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Fractional order PID controller for a shell and tube heat exchanger.


Heat exchanger is widely used in chemical plants because it can sustain wide range of temperature and pressure changes. The function of heat exchanger in a chemical process is to transfer heat from the hot fluid through conduction to a cooler fluid[1]. The temperature control of outlet fluid is of prime importance. In practice, all chemical processes involve production or absorption of energy in the form of heat. Even though there are different types of heat exchangers, the shell and tube type heat exchanger system is mostly used in industries due to its wide range of operating temperature and pressure. Shell and tube heat exchanger provides a comparatively large ratio of heat transfer area to volume and are easy to manufacture in large variety of sizes and flow configurations. They can operate at high pressure and their construction facilitates disassembly for periodic maintenance and cleaning.

A shell and tube heat exchanger consists of a bundle of tubes enclosed within a cylindrical shell. One fluid flows through the tubes and a second fluid flows within the space between the tubes and the shell. Heat is thus transferred from one fluid to the other through the tube walls, either from tube side to shell side or vice versa. They can further be classified according to their flow arrangement. Most shell and tube heat exchangers are 1,2 or 4 pass designs on the tube side depending upon the number of times the fluid in the tubes passes through the fluid in the shell.

Counter-flow and parallel-flow are the two primary flow arrangements in heat exchanger. In Counter current mode, the hot fluid enters from one end of the exchanger and the cold fluid from the opposite end. In Co-current (Parallel) mode the flow of the hot and the cold fluid are taking place in the same direction. The outlet temperature of the shell and tube heat exchanger system has to be kept at a desired set point according to the process requirement. Due to nonlinear nature, shell and tube heat exchanger system is hard to model and control using conventional methods.

The integer order PID controller[7] completely deals with the system dynamics whose behaviors are described by integer order differential equations. The closed system with this controller exhibits poor settling time due to its integer values of control parameters for a system involving non-integer values. Moreover, it has insufficient control parameters for a system such as heat exchanger involving time series of heat transfer. The real physical systems are well characterized by fractional order differential equations involving non integer order derivatives. This gives the option of fractional order dynamic systems and controllers based on fractional order calculus.

In a fractional order PID controller, the I and D-action being fractional, have wider scope of design. Naturally, besides setting the proportional (Kp) derivative (Td) and integral constant Ti respectively, there are two parameters: the power of s in integral and derivative actions-[lambda] and [mu] respectively. Finding these parameters [Kp, Td, Ti, [lambda] and [mu]] as an optimal solution to a given process thus calls for optimization on the fivedimensional space.

The performance of the fractional order PID controller is compared with integer counterpart. Due to its better Time integral performance indices the proposed design FOPID controller will find extensive applications in real industrial processes

Modelling And Process Description:

The present work is carried out in the process control lab of Instrumentation Engineering department in Annamalai University, Chidambaram.

The Shell and tube heat exchanger experimental set up consists of 37 copper tubes of 750 mm length with a single pass arrangement. The two fluid streams can be arranged both in co-current and counter current fashion. The experiment is carried out using water as a single phase medium. In the process tank, water is heated to a particular operating temperature. The hot fluid (water) then flows from the process tank and passes through the tube-side of the heat exchanger. Cold fluid (water) flows from the reservoir tank into the shell side of the heat exchanger. The disturbance tank is provided to study the performance of designed controllers for disturbance rejection. There are two thyristor drives that regulate the voltage and current to the heaters in order to regulate the temperature of the water in process tank and disturbance tank. The cold and hot water inlet flow to the shell and tubes respectively are manipulated using pneumatic control valves.

The experiment is carried out in co-current mode. The hot water outlet temperature is considered as the controlled variable whereas the cold water flow rate to the shell side is treated as the manipulated variable. The flow rate of the hot water is treated as disturbance variable. The hot water inlet temperature (60oC) is maintained with [+ or -] 0.50C variation using an in built digital PID controller. The cold water is supplied at the room temperature. The inlet and outlet temperatures of the shell and tube side fluid are measured using the Resistance Temperature Detectors (RTDs). A Differential Pressure Flow Transmitter (DPT) is used to measure the cold water flow rate. The inlet flow of the cold water can be varied in the range of 0-350 LPH and that of hot water is between 0-250 LPH. All the sensors are interfaced with a 16 bit data acquisition system (Advantech ADAM 5000 series hardware).The process parameters are obtained from real time. A PC MATLAB scientific package is used to log the data and also perform the functions of the controller.

