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Optimal Operation of an Energy Integrated Optimal Operation of an Energy Integrated Optimal Operation of an Energy Integrated Optimal Operation ofFeed an Energy Integrated Batch Reactor Effluent Heat Batch Reactor Feed Effluent Heat Batch Reactor Feed Effluent Heat Batch Reactor Feed Effluent Heat Exchanger System Exchanger System Exchanger System System Exchanger

∗ ∗∗ Sujit S. Jogwar ∗ Prodromos Daoutidis ∗∗ Sujit S. Jogwar ∗ Prodromos Daoutidis ∗∗ Sujit Sujit S. S. Jogwar Jogwar ∗ Prodromos Prodromos Daoutidis Daoutidis ∗∗ ∗ ∗ Department of Chemical Engineering, Institute of Chemical ∗ Department of Chemical Engineering, Institute of Chemical ∗ Department of Chemical Chemical Engineering, Institute of Chemical Chemical Technology, Mumbai, 400019, India (e-mail: Department of Engineering, Institute of Technology, Mumbai, 400019, India (e-mail: Technology, Mumbai, 400019, India (e-mail: [email protected]). Technology, Mumbai, 400019, India (e-mail: [email protected]). ∗∗ Chemical Engineering and Materials Science, ∗∗ Department [email protected]). [email protected]). ∗∗ Department of Chemical Engineering and Materials Science, ∗∗ Department of Engineering and Materials Science, University of Minnesota, Minneapolis, USA (e-mail: Department of Chemical Chemical EngineeringMN and55455, Materials Science, University of Minnesota, Minneapolis, MN 55455, USA (e-mail: University Minneapolis, MN 55455, USA (e-mail: [email protected]) University of of Minnesota, Minnesota, Minneapolis, MN 55455, USA (e-mail: [email protected]) [email protected]) [email protected]) Abstract: Energy integration integration in in batch batch reactors reactors offers offers significant significant savings savings but but at at the the cost cost of of Abstract: Energy Abstract: Energy integration in batch reactors offers significant savings but at the cost of additional operational constraints. In this paper, optimal operation of a batch reactor-feed Abstract: Energy integration in batch reactors offers significant savings but at the cost of additional operational constraints. In this paper, optimal operation of aa batch reactor-feed additional operational constraints. In paper, optimal of reactor-feed effluent heat exchanger system is pursued aa production campaign. coupling between additional operational constraints. In this this for paper, optimal operation operation ofThe a batch batch reactor-feed effluent heat exchanger system is pursued for production campaign. The coupling between effluent heat exchanger system is pursued for a production campaign. The coupling between subsequent batches due to energy integration is exploited to predict and thus plan future batches effluent heatbatches exchanger system isintegration pursued for a production campaign. Theplan coupling between subsequent due to energy is exploited to predict and thus future batches subsequent batches due to energy integration is exploited to predict and thus plan future batches to achieve desired product purity at the end of the production campaign in the presence of subsequent batches due to energy integration is exploited to predict and thus plan future batches to achieve desired product purity at the end of the production campaign in the presence of to achieve desired product purity at the end of the production campaign in the presence of disturbances and time-dependent energy prices. The proposed solution leads to better operation to achieve desired product purity at the end of the production campaign in the presence of disturbances and time-dependent energy prices. The proposed solution leads to better operation disturbances and time-dependent energy prices. The proposed solution leads to better operation compared to a controller with disturbance rejection. disturbances and time-dependent energy prices. The proposed solution leads to better operation compared to a controller with disturbance rejection. compared to to a controller controller with with disturbance disturbance rejection. rejection. compared © 2015, IFAC a(International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Optimization, Batch Process Modeling Keywords: Optimization, Optimization, Batch Batch Process Process Modeling Modeling and and Control, Control, Energy Energy Processes Processes and and Control Control Keywords: and Control, Energy Processes and Keywords: Optimization, Batch Process Modeling and Control, Energy Processes and Control Control 1. MOTIVATION optimal operation over a production campaign. In this 1. MOTIVATION