Controllability and Resiliency Analysis in Heat Exchanger Networks

609 A ublication of CHEMICAL ENGINEERING RANSACIONS VOL. 6, 07 Guest Editors: Petar S Varbanov, Rongxin Su, Hon Loong Lam, Xia Liu, Jiří J Klemeš Coyright 07, AIDIC Servizi S.r.l. ISBN 978-88-95608-5-8; ISSN 83-96 he Italian Association of Chemical Engineering Online at www.aidic.it/cet DOI: 0.3303/CE7666 Controllability and Resiliency Analysis in Heat Exchanger Networks Camila B. Miranda, Caliane B. B. Costa, Cid M. G. Andrade, Mauro A. S. S. Ravagnani* Chemical Engineering Deartment, State Univerity of Maringá, CEP 8700900, Maringá, Paraná, Brazil massravagnani@uem.br Synthesis of heat exchanger network (HEN) is a subject of great industrial interest. In an industrial rocess, it is ossible to take advantage of the existing energy in its own rocess streams and, consequently, to reduce the utilities consumtion. Oerability considerations are imortant for the HEN design due to the uncertainties in streams temeratures and flow rates, which are very common in real oerations. herefore, the HEN design, in addition to economic interests, must consider flexibility, controllability and resiliency asects. his aer erforms the analysis of controllability and resiliency (C&R). For the configuration of the HEN control system, it is used the Relative Gain Array and Disturbance Cost index. his analysis of oerability is alied in a literature examle. A modification in the literature HEN is then suggested to imrove network resiliency.. Introduction An industrial rocess system consists of three main interactive comonents: the industrial rocess itself, the utilities system and the energy recovery system. Hot and cold utilities are used for rocess streams heating and cooling. he synthesis of heat exchanger network (HEN) is a subject of great industrial interest because it is ossible to avoid excessive use of utilities, which contributes to reduce oerating and environmental costs. However, if the network is designed for a single oeration eriod, this can increase oerating costs or make the rocess unviable in a real roduction environment under variations in certain eriods of the year. hus, a flexible HEN is a network able to suort changes in oerating conditions such as variations in inlet and outlet temeratures and stream flowrates. While flexibility is related to the steady state behavior of the rocess, the controllability is concerned with short-term answers (dynamic asects). hus, the main goal of a control system in a HEN is to kee, in the easiest ossible way, the outlet temerature of the streams on the reestablished values or even control those that do not have a secific goal. Process control can be achieved through entries that can be maniulated, such as: control of utilities flowrates; use of byass fractions; imlementation of streams slitters fractions; control of the flowrates of inut rocess streams; use of additional areas in heat exchangers and control of recycle flowrate. In a decentralized control system, to eliminate the disturbance caused to the system by external environment, each maniulated variable can be aired with a controlled variable, resulting in a control loo. he set of loos constitutes the control structure (Escobar et al., 03). Controllability is a roerty of the lant itself, which is strongly affected by the network configuration and it is not deendent on the controller design. According to Bristol (966), a way of measuring controllability is using Relative Gain Array (RGA), which is a controllability index of the system in steady state. Uzturk and Akman (997) used this index for selecting the aroriate match between the byass and the outlet controlled temerature. Westhalen et al. (003) used the RGA analysis for the selection of the control structure. Bristol (966) introduced RGA with the objective of measuring the interaction between rocess variables. he RGA is a normalized gain matrix that describes the imact of each control variable on each outut. he normalization of these gains is based on the otential imact of each inut and outut air. his controllability index can be calculated exerimentally or by means of stationary rocess gains matrix. In De Leon et al. (06) the disturbance cost (DC) and the relative gain array (RGA) were used to analyse a HEN and to determine if the addition of byasses could imrove the controllability and the stability of the system. A Please cite this article as: Miranda C.B., Costa C.B.B., Andrade C.M.G., Ravagnani M.A.S.S., 07, Controllability and resiliency analysis in heat exchanger networks, Chemical Engineering ransactions, 6, 609-6 DOI:0.3303/CE7666

