Article pubs.acs.org/IECR
Cite This: Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX
Tuning Strategies for Overcoming Fouling Effects in Proportional Integral Derivative Controlled Heat Exchangers Carolina Borges de Carvalho,† Esdras P. Carvalho,‡ and Mauro A. S. S. Ravagnani*,† Department of Chemical Engineering and ‡Department of Mathematics, State University of Maringá, Maringá 87020900, Brazil
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†
ABSTRACT: Fouling is one of the main causes of industrial problems regarding operation and control of shell and tube heat exchangers. Although fouling is a time dependent phenomenon, most papers in the literature focus on fouling mitigation in steady-state heat recovery. In this work, a lumped parameter model to describe timevarying conditions and their influence in controller tuning is presented. For each modeling cell, there are four input variables (inlet temperatures and flow rates for shell and tube sides), which generate two output signals (outlet temperatures for shell and tubes). The influence of fouling in process control is evaluated by considering intermediate values of thermal resistance of fouling (Rf), simulating its variation with time. The model was implemented in MATLAB/Simulink, and simulations have been carried out for different periods of operation. A step change was applied in the shell flow rate to evaluate the response in the tube outlet temperature. Results show that periodic fitting in proportional integral derivative (PID) parameters are needed to keep the tube outlet stream at the desired temperature. Moreover, two optimization strategies are presented for tuning controller gains constrained by some step response performance indicators. Three case studies have been taken as benchmarks, and results show that this strategy is promising.
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flow, or dead zones, and also designing the equipment for easy cleaning.5 In this context, a procedure to link a heat exchanger design algorithm to a dynamic simulation of fouling rate was recently proposed by Nakao et al.6 Computational tests using a modified rate model7 was able to modify the heat exchanger design, exploring the predictions of fouling rate and incorporating fouling modeling into STHX design. By the definition of heat exchanger network (HEN) synthesis, fouling effects are seldom considered or are modeled as a fixed value. A methodology proposed by Ravagnani et al.8 includes the detailed design of heat exchangers in a HEN optimization by calculating fouling factors from heat transfer coefficients and also limiting the pressure drop. In another approach, STHX design was formulated as an optimization problem, rigorously following the standards of TEMA with the Bell−Delaware method for the shell side and including overall heat transfer coefficient calculations.9 Particle swarm optimization (PSO) was used to solve the problem considering industrial international parameters standards. Moreover, although fouling is a time dependent phenomenon, most papers in the literature focus on fouling-induced reduction in steady-state heat recovery, and heat exchangers affected by fouling in transient states do not have a wide range of study, especially regarding process control.
INTRODUCTION Almost all fluids used in shell and tube heat exchangers (STHX) have solid particles or microorganisms originally suspended or dissolved, often in a very low concentration, deposition of which on heat transfer surfaces causes several problems regarding operation, efficiency, and process control. Those undesired deposits are callled “fouling” and are the main cause of gradual decline in the performance of heat exchangers.1 The fouling layer increases the overall resistance to heat transfer, and it is commonly expressed in numerical terms by Rf. Operational evidence of fouling resistance values has been provided since the 1950s by a compilation published by the Tubular Exchanger Manufacturers Association (TEMA), and those values are still the basis for designing most heat exchangers worldwide.2 Due to the importance of heat exchangers in thermal efficiency processes, several approaches for fouling mitigation are presented in the literature, representing a subject widely investigated. As complete elimination of fouling is rarely achieved in practice, cleaning of fouled units is a regular task in the process industries.3 Methodologies for managing the cleaning schedule problem in the design of heat exchangers, optimizing heat transfer areas or cleaning intervention periods,4 and mathematical formulation of the optimum cleaning cycle undergoing fouling3 are frequently found in the literature. Fouling may effectively be mitigated at the design step of the heat exchanger by selecting a suitable equipment type and geometry, avoiding operational conditions (velocities and temperature) that promote fouling, such as hot spots, bypass © XXXX American Chemical Society
Received: Revised: Accepted: Published: A
February 28, 2018 May 17, 2018 July 16, 2018 July 16, 2018 DOI: 10.1021/acs.iecr.8b00906 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX
Article
Industrial & Engineering Chemistry Research
Figure 1. Scheme of the division and cell interdependency in a two-pass heat exchanger. Dashed and solid lines represent the tube side and shell side, respectively.
