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Abstract - In this paper a new extended model of a heat

exchanger and a comparison between two temperature control

methodologies, standard PID and Predictive Control, are

presented. To the authors’ knowledge the proposed model is

one of the most general which has been used for control

purposes. The PID and Predictive Control have been designed

by means of a reduced system representation obtained both

by identification and model reduction techniques applied to

the extended model. The two control methodologies are tested

by using a 200 order model which can be considered as the

real system. The simulation results have shown bettercharacteristics concerning both set point tracking and

disturbance robustness for Predictive Control.

Keywords - Heat Exchanger, Model Reduction, Identification,

PID control, Predictive control.

I. I NTRODUCTION

A heat exchanger is a thermal device with the function of

transmitting the heat between two fluids characterized by

different temperatures and situated into two adjacent rooms.There are two kinds of heat exchangers which are different

for the geometry of the thermic exchange surface: double pipe and plate heat exchangers. This work deals with the

first kind, although the obtained results could easily be

extended to the second one.

Figure 1. Double pipe heat exchanger.

The double pipe heat exchanger consists of two tube: the

external one in which the service fluid flows and theinternal one in which the process fluid flows. A double pipe

heat exchanger scheme is depicted in Figure 1.

C. Bonivento, P. Castaldi ([email protected]), D. Mirotta are withthe Department of Electronics, Computer Science and System of the

University of Bologna.

A. Modelling

Modelling a heat exchanger is a difficult task because of its

complex dynamics characterized by distribuited parameters

and non-linearity. Furthermore the model proposed in this

work is more detailed with respect to those already present

in the literature [1, 2, 3]. In fact, as will be shown in thefollowing, it can be used in the case of two different fluids

characterized by two different velocities, it gives the

possibility to consider the geometry, it takes into accountthe overall heat transfer coefficient and, finally, it enables

determining state space representations of any order. A statespace model of the distributed parameter system is obtained

by using the “Direct lumping of the Process” technique.

This methodology is based on a subdivision of the thermal

exchange surface in ∆x sections (lump), so that the state

vector is defined by the temperature of the sections. The

temperature has been considered as constant in every lump,

a hypothesis which always holds for small lumps. Without

this assumption the relations are extremely complicated and a model can be obtained only for a very small number of

lumps [4]. To obtain a model approaching the real case, it

has been necessary to consider a large number of lumps (in this work the system order is 200). For this reason the

project of the regulators has been developed on the basis of

reduced order models obtained by means of identification

and/or model reduction techniques.

B. Predictive Control vs PID Control

The standard PID and Generalized Predictive Control

(GPC) have been used to regulate the process fluid

temperature (cold) by means of the flow rate of hot fluid

(control signal). The actuator is a solenoid valve. It will be

shown that the GPC offers better disturbance robustness

and better set point tracking characteristics. It is important

to observe that the proposed GPC scheme can be

advantagiously implemented by means of the so-called

“predictive PID” control law [9], which is characterized by

the same performance without employing antiwindup

device and specialized software.

The contents of the work are the following. Section II dealswith modelling of a heat exchanger. In section III some

model reduction techniques are given. In section IV the

standard PID control and GPC are applied. In Section V a

comparison between PID control and GPC on the basis of

set point tracking performance and disturbance robustness

is proposed.

Predictive Control vs PID Control of an

Industrial Heat Exchanger

C. Bonivento, P. Castaldi and D. Mirotta

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II. MODELLING

The aim of the modelling phase is to obtain a linear discrete

time, state space representation of the system starting froma non-linear, continuous time model obtained by means of

the “Direct Lumping of the Process” technique. Therefore

linearization and discretization are basic steps of the whole

modelling procedure.

The following hypotheses are introduced:

1. The physical and chemical properties of the fluids are

constant;

2. The radial variations of the fluid velocity and

temperature are negligible;

3. The heat leakage is negligible;4. The overall heat transfer coefficient is constant.

The starting point consists in the application of the EnergyConservation Principle to every lump (see Fig. 1).