60 thermodynamic analysis was also rovided. he resiliency describes the ability of a rocess to avoid external disturbances and measures the degree of maintenance of the oututs of multivariable rocesses in their setoints (Morar 983). According to Lewin (996), a way of evaluating the resiliency is using Disturbance Cost (DC). his index indicates the settling time for disturbance rejection and the limitations due to actuator constraints. Seider et al. (009) resented the imact of rocess design decisions on oerability in four case studies, using RGA and DC index. It is essential that decisions about rocess design consider controllability and resiliency (C&R) due to uncertainties and disturbances. All of these aers treat the cases of synthesizing HEN for a single eriod. In the resent aer, the literature rocedure for the C&R analysis, using RGA and DC index is used to synthesize a multieriod HEN. he HEN here resented has imroved resiliency.. Controllability and Resiliency Analysis In industrial rocesses it is a very imortant feature to consider the C&R of a HEN during the design task, due to uncertainties and disturbances in the rocess model. Steady state C&R analysis rovides useful information for HEN assessment, describing the effects of the control variables and disturbances on the rocess oututs, and it requires less work than dynamic analysis. hus, a rocedure roosed by Seider et al. (009) for assessing steady state C&R of a rocess is described. First ste: generate the rocess model and select the controlled variables (y{t} = [y, y, y3...]), the maniulated variables (u{t} = [u, u, u3...]) and the disturbances (d{t} = [d, d, d3...]). Second ste: linearize the model according to Eq(). Δy { 0} P{0} Δu {0} Pd {0} Δnd () he rocedure to linearize the model is: Solve the model equations using a numerical method, like Newton-Rahson, for the nominal values of the disturbance and maniulated variables to determine the state variables (x). Aly ositive () and negative (n) small disturbances ( ui) to each maniulated variable (ui), one at a time, and recalculate the values of the controlled variables. hese values should be stored in the variables y,j and yn,j. Index j indicates the controlled variable. Column i of the steady state gain matrix (P{0}) can be calculated using the central finite difference as shown in Eq(). Parameter u i is the nominal value of the maniulated variable. y y u ji{ 0} ui, j n, j () i d Aly ositive and negative small erturbations ( di) to each disturbance variable (di), one at a time and recalculate the values of the controlled variables. hese values also must be stored in the variables y,j and yn,j. Column i of the steady state gain matrix (Pd{0}) also should be calculated using the central finite difference as shown in Eq(3). Parameter of the disturbance variable. y y d ji{ d i, j n, j 0} i d i is the imum erturbation hird ste: calculate the measures of controllability and resiliency using the matrix P{0} and Pd{0}. he controllability is analysed by the RGA index (Bristol, 966), which is calculated by multilying the matrices P{0} and (P - {0}) element-by-element, as in Eq(). In this ste, the controlled and maniulated variable are aired. RGA P{0} P {0} () he resiliency of the HEN is examined with the DC calculation (Lewin, 996) for re-determined disturbances. hese disturbances are normalized and arranged in a vector ( nd). he action required to comletely reject the disturbance, given the linearized model and assuming erfect control, is calculated by Eq(5). (3) Δu {0} P {0} Pd {0} Δnd (5) A quantitative measure of the control effort to reject a given disturbance vector is the Euclidean norm. he -norm is the DC. Its value indicates the settling time for disturbance rejection and the limitations due to actuator constraints. Its result is indeendent of controller tuning, since the DC is based on the assumtion of erfect control.

