MATHEMATICAL MODELLING OF MASS TRANSFER
IN MULTI-STAGE ROTATING DISC CONTACTOR COLUMN
NORMAH MAAN
UNIVERSITI TEKNOLOGI MALAYSIA
MATHEMATICAL MODELLING OF MASS TRANSFER
IN A MULTI-STAGE ROTATING DISC CONTACTOR COLUMN
NORMAH MAAN
A thesis submitted in fulfilment of the
requirements for the award of the degree of
Doctor of Philosophy
Faculty of Science
Universiti Teknologi Malaysia
OCTOBER 2005
iii
To
my beloved husband, Zainidi and
my loving daughters, Nur Ezzaty and Nur Amira
and
especially my loving
and supportive parents, Hj. Maan and Hjh. Eashah
iv
ACKNOWLEDGEMENT
First of all, I thank ALLAH (SWT), the Lord Almighty, for giving me the
health, strength and ability to write this thesis.
I wish to express my deepest gratitude to my supervisor, Assoc. Prof. Dr.
Jamalludin Bin Talib, who suggested the research topic and directed the research.
I thank him for his enduring patience. My special thanks are also due to my cosupervisor, Dr. Khairil Anuar Arshad, and to Assoc. Prof. Dr. Tahir Bin Ahmad for
contributing ideas, discussing research plans, and encouragement.
I am forever indebted to my employer Universiti Teknologi Malaysia (UTM) for
granting me the study leave and providing the facilities for my research.
Finally, I am grateful for the help in different ways from a number of individuals.
Among them are Dr. Rohanin, Dr. Zaitul, Pn. Sabariah and other friends.
v
ABSTRACT
In this study, the development of an improved forward and inverse models
for the mass transfer process in the Rotating Disc Contactor (RDC) column were
carried out. The existing mass transfer model with constant boundary condition does
not accurately represent the mass transfer process. Thus, a time-varying boundary
condition was formulated and consequently the new fractional approach to equilibrium
was derived.
This derivation initiated the formulation of the modified quadratic
driving force, called Time-dependent Quadratic Driving Force (TQDF). Based on this
formulation, a Mass Transfer of A Single Drop (MTASD) Algorithm was designed,
followed by a more realistic Mass Transfer of Multiple Drops (MTMD) Algorithm
which was later refined to become another algorithm named the Mass Transfer Steady
State (MTSS) Algorithm. The improved forward models, consisting of a system of
multivariate equations, successfully calculate the amount of mass transfer from the
continuous phase to the dispersed phase and was validated by the simulation results.
The multivariate system is further simplified as the Multiple Input Multiple Output
(MIMO) system of a functional from a space of functions to a plane. This system
serves as the basis for the inverse models of the mass transfer process in which fuzzy
approach was used in solving the problems. In particular, two dimensional fuzzy number
concept and the pyramidal membership functions were adopted along with the use of
a triangular plane as the induced output parameter. A series of algorithms in solving
the inverse problem were then developed corresponding to the forward models. This
eventually brought the study to the implementation of the Inverse Single Drop Multistage (ISDMS)-2D Fuzzy Algorithm on the Mass Transfer of Multiple Drops in Multistage System. This new modelling approach gives useful information and provides a
faster tool for decision-makers in determining the optimal input parameter for mass
transfer in the RDC column.
vi
ABSTRAK
Dalam kajian ini, pembentukan model ke depan yang lebih baik dan model
songsangan bagi proses peralihan jisim di dalam Turus Pengekstrakan Cakera Berputar
(RDC) telah dijalankan. Model yang sedia ada dengan syarat sempadan tetap tidak
mewakili proses peralihan jisim dengan tepat. Dengan itu, syarat sempadan yang
merupakan suatu fungsi masa berubah telah dirumuskan dan seterusnya pendekatan
pecahan untuk keseimbangan yang baru diterbitkan. Penerbitan ini telah memulakan
perumusan daya pacu kuadratik ubahsuai, yang dipanggil daya pacu kuadratik
bersandaran masa (TQDF). Berdasarkan perumusan ini, satu Algoritma Peralihan
Jisim untuk Sebutir Titisan (MTASD) telah direkabentuk , diikuti oleh satu algoritma
yang lebih realistik algoritma Peralihan Jisim untuk Multi Titisan (MTMD) yang mana
kemudiannya, telah diperbaiki dan dinamakan Algoritma Peralihan Jisim Berkeadaan
Mantap (MTSS). Model ke depan yang telah diperbaiki, terdiri daripada satu sistem
persamaan berbilang pembolehubah yang mana kemudiannya dipermudahkan sebagai
sistem berbilang input berbilang output (MIMO) yang merupakan satu rangkap
dari satu ruang fungsi-fungsi kepada satu satah. Sistem ini merupakan satu asas
pembentukan model songsangan bagi proses peralihan jisim dan pendekatan kabur
telah digunakan untuk menyelesaikannya. Secara khususnya, konsep nombor kabur dua
matra dan fungsi keahlian piramid digunakan, diikuti dengan penggunaan satu satah
segitiga sebagai parameter output teraruh. Satu siri algoritma dalam menyelesaikan
masalah songsangan ini kemudiannya telah dibentuk berpadanan dengan model ke
depan masing-masing. Kajian ini akhirnya membawa kepada implementasi Algoritma
Songsangan Sebutir Titisan Multi-tingkat (ISDMS)-2D Kabur ke atas sistem peralihan
jisim Multi Butiran dalam Multi-tingkat.