For representative industrial applications, quantitative technical and economic effects of the dynamics caused by fouling were highlighted by Coletti et al.,10 showing the need to include such dynamics in STHX design and retrofit of heat exchanger networks. Dynamic analysis of cross-flow heat exchangers can also be used to improve cooling unit performance and aid in the design of thermal control systems, as shown by Gao et al.11 In their work, the authors modeled dynamic behavior of heat exchangers and detailed analyses of transient response were conducted for fixed fouling factors. In the same field, numerical investigation of the temperature transient response for perturbations provided in both inlet temperature and flow was studied.12 The effect of fouling resistance was not considered in the examples presented by the authors, which concluded that the mean exit temperature of both fluids increases or decreases with the simultaneous increase or decrease in flow rate of the two fluids. In a recent work, a complete dynamic analysis related to fouling buildup effects in the design and operation of heat exchangers was performed.13 The authors concluded that, based on open loop responses, a greater utility demand would be required and the fouling accumulation would lead to considerable losses in the quality indicator of the output signals. However, aiming to investigate the practical effect of this fouling accumulation, a control study must be performed. It is known that inefficient control of both individual heat exchangers and entire networks may also have detrimental effects on the overall plant performance.14 On the one hand, the proportional integral derivative (PID) controller is often used in industries because of its simplicity and wide range of applicability.15 On the other hand, publications devoted to the effect of fouling in control issues are scarce and limited in scope. Also, no systematic tuning procedures are available for about 80% of PID-controlled heat exchangers that are either operated in manual tuning mode or operated using default values of tuning parameters.16 In this context, a tuning method for a nonlinear model predictive controller (MPC) in district heating networks to reach the best performance using a cell model to describe the dynamic behavior of heat exchangers was shown to be efficient by using an optimal tuning parameter method for the controller.17
The PID controller, however, remains the most popular for industrial applications. There are several design schemes in the literature for optimally tuning PID controllers. They can be based on time-domain, frequency-domain, or multiobjective optimal tuning.18 Hence, the aim of this work is to reduce the undesired effects of fouling in PID-controlled STHX by applying a proper controller tuning strategy, contributing to overcome the shortcomings in this important subject. This study can be considered an extension of our previous work.13 Tube outlet temperature responses are simulated by applying a step change in the shell side flow rate for three case studies in five periods of operation (Rf as a function of time). Therefore, two optimization strategies are applied, aiming to achieve the best values for PID parameters which keep quality control indicators stable. Further, to assess the robustness of the controller tuned, step changes are applied to the inlet temperature.
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METHODOLOGY Dynamic Modeling. Aiming to get the proper dynamic model to describe the heat exchanger behavior, an approach using a lumped parameter model was chosen. Those kinds of models introduce the concept of modeling cell (i, j), defined as two perfectly stirred tanks, exchanging heat only with each other through a dividing wall.19 Figure 1 depicts the division into n cells of a two-pass STHX.13 Therefore, the dynamic model based on heat exchanger cells leads to a set of ordinary differential equations (ODE) with regard to time only. The representativeness of this approach for STHX was presented by several authors working on simulation and control of heat exchangers, showing satisfactory results to predict heat exchanger dynamic behavior allied to computational simplicity.20 As detailed in cell (1, 2) of Figure 1, each modeling element is represented by four inputs (inlet temperatures and mass flow rates for shell and tube sides, Tsi(i, j), Tti(i, j), Ms(i, j), and Mt(i, j)) and two outputs (outlet temperatures for shell and tube, Tso(i, j) and Tto(i, j)). The interdependency of those variables depends on the equipment flow arrangement. In this work, three examples of two-pass shell and tube heat exchangers were considered. Based on similar models, some modeling simplifications are used for fully developed turbulent flows when convective heat transfer is greater than heat B
