Copyright © IFAC Low Cost Automation, Buenos Aires, Argentina, 1995
SUPERVISOR CONTROL OF HEAT EXCHANGER NETWORKS Nestor Aguilera and Jacinto L. Marchetti Institute of Technological Development for the Chemical Industry, CONICET  Universidad Nacional del Litoral, Giiemes 3450, 3000 Santa Fe, Argentina.
Abst.ract. TillS work presents a simple modcling technique for commanding the optimal operation of heat exchanger networks. The proposed representation provides all the necessary relationships to define an optircizing supel'\'isory system aimed to determine the most convenient set points of a multiloop control structure at any instant. The modeling approach used for the optimization emphasizes the computation of heat exchanger duties rather than net work intermediate temperatures, this simplifies the modeling step since all the conditions can be written by simple inspection of the network stnlctw'e, and reduces the computational work. Key Words.
Hierarchical control, Heat exchangers, Control applications
1. INTRODUCTION
ot.hers. Interaction between design and process control concepts during the configuration step of heat exchanger networks has been arising as an unavoidable Ijeed. The operability of these systems have to be carefully analyzed dnring the design stage without excluding a preliminary synthesis of the control structure capable of achieving the operational objectives. The following papers are examples of the contribution made in that direction: Calandranis and Stephanopoulos (1988), IIuang and Fan (H)92) and Mathisen et al., (1992). IIowever, for cases in which the network is already designed or in operation, no much has been said about online optimization methods to care for the best or the more efficient manner to operate these supposedly flexible structures without resigning the control objectives. In this sense, an important coniributioll has been given by I\Iathisen et al., (1094). But a convenient and practical way of modeling heat exchanger networks, showing all t.Ite specific relatiollships to be included for online optimization has not been prcsented yet.
1.1. Flexible Network Design The design, operation, and control of heat exchanger networks has been onc of the central subjects in the field of tllermally integrated c1lemical processes during the last 15 or 20 years. Initially, the increasing cost of energy has served as a driving force for research and design in industry and academic environments (Nisenfeld, 1973; IIuang and Elshout, 1976) . Lately, the concern arises when these thermally integrated processes have to deal with significant changes in the operating conditions. lIeatexchangernetwork systems, t.he central core of many thermally integrated plants, have to be robustly designed to absorb changes which are not just load disturbances with all almost normal distribution around Ilominal statiollary values, l\1any times, it is desirable t.o handle large and stable perturbatiolls 011 inlet. stream conditions or to have the capability to change the stream temperature targets significantly. Nowadays, a realistic design problem includes flexibility as a necessary attribute for heat exchanger networks, particula.rly when they a.re leqnired \.0 cover wide opera.ting spaces. This operability concept, which is closely related to controllability, has been discussed by speciilli::;ts like CaJandranis aud SLephanopoulos (1986), Kotjabasakis ;tIlt! LinnhofT (198G), Galli and Cp.rcbj (1901) and Papalexanclri and Pistikopoulos (1992), cHllOllg
1.2. The objeclives of this work
This paper is aimed at providing a simple modcling procedure to give the basis for online optimization of any kind of heat exchanger network. 111 this dcvelopmellt, it is assl!l11ed that all the necessary exchangers, senice equipmcnts, connec
69
tions and splits are already defined, as well as the sizes of the heatexchange areas. The model equations presented here are used for searching a convenient operating point attending to the specific network structure, and keeping a complete independence from the eventual presence of a pinch condition or the flexibility intended during the design stage. The idea is to determine the best set of actions that can be done using all what is available in the network for each working condition at any time.
whole set of hot streams. Similarly, for a general cold stream j,
(2)
where the adopted nomenclature is similar to the explained above, being qhj the heat taken at a final heating service. The duties to be performed at any instant on hot process streams are defined in this work as follows:
Simultaneous online optimization and regulating control might create conflicting actions. This is particularly true when moving the system from one operating point to another one implies starting up or shutting down one or more equipments used in the regulation task. The eventual saturation of one or more manipulated variables when searching for a new temporary optimal operating point is a problem to be solved if it is desired to keep closedloop performance. In this paper, a practical answer is given to this problem by including additional statements in the optimization problem, however, the development of flexible loop configurations could be a future improvement.