6 DC Δu (6) Finally it is ossible to determine the variables that should be aired and if the resiliency of the network is satisfactory or not. RGA values near one indicate the maniulated variables that must be aired with the controlled variables. he closer to one, the more insensitive to interaction will be the selected air. Pairs with negative elements of RGA index are an indication of resence of destabilizing ositive feedback due to unfavourable rocess interactions. hese airs should be avoided, because it is not ossible to assure stability. In other words, the single inut, single outut (SISO) controllers should be aired in such a way that the RGA elements corresonding to the airings are ositive, so that closed-loo stability in the control loo can be assured. For the DC index, a value higher than one indicates that the actuator limitations can be exceeded and HEN should be modified to ensure adequate regulation. In the second ste, it is necessary to calculate the values of the controlled variables for ositive and negative small erturbations of each maniulated and disturbance variable. With these values, the steady state gain matrix must be comuted using the finite difference. Due to the use of ositive and negative erturbations, it is recommended to use central finite difference. 3. Case study his case study was adated from Floudas and Grossmann (987) and used by Miranda et al. (06) for the synthesis of HEN for multile eriods of oeration. able shows the inut data for the examle. Here, the urose is C&R analysis of the HEN obtained for nominal conditions by Miranda et al. (06). he roblem has two hot and two cold streams and the global heat transfer coefficients (U) for all matches are 8 kw/m²k. able : Streams data Stream in (K) out (K) F (kw/k) Nominal conditions H 583 33. H 73 553.0 C 33 393 3.0 C 388 553.0 Period H 593 33.8 H 73 553.0 C 33 393 3.0 C 383 553. Period H 593 33.8 H 73 553.0 C 33 393 3.0 C 393 553.6 Period 3 H 573 33.0 H 73 553.0 C 33 393 3.0 C 383 553. he HEN was synthesized using a MINLP formulation based on the suerstructure roosed by Yee and Grossmann (990), as resented in Miranda et al. (06). Because of sace limitations in the resent aer, Figure (a) shows only the HEN for the nominal conditions. HEN for eriods, and 3 are synthesized in the same manner. For the HEN in nominal conditions, the four outlet temeratures of the streams (h5, h8, c and c5) are variables that must be controlled. he HEN has three heat exchangers (, and 3) and two coolers ( and 5) with areas (Ai) of., 6.7,.0, 30.3 and.0 m². he thermal caacity of the cold utility used to cool streams H and H are.7 kw/k (FC6) and kw/k (FC5). he energy balance for this system involves 7 variables: the thermal caacity and the temeratures of each stream, the heat exchanged in each equiment and the slit fraction in hot stream H: FC, FC, FC3, FC, FC5, FC6, h, h, h3, h, h5, h6, h7, h8, c, c, c3, c, c5, c6, c7, Q, Q, Q3, Q, Q5 and ϕ. With the stream data in able, it can be observed that there are four disturbance variables: the feed thermal caacity and the inlet temerature of

6 hot stream H (FC and h) and the feed thermal caacity and inlet temerature of cold stream C (FC and c3), with the imum erturbation of 8 %, 0 K, 0 % and 5 K, resectively. In this case, C&R is analysed for the HEN synthesized only for nominal conditions, not taking into account the multieriod network, so the slit fraction of hot stream H (ϕ = 0.88) is considered constant. Also, byasses to heat exchangers, and 3 are not considered. Furthermore, the inlet and outlet temeratures of the cold utility (c6 = 303 K and c7 = 33 K) are fixed variables. he network is required to be resilient to disturbances in hot stream H and cold stream C. For constant heat caacities and no hase change, a steady state model for the network consists of three energy balances for each heat exchanger and an energy balance for a mixer located in hot stream H, totalizing 6 equations. For heat exchanger : h h 0 f{ x} Q FC (7) c c 0 f { x} Q FC 3 (8) h c h c h c / h c f 3{ x} Q UA 0 (9) ln[ ] he degrees of freedom analysis for the HEN indicated that four variables could be maniulated for controlling variables h5, h8, c and c5. Several ossible control systems can be investigated, and in this case, variables FC6, FC5, FC3 and FC were selected as maniulated variables. Using u = and d = for all variables, u = [.7,, 3, ] and d = [8 %, 0 K, 0 %, 5 K], the resulting linearized model is reresented by: h5 3.7 h8 c 89.6 c5 5.8 FC FC FC 7.6 FC 6 5 3. 