0

Modelling, Optimization, and control

of a plate heat exchanger

JIAN FU

18/02/2009

Supervised by: Prof. Stratos Pistikopoulos

Thesis presented to Imperial College London in partial fulfilment for the degree of Master of

Science in Advanced Chemical Engineering with Process System Engineering

Department of Chemical Engineering and Chemical Technology

Imperial College London

SW7 2AZ

1

Abstract

This project focuses on building a precise model which describes the dynamic

behaviour of the fluid in a plate heat exchanger and simulating it in a

professional software gPROMS so that a clear temperature profile over both

time and space is obtained. Based on that, PID and MPC controllers are

designed to make sure the system will run in the expected state. The PID design

is completed in gPROMS environment and the MPC should be designed in

Matlab with the help of MPC and system identification toolboxes then tuned in

gO:Simulink environment. Finally, some conclusions and a comparison

between PID and MPC controllers will be made so that the advantages and

disadvantages of those two controllers will become clear.

2

Acknowledgement

I am grateful to my supervisor Prof. Stratos Pistikopoulos，Dr. Kouramas

Konstantinos and Mr. Christos Panos for their excellent supervision during

the course of this project. The research was extremely well supported and

managed by them with patience, guidance, and encouragements. Without

their help, this work would not be possible.

3

Nomenclature

A channel cross-section area, m2

Ap plate cross-section area, m2

Ai heat transfer area for channel i, m2

Fi mass flow rate in channel i

De equivalent diameter, m

Cp,i fluid heat capacity, J/kg K

Cp,p heat capacity of plate p, J/kg K

e internal energy of unit volume element, J

hi local heat transfer coefficient in channel j, J/m2 s K

kp the plate thermal conductivity, J/s m K

m mass flow rate in a channel, kg s-1

p pressure in axial direction

qp,i heat influx the element from plate i

Re Reynolds number

Tp,i temperature of plate p adjacent to channel j, K

Ti temperature of fluid in channel i, K

Ui overall heat transfer coefficient in channel j, W/m2 K

ui fluid velocity in channel i, m/s

ux average fluid velocity, m/s.

Greek letters

ρi fluid density in channel number j, kg/m3

4

ρp density of the plate, kd/m3

λ thermal conductivity of processing fluid, J/s m K

μ viscosity of processing fluid, kg/m s

θ dimensionless temperature

εp

plate thickness, m

5

Content Chapter 1 Introduction ..................................................................................................... 6

1.1 Motivation..................................................................................................................... 6

1.2 Project Objective and Outline ...................................................................................... 7

Chapter 2 literature Review ........................................................................................... 8

2.1 Dynamic Simulation of the Plate heat exchanger ........................................................ 8

2.2 Review on PID and MPC’s theory ............................................................................ 13

Chapter 3 Dynamic Simulation of the target PHE ............................................... 18

3. 1 Mathematical Model Construction ........................................................................... 18

3. 2 Simulation Results and Discussion .......................................................................... 25

Chapter 4 Controller Design ........................................................................................ 33

4.1 PID Control System Design ...................................................................................... 33

4.2 MPC Control System Design .................................................................................... 38

Chapter 5 Work Results Conclusion & Discussion .............................................. 48

5. 1 Simulation Results Conclusion ................................................................................. 48

5. 2 Conclusion of Controller Design ............................................................................. 52

5. 3 Future Work ............................................................................................................. 56

Appendix I Variables and Key Parameters’ Nomenclature in gPROMS

............................................................................................................................................... 59

Appendix II Dynamic Model of the Target Plate Heat Exchanger In

gPROMS .............................................................................................................................. 61

Appendix III Precious Simulation Results ............................................................ 63

Appendix IV Data Used For System Identification............................................ 64

Appendix V Estimated Black Box Model for the Target PHE ..................... 73

Appendix VI Inputs and Output of MPC Controller with Plant Model

PHE_N1, PHE_N1c, PHE_P1, PHE_P1c ................................................................. 74

6

Chapter 1 Introduction

1.1 Motivation

As one of the most common equipment in modern industry, the plate heat exchanger has been

popular for many decades. Its significant advantages, e.g. high heat exchange efficiency, low

fouling coefficient, easy to clean, make it be widely applied in food, petroleum, polymer,

hydrocarbon, vegetable oil and so on. Among all types of plate heat exchangers, the plate-

frame heat exchanger (PHE) is the most common one. Searching a way to run the PHE

efficiently and steadily could be meaningful in this energy-shortage world.

Figure 1.1 PHE Structure （from internet）

The instability of a PHE could be caused by many factors. Besides the disturbance, even

some normal plant operation (e.g. start-up system or reset the desirable output) could make

the system unstable. Thus, controllers are introduced to eliminate the “error” and make the

system stable again. Usually, PID is a good choice for such works. After several decades of

development, the PID has become a proven technology and been used in quite many fields,

especially process industry. It has been developed into packed unit in many kinds of popular

professional software which make it easy to be simulated and test. However, PID always

needs to be re-tuned when desirable working condition changed or emergencies happen. And

sometimes this could be dangerous when the PID controller used for safety management is

not re-tuned in time. Searching for a valid method to improve this weakness is meaningful.

With the development of mathematical technology, Model Predictive Control (MPC) has

shown a strong momentum to replace the traditional controllers. Unlike PID, MPC is mainly

7

based on process modelling and optimization so that on- line control is easily realised. Its new

design concept makes MPC more sensitive and efficient, but, unfortunately more expensive

and time-consuming. A comparison between PID’s and MPC’s effectiveness on the target

PHE appears necessary.

1.2 Project Objectives and Thesis Outline

The key goals of this project are summarized as follows:

1. Develop a dynamic and distributed model for the target PHE and perform a dynamic

simulation to investigate its heat-exchanging behaviour under desirable conditions.

2. Design suitable PID and MPC controller for the PHE respectively, find out their own

characteristics and how they improve the system operation.

3. Compare the PID and MPC controllers and make a conclusion on their advantages

and disadvantages.

This thesis consists of 5 parts.

Chapter 1 give a general introduction for the whole project;

Chapter 2 reviews the relevant existing work on modelling and dynamic simulation of the

plate heat exchanger and the theory of PID and MPC;

Chapter 3 is concerned with the whole process of the mathematical derivation for the PHE

model, and the model is run in gPROMS to perform a dynamic simulation. Finally, the results

obtained are compared with some existing work to justify its correctness;

Chapter 4 has two contents:

1. PID tuning procedures and results: Optimal tuning method is applied on the system with

the help of gPROMS.

2. MPC design. Firstly, the existing non- linear process model is approximate to a linear

state-space model in matlab; secondly, MPC is developed with mpc design toolbox;

Chapter 5 conclude the advantages and disadvantages of the two control method and show

the future work.

Besides, there are some Appendices in the last part to supply additional information about the

project research work.

8

Chapter 2 Literature Review

This chapter summarizes the previous works on dynamic simulation of the plate heat and

MPC theory. Their limitations are highlighted to motivate improvements processing on.

2.1 Dynamic Simulation of the Plate heat exchanger

Since the plate heat exchanger is popular for many decades, a lot of work has been done to

study its properties of heat exchange. With different purpose, the research works or studies

always have different types of model and focus. In general, most previous works can be

summarized into three types below:

1. Dynamic simulation of lumped PHE system. This type of work focuses on the output

of the system and only tries to obtain the temperature profile co response to time.

2. Steady state simulation of distributed PHE system. In contrast with the previous one，

such works try to clearly describe the temperature distribution along channel under

steady state.

3. Dynamic simulation of distributed PHE system. Focusing on temperature profile in

both time and spatial axis, giving out the most precise simulation results.

Now, we’ll discuss some typical works in details

2.1.1 Dynamic simulation of lumped PHE system.

A method to model and simulate a heat exchanger in the software “Dymola” is developed by

Pierre Kauhanen in 2004. Although used on a shell-tube heat exchanger in his paper, the

algorithm is also available for a plate heat exchanger.

Figure 2.1.1 Dymola model algorithm (Pierre Kauhanen, 2004)

9

As the figure shows above, Pierre’s method can be summarized as the key points below:

Ignoring other factors, the model is only concerned with heat convection and conduction

The heat flux cross streams and the wall is considered as a single quantity but not

quantities from many different points along the channel.