Qi =
WiCi Ti
out
'Pin

WiCi.Li
,
i E 1i,
(3)
j E C.
(4)
and for cold streams,
Qj =
WjCj Tj
out

'Pin
WjCj.Lj
,
Notice that these duties, Qi and Qj, include all possible process perturbations to the network, i.e., changes of inlet stream properties like temperature, flow rate or heat capacity, and changes in target temperatures. Qi and Qj can also be written as functions of heat duties performed at exchangers and service equipments,
n.
Qi
This paper is organized as follows: first, energy balances for process streams are written in terms of the tasks to be done on the streams by the network and the individual heat duties to be performed by the equipments; secondly, it is shown that a careful inspection of the components involved in the overall energy balance of the whole network allows to reach important preliminary statements on the network conditions for operability and to define the need of an online optimizing control level. Then, thermodynamic restrictions are discussed and used to complete the model formulation.
= L
qki  qd,
i E 1i,
(5)
k=l and nj
Qj = L qkj k=l
+ qhj,
j E C,
(6)
where ni and nj are the total number of heat exchangers on streams i and j respectively. The relationships given by Eqns. (5) and (6) show the direct dependency between timevariant duties to be performed by the network and, the exchanged heats that must be accommodated by the control system to satisfy the targets.
2. ENERGY BALANCES OF THE NETWORK 2.1. Stream Energy Balances
2.2 . Total Network Energy Balance
The energy balance for a general hot stream i, in a network can be written as follows :
The heat integration achieved in the network implies the equality between the amount of heat given up at the heat exchangers by all the hot streams and the amount of energy received In these equipments by all the cold streams, i.e .,
(1) i E 1i,
ni
nj
LLqkj = LLqkj . iE1i k=l jECk=l
where WiCi is the stream energy capacity per unit time , T;in is the stream temperature at the network inlet, T;0ut is the stream temperature at the network outlet, qki is the heat transferred at the exchanger k (ni is the total number of exchangers in the stream path), qc; is the heat left at a final cooling service and, 1i stands for the
(7)
Consequently, a general energy balance for the whole network can be written in terms of the stream duties Qi and Qj, and the energy handled by the service equipments only; summing Eqns. (5) and (6) for all the process streams and using 70
2.3 . Degrees of Freedom of the Total Network Energy Balance
Eqn. (7) gives,
I:Qi iE1i
+ I:qei = iE1i
I:Qj
+ I:qhj.
jEC
(8)
jEC
Each term on the left side of Eqn. (9) can be understood as a degree of freedom that allows the escape of either positive or negative energy disturbances. This interpretation encourages to define a function for determining the network capacity to accommodate the total energy balance to a given operating point. This function can be easily derived from the above arguments by accounting all the escape path existing in the network.
Cold and hot process streams belonging to a general heat exchanger network can be classified, from an operating point of view , in to streams that have a defined temperature target that can be reached without using utilities, streams that need a service equipment to reach the desired target, and streams that do not need final temperature control. Hence, the following streams sets can be established for a general analysis:
1i 1
Result 1 (Degrees of Freedom): letting s be the total number of service equipments and nO be the total number of process streams without final temperature regulation in a heatexchanger network, the function
hot streams with final temperature targets and without service equipments } : {
1i2 : { hot streams with final temperature targets and with service equipments }
f =
1i3 : {hot streams without final temperature targets}
nO
+s 
1,
(10)
gives the degrees of freedom available in the total energy balance for online optimization.
Cl : { cold streams with final temperature targets and without service equipments }
The 1 appearing in Eqn. (10) comes from the restriction imposed by the relationship (9) itself. Notice that f can be determined by simple inspection of the network and by knowing all the temperature targets to be reached. These degrees of freedom measure the network capability for flexible operation and gives a rapid evaluation about the eventual need of an optimizing supervisory control, as discussed next.