3.9. 0. 35.0 8.5 FC h 0.6 FC 7.7 c3 (0) he RGA comutation is made using matrix P{0}:.0.0 RGA ().0.0 he steady-state RGA indicates that the diagonal airing is referred (h5 FC6, h8 FC5, c FC3 and c5 FC), roviding answers erfectly dissociated. hese airs are recommended because the elements of the main diagonal are equal to one, as shown in Eq(). Since the other elements of the matrix are zero, one can conclude that the rocess interaction is only in one direction and recludes the ossibility of the occurrence of destabilizing feedback control. he resilience of the HEN is examined (able ) calculating the DC at the steady state for disturbances of ±8 % on FC, ±0 K on h, ±0 % on FC and ±5 K on c3. In Figure (b) it is resented the roosed modified HEN for nominal conditions, considering this analysis. he modified HEN has three heat exchangers (, and 3), two coolers ( and 5) and one heater (6) with areas of., 6.7,., 30.3, 6. and 3.6 m², resectively. he thermal caacity of the cold utility used to cool stream H (FC6) is the same as in the original HEN (.7 kw/k) but to cool stream H (FC5) it is modified to 7 kw/k. he thermal caacity of the hot utility used to heat stream C (FC7) is 00 kw/k. With the heater, the energy balance involves 3 variables: FC, FC, FC3, FC, FC5, FC6, FC7, h, h, h3, h, h5, h6, h7, h8, h9, h0, c, c, c3, c, c5, c6, c7, c8, Q, Q, Q3, Q, Q5, Q6 and ϕ. he disturbances variables are the same (FC, h, FC and c3) and in addition to the three revious fixed variables (inlet and outlet temeratures of the cold utility and the slit fraction of hot stream H), also the inlet and outlet temeratures of the hot utility (h9 = 573 K and h0 = 57 K) are fixed variables. Again, the network is required to be resilient to disturbances in hot stream H and in cold stream C. A steady state model for the network consists of 9 equations: three equations relative to each heat exchanger and an energy balance for the mixer located in hot stream H. For the modified HEN for the nominal conditions, the number of degrees of freedom is four.

63 Stage Stage Stage Stage FC H FC H h=583 K h6=73 K 3 φ h7 h ( - φ) h3 h c7 h5=33 K c6 c7 h8=553 K 5 FC H FC H h=583 K h6=73 K 3 φ h ( - φ) h3 h7=63 K h c8 h5=33 K c7 c8 h8=553 K 5 c6 c7 c=393 K c=33 K FC3 C c=393 K c=33 K FC3 C c5=553 K c FC c3=388 K C c6=553 K h9 c5=503 K c FC c3=388 K C h0 (a) (b) Figure : Heat Exchanger Network for (a) nominal conditions and (b) modified able : Changes on variables and Disturbance Cost FC h FC c3 FC6 FC5 FC3 FC DC +8 % 0 0 0-0.300-063 -0.979 0.509.035 +8 % +0 K 0 0-0.300-0.0-0.9766.8576.9680 +8 % +0 K +0 % 0-0.300-0.976-0.976.79.8557 +8 % +0 K +0 % +5 K -0.300-0.0-0.988.7.6396 0 +0 K +0 % +5 K 0-98 -087.763.783 0 0 +0 % +5 K 0 0.03 - -.30.36 0 0 0 +5 K 0 965-8 -.358.378-8 % +0 K -0 % +5 K 0.300-98 0.9699.357.566-8 % -0 K -0 % -5 K 0.300 0.0 0.988 -.7.6396 +8 % -0 K +0 % -5 K -0.300 98-0.9699 -.357.566 hus, four controlled variables (h5, h8, c and c6) can be aired with four maniulated variables (FC6, FC5, FC3 and FC7). Using u = and d = for all variables, u = [.7, 7, 3, 00] and d = [8 %, 0 K, 0 %, 5 K], the resulting linearized model is reresented by Eq(). h5 3.7 h8 c 89.6 c6 356.8 5.8 FC 57.7 FC FC 9.7 FC 6 5 3 7.. 3.9 0.3 0. 9.9.5 FC. h 0.6 FC c3 () he steady state RGA comutation is made using matrix P{0}..0.0 RGA (3).0.0 Again RGA indicates that the diagonal airing is referred (h5 FC6, h8 FC5, c FC3 and c6 FC7), roviding answers erfectly dissociated. he resilience of the HEN is examined calculating the DC for disturbances of ±8 % on FC, ±0 K on h, ±0 % on FC and ±5 K on c3 and it is resented in able 3. It is also resented the variation values in the maniulated variables for the disturbances, assuming erfect control. It is noted in able 3 that the rejection of disturbances is achieved by the maniulated variables with little control effort. herefore, the resilience of the network for nominal conditions is accetable when a heater in stream C is added, because it rovides DC values less than or close to one.

