A Modeling Framework to Investigate the Influence of Fouling on the Dynamic Characteristics of PID-Controlled Heat Exchangers and Their Networks

Marian Trafczynski; Mariusz Markowski; Piotr Kisielewski; Krzysztof Urbaniec; Jacek Wernik

doi:10.3390/app9050824

Trafczynski, Marian;Markowski, Mariusz;Kisielewski, Piotr;Urbaniec, Krzysztof;Wernik, Jacek

2019-02-26 00:00:00

applied sciences Article A Modeling Framework to Investigate the Inﬂuence of Fouling on the Dynamic Characteristics of PID-Controlled Heat Exchangers and Their Networks Marian Trafczynski , Mariusz Markowski , Piotr Kisielewski, Krzysztof Urbaniec and Jacek Wernik * Warsaw University of Technology, Faculty of Civil Engineering, Mechanics and Petrochemistry, Institute of Mechanical Engineering, Department of Process Equipment, Lukasiewicza 17, 09-400 Plock, Poland; Marian.Trafczynski@pw.edu.pl (M.T.); mariusz.markowski@pw.edu.pl (M.M.); kisielewskipiotr@outlook.com (P.K.); krzysztof.urbaniec@pw.edu.pl (K.U.) * Correspondence: Jacek.Wernik@pw.edu.pl; Tel.: +48-24-367-2212 Received: 21 January 2019; Accepted: 20 February 2019; Published: 26 February 2019 Abstract: The present work is an extension of the authors’ previous research, where changes in the dynamic behavior of heat exchangers induced by fouling build-up were studied. In the present work, the authors used the previously elaborated and validated mathematical model of transient heat exchange with the inﬂuence of thermal resistance of fouling taken into account. The behavior of speciﬁc Heat Exchanger Networks (HENs) coupled with a Crude Distillation Unit together with their control loops is simulated using Simulink/MATLAB and the inﬂuence of fouling build-up on speciﬁc indices of quality of operation is investigated. According to the presented results, the higher the number of heat exchangers in the PID control loop and the greater the number of heat exchangers interacting in the network, the smaller the inﬂuence of fouling on the control quality indices, and in the extreme case, this inﬂuence may be negligible. This might be caused by the compensation of the negative effects of fouling build-up when the heat exchangers are interacting in the HEN. Nevertheless, potential adverse effects of fouling on HEN operation can be prevented by periodic adjustments of the optimal values of PID gains. Keywords: dynamic heat exchanger model; crude oil fouling; fouling impact; PID control; shell-and-tube heat exchanger network 1. Introduction Heat Exchanger Network (HEN) fouling is a chronical problem encountered in many process industries. The operation of a Heat Exchanger (HE) may be affected by fouling which builds up on the heat transfer surface. For example, fouling of HENs in the oil reﬁning industry results in increased energy consumption (burning extra fuel to compensate for reduced heat recovery), reduced plant throughput when the exchangers are cleaned, and induced costs of cleaning interventions [1,2]. In recent years, various approaches to the mitigation of fouling effects in industrial HEs and HENs have been reported in the literature [3,4]. Fouling leads to the reduction of steady-state heat recovery [5], but also to changes in the transient states of HEs [6] and inefﬁcient control of HEs [7,8] that may have an adverse effect on the overall performance of the HEN [9,10]. In the literature, publications devoted to the effect of fouling on the dynamic behavior of HEs and the role of fouling in control issues are rare and limited in scope [11,12]. For a more regular introduction please refer to the authors’ previous work [6], where the relevant research ﬁeld has been reviewed and key publications have been cited. Figure 1 shows an example of dynamic characteristics of a HE (output signal y as a function of time t, in response to a step change in an input signal that occurred at t = 0) and indicates possible Appl. Sci. 2019, 9, 824; doi:10.3390/app9050824 www.mdpi.com/journal/applsci Appl. Sci. 2019, 9, x FOR PEER REVIEW 2 of 24 Appl. Sci. 2019, 9, 824 2 of 23 Figure 1 shows an example of dynamic characteristics of a HE (output signal y as a function of time t, in response to a step change in an input signal that occurred at t=0) and indicates possible deformation of the characteristics induced by the deterioration of control quality. The dynamic deformation of the characteristics induced by the deterioration of control quality. The dynamic characteristics are described by the following parameters: gain K0, delay time td and time constant t1. characteristics are described by the following parameters: gain K , delay time t and time constant t . 0 d 1 The gain is a coefficient determining the change of the output signal with respect to the change in the The gain is a coefﬁcient determining the change of the output signal with respect to the change in the input control signal; the larger the gain value, the stronger output signal’s response to the input input control signal; the larger the gain value, the stronger output signal’s response to the input signal. signal. The delay time defines the waiting period between the step change in the input signal (in The delay time deﬁnes the waiting period between the step change in the input signal (in Figure 1, Figure 1, t=0) and the change of the output signal (that is, delay time describes the speed of response t = 0) and the change of the output signal (that is, delay time describes the speed of response on given on given control). The time constant is a measure of the capacity of the process and determines how control). The time constant is a measure of the capacity of the process and determines how long after long after the cessation of the delay does the output signal reach 63.2% of its final value. When fouling the cessation of the delay does the output signal reach 63.2% of its ﬁnal value. When fouling builds up builds up on the heat transfer surface, the dynamic characteristics of the HEs operated in the network on the heat transfer surface, the dynamic characteristics of the HEs operated in the network may be may be changed. In a previous work [6], the present authors studied the influence of fouling changed. In a previous work [6], the present authors studied the inﬂuence of fouling deposition in deposition in the individual HE units on their dynamic characteristics and on the quality of their the individual HE units on their dynamic characteristics and on the quality of their control, with the control, with the aim of describing fouling effects quantitatively. Moreover, four examples of control aim of describing fouling effects quantitatively. Moreover, four examples of control loops with single loops with single HE and without significant interactions from the other HEs were investigated. In HE and without signiﬁcant interactions from the other HEs were investigated. In Appendix A, as a appendix A, as a complement to the mentioned closed-loop characteristics, the corresponding values complement to the mentioned closed-loop characteristics, the corresponding values of control-quality of control-quality indices are shown in Table A1. indices are shown in Table A1. Figure 1. Dynamic characteristics of a properly designed control system (gain value K ), and of systems Figure 1. Dynamic characteristics of a properly designed control system (gain value K0), and of with degraded control quality (gain values K > K or K < K ). 1 0 2 0 systems with degraded control quality (gain values K1>K0 or K2<K0). In the present paper, using the validated dynamic model, the behavior of speciﬁc crude preheat In the present paper, using the validated dynamic model, the behavior of specific crude preheat trains (branches of HENs interacting with each other—Figures 2 and 3) together with their PID-control trains (branches of HENs interacting with each other – Figures 2 and 3) together with their PID- loops is simulated and the inﬂuence of fouling build-up on the speciﬁc indices of the quality of control loops is simulated and the influence of fouling build-up on the specific indices of the quality operation is investigated. Figures 2 and 3 show two considered examples of real-life PID-controlled of operation is investigated. Figures 2 and 3 show two considered examples of real-life PID-controlled HENs coupled with a Crude Distillation Unit (CDU). The graphical form of Figures 2 and 3 was HENs coupled with a Crude Distillation Unit (CDU). The graphical form of Figures 2 and 3 was developed by the present authors on the basis of schemes and process data made available by the developed by the present authors on the basis of schemes and process data made available by the owner of two different CDUs that operated in a Polish oil reﬁnery. owner of two different CDUs that operated in a Polish oil refinery. The conclusions of the previous publication [6] seemed to suggest that fouling build-up in heat The conclusions of the previous publication [6] seemed to suggest that fouling build-up in heat exchangers usually leads to the signiﬁcant deterioration of control-quality indices of PID-based control exchangers usually leads to the significant deterioration of control-quality indices of PID-based loops. However, from the results of the present research in which more complex PID-controlled loops control loops. However, from the results of the present research in which more complex PID- in large HENs have been investigated, different conclusions can be drawn. It was found that the larger controlled loops in large HENs have been investigated, different conclusions can be drawn. It was the number of heat exchangers in PID-control loops and the larger the number of interacting heat found that the larger the number of heat exchangers in PID-control loops and the larger the number exchangers in the HEN, the less pronounced is the inﬂuence of fouling on the indices of control quality. of interacting heat exchangers in the HEN, the less pronounced is the influence of fouling on the This observation, being new in the pertinent literature, is supported by the presented simulation results indices of control quality. This observation, being new in the pertinent literature, is supported by the and the discussion of a case study. It may be ascribed to the compensation of adverse effects of fouling presented simulation results and the discussion of a case study. It may be ascribed to the build-up in HENs characterized by strong interactions between the heat exchangers. compensation of adverse effects of fouling build-up in HENs characterized by strong interactions The generated knowledge could lead to the development of methods and techniques to prevent between the heat exchangers. heat-recovery reduction that may occur when HEN control is affected by fouling build-up in the exchangers. Appl. Sci. 2019, 9, x FOR PEER REVIEW 3 of 24 The generated knowledge could lead to the development of methods and techniques to prevent heat-recovery reduction that may occur when HEN control is affected by fouling build-up in the Appl. Sci. 2019, 9, 824 3 of 23 exchangers. Appl. Sci. 2019, 9, x FOR PEER REVIEW 4 of 24 Figure 2. Scheme of the HEN with the PID control loops numbered 1 and 2. Figure 2. Scheme of the HEN with the PID control loops numbered 1 and 2. Figure 3. Scheme of the HEN with the PID control loop number 3. Figure 3. Scheme of the HEN with the PID control loop number 3. 2. Materials and Methods Using multi-cell steady-state model of a HE, a control-theory based approach was proposed for the identification and evaluation of the influence of fouling on the dynamic behavior of the heat exchangers and on the quality of their control. A prerequisite for reliable monitoring of the quality of HEN control under fouling conditions is that acquisition and processing of operation data are well organized. Figure 4 illustrates a simplified scheme of the necessary data flow. It is believed that the proposed approach can be applied to the HENs used in continuously operated process plants of oil refining, chemical, food processing and other industries. This is illustrated by a case study in which HENs coupled with crude distillation units are investigated. The scheme of data flow and its details needed for the minimization of uncertainty margins of the monitoring of HEN control are presented below in four stages. Appl. Sci. 2019, 9, 824 4 of 23 2. Materials and Methods Using multi-cell steady-state model of a HE, a control-theory based approach was proposed for the identiﬁcation and evaluation of the inﬂuence of fouling on the dynamic behavior of the heat exchangers and on the quality of their control. A prerequisite for reliable monitoring of the quality of HEN control under fouling conditions is that acquisition and processing of operation data are well organized. Figure 4 illustrates a simpliﬁed scheme of the necessary data ﬂow. It is believed that the proposed approach can be applied to the HENs used in continuously operated process plants of oil reﬁning, chemical, food processing and other industries. This is illustrated by a case study in which HENs coupled with crude distillation units are investigated. The scheme of data ﬂow and its details needed for the minimization of uncertainty margins of the monitoring of HEN control are presented below in four stages. Appl. Sci. 2019, 9, x FOR PEER REVIEW 5 of 24 Figure 4. Scheme of the identification and evaluation of fouling influence on the control quality in an Figure 4. Scheme of the identiﬁcation and evaluation of fouling inﬂuence on the control quality in an industrial HEN. industrial HEN. 2.1. Stage 1 – Acquisition and Pre-processing of the Data Base A prerequisite for identification of the effect of HE fouling on the quality of HEN control is the availability of geometrical data of all the relevant HEs – both those included in the control loop and other ones that may interact with loop components. Equally important is continuous availability of the data on HEs operation, that is, mass flows, temperatures and chemical compositions of the involved process streams, that are necessary for the determination of physico-chemical properties of the media flowing through HEN components. However, as raw process data may also reflect inaccurate measurements, errors in data transmission and recording, as well transient states of the HEN, it is necessary to apply data pre-processing by filtering, averaging and reconciliation. While filtering is aimed at the elimination of gross errors in the recorded data, averaging (over representative time intervals) is needed for the determination of parameter values that enable Appl. Sci. 2019, 9, 824 5 of 23 2.1. Stage 1—Acquisition and Pre-processing of the Data Base A prerequisite for identiﬁcation of the effect of HE fouling on the quality of HEN control is the availability of geometrical data of all the relevant HEs—both those included in the control loop and other ones that may interact with loop components. Equally important is continuous availability of the data on HEs operation, that is, mass ﬂows, temperatures and chemical compositions of the involved process streams, that are necessary for the determination of physico-chemical properties of the media ﬂowing through HEN components. However, as raw process data may also reﬂect inaccurate measurements, errors in data transmission and recording, as well transient states of the HEN, it is Appl. Sci. 2019, 9, x FOR PEER REVIEW 6 of 24 necessary to apply data pre-processing by ﬁltering, averaging and reconciliation. While ﬁltering is aimed at the elimination of gross errors in the recorded data, averaging (over representative time application of mathematical models of steady-state heat transfer. Finally, data reconciliation makes intervals) is needed for the determination of parameter values that enable application of mathematical it possible to minimize uncertainties induced by measurement errors and deviations from steady models of steady-state heat transfer. Finally, data reconciliation makes it possible to minimize state of the HEN. Appropriate methods of data filtering, averaging and reconciliation are presented uncertainties induced by measurement errors and deviations from steady state of the HEN. Appropriate elsewhere [13,14]. methods of data ﬁltering, averaging and reconciliation are presented elsewhere [13,14]. 