of purchasing area A be given by a function aAb. Then the relative term in the objective function will be

a(NAijs)b+ NEn,

=1

NA,

+ (1- nijs)UBND2 A ,

N A , - n,UBND 5 0

i

i E HT, j E Ri, s E IS (A.37)

E HT, j E Ri, s E IS (A.38)

define the total area NAijs of an exchanger that needs to be purchased for the match (i,j)s. If nijs = 1,then (A.37) gives a lower bound to the area to be purchased, which is minimized due to its cost in the objective function, while if nijs = 0 (A.38) forces NAijs = 0 and (A.37) is relaxed. Constraint (A.39) gives a lower bound to the additional area that needs to be purchased for exchanger n if this is assigned to match ijs (when d$ = l),while if dB = 0 it is relaxed. AA,

+ AEA, + (1 - d?JUPBND 1 Aij8 n

E X , i E HT, j E R , s E IS (A.39)

3. (A.40) ensures that the area of an existing exchanger n will either be assigned to the location of the exchanger nor to the location of another existing exchanger or it will be removed from the network.

.,

A

CV,,Il

IEX

(A.40)

nil

4. (A.41) ensures that exchanger n is assigned to a t most one match; heat exchanger or ita area is assigned to the location of existing exchanger 1 or it is removed from the network per period of operation.

3. The cost of purchasing additional area for an existing exchanger, which will be activated if the exchanger is assigned to a match, taken into account. It includes a standard installation cost NEA and the cost of the required area:

aAA,b

+ NEAd!s

4. The cost of reassigning an existing exchanger to a different match, or the same match in a different subnetwork, over the periods of operation is given by

5. Assigning the area of an existing exchanger to the location of another existing exchanger is penalized with a cost CAAI,:

6. Making a new match is penalized with a cost CNMijst, to account for potential stream reallocation. 7. It is assumed that repiping cost is independent of exchanger reassignments. The contribution of repiping cost to the objective function is considered to be that proposed by Yee and Grossmann (1991). Therefore, the total cost of structural modifications including operating cost to be minimized will be

Ind. Eng. Chem. Res., Vol. 33, No. 7, 1994 1733

H1 p 4 C l:rc=mwrc

NEn,,

+

{ u A A , ~+ (NEA + C;)d$

QS

H2

+

n

+

CAAInvln}+ CNMij8+ [CP][W;.~ + #n

k'€w

xi?' s'€

s

+ z? + dl]yijs)(A.43)

ex1

TCl

X X

X X X X

X X

X X X

w2

w3

w4

ex 2

Constraints (A.l)-(A.42) along with the objective function in (A.43) define the mixed integer nonlinear programming (MINLP) model for the multiperiod retrofit problem without decomposition.

P$

TC2

.-

q :

x x

ex 3 w1

el

-~

Gf

-

u1

uz V3

u4 X w6

V5

w6

Appendix B. Disturbance Propagation in HEN B.l. Disturbance Propagation Analysis. Network Representationthrough Structural Matrices. A heat exchanger network can, in general, be described by a set of differential and algebraic equations, of the form

x = Ax + Bu + E,g + D,d 0 = Cx

+ Fu + E,g + D,d

where x is the vector of the state variables, u is the vector of manipulated variables, g is the vector of algebraic variables, and d is the vector of disturbances. For the steady-state operation, which is considered at the design level, x = 0. In general, the entries of the matrices, Anxn, Bnxm,Elnxr,Dlnxp, CsXn,Fsxm,E2sxnand b x p are nonlinear functions of the variable vector ( x , u,g, d). On the basis of such a representation, astructural matrix M is defined as the (n+ s) X (n + m + r p ) matrix the entries of which are fixed to zero where the corresponding entries of the matrix

+

max

Generic Rank. Generic rank of a structural matrix Mnx, is the maximum rank that achieves the matrix as a function of its arbitrary (nonzero) elements. The determination of the genericrank can be formulated as a linear programming problem by defining three types of variables (Georgiou and Floudas, 1989):