MOTIVATION optimal operation over aa production campaign. In this 1. optimal operation over production campaign. In this paper, we address the problem of optimal operation of 1. MOTIVATION optimal operation over a production campaign. In this paper, we address the problem of optimal operation of paper, we address the problem of optimal operation of energy integrated batch systems with the help of a batch Energy integration in batch process systems is achieved by paper, integrated we addressbatch the problem of optimal operation of energy systems with the help of a batch Energy integration in batch process systems is achieved by energy integrated integrated batch systems with(BR-FEHE) the help help of of a asystem batch reactor-feed effluent heat exchanger Energy integration in batch process systems is achieved by (simultaneous or sequential) optimization of batch schedenergy batch systems with the batch Energy integration in batch process systemsofisbatch achieved by reactor-feed effluent heat exchanger (BR-FEHE) system (simultaneous or sequential) optimization schedreactor-feed effluent heat (BR-FEHE) system as aa representative Control and optimization of (simultaneous orconsumption sequential) ofDeakin, batch ule and energy (Kemp and 1989; effluentexample. heat exchanger exchanger (BR-FEHE) system (simultaneous sequential) optimization optimization batch schedschedas representative example. Control and optimization of ule and energy energyorconsumption consumption (Kemp and and ofDeakin, Deakin, 1989; reactor-feed as a representative example. Control and optimization of repetitive batch systems, in the absence of any energy ule and (Kemp 1989; Zhao et al., 1998; Majozi, 2006; Halim and Srinivasan, as a representative example. Control and optimization of ule and energy consumption (Kemp and Deakin, 1989; repetitive batch systems, in the absence of any energy Zhao et al., 1998; Majozi, 2006; Halim and Srinivasan, repetitive batch systems, in the absence of any energy integration, has been pursued within the framework of Zhao et al., 1998; Majozi, 2006; Halim and Srinivasan, 2008). While such designs promise economic benefits, their repetitive batch systems, in the absence of any energy Zhao et al., 1998; Majozi, 2006; Halim and Srinivasan, integration, has been pursued within the framework of 2008). While While such such designs designs promise promise economic economic benefits, benefits, their their adaptive integration, has been been pursued within within the framework framework of control (Flores-Cerrillo and MacGregor, 2003), 2008). operation is challenging compared to their continuous integration, has pursued the of 2008). While such designs promise economic benefits, their adaptive control (Flores-Cerrillo and MacGregor, 2003), operation is challenging compared to their continuous adaptive control (Flores-Cerrillo and MacGregor, 2003), iterative learning control (Lee and Lee, 2007) and optimal operation is challenging compared to their continuous counterparts. Due to the dynamic nature of batch systems, adaptive control (Flores-Cerrillo and MacGregor, 2003), operation is Due challenging compared to of their continuous learning control (Lee and Lee, 2007) and optimal counterparts. to the the dynamic dynamic nature batch systems, iterative iterative learning (Lee and Lee, 2007) optimal (Srinivasan et al., 2003). of this work counterparts. Due nature of systems, energy integration is constrained temperature iterative learning control control (Lee andThe Lee,novelty 2007) and and optimal counterparts. Due to to the dynamicby nature of batch batch (thermal systems, control control (Srinivasan et al., 2003). The novelty of this work energy integration is constrained by temperature (thermal control (Srinivasan et al., 2003). The novelty of this lies in the efficient utilization of the natural inter-batch energy integration is constrained by temperature (thermal equilibrium) as well as time (availability of cold and hot control (Srinivasan et al., 2003).ofThe novelty ofinter-batch this work work energy integration is constrained by temperature (thermal lies in the efficient utilization the natural equilibrium) as well as time (availability of cold and hot lies in in the theresulting efficient from utilization of integration. the natural natural Specifically, inter-batch coupling energy equilibrium) as time (availability of and hot streams) constraints (Wang Smith, 1995). efficient utilization of the inter-batch equilibrium) as well well as as time and (availability of cold coldThis and has hot lies coupling resulting from energy integration. Specifically, streams) constraints (Wang and Smith, 1995). This has coupling resulting from energy integration. Specifically, we explicitly utilize aa model of the slow batch-to-batch streams) constraints (Wang and Smith, This has resulted two major integration for resulting from energy integration. Specifically, streams)in and strategies Smith, 1995). 