Ind. Eng. Chem. Res. 1994,33, 1738-1755
1738
Synthesis and Retrofit Design of Operable Heat Exchanger Networks. 2. Dynamics and Control Structure Considerations Katerina P. Papalexandri and Efstratios N. Pistikopoulos' Centre for Process Systems Engineering, Department of Chemical Engineering h Chemical Technology, Imperial College of Science, Technology and Medicine, London SW7 2BY, United Kingdom Dynamic controllability of heat exchanger networks is considered in this paper within a synthesis/ retrofit framework to improve the operability of HENS (flexibility and controllability). Based on a hyperstructure network representation, where all structural and operational alternatives are included, a variable delay matrix and a variable gain matrix are introduced, to explore potential control structures simultaneously with network structures. Simple controllability criteria (diagonal dominance of the gain matrix, dynamic decoupling, time effective control) are postulated employing graph theory concepts and included within the synthesislretrofit model. A modified generalized Benders decomposition is applied to solve the resulting mixed integer nonlinear programming synthesis/retrofit model and to derive a cost optimal network and a favorable control scheme. 1. Introduction
In part 1 (Papalexandri and Pistikopoulos, 1994) an iterative procedure was presented to address the heat exchangernetwork (HEN) synthesis and retrofit problem, incorporating flexibility and structural controllability considerations (totaldisturbance rejection). In particular, the multiperiod retrofit scheme for flexibility (Papalexandri and Pistikopoulos, 1993) was coupled with explicit total disturbance rejection requirements at the level of structure. However, structural controllability considerations may not reveal control potentials and control system design requirements; this may result in complex and economically unfavorable control and network configurations. Consider for example the network of Figure 1,where the outlet temperatures of H1 and C1 must be maintained at their specified values under the fouling of exchanger 1 and the disturbance variations of H2. The retrofit procedure proposed in part 1, for flexibility and total disturbance rejection with respect to resulted in an area increase of exchanger 3 and a bypass placement on the hot side of exchanger 3. The outlet temperatures of H1 and C1 are not disturbable from the inlet temperature of H2, and the network is fully flexible. However, in the network structure of Figure 1there is only one potential manipulating variable, the split fraction at the splitter of exchanger 3. Although the outlet temperatures of interest are not disturbable from !Z"&, the question remains, what control structure to implement and what would have been the impact on network economics if further controllability requirements had been considered during retrofit. Dynamic controllabilityof heat exchanger networks and process flowsheets, in general, has mostly been considered after the synthesis procedure has resulted in a steadystate economical design and mostly through an analysis scope. Morari (1983) proposed a general framework for the controllability analysis and comparison of alternative flowsheets and showed that dynamic controllability is an inherent feature of the design, independent of the control policy that is applied. Holt and Morari (1985) pointed out that decreasing the time delays in the transfer function may or may not result in improved performance of the process, but they did not propose any guidelinesto improve the controllability characteristics of the system. Jensen et al. (1986) reviewed proposed interaction measures for
* Author to whom correspondence should be addressed. 0888-5885/94/2633-1738~04.5~l0
90%
I15 C
Ind. Eng. Chem. Res., Vol. 33, No. 7,1994 1739 optimization problems where the synthesis objectives are the minimization of the cost and the derivation of a control scheme, through screening various controllability measures. Although the economic trade-offs between controllability and cost are addressed, their proposed synthesis approach is limited to fixed-match networks and lacks a systematic incorporation of controllability requirements within the synthesis procedure, where all alternatives are included. Varga and Hangos (1993),in an attempt to connect HEN controllability to structural properties that can be exploited during synthesis, explore HEN structures embedded within the superstructure representation through dynamic considerations, either via numerical simulationor qualitative prediction. They do not propose, however, a systematic method to connect even qualitative information on dynamics to the synthesis procedure. Considering interactions between process design and process control, Lenhoff and Morari (1982)proposed a design approach that takes into consideration economic and dynamic aspects simultaneously, within a multiobjective framework. A multiobjectiveformulation is also proposed by Palazoglu and Arkun (1986)to address the design of chemical plants with robust operability characteristics, where robustness indices are used to describe the dynamic operability of fixed-structure flowsheets. A multiobjective optimization approach is proposed by Luyben and Floudas (1992),to include process operability considerations, based on open-loop indicators, within the process synthesis model simultaneously with process