Neglecting the metal effects between the hot and the cold

sections (they are thin) and assuming the fluid average

velocity across the tube as constant, the balance referred at

one room ( for the time t , in the point x-∆x) is given by thefollowing relation:

( ) [ ] T U AT T cmT c M t

x x x p p ∆⋅⋅−−⋅⋅=⋅⋅

∂∂

∆−! (1)

with ρ ⋅∆⋅= xa M (Kg) the lump mass; ρ ⋅⋅= vam! (Kg/s)

the flow rate; a (m2) the section of the room where the fluid

flows; ρ (Kg/ m3 ) the fluid density; c p (Cal/Kg⋅K) the heatcapacity; T (K) the temperature of the fluid considered; t

(s) the time; v (m/s) the fluid velocity; ∆T (K) the change in

temperature; ∆ x (m) the incremental distance; A (m2) the

surface of the room included in the lump considered;

U , the constant overall heat transfer coefficient, represents

the heat transmission between the two fluids and depends

on the geometry. For the double pipe heat exchanger:

22

1

1

21

1

ln1

1

α λ α ⋅+⋅+

=

r

r

r

r r U

xr A ∆⋅⋅⋅= 12π

For the exterior tube:

( )2

1

2

22 r r a −⋅= π ρ ⋅⋅=222 vam! ρ ⋅∆⋅= xa M 22

2r (m) is the exterior tube radius; for the internal tube:2

11 r a ⋅= π ρ ⋅⋅= 111 vam! ρ ⋅∆⋅= xa M 11

1r (m) is the internal tube radius.

A. Non- linear model

The heat exchanger is divided into N lumps so that the statespace model of the system refers to a state vector T

representing the temperature of each lump. This vector can

be subdivided into two parts, the first representative of fluid 1 (hot) and the second of fluid 2 (cold):

[ ]T

N N T T T T T T T T T 22322211131211 ,...,,,,...,,,=

where T pj ( p = 1, 2; j = 0, 1, … , N ) is the jth and pth pointroom temperature. T 1,0 e T 2,0 are considered as part of the

inputs. The system order is 2 N . The Energy Conservation

Principle applied to the two sections leads to the following

relations:

( )

( ) ( ) ( )[ ] ( )t T U At T t T ct mdt

t dT c M j j p

j

p ∆⋅⋅−−⋅⋅=⋅⋅ − ,11,111

,1

11 !

(2)

( )

( ) ( ) ( )[ ] ( )t T U At T t T ct mdt

t dT c M j jcp

j

p ∆⋅⋅−−⋅⋅=⋅⋅ − ,21,222

,2

22 !

which can be rewritten as:

1,1

1

1,21,11

1

1,1 −⋅+⋅+⋅

+−= j j j J T

M

mT bT b

M

mT

!!!

(3)

1,2

2

2,12,22

2

2,2 −⋅+⋅+⋅

+−=

j j j J T M

mT bT b

M

mT

!!!

11

1

pc M

U Ab

⋅⋅

=22

2

pc M

U Ab

⋅⋅

=

Relation (3) corresponds to the general form [1]:

( ) ( ) uuGT uF T ⋅+⋅= ,ϑ !

where [ ]T T mT mu 0,220,11 ,,, !!= is the input vector and ϑ

denotes the heat relocation parameters.Considering the general case of N lumps, with j = 1,…, N ,

the following model can be obtained:

⋅

+

+

⋅

=

−

−

−

−

20

2

10

1

4

2

2

12

22

21

1

11

12

11

34

34

3

2

2

2

1

1

1

12

12

1

2

12

22

21

1

11

12

11

0000

0000

000

0000

0000

000

T

m

T

m

c

c

T

T

T

T

T

T

T

T

cc

cc

c

b

b

b

b

b

b

cc

cc

c

T

T

T

T

T

T

T

T

N

N

N

N

N

N

N

N

!

!

""

""

""

""

"

"

##

##

#

#

#

#

##

##

!

!

"

!

!

!

!

"

!

!

(4)

1

1

11 b M

mc −−=

!

1

12 M

mc

!

= 2

2

23 b M

mc −−=

!

2

24 M

mc

!

=

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where the output y corresponds to N T ,2

(i.e. the output

temperature of cold fluid). Hence the output is given by the

relation u D xC y ⋅+⋅= with [ ]T D 0....00=

[ ]T C 10....0= . The input vector is defined by relation

(4), in fact the physical heat exchanger inputs are fluids 1

and 2 which are characterized by their flow rate and their temperature. In practice the system can be described by

means of a S.I.S.O. model because the unique control input

is1m! , the mass flow of fluid 1, while the other elements of

the vector u are constant inputs setting the working point of

the system.