1995). This prohas coupling we explicitly utilize model of the slow batch-to-batch resulted inconstraints two major major (Wang integration strategies for batch batch prowe explicitly utilize a model of the slow batch-to-batch dynamics for the solution of the optimization problem resulted in two integration strategies for batch process systems direct energy integration and indirect energy we explicitly utilize a model of the slow batch-to-batch resulted in two major integration strategies for batch prodynamics for the solution of the optimization problem cess systems systems -- direct direct energy energy integration integration and and indirect indirect energy dynamics for the solution of the optimization problem that determines the optimal operating conditions for the cess energy integration (Fernandez et al., 2012). dynamics for the solution of the optimization problem cess systems direct energy integration and indirect energy that determines the optimal operating conditions for the integration (Fernandez (Fernandez et et al., al., 2012). 2012). that determines the optimal operating conditions for the production campaign. integration that determines the optimal operating conditions for the integration (Fernandez et al., 2012). production campaign. The strong dependence between energy integration and production campaign. The strong dependence between energy integration and production campaign. rest of the paper is organized as follows. Section The strong dependence between energy integration and batch schedule means economic benefits are highly The dependence between energy integration and The The rest of the paper is organized as follows. Section batchstrong schedule means that that economic benefits are highly The rest of the paper is organized as follows. Section 2 provides a brief description of the system. Section batch schedule means that economic benefits are highly sensitive to any disturbances. Such disturbances, espeThe rest of the paper is organized as follows. Section3 batch schedule means that economic benefits are highly 22 provides a brief description of the system. Section 3 sensitive to any disturbances. Such disturbances, espeprovides a brief description of the system. Section 3 presents a prediction model to capture the slow intersensitive to any disturbances. Such disturbances, especially the ones changing batch processing times, can reduce 2 provides a brief description of the system. Section 3 sensitive to any disturbances. Such disturbances, espepresents a prediction model to capture the slow intercially the ones changing batch processing times, can reduce presents a prediction model to capture the slow interbatch dynamics. This model is then used in Section 4 for cially the ones changing batch processing times, can reduce the effectiveness of energy integration, and in some cases, presents a prediction model to capture the slow intercially the ones changing batch processing times, can reduce batch dynamics. This model is then used in Section 4 for the effectiveness effectiveness of of energy energy integration, and and in some some cases, the batch dynamics. model is in Section 4 optimization of energy consumption BR-FEHE the can render such integration integration infeasible. and in batch dynamics. This This model is then then used usedin inthe Section 4 for for the effectiveness of energy integration, integration, in some cases, cases, the optimization of energy consumption in the BR-FEHE can render such infeasible. the optimization of energy consumption in the BR-FEHE system for two sets of disturbance studies. can render such integration infeasible. the optimization of energy consumption in the BR-FEHE can render such integration infeasible. system for two sets of disturbance studies. In previous work (Jogwar and Daoutidis, 2014), we have In previous previous work work (Jogwar (Jogwar and and Daoutidis, Daoutidis, 2014), 2014), we we have have system system for for two two sets sets of of disturbance disturbance studies. studies. In shown that the coupling between subsequent batches due In previous work (Jogwar and Daoutidis, 2014), we have shown that the coupling between subsequent batches due 