Untuk rumusan yang lebih definitif,
pedekatan baru pemodelan ini memberi maklumat yang berguna dan menyediakan
suatu alat yang cepat kepada pembuat-keputusan dalam menentukan parameter
optimum input untuk peralihan jisim dalam turus RDC.
vii
TABLE OF CONTENTS
CHAPTER
TITLE
PAGE
THESIS STATUS DECLARATION
SUPERVISOR’S DECLARATION
1
TITLE PAGE
i
DECLARATION
ii
DEDICATION
iii
ACKNOWLEDGEMENT
iv
ABSTRACT
v
ABSTRAK
vi
TABLE OF CONTENTS
vii
LIST OF TABLES
xii
LIST OF FIGURES
xv
LIST OF SYMBOLS
xvii
LIST OF APPENDICES
xix
INTRODUCTION
1
1.1 Preface
1
1.2 Motivation
2
1.3 Objectives of the Research
4
1.4 Scope of Study
4
1.5 Significance of the Findings
5
1.6 Thesis Organization
5
1.7 Summary
6
viii
2
LITERATURE REVIEW
8
2.1 Introduction
8
2.2 Liquid-liquid Extraction
8
2.2.1 Rotating Disc Contactor Column
2.3 Hydrodynamic
10
12
2.3.1 Terminal Velocity
12
2.3.2 Slip and Characteristic Velocity
13
2.4 Drop Breakage Phenomena
14
2.4.1 Drop Size
15
2.4.2 Maximum Drop Size
15
2.4.3 Drop Breakage
16
2.4.4 Critical Drop Size and Critical Rotor Speed
16
2.4.5 Initial Number of Drops
17
2.4.6 Probability of Breakage
17
2.4.7 Mean Number of Daughter Drops Produced
18
2.5 Mass Transfer
18
2.5.1 The Whitman Two-film Theory
19
2.5.2 The Penetration Theory
21
2.5.3 Dispersed Phase Mass Transfer Coefficient
22
2.5.4 Continuous Phase Mass Transfer Coefficient
23
2.5.5 Overall Mass Transfer Coefficient
24
2.6 The Existing Forward Mathematical Models of the
Processes in the RDC Column
24
2.6.1 Talib’s work
24
2.6.2 Ghalehchian’s work
26
2.6.3 Mohamed’s work
26
2.6.4 Arshad’s work
27
2.7 Inverse Modelling
28
2.7.1 Introduction
28
2.7.2 Inverse Problem in Sciences and Engineering
30
2.7.3 Classes of Inverse Problem
32
2.7.4 Solution of Inverse Problem
33
2.8 Fuzzy Logic Modelling
2.8.1 The Basic Concepts of Fuzzy set Theory
34
35
ix
3
2.8.2 Fuzzy System
37
2.8.3 Fuzzy Modelling
37
2.8.4 Remarks
38
2.9 Summary
39
THE FORWARD MASS TRANSFER MODEL
41
3.1 Introduction
41
3.2 The Forward Mass Transfer Model
41
3.2.1 Diffusion in a Sphere
3.3 The Modified Model
4
42
45
3.3.1 The Analytical Solution
49
3.4 Simulations for Different Drop Sizes
55
3.5 Discussion and Conclusion
56
MASS TRANSFER IN THE MULTI-STAGE
RDC COLUMN
58
4.1 Introduction
58
4.2 The Diffusion Process Based On The Concept
Of Interface Concentration
59
4.2.1 Flux Across The Drop Surface Into The
Drop
60
4.2.2 Flux in The Continuous Phase
61
4.2.3 Process of Mass Transfer Based on Timedependent Quadratic Driving Force
62
4.3 Mass Transfer of a Single Drop
65
4.3.1 Algorithm 4.1: Algorithm for Mass Transfer
Process of a Single Drop (MTASD Algorithm)
66
4.3.2 Simulation Results
67
4.4 Mass Transfer of Multiple Drops
67
4.4.1 Basic Mass Transfer(BMT) Algorithm
72
4.4.2 Algorithm for the Mass Transfer Process
of Multiple Drops in the RDC Column
(MTMD Algorithm)
73
4.4.3 Simulation Results
74
x
4.5 The Normalization Technique
4.5.1 Normalization Procedure
76
4.5.2 De-normalization Procedure
80
4.6 Algorithm 4.4: Forward Model Steady State
Mass Transfer of Multiple Drops
5
83
4.6.1 Algorithm To Find The Drop Concentration
of a Steady State Distribution in
23 Stages RDC Column (MTSS Algorithm)
86
4.6.2 Updating Mechanism Algorithm
91
4.6.3 Simulation Results
92
4.7 Discussion and Conclusion
92
THE INVERSE MODEL OF MASS
TRANSFER: THEORETICAL DETAILS
AND CONCEPTS
96
5.1 Introduction
96
5.2 Inverse Modelling in RDC Column
97
5.2.1 Formulation of the Inverse Problem
5.3 Inverse Modelling Method
98
100
5.3.1 Fuzzy Flow Chart
101
5.3.2 Fuzzification Phase
102
5.3.3 Fuzzy Environment Phase
103
5.3.4 Defuzzification Phase
104
5.3.5 Numerical Example
105
5.4 Inverse Modelling of the Mass Transfer Process
of a Single Drop in a Single Stage RDC Column
Fuzzy-Based Algorithm(ISDSS-Fuzzy)
5.4.1 Inverse Fuzzy-Based Algorithm
(ISDSS-Fuzzy)
6
75
112
112
5.5 Simulation Results
113
5.6 Discussion and Conclusion
114
INVERSE MODEL OF MASS TRANSFER
RDC IN THE MULTI-STAGE COLUMN
118
6.1 Introduction
118
xi
6.2 Theoretical Details
119
6.2.1 Relation
119
6.2.2 Fuzzy Relation
121
6.3 Fuzzy Number of Dimension Two
6.3.1 Alpha-level
6.4 Inverse Modelling of the Mass Transfer Based
on Two Dimensional Fuzzy Number
123
125
6.4.1 The ISDSS-2D Fuzzy Algorithm
126
6.4.2 Numerical Example
131
6.4.3 Simulation Results
137
6.5 Inverse Model of the Mass Transfer of a Single
Drop in a Multi-Stage RDC Column Based on
Two Dimensional Fuzzy Number
138
6.5.1 The Inverse of Single Drop Multi-stage-2D
Fuzzy (ISDMS-2D Fuzzy) Algorithm
141
6.5.2 Simulation Results
144
6.6 Implementation of ISDMS-2D-Fuzzy Algorithm
on the Mass Transfer of Multiple Drops in the
Multi-stage System
6.6.1 Simulation Results
7
121
145
146
6.7 Discussion and Conclusion
147
CONCLUSIONS AND FURTHER RESEARCH
152
7.1 Introduction
7.2 Summary of the Findings and Conclusion
7.3 Further Research
152
152
157
REFERENCES
159
APPENDIX A
165
APPENDIX B
167
APPENDIX C
169
APPENDIX D
171
xii
LIST OF TABLES
TABLE NO.