Mathematical Modeling Of Shell And Tube Heat Exchanger:

Figure1 shows the photographic view of Shell and Tube Heat exchanger. The shell and tube sections are further divided into control volumes. The following assumptions are made while designing the mathematical model of shell and tube heat exchanger[2]. The control volumes are small and assumed to have a constant temperature. The heat exchanger is insulated and there is no heat loss from the heat exchanger to the surrounding. Rate of energy stored in the control volume is equal to the rate of gain of energy from the neighboring control volume.

The energy balance equation on the shell control volume is given by

[[rho].sub.s][C.sub.s][V.sub.s]/N * d[]/dt = [[??].sub.s] [c.sub.s] ([] - []) + [h.sub.s][A.sub.s]/N ([T.sub.ho] - []) (1)

The energy balance equation on the tube control volume is given by

[[rho].sub.t][C.sub.t][V.sub.t]/N * d[]/dt = [[??].sub.t] [c.sub.t] ([] - []) + [h.sub.t][A.sub.t]/N ([T.sub.ho] - []) (2)

In Figure 2, the direction of Coldwater inflow and that of Hotwater inflow describes that the operation of STHE process is in Co-current mode. The flow of the hot and cold fluid is taking place in the same direction in this case. The bundle of tubes is enclosed within a cylindrical shell. Hotwater flows through the tubes and Coldwater flows through the shell. The tube pitch is 1.25 times the tube diameter. It is the shortest distance between centers of two adjacent tubes. There are baffles which are responsible for obstructing and redirecting the flow of fluid in the shell side of an exchanger. The heat is exchanged between shell and tube column. As mentioned in Table 1, the shell side volume is 2.64x10-4m3 and that of Tube side is 1.43x10-4 [m.sup.3]. In the experimental set up only hotwater and cold water are used for analysis

The details of physical quantities of the heat exchanger experimental set up are shown in Table 1. These values are used in the MATLAB simulink block diagram to obtain the Transfer function model of the process and it is shown in Figure 3.

Average Transfer Function:

The process reaction curve can be obtained by giving a sudden step change in the manipulated variable. It is one of the widely followed process identification technique and this method is used for identifying parameters of the shell and tube heat exchanger at each region. For each of these regions a step change in cold water inflow rate is given in both positive and negative directions as shown in Figure 4 and the corresponding reaction curves are obtained as shown in Figure 3. Using the process reaction curves the transfer functions and the average transfer functions for various regions are obtained and related terms are tabulated (Table 2). Using average transfer function, the controller parameters are obtained based on Zeigler-Nichols tuning method, and tabulated (Table 3).


The values of td and t be selected such that the model and the actual responses coincide at two points[8] in the region of high rate of the change.

t1 = 28.2% of final value.

t2 = 63.2% of final value

[tau] = 1.5(t2-t1)

td = t2-[tau]

K = Change in process output at steady state([DELTA][])/Step change in process input([DELTA]u)

where td = the effective process dead time.

[tau] = the effective process time constant.

K = process steady state gain.

The transfer functions are developed for three different regions and are shown in Table 2.

Controller Tuning:

4.1 Tuning Pid Control Parameters:

From the average transfer function, the model parameters are obtained and the controller parameters are obtained based on Zeigler-Nichols tuning method[2] as given in the equations 4,5and 6.The control parameters are tabulated (Table 3).

Proportional Gain [K.sub.c] = 1.2 [tau]/[t.sub.d]K (4)

Integral Gain [K.sub.i] = [K.sub.c]/2[T.sub.d] (5)

Derivative gain Kd = (6)

4.2 Design Of Fopid Controller:

Valerio and Costa introduced Ziegler-Nichols-type tuning rules[3] for FOPID[10] controller parameters. Tuning rules of this method are applicable only for systems that have S-shaped reaction curve. The simplest system to have S-shaped response can be described by equation 3.Valerio and Costa[9]&[11] have employed the minimization tuning method to processes given by equation (3) The parameters of FOPIDs thus obtained based on [tau] and Td, where t is the time constant and Td is the delay time, vary in a regular manner. The value of the t in the transfer function of the proposed system gives the choice of first set of FOPID tuning rule with values shown in the table 4.

To calculate FOPID control for parameters such as KC, KI, [lambda], KD, and p the equations (7), (8), (9), (10) and (11) respectively can be written. The numerical values mentioned in the equations are the multiplying factors of time response specifications and are tabulated in Table 4 to respective control parameter.