2. ENERGY INTEGRATED BR-FEHE SYSTEM shown that the between subsequent batches due 2. ENERGY INTEGRATED BR-FEHE SYSTEM to energy in batch process systems results in shown thatintegration the coupling coupling between subsequent batches due 2. ENERGY INTEGRATED BR-FEHE SYSTEM to energy integration in batch process systems results in 2. ENERGY INTEGRATED BR-FEHE SYSTEM to energy integration in batch process systems results in slow network-level dynamics. This slow dynamics captures to energy integration in batch process systems results in slow network-level dynamics. This slow dynamics captures One of the frequently used strategy employed in energy slow network-level dynamics. This slow dynamics captures the batch-to-batch evolutionThis of process variables. Such One of the frequently used strategy employed in energy slow network-level dynamics. slow dynamics captures the batch-to-batch evolution of process variables. Such One of the frequently used strategy employed in integrated reactors, especially the ones operating at eleOne of thereactors, frequently used strategy employed in energy energy the evolution of process variables. Such slow dynamics can be triggered aa local (to aa batch) especially the ones operating at elethe batch-to-batch batch-to-batch evolution of by process variables. Such integrated slow dynamics can be triggered by local (to batch) integrated reactors, especially the ones operating at elevated temperature, is to utilize the energy available with integrated reactors, especially the ones operating at eleslow dynamics can be triggered by a local (to a batch) disturbance. Furthermore, it has been shown that simple vated temperature, is to utilize the energy available with slow dynamics can be triggered a shown local (to a batch) disturbance. Furthermore, it has has by been that simple vated temperature, is to utilize the energy available with the reactor effluent (hot stream) to preheat the feed (cold vated temperature, is to utilize the energy available with disturbance. Furthermore, it been shown that simple rejection of such a local disturbance may not lead to the reactor effluent (hot stream) to preheat the feed (cold disturbance. Furthermore, it has been shown that simple rejection of of such aa local local disturbance may may not lead lead to stream). the reactor effluent (hot to the This is achieved by installing aa process-to-process reactor effluent (hot stream) stream) to preheat preheat the feed feed (cold (cold rejection stream). This is achieved by installing process-to-process rejection of such such a local disturbance disturbance may not not lead to to the stream). This is achieved by installing a process-to-process heat exchanger (FEHE) as shown in Figure 1. This system Partial financial support for this work by the National Science stream). This is(FEHE) achievedas byshown installing a process-to-process heat exchanger in Figure 1. This system Partial financial support for this work by the National Science heat (FEHE) as in 1. system uses direct energy integration strategy (direct transfer of Foundation, grant support CBET-1133167 India for work by the Science heat exchanger exchanger (FEHE) as shown shown in Figure Figure 1. This This system Partial uses direct energy integration strategy (direct transfer of Partial financial financial for this this and workthe by Government the National Nationalof Foundation, grant support CBET-1133167 and the Government ofScience India uses direct energy integration strategy (direct transfer of energy from the hot stream to the cold stream). This Department of Science and Technology (DST) INSPIRE & SERBFoundation, grant CBET-1133167 and the Government of India uses direct energy integration strategy (direct transfer of Foundation, of grant CBET-1133167 and (DST) the Government India energy from the hot stream to the cold stream). This Department Science and Technology INSPIRE &ofSERBenergy from the hot stream to the cold stream). This requires the hot stream (reactor effluent) to be available SB/S3/CE/090/2013 grant is gratefully acknowledged. Department of Science and Technology (DST) INSPIRE & SERBenergy from the hot stream to the cold stream). This Department of Science and Technology (DST) INSPIRE & SERBrequires the hot stream (reactor effluent) to be available SB/S3/CE/090/2013 grant is gratefully acknowledged. requires SB/S3/CE/090/2013 requires the the hot hot stream stream (reactor (reactor effluent) effluent) to to be be available available SB/S3/CE/090/2013 grant grant is is gratefully gratefully acknowledged. acknowledged.