TITLE
PAGE
2.1
The ill-posed and well-posed problems
29
3.1
Normalized dispersed and continuous phase
concentration
46
The Values of resident time and the slip velocity for
each drop size
48
3.3
The values of a1 and b1
49
4.1
The concentration of the drops along the column
69
4.2
Experiment 1-Continuous phase (aqueous) and
dispersed phase (organic) concentrations
77
Experiment 2-Continuous phase (aqueous) and
dispersed phase (organic) concentrations
78
Experiment 1-Normalized continuous and dispersed
phase concentrations
79
4.5
Experiment 1-De-normalized continuous concentrations
81
4.6
The error By Quadratic fitting
83
4.7
The concentration of the dispersed and continuous
phase according MTMD and MTSS Algorithm
95
5.1
Design parameters
102
5.2
Preferred input values
105
5.3
Preferred output values
105
5.4
α-cuts values for input parameters
107
5.5
α-cuts values for output parameters
108
5.6
The combination for each α-cuts values parameters
108
5.7
The output of each combination of each α-cuts
108
5.8
The min and max of the combination for each α-cuts
values
109
5.9
Input combination with fuzzy value z = 0.8377
110
5.10
Input combination with fuzzy value z = 0.9561
111
3.2
4.3
4.4
xiii
5.11
Optimized input parameters
111
5.12
Calculated output parameters
111
5.13
Simulation 1: The results of input domains [30.8, 57.6]
and [9.2, 17.5]
114
Simulation 2: The results of input domains [28.8, 59.6]
and [7.2, 19.5]
114
The errors between the calculated input values and
preferred values for different input domain
115
The errors between the calculated output values and
preferred values for different input domain
115
6.1
The values of alpha-level set
133
6.2
Input combination with fuzzy value z = 0.7697
137
6.3
Error of input and output parameters
137
6.4
The errors between the calculated input values and
the preferred values for different input domain
138
The errors between the calculated output values and
the preferred values for different input domain
138
The errors of the output solution of ISDSS and
ISDSS-2D-Fuzzy Algorithms
139
The input data for simulations of ISDMS-2D-Fuzzy
Algorithm
145
6.8
The results of ISDMS-2D-Fuzzy Simulations
145
6.9
ISDMS-2D-Fuzzy: The errors between the calculated
Input values and preferred values for different input
domain
146
ISDMS-2D-Fuzzy: The errors between the calculated
output values and preferred values for different Input
Domain
146
Set Data 1: The input data for simulations of
IMDMS-2D-Fuzzy Algorithm
147
IMDMS-2D-Fuzzy: The errors between the calculated
Input values and preferred values
147
IMDMS-2D-Fuzzy: Errors of the calculated output
against preferred values and Experimental Data 1
148
Set Data 2: The input data for simulations of
IMDMS-2D-Fuzzy Algorithm
148
IMDMS-2D-Fuzzy: Errors between the calculated
input values and preferred values
149
5.14
5.15
5.16
6.5
6.6
6.7
6.10
6.11
6.12
6.13
6.14
6.15
xiv
6.16
IMDMS-2D-Fuzzy: Errors of the calculated output
against preferred values and Experimental
Data 2
149
xv
LIST OF FIGURES
FIGURE NO.
TITLE
PAGE
2.1
Single contacting stage
9
2.2
Schematic diagram of RDC column
11
2.3
Mass transfer at interface
20
2.4
Stage wise back-flow for mass transfer process
27
2.5
Forward problem
29
2.6
Inverse problem
29
2.7
F1 = {x ∈ |x is about a2 }
36
2.8
The fuzzy logic modelling
38
3.1
The velocity of 10 different sizes of drops in the RDC
column
47
Sorption curve for sphere with surface concentration
a1 + b1 t
55
3.3
Fractional approach to equilibrium vs. time
56
4.1
Schematic diagram to explain the mass balance process
64
4.2
Flow chart of mass transfer process in the 23-stage
RDC column for MTASD Algorithm
68
The profile of the medium and drop concentration along
the column with respect to the new fractional approach
to equilibrium
70
The profile of the medium and drop concentration along
the column with respect to the new fractional approach
to equilibrium and Crank solution
71
The concentration of the continuous and dispersed
phase of new model, Talib model and experimental
76
The continuous phase concentration along the column:
Experiment Data 1
80
The error between the continuous phase concentration
of Experiment Data 1 with and without de-normalized
values
82
3.2
4.3
4.4
4.6
4.7
4.8
xvi
4.9
The error is fit to Quadratic-like curve
83
4.10
The continuous phase concentration along the column
with corrected value : Experiment Data 1
84
4.11
Flow chart for mass transfer process at itr=1
85
4.12
Flow chart for mass transfer process at itr=2
86
4.13
Flow chart for mass transfer process at itr=3
87
4.14
Flow chart describing the mass transfer process
itr=4,5 6,...,n
88
Schematic diagram of the mass transfer process
in the 23-stage RDC column
89
4.16
Flow Chart For Mass Transfer Process
90
4.17
The concentration of continuous and dispersed
phase of MTMD, MTSS Algorithm and Experimental
93
5.1
The MIMO System
98
5.2
Schematic diagram of the forward model in a
Multi-stage RDC column
99
5.5
Fuzzy Algorithm
106
5.4
The view of the input and output parameters of the
system
103
5.5
Triangular fuzzy number of the input parameters
106
5.6
Triangular fuzzy number of the output parameters
107
5.8
Intersection between induced and preferred output
for continuous phase concentration
110
5.9
The MIMO system is separated into 2 MISO systems
116
6.1
Pyramidal fuzzy number
123
6.2
Pyramidal fuzzy number from Cartesian product
of two triangular fuzzy numbers
124
6.3
The Induced Plane
132
6.4
The Preference Output
134
6.5
(a) The Intersection Between Preferred and
Induced Output
(b) Level Curve of (a)
135
6.6
ISDMS-2D Fuzzy model
140
6.7
The flow chart representing the three phases
143
4.15
xvii
LIST OF SYMBOLS/NOTATIONS
a
A
C
d
dc
dcr
dmax
do
d32
dav
Dc
Dd
De
Doe
Dr
Ds
e
E
Eam
Ec
Eo
Fd
fr
g
gi
h
hc
H
kd
Kodi
m
M
Nr
Ncr
Ncl
-
radius of a sphere
Column cross sectional area (m2 )
Concentration (kg/m3 )
Drop diameter (m)
Column diameter (m)
Critical drop diameter for breakage (m)
Maximum stable drop diameter (m)
Initial drop diameter (m)
Sauter mean drop size (m)