6 able 3: Changes on variables and Disturbance Cost for modified Heat Exchanger Network FC h FC c3 FC6 FC5 FC3 FC7 DC +8 % 0 0 0-0.300-03 -0.979 000.03 +8 % +0 K 0 0-0.300-0 -0.9766 000.07 +8 % +0 K +0 % 0-0.300 36-0.976-77.08 +8 % +0 K +0 % +5 K -0.300 303-0.988-77.0359 0 +0 K +0 % +5 K 0 335-087 -77 85 0 0 +0 % +5 K 0 35 - -77 85 0 0 0 +5 K 0-058 -8 000 3-8 % +0 K -0 % +5 K 0.300-39 0.9699 77.09-8 % -0 K -0 % -5 K 0.300-303 0.988 77.0359 +8 % -0 K +0 % -5 K -0.300 39-0.9699-77.09. Conclusions It is necessary to consider the concets of flexibility, controllability and resiliency for designing a heat exchanger network due to the uncertainties and disturbances in industrial rocesses. his was the motivation to describe the C&R analysis, using RGA and DC index. his aer resented the rocedure used by Seider et al. (009) to analyze the C&R in steady state. A literature examle, which only aimed to synthesize a multieriod HEN, was used to aly the C&R analysis. It was resented the diagonal airing as referred. Furthermore, the analysis erformed in the resent aer indicated that a HEN modified with the addition of a heater in stream C exhibits suerior erformance than the original HEN from the controllability oint of view. References Bristol E. H., 966. On a new measure of interactions for multivariable rocess control. IEEE ransactions on automatic control, AC-,, 33-3. De Leon, C., Miranda, C. B., Andrade, C. M. G. and Ravagnan M. A. S. S., 06. hermodynamically-based Resonse ime as Controllability Indicator in Heat Exchanger Networks. he Canadian Journal of Chemical Engineering, doi:0.00/cjce.770. Escobar, M., rierweiler, J. O., Grossmann, I. E., 03. Simultaneous synthesis of heat exchanger networks with oerability considerations: Flexibility and controllability. Comuters and Chemical Engineering, 55, 58-80. Floudas, C. A., Grossmann, I. E., 987. Synthesis of flexible heat exchanger networks with uncertain flowrates and temeratures. Comuters and Chemical Engineering, (), 39-336. Lewin, D. R., 996. A simle tool for disturbance resiliency diagnosis and feedforward control design. Comuters and Chemical Engineering, 0 (), 3-5. Lewin, D. R., Seider, W. D., Seader, J. D., 00. owards integrated design and control for defect-free roducts. Chater. In: Seferlis, P. & Georgiadis, M. C. he Integration of Process Design and Control, Elsevier, Amsterdam, he Netherlands, ISBN: 0--5557-7,. 533-55. Miranda, C. B., Costa, C. B. B., Caballero, J. A., Ravagnan M. A. S. S., 06. Heat Exchanger Network Otimization for Multile Period Oerations. Industrial & Engineering Chemistry Research, 55, 030-035. Morar M., 983. Design of resilient rocessing lants III: A general framework for the assessment of dynamic resilience. Chemical Engineering Science, 38 (), 88-89. Seider, W. D., Seader, J. D., Lewin, D. R., Widagdo, S. (3 Ed), 009. Product and Process Design Princiles: Synthesis, Analysis and Evaluation, Wiley, USA, 766 s. ISBN: 978--9-863-. Uzturk, D., Akman, U., 997. Centralized and decentralized control of retrofit heat-exchanger networks. Comuters and Chemical Engineering,, S373-S378. Westhalen, D. L., Young, B. R., Svrcek, W. Y., 003. A Controllability Index for Heat Exchanger Networks. Industrial & Engineering Chemistry Research,, 659-667. Yee,. F., Grossmann, I., E., 990. Simultaneous otimization models for heat integration. II. Heat exchanger network synthesis. Comuters and Chemical Engineering,, 65-8.