DOI: 10.1021/acs.iecr.8b00906 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX
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Industrial & Engineering Chemistry Research conduction between cells, such as perfect mixing in all cells; the wall temperature is uniform within the cell volume and heat losses or heat conduction between cells is negligibly small.19 Also, a realistic model for STHX mechanical design is desirable, avoiding heat exchanger parameter adjustment after the design task. Therefore, an optimal design problem that was proposed by Ravagnani and Caballero21 and latter applied by Borges de Carvalho et al.13 is used to model the heat exchangers, considering rigorously standard mechanical variables published by TEMA. This kind of approach aims to obtain more realistic values for the STHX design when compared with the literature. Detailed information about the used model and about the optimization procedure can be found in Borges de Carvalho et al.13 This model was used in the case studies in the present work. Lumped Parameter Energy Balances. For each modeling cell (i, j), three equations derived from the energy balances of a control volume are proposed (cold fluid, hot fluid, and wall). The number of cells (Nc) directly indicates the complexity of the model and the minimum number that accurately describes the system is20 Nc = (Nbf + 1)Ntp
(6)
dTso = a5Ms(Tsi − Tso) + a6(Tsw − Tso) dt
(7)
Vt =
πD12lnb 4
Vw =
π (D2 2 − D12)lnb 4
(14)
(15)
(16)
∫0
t
e(t ) dt + τd
de(t ) dt
(17)
In eq 17, it is assumed that e(t) is the difference between the desired set point and the measured process variable (error) and u(t) is the controller output. In the frequency domain, eq 17 is expressed by G (s ) =
τ U (s ) = K p + i + τds E (s ) s
(18)
where Kp is the controller gain, τi is the integral time, and τd is the derivative time. It is convenient, then, to define the process dynamic system using operator transmittances, defined as the ratio between output and input signal Laplace transforms. It is assumed that initial conditions equal zero. First, eqs 5 and 7 are linearized by Taylor series expansions in the neighborhood of the steadystate point value. After that, the Laplace transform is applied to the equations resulting in an algebraic linear system in the frequency domain. The primary variable of interest regarding fouling accumulation is the temperature of process streams, and it varies not only in time but also spatially within the unit. Therefore, all thermophysical parameters, such as density, heat capacity, viscosity, and thermal conductivity, assume different values in each cell. Several commercial software programs are available to assist engineers in heat exchanger rating and design. The most popular ones are HTRI and Aspen Exchanger Design & Rating (EDR) from Aspen Plus. Although Aspen EDR does
where (8)
t
(13)
where nb is the number of tubes. PID Controller. The PID controller is designed to stabilize the system and to provide good set point tracking and good disturbance rejection with robustness to the parameter uncertainty.15 Also, PID controllers are the most popular controllers used in industrial systems for heat exchanger control, mainly due to their simplicity and robustness, and the following parallel form will be assumed:
(4)
dTw = a3(Tto − Tw ) + a4(Tso − Tw ) dt
nbπD1lhft ρt C p Vt
nbπD2lhfs ρs C pVs
u(t ) = K pe(t ) + τi
(5)
a2 =
a6 =
(2)
dTto = a1M t(Tti − Tto) + a 2(Ttw − Tto) dt
1 ρt Vt
(12)
2 πD2 2nb yzz ji πD zzl Vs = jjjj 3 − 4 z{ k 8
Adopting mechanical constants defined by a1−a6, eqs 2−4 can be transformed to the following form:
a1 =
1 ρs Vs
As each cell assumes the shape of a half-cylinder, shell, tube, and tube wall material volumes can be calculated as follows, respectively:
(3)
s
a5 =
s
w
dTso M h n πD l = s (Tsi − Tso) + fs b 2 (Tsw − Tso) dt ρs Vs ρs VsC p
(11)
w
h n πD l h n πD l dTw = ft b 1 (Tto − Ttw ) + fs b 2 (Tso − Tsw ) dt ρw VwC p ρw VwC p w
nbπD2lhfs ρw C p Vw
a4 =
where Nbf is the number of baffles and Ntp is the number of tube passes. Assuming negligibly small kinetic and potential energies and no work generation, the following energy balances can be written for each cell (i, j) regarding tube side fluid, tube walls, and shell side fluid (“t”, “w”, and “s” subscripts, respectively):
t
(10)
w
(1)
dTto Mt h n πD l = (Tti − Tto) + ft b 1 (Ttw − Tto) dt ρt Vt ρt VtC p
nbπD1lhft ρw C p Vw
a3 =
(9) C
DOI: 10.1021/acs.iecr.8b00906 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX
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Industrial & Engineering Chemistry Research
Figure 2. Closed-loop model implemented in Simulink.