Average diameter of drop (m)
Molecular diffusivity in continuous phase (m2 /s)
Molecular diffusivity in dispersed phase (m2 /s)
Eddy diffusivity (m2 /s)
Overall effective diffusivity (m2 /s)
Rotor diameter (m)
Stator diameter (m)
Back-flow ratio
Power consumption per unit mass (Eq. (2.9)) (w/kg)
Axial mixing coefficient (m2 /s)
Continuous phase axial mixing coefficient (m2 /s)
Eotvos number
Flowrate of dispersed (cm3 /s)
Fraction of daughter drop
Acceleration due to gravity (m2 /s)
Dynamic volume fraction of drops with size di
Height of column (m)
Height of an element of compartment (m)
Column height (m)
Drop film mass transfer coefficient (m/s)
Overall dispersed phase based mass transfer coefficient for
-
drop with size di (m/s)
Exponent in the equation of slip velocity
Morton number in terminal velocity
Rotor speed (s−1 )
Critical rotor speed for drop breakage (s−1 )
Number of classes
xviii
Nst
P
PR
Re
Rek
ReD,ω
Sc
Sh
tr,i
V
Vc
Vd
Vk
Vs
Vt
We
W eD,ω
xm
X
-
Number of stages
Probability of breakage
Power consumption per disc (w/m3 )
Drop Reynolds number
Drop Reynolds number using Vk
Disc Reynolds number based on angular velocity
Schmidt number
Sherwood number
Resident time of drops with size di in a stage (s)
Drop volume (m3 )
Continuous phase superficial velocity (m/s)
Dispersed phase superficial velocity (m/s)
Drop characteristic velocity (m/s)
Slip velocity (m/s)
Drop terminal velocity (m/s)
Weber number for drop
Disc angular Weber number
Mean number of daughter drops
Hold-up
Greek symbols
Φ
γ
βn
µc , µd
ρc , ρd
∆ρ
κ
ω
ωcr
-
Equilibrium curve slope (dCd /dCc
Interfacial tension (N/m)
Eigenvalues
Continuous and dispersed phase viscosities (mP as)
Continuous and dispersed phase densities (kg/m3 )
densities differences (kg/m3 )
Viscosity ratio
Angular velocity (s1 )
Critical angular velocity (s1 )
Supercripts
∗
s
u
-
dimensional variables
differentiation with respect to η
denotes steady part of the solution
denotes unsteady part of the solution
Subcripts
c, d
i
n
av
-
Continuous and dispersed phase
drop size classes
Stage number
average value
xix
LIST OF APPENDICES
APPENDIX
A
TITLE
PAGE
GEOMETRICAL AND PHYSICAL
PROPERTIES OF RDC COLUMN
165
B
GLOSSARY
167
C
PAPERS PUBLISHED DURING THE
AUTHOR’S CANDIDATURE
169
MATLAB PROGRAM : INVERSE ALGORITHM
171
D
CHAPTER 1
INTRODUCTION
1.1
Preface
The study of liquid-liquid extraction has become a very important subject to be
discussed not just amongst chemical engineers but mathematicians as well. This type
of extraction is one of the important separation technology in the process industries
and is widely used in the chemical, biochemical and environmental fields. The principle
of liquid-liquid extraction process is the separation of components from a homogeneous
solution by using another solution which is known as a solvent [1, 2]. Normally, it is
used when separation by distillation is ineffective or very difficult. This is due to the
fact that certain liquids cannot withstand the high temperature of distillation.
There are many types of equipments used for the processes of liquid-liquid
extraction. The concern of this research is only with the column extractor type, namely
the Rotating Disc Contactor Column (RDC). Modelling the extraction processes
involved in the RDC column is the major interest in this work. Modelling can be
divided into two categories. One is the forward modelling and the other is the inverse
modelling.
From mathematical and physical point of view, it is generally easier to calculate
the “effect of a cause” or the outputs of the process than to estimate the “cause of
an effect” or the input of the process. In other words, we usually know how to use
mathematical and physical reasoning to describe what would be measured if conditions
were well posed. This type of calculation is called a forward problem. The resulting
2
mathematical expressions can be used as a model and we call the process in obtaining
the values of outputs as forward modelling.
On the other hand, inverse problems are problems where the causes for a desired
or an observed effect are to be determined. Inverse problem come paired with direct
problems and of course the choice of which problem is called direct and which is called
inverse is, strictly speaking, arbitrary [3]. Before an inverse problem can be solved, we
first need to know how to solve the forward problem. Then the appropriate steps or
algorithms need to be determined in order to get the solution of the inverse model.
Apart from producing an improved mathematical model for the mass transfer
process, another concern of this research is to develop the inverse models that can
determine the value of the input parameters for a desired value of the output parameters
of the mass transfer process in the multi-stage RDC column.
1.2
Motivation
Several models have been developed for the modelling of RDC columns. The
modelling shows that the drop size distribution and the mass transfer processes are
important factors for the column performances. Since the behavior of the drop breakage
and the mass transfer process involve complex interactions between relevant parameters,
the need to get as close as possible to the reality of the processes is evident.
Several
researchers
namely
Korchinsky
and
Azimzadih[4],
Ghalehchian[6] and Arshad[7] had been working in this area.
Talib[5],
Korchinsky and
Azimzadih[4] introduced a stage wise model for mass transfer process, which was
furthered by Talib[5] and Ghalehchian[6]. The unsteady-state models developed by
Talib[5] are referred to as the IAMT (Initial Approach of Mass Transfer) and BAMT
(Boundary Approach of Mass Transfer). To get closer to reality, Ghalehchian[6] had
developed a new steady-state model of mass transfer by including the idea of axial
mixing into the simulation of the mass transfer process. Then Arshad[7] developed a
new steady state model for hydrodynamic process, which updates the current hold up
and drops velocities in every stage after certain time intervals until the system reaches
steady state.