economics. Narraway and Perkins (1993) combined conventional nonlinear optimal control and integer programming techniques and proposed an optimization scheme for the selection of control structures, based on their economic performance. In an attempt to apply integrated process design and control techniques to chemical wastewater treatment, Walsh and Perkins (1992)employ controllability testa, based on the required time before a feedback control system begins to counteract a disturbance (to screen process/control structures) and worst case design techniques to result in a process design that can handle parametric uncertainty. To account for dynamics at the HEN synthesis procedure, a proper dynamic model for the heat exchanger network is required that does not involve detailed information (not available at the synthesis level) but suffices to identify control limitations. Dynamic models that account better for the dynamic behavior of a heat exchanger have been investigated by Steiner (1987). More recently, Mathisen et al. (1993)explore the effect of the dynamic model for heat exchanger network on proper assessment of HEN controllability characteristics. In particular, they investigate thorough representation of dynamics in relation to heat transfer coefficients, model order, temperature driving force, wall capacitance, flow configuration, and pipe dynamics and propose a lumped model, where the accuracy depends on the number of heat exchanger cells considered. As thorough dynamic considerations require detailed information, Morari (1992),in a recent review of synthesis attempts with controllability considerations, points out that simple controllability criteria are needed to be included within synthesis procedures and trade-off controllability and economics. The objective of this paper is to propose a systematic framework for the synthesis and retrofit design of operable heat exchanger networks, where simple controllability criteria will drive HEN synthesis toward network struc-
tures with favorable controllability characteristics and eliminate structures that would require complex and expensive control policies. In a first attempt, steady-state gains and dynamic decoupling will be investigated within the synthesis procedure so that structural and operational alternatives are screened and optimized, based not only on economics but also on potential control configurations. 2. Problem Statement
The problem to be addressed in this paper can be stated as follows: Given are 1. Stream integration data: (i) In the case of synthesis, these involve a set H of hot process streams, a set C of cold process streams, and seta of hot and cold utilities HU and CU, respectively. (ii) In the case of retrofit design, the existing network is also given, in terms of heat exchangers (a set X of existing exchangers with known area EA)and network topology. The heat capacity flow rates F, inlet temperatures TI, of process streams and/or the heat transfer coefficients U are not necessarily fixed. They either vary within a range of values, {FLO,FUPJ, rc~Lo,~w), respectively, or are described by discrete sets of values F ( n , T*(T), U(T) defining multiperiod operation (in T periods). The outlet stream temperatures To are fixed (either for each period of operation or uniquely) or constrained through some specification. A specified range for the uncertain parameters, where flexibility is desired is also given. 2. A set of controlled variables (outlet or intermediate stream temperatures). 3. A set of disturbance inputs (usually inlet stream temperatures). 4. A minimum temperature approach AT- to determine feasible heat exchange in each heat exchanger (not utility consumption as HRAT). The objective is then: (i) in the synthesis case, to synthesize a heat exchanger network; (ii) in the retrofit case, to determine a retrofit design at minimum total (synthesis or retrofit) annualized cost (operating and capital investment cost, TAC), able to operate feasibly under the specified uncertainty, and with a favorable control scheme. The following assumptions are made, in general: 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 can be relaxed through the approach of Trivedi et al. (1988). Dynamic controllability characteristics will be assessed through a dynamic model for the heat exchanger network. As will be shown later, alternative dynamic models can also be utilized to describe the dynamic behavior of heat exchangers. Then, synthesis and/or retrofit are limited by the assumptions that govern the application of the employed model. Flexibility aspects have been described in part 1 of this series (Papalexandri and Pistikopoulos, 1994) together with structural controllability considerations for the synthesis/retrofit design problem. The proposed iterative strategy will be revisited here with the main objective to include other steady-state and dynamic controllability criteria at the synthesis/retrofit design level, which may
{co,VUP],
1740 Ind. Eng. Chem. Res., Vol. 33, No. 7, 1994 Table 3. Areas of Min-TAC Network exchanger 1 2 3 area (m2) 131.33 41 176.75
7(
$
L
;
&
K
I 3 kWK =366 K
I T,
4. Hyperstructure Representation of Structural,
0
Figure 2. Minimum TAC network. Table 1. Stream Data for Motivating Example stream F e p (kW/"C) 9 ("0 H1 16 249 H2 14 205 c1 10 38 c2 12 126 c3 13 93
Operational, and Control Alternatives Put
("C)
121 66 130 182 195
Table 2. Cost Data for Motivating Example cost Steam