2.2. Stage 2 – Identification of fouling effects in HEs in the steady state 2.2. Stage 2—Identiﬁcation of fouling effects in HEs in the steady state For each heat exchanger in the studied HEN, the data base established in Stage 1 is used for For each heat exchanger in the studied HEN, the data base established in Stage 1 is used for determining the characteristics of fouling. The existence of fouling and its time behavior in the HE determining the characteristics of fouling. The existence of fouling and its time behavior in the HE (see example in Figure 5) are represented by the evolution of coefficient Rf of the total thermal (see example in Figure 5) are represented by the evolution of coefﬁcient R of the total thermal resistance resistance of fouling layers on both sides of the heat transfer surface. Parameter Rf is calculated as the of fouling layers on both sides of the heat transfer surface. Parameter R is calculated as the difference difference between thermal resistances of fouled and clean heat transfer surface. between thermal resistances of fouled and clean heat transfer surface. Figure 5. Fouling factor versus time during three years of operation of a selected HE. Figure 5. Fouling factor versus time during three years of operation of a selected HE. The mathematical model includes widely known relationships describing heat transfer The mathematical model includes widely known relationships describing heat transfer phenomena and energy balance in the heat exchanger, as well as multi-cell representation of phenomena and energy balance in the heat exchanger, as well as multi-cell representation of steady- steady-state operation of shell-and-tube HE [15]. It also includes the algorithm of least-squares state operation of shell-and-tube HE [15]. It also includes the algorithm of least-squares based based determination of the heat transfer coefﬁcient that has been presented in earlier publications by determination of the heat transfer coefficient that has been presented in earlier publications by the the present authors [16,17]. Details of the elaborated model are valid for shell-and-tube HEs only, but present authors [16,17]. Details of the elaborated model are valid for shell-and-tube HEs only, but by by adapting the relationships describing the heat transfer phenomena and energy balance for other adapting the relationships describing the heat transfer phenomena and energy balance for other types types of heat exchangers (e.g. spiral or plate HEs), the model can be generalized. of heat exchangers (e.g. spiral or plate HEs), the model can be generalized. 2.3. Stage 3—Development and Validation of the Dynamic HE Model 2.3. Stage 3 – Development and Validation of the Dynamic HE Model The planning of efﬁcient use of HEs under changing operating conditions (e.g., conditions The planning of efficient use of HEs under changing operating conditions (e.g., conditions resulting resulting from fouling build-up with time) requires the application of adequate dynamic models. from fouling build-up with time) requires the application of adequate dynamic models. Stage 3 is Stage 3 is based on the mathematical model proposed by Trafczynski et al. [6], of transient heat based on the mathematical model proposed by Trafczynski et al. [6], of transient heat exchange with exchange with the inﬂuence of thermal resistance of fouling taken into account. the influence of thermal resistance of fouling taken into account. According to the scheme shown in Figure 4, the three main steps of Stage 3 are: According to the scheme shown in Figure 4, the three main steps of Stage 3 are: • Determination of a cell-based dynamic HE model based on the operator transmittances. By solving the equations of the mathematical model, relationships employing operator transmittances can be obtained between disturbances occurring at cell inlet and changes in temperature at the cell outlet. Operator transmittance G(s) is a widely used tool for describing a dynamic system. This step is extensively described in Section 2 of the authors’ previous work [6]. • Implementation of the dynamic HE model. Starting from HEN block diagram in which the role of operator transmittances was visualized and using MATLAB/Simulink program package, a software module was developed to simulate the performance of HEN control. In order to make simulation possible, a database is needed for providing the values of relevant parameters in all the cells at steady state (from stage 2), of all the HEs in the HEN. This step is extensively described in Section 2.2 of the authors’ previous work [6]. Appl. Sci. 2019, 9, 824 6 of 23 Determination of a cell-based dynamic HE model based on the operator transmittances. By solving the equations of the mathematical model, relationships employing operator transmittances can be obtained between disturbances occurring at cell inlet and changes in temperature at the cell outlet. Operator transmittance G(s) is a widely used tool for describing a dynamic system. This step is extensively described in Section 2 of the authors’ previous work [6]. Implementation of the dynamic HE model. Starting from HEN block diagram in which the role of operator transmittances was visualized and using MATLAB/Simulink program package, a software module was developed to simulate the performance of HEN control. In order to make simulation possible, a database is needed for providing the values of relevant parameters in all Appl. Sci. the 2019 cells , 9 at , x FO steady R PEER state REVIEW (from stage 2), of all the HEs in the HEN. This step is extensively described 7 of 24 in Section 2.2 of the authors’ previous work [6]. • Validation of the dynamic HE model using operational data of a real-life HEN coupled with a Validation of the dynamic HE model using operational data of a real-life HEN coupled with a CDU. The values of simulated and real temperature at heat exchanger outlet in transient states CDU. The values of simulated and real temperature at heat exchanger outlet in transient states of the exchanger were compared and found to be in close agreement. This step is in more detail of the exchanger were compared and found to be in close agreement. This step is in more detail described in Section 3 of the authors’ previous work [6]. described in Section 3 of the authors’ previous work [6]. 2.4. Stage 4 – Identification and Evaluation of the Influence of Fouling on the Dynamic Behavior of PID- 2.4. Stage 4—Identiﬁcation and Evaluation of the Inﬂuence of Fouling on the Dynamic Behavior of PID-Controlled HEs and on the Control-Quality Indices Controlled HEs and on the Control-Quality Indices The ﬁrst step is to study the open-loop unit step responses simulated at different periods of The first step is to study the open-loop unit step responses simulated at different periods of fouling build-up for all controlled HEs. fouling build-up for all controlled HEs. When parameter R is increased, the thermal inertia of the HE is changed, leading to changes in When parameter Rf is increased, the thermal inertia of the HE is changed, leading to changes in its dynamic behavior. Such changes can be detected by studying the open-loop unit step responses its dynamic behavior. Such changes can be detected by studying the open-loop unit step responses simulated at different stages of fouling build-up. A typical response of a HE system is illustrated in simulated at different stages of fouling build-up. A typical response of a HE system is illustrated in Figure 6a. In Figure 6b, open-loop responses of the HE model are plotted for a step upset +5% in the Figure 6a. In Figure 6b, open-loop responses of the HE model are plotted for a step upset +5% in the shell-side ﬂowrate M . As can be seen in these responses, fouling build-up on the exchanger ’s heat shell-side flowrate M ss. As can be seen in these responses, fouling build-up on the exchanger’s heat transfer surface induces changes in the values of gain K , delay time t and time constant t . transfer surface induces changes in the values of gain Ko, delay time td and time constant t1. o d 1 Figure 6. A typical open-loop unit step response of a thermal system (a) and the step responses of a Figure 6. A typical open-loop unit step response of a thermal system (a) and the step responses of a HE HE u under nder fou fouling ling conditions conditions ( ( bb ).). The second step is modeling of a control-loop unit including the dynamic exchanger model The second step is modeling of a control-loop unit including the dynamic exchanger model together with PID control conﬁgurations in Matlab-Simulink (see Figure 7a). In the next two steps, together with PID control configurations in Matlab-Simulink (see Figure 7a). In the next two steps, assuming PID control of the exchanger unit, three gain coefﬁcients (proportional K , integral K and assuming PID control of the exchanger unit, three gain coefficients (proportional K p p, integral Ki and derivative K ) are needed to determine its closed-loop characteristics. The values of gain coefﬁcients derivative K dd) are needed to determine its closed-loop characteristics. The values of gain coefficients can be determined by the Ziegler-Nichols method [18], which is commonly used in industry. can be determined by the Ziegler-Nichols method [18], which is commonly used in industry. Appl. Sci. 2019, 9, x FOR PEER REVIEW 7 of 24 • Validation of the dynamic HE model using operational data of a real-life HEN coupled with a CDU. The values of simulated and real temperature at heat exchanger outlet in transient states of the exchanger were compared and found to be in close agreement. This step is in more detail described in Section 3 of the authors’ previous work [6]. 2.4. Stage 4 – Identification and Evaluation of the Influence of Fouling on the Dynamic Behavior of PID- Controlled HEs and on the Control-Quality Indices The first step is to study the open-loop unit step responses simulated at different periods of fouling build-up for all controlled HEs. When parameter Rf is increased, the thermal inertia of the HE is changed, leading to changes in its dynamic behavior. Such changes can be detected by studying the open-loop unit step responses simulated at different stages of fouling build-up. A typical response of a HE system is illustrated in Figure 6a. In Figure 6b, open-loop responses of the HE model are plotted for a step upset +5% in the shell-side flowrate Ms. As can be seen in these responses, fouling build-up on the exchanger’s heat transfer surface induces changes in the values of gain Ko, delay time td and time constant t1. Figure 6. A typical open-loop unit step response of a thermal system (a) and the step responses of a HE under fouling conditions (b). The second step is modeling of a control-loop unit including the dynamic exchanger model together with PID control configurations in Matlab-Simulink (see Figure 7a). In the next two steps, assuming PID control of the exchanger unit, three gain coefficients (proportional Kp, integral Ki and Appl. Sci. 2019, 9, 824 7 of 23 derivative Kd) are needed to determine its closed-loop characteristics. The values of gain coefficients can be determined by the Ziegler-Nichols method [18], which is commonly used in industry. Appl. Sci. 2019, 9, x FOR PEER REVIEW 8 of 24 Figure 7. Scheme of a HE unit with PID temperature control (a) and closed-loop step responses of HE Figure 7. Scheme of a HE unit with PID temperature control (a) and closed-loop step responses of HE models: typical response (b), without adjustment (c) and after adjustment (d) of PID tuning parameters. models: typical response (b), without adjustment (c) and after adjustment (d) of PID tuning parameters. For speciﬁc dynamic characteristics similar to that shown in Figure 7b, description and evaluation of the control quality can be based on the quality indices including: For specific dynamic characteristics similar to that shown in Figure 7b, description and evaluat Overshoot ion of the cont M —per rol q centage uality caof n be the based maximum on the qudeviation ality indiceof s inc step luding response : y(t) from its steady-state value: • Overshoot Mp – percentage of the maximum deviation of step response y(t) from its steady-state M = ((y y )/y ) 100%, (1) p max ss ss value: where: y y(t) at its maximum, y y(t) at steady state (y y ). The value of overshoot is max ss ss max Mp = ((ymax – yss) / yss) x 100%, (1) determined during control-system design and may be used as a measure of system stability; large where: ymax – y(t) at its maximum, yss – y(t) at steady state (yss ≤ ymax). The value of overshoot is overshoot values are not recommended. determined during control-system design and may be used as a measure of system stability; large Peak time t —time interval to the maximum of y(t), that is, y(t ) = y . p p max overshoot values are not recommended. Delay time t —time interval to the step response reaching 50% of its value at steady state, that is, • Peak time tp – time interval to the maximum of y(t), that is, y(tp) = ymax. y(t ) = 0.5 y . ss • Delay time td – time interval to the step response reaching 50% of its value at steady state, that Rise time t —time interval to the step response reaching 80% of its value at steady state, that is, is, y(td) = 0.5 yss. y(t ) = 0.8 y . r ss • Rise time tr – time interval to the step response reaching 80% of its value at steady state, that is, Settling time t —time interval to the step response staying within the tolerance margin of its y(tr) = 0.8 yss. steady-state value, usually y 5% (see Figure 7b). ss • Settling time ts – time interval to the step response staying within the tolerance margin of its Being steadyeasy -state v toadetermine, lue, usually the yss ±abovementioned 5% (see Figure 7bquality ). indices can be used to evaluate the characteristics of the control system on the basis of its response to step changes of process variables. Being easy to determine, the abovementioned quality indices can be used to evaluate the The dynamic characteristics were simulated at different stages of fouling build-up, that is, after one, characteristics of the control system on the basis of its response to step changes of process variables. two and three years of the continuous operation of the HE unit. Initially, transient responses of an The dynamic characteristics were simulated at different stages of fouling build-up, that is, after one, exemplary HE unit were simulated assuming PID control with constant values of the gain coefﬁcients two and three years of the continuous operation of the HE unit. Initially, transient responses of an that were determined for clean heat exchange surface (that is, without fouling). As can be seen in the exemplary HE unit were simulated assuming PID control with constant values of the gain coefficients responses obtained for the consecutive periods of HE operation (Figure 7c), the build-up of fouling and that were determined for clean heat exchange surface (that is, without fouling). As can be seen in the the increased thermal resistance would lead to oscillations of the controlled temperature, a too slow responses obtained for the consecutive periods of HE operation (Figure 7c), the build-up of fouling response to set-point changes and the risk of signiﬁcant temperature overshoot that may be dangerous and the increased thermal resistance would lead to oscillations of the controlled temperature, a too especially during the execution of start-up procedures. However, adverse changes in control quality slow response to set-point changes and the risk of significant temperature overshoot that may be can be prevented by periodic adjustments of the gain coefﬁcients. This can be seen in Figure 7d, which dangerous especially during the execution of start-up procedures. However, adverse changes in depicts simulated step responses of an exemplary HE with controller tuning parameters adjusted for control quality can be prevented by periodic adjustments of the gain coefficients. This can be seen in Figure 7d, which depicts simulated step responses of an exemplary HE with controller tuning parameters adjusted for the consecutive periods of fouling build-up. These characteristics indicate that if the real-life controller tuning was adjusted to fit the requirements of efficient control, then despite increased values of the thermal resistance, the indices of control quality would not be adversely affected. Overall, the presented results for control loop of a HE (Figure 7) indicate that if the thermal resistance of fouling is increased, unchanged parameters of controller tuning could lead to the deterioration of the indices of control quality. By adjusting the values of proportional-integral- derivative gains Kp, Ki, Kd, these adverse effects of fouling could be prevented. For a given value of the thermal resistance of fouling, appropriate gain values could be determined using the dynamic Appl. Sci. 2019, 9, 824 8 of 23 the consecutive periods of fouling build-up. These characteristics indicate that if the real-life controller tuning was adjusted to ﬁt the requirements of efﬁcient control, then despite increased values of the thermal resistance, the indices of control quality would not be adversely affected. Overall, the presented results for control loop of a HE (Figure 7) indicate that if the thermal resistance of fouling is increased, unchanged parameters of controller tuning could lead to the deterioration of the indices of control quality. By adjusting the values of pr Appl. oportional-integral-derivative Sci. 2019, 9, x FOR PEER REVIEW gains K , K , K , these adverse effects of fouling could be prevented. 9 of 24 i d For a given value of the thermal resistance of fouling, appropriate gain values could be determined model of the heat exchanger and the suitability of these values can be tested by simulation - which is using the dynamic model of the heat exchanger and the suitability of these values can be tested by the last step in the Stage 4 of the proposed procedure (see Figure 4). simulation - which is the last step in the Stage 4 of the proposed procedure (see Figure 4). 3. Case Study—Results 3. Case study – Results In In order order to toinvestigate investigate t the he inﬂuence influenceof offouling fouling build-up build-up on the on the dynamic dynamicbehavior behavior of the HENs of the HENs and on the quality of their control, two cases (the real-life HENs coupled with a CDU plant—see and on the quality of their control, two cases (the real-life HENs coupled with a CDU plant – see Figur Figures es 22 and and 3) we 3) werere co consider nsider ed. ed. F Fractional ractional distillation distillation of cr of crude ude oil oil isis aa hig highly hly ener energy-intensive gy-intensive process that requires the crude to be heated from ambient temperature to around 370 C. The required process that requires the crude to be heated from ambient temperature to around 370°C. The required heat heatis is prov provided ided t thr hrough ough aa set setof of HE HEss in in which which heat heat fr fr om om t the he dist distillation illation prod products uctsand and p pump-ar ump-around ound streams of the distillation columns is recovered, and a furnace fuelled by heavy fuel oil. The crude streams of the distillation columns is recovered, and a furnace fuelled by heavy fuel oil. The crude is is pumped thro pumped thr ugh ough the the first p ﬁrst art o part f the of t HE he N to HEN a desalting to a desalting unit where i unit wher t is washed e it is washed with wa with ter to water remove to remove inorganic water-soluble impurities. After that, the crude ﬂows through the second HEN part, inorganic water-soluble impurities. After that, the crude flows through the second HEN part, and and further further to the to the furnac furnace e whe wher re it e is he it is heated ated up up to to the t the temperatur emperaturee n needed eeded for for ent entering ering the the fractional fractional distillation column. distillation column. Using Using opera operational tional da data ta a available vailable from the peri from the period od o off three three years years of c of continuous ontinuous HEN HEN operation operation,, exchanger characteristics were studied at different stages of fouling build-up, that is, after 1, 2 and exchanger characteristics were studied at different stages of fouling build-up, that is, after 1, 2 and 3 3year years s (p (passed assed from from op operation eration st start-up art-up w when hen HE HEsur surfaces faces h had ad been been clean clean). ). 3.1. Case No. 1 3.1. Case No. 1 In case no. 1 four branches ABCD (the crude preheat trains) were selected from a real-life HEN In case no. 1 four branches ABCD (the crude preheat trains) were selected from a real-life HEN coupled with a CDU rated 110 kg/s of crude oil. Twenty-six shell-and-tube, two-pass HEs with straight coupled with a CDU rated 110 kg/s of crude oil. Twenty-six shell-and-tube, two-pass HEs with tubes and ﬂoating heads are connected as schematically shown in Figures 2 and 8. straight tubes and floating heads are connected as schematically shown in Figures 2 and 8. Figure 8. Scheme of the HEN with PID-control loops 1 and 2 implemented in Simulink/MATLAB. Figure 8. Scheme of the HEN with PID-control loops 1 and 2 implemented in Simulink/MATLAB. Owing to limited measurement data, it was not possible to determine the relationship between the thermal resistance of fouling Rf (fouling factor) and time t, for each HE. The measurements of temperature and mass flow were performed only at the inlet and outlet of the studied HEN but no temperature measurements were available between the HEs. In order to resolve this issue, the Rf values of HEs that were used in the simulation studies had been postulated by the authors on the basis of values recommended by TEMA standards [19] (see Table 1). As demonstrated in reference [6], such thermal resistance values may significantly affect the performance of HE control. As shown in the HEN scheme in Figure 2, the crude-oil feed stream is split in parallel branches A and B before the desalting unit and in parallel branches C and D after the Appl. Sci. 2019, 9, 824 9 of 23 Owing to limited measurement data, it was not possible to determine the relationship between the thermal resistance of fouling R (fouling factor) and time t, for each HE. The measurements of temperature and mass ﬂow were performed only at the inlet and outlet of the studied HEN but no temperature measurements were available between the HEs. In order to resolve this issue, the R values of HEs that were used in the simulation studies had been postulated by the authors on the basis of values recommended by TEMA standards [19] (see Table 1). Table 1. Values of the R and heat transfer coefﬁcient for the studied HEN in case no. 1. 3 3 Total Heat Total Heat Fouling Factor R 10 Fouling Factor R 10 HE HE f f 2 2 Transfer Transfer no. no. (m K/W) after Period of (m K/W) after Period of Coefﬁcient for Coefﬁcient Operation: Operation: Clean HE for Clean HE 1 2 3 1 2 3 2 2 U (W/m K) U (W/m K) year years years year years years E1-11AB 0.903 1.118 2.235 728 E2-11 0.608 1.115 2.235 691 E1-12 0.681 1.116 2.235 572 E2-12AB 0.701 1.116 2.235 698 E1-13 1.011 1.118 2.235 363 E2-13 1.332 2.574 5.160 56 E1-14 0.846 1.117 2.235 831 E2-14 0.913 1.374 2.752 346 E1-15 0.869 1.117 2.235 633 E2-15AB 0.789 1.116 2.235 436 E1-16 0.834 1.117 2.235 636 E2-16 0.900 1.374 2.752 518 E1-21 1.104 1.376 2.752 546 E2-21 0.678 1.116 2.235 858 E1-22 0.700 1.116 2.235 810 E2-22 0.828 1.117 2.235 467 E1-23 0.753 1.117 2.235 417 E2-23 0.592 1.115 2.235 891 E1-24 0.679 1.116 2.235 648 E2-24 0.926 1.117 2.235 502 E1-25 0.951 1.375 2.752 649 E2-25AB 0.842 1.373 2.235 673 E1-26 0.992 1.118 2.235 898 E2-26 0.931 1.375 2.752 280 E1-27 0.913 1.373 2.752 678 E2-27 0.900 1.374 2.752 197 As demonstrated in reference [6], such thermal resistance values may signiﬁcantly affect the performance of HE control. As shown in the HEN scheme in Figure 2, the crude-oil feed stream is split in parallel branches A and B before the desalting unit and in parallel branches C and D after the desalting unit. For the two control loops with PID controllers 1 and 2 as indicated in Figure 2, the split ratios in branch pairs AB and CD are adopted as manipulated variables (which can be changed by the action of control valves 1AB and 2AB), while the controlled variables are deﬁned as the differences between the studied outlet temperatures: CV = T T , CV = T T . The control objective is 1 A B 2 C D to maximize heat recovery, understood as total heat ﬂow Q transferred in the HEN, and the setpoint values of the controlled variables should be CV = CV = 0. In other words, when process disturbances 1 2 occur, the controllers installed in the HEN are required to adjust the split ratios in network branches AB and CD to ensure that the values of the controlled variables return to zero. 3.2. Case No. 2 In case no. 2 the crude preheat trains were selected from another real-life HEN coupled with a CDU rated 220 kg/s of crude oil. Fourteen shell-and-tube, two-pass HEs with straight tubes and ﬂoating heads are connected as schematically shown in Figures 3 and 9. For each HE, the relationship between the fouling factor R and time t, was determined using method described in the work [16]. In this case, all measurements of temperature and mass ﬂow at the inlet and outlet of the studied HEs were available. Obtained R values of HEs that were used in the simulation studies are presented in Table 2. As shown in the HEN scheme in Figure 3, the desalted crude-oil stream is split in parallel branches and after exchangers E3-14AB and E3-16ABC the branches are connected again into the one preheat train. There is one simple control setup with PID controller 3. In the control loop 3 with exchanger E3-18AB as indicated in Figure 3, the controlled variable is the tube-side outlet temperature T before the preﬂash column, while the manipulated variable is shell-side by-pass mass ﬂow rate (which can be changed by the action of control valve 3). The other process variables are the disturbances. Appl. Sci. 2019, 9, x FOR PEER REVIEW 11 of 24 Appl. Sci. 2019, 9, 824 10 of 23 Figure 9. Scheme of the HEN with PID-control loop 3 implemented in Simulink/MATLAB. Figure 9. Scheme of the HEN with PID-control loop 3 implemented in Simulink/MATLAB. Table 2. Values of the R and heat transfer coefﬁcient for the studied HEN in case no. 2. Table 2. Values of the Rf and heat transfer coefficient for the studie Td otal HEN Heat in case no. 2. HE Fouling Factor 3 2 Transfer no. R 10 (m K/W) HE Fouling Factor Total Heat Coefﬁcient after Period of Operation: −3 2 no. Rf×10 (m K/W) Transfer for Clean HE 1 2 3 after Period of Operation: Coefficient U (W/m K) year years years 1 2 3 for Clean HE E3-11AB 1.869 year 2.672 years years 3.611 U (W/m K) 429 E3-12AB E3-11AB 1.486 1.869 2.478 2.672 3.611 3.437 429 434 E3-13AB E3-12AB 0.957 1.486 1.522 2.478 3.437 2.589 434 237 E3-14AB E3-13AB 1.563 0.957 2.587 1.522 2.589 3.523 237 443 E3-15AB E3-14AB 0.982 1.563 1.207 2.587 3.523 1.694 443 294 E3-16ABC E3-15AB 1.234 0.982 1.894 1.207 1.694 2.896 294 295 E3-16ABC 1.234 1.894 2.896 295 E3-17ABCD 1.623 2.543 3.431 810 E3-17ABCD 1.623 2.543 3.431 810 E3-18AB 0.323 0.623 1.196 745 E3-18AB 0.323 0.623 1.196 745 E3-21ABCD 1.587 2.452 3.257 894 E3-21ABCD 1.587 2.452 3.257 894 E3-22AB 0.128 0.273 0.532 327 E3-22AB 0.128 0.273 0.532 E3-23AB 2.077 4.448 6.075 383 E3-23AB 2.077 4.448 6.075 383 E3-24AB 0.444 0.699 1.259 623 E3-24AB 0.444 0.699 1.259 623 E3-25AB 0.677 0.823 1.647 603 E3-25AB 0.677 0.823 1.647 603 E3-26ABC 0.279 1.116 1.628 410 E3-26ABC 0.279 1.116 1.628 410 3.3. Dynamic analysis of the HEN 3.3. Dynamic analysis of the HEN 3.3.1. Study the Open-loop Step Responses in Case No. 1 3.3.1. Study the Open-loop Step Responses in Case No. 1 For the different periods of HEN operation during which fouling was building up, simulations For the different periods of HEN operation during which fouling was building up, simulations have been carried out in Simulink. According to the obtained results, when