Copyright © 2015, 2015 IFAC 1193Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2015 IFAC 1193 Copyright 2015 IFAC 1193 Peer review© of International Federation of Automatic Copyright ©under 2015 responsibility IFAC 1193Control. 10.1016/j.ifacol.2015.09.130

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Q Source tank Tin

FEHE Fin

Fin

TC

Texit

Theater

TH

Fheater

Fired heater

Tout

Treactor

α

Product tank

Freactor BSTR

Fig. 1. BR-FEHE system with direct energy integration Table 1. Nominal values of the process parameters for the BR-FEHE system Reactor

L

L

U

L

U O Batch 1

L

O Batch 2

U

Variable/ Parameter Fin Fheater Freactor Tin TCout,F EHE

O

O

FEHE Heater

O Batch 1

O Batch 2

U

L

O

U

Batch 3

time

Fig. 2. Gantt chart representing schedule for the BRFEHE system (L: loading, O: operating, U: unloading)

THout,F EHE Theater,end Treactor,end CA0 CB0 CA,reactor,end CB,reactor,end CB,out Qheater α k10

at the same time as the cold stream (reactor feed). Thus, such a direct strategy cannot be employed within a batch and typically integration is pursued between hot and cold streams of successive batches. Figure 2 shows the Gantt chart for this system. The FEHE, which is a countercurrent exchanger, operates continuously over specific time intervals. Additional details about the system can be found in Jogwar et al. (2014). Detailed dynamic model of the BR-FEHE system can be obtained via unsteady state material and energy balance equations. A consecutive reaction system (A → B →) is considered wherein the intermediate component is the main product. For such a system, the trade-off between product yield and energy consumption becomes a key issue. We are specifically interested in developing an optimal operation strategy which will allow for (energy) efficient operation in an event of a local disturbance (increased/reduced heating capacity) and/or time-dependent energy price. For reference, the detailed dynamic model is presented in the Appendix section. Table 1 gives definitions and the nominal values for the major process variables. 3. PREDICTION MODEL FOR SLOW BATCH-TO-BATCH DYNAMICS In Jogwar and Daoutidis (2014), it was shown that the BR-FEHE in Figure 1 exhibits dynamics over two time scales. For example, Figure 3 depicts the evolution of the reactor temperature for a disturbance of -20% in the heater duty over the production campaign. The fast time scale captures the dynamics of the process variables in a batch whereas the slow time scale captures the batch-

k20 E1 E2 ∆H1 ∆H1 U A

Description

Value

Furnace feed rate Furnace unloading rate Reactor unloading rate Feed inlet temperature FEHE cold side exit temperature FEHE cold side exit temperature Heater exit temperature Reactor exit temperature Feed concentration, A Feed concentration, B Reactor concentration, A Reactor concentration, B Product concentration, B Heater duty FEHE bypass ratio Kinetic Pre-exponent for reaction A → B Kinetic Pre-exponent for reaction B → C Activation energy for reaction A → B Activation energy for reaction B → C Heat of reaction for reaction A → B Heat of reaction for reaction B → C Heat transfer coefficient Hear transfer area

5.77×10−4 m3 /s 2.31×10−3 m3 /s 5.77×10−4 m3 /s 300.00 K 837.84 K 302.01 K 878.57 K 889.96 K 1000 mol/m3 0 mol/m3 21.66 mol/m3 913.29 mol/m3 913.29 mol/m3 393.05 kW 0.1000 5.35 ×105 s−1 9.22 ×1012 s−1 150.72 kJ/mol 301.44 kJ/mol -44307 J/mol -65938 J/mol 209.2 kW/m2 /K 0.4 m2

to-batch evolution of the process variables. This interbatch dynamics can be effectively used to predict the effect of a local disturbance on the system outputs over a production campaign. Furthermore, it can also be used for efficient utilization of costly resources by manipulating the batches of a production campaign. In this section, we develop a simplified model to capture the batch-to-batch dynamics. The main objective of the model is to capture the effect of the heater duty in a batch (Qheater,k ) which is considered the main disturbance on the purity of the desired product in the tank (CB,out,k ) in the presence of inter-batch coupling through the FEHE. The subscript k represents the batch number. It is assumed that the batch processing times are kept constant.

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CB,reactor,end [mol/m3]

1000

data prediction

800 600 400 200 0 800

850

T

900

reactor,0

[K]

950

1000

(a) CB,reactor,end

Fig. 3. Response of the reactor temperature over the production campaign for a -20% disturbance in heater duty for batch # 2

1050 1000

Reactor The hot reactant is completely transferred from the fired heater to the reactor. This gives Vreactor,0,k = Vheater,0,k and Treactor,0,k = Theater,end,k . For the operating phase of the reactor, Eq. (14) and (15) cannot be integrated analytically. To this end, Eq. (14) is numerically integrated for a variety of Treactor,0,k values in the range of 800-1000K. The resulting values of Treactor,end,k and CB,reactor,end,k are fitted in the following Gaussian functions: Treactor,end,k =

3

aT,i ∗ exp −1 ∗ ((Treactor,0,k − bT,i )/cT,i )2

4

aC,i ∗ exp −1 ∗ ((Treactor,0,k − bC,i )/cC,i )2

i=1

+T0 CB,reactor,end,k =

Treactor,end [K]

Fired heater Given the heater temperature at the beginning of the operating phase (Theater,0,k ), the temperature at the end of the operating phase (Theater,end,k ) can be obtained by integrating Eq. (11) in Appendix with Fin = Fheater = 0 and Vheater,0,k = Vheater,min + Fin ×FEHE runtime: Qheater,k × Heater runtime Theater,end,k = Theater,0,k + Vheater,end,k ρCp (1) where the subscript 0 represents the value at the beginning of the operating phase.