(id

yij - vi =

o

i = 1, ...,n

(B.1)

i = 1,..., m

03.2)

j = l , ...,m

i = l ,...,n

yii- wj I O

ui=GR i=l,...,n

1: S:l are zero or otherwise take arbitrary values (Georgiou and Floudas, 1989a). M* is the (n + s) X (n + s) structural matrix with (i) its diagonal elements arbitrary (nonzero) entries and (ii) its nondiagonal elements equal to zero or nonzero at the same locations as the nondiagonal zero or nonzero elements of M, respectively. The structural matrix that remains after the elimination of the cth column of M* is denoted as M*J,. Similarly, matrix D is defined as the structural matrix corresponding to

yij

ij

j=l,...,m

w j = GR

In the above formulation, the coefficients of variables, a totally unimodular matrix, as proven by Georgiou and Floudas (1989) and, thus, the variables can be relaxed to continuous, as they will have an integer value at the optimum solution. Hence, the problem can be solved as an LP. Disturbability. A controlled variable x is called disturbable from the input disturbance d if there is a pathway connecting these variables in the bipartite representation of the system. Georgiou and Floudas have proved (1988) that this is equivalent to the generic rank of (M*lcix, D) being equal to the number of active rows of the system, i.e., the generic rank of (M*, D). Consider for example the network of Figure 17. The inlet temperature of stream H1 is considered as disturbance, whereas the outlet temperature of stream H2 is the controlled variable. The augmented structural matrix of the network is illustrated in Table 15. It must be noted that the process stream exchangers are represented by two similar rows in the structural matrix, corresponding vi, W j and yij form

1734 Ind. Eng. Chem. Res., Vol. 33, No. 7,1994 ,,cOiju + ,.hOijs + ,.cIijs + ,.hIijs - pece ece ece VS ece ece (ifiEHP) (ifiEHP)

=0

j E CP, i E Rj,s E IS (B.lO) $rOijs+ jJii=l

?z

6 -pp =o

rg+

i EHP

(B.II)

(if bi=1)

:T

Figure 18. Retrofitted network.

Table 16. Structural Matrix for Network of Figure 18 TCl G' TC2 ! I $ $

zi

X

X X

I ex21

:

X X

X X

X

JEi I

to the two differential equations for the outlet temperatures of the process streams. Manipulated variables, as utility loads, and constant inputs (for example the inlet temperatures of H2 and C1) are not included. The generic rank of matrix (M*, D)is found equal to 5, whereas the generic rank of matrix (M*L,s, D)is 5 as well. Thus in the existing network, the outlet temperature of H2 is disturbable from the input PA. Consider then the retrofitted network of Figure 18. The augmented structural matrix, (M1*, D),is illustrated in Table 16, where P'; is not included since it can be controlled with the bypass at exchanger 1. The generic rank of (Ml*, D)equals 5, whereas the generic rank of (M1*Ic,3, D)is equal to 4; this implies that PGi is not disturbable from the input T&. B%. Generic Rank of a V a m b l e Structural Matrix. For a certain vector of the structural binary variables, the generic rank of the variable structural matrix can be evaluated through the following maximization problem (see also Table 3):

iEHP

r E - q c I O

(B.15)

i E HP, j E Ri,s E IS (B.17) ,.cIije + ,.cIijs ehe ece (ifiEHP)

cIij8 + ,.cIij8 ecpe (if d*=U

+ recae

- qt;J,j 5 0

j E CP, i E Rj,s E IS (B.18) ,.hOiju + ,.hOijs ehe

(if&)

+ ,.hOijs

ehpe

- qt20,i 5 0 i E HP, j E Ri, s E IS (B.19)

,cOijs + ,.cOijs ehe ece (ifiEHP)

+ ,.cOijs

ecpe

-qtzOj

5

0

j E CP, i E Rj,s E IS (B.20)

max GR ijs

subject to (ift'j,=l)

i E HP, j E Ri, s E IS (B.21) i E HP, j E R , s E IS (B.5) (i',a')

C% +

,.cFOi' ecaejd

+

(i'&

f ecae j (ifdipl)

&g,= 1

p ! y =0

j E CP, i E Rj,s E IS (B.22)

Ji'u''1

j E CP, i

E

Rj,

s E IS (B.6)

i E HP, j E Ri, s E IS (B.7) ,.cIije cOija + ,.cFOije eFpe + recpe ecpe (ifdpl)

- peer = 0 j

E CP, i E Rj,s E IS (B.8)

,.hOib + ,.cOija + ,.hIijs + ,.cIije ehe ehe ehe ehe (ifjECP) (ifjECP)

-p;y = 0

i E HP,jE R , s E IS (B.9)