the thermal resistance of have been carried out in Simulink. According to the obtained results, when the thermal resistance of fouling is increased, the thermal inertia of every HE is changed leading to changes in the dynamic fouling is increased, the thermal inertia of every HE is changed leading to changes in the dynamic behavior of the interacting A, B, C, D branches shown in Figure 2. Such changes can be detected by behavior of the interacting A, B, C, D branches shown in Figure 2. Such changes can be detected by studying the open-loop step responses of the end temperatures (after parallel branches T , T ) ABend CDend studying the open-loop step responses of the end temperatures (after parallel branches TABend, TCDend) Appl. Sci. 2019, 9, x FOR PEER REVIEW 12 of 24 Appl. Sci. 2019, 9, 824 11 of 23 Appl. Sci. 2019, 9, x FOR PEER REVIEW 12 of 24 simulated at the different stages of fouling build-up. The features of a typical response of a heat exchanger system (network branch) are illustrated in Figure 6a. simulated at the different stages of fouling build-up. The features of a typical response of a heat simulated at the different stages of fouling build-up. The features of a typical response of a heat In Figure 10a,b, simulated open-loop responses of the studied HEN models are plotted for a step exchanger exchanger system system (network (network branch) branch) are illustr are illustrated ated in Fig in Figur ure 6a. e 6a. upset +1°C in the tube-side inlet temperatures of the branches A (exchanger E1-11AB) and B In Figure 10a,b, simulated open-loop responses of the studied HEN models are plotted for a In Figure 10a,b, simulated open-loop responses of the studied HEN models are plotted for a step (exchanger E1-21). The open-loop responses of the branches to a step upset +1°C in the shell-side inlet step upset upset +1°C +1in the tube- C in the tube-side side inle inlet t tem temperatur peratures of es of the bra the branches nches AA ((exchanger exchanger E1 E1-11AB) -11AB) a and nd B B temperatures of the exchangers (E1-13, 14, 27 and E2-14, 15AB, 25AB, 26, 27), are presented in Figure (exchanger E1-21). The open-loop responses of the branches to a step upset +1 C in the shell-side (exchanger E1-21). The open-loop responses of the branches to a step upset +1°C in the shell-side inlet 11a,b. Next, Figure 12a,b shows the open-loop responses of the branches to +10% step change in the inlet temptemperatur eratures of t es he exchan of the exchangers gers (E1-13 (E1-13, , 14, 2714, and 27 E2 and -14, 15AB, 25A E2-14, 15AB, B, 25AB, 26, 27), 26, are presented 27), are presented in Figuin re shell-side flowrate of the HEs (+8.61 kg/s in E1-14, +3.75 kg/s in E1-27, +1.22 kg/s in E2-11, +2.55 kg/s Figur 11a,b. e Next, Fi 11a,b. Next, gure 12 Figur a,b shows the open- e 12a,b shows theloop responses of open-loop responses the bra of the nches to +10% step cha branches to +10% step nge i change n the in E2-12AB, +3.46 kg/s in E2-23 and +1.59 kg/s in E2-24). Finally, Figure 13a,b depicts the open-loop in shel the l-sshell-side ide flowrat ﬂowrate e of the HEs of the (+HEs 8.61 k (+8.61 g/s in E kg/s 1-14, in+3 E1-14, .75 kg/s i +3.75 n E1- kg/s 27, in +1.2 E1-27, 2 kg/s +1.22 in E2-1 kg/s 1, +2.5 in E2-11, 5 kg/s responses of the HEN models to +10% and -10% step change in the tube-side flowrates of branches +2.55 in E2-12 kg/s AB, in+3 E2-12AB, .46 kg/s +3.46 in E2-23 kg/s anin d +1 E2-23 .59 kg/s and i+1.59 n E2-24 kg/s ). Fi in naE2-24). lly, Figure 13a Finally,,b Figur depi ects the open- 13a,b depictsloop the A, C (+6.11 kg/s) and B, D (-6.11 kg/s), respectively. open-loop responses o responses f the HEN models to +10% of the HEN models and -10% ste to +10% and p -10% change in t step change he tube-side flowrate in the tube-sides of b ﬂowrates ranches of branches A, C (+6.11 A kg/s) , C (+6.11 and kg/s) B, D (-6 and .11 k B, g D /s) ( , respecti 6.11 kg/s), velyr . espectively. Figure 10. Open-loop responses of the HEN models ((a) – the end temperature TABend and (b) – the end temperature TCDend, after parallel branches) to +1 °C step change in the tube-side inlet temperature of Figure 10. Open-loop responses of the HEN models ((a)—the end temperature T and (b)—the end ABend Figure 10. Open-loop responses of the HEN models ((a) – the end temperature TABend and (b) – the end the branches A and B. temperature T , after parallel branches) to +1 C step change in the tube-side inlet temperature of CDend temperature TCDend, after parallel branches) to +1 °C step change in the tube-side inlet temperature of the branches A and B. the branches A and B. Figure 11. Open-loop responses of the HEN models ((a)—the end temperature T and (b)—the ABend Figure 11. Open-loop responses of the HEN models ((a) – the end temperature TABend and (b) – the end end temperature T , after parallel branches) to +1 C step change in the shell-side inlet temperature CDend temperature TCDend, after parallel branches) to +1°C step change in the shell-side inlet temperature of of Figure 11 selected . Ope exchangers. n-loop responses of the HEN models ((a) – the end temperature TABend and (b) – the end selected exchangers. temperature TCDend, after parallel branches) to +1°C step change in the shell-side inlet temperature of selected exchangers. Appl. Sci. 2019, 9, x FOR PEER REVIEW 13 of 24 Appl. Sci. 2019, 9, 824 12 of 23 Appl. Sci. 2019, 9, x FOR PEER REVIEW 13 of 24 Figure 12. Open-loop responses of the HEN models ((a) – the end temperature TABend and (b) – the end Figure temperature 12. Open-loop TCDend, after responses parallel branches) of the HEN to models +10% step ((a)—the change end in the shell-side temperature T flowrate and of se (b)—the lected ABend Figure 12. Open-loop responses of the HEN models ((a) – the end temperature TABend and (b) – the end end exchangers. temperature T , after parallel branches) to +10% step change in the shell-side ﬂowrate of CDend temperature TCDend, after parallel branches) to +10% step change in the shell-side flowrate of selected selected exchangers. exchangers. Figure 13. Open-loop responses of the HEN models ((a)—the end temperature T and (b)—the ABend Figure 13. Open-loop responses of the HEN models ((a) – the end temperature TABend and (b) – the end end temperature T , after parallel branches) to +10% step change in tube-side ﬂowrate of branch A CDend temperature TCDend, after parallel branches) to +10% step change in tube-side flowrate of branch A and Figure 13. Open-loop responses of the HEN models ((a) – the end temperature TABend and (b) – the end and C, and to 10% step change in ﬂowrate of branch B and D. C, and to -10% step change in flowrate of branch B and D. temperature TCDend, after parallel branches) to +10% step change in tube-side flowrate of branch A and C, and to -10% step change in flowrate of branch B and D. It can be seen in the open-loop responses that in each of the studied branches, variations induced It can be seen in the open-loop responses that in each of the studied branches, variations induced by fouling build-up on the exchangers’ heat transfer surfaces are visible in the values of gain K , delay by fouling build-up on the exchangers’ heat transfer surfaces are visible in the values of gain K0, delay It can be seen in the open-loop responses that in each of the studied branches, variations induced time t and time constant t . In practical terms, the changes in the delay time in the most open-loop step d 1 time td and time constant t1. In practical terms, the changes in the delay time in the most open-loop by fouling build-up on the exchangers’ heat transfer surfaces are visible in the values of gain K0, delay responses are insigniﬁcant but the increased/decreased time constants and reduced/increased gain step responses are insignificant but the increased/decreased time constants and reduced/increased time td and time constant t1. In practical terms, the changes in the delay time in the most open-loop values may impair the quality of PID control considerably. In order to prevent that from happening, it gain values may impair the quality of PID control considerably. In order to prevent that from step responses are insignificant but the increased/decreased time constants and reduced/increased is advisable to investigate all the three components of the tuning of each PID controller (K , K , K ) p i d happening, it is advisable to investigate all the three components of the tuning of each PID controller gain values may impair the quality of PID control considerably. In order to prevent that from that is, gain values in the proportional, integral and derivative components) and to check the resulting (Kp, Ki, Kd) that is, gain values in the proportional, integral and derivative components) and to check happening, it is advisable to investigate all the three components of the tuning of each PID controller transient responses. the resulting transient responses. (Kp, Ki, Kd) that is, gain values in the proportional, integral and derivative components) and to check PID controllers for loops 1 and 2 (see Figure 2) were separately tuned according to the Skogestad PID controllers for loops 1 and 2 (see Figure 2) were separately tuned according to the Skogestad the resulting transient responses. tuning rules [20], by assuming step (10%) increases in the crude oil mass ﬂows (+6.11 kg/s in M and At tuning rules [20], by assuming step (10%) increases in the crude oil mass flows (+6.11 kg/s in MAt and PID controllers for loops 1 and 2 (see Figure 2) were separately tuned according to the Skogestad M ) in each of the branches A and C. The control variable responses for each of the selected operation Ct MCt) in each of the branches A and C. The control variable responses for each of the selected operation tuning rules [20], by assuming step (10%) increases in the crude oil mass flows (+6.11 kg/s in MAt and periods are shown in Figure 14a,b and the resulting values of the tuning parameters for PID controllers periods are shown in Figure 14a,b and the resulting values of the tuning parameters for PID MCt) in each of the branches A and C. The control variable responses for each of the selected operation 1 and 2 are presented in Table 3. controllers 1 and 2 are presented in Table 3. periods are shown in Figure 14a,b and the resulting values of the tuning parameters for PID controllers 1 and 2 are presented in Table 3. Appl. Sci. 2019, 9, x FOR PEER REVIEW 14 of 24 Appl. Sci. 2019, 9, 824 13 of 23 Figure 14. Open-loop step responses under fouling conditions of the control variable CV (a) and CV 1 2 Figure 14. Open-loop step responses under fouling conditions of the control variable CV1 (a) and CV2 (b) on a 10% increase in the inlet mass ﬂow M and M . At Ct (b) on a 10% increase in the inlet mass flow MAt and MCt. Table 3. Values of PID controller parameters obtained using the Skogestad method [20] in case no. 1. Table 3. Values of PID controller parameters obtained using the Skogestad method [20] in case no. 1. HEN Control Loop 1 Control Loop 2 PID Operating HEN K Controtl Loop 1 K Control Loop 2 t PID 1 PID 2 0 1 0 1 Parameters Condition Op ( Cs/kg) erating K0 (s) t1 ( Cs/kg) K0 (s) t1 PID PID 1 PID 2 Condition (°Cs/kg) (s) (°Cs/kg) (s) Parameters K 0.8790 0.6261 R = 0 Kp −0.8790 −0.6261 1.52 110 2.13 96 K 0.0080 0.0065 Rf=0 (clean) −1.52 110 −2.13 96 Ki −0.0080 −0.0065 K 0 0 (clean) Kd 0 0 K 0.8175 0.6042 Kp −0.8175 −0.6042 R after Rf after 1.63 99 2.21 89 K 0.0082 0.0068 −1.63 99 −2.21 89 Ki −0.0082 −0.0068 1 year 1 year K 0 0 Kd 0 0 Kp −0.8036 −0.5954 K 0.8036 0.5954 R after Rf after 1.66 −1.66 97 97 2.24−2.24 87 87 Ki K −0.008 30.0083 −0.0069 0.0069 2 years 2 years Kd K 0 0 0 0 Kp −0.7522 −0.5723 K 0.7522 0.5723 Rf after R after f −1.77 88 −2.33 79 Ki −0.0086 −0.0072 1.77 88 2.33 79 K 0.0086 0.0072 3 years 3 years Kd 0 0 K 0 0 3.3.2. Study the Open-loop Step Responses in Case No. 2 3.3.2. Study the Open-loop Step Responses in Case No. 2 In case no. 2, the changes in the dynamic behavior of the E3-18AB HE unit operated in HEN (see In case no. 2, the changes in the dynamic behavior of the E3-18AB HE unit operated in HEN Figure 3) can be detected by studying the open-loop step responses of the outlet temperature T3 (see Figure 3) can be detected by studying the open-loop step responses of the outlet temperature T simulated at the different stages of fouling build-up. simulated at the different stages of fouling build-up. Figure 15a shows the open-loop responses of the studied outlet temperature T3 to +1% step Figure 15a shows the open-loop responses of the studied outlet temperature T to +1% step change change in the shell-side flowrate (+0.47 kg/s) of the HE. Next, Figure 15b depicts the open-loop in the shell-side ﬂowrate (+0.47 kg/s) of the HE. Next, Figure 15b depicts the open-loop responses of responses of the studied outlet temperature T3 to -1% step change in the tube-side flowrate (-0.73 the studied outlet temperature T to 1% step change in the tube-side ﬂowrate (0.73 kg/s) of the HE. kg/s) of the HE. In Figure 15c, simulated open-loop responses of the studied outlet temperature T3 In Figure 15c, simulated open-loop responses of the studied outlet temperature T are plotted for a are plotted for a step upset +1°C in the shell-side inlet temperature of the HE. Finally, the open-loop step upset +1 C in the shell-side inlet temperature of the HE. Finally, the open-loop responses of the responses of the studied outlet temperature T3 to a step upset +1°C in the tube-side inlet temperature studied outlet temperature T to a step upset +1 C in the tube-side inlet temperature of the exchanger of the exchanger E3-18AB, are presented in Figure 15d. E3-18AB, are presented in Figure 15d. Appl. Sci. 2019, 9, x FOR PEER REVIEW 15 of 24 Appl. Sci. 2019, 9, 824 14 of 23 Figure 15. Open-loop step responses under fouling conditions of the outlet temperature T of E3-18AB Figure 15. Open-loop step responses under fouling conditions of the outlet temperature T3 of E3-18AB exchanger: (a) on a +1% step change in the shell-side ﬂowrate, (b) on a 1% step change in the tube-side exchanger: (a) on a +1% step change in the shell-side flowrate, (b) on a -1% step change in the tube- ﬂowrate, (c) on a +1 C step change in the shell-side inlet temperature and (d) on a +1 C step change side flowrate, (c) on a +1 °C step change in the shell-side inlet temperature and (d) on a +1 °C step in the tube-side inlet temperature. change in the tube-side inlet temperature. Because of the fouling build-up on the exchangers’ heat transfer surfaces, the visible changes in Because of the fouling build-up on the exchangers’ heat transfer surfaces, the visible changes in the delay time t , time constant t and gain K values in the studied open-loop step responses, may d 1 0 the delay time td, time constant t1 