B. Linearization

From equations (3) it follows that:

( ) j j j j J

T T x

vT bT bT

,11,1

1

,21,11,1

−⋅∆

+⋅+⋅−= −

!

(5)

1,2

2

2,12,22

2

2,2 −⋅+⋅+⋅

+−=

j j j J T M

mT bT b

M

mT

!!!

Now if x∆ is small enough we can approximate the

derivative by means of the incremental ratio:

=∆

−−

x

T T j j ,11,1

x

T J

∂∂ 1 .

Furthermore the derivative can be approximated with the

steady state [3]:

ss

J J

x

T

x

T

∂∂

=∂∂ 11 .

Hence:

ss

J j j J

x

T vT bT bT

∂∂

⋅+⋅+⋅−= 11,21,11,1

! ρ ⋅

=a

mv 1

1

!

(6)

1,2

2

2,12,22

2

2,2 −⋅+⋅+⋅

+−= j j j J T

M

mT bT b

M

mT

!!!

which correspond to the general form:

( ) ( ) uuGT F T ⋅+⋅= ϑ ! Considering the general case of N

lumps, we obtain the following model:

+

⋅

−

−−

=

−

−

−

−

N

N

N

N

N

N

N

N

T

T

T

T

T

T

T

T

cc

cc

c

b

b

b

b

b

b

b

b

b

T

T

T

T

T

T

T

T

2

12

22

21

1

11

12

11

34

34

3

2

2

2

1

1

1

1

1

1

2

12

22

21

1

11

12

11

"

"

##

##

#

#

#

#

#

#

!

!

"

!

!

!

!

"

!

!

(7)

2

2

222

2

23 b

M

avb

M

mc −

⋅⋅−=−−=

ρ !

2

22

2

24

M

av

M

mc

ρ ⋅⋅==

!

C.

Discretization of the linear model

From relations (6) and using the backward difference

( ) ( )

1

1

h

k T k T T

−+=! , it follows that:

( ) ( ) ( )ss

J j j j

x

T hvk T bk T ck T

∂∂

⋅⋅+⋅+⋅=+ 1

1123151 1

(8) ( ) ( ) ( ) ( )k T ck T bk T ck T j j j j 12814272 1 −⋅+⋅+⋅=+

where h1 is the sample period. Hence the discrete time

model is:

( )

( )

( )( )

( )

( )

( )

( )

( )

( )

( )( )

( )

( )

( )

( )

⋅

∂∂

⋅⋅

∂∂

⋅⋅

∂

∂⋅

⋅

+

+

⋅

=

++

++++

++

−

−

−

−

20

2

10

1

8

1

1

1

12

1

1

11

1

1

2

12

22

21

1

11

12

11

78

78

7

4

4

4

3

3

3

5

5

5

2

12

22

21

1

11

12

11

0000

0000

000

000

000

000

1

1

1

1

1

1

1

1

T

m

T

m

c

x

T

a

h

x

T

a

h

x

T

a

h

k T

k T

k T

k T

k T

k T

k T

k T

cc

cc

c

b

b

b

b

b

b

c

c

c

k T

k T

k T

k T

k T

k T

k T

k T

ss

N

ss

ss

N

N

N

N

N

N

N

N

!

!

""

""

""

""

"

"

##

##

#

#

#

#

#

#

"

"

ρ

ρ

ρ

(9)

1115 +⋅−= hbc 1137 +⋅= hcc 148 hcc ⋅=

124 hbb ⋅=113 hbb ⋅=

⋅

∂∂

⋅⋅

∂∂

⋅⋅

∂∂

⋅⋅

+

20

2

10

1

4

1

1

12

1

11

1

0000

0000

000

0001

0001

0001

T

mT

m

c

x

T

a

x

T

a

x

T

a

ss

N

ss

ss

!

!

""

""

""

""

ρ

ρ

ρ

7/23/2019 TLTK 4.pdf


The chosen parameters for the double pipe heat exchanger

are:

100= N m x 001,0=∆ K T 40010 = K T 30020 =

mr 02,01 = mr 03,02 = 3/1000 mKg=ρ

smv /3,01 = smv /3,02 = )2

/100 mK W U ⋅=( )KgK J cc p p ⋅== /187,421

i.e., the fluids are water and the length of the heat

exchanger is 10 cm, the external diameter is of 6 cm.