cooling water area coat heater cost piping coat
be for one or more periods of operation. This synthesis step will be considered in detail in the following sections. 3. Working Example
In order to present the proposed synthesishetrofit framework, consider the integration of two hot steams H1 and H2, and three cold streams, C1, C2, and C3 (example taken from Georgiou and Floudas (1989)). Stream data and specifications for the outlet temperatures are given in Table 1. Heating and cooling requirements may be covered by steam, available at 300 O C , and cooling water, at 30-40 OC, respectively. Feasible heat exchange is determined by a AT,, = 10 "C. Due to process specifications the outlet temperatures of H1 and C3 must be maintained at their target values (controlobjectives). The inlet temperatures of C1 and C2 may vary within a small range and at high frequency, constituting, thus, "disturbance inputs". Process conditions prevent from integrating streams H2 with C1 and C2. Parameter variations at low frequencies that would require flexibility considerations are not examined since dynamic aspects are of interest. However, any parameter uncertainty can be addressed through the iterative synthesis/retrofit design framework of part 1. A minimum total annualized cost (TAC) network is to be synthesized, which features a control structure that can be easily implemented to lead to effective control. Cost data for the synthesis of the heat exchanger network are given in Table 2. When only the operating and investment cost of the network are considered, the simultaneous minimization of the total annualized synthesis cost (Ciric and Floudas, 1991) led to the network structure of Figure 2. This network features a total annualized cost of TAC = $120268 yrl. The areas of the exchangers are given in Table 3. The cooling water consumption is QCW= 1076 kW. However, in the network structure of Figure 2, the outlet temperatures of H1 and C3 are disturbable from (F;: is also disturbable from ?d$). Furthermore, the only potential manipulating
c2
5 100.6
variable is the utility load of exchanger 5 , which does not have any effect on the specified control objectives. In the following, the synthesis problem will be solved, whereas easily relizable control structure will be sought, validated, and traded-offsimultaneously with annualized cost.
455 K
HZ 4I4 kWK 7
4 122.3
The generalized match-network hyperstructure representation for multiperiod operation presented in part 1 (Papalexandri and Pistikopoulos, 1994) is adopted to represent all structural alternatives. As utility consumption is not prespecified, neither are the stream flows and intermediate temperatures; therefore, such a representation includes all operational alternatiues as well. The multiperiod hyperstructure for a stream k is shown in Figure 3. It will be first shown that all control alternatives are included within the considered hyperstructure and can be generated automatically for each embedded network structure. The multiperiod hyperstructure model consists of mass and energy balances at the splitters, mixers and exchangers of the hyperstructure network representation, design equations for the exchanger units, and feasibility constraints. For the retrofit case, match-exchanger assignments are also considered. Then, the synthesis/retrofit problem is formulated as a mixed integer nonlinear programming (MINLP) problem, described in detail in Appendix A of part 1 (Papalexandri and Pistikopoulos, 1994). Split fractions and utility loads are in general potential manipulating variables in a heat exchanger network. The manipulating splitters can be located at the inlet of a stream (stream splitting and overall bypass), before an exchanger (exchanger bypasses), and at the outlet of a stream from an exchanger ("multibypasses"). For the selection of the control scheme, pairing binary variables are introduced for all the potential manipulating variables and the desired controlled variables (outlet or intermediate stream temperatures, e.g., the temperature of a cold stream after and if it exchanges heat with a hot stream). In particular, a pairing variable denotes that a particular stream temperature is controlled through the split fraction of (i) a stream flow leaving the initial splitter of a stream (towarda heat exchanger or the final mixer of the stream), (ii) a bypass flow to an exchanger, (iii) a stream flow leaving the splitter of a stream after an exchanger (outlet flow or flow towards another exchanger), or (iv) a utility load (hot or cold). The complete set of pairing variables that is considered in the network hyperstructure, when only the outlet temperatures are of interest, is described in detail in Appendix A. The pairing variables are interconnected, so that for each controlled outlet temperature one manipulating variable is selected and each manipulating variable is assigned to at most one controlled output. The selection of the control scheme depends on the actual network structure. Thus, a number of constraints are introduced to connect the pairing variables to the structural variables. The logic of these constraints is as follows: 1. A split fraction cannot be selected to control a temperature if the corresponding flow does not exist in the network. The same holds for utility loads.
Ind. Eng. Chem. Res., Vol. 33, No. 7, 1994 1741
Figure 3. Hyperstructure network representation through binary variables.