and gain K0 values in the studied open-loop step responses, may impair the quality of PID control considerably. In order to prevent that from happening, it is also impair the quality of PID control considerably. In order to prevent that from happening, it is also advisable to investigate the components of the tuning PID controller 3 (K , K , K ) and to check the p i d advisable to investigate the components of the tuning PID controller 3 (Kp, Ki, Kd) and to check the resulting transient responses. resulting transient responses. PID controller for loop 3 (see Figure 3) was tuned according to the Ziegler-Nichols method [18]. PID controller for loop 3 (see Figure 3) was tuned according to the Ziegler-Nichols method [18]. The values of the parameters of the open-loop characteristics (shown in Figure 15a) for each of the The values of the parameters of the open-loop characteristics (shown in Figure 15a) for each of the selected operation periods and the resulting values of the tuning parameters for PID controller 3 are selected operation periods and the resulting values of the tuning parameters for PID controller 3 are presented in Table 4. presented in Table 4. Table 4. Values of PID controller parameters obtained using the Ziegler-Nichols method [18] in case no. 2. HEN Control Loop 3 PID 3 Operating K0 td t1 Parameters Condition (°Cs/kg) (s) (s) Kp 60.32 Rf = 0 0.1566 8 63 Ki 3.771 (clean) Kd 193.1 Kp 66.24 Rf after 0.1494 8.5 70.1 Ki 3.897 1 year Kd 225.2 Kp 78.72 Rf after 0.1394 8.2 75 Ki 4.801 2 years Kd 258.2 Rf after Kp 87.27 3 years 0.1288 8.7 81.5 Ki 5.016 Appl. Sci. 2019, 9, 824 15 of 23 Table 4. Values of PID controller parameters obtained using the Ziegler-Nichols method [18] in case no. 2. HEN Control Loop 3 t PID 3 Operating K t 0 d (s) Parameters Condition ( Cs/kg) (s) K 60.32 R = 0 0.1566 8 63 K 3.771 (clean) K 193.1 K 66.24 R after 0.1494 8.5 70.1 K 3.897 1 year K 225.2 K 78.72 R after 0.1394 8.2 75 K 4.801 2 years K 258.2 K 87.27 R after Appl. Sci. 2019, 9, x FOR P f EER REVIEW 16 of 24 0.1288 8.7 81.5 K 5.016 3 years K 303.7 Kd 303.7 3.4. Closed-loop Control Analysis 3.4. Closed-loop Control Analysis Using the dynamic HE model outlined in Section 2.3, the entire HENs together with control Using the dynamic HE model outlined in Section 2.3, the entire HENs together with control conﬁgurations were modelled employing Simulink software; the block diagram of the HEN model is configurations were modelled employing Simulink software; the block diagram of the HEN model is presented in Figures 8 and 9. presented in Figures 8 and 9. For case no. 1, the simulations of transient responses were carried out and their results shown For case no. 1, the simulations of transient responses were carried out and their results shown in in Figure 16a,b (for control loops 1 and 2) demonstrate that fouling build-up induces insigniﬁcant Figure 16a,b (for control loops 1 and 2) demonstrate that fouling build-up induces insignificant changes in CV rise time t and settling time t . This can be seen as an indication that in the studied r s changes in CV rise time tr and settling time ts. This can be seen as an indication that in the studied case, no adjustments of K and K values are needed and the indices of control quality would not be p i case, no adjustments of Kp and Ki values are needed and the indices of control quality would not be adversely affected by fouling of heat-exchanger surfaces—see Table 5. adversely affected by fouling of heat-exchanger surfaces – see Table 5. Figure 16. Closed-loop step responses (case no. 1) under fouling conditions of the control variable (a) Figure 16. Closed-loop step responses (case no. 1) under fouling conditions of the control variable (a) CV and (b) CV to setpoints step change. 1 2 CV1 and (b) CV2 to setpoints step change. Table 5. Values of the control-quality indices for PID tuned in clean conditions for different periods of operation (case no. 1). Closed- Control-quality Indices Rf after Base loop Mp tp td tr ts Period of PID Responses (%) (s) (s) (s) (s) Iperation Parameters for Case (years) (Kp/Ki/Kd) No. 1 for control 0 0 - - 139 265 loop 1 1 0 - - 126 263 −0.8790/-0.0080/0 (Figure 2 0 - - 123 262 16a) 3 0 - - 108 260 for control 0 0 - - 115 214 loop 2 1 0 - - 110 223 −0.6261/-0.0065/0 (Figure 2 0 - - 108 224 16b) 3 0 - - 102 229 transient responses were simulated assuming constant values of the PID parameters obtained using the Skogestad method [20] for clean HEN For case no. 2 (HEN with control loop 3), the closed-loop step responses under fouling conditions were simulated with three different sets of the PID parameters: 1. Assuming constant values of the base PID parameters obtained using Ziegler-Nichols method [18] for clean HEN – Figure 17a 2. With the adjusted PID parameters, for the consecutive periods of fouling build-up, in accordance with the data shown in Table 4 – Figure 17b 3. With the optimal PID parameters obtained using Signal Constraint toolbox in SIMULINK [21] under fouling conditions – Figure 17c In case no. 2, control loop 3 comprises heat exchangers E3-18AB whose operation is affected by the interactions with the remaining exchangers in the studied HEN (see Figure 3). Qualitative evaluation of the obtained dynamic closed-loop characteristics (Figure 17abc) can be complemented by the values of quality indices – Table 6. Appl. Sci. 2019, 9, x FOR PEER REVIEW 22 of 24 Figure A4. Closed-loop step responses (case in work [6]) under fouling conditions of the HE E35AB models: without adjustment (a) after adjustment (b) and after adjustment of optimal (c) PID tuning Appl. Sci. 2019, 9, 824 16 of 23 Appl. Sci. 2019, 9, x FOR PEER REVIEW 22 of 24 parameters. Figure A4. Closed-loop step responses (case in work [6]) under fouling conditions of the HE E35AB Table A1. Values of the control-quality indices for used sets of the PID parameters in different periods Table 5. Values of the control-quality indices for PID tuned in clean conditions for different periods of models: without adjustment (a) after adjustment (b) and after adjustment of optimal (c) PID tuning of operation (cases in work [6]). operation (case no. 1). parameters. Closed-loop Optimal Control-quality Indices Rf after Base Adjusted Base Control-Quality Indices R after Period of Closed-Loop Responses PID Mp tp td tr ts Period of PID PID PID Table A1. Values of the control-quality indices for usedM sets of thet PID para t meters in t differetnt periods Iperation p p d r s Responses of the for Parameters (%) (s) (s) (s) (s) 1 1 2 Operation Parameters Parameters Parameters (%) (s) (s) (s) (s) of operation (cases in work [6 (years) ]). Case Stu No. die 1 d (Kp/Ki/Kd) (years) (Kp/Ki/Kd) (Kp/Ki/Kd) (K /K /K ) p i d HEs [6,21] Closed-loop Optimal Control-quality Indices Rf after Base Adjusted 0 0 - 47.3 - 18.7 139 6.1 2658.7 61.3 Responses for HE PID Mp tp td tr ts for control Period of PID PID 1 0 - 67.- 1 26.2 126 9.8 26312.8 157 E of the 11AB 69.8/4.58/2 0.8790/55 - 0.0080/0 Parameters- (%) (s) (s) (s) (s) loop 1 1 2 Operation Parameters Parameters 0 - - 123 262 2 68.9 27.5 10.3 13.4 193 Studied (Kp/Ki/Kd) (Figure A1a) (Figure 16a) (years) 3 (Kp/Ki/Kd) (Kp/Ki/Kd) 0 - 73.-8 31.3 108 11.5 26014.8 >300 HEs [6,21] 0 69.8/4.58/255 47.3 18.7 6.1 8.7 61.3 0 - - 115 214 for HE 0 47.3 18.7 6.1 8.7 61.3 for HE 1 70.7/3.59/334 47.1 24.5 9.4 12.4 60.6 for control 1 0 - 67.- 1 26.2 110 9.8 22312.8 157 E11AB - - E11AB 2 69.8/4.58/2 0.6261/55 - 73. 0.0065/0 9/3.6 6/358 - 47.6 25 9.6 12.6 78.8 loop 2 2 0 - 68.- 9 27.5 108 10.3 22413.4 193 (Figure A1b) (Figure A1a) 3 88.7/4.18/452 51.3 25.8 9.6 12.8 92.1 (Figure 16b) 3 0 - 73.-8 31.3 102 11.5 22914.8 >300 0 53.8/1.78/260 20 19.4 6.8 10 38.1 0 69.8/4.58/255 47.3 18.7 6.1 8.7 61.3 1 for HE transient responses were simulated assuming constant values of the PID parameters obtained using the Skogestad for HE 1 53.8/1.42/260 20 26.9 10.9 14.9 56.2 E11AB 1 - - 70.7/3.59/334 47.1 24.5 9.4 12.4 60.6 method [20] for clean HEN. E11AB - - 2 53.8/1.39/260 20 28.5 11.5 15.7 63.3 (Figure A1c) 2 73.9/3.66/358 47.6 25 9.6 12.6 78.8 (Figure A1b) 3 50.5/1.45/286 20 34.9 12.8 18 75 3 88.7/4.18/452 51.3 25.8 9.6 12.8 92.1 0 46 29.3 10.9 14.6 70.9 For case no. 2 (HEN with control loop 3), the closed-loop step responses under fouling conditions for HE 0 53.8/1.78/260 20 19.4 6.8 10 38.1 for HE 1 44.6 36.2 12.9 17.3 94.9 were simulated E15Awith B three1 different2.44/0. sets10 of/13. the7 - PID parameters: 53.8/1.-4 2/260 20 26.9 10.9 14.9 56.2 E11AB - - 2 45.4 38.1 13.4 18 99.8 (Figure A2a) 2 53.8/1.39/260 20 28.5 11.5 15.7 63.3 (Figure A1c) 3 47.4 41.5 14.5 19.5 145 3 50.5/1.45/286 20 34.9 12.8 18 75 1. Assuming constant values of the base PID parameters obtained using Ziegler-Nichols method [18] 0 2.44/0.104/13.7 46 29.3 10,9 14.6 70.9 for HE 0 46 29.3 10.9 14.6 70.9 for clean HEN—Figure 17a for HE 1 3.38/0.143/19.2 50.6 29.3 11 14.3 75.3 1 44.6 36.2 12.9 17.3 94.9 E15AB - - E15AB 2 2.44/0.10/13.7 - 3.54/0.148/ 20.4 - 51 29.6 11.1 14.5 76.8 2. With the adjusted PID parameters, for the consecutive periods of fouling build-up, in accordance 2 45.4 38.1 13.4 18 99.8 (Figure A2b) (Figure A2a) 3 3.88/0.157/22.9 51.7 30.4 11.4 14.8 79.6 with the data shown in Table 4—Figure 17b 3 47.4 41.5 14.5 19.5 145 0 2.30/0.03/13.8 20 27.7 11.3 15.5 63.2 0 2.44/0.104/13.7 46 29.3 10,9 14.6 70.9 for HE 3. With the optimal PID parameters obtained using Signal Constraint toolbox in SIMULINK [21] for HE 1 2.48/0.04/13.8 19.8 33.6 13.2 18.1 50 E15AB 1 - - 3.38/0.143/19.2 50.6 29.3 11 14.3 75.3 under fouling conditions—Figure 17c E15AB - - 2 2.76/0.03/13.9 20 32.9 13.3 18.1 71.9 (Figure A2c) 2 3.54/0.148/20.4 51 29.6 11.1 14.5 76.8 (Figure A2b) 3 3.05/0.04/17 20 33.2 13.3 18.1 74.6 3 3.88/0.157/22.9 51.7 30.4 11.4 14.8 79.6 In case no. 2, control loop 3 comprises heat exchangers E3-18AB whose operation is affected 0 45.5 24.6 8.6 12 76.4 for HE 0 2.30/0.03/13.8 20 27.7 11.3 15.5 63.2 for HE by the interactions with the 1 remaining exchangers in the studied HEN (see 49.4 Figur 27.e 7 3). 10. Qualitative 1 13.6 94.5 E30AB 1 38.2/1.924/182 - 2.48/0.- 0 4/13.8 19.8 33.6 13.2 18.1 50 E15AB - - 2 51.3 29.8 11 14.6 105 evaluation of the obtained dynamic closed-loop characteristics (Figure 17a–c) can be complemented by 2 2.76/0.03/13.9 20 32.9 13.3 18.1 71.9 (Figure A3a) Appl. Sci. (Figure A 2019, 9, x FO 2c) R PEER RE 3 VIEW 54.3 34 12.6 17 of 16. 24 6 152 3 3.05/0.04/17 20 33.2 13.3 18.1 74.6 the values of quality indices—Table 6. 0 38.2/1.924/182 45.5 24.6 8.6 12 76.4 for HE 0 45.5 24.6 8.6 12 76.4 for HE 1 41.5/1.975/209 47.2 26.2 9.5 12.9 65.2 1 49.4 27.7 10.1 13.6 94.5 E30AB - - E30AB 2 38.2/1.924/182 - 44.1/2.0 29/230 - 48 27.1 10 13.4 86.3 2 51.3 29.8 11 14.6 105 (Figure A3b) (Figure A3a) 3 49.7/2.157/275 49.2 28.5 10.7 14.2 98.8 3 54.3 34 12.6 16.6 152 0 31.6/0.814/184 20 25.1 9.3 13.4 52.9 0 38.2/1.924/182 45.5 24.6 8.6 12 76.4 for HE for HE 1 31.6/0.808/184 20 28.2 10.9 15.2 55.7 E30AB 1 - - 41.5/1.975/209 47.2 26.2 9.5 12.9 65.2 E30AB - - 2 31.6/0.808/184 20 30.5 11.8 16.5 63.7 (Figure A3c) 2 44.1/2.029/230 48 27.1 10 13.4 86.3 (Figure A3b) 3 33.3/0.859/213 19.2 33.5 12.9 18 75.1 3 49.7/2.157/275 49.2 28.5 10.7 14.2 98.8 0 51.2 16.2 5.1 7.3 70.9 for HE 0 31.6/0.814/184 20 25.1 9.3 13.4 52.9 for HE 1 76.6 25.1 8.9 11.6 232 E35AB 1 32.5/2.371/107 - 31.6/0.8- 08/184 20 28.2 10.9 15.2 55.7 E30AB - - 2 88.5 34.9 12.2 15.8 >300 (Figure A4a) 2 31.6/0.808/184 20 30.5 11.8 16.5 63.7 (Figure A3c) 3 94.5 40.8 14.3 18.5 ~ 3 33.3/0.859/213 19.2 33.5 12.9 18 75.1 0 32.5/2.371/107 51.2 16.2 5.1 7.3 70.9 for HE 0 51.2 16.2 5.1 7.3 70.9 for HE 1 35.8/1.945/158 53.4 22.5 8 10.7 78.2 1 76.6 25.1 8.9 11.6 232 E35AB - - Figure 17. Closed-loop step responses (case no. 2) under fouling conditions of the HE models: without Figure 17. Closed-loop step responses (case no. 2) under fouling conditions of the HE models: without E35AB 2 32.5/2.371/107 - 51.1/2.3 98/262 - 54.9 25.9 9.1 12.1 92.6 2 88.5 34.9 12.2 15.8 >300 (Figure A4b) adjustment (a) after adjustment (b) and after adjustment of optimal (c) PID tuning parameters. (Figure A4a) 3 62.2/2.714/342 55.4 27.7 9.6 12.8 99.8 adjustment (a) after adjustment (b) and after adjustment of optimal (c) PID tuning parameters. 3 94.5 40.8 14.3 18.5 ~ 0 25.9/0.395/111 18.4 16.4 5.6 8.4 39.9 0 32.5/2.371/107 51.2 16.2 5.1 7.3 70.9 for HE for HE 1 25.9/0.386/111 20 25.5 9.9 13.7 58.1 Table 6. Values of the control-quality indices for used sets of the PID parameters in different periods 1 35.8/1.945/158 53.4 22.5 8 10.7 78.2 E35AB - - E35AB - - 2 30.4/0.416/139 20 34.2 12.7 17.9 73.9 of operation (case no. 2). (Figure A4c) 2 51.1/2.398/262 54.9 25.9 9.1 12.1 92.6 (Figure A4b) 3 39.5/0.479/216 20 34.7 12.7 17.9 74.9 3 62.2/2.714/342 55.4 27.7 9.6 12.8 99.8 Rf after Base Adjusted Optimal Control-quality Indices characteristics were simulated assuming constant values of the PID parameters obtained using Ziegler-Nichols method [18] for clean HEs Closed-loop 0 25.9/0.395/111 18.4 16.4 5.6 8.4 39.9 characteristics with the adjusted PID parameters, for the consecutive periods of fouling build-up, using Ziegler-Nichols method [18] Period of PID PID PID Mp tp td tr ts for HE Responses for 3 1 25.9/0.386/111 20 25.5 9.9 13.7 58.1 1 2 3 characteristics with the optimal PID parameters obtained under fouling conditions using Signal Constraint toolbox in SIMULINK [21] Operation Parameters Parameters Parameters (%) (s) (s) (s) (s) E35AB - - Case No. 2 non-expiring oscillations detected 2 30.4/0.416/139 20 34.2 12.7 17.9 73.9 (years) (Kp/Ki/Kd) (Kp/Ki/Kd) (Kp/Ki/Kd) (Figure A4c) 3 39.5/0.479/216 20 34.7 12.7 17.9 74.9 0 58.9 21.2 8.1 10.6 124.6 for control characteristics were simulated assuming constant values of the PID parameters obtained using Ziegler-Nichols method [18] for clean HEs 1 59.8 23.4 9.1 11.9 133.6 2 loop 3 60.3/3.77/193 - - characteristics with the adjusted PID parameters, for the consecutive periods of fouling build-up, using Ziegler-Nichols method [18] 2 59.5 24.6 9.7 12.5 134.2 (Figure 17a) characteristics with the optimal PID parameters obtained under fouling conditions using Signal Constraint toolbox in SIMULINK [21] 3 58 27.5 10.8 14 173.2 non-expiring oscillations detected 0 60.3/3.77/193 58.9 21.2 8.1 10.6 124.6 for control 1 66.2/3.90/225 58.3 22.2 8.6 11.2 126.4 loop 3 - - 2 78.7/4.80/258 63.3 21.9 8.5 11 142.6 (Figure 17b) 3 87.2/5.02/304 65.3 23.1 9.0 12 129.6 0 56.8/0.82/237 20 19.1 8.0 11 54.5 for control 1 68.1/0.81/359 20 18.8 7.7 10.6 50.2 loop 3 - - 2 71.0/0.85/377 20 18 8.1 11 54.3 (Figure 17c) 3 76.8/0.90/553 16 18 8.0 10.9 50.9 characteristics were simulated assuming constant values of the PID parameters obtained using Ziegler-Nichols method [18] for clean HEN characteristics with the adjusted PID parameters, for the consecutive periods of fouling build-up, in accordance with the data shown in Table characteristics with the optimal PID parameters obtained under fouling conditions using Signal Constraint toolbox in Simulink [21] In this case, judging from the information presented in Figure 17 and Table 6, the effect of fouling that builds up during HEN operation is more pronounced than that observed in case no. 1. At unchanged PID-controller settings, settling time ts is increased from 124.6 s for the clean exchanger to 173.2 s for the exchanger fouled after 3 years of HEN operation. While some quality indices including peak time tp, delay time td and rise time tr, are changed, overshoot Mp remains nearly constant (Figure 17a). If adjustments of PID-controller settings were applied in reaction to a fouling build-up, then the resulting dynamic characteristics and control-quality indices would not deteriorate (Figure 17b and Table 6). Using the Ziegler-Nichols method [18] or the Signal Constraint toolbox in Simulink [21], optimal controller settings can be determined to eliminate excessively large values of overshoot Mp (Figure 17c). In this context, a more advanced approach recently introduced by Oravec [12] in cooperation with Trafczynski and Markowski can be mentioned. In their work, robust Model Predictive Control - MPC with integral action is used for optimizing the control performance when the operation of heat exchangers has been affected by fouling that induces changes of the exchangers' parameters. 