C2 : { cold streams with final temperature targets and with service equipments }
C3 : { cold streams without final temperature targets}
Now, Eqn. (8) can be written using terms that imply different meanings from the operability point of view,
1.
f < 0, means that there are no service
equipments and that all the process streams need to be controlled to reach a given target. It does not matter how the structure is defined, the network cannot be operated since there is no way out to any eventual energy disturbance. 2 . f = 0, there is one service stream or one outletfree process stream which allows the operation of the network. The thermal condition of this stream is subordinated to the needs of the regulating system, i.e., all the energy disturbances go out through it, which could make the system very interacting and reduce regulatory closedloop performance . Once the regulating system achieves the control targets, the energy integration is completely determined. 3. f > 0, there are more ,than one stream path to release energy disturbances. This possibility, which is a necessary condition for an online supervisor , presents three sub cases to the analysis : a) All the stream paths allowing to release energy disturbances are associated to 1i2 and 1i 3 . This means that all the ser
(9)
All the right side terms ofEqn. (9) are completely determined by inlet conditions and temperature targets. Actually, the network and its regulating system must be designed to ensure that these amounts of energy will be achieved by the process streams, they are no more than an alternative expression of the control targets. The terms on the left side , are quantities that can be accommodated to satisfy the conditions imposed by the right terms or to improve the optimality of the operation. Notice that any energy disturbance entering the network must be released through one or more outletfree process or service streams implied in the left side terms.
71
vice equipments are for cooling operations only and eventually there are hot process streams without temperature targets. If these utilities have the same cost, and there are no freeoutlet streams (which is equivalent to a null cost service) this case is similar to f = O. On the contrary, if the cooling services have different costs, then an optimizing system will be advisable . b) All the stream paths allowing to release energy disturbances are associated to C2 and C3 • This means that all the service equipments are for heating operations only and eventually there are cold process streams without temperature targets. Similarly to the previous case a), an optimizing supervisor would be advisable only if the heating services have different costs or if a freeoutlet cold process stream is available, or both . e) The network has available heating and cooling services and eventually process streams without temperature targets. This is the case for which an optimizing supervisor control could yield immediate benefits since there exist enough operative degrees of freedom to search for an optimal operating point .
through the relationship q
1{3 /={3' 0'
NTU / = 1 + NTU'
0'#{3#1
(13)
if
0' = {3 = 1.
(14)
Heatexchanger parameters involved in the last two equations are: the number of transfer units NTU, NTU
=
U A,
i E 1i
(15)
WiCi
the heatcapacityflow rate ratio 0', WiCi
0'= ,
iE1i,
jEC
(16)
WjCj
and a nonlinear relationship {3,
{3 = exp {NTU (1  a)}.
(17)
Note that factor / gives a limit to the total amount of thermal potential that can be used when a finite exchange area is available at the stream match under consideration. An analysis of relationships in Eqns. (12) to (17) allows to determine that / takes values in the interval [0,1) only. Furthermore, if any of the flow rates, Wi or Wj, vary from a nominal value to zero the heatexchanger duty changes in a similar manner, i.e. , • If Wi > 0, L > 0, / > 1 and q > O. • If Wj > 0, L # 0 and bounded, / > 0 and q > O.
These operational limits are found in service equipments where the total flowrate of a utility stream is manipulated , or in heatexchangers with a bypass used for temperature regulation or heattransfer control. In both cases, one of the heatexchanger streams might change from a maximum value to zero to define different points of the operation space.
3.1. Single Heat Exchangers
Let us define , for convenience , the thermodynamics maximum heattransfer capability or more simply, the thermal potential of a stream match as the energy that could be transferred by the hot stream to a sink having the temperature of the cold stream, i.e.;
(TP  T/) ,
if
and
3. HEAT TRANSFER OPERATION CONSTRAINTS
WiCi
(12)
where / is an effectiveness factor computed using the following definitions:
This analysis, as presented so far, does not include the thermodynamics conditions that must be accomplished at each matching stream node . Consequently, the information provided by Eqn. (10) must be taken as a necessary condition for the operability of a network , not as a sufficient one. It gives a simple and rapid way to determine the usefulness or the potential need of an online optimization system for a given heat exchanger network.