Table 4. Control Alternatives for the Illustrative Example
2. For each splitter n - 1 split fractions can be manipulated, where n is the number of the outlet flows of the splitter. The complete set of constraints that connect the pairing scheme to the network hyperstructure is also described in detail in Appendix A. Via this representation of the structural and control alternatives, a variable time delay matrix and a variable gain matrix can be developed for the network hyperstructure, where time delays and gains are functions of the network structure, operating point, and control structure. This then allows the incorporation of controllability criteria based on time delays and gains within the synthesis/retrofit model, as will be discussed in sections 5 and 6. On the Example. For presentation reasons the network hyperstructure for the illustrative example is restricted to the case of fixed matches, as they have been predicted by the minimization of the total annualized cost. For simplificationreasons only the superstructures of streams H1 and C3 are considered and stream H2 enters first exchanger 4 and then is cooled in exchanger 5 to ita final temperature. For the considered network superstructure of the illustrative example, the potential manipulating variables are (see Table 4)
1. Initial splitter of H1: split fractions of H1 flows toward the exchangers 1,2,3 and the overall bypass of H1, SLY,s c - : si?, and ssJil, respectively. 2. Initial splitters of C3: split fractions of C3 flows toward the exchangers 3 and 4 and the overall bypass of C3, s$y3, s$i3, and sStc3, respectively. 3. Splitter prior to exchangers: hot and cold bypass streams in exchangers 1, 2, 3, 4, s&, sgfcl, CE HE CE HE CE sH1C29 SH!C39 sH1C3' sH2c31 and $H2C3. 4. Splitter of H1 after exchangers 1,2,3: split fractions of the outlet flows of H1 (from each exchanger to ita final mixer) and flows to the other exchangers of H1, ,O.Hl sB,Hl 031, sB,H1 sB,H1 O m ,B,Hl C1C29 C3C2, 'C3 , C2C39 and gbl' SB,Hl C2C19 C3Cli sC2
sgm,
SClC3.
5. Splitters of C3 are exchangers 3 and 4 split fractions of the outlet flows of C3 and flows to the other exchanger o c 3 B,C3 oc3 B,C3 of c 3 , si1 s m l , t and sHl@' The control scheme selection in this example involves 58pairing binary variables. These variables are connected to the 29 (in number) structural variables of the network, denoting the existence of the stream flows. f
5. Variable Time Delay Matrix Dynamics will be investigated through time delays in
the network. A variable delay matrix will be developed,
1742 Ind. Eng. Chem. Res., Vol. 33, No. 7, 1994
P
Cold mixer prior to utchanga
cold mixer after exchanga
4-:
........
Cold spliaa afta exchanger
Figure 4. Time-delay graphhetwork representation.
where the time response of an output to an input is given as a function of the network structure and design parameters to be determined, as stream flows, exchanger areas, pipe characteristics, etc. To each hyperstructure element a dead time can be associated as a function of stream flows, areas, hold-ups, etc. The actual time delay contribution of each element depends on the network operating point, which, even when the structure is known, is to be optimized on a TAC basis. However, the network paths that determine the response of an output to an input (disturbance or manipulating) can be easily identified when the structure is known and consequently can be expressed as a function of the network structural variables. When the network structure is specified, the timeresponse paths of an output to an input can be determined through the solution of linear programming (LP) shortest path problems. The network in this case is represented by a directed graph with variable vertices (splitters,mixers, exchanger units) and arcs (see Figure 4). The existence of each piping segment and vertex in the directed graph is represented by a corresponding set of hyperstructure structural variables; i.e., inlet pipes of hot streams (hi in Figure41 are represented by wh,bypass pipes of hot streams (hb in Figure 4) by xh, etc. To account for the dead time of an exchanger, each exchanger unit is represented by a set of arcs and vertices as shown in Figure 4. The dead time that corresponds to the hot side of the exchanger is associated with the arcs hea and heb, whereas the dead time for the cold side is associated with the arcs cea and ceb. Exchanger wall can be represented by the arc e. When a lumped parameter model is employed to describe the exchanger behavior (Mathisen et al., 1993), the set of arcs hea, heb, cea, ceb, and e can be considered for each cell. The vertices of interest, when time responses are considered, are the initial splitter of each stream (where the disturbance inputs enter the network), the splitters of the hyperstructure (where the manipulating split fractions are located), the utility exchangers, and the final mixers
of the streams (outlet temperatures). When the control objective is an intermediate temperature, the previous mixer of the stream (in the general hyperstructure representation) is also considered. An estimation of the dead time of each network element (based on the operating point which results from total cost optimization without controllability considerations, for each set of structural variables, during the iterations in the MINLP synthesis/retrofit problem) is associated to the corresponding arc of the network-directed graph. The actual LP shortest path formulations to determine the time-response paths of the considered outputs to the inputs of interest are described in Appendix C. The LP shortest path problem is solved for each pair of input-output vertices that are considered when fixing temporarily the network and control structure. When all the propagation paths are of interest, the LP shortest path problem is solved iteratively for each input-output pair, until infeasibility, bounding in each iteration the fastest time response. After the response paths have been identified, the time response of an output to an input can be expressed in terms of stream flow rates and design variables of the network to be determined (eg exchanger areas). Thus, the delay matrix for the current network and control structure will be expressed as a function of the network operating point. Controllability requirements can then be formulated in terms of time delays (for the current network structure and control scheme) and included within a primal synthesis/retrofit model. The solution of the synthesis model will provide an optimal operating point and design parameters which will satisfy the contro2lability criteria. In this work, two time-delay based criteria will be used, based on the shortest time response of an outlet temperature to a manipulating or disturbance input: Dynamic decoupling: The off-diagonal elements of the delay matrix, when the manipulating variables are considered, should be greater than the diagonal elements:
Ind. Eng. Chem. Res., Vol. 33, No.7, 1994 1743
+ a I0
tij- tiY- (1 - mij)M
V i, j , j'
where t is time response, i is a control objective, j and are j' manipulating variables, mij is the pairing variable that defines the control of i through j , M is a large positive number, and ct is a positive variable which is maximized. Whenthecontrolpairijisselectsd(mij= 11,thecriterion is activated and the time response of i to the manipulating input j must be as small as possible compared to the time response to any other manipulating input .'j This is achieved through the maximization of ct which is considered in the objective function with a (large) negative coefficient. M is properly selected so that M > Cru (where Cru is a valid upper bound of a) and when mij = 0 the criterion is a redundant constraint, even when a is maximized. Timeeffective control The time response of a control objective to the corresponding manipulating input should be less than the time response to the network disturbances. tij - t ,
- (1- m,)M + CY 5 0
V i, j , k
where i is a control objective,j is a manipulating variable, k is disturbance input, mij the pairing variable that defines the control of i through j , M is a large positive number, and CY is a positive variable which is maximized. M is selected as described above. When the pair i j is selected, the above criterion is activated and the time response of the output i to the selected manipulating input should be less than the time response to all the disturbance inputs. This criterion is realistic only in the case of feedforward control. Nevertheless, quick response of a controlled output to the corresponding mainpulating input, in relation to the response to disturbances, is a desired feature of a control structure. The relative importance of such a criterion can be manipulated through the weighting factor in the maximization of a. On the Example. Consider as potential network structure the one illustrated in Figure 5. The cost optimal areas and stream flows are the aame with the network in Figure 2. A potential pairing scheme is when the hot bypass of H2 in exchanger 4 is used to control the outlet temperature of C3 and the hot bypass of H1 in exchanger 1 to control the outlet temperature of H1. When the disturbance 7& is concerned, only C1is disturbable. The shortest path, in terms of time response, is defined by the inlet pipe of C1 to exchanger 1, the exchanger 1itself (hot and cold side), and the outlet pipe for all pipes of H1. Assuming a holdup T = l(kW/K)-ls-l and a pipe time delay proportional to the stream flow (see to the Appendix A), the shortest time response of T', is found (see Appendix B) ti?: = 103.781 8. dimilarly the shortest path, in terms of time response, from T',2to pmis defined by the inlet pipe of C2 to exchanger 2, exchanger 2 (hot and cold side), the bypass stream of H1 from exchanger 2 to exchanger 3,exchanger 3 itself, the bypass of H1 from exchanger 3 to exchanger 1,exchanger 1,and the outlet pipe of H1.These network elements result in a time response t? = 285.284 8. In the sameway, for the structure andthe controlscheme of Figure 5 the time responses of Table 5 result. Based on the time-response criteria, described above, the proposed control scheme is acceptable. Analytical calculation of the gains of the controlled outputs with respect to the selected manipulating inputs in the network of Figure 5 results in the gain matrix of Table 6.
fi1
I2kWlK
1%-
I3 kWK
a
Figure 5. Potential network structure and control =heme. Table 5. Fastest Time Response of Controlled Outputs in Propoaed Structure of Figure 5 invut
Pa
output
7%
103.781
G
m
Pm 286.244 189.143
ea 48 m
HE
hca
226.93 130.788
Table 6. Gain Matrix for Proposed Control Scheme in Network of Figure 6 input output
7%
HE 8HCl
11.392 0
-6.627 -4.666
One can note that the hot bypass in exchanger 4,chosen to control T&, has a strong gain effect on This implies that the network and control scheme of Figure 5 may not be preferable when minimal interactions are of interest, as will be discussed in the next section.
el.