950 data prediction

900 850 800 800

850

T

900

reactor,0

[K]

950

1000

(b) Treactor,end

Fig. 4. Comparison between the actual and fitted data for the batch reactor the FEHE Eq. (8) and (9) were numerically integrated. During the operating phase of the FEHE, the fired heater is loaded with the heated feed. For the various values of the reactor effluent temperature Treactor,end,k , the corresponding temperature of the heater at the end of the loading phase was fitted in the following form: Theater,0,k+1 = 0.8847 × Treactor,end,k + 50.91

(3)

Product tank The product formed in the reactor in the k th batch is added to the product tank where it mixes with the product with T T aT = [328.1 5.063 856.3] , aC = [385 570.8 182.5 468.9] , formed in the earlier batches. The overall and component T T bT = [1135 948.8 930.7] , bC = [902.2 870.8 924.6 829.4] , material balance gives the following evolution equations: T T cT = [180.6 28.94 470.6] , cC = [27.2 37.61 20.35 50.86] , VproductT ank,k = VproductT ank,k−1 T0 = 0.366 and C0 = 0. +Freactor × FEHE runtime Figure 4 shows the comparison between the actual and the Freactor × FEHE runtime fitted data. CB,reactor,end,k CB,out,k = VproductT ank,k FEHE VproductT ank,k−1 The reactor effluent stream and the cold feed are the two + CB,out,k−1 (4) inputs to the FEHE. Assuming constant feed conditions, VproductT ank,k i=1

+C0

(2)

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Table 2. Values of parameters used for optimization Parameter p1 to p20 p21 to p50 CB,out,desired Qheater,min Qheater,max

Value 80 units/kW h 40 units/kW h 913.3 mol/m3 275.13 kW 510.96 kW

Eq. (1), (2), (3) and (4) together capture the evolution of process variables over multiple batches. Specifically, they can predict the effect of a local disturbance in Qheater,k on the purity of the product B (CB,out,k ) over the production campaign. 4. OPTIMAL OPERATION OF THE BR-FEHE SYSTEM Let us now consider optimal operation of this BR-FEHE system in the presence of unanticipated disturbances. Optimal operation for this system is achieved by meeting the desired product purity in the product tank at the end of the campaign (batch # 50) by consuming minimum external energy. It is considered that there is a disturbance in the heater duty during batch # 2 (which triggers the batch-to-batch dynamics). Additionally, it is considered that energy price is time-dependent. The following optimal operation problem is set up: 50 J= pi Qheater,i minimize Qheater,k (2

i=1

subject to

CB,out,k = f (CB,out,k−1 , Qheater,k ) CB,out,50 ≥ CB,out,desired Qheater,min ≤ Qheater,k ≤ Qheater,max (5) where the objective function J represents the total cost of utilities, pi represents the cost of utility during the ith batch, CB,out,desired is the desired product purity at the end of the campaign, and Qheater,min and Qheater,max represent the minimum and maximum duty possible for the fired heater. The function f (·) captures the evolution of CB,out,k over multiple batches and is a combination of Eq. (1), (2), (3) and (4). The values of the various parameters used are tabulated in Table 2.

In the first simulation case study, the disturbance in Qheater,2 is -20% of the nominal value. The optimization problem in Eq. (5) is solved using the NLPSQP solver in gPROMS. After 97 iterations, an optimal profile of heater duty is found for batches 3 through 50. The CPU time taken is only 6.9895s. The BR-FEHE dynamic model in Appendix is then simulated with this optimal input profile for the considered local disturbance. Additionally, this performance is compared with a case where the local disturbance is regulated using a controller of the following form: Vheater,0,k Cp Qheater,k = (Theater,end,set − Theater,0,k ) heater runtime (6) wherein Theater,end,set is the desired set point. Figure 5 depicts the responses of the uncontrolled, controlled and optimal cases. The values of the objective function for