3
The mass transfer models are based on a radial diffusion equation with a
constant boundary condition. However a mass transfer model with varied boundary
condition has yet to be developed. The development of the model will enhance the
understanding of the real process. This is because in reality the concentration of the
drops in each compartment in the RDC column is not constant.
The mathematical simulation models of the processes in the RDC column are
very complex and need excessive computer time, particularly in predicting the values
of output parameters. The determination by trial and error of the input parameter
values in order to produce the desired output need excessive computer time and it will
be costly if the actual processes are involved. This type of problem is known as inverse
problem. Therefore, to overcome these difficulties, an alternative approach based on
fuzzy logic is considered.
Fuzzy logic is a well-known method for modelling such uncertain systems of
great complexity. They have been approved and demonstrated by many researchers in
other disciplines of study to have the capability of modelling a complicated system as
well as predicting the actual behavior of a system. So, this study will adopt this method
for assessing inverse modelling of the mass transfer process in the RDC column.
A few researchers for examples Ahmad et.al. [8, 9] and Ismail et. al. [10] have
been using this approach in their works. Ahmad developed an algorithm which was used
in determining the optimized electrical parameters of microstrip lines. The problem was
presented as multiple input single output (MISO) system of some algebraic equations.
Whilst, the problem involved in Ismail’s work is a multiple input multiple output
(MIMO) system of a crisp state-space equation. Both works used a one dimensional
fuzzy number concept and a triangular membership function.
The forward model of the mass transfer process in the RDC column consists of
Initial Boundary Value Problem (IBVP) of diffusion equation, a nonlinear and a few
of linear algebraic equations. The details of the equations will be found in Chapter 3
to 6. Thus the multivariate system modelled by these equations can be simplified as
MIMO system. In this work, a two dimensional fuzzy number concept will be used. A
pyramidal membership function will be also implemented in this work.
4
1.3
Objectives of the Research
1. To investigate an equation that will be used as the boundary condition of the
IBVP.
2. To formulate a new fractional approach to equilibrium based on the IBVP of
time-dependent boundary condition.
3. To formulate a modified driving force based on the new fractional approach to
equilibrium.
4. To develop an algorithm for the mass transfer of a single drop in the multi-stage
RDC column.
5. To develop algorithms for the mass transfer of the multiple drops in the multistage RDC column.
6. To establish a technique for assessing the inverse models of the corresponding
new forward mass transfer models.
1.4
Scope of Study
This study will be based on a radial diffusion equation with varied boundary
value problem for mass transfer process and a few algebraic equations governed by
experiments carried out by a previous researcher for the process of hydrodynamics in
the RDC column. The study will also be based on the experimental data obtained by
the researchers at the University of Bradford under contract to Separation Processes
Service, AEA Technology, Harwell.
In this study, the development of inverse model will be based on the concept
of fuzzy algorithm.
In this development, the mathematical equations used in
mathematical forward modelling are also being considered. This model is a structure
based model.
5
1.5
Significance of the Findings
This study achieves a new development of the forward model which will provide
a better simulation and hence get a better control system for the RDC column. This
study also give a significant contribution in the form of algorithms. These algorithms
are able to calculate the optimal solution of the inverse model for the mass transfer
process in the RDC column. The inverse model will give a new paradigm to the
decision maker or to the engineer in making decision to decide approximate values of
input concentrations of continuous and dispersed phases for desired values of output
concentrations of continuous and dispersed phases.
1.6
Thesis Organization
Chapter 2 gives a literature review on liquid-liquid extraction in general. It
is then followed by a review on the RDC columns including the important processes
involved. The theoretical details on the drop distribution, breakage phenomena and the
mass transfer process are also included. The existing forward mathematical modelling
by the most recent researchers are presented. These reviews are significantly used as a
background in order to develop a new mass transfer model; which will be described in
Chapter 3.
To achieve Objective 6 of the research, the review on the inverse problem in
general, including the definition, the examples of real world problems, the classes of
the inverse problem and the steps involved in solving the problem are given. Whilst
Section 2.8 will provide the reviews on the Fuzzy Concepts. These concepts will be
applied in Chapters 5 and 6 to develop an algorithm for solving the corresponding
inverse problem.
Chapter 3 provides the formulation of the varied boundary function from the
experimental data in [5]. The details of the exact solution of the IBVP with the time
depending function boundary condition will be shown which is then followed by the
derivation of the new fractional approach to equilibrium. The comparison between the
new fractional approach to equilibrium and the one introduced in [11] will be made in
the last section of Chapter 3.
6
Chapter 4 comprises the development of the forward models of the mass transfer
in the multi-stage RDC column. Prior to the development, the formulation of the
modified quadratic driving force which is called Time-dependent Quadratic Driving
Force(TQDF) will be given. Based on this formulation, a Mass Transfer of A Single
Drop Algorithm is designed and this is then followed by a more realistic Mass Transfer
of Multiple Drops Algorithm. An alternative method for calculating the mass transfer
of a Multi-Stage System will also be presented in the form of an algorithm named as
the Mass Transfer Steady State Algorithm.
Chapter 5 discusses the formulation of the inverse model for mass transfer
process in the RDC column. The mappings which represent the forward model involved
will also be given. Basically this chapter introduces an Inverse Single Drop Single StageFuzzy (ISDSS-Fuzzy) model which represents the mass transfer process of a single drop
in a single stage RDC column. This model is a base for the inverse model of the mass
transfer process in the real RDC column.
Chapter 6 provides the theoretical details which become the basis for
accomplishing the task of the thesis. The details are about the relation of two crisp
sets and this is followed by the relation of two fuzzy sets. We also include some
examples which can explain the concept more clearly. From fuzzy relation we extend
the concept of fuzzy number of dimension one to dimension two. Section 6.4 discusses
the development of the Inverse Model of Mass Transfer Process of a Single Drop in
a Single Stage RDC Column based on the two dimensional fuzzy number. We then
implement the algorithm to the mass transfer process in the multi-stage RDC column.
We then summarize the findings and suggest areas for further research in
Chapter 7.
1.7
Summary
In this introductory chapter, a short introduction on the liquid-liquid extraction
process particularly on the RDC column has been presented. The deficiency of the
existing mass transfer models in the multi-stage RDC column has also briefly discussed.
Next, come the research objectives and scope, and the contributions of the work
7
described in the thesis. Finally, the outline of the thesis is presented.