A mean value is used to represent the temperature of the entire volume element

These three points above are always the typical properties of dynamic simulation of lumped

heat exchanger system. On the other hand, the simulation result of Pierre’s work also gives

out a standard form of this type of simulation:

Figure 2.1.2 A Dynamic Simulation Result of Heat Exchanger ((Pierre Kauhanen, 2004)

Both the advantages and limitations of this algorithm are significant. It is easy to build the

model and perform the simulation, results are also clear. However, one cannot judge whether

the model has reflected the actual situation of the system unless the results are compared with

the measured data. Even worse, when disturbance exists, finding out the source of the

disturbance could be nearly impossible. Thus, a distributed model is usually necessary.

2.1.2 Steady state simulation of distributed PHE system

C.P. Ribeiro Jr. and M.H. Ca~no Andrade developed a method for steady-state simulation

of plate heat exchangers in 2001. In contrast with Pierre’s work, they tried to clarify mass and

temperature distribution of the streams in PHE channels. Their results are showed in Figure

2.1.3.

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Figure 2.1.3 Steady state simulation result of a PHE (C.P. Ribeiro Jr. and M.H. Cano

Andrade’s, 2001)

The figure clearly describes how the temperatures of different channels change along the

flow direction. It has become a quite standard form to validate the distributed models of PHE

system. However, the change of overall heat transfer coefficient is not taken into account here.

Jorge A.W. Gut, Jose M. Pinto improved the work.

Figure 2.1.4 Heat Transfer Coefficient Computation (Jorge and Jose, 2003)

11

As Figure 2.1.4 shows, the overall heat transfer coefficient U of a plate relates with the local

heat transfer coefficients of the streams on the plate’s both side. Usually, the local heat

transfer coefficient h is a function of the fluid physical properties Cp (heat capacity), μ

(viscosity) and λ (heat conductivity), which all changes with the temperature. Thus, there

should also be a profile of the overall heat transfer coefficient U corresponds to the

temperature distribution. Gut and Pinto did such a job in their research and their result is

shown below

Figure 2.1.5 Overall Heat Transfer Coefficient Distribution Profile (Jorge and Jose, 2003)

Although precisely derided the PHE system, distributed modelling under steady state can’t

predict how the system will go on in the future time. Its limitation is just the dynamic

simulation’s advantage. Thus a method combining these two algorithms comes out.

2.1.3 Dynamic simulation of distributed PHE system

The method used in Michael C. Georgiadis and Sandro Macchietto ( 2001)’s research work in

which focuses on dynamic simulation of a PHE under milk fouling condition is a standard

algorithm to model a PHE as both distributed and dynamic system. Compared with Pierre ’s

work, the fluid in a channel is no longer considered as an integral whole. Instead, a

differential volume element is taken out of the stream and dynamic modelling is performed.

As the volume of the tiny element approach to zero, a distributed dynamic system is

constructed.

Michael and Sandro used some same assumptions with C.P. Ribeiro Jr. and M.H. Cano

Andrade’s. The heat fluxes of the fluids in axial direction in these two works are both ignored.

12

They got a very similar result on temperature distribution. On the other hand, they tested the

overall heat transfer coefficient’s change over time. The result is shown below. It can be

easily seen that, as the heat exchange goes on, the overall heat transfer coefficient keeps

decreasing.

Figure 2.1.6 Unit volume element (Michael C. Georgiadis and Sandro Macchietto’s, 2001)

Figure 2.1.7 Overall heat transfer coefficient’s change over time (Michael and Sandro, 2001)

In summary of the three types of work above, we see that it is more complex to build a

distributed system than a lumped system. At least the two points below need to be considered

seriously:

1. How to deal with the heat conductivities of both fluids and plates in the axial direction?

13

2. How should the overall heat transfer coefficients be calculated?

All these questions will be discussed in details in Chapter 3.

2.2 Review on PID and MPC’s theory

As mentioned above, controllers are playing more and more important roles in modern

process industry these days. For PID, due to its nearly perfect theoretical basis, most research

works are on the tuning method development. Instead, there are still many s tudies are on

MPC mathematical formulation.

2.2.1 General PID formulation and tuning method

As implied by the name, a PID (proportional-integral-derivative) controller consists of three parts:

proportional part, Integral part and Derivative part. The weighted sum of these three parts is used

Figure 2.2.1 PID control system (form Wikipedia)

to adjust the process via a control element such as the position of a control valve or the power

supply of a heating element. Usually, a PID is formulated as below:

oytoutout DIPtMV )(

Where )(teKP pout

deKIt

iout )(0

dt

deKD dout

14

Here, Kp, Ki, and Kd are called proportional gain, integral gain and derivative gain. They are

the key parameters of a PID controller. The so-called “tuning” is to adjust all or part of these

three parameters. Any incorrect choice of the gains could lead to process instability,

oscillation or insensitivity. To solve these problems, firstly, one should decide which of the

three parts are necessary according to the actual operation condition; secondly, the control

parameters must be tuned so that desired control responses can be obtained. Usually, for

different applications there are different optimal behaviours of the process. For some

processes, overshoot can cause the system to become unsafe and must be kept off; for others,

oscillation may be the serious problem. For most heat exchangers, both stability of response

and optimality are required, and an effective tuning method is always necessary when

performing simulation of a PHE system.

Ziegler-Nichols closed loop tuning method is a simple way to tune PID controllers and can

be refined to give better approximations of the controller (Wikipedia, 2008). Its main idea is to

use the ultimate gain value Ku and the ultimate period of oscillation ρu(the time required to

complete one full oscillation while the system is at steady state) to calculate the “critical gain”

Kc. Ku can be found by setting the gains from I and D controller to zero, which totally reflect

the proportional influence on the system, and adjust the proportional gain until the control

loop oscillates indefinitely at steady state. This tests the robustness of the Kc value so that it is

optimized for the controller.

Although Ziegler-Nichols method is easy to operate, it is not the best because it only focuses

on the system’s stability and doesn’t perform any optimization to meet higher demand.

Nowadays, it has already been possible for any given arbitrary process model to use standard

computer programs to find out the optimal tuning parameters Kp, τI, τD for any mathematical

objective.

The most common optimal tuning method is Integral Performance Criteria. M. Zhang and

D. P. Atherton (1993) formulated this criterion as:

0

2

,,

)}({

.min

deJ

J

n

KdkiKp

It is believed that minimization of J always gives minimisation of the integral of time

absolute error. But, the tuning result using this criterion seems not accurate enough and a

further improvement may be needed.

15

2.2.2 MPC controller

Model predictive control (MPC) is a form of controller whose current control action is

obtained by solving a finite horizon open- loop optimal control problem at each sampling

instant on- line( as Figure 2.2.2 shows) . The optimal control problem is initialized with

current state of the plant, and yields an optimal control sequence, from which only the first

control is applied to the plant. This is the main difference between MPC and conventional

controllers which use a pre-computed control law. An important advantage of this type of

controller is that it is able to cope with hard constraints on controls and states and therefore it

has been wildly used in chemical and process industries where satisfaction of constraints is

particularly important because efficiency demands operating points on or close to the

boundary of the set of admissible states and controls. Besides, due to its special method to

calculate the manipulated variables, no tuning is needed during the whole process.

Figure 2.2.2 MPC control system (Kouramas Konstantinos’ Lecture material)

1. Mathematical formulation

Generally, an MPC model consists of three parts :

Objective function: this part determines how the controller meets the specification by

optimizing the manipulated variables. Usually, there is another term after the objective

function which is called terminal cost function. It is used to ensure the system’s stability

because the objective function could make the plant unstable when it leads the system

approaching to a optimal point. In another word, the terminal cost function can inhibit

the terminal state from going too far (James B. Rawlings, 2000).

Process model: the process model here is always no longer the original one. It should be

modified (linearization or approximation, etc.) according to the MPC types. This will be

discussed in details later.

Constraints: no matter what kind of the problem is, constraints always exist in actual

process system. Constraints describe how the actual operational, physical and safety

16

conditions restrict the system running. Meanwhile, constraints could appear in any part

of the system. There could be input constraints, state constrain, and also output

constraints.