not provide a detailed calculation, it has a reliable database that accurately calculates the local temperature distribution.22 Aiming to simulate the heat exchanger designed according to TEMA recommendations, Aspen was chosen due to its capability to simulate all TEMA types and mechanical features. With all STHX design variables already calculated,13 those values were inputted in Aspen EDR to provide a temperature distribution along the unit for each case. After that, a database containing density, heat capacity, viscosity, and thermal conductivity values for each cell temperature was composed. Those cell temperatures are the steady-state point values, and the subscript “e” is used to distinguish the steady-state properties from transient ones. The linear system solution provides eight operator transmittances, G(s), written in terms of deviation variables denoted with an overbar, for each cell (i, j) as follows:13 G1(s) =
G5(s) =
{s + a1M te + a 2 − a 2a3(s + a5Mse + a6) −1
[(s + a3 + a4)(s + a5Mse + a6) − a4a6]−1 } G6(s) =
Tto̅ = [a 2a4a5(Tsie − Tsoe)] M̅ s
+ a6) − a4a6] − a 2a3(s + a5Mse + a6)}−1 G7(s) =
(24)
Tto̅ = [a 2a4a5Mse] Tsi̅
{(s + a1M te + a 2)[(s + a3 + a4)(s + a5Mse
Tso̅ = [a1a3a6(Ttie − Ttoe)] M̅ t
+ a6) − a4a6] − a 2a3(s + a5Mse + a6)}−1
+ a6) − a4a6] − a 2a3(s + a5Mse + a6)}−1
G8(s) =
(19)
(25)
Tto̅ = [a1M te] Tti̅
{s + a1M te + a 2 − a 2a3(s + a5Mse + a6)
T̅ G2(s) = so = [a5(Tsoe − Tsie)] M̅ s
−1
[(s + a3 + a4)(s + a5Mse + a6) − a4a6]−1 } −1 −1
[(s + a3 + a4)(s + a1M te + a 2) − a 2a3] }
(20)
Tso̅ = [a5Mse] Tsi̅
{(s + a5Mse + a6) − a4a6(s + a1M te + a 2) −1
[(s + a3 + a4)(s + a1M te + a 2) − a 2a3]−1 }
(21)
Tso̅ = [a1a3a6M te] Tti̅
{(s + a1M te + a 2)[(s + a3 + a4)(s + a5Mse −1
+ a6) − a4a6] − a 2a3(s + a5Mse + a6)}
(26)
Dynamic Simulation. The designed flow arrangement for each STHX and operator transmittances calculated in eqs 19−26 were implemented in the MATLAB/Simulink environment. The entire arrangement varies according to the cell numbers and assumes a further complex arrangement. A simplified scheme for the model implemented as a closed-loop system equipped with a PID controller is detailed in Figure 2. The effect of varying the fouling conditions (Rf) for shell and tube sides affects the film transfer coefficients and, consequently, the mechanical constants a3, a4, and a6. The following relationships are used to vary Rf and, consequently, hfs and hft for different periods of operation (subscript “f” denotes fouled conditions):
{s + a5Mse + a6 − a4a6(s + a1M te + a 2)
G4(s) =
(23)
{(s + a1M te + a 2)[(s + a3 + a4)(s + a5Mse
{(s + a1M te + a 2)[(s + a3 + a4)(s + a5Mse
G3(s) =
Tto̅ = [a1(Ttie − Ttoe)] M̅ t
hfs =
(22) D
hs hsR fs + 1
(27) DOI: 10.1021/acs.iecr.8b00906 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX
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Industrial & Engineering Chemistry Research hft =
ht htR ft + 1