KC = -0.0048 + 0.2664Td + 0.4982([tau]/K) + 0.0232(Td2) - 0.0720([[tau].sup.2]/[K.sup.2]) - 0.0348 ([tau][T.sub.d]/K) (7)

KI = 0.3254+0.2478(Td) + 0.142 ([tau]/K) - 0.1330(Td2) + 0.0258([[tau].sup.2]/[K.sup.2])- 0.0171 ([tau][T.sub.d]/K) (8)

[lambda] = -1.5766-0.2098Td + 0.1313([tau]/K) + 0.0713(Td2) - 0.0016 ([[tau].sup.2]/[K.sup.2]) -0.0114 ([tau][T.sub.d]/K) (9)

KD = 0.0662-0.2528Td + 0.1081([tau]/K)+0.0702(Td2) - 0.0328([[tau].sup.2]/[K.sup.2]) - 0.2202 ([tau][T.sub.d]/K) (10)

[mu] = 0.8736 + 0.2746Td + 0.1489([tau]/K) - 0.1557(Td2) -0.0250([[tau].sup.2]/[K.sup.2]) -0.0323([tau][T.sub.d]/K) (11)

Tuning parameters of FOPID are tabulated in table 5. The values are substituted in the FOPID controller toolbox[5]. Figure 7 shows the block diagram of heat exchanger system with ZN and FOPID controller.

Figure 9 shows the responses of FOPID and integer order PID controllers. The controlled variable of system with FOPID reaches the setpoint with less time duration and with acceptable error indicies. The controller parameters [lambda] and [mu] makes the controller effective and the response of STHE with FOPID is better than that of integer order PID. The %peak overshoot, (%Mp) is less for two regions namely Region I and Region II, and rise time(tr) of system with FOPID is less for two regions namely Region II and Region III.

The regulatory response of STHE system with IOPID and FOPID is shown in Figure 10. The response of Manipulated variable, coldwater inflow rate is shown in Figure 11. The flow direction of the manipulated variable is dependent upon the deviation of controlled variable from the setpoint. Table 5 describes the evaluation of Performance criteria and Time response analysis of integer order PID and FOPID controller based STHE.

Figure 10 describes the comparative regulatory response of Integer order PID and FOPID based system. FOPID based system is good in setpoint tracking and overcomes disturbances effect quickly. Table 6 shows the error indices of integer order PID and FOPID controller for regulatory response. The error values of FOPID based system are lesser than that of Integer order PID based system.


The integer order and Fractional order PID controller for Shell and Tube heat exchanger is designed, implemented and the responses are analyzed. The transfer functions are obtained for various regions. Performance indices are calculated from the servo response for various ranges and Regulatory response with load disturbance. In the proposed STHE system with integer order PID controller, the Error index ITAE increases for every unit rise of set point. In the system with FOPID controller, the error indices are at acceptable level for all the set points.


[1.] Ahilan, C., S.Kumanan, N.Sivakumaran, 2011. Online performance assessment of heat exchanger using artificial neural networks,International Journal of Energy and Environment.

[2.] Dinesh, B., E.Sivaraman, 2014. Fuzzy C-means Modeling for Shell and Tube Heat Exchanger, International JournalocComputer Applications.

[3.] On Fractional-Order PID Design, Mohammad Reza Faieghi and Abbas Nemati, Department of Electrical Engineering, Miyane Branch, Islamic Azad University, Miyaneh, Iran.

[4.] Padula, F.,A.Visioli, 2010a. Tuning rules for optimal PID and fractional-order PID controllers, Journal of Process Control.

[5.] Patrick Lanusse, Richard Maltiand Pierre Melcior, 2013. CRONE control system design toolbox for the control engineering community:tutorial and case study , Philosophical transactions of The Royal Society.

[6.] Tushar Verma Akilesh Kumar Mishra, 2014. Comparitive Study of PID and FOPID controller resposnse for Automatic Voltage Regulation, International organization of Scientific Research.

[7.] A Novel Evolutionary Tuning Method for Fractional Order PID Controller, 2011. International Journal of Soft Computing and Engineering (IJSCE), 1: 3, Subhransu Padhee, Abhinav Gautam, Yaduvir Singh, and Gagandeep Kaur

[8.] Wayne, B., 1998. Bequette Process Control: Modeling, Design, and Simulation.Prentice Hall.

[9.] Realization of Fractional Order Controllers by Using Multiple Tuning-Rules, Zhe Yan, Jing He, Yingyan Li, Kai Li and Changqi Song, 2013. International Journal of Signal Processing, Image Processing and Pattern Recognition, 6: 6.

[10.] Serdar, E., Hamamci An algorithm for stabilization of fractional-order time Delay systems using fractional-order pid controllers.

[11.] Valerio,D.and J. da Costa, 2005. "Time-domain implementation of fractional order controllers," in Control Theory and Applications, IEEProceedings-, 152(5): 539-552.