The current chapter serves as a defining point of the thesis. It gives direction
and purpose to the research and the discussions presented here are the basis for the
work done in the subsequent chapters.
CHAPTER 2
LITERATURE REVIEW
2.1
Introduction
This chapter is divided into six sections. The first section will discuss the liquid-
liquid extraction and the RDC column in general. The subsequent two sections will
give a brief information about the theoretical concepts and the mathematical equations
used in governing the mathematical models of the processes in the RDC column. The
fourth section will give a summary of the existing forward mathematical models. A
general overview of inverse modelling will be given in the fifth section. Finally the last
section will give a review on fuzzy logic concepts and a brief information about the
fuzzy modelling of Multiple Input Single Output(MISO) and Multiple Input Multiple
Output(MIMO) systems by previous researchers.
2.2
Liquid-liquid Extraction
Liquid-liquid extraction is an operation that affects the transfer of a solute
between two immiscible or partially immiscible liquids. The two liquids are called
the feed and extraction solvent. In simple words, this is the process of the removal
of the solute, say C, from the feed, say solution A, by the extraction solvent B. The
solvent containing the solute C, after the extraction process is completed, is known as
the extract and the solution A from which the solute C has been removed is called
raffinate.
In this operation, the feed and extraction solvent, are brought into intimate
contact with each other in order to extract the solute from the feed. This is actually
9
a mass transfer operation based on the difference in concentration between two phases
(the feed and the solvent) rather than the difference in physical properties.
This principle and some of the special terminology of a single contacting stage
are illustrated in Figure 2.1. If equilibrium is established after contact, the stage is
defined as an ideal stage. On the laboratory scale this can be achieved in a few minutes
simply by hand agitation of two liquid phases in a stoppered flask or separatory funnel
[12].
Feed flow, F
Solvent A
+
Solute C
Raffinate flow, R Extract flow, F
Solvent flow, S
+
Solvent B
=
Solvent A
+
Solute C
Solvent B
+
Solute C
Phase in equilibrium, one ideal stage
Figure 2.1: Single contacting stage
There is a wide range of applications of liquid-liquid extraction, among them are
in petroleum industries, food processing, separation and purification of pharmaceutical
and natural products, etc [13, 14].
Liquid-liquid extraction equipment [15, 16] can be classified as
Mixer settlers consisting of an agitated tank for mass transfer followed by a settling
tank to separate both phases, due to a density difference between the two liquids.
Usually it requires a series of mixer settler for a desired separation. This type of
equipment is used when there will only be one equilibrium stage in the process.
Column Extractor consisting of a vertical column where the more dense phase enter
at the top and flows downwards whilst the less dense phase enters at the bottom
and flows upward. One of the phases can be pumped through the column at any
desired flow rate, while the maximum rate of the other phase will be limited by
the rate of the first phase and also the physical properties of both phases. There
is a maximum rate at which the phases can flow through the column and at this
rate the dispersed phase will be rejected at its point of entry and the column
is said to be flooded. Thus for a particular set of process conditions, the cross
sectional area of the column must be sufficiently large so that flooding does not
occur. The height of the column will be set by the rate of mass transfer and the
quantity of material that is required to be extracted.
10
There are two different types of column for the latter classification, which are
non-agitated and agitated columns. For the non-agitated column such as packed and
spray extraction column give differential contact, where mixing and settling proceed
continuously and simultaneously. In particular for the packed column, to make it more
effective, the column is filled with packing such as Rascing or Berl saddles, which cause
the droplets to coalesce at frequent intervals. For the agitated column, there is a series
of disc or turbine agitators mounted on the central rotating shaft. Each agitator is
separated from the next agitator by a calming section, either a mesh of wire or a stator
ring, or perforated plate, that will encourage coalescence of the drops.
For the latter type of columns, such as Rotating Disc Contactor (RDC),
Oldshue-Rushton Contactor, Scheibel extractor and Rotary Annular Column are widely
used for liquid-liquid extraction[17].
The performance of these column contactors
indicates that they are more efficient and provides better operational flexibility than
non-agitated column type. In this study, we only concentrate on the RDC column.
Therefore, the details of this type of column are provided in the following section.
2.2.1
Rotating Disc Contactor Column
The rotating disc contactor column is one of the agitated mechanical devices
that is being widely used in the study of liquid-liquid extraction. It was initially
developed in the Royal Dutch/Shell laboratories in Amsterdam by Reman in 1948-52.
Some hundreds of RDCs are at present in use world-wide, ranging from less than 1m
to 4.5m in diameter [18]. There is another column with 2.8 meters in diameter and
100 actual stages along the column, which is used to remove colour bodies from high
molecular weight hydrocarbons (see [6]).
The mechanical layout of the RDC column is very simple and ideal for processing
liquids with different densities. According to Reissinger[19] , RDCs are preferable
compared to other extractor columns in the case of high through-puts and large capacity
range. The RDC column consists of a vertical cylindrical column in which horizontal
stator rings are installed. These rings are purposely fixed so that several compartments
are formed in the column. In the middle of the compartment, flat rotating disc plates
are installed, attached to a common rotating long shaft which is driven by an electric
11
Figure 2.2: Schematic diagram of RDC column
12
motor. The diameter of the rotor discs are smaller than the diameter of the stator
opening, thus facilitating construction and maintenance. Above the top stator ring
and below the bottom stator ring, settling compartments are installed. Wide -mesh
grids are used between the agitated section and the settling zones to nullify the liquid
circular motion, thus ensuring optimum settling conditions.
According to Korchinsky[20], an RDC’s performance is affected by its column
diameter, rotor disc diameter, stator ring opening, compartment height, number of
compartments and disc rotational speed. Careful consideration must be given to these
parameters in designing a satisfactory and efficient RDC column.
There are two important processes involved in the RDC column which are drop
size distribution and mass transfer process [17]. Beside the parameters mentioned
above, work on extraction has also shown that drop size distribution need to be
determined if improvements in design are to be made. There are two factors that
influence the drop size distribution which are the hydrodynamics of the drops and also
the breakage process of the drops in the column. Therefore in the following section,
some of the terminologies involved in the hydrodynamic process are given.
2.3
Hydrodynamics
The study of drops is more complex than that of solid particles or bubbles,
because of the internal motion and the drag coefficients involved. This section begins
with the review of terminal velocity of a single drop, leading to the ideas of slip velocity
and characteristic velocity.