implemented in the Simulink environment, being able to specify multiple tuning objectives, such as reference tracking, disturbance rejection, and stability margins. Strategy 1. Aiming to achieve a practical strategy to tune PID controller parameters, the first alternative is to use the PID Tuner Algorithm provided by the Simulink Control System. This algorithm bases the initial design upon the open loop frequency response of the linearized plant. By default, the algorithm designs for a 60° phase margin. Due to the plant model nonlinearities, the automatic tuner computes a linearized approximation of the plant using the initial conditions specified in the Simulink model as the operating point. The user is able to refine the performance (response time, bandwidth) through a graphical interface, balancing the focus to reference tracking or disturbance rejection avoiding classical trial-and-error processes, such as Ziegler and Nichols.23 Strategy 2. The second alternative is to implement a numerical algorithm based on sequential quadratic programming (SQP) using Signal Constraint Block in Simulink.14 The optimization problem consists of finding the optimal values of Kp, τi, and τd constrained to the maximum acceptable values of tr, ta, and Mp. With all those values well-defined, the algorithm minimizes the deviation between reference values and calculated ones. The basic idea of SQP is to model nonlinear programming (NLP) at a given approximate solution by a quadratic programming subproblem and, then, use the solution of this subproblem to construct a better approximation, creating an iterative process. The method can be viewed as an extension of quasi-Newton methods for constrained optimization. Also, this algorithm is widely used as a local search method and is highly fast and effective to find optimal solutions, although there is no guarantee of finding global optimum solutions for nonconvex problems.24 In this strategy, the influence of the user in the tuning procedure is minimal and external factors are eliminated by the optimization problem.
(28)
Five periods of operation are considered, varying Rf since the clean condition until the maximum fouled condition provided by the literature for each fluid, simulating the dynamic behavior in different stages of uninterrupted operation.13 For the simulation, a step change in the shell fluid flow rate (Ms) was applied to evaluate the response in the tube outlet temperature (Tto) at different periods of operation. The initial operational conditions occur when the heat exchanger surfaces are clean, that is, Rf = 0. The controller parameters (Kp, τi, and τd) are first tuned automatically for a step change in the shell fluid flow rate at this condition, and then Rf increases gradually until the value available in the literature (Table 3). Related to its time behavior, the system performance specifications to a step response can be described by the rise time (tr), settling time (ta), and overshoot (Mp). The swiftness of the response is measured by tr (time to achieve 80% of the step, in this case). The time required for the system to settle within an error of 3.5% of the input is defined by ta, and Mp is the maximum value (peak) achieved after the input signal (Figure 3).
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CASE STUDIES AND DISCUSSION Three cases studied for optimal design problems were simulated.25 Table 1 shows the original design specifications. For those case studies, the designer defines the fluid allocation before initializing the optimization routine. In Table 1, the left column of each case corresponds to the shell side while the right column refers to the tube side. The optimal mechanical designs for the exchangers were calculated by previous work, and Table 2 shows the results obtained.13
Figure 3. System performance specifications (tr, ta for a given error ϵ̅ and Mp in peak time tp).