4. Discussion Using Simulink software, a validated multi-cell dynamic model of a shell-and-tube HE was applied in simulating the operation of PID-controlled HEs (see Figures 8 and 9). A control-theory based approach was proposed for the identification and evaluation of the influence of fouling on the Appl. Sci. 2019, 9, x FOR PEER REVIEW 22 of 24 Figure A4. Closed-loop step responses (case in work [6]) under fouling conditions of the HE E35AB Appl. Sci. 2019, 9, 824 17 of 23 models: without adjustment (a) after adjustment (b) and after adjustment of optimal (c) PID tuning parameters. Table 6. Values of the control-quality indices for used sets of the PID parameters in different periods of Table A1. Values of the control-quality indices for used sets of the PID parameters in different periods of operation (cases in work [6]). operation (case no. 2). Closed-loop Optimal Control-quality Indices Rf after Base Adjusted Base Adjusted Optimal Control-Quality Indices Responses PID Mp tp td tr ts R after Period Closed-Loop f Period of PID PID PID PID PID 3 M t t t t p p d r s of the Parameters (%) (s) (s) (s) (s) of Operation Responses for 1 1 2 2 3 Operation Parameters Parameters Parameters Parameters Parameters (%) (s) (s) (s) (s) Studied (Kp/Ki/Kd) (years) Case No. 2 (years) (Kp/Ki/Kd) (Kp/Ki/Kd) (K /K /K ) (K /K /K ) p (K /K /K ) i d p i d p i d HEs [6,21] 0 - 47. - 3 18.7 58.9 6.1 8. 21.2 7 61.3 8.1 10.6 124.6 for contr fool r HE 1 67.1 26.2 59.8 9.8 12. 23.4 8 1579.1 11.9 133.6 E11AB 69.8/4.58/255 - - loop 3 60.3/3.77/193 2 68.9 27.5 59.5 10.3 13. 24.6 4 1939.7 12.5 134.2 (Figure A1a) (Figure 17a) 3 73.8 31.3 58 11.5 14. 27.5 8 >300 10.8 14 173.2 0 69.8/4.58/255 47.3 18.7 6.1 8.7 61.3 0 - 60.3/3.77/193 - 58.9 21.2 8.1 10.6 124.6 for HE for control 1 70.7/3.59/334 47.1 24.5 9.4 12.4 60.6 E11AB 1 - 66.2/3.90/225 - 58.3 22.2 8.6 11.2 126.4 loop 3 2 73.9/3.66/358 47.6 25 9.6 12.6 78.8 (Figure A1b) 2 78.7/4.80/258 63.3 21.9 8.5 11 142.6 3 88.7/4.18/452 51.3 25.8 9.6 12.8 92.1 (Figure 17b) 3 87.2/5.02/304 65.3 23.1 9.0 12 129.6 0 53.8/1.78/260 20 19.4 6.8 10 38.1 for HE 0 - - 56.8/0.82/237 20 19.1 8.0 11 54.5 1 53.8/1.42/260 20 26.9 10.9 14.9 56.2 for contr E11A ol B - - 1 68.1/0.81/359 20 18.8 7.7 10.6 50.2 2 53.8/1.39/260 20 28.5 11.5 15.7 63.3 loop (Figure A 3 1c) 3 2 50.5/1.45/2 71.0/0.85/377 86 20 34.9 20 12.8 18 18 75 8.1 11 54.3 (Figure 17c) 3 76.8/0.90/553 16 18 8.0 10.9 50.9 0 46 29.3 10.9 14.6 70.9 for HE 1 44.6 36.2 12.9 17.3 94.9 characteristics E15AB were simulated2.44/0.1 assuming 0/13.7 - constant values of- the PID parameters obtained using Ziegler-Nichols 2 45.4 38.1 13.4 18 99.8 method (Figure A [18 2a) ] for clean HEN characteristics with the adjusted PID parameters, for the consecutive periods of fouling 3 47.4 41.5 14.5 19.5 145 build-up, in accordance with the data shown in Table 4 characteristics with the optimal PID parameters obtained 0 2.44/0.104/13.7 46 29.3 10,9 14.6 70.9 for HE under fouling conditions using Signal Constraint toolbox in Simulink [21] 1 3.38/0.143/19.2 50.6 29.3 11 14.3 75.3 E15AB - - 2 3.54/0.148/20.4 51 29.6 11.1 14.5 76.8 (Figure A2b) 3 3.88/0.157/22.9 51.7 30.4 11.4 14.8 79.6 In this case, judging 0 from the information 2.pr 30/0. esented 03/13.8 20 in Figur 27.7 e11. 17 3 and 15.5 Table 63.2 6, the effect of for HE 1 2.48/0.04/13.8 19.8 33.6 13.2 18.1 50 fouling that E15AB builds up during HEN- - operation is more pronounced than that observed in case no. 1. 2 2.76/0.03/13.9 20 32.9 13.3 18.1 71.9 (Figure A2c) 3 3.05/0.04/17 20 33.2 13.3 18.1 74.6 At unchanged PID-controller settings, settling time t is increased from 124.6 s for the clean exchanger 0 45.5 24.6 8.6 12 76.4 for HE to 173.2 s for the exchanger fouled after 3 years of HEN operation. While some quality indices 1 49.4 27.7 10.1 13.6 94.5 E30AB 38.2/1.924/182 - - 2 51.3 29.8 11 14.6 105 including peak time t , delay time t and rise time t , are changed, overshoot M remains nearly p r p (Figure A3a) 3 54.3 34 12.6 16.6 152 constant (Figure 17a). If adjustments of PID-controller settings were applied in reaction to a fouling 0 38.2/1.924/182 45.5 24.6 8.6 12 76.4 for HE 1 41.5/1.975/209 47.2 26.2 9.5 12.9 65.2 build-up, then the resulting dynamic characteristics and control-quality indices would not deteriorate E30AB - - 2 44.1/2.029/230 48 27.1 10 13.4 86.3 (Figure A3b) (Figure 17b and Table 6). Using the Ziegler-Nichols method [18] or the Signal Constraint toolbox in 3 49.7/2.157/275 49.2 28.5 10.7 14.2 98.8 0 31.6/0.814/184 20 25.1 9.3 13.4 52.9 Simulink [21], optimal controller settings can be determined to eliminate excessively large values for HE 1 31.6/0.808/184 20 28.2 10.9 15.2 55.7 E30AB - - of overshoot M (Figure 17c). In this context, a more advanced approach recently introduced by 2 31.6/0.808/184 20 30.5 11.8 16.5 63.7 (Figure A3c) 3 33.3/0.859/213 19.2 33.5 12.9 18 75.1 Oravec [12] in cooperation with Trafczynski and Markowski can be mentioned. In their work, robust 0 51.2 16.2 5.1 7.3 70.9 for HE Model Predictive Contr 1 ol - MPC with integral action is used 76. for 6 optimizing 25.1 8.9 11. the 6 contr 232 ol performance E35AB 32.5/2.371/107 - - 2 88.5 34.9 12.2 15.8 >300 (Figure A4a) when the operation of heat exchangers has been affected by fouling that induces changes of the 3 94.5 40.8 14.3 18.5 ~ exchangers’ parameters. 0 32.5/2.371/107 51.2 16.2 5.1 7.3 70.9 for HE 1 35.8/1.945/158 53.4 22.5 8 10.7 78.2 E35AB - - 2 51.1/2.398/262 54.9 25.9 9.1 12.1 92.6 (Figure A4b) 4. Discussion 3 62.2/2.714/342 55.4 27.7 9.6 12.8 99.8 0 25.9/0.395/111 18.4 16.4 5.6 8.4 39.9 for HE Using Simulink 1 software, a validated multi-cell 25.9/0.386/ dynamic 111 20 model 25.5 9.9 of a 13.shell-and-tube 7 58.1 HE was E35AB - - 2 30.4/0.416/139 20 34.2 12.7 17.9 73.9 (Figure A4c) applied in simulating the operation of PID-controlled HEs (see Figures 8 and 9). A control-theory 3 39.5/0.479/216 20 34.7 12.7 17.9 74.9 characteristics were simulated assuming constant values of the PID parameters obtained using Ziegler-Nichols method [18] for clean HEs based approach was proposed for the identiﬁcation and evaluation of the inﬂuence of fouling on characteristics with the adjusted PID parameters, for the consecutive periods of fouling build-up, using Ziegler-Nichols method [18] the dynamic characteristics behavior with the optimal PI of D pa the rameter HEN s obtained and under fo on uling co the quality nditions using Sign of its al Constr contr aint tool ol box (Figur in SIMULINK [2 e 4). 1] The dynamic model non-expiring oscillations detected was applied to a case study on the HEs and HENs operated in the crude distillation unit under fouling conditions. In case no. 1, control loops no. 1 and 2 include all the HEs operated in the HEN and interactions between the HEs are signiﬁcant (Figure 2). The simulated step responses prove that as fouling was building up, the quality indices of network control remained nearly unchanged even if the tuning of PI controllers was not adjusted (see Figure 16 and Table 5). In case no. 2, control loop no. 3 includes a set of HEs that interact with other exchangers present in the HEN (Figure 3). From the qualitative and quantitative estimates presented above, it can be inferred that the effect of fouling on HEN operation is more pronounced than that observed in case no. 1. Although most indices of control quality remain unchanged as the fouling increases, the settling time becomes longer. Periodic adjustments of PID controller tuning are required in the consecutive Appl. Sci. 2019, 9, 824 18 of 23 stages of fouling build-up because the value of parameter t is signiﬁcantly increased (see Figure 17 and Table 6). In previous publications [6,21], the present authors evaluated dynamic characteristics of four different sets of heat exchangers operated in simple control loops according to the scheme shown in Figure 7a, that is, without meaningful interactions with other exchangers in the HEN. The performance of those control loops have now been simulated and their closed-loop step responses are presented in Appendix A, Figures A1–A4, while the corresponding values of control quality indices are shown in Table A1. In each of the studied PID-controlled HEs, it was found that increased fouling led to the deterioration of all the indices of control quality so that periodic adjustments of PID-controller tuning appeared necessary. It can be mentioned that the dynamic model of shell-and-tube heat exchangers developed by the present authors found application in the work done by Borges de Carvalho et al. [22], who performed the dynamic analysis of fouling build-up in the HEs designed according to TEMA standards. The same author team also tested several tuning strategies for the PID-controlled HEs under fouling conditions [23] and arrived at results that appear to be consistent with those of the present authors. Overall, according to the results of the mentioned case studies, the higher the number of heat exchangers in the PID control loop and the more interactions occur between heat exchangers in the network, the weaker the inﬂuence of fouling on the control quality indices (see Table 7). This observation may be attributed to underestimated values of R (calculated according to TEMA standards) and/or to the compensation of the negative impacts of fouling on the heat transfer in the HEs. Such a compensation is possible only in the network where signiﬁcant interactions occur between the HEs (that is, if antecedent exchangers are operated on the process streams—Figure 18). As previously observed by the present authors [17,24], the larger the number of interacting exchangers, the better the compensation of the detrimental effects of fouling. Fouling on the heat transfer surface of a HE operated in the HEN brings about a change in the exchanger capacity as well as changes in the outlet temperatures of process streams. However, the operation of the HE can be affected by other exchangers serving the same process streams (antecedent exchangers); examples of such exchangers in the HENs can be found in Figures 2 and 3. As fouling builds up on the heat transfer surfaces of the antecedent exchangers, temperatures of process streams at HE inlet are increased. Due to that, although heat transfer intensity has been reduced by fouling, the thermal power of the HE may remain unchanged. Similarly, the indices of control quality in the associated control loop may also remain unchanged (Figure 16a,b). Table 7. The number of HEs in a control loop and the number of antecedent HEs compared to the level of fouling inﬂuence on the control-quality indices. The Number of The Number The Level of Case Antecedent HEs of HEs in a Fouling Inﬂuence on the Study on Hot on Cold Control Loop Control-quality Indices. Stream Stream Case no. 1 (Figure 2) (see Table 5) for control loop 1 13 9 11 negligibly low (Figure 16a) for control loop 2 13 2 24 negligibly low (Figure 16b) Case no. 2 (Figure 3) (see Table 6) for control loop 3 1 7 7 low (Figure 17a) Cases in work [6] (Figure 18) (see Table A1) for HE11 control loop 1 0 0 high (Figure A1a) for HE15 control loop 1 2 0 medium (Figure A2a) for HE30 control loop 1 2 2 medium (Figure A3a) for HE35 control loop 1 0 0 high (Figure A4a) Appl. Sci. 2019, 9, x FOR PEER REVIEW 19 of 24 for control loop 1 13 9 11 negligibly low (Figure 16a) for control loop 2 13 2 24 negligibly low (Figure 16b) Case no. 2 (Figure 3) (see Table 6) for control loop 3 1 7 7 low (Figure 17a) Cases in work [6] (Figure 18) (see Table A1) for HE11 control loop 1 0 0 high (Figure A1a) for HE15 control loop 1 2 0 medium (Figure A2a) Appl. Sci. 2019, 9, 824 19 of 23 for HE30 control loop 1 2 2 medium (Figure A3a) for HE35 control loop 1 0 0 high (Figure A4a) Figure 18. Schematic diagram of a PID-controlled HE with the antecedent HEs. Figure 18. Schematic diagram of a PID-controlled HE with the antecedent HEs. 5. Conclusions 5. Conclusions In conclusion, insufﬁcient quality of HEN control may lead to excessive oscillations (increased In conclusion, insufficient quality of HEN control may lead to excessive oscillations (increased settling time and overshoot) of process parameters, as well as to excessive consumption of energy settling time and overshoot) of process parameters, as well as to excessive consumption of energy and raw materials, resulting in increased production costs. It may also generate the risk of dangerous and raw materials, resulting in increased production costs. It may also generate the risk of dangerous process perturbations such as, exceeding safety margins of temperature values. In order to prevent such process perturbations such as, exceeding safety margins of temperature values. In order to prevent situations from occurring, various approaches to the determination of controller-tuning parameters can such situations from occurring, various approaches to the determination of controller-tuning be applied such as the trial-and-error procedure, the Ziegler-Nichols method, or the MPC methodology. parameters can be applied such as the trial-and-error procedure, the Ziegler-Nichols method, or the Appropriate PID-gain values can be determined using the dynamic model of the heat exchanger MPC methodology. Appropriate PID-gain values can be determined using the dynamic model of the network and the suitability of these values can be tested by