In conclusion, the MPC is formulated as the form below:

Nj

jtNttjttjttjt

uxuuxLujxJ

0

)(),,(min),,(

)(

1,......,0),,(

1,......,0),,(.1

txx

uuu

uuu

yyy

Njuxgy

Njuxfxts

tt

tjt

tjt

tjt

tjttjttjt

tjttjttjt

Where L: objective function

φ: terminal cost

N: prediction horizon

t: initial time

yt+j|t, ut+j|t: output and input j times

△u: Limit actuator spead

2. MPC with linear process model

As mentioned above, the process model used for MPC controller may not be in the original

form. According to the type of the process model, James B. Rawlings (2000) defined two

kinds of MPC controller: MPC with linear model and MPC with nonlinear model. And the

former is more common than the later.

1) MPC with State-space model

James made very detailed discussion on MPC with state-space model in his paper” Tutorial:

Model Predictive Control Technology” in 1999. He highlighted that state-space is very

applicable to MPC controller because its significant advantages including easy generation

and analysis for multivariable system and online computation made the operation more

convenient. Even further, James also thought using state-space model is a good way to make

use of a series of linear system theory tools (e.g. the linear quadratic regulator theory and

17

Kalman filtering theory). In spite of this, James opposed to formulate all control problems in

such easy methods because it would lead one neglecting other possibilities, which could be

very important sometimes

As showed below (from James B. Rawlings, 2000), there are two common types of linear

state-space model. The left set of equations describes continues system and the right set is

used for discrete system. However, actually there is not that a clear line between these two

kinds of model. It is common to approximate continues system into a discrete one, all

depending on the actual situation.

hHxhHx

dDudDu

CxyCxy

BuAxxBuAxdt

dx

j

j

jj

jjj

1

2) MPC with Linear quadratic model

Based on the state-space model mentioned in last section, an MPC controller can also be

formulated with a feedback control law uj=u (xj) as below:

jTjjT

j

j

j

T

i uSuuuRuuyyQyyL

02

1

Where 1 jjj uuu ;

y desired output

u desired input

Q, R, and S parameter matrices which are assumed to be

symmetric positive definite;

(From Kouramas Konstantinos’ Lecture material)

Here, the vector y and u are always calculated from a steady-state economic optimization and

assumed to be time invariant. But, usually it is not possible to reach the idea steady a state (xs,

ys, us) for the actual constrained system. Instead, a value very closed to the idea steady state is

chosen as long as it can satisfy the condition. More details about this will be discussed in

Chapter 4.

18

Chapter 3 Dynamic Simulation of the target PHE

This chapter is concerned with modeling of the target PHE system and dynamic simulation

result analysis. Firstly, detailed model procedures will be showed and then, the model is to be

performed in gPROMS. Some tests will be done to validate the model and comparisons

between the simulation results and some existing work will be made.

3. 1 Mathematical model construction

3. 1. 1 Problem description

The PHE system has a structure of 3 passes which contains 36 channels each. A simplified

pass arrangement is showed below:

1 1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

1 2 3 4 5 6 7 8 9 10 1112 13 14 15 16 17 18

1 1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

1 2 3 4 5 6 7 8 9 10 1112 13 14 15 16 17 18

1 1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

1 2 3 4 5 6 7 8 9 10 1112 13 14 15 16 17 18

1

2

3

Cold

stream

hot

stream

Figure 3.1 Flow structure of the PHE in this research

In Figure 3.1, hot streams and cold streams are denoted by red and blue arrows respectively.

The hot stream flows through the PHE in the order of pass 3, 2 and 1 and the cold stream

flows right in the opposite direction. This kind of arrangement can improve the efficiency of

heat transfer between hot and cold streams.

19

Besides, some necessary parameters of the plant and operation conditions are listed in the

table below:

Channel length / m 0.71856

Channel height/ m 0.025

Gap width /m 0.033

Plate thickness / m 0.005

Plate thermal conductivity / W/m2s 16

Plate density/ kg/m3 7800

Plate heat capacity / W/kgK 502

Operation Pressure/bar 6

Table 3.1 technical details of the target plate heat exchanger

3. 1. 2 Mathematical Model Derivations

1. Assumptions

Many physical properties of the fluids and plates can affect the heat transfer result. Some of

them could be always very effective, some may not, and some may only remarkably

influence the result under certain special condition. Generally, we can ignore those

ineffective factors and especially point out the special cases.

To model a PHE system, the most important factors which mostly determine the results are:

Mass transport, Energy transport, and momentum transport in plug flow and the heat

transport between plates and fluid. These processes can be described and linked with each

other by conservation law equations. Any change in one of them could influence the output,

e.g. decrease in the inlet flow rate of hot fluid will cause the outlet temperature of cold fluid

to drop. These equations form the main body of the model.

However, for specific system, some of the factors mentioned above might be not that

important. In this project, we can ignore some of them:

Mass Transport: For the target system, we focus on the energy transport between cold

and hot streams. On the other hand, in gPROMS’s physical property database IPPFO,

the density of water doesn’t change significantly with temperature. Thus we can

assume the streams as incompressible fluids.

Momentum Transport: here we assume the constant pressure, thus there is no need to

consider the momentum equation.

Phase change: quite limited phase change in the channels.

20

Fouling, transports in boundary layer: These phenomena are ignored in this modelling

because the composition is simply water.

In conclusion, the assumptions used for this project are as follows:

(1) One-dimension flow;

(2) Non-fouling condition;

(3) No phase change in the channel;

(4) Constant pressure

(5) Incompressible fluid

(6) Ignore the heat transport in axial direction

2. Mathematical Derivation

To analyze the distributed fluid in a channel, we need to take a differential volume element

with the length of δx at position x first (as figure 3.2 shows).

Figure 3.2 Analysis on a differential fluid element in a channel

1) Mass balance

Since the fluid is assumed to be incompressible, there is no need to use continuous equation

here. So the mass balance in a channel is simply in the form below:

ixi uAF

2) Energy balance

x=0 x=L x

Pi-1

Pi

qp,i

x+δx

qp,i-1

Fluid in Fluid out

21

To describe the temperature profile, we need to consider the energy balance on the

differential element. In doing this, we should consider both internal and kinetic energy

contributions:

)(|)2

1([

|)2

1([)

2

1(

1,,

2

22

ipipxxi

iii

xi

iiiiii

qqx

TAuhAu

x

TAuhAuxuAxeA

t

Here, heat diffusion caused by temperature gradient along the axial direction x

TA i

is

considered, and the heat Q from plates will be discussed separately.

Dividing through by Aδx and taking the limit as δx→0, we obtain:

)()()(2

1)()(

2

1)( 1,,

22

ipip

iiiiiiiii qq

x

T

xuu

xhu

xu

te

t

Now, we can use the thermal dynamic relation:

/peh

We get :

)()]ln

ln()()()

)/1(( 1,,,

ipip

i

iiii

iiiipi qq

Dt

Dp

Tx

T

xA

Cufu

x

u

Dt

Dp

Dt

DTC

However, this is not the final version. As mentioned above, we should ignore some factors.

According to the assumptions, we can ignore all terms on the right hand side except the last

one, which gives out:

1,,, ipipipi qqDt

DTC

Introducing )( ,, iiPP

iip TTUq , we can have the final version of the energy balance equations

of the fluids:

1,...,3,2

)()( 111

ni

TTAUTTAUx

Tun

t

TCpA Piii

P

i

P

iii

P

ii

iii

iix

)( 21221

111

11

PP

x TTAUx

Tun

t

TCpA

22

)( PnnnP

n

n

nn

n

nnx TTAUx

Tun

t

TCpA

Similarly, the energy balance of the plate is formulated as:

nj

x

TTTTTAU

t

TCpA

P

jP

jj

P

ijj

P

j

P

jPPP

x

,...,3,2

)]()[(2

2

1

Note: since the thermal conductivity of the metal is very high, we should not neglect the heat

transport inside the plate.

3) Heat transfer coefficient

In this project, Jorge and Jose’s method is use. The heat transfer coefficient is treated as a

distributed variable.

ni

khhU P

P

ii

P

i

,...,3,2

111

1

ni

Deu

Cp

Nu

DehNu

i

iii

i

iii

ii

i

ii

,...,2,1

Re

Pr

Pr)2.3(Re214.0 4.0662.0

In all, the one-pass model is formulated as Table 3.2 shows.

4) Model modification

The model is modified as explained in the following, in order to make its implementation in gPROMS easier.

The basic idea is that all variables should be re-named according to hot and cold streams and

the set of the equations should be rearranged. For example, the hot temperature of the hot

stream in channel i is named as T_hoti, and for the cold stream is T_coldi. A completed

modified model is available in Appendix 1 and 2.

23

Equations No. New Variables No

ni

nFF

uAF

i

ixi

,...,2,1

/

n iu n

1,...,3,2

)()( 111

ni

TTAUTTAUx

Tun

t

TCpA Piii

P

i

P

iii

P

ii

iii

iix

n-2 Pi

P

iii UTCpT ,,, 4n-6

nj

TTTTAUt

TCpA Pjj

P

ijj

P

j

P

jPPP

x

,...,3,2

)]()[( 1

n-1

nTT ,1 2

)( 21221

111

11

PP

x TTAUx

Tun

t

TCpA

1

1Cp 1

)( PnnnP

n

n

nn

n

nnx TTAUx

Tun

t

TCpA

1

nCp 1

ni

khhU P

P

ii

P

i

,...,3,2

111

1

n-1 ih n

ni

Deu

Cp

Nu

DehNu

i

iii

i

iii

ii

i

ii

,...,2,1

Re

Pr

Pr)2.3(Re214.0 4.0662.0

4n ii

iii Nu

,Pr

,Re,, 5n

)( iii TCpCp

)( iii T

)( iii T

ni ,...,2,1

3n 0

Totally 11n-2 11n-

2

D.O.F=0

Table 3.2 Completed one-pass model for the target PHE

24

(5) Boundary conditions and initial conditions

Initial conditions:

It is assumed that all steams in the three passes start at the inlet temperature which is given as:

18...3,2,1__

18...3,2,1__

18...3,2,1__

0

3

0

3

0

2

0

2

0

1

0

1

iTcoldTThotT

iTcoldTThotT

iTcoldTThotT

cold

int

pass

i

hot

int

pass

i

cold

int

pass

i

hot

int

pass

i

cold

int

pass

i

hot

int

pass

i

For the plate, it can be considered to start from steady state:

;35...3,2,10_

;35...3,2,10_

;35...3,2,10_

3

2

1

jt

plateT

jt

plateT

jt

plateT

Pass

j

Pass

j

Pass

j

Boundary conditions:

Based on the modified model and the pass arrangement, the boundary conditions are listed

below:

Pass I:

coldF

coldFcoldT

coldT

ix

coldTTcoldT

ix

hotThotThotT

i

pass

ix

pass

i

Pass

out

x

pass

iin

coldx

pass

i

x

pass

iPass

outx

pass

i

_

__

_

;18...3,2,10_

_

;18...3,2,10_

__

18

1

1

0

1

1

0

1

1

1

1

1

2

0

1

Pass II:

coldF

coldFcoldT

coldT

hotF

hotFhotT

hotT

ix

coldTcoldTcoldT

ix

hotThotThotT

i

pass

ix

pass

i

Pass

out

i

pass

ix

pass

i

Pass

out

x

pass

ipass

outx

pass

j

x

pass

iPass

outx

pass

i

_

__

_

_

__

_

;18...3,2,10_

__

;18...3,2,10_

__

18

1

2

0

2

2

18

1

2

1

2

2

0

2

1

1

2

1

2

3

0

2

25

Pass III:

hotF

hotFhotT

hotT

ix

coldTcoldTcoldT

ix

hotTThotT

i

pass

ix

pass

i

Pass

out

x

pass

ipass

outx

pass

i

x

pass

iin

hotx

pass

i

_

__

_

;18...3,2,10_

__

;18...3,2,10_

_

18

1

3

1

3

3

0

3

2

1

3

1

3

0

3

3.2 Simulation results and discussion

Based on the constructed model, a dynamic simulation is performed in gPROMS. Here the

solver chosen to solve the PDAE system is BFDM, and the whole length is discrete into 15

points.

1) Topology arrangements

Figure 3.2 shows how the whole process is connected. In the picture, the blue and red lines

represent cold and hot streams respectively. We see that before going into the heat exchanger,

the hot stream firstly flow through a valve which is used to manipulated the flow rate. Then it

goes into the plant and exchange heat with the cold stream. Finally, all the output data we

need can be collected from the port_cold and port_hot.

Figure 3.2 Topology of the whole process

26

2) Outlet temperature analysis

Figure 3.3(a) and 3.3(b) shows the outlet temperature of hot and cold streams from port_hot

and cold_port in 100 hrs. We see that the temperature of cold stream keeps increasing and

approached to 335K. In contrast, the hot stream’s outlet temperature sharply decreases to

around 335K in the first 10 hrs and then gradually rises back to original level. Compare them

with the previous works did before (see Appendix III), we found that those two models shows

very similar dynamic behaviour, which prove that the profile we obtain is relatively normal.

Figure 3.3(a) Outlet temperature of the cold stream

Figure 3.3(b) Outlet temperature of the hot stream

27

3) Temperature distribution along the axial direction

Temperature distribution of hot streams

Figure 3.4(a) Temperature distributions of hot streams at 20.0th second

Figure 3.4(b) Temperature distributions of hot streams at 3.0th second

336

337

338

339

340

341

342

343

344

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

PH

E.P

ass_3001.T

_hot

Axial

PHE.Pass_3001.T_hot(20,3,) PHE.Pass_2001.T_hot(20,3,) PHE.Pass_1001.T_hot(20,3,)

339

340

341

342

343

344

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

PH

E.P

ass_3001.T

_hot

Axial

PHE.Pass_3001.T_hot(3,3,) PHE.Pass_2001.T_hot(3,3,) PHE.Pass_1001.T_hot(3,3,) PHE.Pass_1001.T_hot(3,6,)

28

Figure 3.4 shows how the temperature changes when the hot stream flow through the 3 passes.

In the picture, the black, red and blue lines denote the temperature profiles of the third

channels of Pass1, Pass 2 and Pass 3 at the twentieth hour respectively. It is obviously that

the temperature is lowed pass by pass and keeps decreasing all along the axial direction.

However, this could not be that clear at some time:

Compared with last picture, we found that the temperature drops in pass 1 and Pass 2 are not

as clear as the one in Pass three. However, Figure 3.3(b) showed there is a very significant

temperature drop in the first 10 hours. Therefore we can make a conclusion that, for Pass 1,

the hot stream’s temperature doesn’t change gradually at the beginning time (first 10 hours

more or less), instead, sudden changes make it drop fast, which reflects that the system is

very unstable during the beginning period.

On the other hand, hot streams in same pass have very similar temperature profiles. Take

Pass 3 as an example. As figure 3.5 shows, the fluids in channel 1 to channel 17 almost have

a same behaviour on temperature changes. Only the fluid in the last channel’s performance is

somewhat different. This could because there is only one plate exchanging heat with it so that

less heat is lost than other fluids.

Figure 3.5 Temperature Distributions of hot streams in Pass 3

29

Temperature distributions of cold streams

In contrast with hot streams, cold streams’ temperatures do not change significantly along the

axial direction. Instead, the temperature gradient is very subtle and even worse sometimes it

could be zero (see figure 3.6). This result is quite different from the previous work we

mentioned in Chapter 2. The main cause such differences will be discussed in the next section.

Similarly, the cold streams are heated by hot streams pass by pass. This kind of behaviour is

relatively clear in Figure 3.6(a). In the graph, the black, red and blue lines separately denote

the temperature of the fluids in Pass 1, Pass 2 and Pass 3. It is obvious that the first pass plays

the most important role in raising the temperature level. This is especially clear at the

beginning period (see Figure 3.6(b)).

This phenomenon also appears on hot streams (see Figure 3.4(b)). It can be easily explained:

when a hot or cold stream flows into the heat exchanger, the temperature difference between

the fluids and plates are very significant. As the heat exchange goes on, it gets smaller and

smaller. However, finally the fluids in different passes will stay in temperature levels which

are much closed to each other, and the temperature difference will then become uniform,

which leads the different passes approaching to same effectiveness.

Figure 3.6(a) Temperature distributions of cold streams at 20.0th second

278

279

280

281

282

283

284

285

286

287

288

289

290

291

292

293

294

295

296

297

298

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

PH

E.P

ass_1001.T

_cold

Axial

PHE.Pass_1001.T_cold(20,3,) PHE.Pass_2001.T_cold(20,3,) PHE.Pass_3001.T_cold(20,3,)

30

Figure 3.6(b) Temperature distributions of cold streams at 5.0th second

Figure 3.7 Temperature distributions of cold streams in first channel and second channel of

Pass 2 at 5.0th second

Another interesting situation appears in cold stream temperature profile in Pass 2 and Pass 3.

Figure 3.7 shows the phenomenon. It can be clearly seen that at the fifth hour, the

278

279

280

281

282

283

284

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

PH

E.P

ass_1001.T

_cold

Axial

PHE.Pass_1001.T_cold(5,3,) PHE.Pass_2001.T_cold(5,3,) PHE.Pass_3001.T_cold(5,3,)

283

284

285

286

287

288

289

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

PH

E.P

ass_2001.T

_cold

Axial

PHE.Pass_2001.T_cold(10,1,) PHE.Pass_2001.T_cold(10,2,)

31

temperature of the fluid in Pass 2’s first channel is lower than entrance temperature so that it

looks like the temperature is “decreasing” along the channel. However, the truth is not like

what we see and his is due to the following reasons:

For the first channel, there is only one plate exchanging heat with the fluid in it, therefore

much less heat is transported to it than other fluids. Besides, the initial temperature is also

much lower than the entrance temperature. Thus the fluid in that channel could not approach

to the same temperature level in short time, and this leads to the phenomenon we see.

Analysis on the overall heat transfer coefficient

As mentioned before, the cold stream temperature distributions are very different from the

results of previous work showed in Figure 2.1.3. The reason could be found in the overall

heat transfer coefficients. Figure 3.8(a) show how some plates of Pass 1’s overall heat

transfer coefficients change over time. The black line denotes Plate 1 and the lines of other

colours denote plates at some other positions. We can see, as time goes on, the values of

those coefficients are getting smaller. And the first one decrease a bit slowly. This is same as

Sandro and Michael’s conclusion in 2000; however, the drop is not as significant as their

result. Similarly, the axial distributions of those coefficients are very stable, too. The values

of the differences between positions are very small, which makes the curve almost a

horizontal (see figure 3.8(b)).

Figure 3.8(a) overall heat transfer coefficient changes over time

3530

3540

3550

3560

3570

3580

0 10 20 30 40 50 60 70 80 90 100

PH

E.P

ass_2001.U

Time

PHE.Pass_2001.U(,1,0.466667) PHE.Pass_2001.U(,2,0.466667) PHE.Pass_2001.U(,35,0.466667) PHE.Pass_2001.U(,18,0.466667)

32

The reason why these heat transfer coefficients are so different is, the platform we chose to

perform the simulation is gPROMS, of which the data base IPPFO is very different with the

normal physical property data. The IPPFO has simplified those data and make the physical

properties very insensitive to temperature. Sometimes the change of temperature even

couldn’t cause those parameters to change. Therefore, the heat transfer coefficients we

obtained become so stable, and even further we got very different results of cold streams

from previous work.

Figure 3.8(b) overall heat transfer coefficients of Pass 1 at 50.0th second

33

Chapter 4 Controller Design

4.1 PID Control System Design

With the help of gPROMS, we can now design the PID controller more easily. A completed

PID unit model is inserted into the PML (process model library) and can be added on the

plant simply by topology function. What we need to do is just to tune and determine its gains.

The tuning method is chosen to be Integral Performance Criteria which is a kind of

optimization tuning method. Thus the PID design work becomes an optimization problem.

Firstly, the PID and a valve are connected to the plant as the figure shows below:

Figure 4.1 topology of PID control system

It is clearly seen that, the output of the plant is first detected by a sensor and transformed into

electrical signal then sent to PID controller. After calculation based on the current “error”, a

control action is sent to the valve which is manipulated by the control signal so that the inlet

flow rate of hot water is changed. By this means, the output is led approaching to the set point.

34

Since differential operation is much easier to realise than integral operation in gPROMS, here

we formulate the objective function in differential form as below:

2

.min

edt

dJ

J

Where e is the error

Note: the error here is not calculated as the difference between measured value and reference

signal. More details about it can be found in Appendix II.

Figure 4.2 (a) PID controller output with proportional gain of 1

Figure 4.2 (b) PID controller output with proportional gain of 5

35

Figure 4.2 (c) PID controller output with proportional gain of 10

As mentioned before, a too large proportional gain could lead the system unstable. But, there

is not a very strict rule to follow for a particular system. The normal way is to use manual

tuning method firstly so that a general range of the gain is obtained. Figure 4.2 (a), 4.2 (b)

and 4.2(c) show the controller outputs with proportional gain of 1, 5 and 10 respectively.

It is obvious that a relatively big proportional gain makes a sudden change happen. When the

gain increases to 10, the change is even more shape. Thus, we can preliminary estimate that

the value of proportional gain should be no more 10. With such estimation, we perform the

optimization with an initial setting as Figure 4.3 shows.

Figure 4.3 PID initial settings

36

The time horizon is set to 500 seconds but not 100 seconds so that the optimization results

will more suitable for the system when running long period. On the other hand, since the

physical and operation limitations have already been taken into account in the process model,

none extra constrains is needed to be included in the optimization problem. Therefore, we

obtain the optimal control action as the figures and tables show below:

Figure 4.4 (a) First PID optimization tuning results

With the parameters above, the system simulation result of 100 sconds is showed below:

Figure 4.4 (b) simulation results of the system with tuned PID

Bias 0.01642078

Gain 5.4744306

Rate 10

Rate limit 100

Reset time 95.75582

37

However, if we narrow down the scope of the proportional gain to (0, 5), will the

optimization certainly give out a very different set of results and how is the simulation results

to change? The answer is easy to find from figure 4.5. Compare Figure 4.5 with Figure 4.4,

we find that, those two set of results don’t vary a lot from one to another. However, the

output curve in figure 4.5 (b) shows a more obvious decreasing trend just after the output

value reaching the peak which means the system become less stable with the later set of key

parameters. Even further, the proportional gain’s value in this optimization hits its upper

bound which indicates an active bound. In sum of all these factors, we can conclude that the

first set of optimization results is the best choice for this system

Figure 4.5(a) Second PID optimization tuning result

Figure 4.5 (b) simulation results of the system with tuned PID

Bias 0.014700688

Gain 5

Rate 10

Rate limit 91.99628

Reset time 96.28032

38

4.2 MPC Control System Design

As the latest control system in the world, model predictive control (MPC) has a series of

advantages compared with the traditional controllers. By now, the application of MPC in

many different regions has shown its advancement and robustness. However, due to its high

cost, it has not been wildly used. That’s why there is not such completed model unit in

gPROMS. To design an MPC controller, we need the help of MATLAB.

4. 2. 1 Model export from gPROMS to MATLAB

No matter what we what to do with the controller, we must connect it with our target system

together to simulate and tune the whole process. Since the MPC must be programmed in

MATLAB environment, the only way to add the controller on the system is to export one part

to another so that they will run in same platform. Fortunately, gPROMS offered us a tool to

do this, and the tool is gO:Simulink.

Figure 4.6 topology of the composite model

gO:simulink is a software included in standard gPROMS installation. It allows complex non-

linear gPROMS process models to be incorporated directly into The MathWorks MATLAB

& Simulink environment. The gPROMS model is run from a gO:Simulink block that can be

added to any Simulink model[10]. gO:Simulink offers us two methods for exchanging data

between gPROMS and Simulink. One is exporting simulink blocks into gPROMS as a

39

foreign objective; the other is exporting a gPROMS unit into simulink as an simulink block.

Here, we choose the later one because the former method needs Visual C++ as accessory

appliance.

The first step to export a unit is to pack all units you want into one gPROMS unit. The reason

to do so is gOSimulink can only export one gPROMS unit a time. In another word, all the

component models (three passes and liquid source) must be put together to form a composite

model “PHE” which include a topology. The topology is showed in Figure 4.6 below.

(Note: here we delete the valve because in MPC controller design the inlet flow rate of hot

water is chose to be input directly)

Figure 4.7 Flow sheet in simulink

From the original topology we know that all the inputs for the system are supplied from the

unit named “liquid source”. But, this function must be distributed to other four independent

units. The reason to do so is that, the model export is carried out with the help of

“gOSimulink library”. The input data we need must be generated in a gOSimulink library

model so that they can be transferred to Simulink with the composite model together. In this

case, the “liquid source” unit only works as a channel which combining all the inputs.

40

Finishing all the preparatory work, the composite model is transferred into Simulink and the

whole process is represented as the figure shows below:

4. 2. 2 MPC controller design

1. System identification

Because of the MPC controller’s specific characteristics, the non- linear process model

constructed in gPROMS cannot be used for controller design directly. In all, the PHE model

includes more than 600 equations and a series of complex connection. It is quite obvious such

a detailed high-fidelity PDAE system is impossible to be used with the current MPC methods.

Therefore linearization and approximation are necessary to obtain a reduced order model.

Since the PHE in this project is a relatively easy SISO system, it is convenient to finish the

design work in MPC tool box. Thus, the process model can be only LTI (linear-time- invariant)

model which is the only form available for MPC tool box. Three formats of model can be

specified as LTI model: transfer function model (TF), Zero-pole-gain models (ZPK) and

state-space model (SS) (From MATLAB Product introduction). Here we choose the state-

space model as the plant model we want to estimate from the developed nonlinear model.

State space model is a mathematical model of a physical system as a set of input, output and

state variables related by first-order differential equations (From Wikipedia). For a state space

model, all the process inputs, outputs and states are all expressed as vectors and the all the

equations are written in matrix form. That is why it is convenient to use a state space model

for MPC controller design. With the help of MATLAB’s system identification tool box, we

can transform the PHE model we have construct into a state space model easily. And the

detailed procedures are state in following:

The first step is preparing the data. Since a dynamic simulation is successfully run in

gPROMS, we can collect any data easily as required. S ince for a plate heat exchanger, the

system input (inlet flow rate of hot water) keeps constant for the whole simulation process,

we need to collect more than one set of data with different inputs. Then we should use one of

them to estimate model (working data) and others to validate it (validation data) so that the

model will be confirmed to be suitable for different inputs.

After exported into matlab, the set which is chosen to be working data is pre-processed

(remove means) and stored in an M-file. Using the tool box’s plot function we can easily see

the pre-processed working data Figure 4.8 shows:

41

Figure 4.8 input and output data used for estimation time-plot

Finishing data preparation, the model estimation can be preceded now. There are two

common algorithms for state-space model estimation: N4SID and PEM. It is hard to say

which method will give us a better result unless try them both.

N4SID method

The numerical algorithms for subspace state space system identification is usually

known as N4SID. They are always convergent (non- iterative) and numerically stable

since they only make use of QR and singular value decompositions [10]. The N4SID

methods have very significant advantages as follows:

Firstly, with N4SID algorithms, since the parameter matrices for the state space model

are not calculated in their canonical forms but as full state space matrices in an

optimal conditioned basis, a priori parameterization can always be avoid [11]. The only

thing needs to be determined is the model’s order (the number of states), which is

easy to obtained by computation.

Secondly, because N4SID algorithms are non- iterative, there is no non- linear

optimization computation involved in the identification so that they are numerically

0 50 100 150 200 250 300 350 400 450 500-100

-50

0

50

y1

Input and output signals

0 50 100 150 200 250 300 350 400 450 500-1

-0.5

0

0.5

1

Time

u1

42

stable and do not suffer some typically problem of iterative algorithms (e.g. no

guaranteed convergence, sensitive to initial estimates, etc.)

Best fit: PHE_N1 80.14% & PHE_N2 76.9%

Figure 4.9 (a) simulated N4SID model outputs Vs data_V1

Best fit: PHE_N1 88.67% & PHE_N2 88.45%

Figure 4.9 (b) simulated N4SID model outputs Vs validation data 2

0 50 100 150 200 250 300 350 400 450 500260

270

280

290

300

310

320

330

340

350

360

Time

Measured and simulated model output

0 50 100 150 200 250 300 350 400 450 500260

270

280

290

300

310

320

330

340

350

360

Time

Measured and simulated model output

43

Finally, as mentioned above, the N4SID algorithms are not sensitive to initial state so

that no extra parameterization is needed.

The first step of estimating state-space model with N4SID is to fix the model’s order.

The toolbox set the order to 4 as default value. But, using the “order selection”

function, we are recommended to set the order to 2. Estimated the model with both

orders of 2 and 4, we get two different model plots as Figure 4.9 shows.

(The deep blue line denotes model PHE_N14 which is a model of order 2; The yellow

line denotes model PHE_N2 which is a model of order 4)

PEM method

Iterative prediction-error minimization method is known as PEM in matlab

environment. Unlike N4SID algorithms, PEM is a typically iterative method thus it

may not grantee the convergence of the model (as Figure 4.10 (a) and (b) show).

When using PEM to estimate a state space model, an initial estimation is proceeded in

advance with a N4SID algorithm.

With PEM, an order selector cannot be use. According to experience of last estimate,

we set 2 and 4 as the model order, estimate the model focus on prediction and obtain

the model plots as below:

(The blue line denotes model PHE_P2 which is a model of order 4; the red line

denotes model PHE_P1 which is a model of order 2)

Best fit: PHE_P1 79.03% & PHE_P2 -671.8%

Figure 4.10 (a) Simulated PEM model outputs Vs validation data 1

0 50 100 150 200 250 300 350 400 450 5000

50

100

150

200

250

300

350

400

450

500

Time

Measured and simulated model output

44

Best fit: PHE_N1 88.69% & PHE_N2c -137.9%

Figure 4.10 (b) Simulated PEM model outputs Vs validation data 2

From all the four figures above, we can tell that, all the models estimated have good fit with

the measured data except model PHE_P2 which is not convergent. Among all the models,

Figure 4.11 Refined models PHE_N1c and PHE_P1_c

0 50 100 150 200 250 300 350 400 450 500200

220

240

260

280

300

320

340

360

380

400

Time

Measured and simulated model output

0 50 100 150 200 250 300 350 400 450 500260

270

280

290

300

310

320

330

340

350

360

Time

Measured and simulated model output

45

PHE_N1 has the best fit of all. Even further, PHE_P1 and PHE_N1 can be refined to better

models PHE_P1c and PHE_N1c which are both with best fits of 88.69% (showed in Figure

4.11 above). Instead, the iterative refine process makes PHE_N2 lose convergence. Thus,

there are five models which can be used for MPC controller design (PHE_N1, PHE_N1c,

PHE_P1, PHE_P1c and PHE_P2). However, the fitness of those five models doesn’t actually

vary a lot from each other. Although one of them is the best, the others are not too much

worse than it. On the other hand, whether a model can be suitable for a MPC controller

doesn’t depend on its fitness, only practise can tell which one should be chosen.

2. MPC Preliminary Design

Because the plant model is a relatively simple SISO system, there is no need to formulate

complex MPC controller. The MPC library in simulink can totally satisfy the demands of the

work. What we need to do is simply to determine some key parameters of it with the help of

MPC tool box and MPC block.

Figure 4.12 MPC output

Once defined a MPC block by simply copying a MPC library block to a new utility, a new

MPC controller is defined in matlab work space. Import all the estimated plant model and the

defined controller into the MPC design tool box and run the simulation of 5 different

collocations, we find that except the model PHE_N2, all the other four plant model cannot

0 50 100 150 200 250 300 350 400 450 5000

50

100

150

200

250

300

350

Plant Output: y1

Time (sec)

46

work well with the MPC controller and give out very bad behaviour (see Appendix VI) .