2.3.1
Terminal velocity
The terminal velocity of the drop in an infinite unhindered medium is the
maximum speed of the drop motion obtained by balancing buoyancy and drag forces.
The factor determining terminal velocity are drop size, drop shape and the physical
properties of the system.
13
Based on the study of the movement of a single drop of various sizes, Grace et
al. (see Talib[5] ) built their own equation of terminal velocity. From these observation
they found that every drop has its own terminal velocity, in which the equation involves
the dimensionless numbers i.e the Morton number, the Eotvos number and the Reynolds
number. The terminal velocity equation that they proposed is
Vt =
µc −0.149
(J − 0.857),
M
ρc d
(2.1)
where
M=
gµ4c ∆ρ
’
ρ2c γ
J = 0.94H 0.757
for 2 < H < 59.3,
J = 3.42H 0.441
for H ≥ 59.3,
H = 43 E0 M −0.149
E0 =
µc −0.14
,
0.0009
gd2 ∆ρ
γ ,
where Vt is the drop terminal velocity, M is the Morton number, E0 is the Eotvos
number, ρc and µc are the continuous phase density and viscosity respectively, ∆ρ is
the densities differences and γ is the interfacial tension.
For low values of H (i.e H ≤ 2), terminal velocity follows the Stokes’ law which is
Vt =
2.3.2
g∆ρd2
.
18µc
(2.2)
Slip and Characteristic Velocity
Characteristic velocity is the velocity of single drop isolated from other drops
but influenced by column internal, geometry and agitation.
In the 1980’s many researchers made a correlation between slip velocity and
other terms such as hold-up or Reynolds number according to their own experiments
[21]. This led to a new development of the slip velocity and characteristic equation.
14
Based on 117 data points Godfrey and Slater [22] obtained the characteristic velocity
equation of single drop in RDC which is
0.766
Vk
d
= 1.0 − 1.443(Nr3 Dr5 )0.305 − 0.494
,
Vt
Ds − D r
(2.3)
where Vk is the characteristic velocity, Nr is the rotor speed, Dr is the rotor diameter
and Ds is the stator diameter.
However, according to Weiss et.al. [23], although this characteristic velocity is
a function of drop diameter, it is not the actual velocity of the drops. Then in 1996
Ghalehchian[6] extended the equation to become
0.766
d
Vc
Vk
= 1.0 − 1.443(Nr3 Dr5 )0.305 − 0.494
− 4.08 .
Vt
Ds − D r
Vt
(2.4)
The slip velocity is the relative velocity of drops with respect to the continuous
phase. Under counter current flow as in RDC, the slip velocity is given by
Vs =
Vc
Vd
+
.
X
1−X
(2.5)
Godfrey and Slater [22] showed that the slip velocity can also be represented as a
function of characteristic velocity and hold-up, X, that is
Vs = Vk (1 − X)m ,
√
where Vk is the characteristic velocity, m = 0.129 Rek and Rek =
(2.6)
ρc Vk di
muc .
Applying the equation of slip velocity for each drop fraction, with assumption
that the amount of hold-up is uniform around all drop fractions, the basic hydrodynamic
equation may be written as
Vd = Vk (1 − X)m −
2.4
Vc X
.
1−X
(2.7)
Drop Breakage Phenomena
In an RDC column, the drops are dispersed into the column through a
distributor. This distributor is located at the bottom of the column. These drops will
rise up the column and will break into smaller drops of different sizes as they hit the
rotating discs. In the following subsections the important terms of the drop breakage
phenomena are given in order to understand the factors that affect the breakage of
drops in the column.
15
2.4.1
Drop Size
Drop size is a very important variable affecting the hydrodynamic and mass
transfer processes. Several researchers studied the drop breakage in liquid-liquid system
in the RDC column [24, 25, 26, 20]. From these studies, they concluded that Weber
and Reynolds numbers are required to correlate the parameters involves for the drop
breakage factors in the column. According to Korchinsky [20], the smaller drop required
larger column diameter, because of lower slip velocities relative to the continuous phase
but provide larger specific interfacial surface area.
Knowledge on the prediction of column drop size is an important factor in
performance prediction or designing of RDC column.
Large number of relatively
stationary small drops will decrease the column capacity. Larger drops will have larger
volume, low surface area per unit volume, higher slip velocity which means that the
column height must be increased to effect satisfactory extraction efficiencies for such
drops.
2.4.2
Maximum Drop Size
In a dispersion process there is a maximum drop diameter above which no drop
can exist in a stable condition. Kolmogrov’s theory of local isotropic turbulence was
used by Hinze [27] in 1955 to describe this maximum drop size. There is a relationship
between the power consumption per unit mass and the power consumption per disc in
a compartment with the later depending on disc Reynold number. Later more work
were done by Strand et al. [28], Zhang et al. [29], Slater et al. [30] and Chang-Kakoti
et al. [31]. Using several sets of data from different sources, Chang-Kakoti et al. [31]
found a correlation between the maximum drop size and the Sauter mean drop size, to
be
dmax = 2.4d0.8
32 ,
where
(2.8)
16
d32 = 3.6 × 10−5
P =
ρ2c γ 3
,
gµ4c ∆ρ
E=
4PR
πd2c hc ρc
hc 0.18 0.13 0.21
P
E .
Dr
and
−0.658
ρc Nr3 Dr5
PR = 6.87 × ReD
ReD < 6 × 104
PR = 0.069 × Re−0.155
ρc Nr3 Dr5
D
2.4.3
(2.9)
ReD ≥ 6 × 104
Drop Breakage
Drop breakage in liquid-liquid system is induced by the effect of high shear
stress or by the influence of turbulent inertial stress. However as the diameter of the
drop decreases, the deforming stress across it also decreases. Due to these phenomena
a diameter is finally reached where the deforming stress is unable to break the drop.
2.4.4
Critical Drop Size and Critical Rotor Speed
Critical drop size is the maximum drop size below which drop do not break for
a given rotor speed. Correspondingly, for a given drop size in the column, the minimum
rotor speed below which no drop will break is called the critical rotor speed. This rotor
speed was given by Cauwenberg et al. [32],
Ncr = 0.802
γ 0.7
0.4 0.59 D 0.71
ρ0.3
c µc di
r
for both laminar and turbulent regions.