Controller Tuning Strategies. Two different approaches were simulated to reduce the adverse effects of fouling in the control task. Both are based on automated algorithms
Table 1. Process Input and Physical Properties for Different Case Studies case 01 M (kg/s) Tin (°C) Tout (°C) ρ (kg/m3) Cp (kJ/kg·K) μ (Pa·s) k (W/m·K) Rf (m2 K/W)
case 02
case 03
methanol
seawater
kerosene
crude oil
distilled water
industrial water
27.80 95.0 40.0 750.0 2.84 0.00034 0.19 0.00033
68.90 25.0 40.0 995.0 4.20 0.00080 0.59 0.00020
5.52 199.0 93.3 850.0 2.47 0.00040 0.13 0.00061
18.80 37.8 76.7 995.0 2.05 0.00358 0.13 0.00061
22.07 33.9 29.4 995.0 4.18 0.00080 0.62 0.00017
35.31 23.9 26.7 995.0 4.18 0.00092 0.62 0.00017
E
DOI: 10.1021/acs.iecr.8b00906 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX
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Industrial & Engineering Chemistry Research Table 2. Mechanical Configurations Obtained for the Three Study Cases design variable
case 01
case 02
case 03
heat exchanger area (m2) tube length (m) no. of tube passes no. of baffles no. of tubes tube inside diam (mm) tube bundle diam (m) shell external diam (m) pitch (mm)
272.36 2.438 2 5 1400 23 1.372 1.422 31.75
61.00 6.706 2 8 152 19.05 0.356 0.387 25.4
66.35 6.706 2 8 124 23 0.406 0.438 31.75
Dynamic Simulation Analysis. The three heat exchangers were simulated in Simulink, and a step change in shell-side stream flow rate (Ms) was applied in their initial condition (Rf = 0). Due to the transient profiles of linear control systems, their response behaviors are frequently evaluated by applying a unit input signal 1(t). The controllers were first tuned based on this clean condition using the PID tuner provided by Simulink. The plant comprehends all input and output block combinations, and the tuner computes the linear model of the plant to calculate Kp, τi, and τd values. To simulate the different periods of operation, intermediate values of Rf were chosen. The upper limits of thermal resistance values are provided by the design results and assumed as the 100% fouled condition. Table 3 shows Rf values for each case in the five periods. Tube temperature outlet (Tto) response behaviors due to a mass flow rate step were first simulated (Figures 4, 5, and 6) without changing controller parameters. As can be seen, unadjusted PID gains affect directly the step response quality as the fouling buildup increases in all cases. In regard to performance specifications (tr, ta, and Mp), all values were deteriorated (Table 4). More specifically, for the first case (Figure 4) the most relevant change occurs in the settling time. It increased by almost 4 times when comparing the initial condition with the final simulated condition. In addition, the rise time increased significantly, leading to a delay for the controller response. Figure 5 shows the overshoot increasing from about 18% to more than 75%, also accompanied by a settling time about 300% greater than the initial condition. In the third case, a similar analysis can be performed, indicating that if the controllers are not adjusted, this situation could lead to an instability in the entire control system ahead or be dangerous due mainly to the high overshoot. Moreover, those simulations suggest that a cleaning intervention would be performed to compensate the adverse effects caused by fouling buildup, aiming to mitigate loss of production caused by poor quality indexes.
Figure 4. Step response without PID adjustment for the first case.
Figure 5. Step response without PID adjustment for the second case.
Tuning Strategy 1. The first strategy to compensate those fouling effects in controller performance is to periodically retune PID applying the Simulink Tuner Algorithm (Figures 7, 8, and 9). Table 5 indicates the new PID values obtained after all intermediate periods of operation. Significant changes have been made to compensate the effects of fouling in all periods of operation, as can be seen in Table 6. As previously mentioned, the PID tuner in MATLAB, for a given robustness, tunes the controller to balance reference tracking and disturbance rejection. Depending on the dynamic behavior of the system, further adjustments are necessary. In case 02 (Figure 8) when fouling achieves its designed value (100% Rf), this tuning approach is not able to keep overshoot
Table 3. Rf Shell and Tube Values for Each Case case 01 period clean 35% fouled 50% fouled 85% fouled 100% fouled
Rfs × 10
−4
2
(m K/W)
0.00 1.16 1.65 2.81 3.30
case 02 −4
Rft × 10
2
(m K/W)
0.00 0.70 1.00 1.70 2.00
−4
Rfs × 10
2
(m K/W)
0.00 2.14 3.05 5.19 6.10
Rft × 10
case 03 −4
(m K/W)
0.00 2.14 3.05 5.19 6.10 F
2
−4
Rfs × 10
2
(m K/W)
0.00 0.60 0.85 1.45 1.70
Rft × 10−4 (m2 K/W) 0.00 0.60 0.85 1.45 1.70
DOI: 10.1021/acs.iecr.8b00906 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX
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Industrial & Engineering Chemistry Research
Figure 8. Step response after PID adjustment for the second case.
Figure 6. Step response without PID adjustment for the third case.