(B.24)

Ind. Eng. Chem. Res., Vol. 33, No. 7, 1994 1735 h(x) = 0

g(x) 5 0

GR = max GR I AR - 1 Par

GR = c r i j = 0 51

+

(ifWib=l)

,.cIijs ecpe (if a$,=l)

eOija + rcFOij8 + rc0ij8 + + recpe ecpe eee

ece

Crij- qj Io

+

t

(if iEHP)

C r i j - p i= 0 I

0 Irij I1 OIp,Il

oIqjIl Appendix C. On the Equivalence of Problems (P) and (PI) The multiperiod synthesis/retrofit model (P)can be written, in general as follows:

F’@,q,r) I0 Introducing the penalty parameter M‘ in the objective function to overcome the inner maximization of GR, the corresponding primal subproblem of (P’)can be written as min(f(x) X,U - M’u)

subject to

(P’primal)

subject to h(x) = 0

GRlu

Crij- qj Io Crij- pi = 0

GR-grij=O

1

tl

C r i j- qj Io

I

I

crij-

0 Irij I1

pi = o

1

OIPiI1

0 Irij I1

oIqjI1

OIPiI1

F@,p,q,r) I0 where x is the continuous variables, y is the structural binary variables, p, q, and r are the variables of the structural matrix, f is the cost objective function, h and g are the equality and inequality constraints of the hyperstructure model, andFis the constraints that connect the variable structural matrix to the structural variables (1)-(8). When the binary structural variables are fixed, at the primal subproblem, AR is constant calculated through (7) and F becomes simple bounds to p, q, and r variables; Le., the primal subproblem can be written as min f ( x )

oIqjI1 F’@,q,r) I0 Suppose that X = (z, t,p, ij) is a local optimal solution of (P’ primal) and (p’,A’, a’,p’ r , b’, k’, Z’, y’, c’, a’, d’, e’, e’, t’) the corresponding Lagrange multipliers of the above set of constraints. Thus, the Kuhn-Tucker optimality conditionss (Bazaraa and Shetty, 1979) of problem (P’ primal) are

(Pprimal)

X

subject to

-M‘

+ 5’ = 0

1736 Ind. Eng. Chem. Res., Vol. 33, No. 7,1994

+ b‘ = 0 -b’ + k‘ + 1‘ - y’ + c‘ + t’V,.F’@,q,r) = 0 -k’- c + e’ + t’V$“@,q,r) = 0 a‘-

(1)

-1‘ - 6‘ + d’ + t’Vd;v@,q,r) = 0 For X to be a local optimal solution of (Pprimal) there must exist corresponding multipliers so that the KuhnTucker conditions for (Pprimal) and the inner maximization problem hold; i.e.; for P primal

and for the inner maximization: l+b=O

-b

+ k + 1-7 + c + tV,.F’@,q,r) = 0 -k - e + e + tV$“@,q,r) = 0 -1 - 6 + d + t V F @ , q , r ) = 0

where (p,A, a,b, k,1, y,c, 6,d , e, e, t ) are the corresponding multipliers of problem (Pprimal). If M = (‘ = 1 - a’ then X is a local minimum of (P primal) (from (11)). Now, if X is a local minimum of (P primal), then conditions (11)hold. Define M = 1- a’ = p ;then is also a local minimum of (P’ primal) (since (I) are satisfied). Therefore, a critical value of M’can be found, so that the primal subproblems of (P)and (P’)are equivalent. This can be achieved by solving first the inner maximization problem with fixed structure; then a critical value of u is found for constraint GR 1 0 to be active. From the multiplier of that constraint in the (P’)problem, a proper value of M’can be found for the optimization to continue and extract the information of the multipliers of the rest constraints in order to construct the Lagrangian cut. Literature Cited Arkun, Y.; Manousiouthakis, B.; Palazoglu, A. Robustness Analysis of Process Control System-A Case Study of Decoupling Control in Distillation. Znd. Eng. Chem. Process Des. Dev. 1984,23 (1). Brietol, E. H. On a New Measure of Interaction for Multivariable Process Control. ZEEE Trans. Autom. Control 1966,ll. Brooke, A.; Kendrick, D.; Meeraus, M. GAMS A User’s Guide; Scientific Press: Redwood City, 1988. Calandranis, J.; Stephanopoulos, G. Structural Operability Analysis of Heat Exchanger Networks. Chem. Eng. Res. Des. 1986,M. Cerda, J.; Galli, M. R. Synthesis of Flexible Heat Exchanger Networks-11. Nonconvex Networks with Large Temperature Variations. Comput. Chem. Eng. 1990,14(2). Cerda, J.; Galli, M. R.; Camuesi, N.; Isla,M. A. Synthesis of Flexible Heat Exchamer Networks-I. ConvexNetworks. Commt. Chem. Eng. 1990, li (2). Colberg, R.D.; Morari, M. Analysis and Synthesis of Resilient Heat Exchamer Networks. Adu. Chem. End. 1988.14. Daoutidis,-P.; Kravaris, C. Structural Ev&ation of Control Configurations for Multivariable Nonlinear Processes. Chem. E n-s S k 1992,47 (5). Dovle. J. C.: Stein. G. ZEEE Autom. Control 1981.26 (1). Floudb, C.’ A. OASZS Discrete/ Continuous Optimization Approaches in Process Systems; Computed Aided Systems Laboratory, Department of Chemical Engineering, Princeton University: Princeton, NJ, 1990. Floudas, C. A.; Grossmann, I. E. Synthesis of Flexible Heat Exchanger Networks for Multiperiod Operation. Comput. Chem.Eng. 1985, 10 (2).