L =
= /L,
Result 2 (Single Equipment Constraint): operative limits of a single heat exchanger, where a total flow rate or a bypass ratio is manipulated, can be written in terms of the streammatch thermalpotential L given by Equation (11) and the heat exchanger effectiveness factor /, given by Eqns . (13) and (14) , as follows :
(11)
where i E 1i , j E C and the superscript 0 stands for heatexchanger inlet conditions . Notice that the second law of thermodynamics requires L to be a positive quantity always. This virtual amount of exchangeable energy is useful for determining the heat transferred by an actual heat exchanger
O:S q :S /L ,
72
(18)
=
=
where the extremes q 0 and q I L implies fully closed and fully open control valve, or fully closed and fully open bypass, respectively. All other intermediate condition represents an operating point where the control valve, or bypass, is partially open
and (14).
4. THE OPTIMAL OPERATION PROBLEM The aim of this section is to provide modeling equations for online optimization of any kind of heat exchanger network. All necessary exchangers, service equipments, piping connections and splits are assumed known, as well as the heatexchange areas. The network structure is fix (a given equipmen t cannot be connected to different streams), but some exchangers might be completely bypassed or some services might be started up or shut down depending on the optimizer commands. This framework allows to use Eqns. (5) and (6) as equality constraints; since they define the main tasks required by the operator to the network. Conditions given by Eqn . (22) provide unequally constraints that limit individual heatexchanger capacities. The objective function is formulated in terms of the minimization of utilities or the maximization of the energy integration.
3.2. Operative Restriction for a Heat Exchanger in the Network
A heat exchanger which is embedded in a network is represented also by the above relationships, the only difference being the form in which the thermal potential of the streams reaching to that particular node is calculated. Once the network is given, a convenient and efficient way to completely describe its structure is by determining all previous exchanges made on both streams reaching each heat exchanger in the network . This can be accomplished by defining the following sets : prej(k) = { Heat exchangers on hot stream z previous to the heat exchanger k } , prej (k) = { Heat exchangers on cold stream j previous to the heat exchanger k } .
Result 4 (Optimization Problem): the optimal operating point of an alldefined heatexchanger network can be obtained by solving the following maximization problem:
where i E 1i and j E C. Hence, the temperatures of streams reaching the heat exchanger k can be written as follows:
o 1'; (k) =
Tl.n

1
~
WjCj
I.

k E {I, ne}
L.;ql., Ij E prej(k),
(23)
(19) subject to
n.  Lqkj  qcj = Qi ,
i E 'H. ,
(24)
j E C,
(25)
qk ::; IkLk,
kE{I,ne+s},
qk ::; 0,
k E {I , ne + s} ,
(26) (27) (28) (29)
k=l nj
where the superscript 0 stands for the inlet to the exchanger k. Based on Eqns. (19) and (20), the thermal potential for the exchanger k is written now as
L
qkj
+ qhj = Qj ,
k=l
0, i E 'H., qhj ::; 0, j E C.
qci ::; I.
Ij
Hence, the restriction in Eqn . (18) can be extended to each heat transfer equipment in the network.
This is a typical LP problem to be solved in terms of the individual heat exchanger duties, qk for k E {I, ne} only, since service utilities are fix once the integration is defined. The weights assigned to the heats in the objective function in Eqn. (23) , might have just a formal sense, however, if minimum utility is required, i .e~
Result 3 (Network Equipment Constraint): operative limits of a heat exchanger in a network can be represented by
.
:~.
(30) where ne and s are the total number of heat exchangers and service equipments in the network respectively, Lk is the streammatch thermal potential given by Eqn. (21) and Ik the heatexchanger effectiveness factor given by Eqns . (13)
a selective importance can be given to the different services accordingly to its availability or actual cost. Any numerical method that provides a solution to the above problem can be used for 73
configuration of the regulating system. The main guidelines used to synthesize loop configurations are: i) avoid intermittent equipments as source of manipulated variables for the regulating system even if they are final equipments on the streams and, ii) consider the high performance achievable using an integral controller when manipulating a bypass on the controlled stream (Rotea and Marchetti, 1995) . Furthermore, some energy integration have to be eventually sacrificed to preserve regulation in the network .