6. Variable Gain Matrix
A generalized dynamic model for a heat exchanger network is utilized (Groep, 1993) in this study to derive the gain matrix of the controlled outputs with respect to the potential manipulating inputs. This dynamic model consists of modules that describe the dynamic behavior of the network elements, linearized at a steady-state operating point, to be determined through theoptimization of the network synthesis or retrofit cost. The elements of HEN that determine its dynamic behavior are the heat exchangers, splitters, mixers, and pipes. To account for the effects on the dynamics of the heat exchanger network,a linearized transfer function has been developed, where the input to each network element is assumed to consist of a steady-state component plus a dynamically varying componentimposed on this. Steadystate inputs and outputs are evaluated through the steadystate model included in the synthesis model (Appendix A of part 1). The steady-state parameters are used to construct a linearized model that describes the relation between dynamic input signal variations and the corresponding dynamic output signal variations. The analytical linearized modele for the network elements are described in detail in Appendix D. Within the hyperstructure network representation the linearized models are developed for each potential network element. Elements are connected through output to input assignments, where signal addition points for the heat capacity streams are implemented by introducing the mixer as a linearized elementary model. Thus, a generalized dynamic model is developed for the heat exchanger network hyperstructure and a variable transfer function is introduced, where the outputs are the stream outlet
1744 Ind. Eng. Chem. Res., Vol. 33, No. 7, 1994
temperatures and the inputs are potential manipulating variables (split fractions of the several hyperstructure splitters and utility loads). The variable transfer function is connected to the structural variables (defining the network topology) and the steady-state operating and design parameters (areas, stream flows, temperatures), which are variables to be optimized. When a hyperstructure element (splitter, exchanger, mixer, pipe) does not exist in the network (corresponding structural variable in zero), then the corresponding steady-state and dynamic variations of related inputs and/or outputs due to this element are forced to zero, through logical constraints of the type -(structural variable) U 5 (variation) I (structural variable)U where U is a large positive number. When an input si is selected as manipulating (i.e., to control k control objective),an input variation 3.i is imposed whereas all other inputs are forced to be constant: -3.. - (1 -
E
rn,)U+cIO
k;control obj
-
rnikU5JkI
Iqcontrol obj
E
mikU
k;controlobj
Thus, when a path exists from the manipulating input under consideration to a controlled output (and this is determined through an LP shortest path problem) the gain gik of the output is evaluated: (0
if (path), = 0 if (path), # 0
A variable gain matrix is derived in this manner for each potential control scheme, where the gains are included explicitly within the MINLP synthesis or retrofit model as a function of the network structure and the design and operating parameters, which are to be optimized. The selection of a favorable control scheme is driven by minimal interactions. For a potential control scheme ( i j ) E CM = {(ij);i is manipulating input, j is controlled output and mij = 1) to be selected, the following criteria are explicitly included within the synthesis model: (gainkjl- Igainij)- (1 - mij)U + a' 5 0 V manipulating 12 # i In the above constraint, U is a large positive number and a' a positive variable to be maximized. When the pair (id)is selected (mij= l), then the criterion becomes active and the absolute value of gainij must be greater than the absolute value of the gain of output j to any other considered manipulating input k. Otherwise (mi, = 0) the criterion is redundant. On the Example. Consider the network structure in Figure 5 and the proposed control scheme m ~ ~ =, 1, l m~,,= ~ 1,where s l and s2 are the split fractions at the splitters prior to exchanges 1 and 4, respectively. Since there is no path from s l to T&,gpO,,,3 = 0. Considering the gain effect of sl to
cl:
81' 20.01
$2' = 0 fl' = 81'FH1
where A1 and U1 are the area and heat transfer coefficient in exchanger 1 and 2' is the variation of variable x due to input signal 3.1'. The above equations, as well as similar equations for the evaluation of the gains of and T& with respect to s2, are included in the synthesis model. From the gain matrix in Table 6 one can easily see that the criterion
Gl
cannot be satisfied. Thus, the network structure of Figure 5 and the aforementioned control scheme would lead to an infeasible synthesis subproblem and would be effectively rejected. 7. Synthesis and Retrofit Design with Controllability Requirements
The hyperstructure synthesis/retrofit model described in Appendix A of part 1 (Papalexandri and Pistikopoulos, 1994), can be extended to include the variable time delay matrix and the variable gain matrix, which are explicitly expressed as functions of the network structural variables, design variables, and operating parameters. Explicit pairing criteria based on the time delay matrix and the gain matrix can be included within the MINLP model. For the solution of the resulting MINLP synthesis and/or retrofit model a modified generalized Benders decomposition (GBD) is applied. In particular the following procedure is followed (see Figures 6 and 7): Step 0: Select an initial network and control structure, i.e., avector (yi,wi, zi, ai,xi, b', mi) for the case of synthesis of (yi, wi, zi, ai, xi, bi, di, vi, ni, mi) for the case of retrofit design. Set k = 0. Step 1: Solve the minimum total (synthesis/retrofit) cost primal subproblem P' (see Appendix A, in part 1)) without controllability considerations for the current vector (yk,wk, zk,ak,xk, bk,mk)or ( y k ,wk, zk,ak,xk, bk, dk,vk,nk,mk). a. If P' is feasible, based on the cost optimal network flows estimate the dead times of the network pipes and exchanger units (employinga lumped, distributed, or other dynamic model). Go to step 2. b. If P' is infeasible, set the shortest path of the disturbance inputs to the controlled outputs to zero and of the manipulating variables (to the output) to infinity. Go to step 4b. Step 2: For the disturbance inputs and the currently selected manipulating variables, solve the LP shortest path problems, which involves the structural variables and mixed integer constraints that connect the network structure to network paths (seeAppendix C), to determine the paths that correspond to the fastest response of the controlled outputs to these inputs. The LP shortest path problem can be solved sequentially for each pair of inputoutput variables, to determine all the response paths. a. If such an LP is infeasible, there is no path between the considered input and output. Set the corresponding time response equal to infinity.
Ind. Eng. Chem. Res., Vol. 33,No. 7,1994 1745
b. Otherwise, obtain analytical expressions for the time responses,fixingthe values of the variables that correspond to the existence of an arc in the path (see Appendix C). Step 3: Postulate controllability criteria based on the current delay and gain matrices and include them within the primal synthesis or retrofit problem. In this paper, the aforementioned criteria of dynamic decoupling, time effective control and minimal interactions (described in sections 5 and 6) are used. Step 4: Solve the primal synthesis or retrofit problem P, including the controllability requirements. a. If P is feasible, the solution is a valid upper bound to the solution of the original problem. If this is less than the current upper bound, update this upper bound. On the basis of the information on the Lagrangian multipliers of the mixed integer constraints, which connect the delay matrix and the transfer function to the structure and the pairing criteria to the control scheme, construct the Lagrangian function for the master problem. b. Otherwise, solve the relaxed primal subproblem and construct a relaxed Lagrangian cut to include in the master problem. Step 5: Solve the master MILP problem, which includes the pure integer constraints (for structural and pairing variables) and the Lagrangian cuts, to obtain a new vector (yk+l, Wk+l, Zk+l, g k + l , xk+l, bk+l mk+l) or (yk+l Wk+l &+I, ak+l,xk+l,bk+l,dk+l,vk+l, nk+l,rnk+l) and a lower bound to the MINLP solution. a. If the master problem is feasible, its solution is a valid lower bound. If this is greater than the current lower bound, update this bound. b. If the current lower bound is greater or equal (or within a specific e-range) to the current upper bound, STOP. The solution of the MINLP is the one that corresponds to the current upper bound. c. If the MILP is integer infeasible, STOP. The solution is the current upper bound and the corresponding network. d. Otherwise, fix the current vector (yk+',wk+l,zk+l, g k + l Xk+l, bk+l,mO) or (yk+l,Wk+l, ~ k + l ,ak+l,xk+l,bk+l, dk+l,vk+l, nk+l,mk+l),set k = k + 1 and go back to step 1.
The proposed synthesis/retrofit procedure (schematically depicted in Figures 6 and 7) has been implemented using the modeling language GAMS (Brooke et al., 1988), where MINOS 5.3 (Murtagh et al., 1990)and SCICONIC 2.11 (SCICON, 1990)are used for the solution of the NLP and MILP subproblems, respectively. The described solution procedure refers only to the synthesis/retrofit design model when controllability is investigated through the selection of favorable control structures. The decomposition procedure and iterative scheme of Figure 6 correspond to the multiperiod MINLP synthesis/retrofit block of the iterative framework for operability developed in part 1. On the Example. The proposed synthesis framework with controllability considerations was applied to the small HEN of the illustrative example. The optimal network (obtained in 10 GBD iterations) is shown in Figure 8. It features a total annualized cost of TAC = $248268 yr'. The splitter prior to exchanger 1 is manipulating to control whereas the splitter prior to exchanger 3 is manipulating to control T& The areas, time responses, and gain matrix of the proposed network and control scheme are shown in Tables 7,8,and 9, respectively. As one can note, although the heat exchanger areas remain the same, the selected manipulating bypasses are different.
G1,
Table 7. Areas of Proposed Network exchanger area(m2)
1 131.33
2 41
3 176.75
4 122.3
5 100.6
Table 8. Fastest Time Response of Controlled Outputs in Proposed Structure input
Gl
T',
103.781
285.244 189.143
output
Gl
m
S E I
str
48
140.57 26
a,
Table 9. Gain Matrix for ProDosed Control Scheme ~~
input CE
HE
output
Gl
'HlCl
SHICB
11.392
e !3
0
6.288 -13.023
Table 10. Stream Data for Example on Fouling ! I $ ("C)
stream no. H1 H2 c1 c2 Steam
200 155 20 20 300
Twt ("C) 115 90 175 175 300
FCP(kW/K) 60 60 20 50
Table 11. Retrofit Cost Data for Example on Fouling cost 11.05$ kW-l y r 1 350AA0*s $ yrl
steam cost
additional area coat
Table 12. Exchanger Areas for Example on Fouling exchanger no. 1 2 3 4
exchanger area (m2) 305 600 63 173
u (kW/(m2K))
0.12-0.081 0.225 0.15 0.36
new area new area 1 (m2) 2 (m2) 305 453 600 600 253 63 173 173
8. Example on Fouling
The proposed synthesis framework of Figure 6 can be applied to the retrofit and multiperiod cases as well. When periods of operations are considered, the multiperiod hyperstructure model is used and the variable delay matrix and gain matrix are considered for each period of operation. The controllability criteria (pairing criteria) must hold for all the considered periods. Consider the existing network of the fouling example described in the Introduction (Kotjabasakisand Linnhoff, 1987). The existing network (see Figure 9) involves two hot and two cold streams. Stream data are given in Table 10,whereas the areas of the existing exchangers and the nominal values of heat transfer coefficients are shown in Table 12. The heat transfer coefficient between streams H1 and C1,depending on the temperature of stream C1, varies as shown below:
0.081 kW/(m2 K) IU,,,,
I0.12 kW/(m2K)
The outlet temperatures of the process streams are bounded (upper bounds for the hot streams and lower for the cold are given in Table 10);moreover, those of H1 and C1 must be controlled, thus defining two control objectives, whereas the inlet temperature of H 2 is considered as a disturbance input. A two-period retrofit model is employed, with the two extreme values of UH141 defining the periods of operation.
1746 Ind. Eng. Chem. Res., Vol. 33, No. 7, 1994
V-
V
I
h
I
I
I
1
V
&rive a new (Yk+l,wk+ 1Jk+ la+ 1W+l)
L
I
Figure 6. Synthesis of cost-optimal and controllable HENS.
A control structure is also sought, based on the dynamic decoupling,time effectivecontrol, and minimal interaction criteria described in the previous sections. For the retrofit, steam at 300 "C is considered available as hot utility. Operating cost and investment cost data are given in Table 11. A AT,i, = 5 O C is specified. A high piping cost of 1000 y r l is also considered, so as to account for pressure drop considerations and to avoid extensive stream splitting. When control alternatives are considered, only exchanger bypass placement is taken into account.
The retrofitted network is shown in Figure 10 and was obtained in five GBD iterations. The new exchanger areas are given in Table 12. The area of exchanger 3, which is not the fouling exchanger, is increased to 253 m2, and bypasses are placed on the cold side of exchanger 3 to control T"Gt and on the hot side of exchanger 4 to control T";;. The proposed control structure may not be the beat possible one. However, as can be seen in Tables 13 and 14, it complied with the considered controllability requirements (the time responses to the manipulatinginputs
Ind. Eng. Chem. Res., Vol. 33, No. 7, 1994 1747
.....................................................................................................................................................
i
s
Fkulecriaialpshtbp duam&stholhanaarapauo d a p s l b e t h r l c ~
a
b
.e c
i~ m
c
~
qual to infmiy
Nopsh~~tbobput~a ~amhollcdOutplt
inleausduleasnanrdlhLpah
na
HI 2005
60 KWlK I
ax
c1
I
455 K
HZ 14 kWK 4 7 13 kWK
I
+-I
a
Figure 8. Potential network structure and control scheme. HZ
Q
175 'C
50 KWlK
1
'12%
D' 1 6155 0 Kc WlK
.
205
115
c
co1 0.081 KWM'K 5 ui < 0.12 KWI~II'K Figure 10. Proposed retrofitted network for example on fouling.
Table 13. Time Responses in Proposed Network and Control Structure (Two Periods) input
%
HE
e1
543 251.2
294.9-295.2
Gl
50 KWlK