Finally the chosen system order is 200. Simulations of the

continuous non-linear system and the corresponding linear

discrete time model have shown similar internal

temperature profiles (0.06 % error on the output variable).

So it is possible to conclude that the model described by (9)

is a very good approximation of the real plant.

With reference to the solenoid valve it is straightforward to

obtain the model:

( )11.01.0

101

3

+⋅−=

−

−

z zG M

The linear working range is [0, 1] Kg/s .

III. IDENTIFICATION AND REDUCTION

A. Step Response Standard Identification Method

A first order model can be determined by means of the

standard step response identification procedures. The

identified parameter values are:

5.88=K 7=τ 1=d τ .

and the identified model is:

( ) ( ) ( )sU sGT sY ⋅+= 20where ( ) se

ssG

−⋅⋅+

=71

5.88

K T 30020 = ( ) out T t y 2

= ( ) ( )t mt u 1!=

By comparison, it results that the first order model is a quite

good approximation of the 200th

order model.

B. Least Squares Identification Method

By using the least squares method the identified parameters

are the following:

7564.88=K 8795.4=τ 2129.1=d τ

( ) 2129.1

8795.41

7564.88 ⋅−⋅⋅+

= se

ssG

In Fig. 2 it can be observed that the outputs of the 200th

model (solid line) and of the least squares identified model(dashed line) overlap, whereas there is a difference with the

dash-dot line corresponding to the step response identified

model. Therefore the model identified by means of the leastsquares method can be regarded as an optimum

approximation of the whole system.

Figure 2. Output temperature: extended model, LS

identified model and SR identified model.

C. Reduction

A reduced order model can also be obtained by determining

the reachable and observable subspaces [5]. The model

corresponding to the (numerically) reachable and

observable part of the system results of the second order:

−=

9855.00031.0

0031.09988.0r A

−

=0156.02737.0

0062.02391.0r B

[ ]2741.02392.0=r C [ ]1381.01632.0 −=

r D

Figure 3 shows the comparison between the outputs of the

extended model and the reduced model.

Figure 3. Output temperature: extended model and reduced

order model.

It can easily be proved that this model (without lag)

corresponds to a first order model with lag characterized by

the following parameters:

6561.88=K 14021.5=τ 7833.0=d τ

These values are very similar to those of the LS identified

model, thus confirming the good quality of the LS

identified model.

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IV. CONTROL

A. Standard PID Control

The standard PID regulator design has been applied by

using a step input signal ranging from 330K to 350K .

1) Ziegler –Nichols and Kappa-Tau methods

By means of the Ziegler- Nichols rules and trial-and-error

tuning the following values have been determined:

7.0= pK , 3=iT , 7.0=d T

By using the Kappa-Tau calibration technique [6], in the

case of sensitivity 1.4: K = 0.2 ,iT = 3.32 ,

d T = 0.75 has

been obtained; in the case of sensitivity 2 the values are: K

= 0.4 iT = 2.44

d T = 0.62

2) Antiwindup Controller

We consider a discrete PID regulator:

( ) ( )

( )1

1

1

21

1

211

−

−

−

−−

=−

⋅⋅+⋅

⋅+⋅−

++⋅

= z A

z B

z

zT

T K z

T

T K

T

T

T

T K

zG

d p

d p

d

i

p

c

The scheme of antiwindup used is shown in Figure 4.

Figure 4. Scheme of control with antiwindup device.

Comparing the simulation, of the case with antiwindup

(dashed line) and without antiwindup (solid line), an

improvement on the maximum overshoot of the outputvariable can be observed (Figure 5) without saturation on

the control input (Figure 6). Hence the antiwindup device

has considerably improved the controller performance.

Figure 5: Controlled variable: PID control and PID control

with antiwindup.

Figure 6. Manipulated variable: PID control and PID

control with antiwindup.

B. Generalized Predictive Control

The Generalized Predictive Control [7, 8, 9] has been

implemented on the basis of the LS identified model which, Z -transformed, is given by:

( ) d d z z za

zb zG −−

−

−− ⋅

⋅⋅

=⋅⋅−

⋅=

1-

-1

1

11

z0.9979527-1

z0.1817102

1

so that the reference ARMAX model for GPC is:

( ) ( ) ( ) ( ) ( ) ( )t e zC t u z B zt y z A d ⋅+⋅⋅=⋅ −−−− 111 (10)

where ( )t e is the null signal; ( )t y and ( )t u are the system

output and input respectively. Therefore (10) corresponds to

( ) ( ) ( )11 −−⋅+−⋅= d t ubt yat y .