Floudas, C. A.; Ciric, A. R.Strategies for Overcoming Uncertainties in Heat Exchanger Network Synthesis. Comput. Chem.Eng. 1989, 13 (10). Floudas, C. A.; Visweswaran, V. A Global Optimization Algorithm (GOP) for Certain Classes of Nonconvex NLPs-I & 11. Comp. Chem. Eng. 1990,14. Floudas, C. A.; Visweswaran, V. A Primal-Relaxed Dual Global Optimization Approach. J. Optim. Theory Appl. 1993, 78 (2). Floudas, C. A.; Ciric, A. R.; Grossmann, I. E. Automatic Synthesis of Optimum Heat Exchanger Network Configurations. AZChE J. 1986,32,2. Galli, M. R.; Cerda, J. Synthesis of Flexible Heat Exchanger Networks-111. Temperature and Flowrate Variations. Comput. Chem. Eng. 1991,15. Garfinkel, R. S.; Nemhauser, J. L. Integer Programming; John Wiley New York, 1972. Geoffrion, A. M.Generalized Benders Decomposition. J. Optim. Theory Appl. 1972,lO (4). Georgiou, A,; Floudas, C. A. Simultaneous Process Synthesis and Control: Minimization of Disturbance Propagation in Heat Exchanger Networks. Presented at the International Conference on Foundations of Computed Aided Process Design (FOCAPD), Colorado, 1989a. Georgiou, A.; Floudas, C. A. Optimization Model for Generic Rank Determination of Structural Matrices. Int. J. Control 1989b,49 (5). Georgiou, A.; Floudas, C. A. Structural Analysis and Synthesis of Feasible Control System. Chem. Eng. Res. Des. 1989c,67. Georgiou, A.; Floudas, C. A. Structural Properties of Large Scale Systems. Znt. J. Control 1990,51 (1). Groaemann, I. E.; Floudas, C. A. Active Constraint Strategy for Flexibility Analysis in Chemical Processes. Comput. Chem. Eng. 1987,1 (6). Groaemann, I. E.; Westerberg, A. W.; Biegler, L. T. Retrofit Design of Processes. Presented at the First International Conference on Foundations of Computed Aided Process Operations (FOCAPO), Park City, UT, 1987. Gundersen, T. Retrofit Process Design-Research and Application of Systematic Methods. Preaentedat the International Conference on Foundations of Computer Aided Process Operations (FOCAPO), Colorado, 1989. Gundersen, T.; Naess, L. The Synthesis of Cost Optimal Heat Exchanger Networks. An Industrial Review of the State of the Art. Comput. Chem. Eng. 1988,12. Gundersen, T.; Sagli, B.; Kiste, K. Problems in Sequential and Simultaneous Strategies for Heat Exchanger Network Synthesis. In Computer-Oriented Process Engineering; Elsevier Science Publishers B.V.: Amsterdam, 1991. Huang, Y. L.; Fan, L. T. Strategy for Enhancing Structural Controllability in Heat Exchanger Network Synthesis: A Knowledge Engineering Approach. Presented at the Annual Meeting of the AIChE, Washington, DC, 1988. Johnston, R. D.; Barton, G. W. Design and Performance Assessment of Control Systems Using Singular Value Analysis. Ind. Eng. Chem. Res. 1987,26,830. Johnston, R. D.; Barton, G. W.; Brisk, M. L. Single-Input-SingleOutput Control Systems Synthesis. Part I: Structural Analysis and the Development of Feedback Control Schemes. Comput. Chem. Eng. 1985,9 (6). Kotjabasakis, E.; Linnhoff, B. Better System Design Reduces Heat Exchanger Fouling Costa. Oil Gas J. 1987,85. Lin, C. T. Structural Controllability. ZEEE Trans. Autom. Control 1974,19 (3). Linnhoff, B.; Kotjabasakis, E. Downstream Patha for Operable Process Design. Chem. Eng. Prog. 1986,82. Luyben, M. L.; Floudas, C. A. A Multiobjective Optimization Approach for Analyzing the Interaction of Design and Control-I. Submitted for publication, 1992. Mathisen, K. W.; Skogeatad, S.; Gundersen, T. Optimal Bypasa Placement-in Heat Exchanger Networks. Presented at the National Spring Meeting of AIChE, New Orleans, 1992. McAvoy, T.J. Some Resulta on Dynamic Interaction Analysis of Complex Control Systems. Znd. Eng. Chem. Process Des. Deu. 1983,22( 1 ) . McAvoy, T.J. Interaction Analysis;Instrument Society of America: 1983. Morari, M.; Stephanopoulos, G. Part 11 Structural Aspects and the Synthesis of Alternative Feasible Control Schemes. AZChE J. 1980,26.