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the uncontrolled case, the controlled case and the optimal case are 273.56, 275.10 and 269.09 units respectively. The uncontrolled case, which represents the openloop response, shows the propagation of the local disturbance over the production campaign. It can be seen that this case leads to the highest product purity which can be attributed to the constructive nature of the disturbance. However, energy consumption in this case (as highlighted by the objective function) is higher than the optimal case. The controlled case tends to compensate for the disturbance in the batches following the local disturbance (without considering time-dependent energy prices) and thus performs less efficiently even compared to the uncontrolled case. The optimal solution defers the use of external energy until it is available at a cheaper rate (see pi values in Table 2). The optimizer does not react to the local disturbance directly but keeps the heater duty close to the nominal value for most of the time. Only when the energy is available at a cheaper rate, extra energy is utilized. Furthermore, less energy is consumed towards the end of the campaign as the purity deviations at that time are not going to significantly change the overall composition of the tank. The optimal case presents a 2.2% reduction in the objective function over the controlled case. Note that the final product purity in the optimal case is slightly lower than the desired value. This can be attributed to the mismatch between the detailed hybrid (continuous + discrete) dynamic model used for simulation and the simplified discrete prediction model used for optimization. One way to handle this is by increasing the product purity specification during optimization. For example, if we set the desired product purity as 914.0 during optimization, Figure 6 shows the corresponding responses. In this case, the savings reduce to 2.0% but the final product purity obtained is greater than the desired value. In the next simulation case, the disturbance in Qheater,2 is +20% of the nominal value. In this case, the optimizer took 108 iterations with CPU time of 7.301s. Figure 7 depicts the response of the BR-FEHE system under this local disturbance for the controlled, uncontrolled and optimal case. The values of the objective function for the uncontrolled case, the controlled case and the optimal case are 276.71, 275.15 and 269.52 units respectively. For the uncontrolled case, the disturbance adversely affects the reactor operation and the final product purity is lower than the desired value. The nature of the openloop responses to disturbances of the same magnitude but different directions (±20%) highlights the nonlinear nature of the batch-to-batch dynamics. Due to this nonlinearity, simple rejection of the local disturbance in the controlled case does not lead to the desired product purity for this local disturbance. The optimal case presents a 2.0% reduction in the objective function over the controlled case. 5. CONCLUDING REMARKS In this paper, we have considered the optimal operation of an energy integrated batch process system. It was shown that the inter-batch coupling introduced due to energy integration provides an opportunity to pursue campaignlevel optimization of energy consumption through the manipulation of individual batches. To capture the interbatch dynamics, a simplified model was developed based

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917

917

CB,out [mol/m3]

Uncontrolled Controlled Optimal CB,out,desired

3

CB,out [mol/m ]

916.5 916 915 914 913

916

Uncontrolled Controlled Optimal CB,out,desired

915.5 915 914.5 914 913.5

912 0

10

20

30

40

batch number

913 0

50

10

(a) CB,out

50

4.5

x 10

Uncontrolled Controlled Optimal Qmin

5

Qheater [W]

[W] heater

5.5

Uncontrolled Controlled Optimal Qmin

5

Q

40

5

x 10

Q

max

4 3.5

4.5

Qmax

4 3.5 3

3 2.5 0

30

(a) CB,out

5

5.5

20

batch number

10

20

30

batch number

40

2.5 0

50

10

20

30

batch number

40

50

(b) Qheater

(b) Qheater

Fig. 5. Product purity and heater duty profile for a -20% disturbance in Qheater,2 on the solution of the material and energy balance equations, and mathematical regression. The effectiveness of the proposed framework was demonstrated with the help of case studies on an energy integrated batch reactorfeed effluent heat exchanger network. Specifically, energy savings of the order of 2% were documented over a (controlled) case without optimization. It can be noted that the case considered here tries to achieve the desired product purity at the end of the campaign with upper/lower bounds on the purity of the product formed during the intermediate batches. The operational strategy considered here does not discard the below-par batch (obtained during the local disturbance) but adjusts the remaining batches (as per slow batch-to-batch dynamics) in the campaign to achieve the desired product purity at the end of the campaign. Such a strategy has additional time and energy savings compared to conventional policy of discarding the below-par batch obtained in the case of such local disturbances. The results in this paper highlight opportunities for campaign-wide run-time optimization for energy integrated batch process systems. The framework presented in this paper has general applicability. The challenge remains to systematically derive