Table 4. Quality Indicators for PID Tuned in Clean Conditions for Different Periods of Operation
case 01
case 02
case 03
quality indicators
clean
35% Rf
50% Rf
85% Rf
100% Rf
tr (s) ta (s) Mp (%) tr (s) ta (s) Mp (%) tr (s) ta (s) Mp (%)
12.8 70.5 10.9 7.12 48.8 17.9 9.82 56.6 21.4
14.2 78 15.1 7.15 55.6 31.7 11.3 65.9 25.1
16.7 94.1 21.7 7.45 61.4 39.3 14.2 107 28.9
19.7 143 26.8 7.48 115 67.5 14.4 215 28.5
32.5 257 33.4 8.44 143 75.9 16.1 246 38.4
Figure 9. Step response after PID adjustment for the third case.
Table 5. PID Parameters Tuned after Different Periods of Operation case 01
case 02
case 03
Figure 7. Step response after PID adjustment for the first case.
PID param
clean
35% Rf
50% Rf
85% Rf
100% Rf
Kp τi τd Kp τi τd Kp τi τd
15.61 0.203 1.65 8.98 0.38 53.16 85.35 2.09 65.1
18.10 0.21 2.71 7.95 0.31 49.67 91.92 2.09 65.3
22.45 0.23 8.60 7.95 0.23 52.30 99.50 2.93 162.3
24.39 0.26 12.77 6.40 0.07 44.33 106.4 3.18 193.8
25.23 0.25 13.98 7.54 0.24 57.40 142.03 5.28 201.4
Upon starting the optimization, the routine adjusts the tunable parameters Kp, τi, and τd to satisfy the values of tr ta, and Mp (Table 7). Figures 10, 11, and 12 show the simulated step responses with the optimal values. It is possible to notice a significant improvement in the quality indexes, presented in Tables 8, 9, and 10, showing that the controller was able to meet the process requirements through the proper tuning values up to the maximum heat exchanger fouling condition. Effect of Operating Conditions. The variation of temperature in a hot fluid stream is simulated to evaluate the
low and significantly oscillations remained in the STHX response. Tuning Strategy 2. It is convenient to adopt a more advanced optimization method for tuning when reference values are provided for tr, ta, and Mp. Also, it is desirable that the influence of the user in the tuning procedure be minimal. Based on the initial characteristics of the system response to the step change, Table 7 defines the constraint upper bounds of the problem. G
DOI: 10.1021/acs.iecr.8b00906 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX
Article
Industrial & Engineering Chemistry Research Table 6. Quality Indicators after PID Adjustment in Different Periods of Operation quality indicator
clean
35% Rf
50% Rf
85% Rf
100% Rf
tr (s) ta (s) Mp (%) tr (s) ta (s) Mp (%) tr (s) ta (s) Mp (%)
12.8 70.5 10.9 7.12 48.8 17.9 9.82 56.6 21.4
16.5 73.8 11.7 7.64 57.4 23.6 11.3 65.9 25.1
18.4 76.4 15.3 8.06 61.5 24.9 14.2 107 28.9
21.5 80.8 15.0 8.92 130 27.4 14.4 215 28.5
21.9 81.4 12.5 8.96 88.8 51.3 16.1 246 38.4
case 01
case 02
case 03
Table 7. Response Specifications Limits for Each Case case 01 case 02 case 03
tr (s)
ta (s)
Mp (%)
14.1 8.96 11.0
75 110 65
12 23.5 23.0
Figure 12. Step response after optimal PID adjustment for the third case.
Table 8. Optimal PID Parameters and Step Response Characteristics for the First Case condition
Kp
τi
τd
tr (s)
ta (s)
Mp (%)
clean 35% fouled 50% fouled 85% fouled 100% fouled
15.602 18.103 22.454 24.391 25.231
0.203 0.212 0.228 0.264 0.254
1.652 2.709 8.602 12.765 13.981
14.0 14.1 14.1 14.1 14.1
73.5 74.4 75.0 75.0 74.8
11.9 12.0 12.0 12.0 12.0
Table 9. Optimal PID Parameters and Step Response Characteristics for the Second Case
Figure 10. Step response after optimal PID adjustment for the first case.