Narraway, L. T.; Perkina, J. D.; Barton,G. W.Interaction between Proceaa Deeign and Proceaa Control: EconomicAnalysia of Proceee Dynamics. J. Process Control 1991,l. Papalexandri, K. P.; Pistikopouloe, E. N. An MINLP Retrofit Approach for Improving the Flexibility of Heat Exchanger Networks. Ann. Oper. Res. 1993,42. Pateraon, W. R. A Replacement for Logarithmic Mean. Chem. Eng. Sci. 1984,39 (11). Paulee, G. E.; Eloudae, C. A. APROS Algorithmic Development Methodology for Diecrete-Continuous Optimization Problems. Oper. Res. 1989,37 (6). Perkina, J. D. Interactions between Process Design and Procese Control. Presented at the International Federation of Automatic Control Symposium, Maastricht, The Netherlands, 1989. Perkina, J. D.; Wong, M. P. F. Asseeaing Controllability of Chemical Plants. Chem. Eng. Res. Dev. 1986,63. Pho, T. K.;Lapidus, L. Topica in Computer Aided Design: Part II. Syntheeiaof Optimal Heat Exchanger Networks by TreeSearching Algarithme. AIChE J. 1973,19 (6). Pistikopouloe, E. N.; Grossmann, I. E. Optimal Retrofit Design for Improving Flexibility in Linear System. Comput. Chem. Eng. 1988,12 (7).

Ind. Eng. Chem. Res., Vol. 33, No.7,1994 1737 Pistikopouloe, E. N.; Groaemann, I. E. Optimal Retrofit Mi for Improving Procea Flexibility in Nonlinear System. I & II. Comput. Chem. Eng. 1989,13 (9). Saboo, A. K.; Morari, M.; Woodcock, D. C. D m of Resilient Processing Plants VIII. A Reaiilience Index for Heat Exchanger Networks. Chem. Eng. Sei. 1985,40(8). Skogeetad, 5.;Morari, M. Implications of Large RGA Elements on Control Performance. Znd. Eng. Chem. Res. 1987,26,2323. Swaney, R.E.; Groesmann, 1. E. An Index for Operational Flexibility in Chemical Process Mi.I & II. AIChE J. 1985,31. Trivedi, K. K.; ONeil, B. K.; Roach, J. R.; Wood,R. M. A New Dual-Temperature Deeign Method for the Synthesis of Heat Exchanger Networks. Comput. Chem. Eng. 1988,13 (6). Received for review October 13, 1993 Revised munuscript received March 8,1994 Accepted April 6,19940 Abstract published in Advance ACSAbstracts, June 1,1994.