(1) B.Girirajan and (2) D.Rathikarani

(1) Research scholar, Electronics and Instrumentation Engg., Annamalai University, India-608002

(2) Professor, Electronics and Instrumentation Engg., Annamalai University, India-608002

Received lApril 2017; Accepted 18 June 2017; Available online 2 July 2017

Address For Correspondence:

B.Girirajan, Department of Electronics and Instrumentation Engineering, Annamalai University, Annamalai nagar, Tamil nadu, India-608002

Caption: Fig. 1: Photographic view of experimental set-up

Caption: Fig. 2: Shell and Tube sections of a Heat Exchanger

Caption: Fig. 3: Process reaction curve

Caption: Fig. 4: Step input

Caption: Fig. 7: Block Schematic representation of the process with controllers.

Caption: Fig. 8: Servo Response of Shell and tube heat exchanger with Integer order PID controller and FOPID controller.

Caption: Fig. 9: Integer order PID and FOPID Controller output for various ranges.

Caption: Fig. 10: Regulatory response of Shell and tube heat exchanger with IOPID controller and FOPID controller.

Caption: Fig. 11: Integer order PID and FOPID Controller output for load disturbance
Table1: Physical quantities of the heat exchanger experimental set up
used for mathematical model.

Inputs                         Value       Units

Density of water ([rho]s,      1000        Kg/m3
Specific Heat Capacity         4230        J/Kg oC
  of water (Cs, Ct)
Shell heat transfer area(As)   0.281       m2
Tube heat transfer Area(At)    0.253       m2
Shell side volume(Vs)          2.64x10-4   m3
Tube side volume(Vt)           1.43x10-4   m3
Heat transfer coefficient      2162        W/m2 oC
  of shell(hs)
Heat transfer coefficient      2162        W/m2 oC
  of tube(ht)
Mass flow rate of cold         0-0.11      LPS
Mass flow rate of hot          0.0282      LPS
Cold water inlet               33          oC
Hot water inlet                60          oC
Number of control Volume(N)    10          N/A
Cold water outlet              --          oC
  temperature (TCo)

Table2: Average Transfer function

Region       Process Gain   Time constant   Dead Time
             (o C/LPS)      (sec)           (sec)

Region I     -32.15         0.835           0.182

Region II    -62.421        1.385           0.2325

Region III   -369.914       1.978           0.2826

Table3: PID controller parameters

Region       Proportional gain   Integral gain   Derivative gain

Region I     -0.1714             -0.471          0.0156

Region II    -0.1145             -0.246          0.1163

Region III   -0.0227             -0.402          0.1413

Table 4: Parameters for the first set of tuning rules when 0.1 [less
than or equal to] [tau] [less than or equal to] 5.

Time                    Controller gain Multiplying factors used to
Factors                        determine tuning parameters
                  KCg       KIg       [lambda] g   KDg       [micro]g

1                 -0.0048   0.3254    1.5766       0.0662    0.8736
Td                0.2664    0.2478    -0.2098      -0.2528   0.2746
[tau]             0.4982    0.1429    -0.1313      0.1081    0.1489
[T.sup.2.sub.d]   0.0232    -0.1330   0.0713       0.0702    -0.1557
[tau]2            -0.0720   0.0258    0.0016       0.0328    -0.0250
Td [tau]          -0.0348   -0.0171   0.0114       0.2202    -0.0323

Table5: FOPID controller parameters

Region       Kc        [lambda]   Ki        Kd        M

Region I     -0.4053   1.437      -0.5010   -0.1693   1.0205

Region II    -1.056    1.3897     -0.4846   -0.2948   1.0771

Region III   -0.7566   1.2759     -0.7588   -0.4656   1.1174

Table 5: Results for error indices of PID and FOPID controller for
servo response

Region/                    I         II        III
                           (52-54)   (54-56)   (56-58)
(oC)                       53        55        57

Sampling Instants          0-29      30-49     50-65
PID                 ISE    0.67      2.27      2.16
                    IAE    1.39      2.77      2.72
                    ITAE   43.75     88.24     140.9
                    Tr     2.5       1         0.933
                    %Mp    0.274     0.805     0.83

FO                  ISE    0.04      0.074     0.072
PID                 IAE    0.30      0.18      0.11
                    ITAE   10.59     5.62      5.73
                    Tr     4.368     0.02      0.01
                    %Mp    0.109     0.801     0.99

Table 6: Results for error indices of PID and FOPID controller for
regulatory response

Sl. No   Error Indices   PID     FOPID

1.       IAE             8.05    5.066
2.       ISE             16.08   7.739
3.       ITAE            129.3   88.01
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Title Annotation:proportional-integral-derivative
Author:Girirajan, B.; Rathikarani, D.
Publication:Advances in Natural and Applied Sciences
Article Type:Report
Date:Jul 1, 2017
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