condition
Kp
τi
τd
tr (s)
ta (s)
Mp (%)
clean 35% fouled 50% fouled 85% fouled 100% fouled
8.98 9.07 9.19 9.26 10.76
0.38 0.19 0.11 0.08 0.06
53.16 48.76 61.71 75.71 77.80
7.12 7.83 7.69 7.76 8.91
48.8 62.8 65.0 107 89.0
17.9 18.0 18.2 18.0 23.5
Table 10. Optimal PID Parameters and Step Response Characteristics for the Third Case condition
Kp
τi
τd
tr (s)
ta (s)
Mp (%)
clean 35% fouled 50% fouled 85% fouled 100% fouled
85.35 87.41 89.22 93.70 94.10
2.09 2.09 2.09 3.15 3.41
65.01 65.40 82.31 94.73 101.1
9.82 10.8 10.9 11.0 11.0
56.6 63.5 64.1 64.3 64.9
21.4 22.4 22.6 22.9 23.0
The PID optimum values are kept constant as a step change was applied in the tube inlet temperature at the maximum fouled condition for each case. Figure 13 depicts all time responses. As the temperature is a very active variable, it is possible to evaluate the robustness of the tuned controller by investigating the behavior of shell outlet normalized temperature. As can be seen in Figure 13, overshoot reaches 20% maximum in case 02, with a settling time under 120 s for all cases in the PID optimal tuning. The steady state is achieved is all cases, and particularly, case 03 presents the lower overshoot (about 5%) and the faster settling time (96 s). These results are consistent with the ones presented in Table 10 for case 03, which barely varied between clean and
Figure 11. Step response after optimal PID adjustment for the second case.
tuned controller performance with transients generally observed in practice. H
DOI: 10.1021/acs.iecr.8b00906 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX
Industrial & Engineering Chemistry Research
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Article
ACKNOWLEDGMENTS
The authors acknowledge the financial support from the Coordination for the Improvement of Higher Education Personnel (CAPES, Brazil) and the National Science and Technology Development Council (CNPq, Brazil).
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NOMENCLATURE a1−a6 = auxiliary constants Cp = specific heat D1 = tube inside diameter D2 = tube outside diameter D3 = tube bundle diameter e(t) = error G1−G6 = transfer functions h = heat transfer coefficient Kp = controller gain l = cell length L = tube length M = mass flow rate Mp = overshoot nb = tube number Nbf = baffles number Nc = number of modeling cells Ntp = number of tube passes Rf = thermal resistance of fouling T = temperature ta = accommodation time tr = rise time V = volume
Figure 13. Shell outlet temperature response for a step change in Tti.
fouled conditions with optimal PID parameters. After changing the disturbance, case 03 performed better than the others, possible due to its lower inlet temperature.
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CONCLUDING REMARKS Lumped models are popular in the literature mainly due to their modeling flexibility and computational simplicity, providing a reliable method to describe dynamic systems. For a given value of the thermal resistance of fouling, appropriate PID parameter values can be determined using the dynamic model of the heat exchanger and the suitability of these values can be tested by simulation. The case studies simulated showed that unadjusted PID parameters could lead to step responses out of the process requirements, that is, high oscillation and temperature overshoot, that could be unacceptable for certain stages of processes, product requirements, or safety requirements. Two different alternatives to automatically adjust PID parameters were shown. It was possible to compensate the adverse effects of fouling buildup and stabilize the quality indicators. Depending on the process restrictions, the more advanced method presented in Strategy 2 should be used, in order to find the optimal values for controller parameters with minimal human interference and maximum performance of the control system. Moreover, further investigations regarding cleaning intervention costs and energy losses that would be prevented by the appropriate tuning PID controller are desirable to apply those approaches as a tool for industrial decision makers.
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Subscripts
e = steady state f = fouled i = inlet o = outlet s = shell side t = tube side w = relating to tube wall Greek Symbols
κ = thermal conductivity μ = dynamic viscosity ρ = density τd = derivative time τi = integral time
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REFERENCES
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. ORCID
Carolina Borges de Carvalho: 0000-0003-0152-0641 Esdras P. Carvalho: 0000-0002-6801-8245 Mauro A. S. S. Ravagnani: 0000-0002-2151-1534 Notes
The authors declare no competing financial interest. I
DOI: 10.1021/acs.iecr.8b00906 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX
Article
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DOI: 10.1021/acs.iecr.8b00906 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX
