parametric and nonparametric identification of

PARAMETRIC AND NONPARAMETRIC IDENTIFICATION OF SHELL

AND TUBE HEAT EXCHANGER MATHEMATICAL MODEL

TATANG MULYANA

A Thesis submitted in

Fulfillment of the requirement for the award of the Degree of

Philosophy of Doctoral in Electrical Engineering

Faculty of Electrical and Electronic Engineering

Universiti Tun Hussein Onn Malaysia

AUGUST 2014

v

ABSTRACT

Parametric and nonparametric models of a shell and tube heat exchanger are studied.

Such models are very important because they provide information about controlling a

system operation. Without the model, the control task would be difficult for tuning of

controller. For many years, researchers have studied these models; however, their

models are still less satisfactory since they are not in general form. This problem is

caused by two key issues, namely, multiple unknown parameters and highly

nonlinear structures. Energy balances have been set-up for condition of unknown

parameters which involved, among others, temperature, flow rate, density and heat

capacity. The identification process produces a dynamic model of the heat exchanger

which is developed based on a lumped parameter system. The model developed is

single input single output whereas input signal is hot water flow rate and the output is

cold water temperature. The general form of the model obtained could have

parametric model structures such as auto regressive with external input, average auto

regressive moves with external input, output error or box-jenkins. The study in this

thesis aims to solve the general form through parametric and nonparametric models

which has been proposed as candidate models. Both candidate models have been

implemented and tested by applying several data sets constructed in lab experiments.

The first finding is the derivation of the dynamic model in the general form of the

transfer function in s domain, and it has been proven that it has parametric model

structure. The second finding is the first order without delay time transfer function of

the nonparametric model where they have gain is 35.20C and time constant 7200s.

These have proven to fulfill that the measured experimental data contains calculated

error that is no than more 2%. The third finding is the parametric model obtained has

proven that the measured experimental data contains calculated error level that is

very satisfactory, i.e. less than 1%. This error has been determined based on the final

prediction error for each model structure used. The best model has been chosen, i.e.

bj31131. It has the smallest values of the loss function and final prediction error of

0.0023, and it has high values of the best fits, i.e. 96.84%. Parameter optimization

has been calculated to determine minimization or maximization of functions which

involved the parameter studied. It is used to find a set of design parameters that can

in some way be defined as optimal. The first until the third findings are minor

contribution while the parameter optimization has been a major contribution.

vi

ABSTRAK

Model parametrik dan bukan parametrik sebuah penukar haba sel dan tiub telah dikaji.

Model seperti ini amatlah sangat penting kerana ia menyediakan maklumat berkenaan

dengan kawalan sebuah operasi system. Tanpa model ini, tugas kawalan akan menjadi sukar

bagi penalaan pengawal. Selama bertahun-tahun, ramai penyelidik telah membuat kajian

kepada model-model ini; walau bagaimanapun, model-model yang mereka perolehi masih

kurang memuaskan kerana ianya tidak dalam bentuk am. Masalah ini wujud disebabkan oleh

dua isu utama, iaitu pelbagai parameter tidak diketahui dan struktur yang sangat tak linear.

Kesetimbangan tenaga telah ditubuhkan bagi keadaan pelbagai parameter tidak diketahui

yang melibatkan, antara lain, seperti suhu, kadar aliran jisim, ketumpatan dan kapasiti haba.

Proses pengenalpastian telah menghasilkan sebuah model dinamik penukar haba yang

dibangunkan berdasarkan kepada sistem parameter tergumpal. Model yang telah

dibangunkan ini mempunyai masukan dan keluaran tunggal dimana sebagai isyarat masukan

adalah kadar aliran air panas dan sebagai isyarat keluaran adalah suhu air sejuk. Bentuk am

daripada model yang diperolehi ternyata dapat dijadikan sehingga ianya mempunyai struktur

model parametrik seperti auto regresif dengan masukan luar, purata auto bergerak regresif

dengan masukan luar, ralat keluaran atau box-jenkins. Kajian didalam tesis ini bertujuan

untuk menyelesaikan bentuk am tersebut melalui model parametrik dan bukan parametrik

yang telah dicadangkan sebagai model calon. Kedua-dua model calon ini telah dilaksana dan

diuji dengan menggunakan tiga buah set data dalam percubaan makmal. Sebagai dapatan

pertama adalah model dinamik terhasil dalam bentuk am fungsi pindah dalam domain s, dan

ianya telah terbukti mempunyai struktur model parametrik. Sebagai dapatan kedua adalah

model nonparametrik dalam bentuk fungsi rangkap pindah perintah pertama tanpa kelewatan

masa dimana ianya mempunyai gain sebesar 35.20C dan konstanta masa sebesar 7200 detik.

Model ini telah terbukti memenuhi data percubaan dan mengandungi ralat kiraan yang

besarnya tidak melebihi daripada 2%. Sebagai dapatan ketiga adalah model parametrik yang

mana ianya telah terbukti memenuhi data percubaan dengan ralat yang mempunyai tahap

yang sangat memuaskan, iaitu kurang daripada 1%. Ralat ini telah ditentukan berdasarkan

ralat ramalan akhir bagi setiap struktur model yang digunakan. Model terbaik yang terpilih,

iaitu bj31131, yang mana ianya mempunyai nilai bahagian kerugian dan ralat ramalan akhir

yang paling kecil, iaitu 0.0023, dan juga mempunyai nilai sawan terbaik yang paling tinggi,

iaitu 96.84%. Pengoptimuman parameter telah dikira untuk menentukan minimum atau

maksimum fungsi-fungsi yang melibatkan parameter yang dikaji. Ianya digunakan untuk

mencari satu set parameter rekabentuk yang dalam beberapa cara boleh ditakrifkan sebagai

optimum. Penemuan yang pertama sehingga yang ketiga merupakan sumbangan kecil

manakala pengoptimuman parameter merupakan sumbangan yang besar.

vii

TABLE OF CONTENTS

TITLE i

DECLARATION ii

DEDICATION iii

ACKNOWLEDGEMENT iv

ABSTRACT v

ABSTRAK vi

TABLE OF CONTENTS vii

LIST OF TABLES ix

LILST OF FIGURES x

LIST OF ABBREVATIONS xiii

CHAPTER 1 INTRODUCTION 1

1.1 Research Background 1

1.2 Problem Statement 2

1.3 Benefit of the Study 3

1.4 Objectives 4

1.5 Scope of the Study 4

1.6 Research Methodology 6

1.7 The Contributions of the Study 6

1.8 Thesis Outline 9

CHAPTER 2 LITERATURE REVIEW 11

2.1 Introduction 11

2.2 Mathemtical Modeling Approach 11

2.3 Nonparametric Identification 13

2.4 Parametric Identification 15

2.5 Summary 16

viii

CHAPTER 3 PROCESS DESCRIPTION AND CANDIDATE MODEL 18

3.1 Introduction 18

3.2 Classification of Heat Exchanger 19

3.3 Process Description 23

3.4 Dynamic Model 25

3.5 Nonparametric Model 39

3.6 Parametric Model 44

3.7 Summary 48

CHAPTER 4 SYSTEM IDENTIFICATION AND VALIDATION THEORY 49

4.1 Introduction 49

4.2 Parametric System Identification 50

4.3 Nonarametric System Identification 53

4.4 Validation Theory 58

4.5 Parameter Optimization 87

4.5 Summary 104

CHAPTER 5 DATA COLLECTION AND PROCESSING 106

5.1 Introduction 106

5.2 Plant Description 106

5.3 Data Collection and Processing 113

5.4 Summary 117

CHAPTER 6 RESUL AND ANALYSIS 118


6.2 Nonparametric Model 118

6.3 Parametric Model 125

6.4 Parameter Optimization 139

6.4 Summary 142

CHAPTER 7 CONCLUSION AND RECOMMENDATION 144


7.2 Significance of Findings 145

7.3 Recommendation 147

REFERENCES 148

ix

LIST OF TABLES

Table 2.1: The nonparametric models of shell and tube heat exchanger 13

Table 3.1: Coefficients of Pade-approximations of order and 41

Table 3.2: Type of step response for various value of zeta 43

Table 3.3: Some common parametric models 47

Table 4.1: PEM model structures most frequently used 99

Table 5.1: The condition of data experimental shell and tube heat exchanger 119

Table 6.1: Error calculation nonparametric model and data measured dataexp1 123



Table 6.4: The comparison results of nonparametric models 128

Table 6.5: Estimation model and validation criteria dataexp1 130

Table 6.6: Estimation model and equation dataexp1 131





Table 6.11: The comparison results of parametric best model 143

Table 6.12: The results of assessment measure as parameter optimization 144

Table 6.13: The dataexp2 of assessment measure as parameter optimization 145

Table 6.14: The dataexp3 of assessment measure as parameter optimization 145

Table 6.15: The result of assessment measure as parameter optimization 146

Table 6.16: The heat and disturnace transfer function data experimental 146

Table 6.17: Results of parameter optimization data experimental 147

x

LIST OF FIGURES

Figure 3.1 Classification heat exchanger according to process function 20

Figure 3.2 Classification heat exchanger 21

Figure 3.3 Classification heat exchanger according to construction 22

Figure 3.4 Relationships input and output shell and tube heat exchanger 24

Figure 3.5 Shell and tube heat exchanger construction 24

Figure 3.6 Countercurrent shell and tube heat exchanger schematics 28

Figure 3.7 Divided the length heat exchanger into N equal size regions 28

Figure 3.8 Shell and tube temperature are constant at the center region n 29

Figure 3.9 Step response first order transfer function without time delay 39

Figure 3.10 Step response first order transfer function with time delay 40

Figure 3.11 Step and phases responses first order Pade-approximation 42

Figure 3.12 Step and phases responses second order Pade-approximation 42

Figure 3.13 Block diagram parametric model structure 44

Figure 3.14 Linear time invariant (LTI) system subject to disturbance 46

Figure 3.15 Generalized model structure LTI systems 47

Figure 4.1 Block dynamic system schematic 50

Figure 4.2 Schematic flowcharts parametric system identification 52

Figure 4.3 Step response first order system with time delay 54

Figure 4.4 Step response damped oscillator 55

Figure 4.5 Overshoot versus relative damping ζ damped oscillator 56

Figure 4.6 Determination parameter damped oscillator from the tep response 57

Figure 4.7 Assessment criterion 66

Figure 4.8 Block diagram prediction error method 75

Figure 4.9 Schematic representation of the assumed system structure 90

Figure 4.10 Schematic representation of the prediction error 95

Figure 5.1 the real plant of the Heat Exchanger QAD Model BDT 921 110

xi

Figure 5.2 the boiler drum Tank T11 filled with the hot water 111

Figure 5.3 the Tank T12 filled with the preheat water 111

Figure 5.4 the Tank T13 filled with the cold water 112

Figure 5.5 the power supply controller 112

Figure 5.6 the transmitter displayed of TIC11 and LIC/FIC11 113

Figure 5.7 the recorder LFTR11 that used to record data experiments 113

Figure 5.8 the pump P11 and P14 respectively 114

Figure 5.9 the gauge TG11 and TG12 respectively 114

Figure 5.10 the schematic of the Heat Exchanger QAD Model BDT 921 115

Figure 5.11 the paper data sample from the recorder 117

Figure 5.12 the data transferred in the form excel 118

Figure 5.13 the input and output signals dataexp1 119



Figure 6.1 Nonparametric model output and data measured dataexp1 122

Figure 6.2 Error calculation nonparametric model dataexp1 123





Figure 6.7 Measured and simulated parametric model output cold water

temperature outlet dataexp1 estimation 129


temperature outlet dataexp1 validation 130

Figure 6.9 Measured and simulated parametric models output cold water

temperature outlet dataexp1 measured 131

Figure 6.10 Autocorrelation and cross correlation parametric models dataexp1 134






temperature outlet dataexp2 137


xii






temperature outlet dataexp3 141


xiii

LIST OF ABBREVIATIONS

ARMAX

- auto regressive moving average with exogenous input

ARX

- auto regressive with exogenous input

BIBO

- bounded input bounded output

BJ

- box-jenkins

CFD

- computational fluid dynamics

CUSUM

- cumulative sum

D

- drain

DAE

- differential and algebraic equations

EE

- equation error

FE

- flow emitter

FOPDT

- first order plus dead time

FPDD

- first-principle data-driven

FPE

- final prediction error

FT

- flow transducer

HTC

- heat transfer coefficients

I

- integral

IMC

- internal model control

LCV

- level control valve

LF

- loss function

LHP

- left half plane

LT

- level transmitter

MCHE

- main cryogenic heat exchanger

MDF

- matrix fraction description

MIMO

- multi input multi output

MINLP

- mixed integer non-linear programming

xiv

MSE

- mean square error

ODE

- ordinary differential equations

OE

- output error

OELS

- output error least square

P

- proportional

P&ID

- piping and instrumentation drawing

PB

- proportional band

PDE

- partial differential equations

PEM

- prediction error method

PG

- pressure gauge

PIA

- pressure indicating alarm

PID

- proportional-integral-derivative

PRBS

- pseudo random binary sequence

PT

- pressure transmitter

RTD

- resistance temperature detector

SRIVM

- simplified refined instrumental variable method

T

- tank

TE

- thermocouple element

TEMA

- tubular exchanger manufacturers association

TG

- temperature gauge

TIC

- temperature indicating controller

TS

- temperature sensor

TT

- temperature transducer

TIT

- temperature indicating transducer

1

CHAPTER 1 Introduction

CHAPTER 1

INTRODUCTION

1.1 Research Background

Introducing a mathematical model related to represent the actual system dynamics is very

important to science and technology. Such models can be useful e.g. for simulation and

prediction or for designation of digital control systems. Many industrial processes must be

controlled in order to be run safely and efficiently. To design regulators, some type of model

for the process is needed. The models can be of various types and degrees of sophistication.

Sometimes it is sufficient to know the crossover frequency and the phase margin in a Bode

plot. In other cases, such as the designation of an optimal controller, the designer will need a

much more detailed model that can also describe the properties of the disturbances acting on

the process.

In most application of signal processing in forecasting, the recorded data are filtered

in some ways. In addition, a good design of the filter should reflect the properties of the

signal. To describe such spectral properties, a model of the signal is needed. In many cases,

the primary aim of modeling is to assist in the designing process. In other cases, the

knowledge of a model can be the purpose itself. If the models can explain measured data

satisfactorily then they might also be used to explain and understand the observed

phenomena. However, sometimes this model is not easy to get especially when global

representation system is required. Therefore, in order to solve the difficulties to get the

dynamics model of process system, one can use system identification. The system

2

identification is a technique to estimate the mathematical models of system dynamics based

on data observed from the system (Knudsen, 2004; Ljung, 2011).

There are many usage of system identification. Some of those are for simulations and

predictions for control and diagnosis in many fields such as engineering, economics,

medicine, physiology, and geophysics. Therefore, it is rational to ask questions of where the

model came from. There are currently three methods of system identification: white-box,

gray-box, and black box (Aarts, 2012; Ljung, 2011). White box is an identification performed

without the use of test data, based on the main principles only. Gray box is used both of prior

knowledge process and experimental data for identification. Black box is an identification

made exclusively from test data (Soderstrom, 2001).

Many researchers such as Nithya (2007), Sharma (2011), and Sivakumar (2013) are

interested in identifying the dynamic model of heat exchanger. Modeling and control

designing are not an easy task to do but very important because they are used in many

industrial processes. Heat exchangers are used in the process of heating, cooling, and

economizing processes. Heat exchanger is commonly in shell and tube type.

In this thesis, a parametric and nonparametric identification approach has been

implemented to the shell and tube heat exchanger to produce the parametric and

nonparametric models based on the few set of experimental data. In general, a parametric

method can be characterized as a mapping from the recorded data to the estimated parameter

vector while the nonparametric method is characterized by the property that the resulting

model is a curve or function, which is not necessarily parameterized by a finite dimensional

parameter vector (Soderstrom, 2001).

1.2 Problem Statement

The need for models of dynamic systems frequently arises in several engineering disciplines,

for example in communication and control systems design. The emphasis in this work will be

on parametric and nonparametric identification methods. For many years, researchers have

studied about models of shell and tube heat exchanger. Their models have various

coefficients. These facts can be seen in the Table 2.1. The models shown in this table are the

models in which they are classified as the nonparametric model in first order system with

time delay.

3

The disadvantage of this nonparametric model is the fact that it is not including offset.

The offset means time needed to reach a heat exchanger temperature operation where the

starting of the real heat exchanger operation is room temperature water at approximately

. While the starting of the nonparametric model equals to . Alternatively, one can

handle the offset problem by determining the parametric model. The main problem is, to

determine the parametric model; there is a need to prove that the dynamic model has certain

structure such as auto regressive with external input (ARX), average auto regressive moves

with external input (ARMAX), output error (OE) or box-jenkins (BJ) structures. Therefore,

the next step is to find a model that has best fits above 90% with the smallest values of loss

function and final prediction error (FPE).

The best model of the parametric model depends on the order of polynomial equation.

By choosing high order, it will result in highly nonlinear structure and increase possibility of

best fits. But the resulted model is not practical because it will make the control task difficult.

Therefore one must find a model that has the highest best fits; however, at the same time, it

should not have highly nonlinear structure. In this case, the third order system is sufficiently

high for the model system which is applied in the control design.

1.3 Benefit of the Study

In industrial process, heat exchanger is designed to transfer heat from one fluid to another. It

has many different applications, especially in chemical process, air conditioning, and

refrigeration. Since heat exchanger has a wide variety of applications and is commonly used

in industry, control of the system is essential. A dynamic model may be created to allow the

chemical engineer to optimize and control the heat exchanger.

By utilizing this model, predictions can be made to analyze how altering the

independent variables of the system can change the outputs. There are many independent

variables and considerations to account for in the model. If it is done correctly, accurate

predictions can be made. Today, process engineers are responsible for many project

activities, including conceptual design, revamp studies, and operational troubleshooting.

Increasingly, the process simulator is an essential tool and become the central to these

activities.

4

Process simulators are some very powerful tools for modeling all or parts of a

process. While they are excellent for general purpose process modeling, it is the process

engineer’s responsibility to understand to what extent these tools can be applied, and how

combining their application with more specialized tools might be appropriate. This choice is

ultimately based on the business and technical objectives that want to be achieved. The heat

exchanger models can enhance value derived from process simulation and provide more

accurate results. These applications include conceptual designs of new plants, revamps of

existing facilities, and operations support.

1.4 Objectives

These research objectives consist of:

1. To apply a parametric and nonparametric identification methods of shell and tube heat

exchanger;

2. To collect and process a set data of the shell and tube heat exchanger;

3. To identify the parametric and nonparametric models of the shell and tube heat

exchanger;

4. To validate the parametric and nonparametric models of the shell and tube heat

exchanger.

1.5 Scope of the Study

The scopes of this study are:

1. A series of input-output datasets of the shell and tube heat exchanger are collected in

experimental way. Three experimental datasets will be collected. The input signal is

hot water flow rate inlet from tube side and the output signal is cold water

temperature from shell side. These signals was measured using flow indicator to

measure the hot water flow rate in and temperature indicating transmitter in

. The data was recorded in a graph paper at sampling time used of . The

number of samples is 1000, so the duration times of each experimental data equals to

5

7200s. The real plant to get the data is Heat Exchanger QAD Model BDT 921 which

has been installed in the Control Laboratory at the Universiti Tun Hussein Onn

Malaysia. The data in the graph paper was then imported to workspace in MATLAB

software and then displayed in the graph form. The details about plant description and

how to undertake the data collection and processing can be seen in the Chapter 5.

2. The nonparametric identified method is applied to all data sampling collection to

generate a nonparametric model. This method is determined based on the first-order

plus dead time (FOPDT). Two models can be chosen namely, the first order transfer

function without or with time delay, and the first or second order Pade-approximation.

The best nonparametric model is chosen based on percentage of the error calculation

between the model output and the data measured. The detailed about the candidate of

the nonparametric model can be seen in the Chapter 3, especially in the Section 3.5.

Furthermore, how the nonparametric identified method is applied can be seen in the

Chapter 4, especially in the Section 4.3. The m-file program in the MATLAB

software can be used to obtain the best nonparametric model.

3. The parametric identified method is applied based on a linear structure such as ARX,

ARMAX, BJ, or OE. Before this method is implemented, one important thing is a

need to prove that the dynamic model has parametric model structure. The dynamic

model is formulated in an equation which has highly nonlinear structure and has many

unknown parameters involved. This equation is generated from a description of the

shell and tube heat exchanger process where it is assumed as a lumped parameter

system. In this case, it is difficult to find a solution of the equation using numerical

methods since many parameters involved are unknown values. The details to prove

that the dynamic model has a linear structure can be seen in the Chapter 3, especially

in the Section 3.4. The flowchart of the parametric identified method can be seen in

the Chapter 4 (in the Section 4.2). The data applied in this method is required to be

divided into two parts. The first half part is used to obtain the parameter estimation

and the second half part is used to obtain the model validation.

4. The parameter estimation is determined through the order of polynomial equation in

the parametric model structure. The polynomial equation of each parametric model

structure includes ARX, ARMAX, BJ, or OE model structures can be seen in the

Chapter 3 (Section 3.6). The validation parametric model is determined based on the

loss of function (loss function), a final prediction error (FPE), and the appropriate

percentage of fitting (best fits). The details about the validation theory can be seen in

6

the Chapter 4, especially in the Section 4.4. The m-file program or the toolbox in the

MATLAB software can be used to produce the parameter estimation and the

validation parametric model.

1.6 Research Methodology

The main goal in this thesis is to obtain parametric and nonparametric of shell and tube heat

exchanger mathematical model. The model is produced by solving the dynamic model

equations that has used parametric and nonparametric identification methods. One way to

determine this model is based on the experimental data with implemented parametric and

nonparametric identification methods. Based on the experimental data, it can be concluded

that the nonparametric model is first order with time delay. In using parametric identification

method, one needs to prove that the equations have a certain structure such as ARX,

ARMAX, OE or BJ. In this thesis, it will be proven that the equation can have a parametric

model structure.

The study in this thesis is aiming to solve the problem by proposing the parametric

and nonparametric identification. Of especial importance, the thesis mainly conducts

experiments to find the parametric and nonparametric models of the shell and tube heat

exchanger. The nonparametric model has been found through first-order and second-order

systems. Whilst the parametric model is developed with model structure, and it must be

ARX, ARMAX, BJ, or OE. Based on these structures, one then needs to find the best model.

The criteria to find the best model is based on the corresponding loss functions, final

prediction error (FPE), and a percentage of fitting the model error (best fits) using MATLAB.

1.7 The Contributions of the Study

Thesis contributions are grouped into two parts, i.e. minor and major contributions as

described herewith:

1. First contribution of the minor contribution is the derivation of dynamic model of

shell and tube heat exchanger. The model is derived based on a lumped parameter

7

system where it could be more desirable and structured. The heat exchangers are

divided into five regions of equal size in length with an assumption that, inside each

element, shell and tube temperatures are uniform. For each section, energy balances

can be set-up for shell and tube sides where the sides for inlet and outlet conditions

involving temperature, mass flow rate, density and heat capacity. For each section, the

set equations are given by Flores (2002). These equations can be seen in the Section

3.4 Chapter 3 equations (3.1) through (3.32). In this thesis, the set equations are

developed to obtain a general form of a counter current shell and tube heat exchanger

model in s domain. The general form is a single input single output (SISO) where the

input variable is hot water flow rate inlet in tube side heat exchanger and the output

variable is cold water temperature outlet in shell side heat exchanger. The general

form obtained has been proven that it has a parametric model structure. These

equations can be seen in equations (3.49) through (3.61). The finding of general form

obtained to complete the study of the heat exchanger dynamic model. Therefore, this

finding is the first contribution of this thesis.

2. Second contribution of the minor contribution is the nonparametric model obtained to

solve the complicated equation of the dynamic model that has been derived. It has

been previously verified that the equation is a nonparametric model. The verification

can be seen in the equations (3.42) up to (3.48). The first order transfer function in the

s domain is proposed as candidate model as it is proven to fulfill temperature output

response of the heat exchanger experimental data, which has given satisfactory

tolerable error. The candidate of the nonparametric model is shown in equations

(3.62) up to (3.73) based on the experimental data. They are called dataexp1,

dataexp2 and dataexp3 which have nonparametric model characteristics. These results

can be seen in equation (6.3) for dataexp1, equation (6.4) for dataexp2, and equation

(6.5) for dataexp3 in Chapter 6. The first order transfer function of the nonparametric

model has proven to fulfill the measured experimental data, whilst its estimated error

is decreasing to a zero level after 2600s. This result is the second contribution of this

thesis to complete the study of the heat exchanger dynamic model. It is substantially

different from the results obtained by many researchers as demonstrated in Table 2.1

in the Literature Review in Chapter 2.

3. The third contribution of the minor contribution is the parametric model obtained to

solve for the complicated equation of the dynamic model. It is derived to be simpler

and more structured as shown in equations (3.49) through (3.61) in Section 3.4 of

8

Chapter 3. Equation (3.49) is ARX model structure, equation (3.53) is ARMAX

model structure, whilst equation (3.57) has OE model structure. Furthermore,

equation (3.60) is BJ model structure. The candidate of the parametric model such as

ARX, ARMAX, OE, and BJ, has been proven to fulfill the measured experimental

data, and it has satisfactorily met certain criterion for parameter estimation and model

validation. This criterion has been determined based on the best fits, loss function and

FPE values for each model structure used. The best fits of all experimental data is

more than 90%, the loss function and FPE are less than 0.0400. The resulted

parametric model produces residuals that are within the confidence interval. The best

model chosen is bj31131, it has the smallest value of loss function and FPE, i.e.

0.0023 and high best fits of 96.84%. Lastly, it has also been proven by parameter

optimization through the calculation of an assessment of .

4. Parameter optimization which has been as major contribution is determined to use

minimization or maximization of functions which involved the parameter studied. It is

used to find a set of design parameters that can in some way be defined as optimal. In

a simple case this may be the minimization or maximization of some system

characteristic that is dependent on its parameters. In a more advanced formulation the

objective function to be minimized or maximized, may be subject to constraints in the

form of equality constraints, inequality constraints and/or parameter bounds. An

efficient and accurate solution to this problem is not only dependent on the size of the

problem in terms of the number of constraints and design variables but also on

characteristics of the objective function and constraints. This optimization is used to

find the best or optimal solution to a problem. Steps involved in formulating an

optimization problem: conversion of the problem into a mathematical model that

abstracts all the essential elements, choosing a suitable optimization method for the

problem, and obtaining the optimum solution. Unconstrained optimization has been

used in this thesis. It finds a vector that is a local maximum or minimum to a scalar

function. The term unconstrained means that no restriction is placed on the range of

the vector. Good algorithms exist for solving this optimization problem; such

algorithms typically involve the computation of a Full Eigen System and a Newton

process applied to the secular equation. Such algorithms provide an accurate solution.

However, it requires time proportional to several factorizations. Therefore, for large-

scale problems a different approach is needed. Several approximation and heuristic

strategies have been proposed in the literature. The maximum of percentages of fitting

9

and minimum of a final prediction error have indicated the best model for the

previous models. It gives consistent parameter estimates only under rather

restrictive conditions. Two different ways are given by modifying the method so that

a consistent estimate could be obtained under less restrictive conditions. The

modifications are minimization of the prediction error for other more detailed model

structures and modification of the normal equations associated with the least squares

estimate. A scalar measure has been used to assess the goodness of the model

associated with the true parameter. It is denoted by assessment measure, where the

dependence on the number of data is emphasized. The assessment measure must be

minimized by this parameter. To determine it, Equations (4.96) and (4.98) in the

Chapter 4 could be used. Both equations constitute the relationship between the loss

function and the FPE through the assessment measure. Its simplified form is given as

AIC is Akaike’s information criterion. The result of the best parametric model could

be used to calculate this assessment, and it is given in Table 6.15. It can be seen from

this table that the minimum AIC is -2630.2722. Based on this result, it can be

concluded that the best model which has parameter optimization is bj31131. This

model has the smallest loss function value of 0.0023 and FPE of 0.0023. In addition to

that, it has the highest best fits value of 96.84%. The other method to determine the

parameter optimization of the heat and disturbance transfer functions data

experimental which it is shown in Table 6.16 used the Equation (4.174) and (4.175) as

introduced in the Chapter 4. The results are presented in Table 6.17. From these

results could be observed that the best heat transfer function which it is reached is

237.5094 by dataexp3 bj31131. While the disturbance transfer function for all data

experimental is very satisfying, i.e. zero.

1.8 Thesis Outline

Chapter 1 consists of research background and problem statement in which the reflection of

heat exchanger model is introduced. This model is very important; it provides information

about the nature and the characteristics of the heat exchanger system that is crucial for the

investigation and forecast operational system. The objectives, scopes, benefit and

contribution of this study are also elaborated here.

10

Chapter 2 consists of introduction, mathematical modeling and system identification,

nonparametric identification, parametric identification, and summary. The difference between

mathematical modeling and system identification will be reviewed related to this study. The

system identification method includes parametric and nonparametric are reviewed from many

papers to provide information and to gain the knowledge related to this study.

Chapter 3 consists of process description and candidate model of shell and tube heat

exchanger. It includes the introduction, classification of heat exchangers, process description,

dynamic model, nonparametric model, parametric model, and summary.

Chapter 4 consists of system identification and validation theory. It includes the

introduction, system identification, validation theory, and summary.

Chapter 5 consists of data collection and processing data of heat exchanger. It

includes the introduction, plant description, data collection, processing data, and summary.

The counter current shell and tube used in this study is heat exchanger of QAD Model BDT

921.

Chapter 6 is system identification result and analysis. It includes the introduction,

result, analysis and summary.

The last chapter is the conclusion and recommendation.

11

CHAPTER 2

CHAPTER 2

LITERATURE REVIEW

2.1 Introduction

This chapter consists of literature review related to mathematical modeling and system

identification. The difference between the mathematical modeling approaches will be

presented in Section 2.2. The system identification method, including the parametric and

nonparametric models, is reviewed from many papers to provide information and to gain the

knowledge related to this study. The nonparametric identification and parametric

identification methods as the implementation to shell and tube heat exchanger from many

researchers on journals and proceedings papers can be found in Sections 2.3 and 2.4. The last

section is the summary of this chapter.

2.2 Mathematical Modeling Approach

Basically, there exist two types of applied knowledge or information in describing a system

in terms of a mathematical model. The first information is the experience that experts have

built, including the literature on the topic, and also the laws of physics. The other type of

information is the system itself. Observations from the system and experiments on the system

are the foundation for the description of the system and its properties. In principle, there are

also two different approaches in constructing a mathematical model of a system. The first is

12

to use the well-established relationships, e.g. physical laws that can be applied to the system.

The second approach is to apply observations from the system, and further adjust the

properties of the model to the system. The first approach is often referred to as a deterministic

approach or white box; while the second is often referred to as a stochastic or black box.

When the two approaches are used in a complimentary way, the approach is often referred to

as a grey box. According to Bohlin (1995) the following distinction can be made for different

types of models:

White box: The system identification is performed without the use of experimental

data, e.g. based only on first principles.

Grey box: Both, a priori process knowledge and experimental data are used for

identification, e.g. only a subset of parameters is estimated from experimental data.

Black box: The system identification is performed exclusively from experimental

data.

In the general sense, modeling is used in many branches of science as an aid to

describe and understand the reality. Sometimes it is interesting to model a technical system

that actually is not in existence, but it might be constructed at some time in the future.

Essentially, therefore, the purpose of modeling is to gain insight knowledge of the dynamic

behavior of the system. Mathematical models of dynamic systems are useful in many areas

and applications. Basically, there are two ways of constructing mathematical models i.e.

mathematical modeling and system identification (Soderstrom, 2001).

A mathematical modeling is an analytical approach. Basic laws from physics (such as

Newton’s laws and balance equations) are used to describe the dynamic behavior of a

phenomenon or a process. On the other hand, a system identification is an experimental

approach. Some experiments are performed on the system; a model is then fitted to the

recorded data by assigning suitable numerical values to its parameters. A comparison can be

made of the two modeling approaches: mathematical modeling and system identification. In

many cases the plant processes are so complex that it is not possible to obtain reasonable

models using only physical insight (using first principles, e.g. balance equations). In such

cases one is forced to use system identification techniques. It often happens that a model that

is based on physical insight contains a number of known parameters even if the structure is

derived from physical laws. Identification methods can be applied to estimate the unknown

parameters.

13

Based on Soderstrom (2001), techniques used in a system identification can divided

into two distinct methods, i.e. parametric and nonparametric methods. The parametric method

is built through a model structure determination which is described by a set of parameters

while the nonparametric method is determined based on an output system response in a

function or a graph forms. In general, a parametric method can be characterized as a mapping

from the recorded data to the estimated parameter vector. A typical example of the

nonparametric method is transient analysis where the input can be a recorded step signal and

step response. This response will by itself give certain characteristics (dominating time

constant, damping factor, static gain, etc.) of a process. Nonparametric techniques are often

sensitive to noise and do not give very accurate results. However, as they are easy to apply

they often become useful means of deriving preliminary or crude models.

2.3 Nonparametric Identification

A nonparametric system identification of shell and tube heat exchanger has been developed

by Belinda (2010). The model is taken from some literatures and the plant transfer function of

this model was determined by using first-order plus dead time (FOPDT) model by assuming

that the heat exchanger is well insulated.

Several papers in journals and conferences have undertaken corresponding study

relating to nonparametric identification of shell and tube heat exchanger. Among others are

Dougherty (2003), Nachtwey (2006), Nithya (2007), Srinivasan (2008), Erkan (2009),

Kokate (2009), Ding (2010), Habobi (2010), Khare (2010), Rajasekaran (2010), Pandhee

(2011), Sharma (2011), Venkatesan (2012), Dhakad (2013), Kishore (2013), Saranya (2013),

and Vasickaninova (2013). The dynamics of the shell and tube heat exchanger process in the

nonparametric identification are described by first order plus dead time (FOPDT) model.

Table 2.1 shows the results of nonparametric models of shell and tube heat exchanger based

on FOPDT model.

The FOPDT model parameters are found from the experimental data. The data is used

to identify the three parameters of the models namely, process gain, dead time, and time

constant. Therefore FOPDT model approximation actually does not capture all the features of

higher order processes. Specifically, process gain indicates the size and direction of the

process variable response to a control direction, time constant describes the speed of the

response, and dead time indicates the delay prior to the response begins.

14

Table 2.1: The nonparametric models of shell and tube heat exchanger

No. Researcher Result

1 Dougherty (2003)

2 Nachtwey (2006)

3 Nithya (2007)

4 Srinivasan (2008)

5 Erkan (2009)

6 Kokate (2009)

7 Kokate (2010)

8 Belinda (2010)

9 Ding (2010)

10 Habobi (2010)

11 Khare (2010)

12 Rajasekaran (2010)

13 Pandhee (2011)

14 Sharma (2011)

15 Venkatesan (2012)

16 Dhakad (2013)

17 Kishore (2013)

18 Rajasekaran (2013)

15

2.4 Parametric Identification

Many scientists have also studied extensive research related to parametric identification of

shell and tube heat exchanger to produce its model. Among others are Qingchang (1990), Xia

(1991), Andersen (2001), Bendapudi (2004), Piotrowska (2012), Dostal (2013), Amlashi

(2013), and Sivaram (2013). Qingchang (1990) has applied the experimental system

identification with Pseudo Random Binary Sequence (PRBS) test signals. The result has

shown that it is possible to describe the dynamic behavior of the shell and tube heat

exchanger as a nonlinear distributed parameter system using a linear lumped parametric

model. Both auto regressive with external input model (ARX model) and output error model

(OE model) structures based on the prediction error method (PEM) were used in order to do

this. The identification results were compared with each other.

Xia (1991), on the other hand, studied two models for two different control schemes

that were developed for a parallel flow heat exchanger. Firstly, a spatially lumped heat

exchanger model was developed as a good approximate model which has produced a high

system order. Model reduction techniques were then applied to obtain low order models that

were suitable for dynamic analysis and control design. The simulation method was discussed

to ensure a valid simulation result.

Anderson (2001) later studied a grey box modeling method as a counter-flow heat

exchanger arrangement. This method was characterized by using information from

measurements in conjunction with physical knowledge. The combination of statistical

methods and physical interpretation was exploited in the modeling procedure, from the

design of experiments to parameter estimation and model validation. The presented models

were mainly formulated as state space models in continuous time with discrete time

observation equations. The state equations were expressed in terms of stochastic differential

equations. From a theoretical viewpoint the techniques for experimental design, the

parameter estimation and model validation were considered. From the practical viewpoint,

emphasis was put on how these methods can be used to construct models adequate for

heating system simulations.

Bendapudi (2004) presented in details the formulation of shell and tube evaporators

and condensers using the moving-boundary approach and comparative results of model

execution with a finite-volume approach. Both formulations are developed to capture start-up

and load change transients. The moving-boundary formulation has the ability to handle the

16

discontinuities associated with the exiting phase-boundaries and entering the heat-exchanger

during transient operation. A significant saving in execution time is shown over the finite-

volume approach with comparable accuracy.

Piotrowska (2012) analyzed the dynamics of the shell and tube heat exchanger using

the parametric system identification methods and their implementation in MATLAB. The

character of observed phenomenon is not unique. In some of the cases it shows inertial

character, in the others oscillatory character. It might indicate an appearing of the kind of a

thermal oscillation during the non-stationary states of the heat exchanger.

Dostal (2013) studied the shell and tube heat exchanger which is based on the

approximation of a nonlinear model. The parameter model is determined by continuous-time

external linear models with parameters recursively estimated via corresponding external delta

models. The approximating linear model structures were chosen on the basis of primary

steady-state and the process dynamic analysis.

Amlashi (2013) studied a nonlinear system identification of liquid saturated steam

heat exchanger using artificial neural network model. Heat exchanger is a highly nonlinear

and non-minimum phase process and often its working conditions are variable. Experimental

data obtained from fluid outlet temperature measurement in laboratory environment was used

as the output variable and the rate of change of fluid flow was taken into the system to be the

input. The neural network system identification results and conventional nonlinear models

were compared where neural network model is more accurate and faster, and then

conventional nonlinear models for a time series data. This is because of the independence of

the model assignment.

Sivaram (2013) scrutinized the process parameter estimation of a counter flow shell

and tube heat exchanger. Different flow rates of cold flow for specific time intervals and

corresponding tube outlet temperature were measured. Least square estimation algorithm was

used for parameter estimation based on the experiments.

2.5 Summary

Based on the literature review, there are two ways of constructing mathematical models i.e.

physical modeling (dynamic modeling) and system identification. The physical modeling is

an analytical approach while the system identification is an experimental approach. The

17

system identification can be implemented i.e. white box, black box and gray box. The black

box is parametric and nonparametric system identification. The parametric method is built

through a model structure determination while the nonparametric method is determined based

on an output response system in the function or a graph forms.

18

CHAPTER 3

CHAPTER 3

PROCESS DESCRIPTION AND CANDIDATE MODEL

3.1 Introduction

To develop a candidate model, firstly, it is necessary to describe the classification, process

description, and dynamic model of the system. Therefore, this chapter includes introduction,

classification of heat exchangers, process description, dynamic model, nonparametric model,

parametric model and the summary. Heat exchanger is an important and expensive item of

equipment that is used almost in every industry (oil and petrochemical, sugar, food,

pharmaceutical and power industry).

They are generally designed to meet certain performance requirements under steady

operating conditions. However, transient response of heat exchangers needs to be known for

designing the control strategy of industrial processes. Problems such as start-up, shutdown,

failure and accidents have motivated investigations of transient thermal response in heat

exchangers. It also helps the designer to find a solution of the time dependent temperature

problems, which is essential for its performance requirements.

The heat exchangers are classified according to transfer processes, number of fluids,

heat transfer mechanisms, construction type, flow arrangements, process function,

condensers, and liquid-to-vapor phase-change exchangers. Perhaps, the most common type of

heat exchanger in industrial applications is the shell and tube heat exchanger. Shell and tube

heat exchangers contain a large number of tubes (sometimes several hundred) packed in a

shell with their axes parallel to the shell. Heat transfer takes place as one fluid flows inside

the tubes while the other fluid flows outside the tubes through the shell.

19

Baffles are commonly placed in the shell to force the shell-side fluid to flow across

the shell to enhance heat transfer and to maintain uniform spacing between the tubes. Despite

their widespread use, shell and tube heat exchangers are not suitable for use in automotive

and aircraft applications because of their relatively large size and weight. Note that the tubes

in a shell and tube heat exchanger open to some large flow areas called headers at both ends

of the shell, where the tube-side fluid accumulates before entering the tubes and after leaving

them.

Process description needed to give a heat exchanger model is expressed as a

mathematical equation in the form of a plant transfer function linking output to the input

signal. The system identification is referred to in this study is to solve this equation. There are

two identification methods that will be used for this system which are firstly is the parametric

identification methods and secondly is nonparametric identification methods. These methods

will generate parametric and nonparametric models and these models become candidate

models.

3.2 Classification of Heat Exchanger

A variety of heat exchangers are used in industry and in their products. Starting with a

definition, heat exchangers are classified according to transfer processes, number of fluids,

and degree of surface compactness, construction features, flow arrangements, and heat

transfer mechanisms. A heat exchanger is a device that is used to transfer thermal energy

(enthalpy) between two or more fluids, between a solid surface and a fluid, or between solid

particulates and a fluid, at different temperatures and in thermal contact. In heat exchangers,

there are usually no external heat and work interactions. Typical applications involve heating

or cooling of a fluid stream of concern and evaporation or condensation of single or multi

component fluid streams. In other applications, the objective may be to recover or reject heat,

or sterilize, pasteurize, fractionate, distill, concentrate, crystallize, or control a process fluid.

In a few heat exchangers, the fluids exchanging heat are in direct contact. In most heat

exchangers, heat transfer between fluids takes place through a separating wall or into and out

of a wall in a transient manner. In many heat exchangers, the fluids are separated by a heat

transfer surface, and ideally they do not mix or leak. Such exchangers are referred to as direct

transfer type, or simply recuperators. In contrast, exchangers in which there is intermittent

heat exchange between the hot and cold fluids via thermal energy storage and release through

20

the exchanger surface or matrix are referred to as indirect transfer type, or simply

regenerators. Such exchangers usually have fluid leakage from one fluid stream to the other,

due to pressure differences and matrix rotation/valve switching.

Common examples of heat exchangers are shell and tube exchangers, automobile

radiators, condensers, evaporators, air pre-heaters, and cooling towers. If no phase change

occurs in any of the fluids in the exchanger, it is sometimes referred to as a sensible heat

exchanger. The internal thermal energy sources in the exchangers, such as in electric heaters

and nuclear fuel elements. Combustion and chemical reaction may take place within the

exchanger, such as in boilers, fired heaters, and fluidized-bed exchangers. Mechanical

devices have used in some exchangers such as in scraped surface exchangers, agitated

vessels, and stirred tank reactors. Heat transfer in the separating wall of a recuperator

generally takes place by conduction. However, in a heat pipe heat exchanger, the heat pipe

not only acts as a separating wall, but also facilitates the transfer of heat by condensation,

evaporation, and conduction of the working fluid inside the heat pipe. In general, if the fluids

are immiscible, the separating wall has eliminated, and the interface between the fluids

replaces a heat transfer surface, as in a direct-contact heat exchanger.

Not only heat exchangers are often used in the process, power, petroleum,

transportation, air-conditioning, refrigeration, cryogenic, heat recovery, alternative fuel, and

manufacturing industries, they also serve as key components of many industrial products

available in the marketplace. These exchangers are classified in many different ways. The

classification is conducted according to transfer processes, number of fluids, and heat transfer

mechanisms. Conventional heat exchangers are further classified according to construction

type and flow arrangements. Another arbitrary classification is conducted based on the heat

transfer surface area/volume ratio, into compact and non-compact heat exchangers. This

classification is made because the type of equipment, fields of applications, and design

techniques generally differ. Heat exchangers are classified according to the process function,

as outlined in Figure 3.1. Additional ways to classify heat exchangers are by fluid type (gas–

gas, gas–liquid, liquid–liquid, gas two-phase, liquid two-phase, and so forth) and industrial

application. All these classifications are summarized in Figure 3.2.

21

Figure 3.1 (a) Classification according to process function; (b) Classification of condensers;(c)

Classification of liquid-to-vapor phase-change exchangers (Shah, 2003)

22

Figure 3.2 Classification of heat exchangers (Shah, 2003)

According to Rao (2007), classification of heat exchanger according to its

construction is shown in Figure 3.3. In the simplest exchangers, the hot and cold fluids mix

directly; common are those in which the fluids are separated by a wall. This type which is

23

called a recuperator, it ranges from a simple plane wall between two flowing fluids to

complex confirmations involving multiple passes, fins or baffles. In this case, conduction,

convection and sometimes radiation principles are required to describe the energy exchange

process. Many factors enter into design of heat exchangers, including thermal analysis, size,

weight, structural strength, pressure drop and cost.

Figure 3.3 Classification of heat exchanger according to construction

3.3 Process Description

A very common type of heat exchanger is a shell and tube heat exchanger. It occurs in many

forms, as a separate unit in the form of pre-heaters, coolers and condensers, as an integral part

of some other units. This research addresses a shell and tube counter flow water-water heat

exchanger. Heat exchanger is a device in which energy in the form of heat is transferred into

24

what is usually realized by the confinement of both fluids in some geometry in which they are

separated by a conductive material. The properties of heat exchanger are strongly dependent

on geometry and material as well as on properties of both fluids. It is known that such devices

usually have nonlinear behavior (Kupran, 2000).

The dynamics of a heat exchanger process are complex because of various

nonlinearities introduced into the system. Furthermore, the installed valve characteristic of

the steam may not be linear either. Dead-time, on the other hands, depends on the steam and

water flow rates, the location and the method of installation of the temperature-measuring

devices. To take into account the non-linearity and the dead-time, gain scheduling features

and dead-time compensators have to be added. Also, the process is subjected to various

external disturbances such as pressure fluctuations in the steam header, disturbances in the

water line, and changes in the inlet steam enthalpy and so forth.

In line with the different applications, its mathematical model can be described in

different ways: linear (or nonlinear) partial differential equations, and linear (or nonlinear)

ordinary differential equations. In the case of a heat exchanger, there are four variables to be

considered: hot stream flow-rate, cold stream flow-rate, hot stream temperature, and cold

stream temperature. There are only two output variables, i.e. the temperatures of each stream,

since the mass flow rate cannot change by passing through the exchanger (Davidson, 1995).

Each input will affect the corresponding output; therefore, there will be eight possible

combinations of input-output relationships. These relationships are illustrated in Figure 3.4.

Figure 3.4 Relationships input and output shell and tube heat exchanger

The shell-and-tube heat exchangers are still the most common type of exchanger in

use today. They have larger heat transfer surface area-to-volume ratios than the most of

common types of heat exchangers, and they are easily manufactured for a large variety of

148

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