simulation. In order to ensure a satisfactory heat exchanger network and the suitability of these values can be tested by simulation. In order to performance of PID control when fouling layers build up on the heat-transfer surfaces of the exchangers ensure a satisfactory performance of PID control when fouling layers build up on the heat-transfer in the HEN, periodic adjustments of PID-controller tuning are needed. A more costly alternative is surfaces of the exchangers in the HEN, periodic adjustments of PID-controller tuning are needed. A to apply periodic cleaning of the exchangers. Where the rate of fouling build up is very high and more costly alternative is to apply periodic cleaning of the exchangers. Where the rate of fouling build therefore exchanger cleaning cannot be avoided, the adjustments of controller tuning may help to up is very high and therefore exchanger cleaning cannot be avoided, the adjustments of controller reduce the frequency of cleaning interventions, thus lowering their total cost. tuning may help to reduce the frequency of cleaning interventions, thus lowering their total cost. Author Contributions: Conceptualization, M.T.; Methodology, M.T. and M.M.; Validation, M.T. and M.M.; Formal analysis, M.T.; Investigation, M.T. and P.K.; Data curation, M.T., M.M. and P.K.; Writing—original draft preparation, Author Contributions: Conceptualization, M.T.; Methodology, M.T. and M.M.; Validation, M.T. and M.M.; M.T. and K.U.; Writing—review and editing, M.T., K.U. and J.W.; Visualization, M.T.; Supervision, M.T.; Funding Formal analysis, M.T.; Investigation, M.T. and P.K.; Data curation, M.T., M.M. and P.K.; Writing—original draft acquisition, P.K., J.W. and M.T. preparation, M.T. and K.U.; Writing—review and editing, M.T., K.U. and J.W.; Visualization, M.T.; Supervision, Funding: This research received no external funding. M.T.; Funding acquisition, P.K., J.W. and M.T. Conﬂicts of Interest: The authors declare no conﬂict of interest. Funding: This research received no external funding. Appendix A Conflicts of Interest: The authors declare no conflict of interest. It can be mentioned that in previous publications [6,21], the present authors qualitatively Appendix A evaluated dynamic characteristics of some other components of the studied HEN, namely heat exchangers E11AB, E15AB, E30AB and E35AB—see Figures A1–A4. These HEs were assumed to operate in the control loops similar to that shown in Figure 7a in Section 2.4., that is, in the absence of meaningful interactions with other HEN components. As a complement to the mentioned characteristics, the corresponding values of control-quality indices are shown in Table A1. Regarding exchanger E15AB, the evaluation of its control performance is similar to that discussed for case no. 2 (exchanger E3-18AB) in Section 3.4. As a consequence of fouling build-up at unchanged PID-controller settings (Figure A2a), settling time t increases from 70.9 s to 145 s, for the clean HE and s Appl. Sci. 2019, 9, 824 20 of 23 fouled HE after 3-year operation (Table A1). Concurrently, peak time t , delay time t and rise time t p r are slightly changed, while overshoot M remains nearly constant. The indices of control quality of the other HEs affected by fouling build-up are generally deteriorated, however the extent of change is differentiated. At constant PID-controller settings, the indices of E11AB are changed as follows: overshoot M is increased from 47.3% to 73.8%, peak time t from 18.7 s do 31.3 s, and settling time t from 61.3 s to >300 s, for the clean HE and fouled HE p s after 3-year operation, respectively. The remaining indices, that is, delay time t and rise time t are changed insigniﬁcantly (see Table A1 and Figure A1a). Analogous changes determined for E30AB are: overshoot M from 45.5% to 54.3%, and settling time t from 76.4 s to 152 s, while peak time t , p s p delay time t and rise time t are nearly unchanged (Table A1 and Figure A3a). The situation of heat d r exchanger E35AB is different because at constant controller settings, fouling build-up may lead to drastic deterioration of control-quality indices and unstable control performance (see Figure A4a and Table A1). However, if periodic adjustment of the settings of PID-controllers were applied for all the mentioned heat exchangers (E11AB, E15AB, E30AB and E35AB), then increasing thermal resistances of the fouling layers would not induce deterioration of control characteristics (Figures A1b, A2b, A3b and A4b) and their corresponding quality indices (Table A1). The adjustments of controller settings, Appl. Sci. 2019, 9, x FOR PEER REVIEW 21 of 24 optimized using Ziegler-Nichols method [18] or Signal Constraint toolbox in Simulink [21], would Appl. Sci. 2019, 9, x FOR PEER REVIEW 21 of 24 result in the elimination of too high values of overshoot M (Figures A1c, A2c, A3c and A4c). Figure A1. Closed-loop step responses (case in work F) under fouling conditions of the HE E11AB models: without adjustment (a) after adjustment (b) and after adjustment of optimal (c) PID tuning Figure A1. Closed-loop step responses (case in work F) under fouling conditions of the HE E11AB Figure A1. Closed-loop step responses (case in work F) under fouling conditions of the HE E11AB parameters. models: without adjustment (a) after adjustment (b) and after adjustment of optimal (c) PID models: without adjustment (a) after adjustment (b) and after adjustment of optimal (c) PID tuning tuning parameters. parameters. Figure A2. Closed-loop step responses (case in work [6]) under fouling conditions of the HE Figure A2. Closed-loop step responses (case in work [6]) under fouling conditions of the HE E15AB E15AB models: without adjustment (a) after adjustment (b) and after adjustment of optimal (c) PID models: without adjustment (a) after adjustment (b) and after adjustment of optimal (c) PID tuning Figure A2. Closed-loop step responses (case in work [6]) under fouling conditions of the HE E15AB tuning parameters. parameters. models: without adjustment (a) after adjustment (b) and after adjustment of optimal (c) PID tuning parameters. Figure A3. Closed-loop step responses (case in work [6]) under fouling conditions of the HE E30AB models: without adjustment (a) after adjustment (b) and after adjustment of optimal (c) PID tuning Figure A3. Closed-loop step responses (case in work [6]) under fouling conditions of the HE E30AB parameters. models: without adjustment (a) after adjustment (b) and after adjustment of optimal (c) PID tuning parameters. Appl. Sci. 2019, 9, x FOR PEER REVIEW 21 of 24 Appl. Sci. 2019, 9, x FOR PEER REVIEW 21 of 24 Figure A1. Closed-loop step responses (case in work F) under fouling conditions of the HE E11AB models: without adjustment (a) after adjustment (b) and after adjustment of optimal (c) PID tuning Figure A1. Closed-loop step responses (case in work F) under fouling conditions of the HE E11AB parameters. models: without adjustment (a) after adjustment (b) and after adjustment of optimal (c) PID tuning parameters. Figure A2. Closed-loop step responses (case in work [6]) under fouling conditions of the HE E15AB models: without adjustment (a) after adjustment (b) and after adjustment of optimal (c) PID tuning Figure A2. Closed-loop step responses (case in work [6]) under fouling conditions of the HE E15AB parameters. models: without adjustment (a) after adjustment (b) and after adjustment of optimal (c) PID tuning parameters. Appl. Sci. 2019, 9, 824 21 of 23 Figure A3. Closed-loop step responses (case in work [6]) under fouling conditions of the HE E30AB models: without adjustment (a) after adjustment (b) and after adjustment of optimal (c) PID tuning Figure A3. Closed-loop step responses (case in work [6]) under fouling conditions of the HE Figure A3. Closed-loop step responses (case in work [6]) under fouling conditions of the HE E30AB parameters. E30AB models: without adjustment (a) after adjustment (b) and after adjustment of optimal (c) PID models: without adjustment (a) after adjustment (b) and after adjustment of optimal (c) PID tuning tuning parameters. parameters. Figure A4. Closed-loop step responses (case in work [6]) under fouling conditions of the HE E35AB models: without adjustment (a) after adjustment (b) and after adjustment of optimal (c) PID tuning parameters. Appl. Sci. 2019, 9, x FOR PEER REVIEW 22 of 24 Figure A4. Closed-loop step responses (case in work [6]) under fouling conditions of the HE E35AB Appl. Sci. 2019, 9, 824 22 of 23 models: without adjustment (a) after adjustment (b) and after adjustment of optimal (c) PID tuning parameters. Table A1. Values of the control-quality indices for used sets of the PID parameters in different periods Table A1. Values of the control-quality indices for used sets of the PID parameters in different periods of operation (cases in work [6]). of operation (cases in work [6]). Closed-loop Optimal Control-quality Indices Rf after Base Adjusted Closed-loop R after Base Adjusted Optimal Control-quality Indices Responses PID Mp tp td tr ts Period of PID PID Responses of the Period of PID PID PID 3 M t t t t p p d r s of the Parameters (%) (s) (s) (s) (s) 1 1 2 2 3 Operation Parameters Parameters Parameters Studied HEs [6,21] Operation Parameters Parameters (%) (s) (s) (s) (s) Studied (Kp/Ki/Kd) (years) (Kp/Ki/Kd) (Kp/Ki/Kd) (years) (K /K /K ) (K /K /K ) (K /K /K ) Appl. Sci. 2019, 9, x FOR PEER REVIEW p 22 of 24 i d p i d p i d HEs [6,21] 0 - - 47.3 47.3 18.7 6.18.7 1 8.7 6.161.3 8.7 61.3 for HE Figure A4. Closed-loop step responses (case in work [6]) under fouling conditions of the HE E35AB 1 67.1 67.1 26.2 9.26.2 8 12.8 9.8157 12.8 157 for HE E11AB E11AB 69.8/4.58/255 - - 69.8/4.58/255 (Figure A1a) 2 68.9 68.9 27.5 10. 27.5 3 13.4 10.3 193 13.4 193 models: without adjustment (a) after adjustment (b) and after adjustment of optimal (c) PID tuning (Figure A1a) 3 73.8 73.8 31.3 11. 31.3 5 14.8 11.5 >300 14.8 >300 parameters. 0 69.8/4.58/255 47.3 18.7 6.1 8.7 61.3 0 - 69.8/4.58/255 - 47.3 18.7 6.1 8.7 61.3 for HE 1 70.7/3.59/334 47.1 24.5 9.4 12.4 60.6 E11AB 1 - 70.7/3.59/334 - 47.1 24.5 9.4 12.4 60.6 for HE E11AB Table A1. Values of the control-quality indices for used sets of the PID parameters in different periods 2 73.9/3.66/358 47.6 25 9.6 12.6 78.8 (Figure(Figure A A1b) 1b) 2 73.9/3.66/358 47.6 25 9.6 12.6 78.8 of operation (cases in work [6]). 3 88.7/4.18/452 51.3 25.8 9.6 12.8 92.1 3 88.7/4.18/452 51.3 25.8 9.6 12.8 92.1 0 53.8/1.78/260 20 19.4 6.8 10 38.1 Closed-loop Optimal Control-quality Indices for HE Rf aft 0 er Base - Adjuste- d 53.8/1.78/260 20 19.4 6.8 10 38.1 1 53.8/1.42/260 20 26.9 10.9 14.9 56.2 Responses PID Mp tp td tr ts E11AB - - Period of 1 PID PID 53.8/1.42/260 20 26.9 10.9 14.9 56.2 for HE E11AB 2 53.8/1.39/260 20 28.5 11.5 15.7 63.3 of the Parameters (%) (s) (s) (s) (s) (Figure A1c) 1 2 Operation Parameters Parameters (Figure A1c) 3 2 50.5/1. 53.8/1.39/260 45/286 20 34. 209 12. 28.5 8 18 11.575 15.7 63.3 Studied (Kp/Ki/Kd) (years) 3 (Kp/Ki/Kd) (Kp/Ki/Kd) 50.5/1.45/286 20 34.9 12.8 18 75 Appl. Sci. 2019, 9, x FOR P 0 EER REVIEW 46 29.3 10.9 14.6 22 of 70.9 24 HEs [6,21] for HE 1 44.6 36.2 12.9 17.3 94.9 0 - - 47.3 18. 467 6.29.3 1 8.7 10.9 61.3 14.6 70.9 E15AB 2.44/0.10/13.7 - - for HE 2 45.4 38.1 13.4 18 99.8 Figure A4. Closed-loop step responses (case in work [6]) under fouling conditions of the HE E35AB for HE E15AB 1 67.1 44.6 26.2 9.36.2 8 12.8 12.9 157 17.3 94.9 (Figure A2a) E11AB 69. 2.44/0.10/13.7 8/4.58/255 - - 3 47.4 41.5 14.5 19.5 145 45.4 38.1 13.4 18 99.8 (Figure A2a) 2 68.9 27.5 10.3 13.4 193 models: without adjustment (a) after adjustment (b) and after adjustment of optimal (c) PID tuning (Figure A1a) 0 2.44/0.104/13.7 46 29.3 10,9 14.6 70.9 3 73.8 47.4 31.3 11. 41.5 5 14.8 14.5 >300 19.5 145 for HE parameters. 1 3.38/0.143/19.2 50.6 29.3 11 14.3 75.3 0 69.8/4.58/255 47.3 18.7 6.1 8.7 61.3 E15AB - - 0 - 2.44/0.104/13.7 - 46 29.3 10.9 14.6 70.9 for HE 2 3.54/0.148/20.4 51 29.6 11.1 14.5 76.8 1 70.7/3.59/334 47.1 24.5 9.4 12.4 60.6 (Figure A2b) for HE E15AB E11AB 1 - 3.38/0.143/19.2 - 50.6 29.3 11 14.3 75.3 Table A1. Values 3 of the control-quality in 3. dice 88/0. s for 157/22. us9 ed sets of the PID para 51.7m eters in 30.4 d11. iffer 4 ent period 14.8 79. s 6 2 73.9/3.66/358 47.6 25 9.6 12.6 78.8 2 3.54/0.148/20.4 51 29.6 11.1 14.5 76.8 (Figure(Figure A A2b) 1b) 0 2.30/0.03/13.8 20 27.7 11.3 15.5 63.2 of operation (cases in work [6]). 3 88.7/4.18/452 51.3 25.8 9.6 12.8 92.1 for HE 3 3.88/0.157/22.9 51.7 30.4 11.4 14.8 79.6 1 2.48/0.04/13.8 19.8 33.6 13.2 18.1 50 0 53.8/1.78/260 20 19.4 6.8 10 38.1 E15AB - - Close ford HE -loop Optimal Control-quality Indices 2 2.76/0.03/13.9 20 32.9 13.3 18.1 71.9 Rf aft 0 er Base - Adjuste- d 2.30/0.03/13.8 20 27.7 11.3 15.5 63.2 1 53.8/1.42/260 20 26.9 10.9 14.9 56.2 (Figure A2c) Responses PID Mp tp td tr ts E11AB - - 3 3.05/0.04/17 20 33.2 13.3 18.1 74.6 Period of PID PID for HE E15AB 2 1 53.8/1. 2.48/0.04/13.8 39/260 20 19.8 28.5 11. 33.6 5 15.7 13.2 63.3 18.1 50 of the Parameters (%) (s) (s) (s) (s) (Figure A1c) 0 45.5 24.6 8.6 12 76.4 1 2 Operat 2 ion Parameters Parameters 2.76/0.03/13.9 20 32.9 13.3 18.1 71.9 (Figure A2c) 3 50.5/1.45/286 20 34.9 12.8 18 75 for HE Studied (Kp/Ki/Kd) 1 49.4 27.7 10.1 13.6 94.5 (years) 3 (Kp/Ki/Kd) (Kp/Ki/Kd) 3.05/0.04/17 20 33.2 13.3 18.1 74.6 Appl. Sci. 2019, 9, x FOR P 0 EER REVIEW 46 29.3 10.9 14.6 22 of 70.9 24 E30AB 38.2/1.924/182 - - HEs [6,21] for HE 2 51.3 29.8 11 14.6 105 1 44.6 36.2 12.9 17.3 94.9 (Figure A3a) 0 - - 47.3 45.5 18.7 6.24.6 1 8.7 8.661.3 12 76.4 E15AB 2.44/0.10/13.7 - - 3 54.3 34 12.6 16.6 152 for HE 2 45.4 38.1 13.4 18 99.8 Figure A4. Closed-loop step responses (case in work [6]) under fouling conditions of the HE E35AB for HE E30AB 1 67.1 49.4 26.2 9.27.7 8 12.8 10.1 157 13.6 94.5 (Figure A2a) 0 38.2/1.924/182 45.5 24.6 8.6 12 76.4 E11AB 69. 38.2/1.924/182 8/4.58/255 - - 3 47.4 41.5 14.5 19.5 145 for HE 2 68.9 51.3 27.5 10. 29.8 3 13.4 11 193 14.6 105 (Figure A3 models: without adjustment a) (a) after adjustment (b) and after adjustment of optimal (c) PID tuning 1 41.5/1.975/209 47.2 26.2 9.5 12.9 65.2 (Figure A1a) 0 2.44/0.104/13.7 46 29.3 10,9 14.6 70.9 E30AB - - 3 73.8 54.3 31.3 11.5 34 14.8 12.6 >300 16.6 152 for HE 2 44.1/2.029/230 48 27.1 10 13.4 86.3 parameters. 