The Predictive Control has been implemented as follows:

1. Prediction: the output is predicted at the instant t+d by

means of the following relation:

( ) ( ) ( )d it ubit yait y −−+⋅+−+⋅=+ 11 d i ,....,1= .

2. Control: the control ( )t u optimizes the cost function

( ) ( )[ ]

2

1 Y iiYY

N

i −Σ = where YY and Y are the desired and

the predicted output respectively.

V. EXPERIMENTAL R ESULTS

The simulations have been performed by using as a

reference signal a square wave ranging from 330K to 350K .

The control performances of the GPC with reference to set

point changes are shown in Fig. 7: the GPC have the same

response velocities as the PID without its overshoot.

Moreover the controller output in the case of the GPC

presents less oscillations (Fig. 8).

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Figure 7. System output in noise absence conditions: PID

and GPC control .

Figura 8. Controller output in noise absence conditions: PID

and GPC Control.

The robustness of the control schemes to noises affecting

the output has also been tested. Figures (9, 10, 11, 12) refer

to the case of white output noise (σ2=0.116): in addittion to

previous considerations which are still fulfilled, it can be

observed that the Predictive Control shows better

characteristics concerning the variances of both the

controlled output (0.095 vs 0.133) and the controller output

(3.7⋅10-5 vs 3⋅10-3).

Figure 9. Noisy case with PID: system output.

Figure 10. Noisy case with GPC: system output.

Figure 11. Noisy case: PID output.

Figure 12. Noisy case: GPC controller output

VI. CONCLUSION

A new detailed general model of a heat axchanger has been

presented. This model can be regarded as a very good

approximation of the real system. A standard PID controller

with antiwindup and a Generalized Predictive Control have

been developed on the basis of a reduced order model. The

two control methodologies have been compared by meansof a 200th

degree model, resulting in better characteristics

regarding set point changes tracking and disturbance

robustness for the GPC controller. It is worth observing that

on the basis of the results given in [9] the GPC controller

can be implemented by means of a predictive PID without

antiwindup.

REFERENCES[1] M.H.R. Fazlur Rahman and R. Devanathan, “Modelling and

Dynamic Feedback Linearisation of a Heat Exchanger Model”, Proc.

of the 33rd IEEE CDC, Lake Buena Vista, FL, pp. 1801-1806, 1994.

[2] M.H.R. Fazlur Rahman and R. Devanathan, “Feedback

Linearisation of a Heat Exchanger”, Proc. of the 33rd IEEE CDC,

Lake Buena Vista, FL, pp. 2936-2937, 1994.[3] Lei Xia, J.A. De Abreu-Garcia and Tom T. Hartley, “Modelling

and Simulation of a Heat Exchanger”, IEEE International Conf. on

System Engineering, Ohio, pp. 453-456, 1991.[4] P. Chantre, B.M. Maschke and B. Barthelemy, “Physical

Modeling and Parameter Identification of a Heat Exchanger”, 20 th

IEEE IECON, vol. 3, France, pp. 1965-1970, 1994.[5] B.C. Moore, “Principal Component Analysis in Linear Systems:

Controllability, Observability and Model Reduction”, IEEE Trans.

On AC, vol. AC-27, pp. 17-32, 1989.[6] K.J. Astrom, C.C. Hang, P. Persson and W.K. Ho, “Towards

intelligent PID control”, Automatica, 28:1, pp. 1-13, 1992.

[7] David W. Clarke, “Application of Generalized Predictive Controlto Industrial Processes”, IEEE, American Control Conference,

Minneapolis, Minnesota, pp. 49-55, 1987.

[8] K.W. Lim and K.V. Ling, “Generalized Predictive Control of aHeat Exchanger”, IEEE, International Conference on Industrial

Electronics Control and Instrumentation, Singapore, pp. 9-12, 1988.

[9]

R.M. Miller, S.L. Shah, R.K. Wood and E.K. Kwok, “PredictivePID”, ISA Trans. 38, Elsevier Science Ltd, pp. 11-23, 1999.

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