Thus, we can fix the plant model as PHE_N2 which is present as below:

1.0564)0(x5.4399)0(089104.0)0(x 0.25819- = (0)x

)()(0060106.0)(00011983.0)(4272.1)(66.222)(

)(77886.0)(103433.1)(63944.0)(7702.0)(0027408.0)(0010531.0)1(

)(049.66)(103349.6)(72636.0)(21513.0)(0049731.0)(0057565.0)1(

)(19687.0)(100366.600057857.000818969.0)(99771.0)(0029473.0)1(

)(0033865.0107138.3102409.8)(012403.0)(98494.0)1(

4321

4321

15

43214

13

43213

12

43212

4

5

3

6

211

x

tetxtxtxtxty

tetutxtxtxtxtx

tetutxtxtxtxtx

tetuxxtxtxtx

texxtxtxtx

With the plant model above and after a series tuning (see Chapter 5), we obtain the MPC

controller output as 4.12, which can prove the MPC now can be added in to the system.

(Note: since the MPC controller is created by defined new MPC simulink block, the exactly

formulation is impossible to obtained)

Figure 4.1.3 MPC control system flowsheet arangement

3. MPC control system in Simulink

After setting up the controller, we should run the simulation so that we can know the exact

performance of the whole control system.

47

For the first step, the MPC controller is added into the simulink flowsheet as Figure Figure

4.1.3 shows: the outlet temperature is firstly sent to the MPC block, and then the MPC will

compute an optimal conrol action and sent it out. Timed by a scaling factor in the Gain block,

the control action is finally sent to th pant to make it approach to the set point. At the

beginging, the flowrate is set to be 5 kg/s and the referece signal is left blank for the set point

has already been fixed as 300K in the design tool box.

Unfortunately, due to some unknown reason, gOSimulink cannot run crecotly. This

simulation part is left for the future work and will be discussed in next Chapter.

48

Chapter 5 Work Results Conclusion & Discussion

5. 1 Simulation Results Conclusion

As mentioned before, due to the widely use of the PHE, so many works have been done to

research their behaviour under different conditions. Many of the ideas have been taken and

used in this project and played effective roles to obtain some key results. Meanwhile, because

every project has its own properties, there are also some special methods and ideas which

have shown significant effectiveness.

1. Assumptions and model construction

We have generally summarized how to choose adapted assumptions in Chapter 3. Here, a

more classic and systemic conclusion is made to show how consumptions effect the model

construction and simulation results in this project. Obviously, since assumptions are the bases

of the models, assumptions can great affect a model’s form, solvability, accuracy and so on.

Suitable assumptions can make the work both easy and precise; instead, bad assumptions

always bring unnecessary troubles or lead the work going in a wrong direction.

Assumptions determining a model’s form

A certain model’s form is usually determined by the objective of the project work,

operation conditions and some other key factors. In this work, since we want to know the

system’s both over time and space, the model is constructed in a dynamic and distributed

form.

It is easy to judge whether a model should be dynamic or in steady-state form. However,

it is another thing to determine whether the system should be distributed or lumped.

Usually, a heat exchanger is formulated as a lumped system for we only care about the

outlet temperature profile over time. However, when a precise control system is needed,

clear energy and mass distribution become more important and a distributed model is

necessary. Even further, we should determine the dimensions of the model because any

unnecessary increase in dimensions of the model can significantly enlarge its scale and

made it more difficult to solve or sometimes unsolvable.

For the project we have done, since the model is going to be used for the controller and

one of our objectives is to obtain clear temperature profile inside every channel, we

determine the model as 1-D distributed model(x-direction only) and the total number of

the equations is more than 200. According to the simulation results we got, the 1-D

model has done very good work to show us all the data and profiles we need. If the plant

49

is formulated as 3-D model (in x, y, z three direction), the number of equation could be

more than 700 and the difficulty to solve the system will be highly increased or maybe

the system could not be solved. Even the model can be well solved, the simulation results

will not be greatly improved for the flow behaviour in the other two directions are too

negligible.

Assumption of constant pressure and density

It is well known that pressure is one the key parameters of fluid and is reflected by

momentum equation in a model. The momentum balance formulated for this project is:

x

p

x

uu

t

u

This equation gives out the relationship between velocity and pressure (since gravity is

neglected in this project there is no term describes the body force). It is obvious that if

this balance is added into the system, the model will become much more complex.

However, in practical applications of a PHE, the velocity of fluid in the channels doesn’t

vary a lot from point to point. On the other hand, for any fixed valve position, the flow

rate is constant so that the average velocity u doesn’t change over time, either. Thus, we

have:

0

x

p

x

uu

t

u

Which means the pressure can be assumed to be constant.

Similarly with the pressure, the density can also be assumed to be constant because in

gPROMS, the water density doesn’t change a lot over temperature. Thus a continue

equation is not necessary anymore and the model is more simplified.

Domain of applicability of the PDAEs

Another thing worth to be discussed is the domain of applicability of the model. In this

project, all the equations are applicable on the whole axial domain only except energy

balance equations. These equations are not available at both the entrances and the exits of

the channels because at such positions, sudden changes of flow rate happen and make the

fluid no more continuous so that the heat exchange doesn’t work as the model.

2. Simulation conclusion

Solver choice

Actually, the basic idea to solve a PDAE (partial differential-algebraic equation) system

is to define a grid on the spatial domain so that the PDAE system is discrete into a set of

DAEs. Then, the initial conditions are approximated and the DAEs are solved so that an

50

approximated solution of the original PDAE system is obtained. Based on this idea, there

are three numerical methods for the PDAE system solution in gPROMS: BFDM

(backward finite difference method), CFDM (centered finite difference method), and FFDM

(Forward finite difference method) A finite difference is defined as a mathematical

expression of the form: f(x + b) − f(x + a). There are three kinds of finite differences:

Forward, backward, and central differences, which are expressed as follows:

A forward difference is an expression of the form:

)()()]([ xfhxfxfh .

A backward difference uses the function values at x and x − h, instead of the values at x +

h and x :

)()()]([ hxfxfxfh

A central difference is given by

)2

1()

2

1()]([ hxfxfxfh

Besides, there is also another solver OCFEM (orthogonal collection on the finite elements method) which is based on the idea of polynomial approximation method. It is not

introduced here for it is not suitable for the model we built.

Figure 5.1 Grid on spatial domain (Imperial College lecture material)

We can obtain very different results by using different solver to solve the systems. Figure 5.2 (a)

and (b) show the outlet temperature profile over time of a same channel with different solvers.

Figure 5.2 (a) Outlet temperature of a certain channel solved with BFDM solver

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Figure 5.2 (b) Outlet temperature of a certain channel solved with CFDM solver

According to the previous works, it is quite obvious that the simulation result showed by

Figure 5.2 (b) is definitely wrong. Even worse, gPROMS cannot give out any result

when using the FFDM solver. That is why we finally choose the BFDM. Therefore we

can summarize that, for any constructed model, the solution can be affected great by the

solver and one cannot decide which the best is unless trying all of them.

Simulation results summarization

According to the simulation results and the previous work, the following points of PHE

dynamic simulation can be concluded:

1) When a PHE is used for heating cold fluid, if the inlet flow rate of hot stream keeps

constant, the outlet temperature of the cold stream will keep increasing until it reach a

steady state that has very limited temperature gap with the temperature of the hot

stream. Instead, the outlet temperature decreases very fast at the first time, and very

soon, it starts to pick up in a even higher speed until it reach the steady state. These

two distinct behaviours can be seen from the Figure 3.3 (a) and (b).

2) In each channel, the stream’s temperature should keep decreasing along the axial

direction no matter it is a cold or hot stream. Meanwhile, the heat transfer coefficient

should also show a similar distribution with the temperature. This has been proved by

some previous work mentioned in Chapter 2. However, when physical properties of

the fluid are not sensitive to the temperature or in other words they keep almost

constant, the change of heat transfer coefficient will become inconspicuous and

further leads the cold stream temperature showing constant distribution along the

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axial direction. These phenomena can be seen in Figure 3.6 (a) and Figure 3.8 (b) of

Chapter 3.