Fig. 6. Product purity and heater duty profile for a -20% disturbance in Qheater,2 with updated desired purity constraint the mathematical description of the slow batch-to-batch dynamics, and is currently being pursued in our research. REFERENCES Fernandez, I., Renedo, C.J., Perez, S.F., Ortiz, A., and Manana, M. (2012). A review: Energy recovery in batch processes. Renew. Sust. Energ. Rev., 16, 2260–2277. Flores-Cerrillo, J. and MacGregor, J.F. (2003). Withinbatch and batch-to-batch inferential-adaptive control of semibatch reactors: A partial least squares approach. Ind. Eng. Chem. Res., 42(14), 3334–3345. Halim, I. and Srinivasan, R. (2008). Multi-objective scheduling for environmentally-friendly batch operations. Comput. Chem. Eng., 25, 847–852. Jogwar, S.S. and Daoutidis, P. (2014). Dynamic characteristics of energy-integrated batch process systems: Insights from two case studies. Ind. Eng. Chem. Res., accepted for publication DOI: 10.1021/ie503811p. Jogwar, S.S., Rangarajan, S., and Daoutidis, P. (2014). Reduction of complex energy-integrated process networks using graph theory. Comput. Chem. Eng., submitted for publication.

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Sujit S. Jogwar et al. / IFAC-PapersOnLine 48-8 (2015) 1192–1197

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APPENDIX

915

Dynamic model for the BR-FEHE system as given in Jogwar et al. (2014) 3

[mol/m ]

910

Uncontrolled Controlled Optimal CB,out,desired

C

B,out

905

dVsourceT ank = −Fin dt ∂TH ∂TH U A (TH − TC ) = −vH − ∂t ∂z ρCp VH

10

20

30

40

batch number

dVheater = Fin − Fheater dt

50

Fin TCout,F EHE − Theater dTheater = dt Vheater

5

x 10

Uncontrolled Controlled Optimal Qmin

5 4.5

max

4 3.5 3 20

30

batch number

40

Qheater Vheater ρCp

+

(11)

dVreactor = Fheater − Freactor dt dCA,reactor Fheater CA0 − CA,reactor = dt Vreactor −E1 −k10 exp CA,reactor RTreactor dCB,reactor Fheater CB0 − CB,reactor = dt Vreactor −E1 +k10 exp CA,reactor RT reactor −E2 −k20 exp CB,reactor RTreactor dTreactor Fheater (Theater − Treactor ) = dt Vreactor

Q

10

(10)

(a) CB,out

Qheater [W]

(9)

TC,z=L = Tin

895 0

2.5 0

(8)

∂TC ∂TC U A (TH − TC ) = vC + ∂t ∂z ρCp VC TH,z=0 = Treactor,end

900

5.5

(7)

50

(b) Qheater

+

Fig. 7. Product purity and heater duty profile for a +20% disturbance in Qheater,2

+

Kemp, I.C. and Deakin, A.W. (1989). The cascade analysis for energy and process integration of batch processes. iii: A case study. Chem. Eng. Res. Des., 67(5), 517–525. Lee, J.H. and Lee, K.S. (2007). Iterative learning control applied to batch processes: An overview. Control Eng. Pract., 15(10), 1306–1318. Majozi, T. (2006). Heat integration of multipurpose batch plants using a continuous-time framework. Appl. Therm. Eng., 26(13), 1369–1377. Srinivasan, B., Bonvin, D., Visser, E., and Palanki, S. (2003). Dynamic optimization of batch processes: Ii. role of measurements in handling uncertainty. Comput. Chem. Eng., 27(1), 27–44. Wang, Y.P. and Smith, R. (1995). Time pinch analysis. Chem. Eng. Res. Des., 73(8), 905–914. Zhao, X.G., O’Neill, B.K., Roach, J.R., and Wood, R.M. (1998). Heat integration for batch processes: part 1: Process scheduling based on cascade analysis. Chem. Eng. Res. Des., 76(6), 685–699.

−∆H1 ρCp −∆H2 ρCp

k10 exp k20 exp

−E1 RTreactor −E2 RTreactor

(12)

(13)

(14)

CA,reactor CB,reactor

dVproductT ank = Freactor dt dCB,out Freactor CB,reactor − CB,out = dt VproductT ank

dTout (1 − α)Freactor THout,F EHE − Tout = dt VproductT ank +

αFreactor (Treactor − Tout ) VproductT ank

(15) (16) (17)

(18)

Note that these equations capture all the three (L, O and U) phases of these equipment.

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