1 3.38/0.143/19.2 50.6 29.3 11 14.3 75.3 (Figure A3b) 0 69.8/4.58/255 47.3 18.7 6.1 8.7 61.3 E15AB - - 3 49.7/2.157/275 49.2 28.5 10.7 14.2 98.8 for HE 0 - 38.2/1.924/182 - 45.5 24.6 8.6 12 76.4 2 3.54/0.148/20.4 51 29.6 11.1 14.5 76.8 1 70.7/3.59/334 47.1 24.5 9.4 12.4 60.6 (Figure A2b) 0 31.6/0.814/184 20 25.1 9.3 13.4 52.9 for HE E30AB E11AB 1 - 41.5/1.975/209 - 47.2 26.2 9.5 12.9 65.2 Table A1. Values 3 of the control-quality in 3. dice 88/0. s for 157/22. us9 ed sets of the PID para 51.7m eters in 30.4 d11. iffer 4 ent period 14.8 79. s 6 for HE 2 73.9/3.66/358 47.6 25 9.6 12.6 78.8 1 31.6/0.808/184 20 28.2 10.9 15.2 55.7 2 44.1/2.029/230 48 27.1 10 13.4 86.3 (Figure(Figure A A3b) 1b) 0 2.30/0.03/13.8 20 27.7 11.3 15.5 63.2 of operation ( E30AB cases in work [6]). - - 3 88.7/4.18/452 51.3 25.8 9.6 12.8 92.1 for HE 2 31.6/0.808/184 20 30.5 11.8 16.5 63.7 3 49.7/2.157/275 49.2 28.5 10.7 14.2 98.8 (Figure A3c) 1 2.48/0.04/13.8 19.8 33.6 13.2 18.1 50 0 53.8/1.78/260 20 19.4 6.8 10 38.1 E15AB - - 3 33.3/0.859/213 19.2 33.5 12.9 18 75.1 Closed-loop Optimal Control-quality Indices for HE 2 2.76/0.03/13.9 20 32.9 13.3 18.1 71.9 Rf aft 0 er Base - Adjuste- d 31.6/0.814/184 20 25.1 9.3 13.4 52.9 1 53.8/1.42/260 20 26.9 10.9 14.9 56.2 (Figure A2c) 0 51.2 16.2 5.1 7.3 70.9 Responses PID Mp tp td tr ts E11AB - - 3 3.05/0.04/17 20 33.2 13.3 18.1 74.6 for HE Period of 1 PID PID 31.6/0.808/184 20 28.2 10.9 15.2 55.7 for HE E30AB 2 53.8/1.39/260 20 28.5 11.5 15.7 63.3 1 3 76.6 25.1 8.9 11.6 232 of the Parameters (%) (s) (s) (s) (s) (Figure A1c) 0 1 2 45.5 24.6 8.6 12 76.4 E35AB 32.5/2.371/107 - - Operat 2 ion Parameters Parameters 31.6/0.808/184 20 30.5 11.8 16.5 63.7 (Figure A3c) 3 50.5/1.45/286 20 34.9 12.8 18 75 for HE 2 88.5 34.9 12.2 15.8 >300 Studied (Kp/Ki/Kd) 1 49.4 27.7 10.1 13.6 94.5 (Figure A4a) (years) 3 (Kp/Ki/Kd) (Kp/Ki/Kd) 33.3/0.859/213 19.2 33.5 12.9 18 75.1 0 46 29.3 10.9 14.6 70. 4 9 E30AB 38.2/1.924/182 - - 3 94.5 40.8 14.3 18.5 ~ HEs [6,21] for HE 2 51.3 29.8 11 14.6 105 (Figure A3a) 1 44.6 36.2 12.9 17.3 94.9 0 32.5/2.371/107 51.2 16.2 5.1 7.3 70.9 0 - - 47.3 51.2 18.7 6.16.2 1 8.7 5.161.3 7.3 70.9 E15AB 2.44/0.10/13.7 - - 3 54.3 34 12.6 16.6 152 for HE for HE 2 45.4 38.1 13.4 18 99.8 1 35.8/1.945/158 53.4 22.5 8 10.7 78.2 76.6 25.1 8.9 11.6 232 for HE (Figure A E35AB 2a) 1 67.1 26.2 9.8 12.8 157 0 38.2/1.924/182 45.5 24.6 8.6 12 76.4 E35AB - - E11AB 69. 32.5/2.371/107 8/4.58/255 - - 3 47.4 41.5 14.5 19.5 145 for HE 2 51.1/2.398/262 54.9 25.9 9.1 12.1 92.6 2 68.9 88.5 27.5 10. 34.9 3 13.4 12.2 193 15.8 >300 (Figure A4a) 1 41.5/1.975/209 47.2 26.2 9.5 12.9 65.2 (Figure A4b) (Figure A1a) 0 2.44/0.104/13.7 46 29.3 10,9 14.6 70.9 4 E30AB - - 3 62.2/2.714/342 55.4 27.7 9.6 12.8 99.8 3 73.8 94.5 31.3 11. 40.8 5 14.8 14.3 >300 18.5 ~ for HE 2 44.1/2.029/230 48 27.1 10 13.4 86.3 (Figure A3b) 1 3.38/0.143/19.2 50.6 29.3 11 14.3 75.3 0 25.9/0.395/111 18.4 16.4 5.6 8.4 39.9 0 69.8/4.58/255 47.3 18.7 6.1 8.7 61.3 E15AB - - 3 49.7/2.157/275 49.2 28.5 10.7 14.2 98.8 for HE for HE 0 - 32.5/2.371/107 - 51.2 16.2 5.1 7.3 70.9 2 3.54/0.148/20.4 51 29.6 11.1 14.5 76.8 1 25.9/0.386/111 20 25.5 9.9 13.7 58.1 (Figure A2b) 1 70.7/3.59/334 47.1 24.5 9.4 12.4 60.6 0 31.6/0.814/184 20 25.1 9.3 13.4 52.9 E35AB 1 - - 35.8/1.945/158 53.4 22.5 8 10.7 78.2 for HE E35AB E11AB - - 3 3.88/0.157/22.9 51.7 30.4 11.4 14.8 79.6 for HE 2 30.4/0.416/139 20 34.2 12.7 17.9 73.9 2 73.9/3.66/358 47.6 25 9.6 12.6 78.8 1 31.6/0.808/184 20 28.2 10.9 15.2 55.7 (Figure A4c) (Figure A1b) 2 51.1/2.398/262 54.9 25.9 9.1 12.1 92.6 (Figure A4b) 0 2.30/0.03/13.8 20 27.7 11.3 15.5 63.2 E30AB - - 3 39.5/0.479/216 20 34.7 12.7 17.9 74.9 3 88.7/4.18/452 51.3 25.8 9.6 12.8 92.1 for HE 2 31.6/0.808/184 20 30.5 11.8 16.5 63.7 3 62.2/2.714/342 55.4 27.7 9.6 12.8 99.8 1 1 2.48/0.04/13.8 19.8 33.6 13.2 18.1 50 (Figure A3c) characteristics were simulated assuming constant values of the PID parameters obtained using Ziegler-Nichols method [18] for clean HEs 0 53.8/1.78/260 20 19.4 6.8 10 38.1 E15AB - - 3 33.3/0.859/213 19.2 33.5 12.9 18 75.1 2 for HE characteristics with the adju 2 sted PID parameters, for the consecutive periods of fo 2.76/0. uling b 03/1 u3. ild- 9 up, using Ziegler 20 32.9 -Nic13. hols meth 3 18. od1 [18] 71.9 1 0 - - 53.8/1. 25.9/0.395/111 42/260 20 18.4 26.9 10. 16.4 9 14.9 5.656.2 8.4 39.9 (Figure A2c) 3 0 51.2 16.2 5.1 7.3 70.9 characE teristi 11AcB s with the optimal PID parameter- - s obtained under fouling conditions using Signal Constraint toolbox in SIMULINK [21] 3 3.05/0.04/17 20 33.2 13.3 18.1 74.6 for HE 1 25.9/0.386/111 20 25.5 9.9 13.7 58.1 for HE E35AB 2 53.8/1.39/260 20 28.5 11.5 15.7 63.3 4 1 76.6 25.1 8.9 11.6 232 non-expiring oscillations detected (Figure A1c) 0 45.5 24.6 8.6 12 76.4 E35AB 32.5/2.371/107 - - 2 30.4/0.416/139 20 34.2 12.7 17.9 73.9 (Figure A4c) 3 50.5/1.45/286 20 34.9 12.8 18 75 for HE 2 88.5 34.9 12.2 15.8 >300 1 49.4 27.7 10.1 13.6 94.5 (Figure A4a) 3 39.5/0.479/216 20 34.7 12.7 17.9 74.9 0 46 29.3 10.9 14.6 70. 4 9 E30AB 38.2/1.924/182 - - 3 94.5 40.8 14.3 18.5 ~ for HE 2 51.3 29.8 11 14.6 105 1 44.6 36.2 12.9 17.3 94.9 1 (Figure A3a) 0 32.5/2.371/107 51.2 16.2 5.1 7.3 70.9 characteristics were simulated assuming constant values of the PID parameters obtained using Ziegler-Nichols E15AB 2.44/0.10/13.7 - - 3 54.3 34 12.6 16.6 152 for HE 2 45.4 38.1 13.4 18 99.8 1 2 35.8/1.945/158 53.4 22.5 8 10.7 78.2 (Figure A2a) method [18] for clean HEs characteristics with the adjusted PID parameters, for the consecutive periods of fouling 0 38.2/1.924/182 45.5 24.6 8.6 12 76.4 E35AB - - 3 47.4 41.5 14.5 19.5 145 for HE 2 51. 31/2.398/262 54.9 25.9 9.1 12.1 92.6 build-up, using Ziegler 1 -Nichols method [1841. ] 5/1. characteristics 975/209 with the 47. optimal 2 26.2PID 9. parameters 5 12.9 65. obtained 2 under (Figure A4b) 0 2.44/0.104/13.7 46 29.3 10,9 14.6 70.9 E30AB - - 3 62.2/2.714/342 55.4 27.7 9.6 12.8 99.8 for HE 2 44.1/2.029/230 48 27.1 10 13.4 86.3 fouling conditions using Signal Constraint toolbox in SIMULINK [21] non-expiring oscillations detected 1 3.38/0.143/19.2 50.6 29.3 11 14.3 75.3 (Figure A3b) 0 25.9/0.395/111 18.4 16.4 5.6 8.4 39.9 E15AB - - 3 49.7/2.157/275 49.2 28.5 10.7 14.2 98.8 for HE 2 3.54/0.148/20.4 51 29.6 11.1 14.5 76.8 1 25.9/0.386/111 20 25.5 9.9 13.7 58.1 (Figure A2b) 0 31.6/0.814/184 20 25.1 9.3 13.4 52.9 E35AB - - 3 3.88/0.157/22.9 51.7 30.4 11.4 14.8 79.6 for HE 2 30.4/0.416/139 20 34.2 12.7 17.9 73.9 1 31.6/0.808/184 20 28.2 10.9 15.2 55.7 (Figure A4c) References 0 2.30/0.03/13.8 20 27.7 11.3 15.5 63.2 E30AB 3 - - 39.5/0.479/216 20 34.7 12.7 17.9 74.9 for HE 2 31.6/0.808/184 20 30.5 11.8 16.5 63.7 1 1 2.48/0.04/13.8 19.8 33.6 13.2 18.1 50 (Figure A3c) characteristics were simulated assuming constant values of the PID parameters obtained using Ziegler-Nichols method [18] for clean HEs E15AB - - 3 33.3/0.859/213 19.2 33.5 12.9 18 75.1 characteristics with the adju 2 sted PID parameters, for the consecutive periods of fo 2.76/0. uling b 03/1 u3. ild- 9 up, using Ziegler 20 32.9 -Nic13. hols meth 3 18. od1 [18] 71.9 1. Diaby, A.L.; Miklavcic, S.J.; Bari, S.; Addai-Mensah, J. Evaluation of crude oil heat exchanger network fouling (Figure A2c) 3 0 51.2 16.2 5.1 7.3 70.9 characteristics with the optimal PID parameters obtained under fouling conditions using Signal Constraint toolbox in SIMULINK [21] 3 3.05/0.04/17 20 33.2 13.3 18.1 74.6 for HE 4 1 76.6 25.1 8.9 11.6 232 behavior non-expiring o under scillaaging tions detected conditions for scheduled cleaning. Heat Trans. Eng. 2016, 37, 1211–1230. [CrossRef] E35AB 0 32.5/2.371/107 - - 45.5 24.6 8.6 12 76.4 for HE 2 88.5 34.9 12.2 15.8 >300 1 49.4 27.7 10.1 13.6 94.5 (Figure A4a) 2. Tian, J.; Wang, Y.; Feng, X. Simultaneous optimization of ﬂow velocity and cleaning schedule for mitigating E30AB 3 38.2/1.924/182 - - 94.5 40.8 14.3 18.5 ~ 2 51.3 29.8 11 14.6 105 (Figure A3a) 0 32.5/2.371/107 51.2 16.2 5.1 7.3 70.9 fouling in reﬁnery heat exchanger networks. Energy 2016, 109, 1118–1129. [CrossRef] 3 54.3 34 12.6 16.6 152 for HE 1 35.8/1.945/158 53.4 22.5 8 10.7 78.2 0 38.2/1.924/182 45.5 24.6 8.6 12 76.4 E35AB - - for HE 2 51.1/2.398/262 54.9 25.9 9.1 12.1 92.6 1 41.5/1.975/209 47.2 26.2 9.5 12.9 65.2 (Figure A4b) E30AB - - 3 62.2/2.714/342 55.4 27.7 9.6 12.8 99.8 2 44.1/2.029/230 48 27.1 10 13.4 86.3 (Figure A3b) 0 25.9/0.395/111 18.4 16.4 5.6 8.4 39.9 3 49.7/2.157/275 49.2 28.5 10.7 14.2 98.8 for HE 1 25.9/0.386/111 20 25.5 9.9 13.7 58.1 0 31.6/0.814/184 20 25.1 9.3 13.4 52.9 E35AB - - for HE 2 30.4/0.416/139 20 34.2 12.7 17.9 73.9 1 31.6/0.808/184 20 28.2 10.9 15.2 55.7 (Figure A4c) E30AB - - 3 39.5/0.479/216 20 34.7 12.7 17.9 74.9 2 31.6/0.808/184 20 30.5 11.8 16.5 63.7 (Figure A3c) characteristics were simulated assuming constant values of the PID parameters obtained using Ziegler-Nichols method [18] for clean HEs 3 33.3/0.859/213 19.2 33.5 12.9 18 75.1 characteristics with the adjusted PID parameters, for the consecutive periods of fouling build-up, using Ziegler-Nichols method [18] 0 51.2 16.2 5.1 7.3 70.9 characteristics with the optimal PID parameters obtained under fouling conditions using Signal Constraint toolbox in SIMULINK [21] for HE 4 1 76.6 25.1 8.9 11.6 232 non-expiring oscillations detected E35AB 32.5/2.371/107 - - 2 88.5 34.9 12.2 15.8 >300 (Figure A4a) 3 94.5 40.8 14.3 18.5 ~ 0 32.5/2.371/107 51.2 16.2 5.1 7.3 70.9 for HE 1 35.8/1.945/158 53.4 22.5 8 10.7 78.2 E35AB - - 2 51.1/2.398/262 54.9 25.9 9.1 12.1 92.6 (Figure A4b) 3 62.2/2.714/342 55.4 27.7 9.6 12.8 99.8 0 25.9/0.395/111 18.4 16.4 5.6 8.4 39.9 for HE 1 25.9/0.386/111 20 25.5 9.9 13.7 58.1 E35AB - - 2 30.4/0.416/139 20 34.2 12.7 17.9 73.9 (Figure A4c) 3 39.5/0.479/216 20 34.7 12.7 17.9 74.9 characteristics were simulated assuming constant values of the PID parameters obtained using Ziegler-Nichols method [18] for clean HEs characteristics with the adjusted PID parameters, for the consecutive periods of fouling build-up, using Ziegler-Nichols method [18] characteristics with the optimal PID parameters obtained under fouling conditions using Signal Constraint toolbox in SIMULINK [21] non-expiring oscillations detected Appl. Sci. 2019, 9, 824 23 of 23 3. Liu, R.; Wang, Y.; Feng, X. Mitigation of fouling in crude preheat trains by simultaneous dynamic optimization of ﬂow rate and velocity distribution. Comput. Aided Chem. Eng. 2017, 40, 817–822. [CrossRef] 4. Smith, R.; Loyola-Fuentes, J.; Jobson, M. Fouling in heat exchanger networks. Chem. Eng. Trans. 2017, 61, 1789–1794. [CrossRef] 5. Liu, L.; Fan, J.; Chen, P.; Du, J.; Yang, F. Synthesis of heat exchanger networks considering fouling, aging, and cleaning. Ind. Eng. Chem. Res. 2015, 54, 296–306. [CrossRef] 6. Trafczynski, M.; Markowski, M.; Alabrudzinski, ´ S.; Urbaniec, K. The inﬂuence of fouling on the dynamic behavior of PID-controlled heat exchangers. Appl. Therm. Eng. 2016, 109, 727–738. [CrossRef] 7. Santamaria, F.L.; Macchietto, S. Integration of optimal cleaning scheduling and control of heat exchanger networks undergoing fouling: Model and formulation. Ind. Eng. Chem. Res. 2018, 57, 12842–12860. [CrossRef] 8. Varga, E.I.; Hangos, K.M.; Szigeti, F. Controllability and observability of heat exchanger networks in the time-varying parameter case. Control Eng. Pract. 1995, 3, 1409–1419. [CrossRef] 9. Jäschke, J.; Skogestad, S. Optimal operation of heat exchanger networks with stream split: Only temperature measurements are required. Comput. Chem. Eng. 2014, 70, 35–49. [CrossRef] 10. Bakošová, M.; Oravec, J. Robust model predictive control for heat exchanger network. Appl. Therm. Eng. 2014, 73, 924–930. [CrossRef] 11. Oravec, J.; Trafczynski, M.; Bakošová, M.; Markowski, M.; Mészáros, A.; Urbaniec, K. Robust model predictive control of heat exchanger network in the presence of fouling. Chem. Eng. Trans. 2017, 61, 337–342. [CrossRef] 12. Oravec, J.; Bakošová, M.; Trafczynski, M.; Vasickaninov ˇ á, A.; Mészáros, A.; Markowski, M. Robust model predictive control and PID control of shell-and-tube heat exchangers. Energy 2018, 159, 1–10. [CrossRef] 13. Korbel, M.; Bellec, S.; Jiang, T.; Stuart, P. Steady state identiﬁcation for on-line data reconciliation based on wavelet transform and ﬁltering. Comput. Chem. Eng. 2014, 63, 206–218. [CrossRef] 14. Narasimhan, S.; Jordache, C. Data Reconciliation and Gross Error Detection: An Intelligent Use of Process Data; Gulf Professional Publishing: Huston, TX, USA, 1999. 15. Gaddis, E.S.; Schlünder, E.U. Temperature distribution and heat exchange in multipass shell and tube exchangers with bafﬂes. Heat Transf. Eng. 1979, 1, 43–52. [CrossRef] 16. Markowski, M.; Trafczynski, M.; Urbaniec, K. Validation of the method for determination of the thermal resistance of fouling in shell and tube heat exchangers. Energy Convers. Manage. 2013, 76, 307–313. [CrossRef] 17. Markowski, M.; Trafczynski, M.; Urbaniec, K. Identiﬁcation of the inﬂuence of fouling on the heat recovery in a network of shell and tube heat exchangers. Appl. Energy 2013, 102, 755–764. [CrossRef] 18. Ziegler, J.G.; Nichols, N.B. Optimum settings for automatic controllers. Trans. ASME 1942, 64, 759–765. [CrossRef] 19. Tubular Exchanger Manufacturers Association. Standards of the Tubular Exchanger Manufacturers Association, 8th ed.; TEMA: New York, NY, USA, 1999. 20. Skogestad, S. Simple analytic rules for model reduction and PID controller tuning. J. Proc. Control 2003, 13, 291–309. [CrossRef] 21. Trafczynski, M.; Markowski, M.; Alabrudzinski, S.; Urbaniec, K. Tuning parameters of PID controllers for the operation of heat exchangers under fouling conditions. Chem. Eng. Trans. 2016, 52, 1237–1242. [CrossRef] 22. Borges de Carvalho, C.; Carvalho, E.P.; Ravagnani, M.A.S.S. Dynamic analysis of fouling buildup in heat exchangers designed according to tema standards. Ind. Eng. Chem. Res. 2018, 57, 3753–3764. [CrossRef] 23. Borges de Carvalho, C.; Carvalho, E.P.; Ravagnani, M.A.S.S. Tuning strategies for overcoming fouling effects in proportional integral derivative controlled heat exchangers. Ind. Eng. Chem. Res. 2018, 57, 10518–10527. [CrossRef] 24. Markowski, M.; Urbaniec, K. Optimal cleaning schedule for heat exchangers in a heat exchanger network. Appl. Therm. Eng. 2005, 25, 1019–1032. [CrossRef] © 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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A Modeling Framework to Investigate the Influence of Fouling on the Dynamic Characteristics of PID-Controlled Heat Exchangers and Their Networks

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Publisher: Multidisciplinary Digital Publishing Institute
ISSN: 2076-3417
DOI: 10.3390/app9050824
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A Modeling Framework to Investigate the Influence of Fouling on the Dynamic Characteristics of PID-Controlled Heat Exchangers and Their Networks

A Modeling Framework to Investigate the Influence of Fouling on the Dynamic Characteristics of PID-Controlled Heat Exchangers and Their Networks

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A Modeling Framework to Investigate the Influence of Fouling on the Dynamic Characteristics of PID-Controlled Heat Exchangers and Their Networks

A Modeling Framework to Investigate the Influence of Fouling on the Dynamic Characteristics of PID-Controlled Heat Exchangers and Their Networks

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