3) These behaviours can be explained with the theory of energy balance. At the

beginning time, the temperature gap between hot and cold streams is at its biggest

value. Due to the high heat transfer coefficient, the hot stream gives out big amount

of heat to cold stream and leads its temperature decrease very sharply. As the

temperature of cold stream raises, to keep the gap of temperature, hot stream

temperature must raise up again. Because the physical properties of water in

gPROMS are note very sensitive to temperature, the heat transfer coefficient keeps in

high level and the temperature of both hot and cold streams are forced to rise very

fast. This is one reason that explains why the heat transfers so fast in this project

compared with Michael and Sandro’s work (It takes 25000 seconds for them to cold

the milk from 370K to 367K with their PHE).

5.2 Conclusion of Controller Design

1. PID design conclusion

With decades of years’ accumulated experience, PID controllers have already been very well

known by people and packed in many kinds of professional software as completed units.

Model construction is no longer a part of the so-called “PID design“, the only work left is

PID tuning. Usually, the parameters need to be tuned are: proportional gain, integral gain and

derivative gain. But, for gPROMS special formulations of PID, there are 5 parameters need to

be adjusted in this project (see Chapter 4). The most important parameter is the proportional

gain K (named gain in gPROMS). The proportional gain directly determine the how fast the

system can approach to steady state. This point has already been proved by the simulation

results showed in Figure 4.2 (a), (b) and (c).

However, an excessive proportional gain could also bring instability into the system. F igure

4.2 (c) has not only shown a big proportional gain make the system reach the set point fast,

but also shown sudden decrease happens to the output signal under such a big gain. Therefore,

we need to adjust the other four parameters to balance the effectiveness of proportional gain.

With the Integral Performance Criteria, we found the best set of parameters below:

Bias 0.01642078 Rate limit 100

Gain 5.4744306 Reset time 95.75582

Rate 10

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2. MPC control conclusion

Unlike PID controller, MPC is a type of controller which focuses more on the characteristics

of the controlled system itself. Always, the system model is included in the MPC controller

mathematical model as an integral part. Therefore it is hard to make a standard “MPC unit”

like what has been done to PID in gPROMS. Even so, there are still some tools we can use to

simplify the design work. As the work done in Chapter 4, the system identification tool box

and mpc tool box can help us complete the work much more easily.

For our PHE system, it is not an easy work to formulate it as a state-space model or other

forms that MPC can accept. Thus the non- linear continues model we have constructed should

be transferred into state-space model for the use of MPC controller design. That is why we

need system identification tool box. According to the work done in Chapter 4, the key points

in state-space model re-estimation are estimation method and model order. With different

estimation method and order, we got very different re-estimated process model. It is very

obvious that some of them lose convergence (PHE_P2 and PHE_N2c in Figure 4.10). For all

the other models, no judgement can be made until they have been put into mpc design work.

Most of the MPC controller design work is completed in MPC tool box. The design

procedure can be summarized as follows:

1) Define new mpc controller model by introduce new mpc simulink block.

2) Import both controller model and plant model to the mpc design tool box

3) Set up mpc controller correctly and run the simulation until good controller is obtained

And all the key settings are as below:

Horizon setting

Control interval (time units) 1

Prediction horizon (intervals) 30

Control horizon (intervals) 5

Weight tuning

overall 0.8

Input weight (u1)

Weight 1

Rate weight 0.1

Output weight (y1)

Weight 1×108

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When tuning the MPC controller, weight tuning is a key step. It is very common that the

output doesn’t approach to the set level even if the simulation time is already very long. This

happens especially when the coefficient of the input variable is especially small (like the

plant model for this project). To deal this kind of problem, we need to enlarge the output

weight so that the output will become much more sensitive to the input change. That is we

have a relatively large output weight in the table above.

5.3 Comparison between PID and MPC controller

By now, based on the design work of those two distinct controllers, we can compare them in

some aspects as follows.

1. Control flow sheet

PID MPC

Figure 5.3 PID & MPC control process comparison

Figure 5.3 shows the basic flow sheet of PID and MPC’s control action. From the figure, we

can see that the improvement made by a PID controller is definitely based on the measured

system output or the “error”. For the whole process, the control action has nothing to do with

the pant model. Therefore, it is much easier for it to make the system instable o r overflow.

Instead, for a MPC controller, since the control action is calculated from the current state by

solving the optimization problem in which the plant model is included as constraints, it will

grantee the system run within permitted boundary and keep the system stable. Even further,

the MPC controller can generate control action which also makes the system approach to

Collect system output

Set point- Measured data = error

Send the error to controller to

calculate the control action

Send the control action to the plant to

generate new output approaching to

set point closer

Collect system state

Send the state vector to

controller to calculate the

control action sequence by

solving constrained

optimization problem

Take the first of the control sequence as

the control action and send it to the

plant to generate optimal output

55

certain optimal state very easily. The only thing need to do is reformulate the objective

function. But for a PID controller, this kind of mission means an extra work on re-computing

the set point which could be very time-consuming.

2. Work results comparison

Manipulated variable

Figure 5.4 (a) MPC control system input

Figure 5.4 (b) PID control system input

-5

0

5

10

15

20x 10

9

u1

0 50 100 150 200 250 300 350 400 450 500-1

-0.5

0

0.5

1

v@y1

Plant Inputs

Time (sec)

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In this system, the only one manipulated variable (or system input) is the hot stream flow

rate. Figure 5.4 shows the input of the system with PID and MPC controllers

respectively. For the system with PID, we can see that the input signal keeps a decreasing

trend for most of the time. But for the MPC controller, the input variable shows very

significant fluctuation even under the non-disturbance condition. This means that when

the controller is added to the plant, every reset on the system could bring very significant

instability. On the other hand, the wave amplitude reaches a magnitude order of 1012

therefore an extra scaling in needed to eliminate its effect. Thus, for the moment we can

conclude that PID controller did a better job than MPC on adjusting the input signal.

System Output

Figure 4.12 and 4.4(b) gives out the simulation results of system with MPC and PID

controllers respectively. Those simulation results show, for the same PHE plant, it takes

about 37 seconds with PID controller and 400 seconds with MPC controller to make the

system reach the expected steady state. And similarly with manipulated variable, the

output signal of the system with MPC oscillated to a certain extend before the system

settle down, but not very serious. Thus it seems that PID again shows a better behaviour.

Since the MPC has not been added into the system yet, all the comparison are just

preliminary judging. Further comparison work on the system work results will not be

performed until the PHE-MPC system simulation is successfully run in simulink.

Time price comparison

As mentioned before, due to different properties, PID and MPC have different

complexities and the time required for the design work also varied a lot from each other.

For this research work has been done, with the help of gPROMS, PID design work only

costs around one week time and the time is spent mainly on the non- linear optimization

computation. Instead, the MPC control work needs more than three weeks to complete

all the design work, including system identification, MPC tuning and Simulink data

preparation.

In all, we can summarize that at this stage, PID has done a better job than MPC controller in

all aspects. However, since the completed MPC control system (PHE system with MPC

controller flow sheet in Simulink) has not been simulated, a final judgement is impossible to

make and this will be discussed in future work later.

5. 3 Future Work

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5. 3. 1 MPC control system simulation in Simulink environment

Due to some unknown reason, gOSimulink failed to run correctly which leads the MPC

control system cannot be simulated in Simulink. Even worse, the completed work results of

the MPC controller fail to be obtained. Therefore, finding out the causes that has disabled

gOSimulink and going on the simulation are necessary in the future.

1. Simulation under non-disturbance condition

The MPC’s performance under non-disturbance condition is the controller’s basic prosperity

which mainly affects the system’s work results. Set both measured and unmeasured

disturbance to zero, the test can be easily done by simply running the simulink simulation.

Firstly, we should observe the system output and input behaviour without enforced

constraints. By doing this, the controller’s most actual behaviour is lay before us. According

to the performance of single MPC tuning showed in Figure 4.12, both the system input and

output signal could oscillate greatly. What can be done then to stabilize the plant is reset the

controller weight and enlarge the input scaling. Instead, if the system is very insensitivity,

lowering the scaling level of the input signal should be taken into account. Otherwise, we

should replace the plant model with other better ones.

Secondly, the constraints should be added to the controller. This is very likely to cause the

controller stop working for the strict optimal solution cannot found in the feasible region. To

solve this problem, the co