online search of the convenient operating point of an specific network. This allows to define the best set of actions that can be done by the control system using all what is available in the network for each working condition at any time. Endstream temperatures are frequently regulated through a final or last equipment on the stream, which might go out of operation to satisfy optimal conditions . Flexible control structures could be proposed to overcome the problem, however, this is not an issue to be addressed in this paper. The straightforward alternative is to set minimum values to the duty of every equipment supporting manipulated variables having the possibility of saturation. Hence, to preserve regulation in the network, some energy integration have to be sacrificed; this is done by modifying the conditions of non negativity for the duties of these n r , particular equipments only,  qi ~ emin ,i,
i E {I , n r }.
6. REFERENCES Calandranis, J . and Stephanopoulos, G. (1986). Structural Operability Analysis of Heat Exchanger Networks. Chem. Eng. Res. Des., 64 , pp. 347364. Calandranis, J. and Stephanopoulos, G. (1988) . A Structural Approach to the Design of Control Systems in Heat Exchanger Networks. Comput. Chem. Eng., 12, 7, pp . 651699. Galli, M.R. and Cerdci J . (1989). Synthesis of Flexible Heat Exchanger Networks Ill. Temperature and Flowrate Variations. Comput. Chem. Eng., 15, 1, pp. 724. Huang, F., and Elshout, R. (1976) . Optimizing the Heat Recovery of Crude Units. Chem. Eng. Prog., 72 , July, pp . 6874. Huang, Y.L. and Fan, L.T. (1992) . Distributed Strategy for Integration of Process Design and Control. Comput. Chem. Eng., 16, 5, pp . 497522. Kotjabasakis, E. and Linnhoff, B. (1986). Sensitivity Tables in the Design of Flexible Processes. Chem. Eng. Res. Des., 64, pp. 19721l. Mathisen, K.W., Skogestad, S. and Wolf, E.A. (1992). Bypass Selection for Control of Heat Exchanger Networks. European Sym. Comp . Aided Proc. Eng.l , May 2428, Elsinove , Denmark. Comput. Chem. Eng., pp. s263s272. Mathisen, K.W ., Morari, M. and Skogestad, S. (1994) . Optimal Operation of Heat Exchanger Networks. Paper presented in the PSE '94, Kyungju, Korea, May 30June 3. Nisenfeld, A.E . (1973) . Applying Control Computers to an Integrated Plant . Chem. Eng. Prog., 69 , 9, pp . 4548 . Papalexandri, K.P. and Pistikopoulos, E.N . (1992) . A Multiperiod MINLP Model for Improving the Flexibility of Heat Exchanger Networks. European Sym. Comput. Aided Proc . Eng.2 , October 57, Toulouse , France. Comput. Chem. Eng., pp . s329s334. Rotea, M.A. and Marchetti , J .L. (1995). Integral Control of HeatExchangerPlusBypass Systems . Submitted for publication to the Trans . ASME Journal of Dynamic Systems, Measurement and Control.
(31)
The right side value in Eqn.(31) must be selected for each endstream temperature control loop according to the lowest limit required for the span of the manipulated variable. Certainly, these values can be previously estimated , but the final adjustment will probably occurs at the field .
5. CONCLUSIONS This paper provides a simple modeling procedure that gives the basis for online optimization of heat exchanger networks and suggests guidelines to define the loop configuration of the regulating system. A formula that gives the degrees of freedom of the network total energy balance is derived and used for a preliminary analysis of the network capacity to allow the escape of energy perturbations or the change of any stream target. The final model of the network includes one energy balance equation for each process stream having a final temperature target and one constraint equation for every equipment . It is assumed that the network structure and size of all the exchanger areas are known. The objective function for online optimization is formulated in terms of the minimization of the utility consumptions or the maximization of the energy integration. The operative limits of a heat exchanger in a network is determined by a simple constraint equation where a single factor gives a limit to the amount of energy that can be used when a finite exchange area is available at the correspondent stream match. The model proposed here is useful not only for online optimization of the network operation , it also gives information that helps to define the 74