(2.10)
According to him, this relationship has
shown good agreement with the experimental data in RDC columns of different sizes
(152, 300, 600mm). He also found that the equation of critical drop size is
−1.2
Re0.7
dcr = 0.685W eD,ω
D,ω Dr ,
(2.11)
where the disc angular Weber number and the Reynolds number are of the forms
W eD,ω =
ρc (2πNr )2 Dr3
,
γ
(2.12)
17
and
ReD,ω =
ρc 2πNr Dr2
,
µc
(2.13)
respectively.
2.4.5
Initial Number of Drops
The initial number of drops in an RDC could be controlled by the distributor
but basically it depends on the flow rate of dispersed phase, Fd and the size of initial
drop [7], that is
3Fd (d0 /2)−3
N umber of drops
=
.
unit time
4π
(2.14)
These drops are dispersed through the distributor, which is located at the top
or bottom of the column. The counter current flow of the phases can be achieved by
introducing different densities of liquid. The drops will rise up the column if their
density is less than that of the continuous phase. When the drops move through the
medium in the column, they will hit the rotating discs and will possibly break into
smaller drops until they arrive at the settling compartment.
2.4.6
Probability of Breakage
The probability of breakage for a given drop size is defined as the ratio of number
of broken drops and the total number of drops observed in a large sample number. In
1991, Bahmanyar and Slater [25] had introduced this idea of breakage probability.
Based on this idea, Cauwenberg et al. [32] had developed an equation of breakage
probability, P , for laminar region (ReD,ω < 105 ) and turbulent region (ReD,ω > 105 )
which are
P =
and
P =
0.258W e1.16
D,ω,m
1 + 0.258W e1.16
D,ω,m
,
0.00312W e1.01
D,ω,m
1 + 0.00312W e1.16
D,ω,m
(2.15)
,
(2.16)
where W eD,ω,m represents as a modified Weber number that is
W eD,ω,m =
0.5 1.5 − ω 1.5 )D d
ρ0.5
r i
c µc (ω
cr
,
γ
(2.17)
18
and
W eD,ω,m =
0.2 1.8 − ω 1.8 )D 1.6 d
ρ0.8
i
c µc (ω
cr
r
,
γ
(2.18)
respectively.
2.4.7
Mean Number of Daughter Drops Produced
The breakage of a drop will result in various numbers and sizes of daughter
drops. The number of daughter drops produced depends on the initial mother drop
size, physical properties and agitation speed. Many researches have been carried out by
previous researchers concerning the mean number of daughter drops produced. Starting
from Hancil and Rod in 1981, followed by Eid (see [7]) then finally Bahmanyar and
Slater [25] worked out experiments using different chemical systems. The data obtained
were well represented by the equation
di
−1 ,
Xm = 2 + 0.9
dcr
(2.19)
where di is mother drop size, Xm is number of daughter drops produced and dcr is the
critical drop size at the appropriate agitation speed.
Also, from the work done by Coulaloglou and Tavlarides [33] and Jares and
Prochazka [34] concerning the the drop distribution of daughter drops produced,
they found that the Beta distribution function fits the experimental data in most
column including RDC. The Beta probability density function which is related to this
distribution can be written in the form
f (Xm , y) = (Xm − 1)(1 − y)Xm −2
(2.20)
y is volume ratio of daughter drops to mother drops ie d3d /d3m .
2.5
Mass Transfer
In a single phase system, the mass transfer is defined as the movement of mass
or molecules from an area of high concentration to that of low concentration until a
homogeneous or equilibrium concentration in the system is achieved. Basically, there
are two modes of mass transfer in a single liquid phase:
19
Molecular Diffusion This type of diffusion occurs in the absence of any bulk motion
of the liquid. In this mode, the phase will tend to a uniform concentration as a
result of the random motion of the molecules.
Eddy Diffusion Meanwhile eddy diffusion occurs in turbulent flow processes because
of the existence of bulk motion of the molecules. The phase will tend to a uniform
concentration due to agitation.
The mass transfer process in the RDC column considered in this study involves
liquids in turbulent flow. In this column, mass transfer will occur whenever there is a
concentration gradient between the two phases in direction of decreasing concentration.
The rate of mass transfer of materials from one phase to the other depends on the mass
transfer coefficient. The prediction of mass transfer coefficient have been studied by
many researchers and recently by Slater et al.[30] and Bahmanyar et al.[35] on mass
transfer rates of single drop in short RDCs and non-flowing continuous phase. In the
following subsection we provide some theories associated with mass transfer before the
modelling of mass transfer in the RDC column is discussed.
2.5.1
The Whitman Two-film Theory
This theory is the earliest and simplest theory of mass transfer between two
liquids phases across a plane interface [36] . It assumes that there is a thin layer
on both sides of the interface. In this thin film, resistance to mass transfer exists.
The mass transfer across these films is assumed to take place by molecular diffusion.
Outside these films the bulk concentration of the liquid phases are uniform which is
brought about by the eddy diffusion. The eddy diffusion caused by the turbulence in
the bulk is considered to vanish abruptly at the interface of the films. It is assumed
that equilibrium is established between the two phases at the interfaces. Therefore any
resistance to transfer at the phase boundary is non-existent.
In order to describe the above process explicitly, let two liquids X and Y with
bulk concentrations, xb and yb respectively where the direction of mass transfer is
assumed from the X phase to the Y phase. The schematic diagram in Figure 2.3 will
describe the process. In the X phase, mass transfer at a steady state from the bulk
20
concentration to the interface is described by the equation
Jx = kx (xb − xi ),
(2.21)
and for the Y phase, the mass transfer is from the interface to the bulk concentration
which is
Jy = ky (yi − yb ),
(2.22)
where Jx and Jy are known as flux or rate of mass transfer. Meanwhile kx and ky are
the mass transfer coefficients for the liquids X and Y respectively.
Direction of mass transfer
x
b
x
XïPhase
YïPhase
i
yi
yb
Raffinate Phase
Extract Phase
Interface
Figure 2.3: Mass transfer at interface
Since equilibrium is established between the two phases at the interface, the
fluxes must be the same. Thus
kx (xb − xi ) = ky (yi − yb ).
(2.23)
The expression governing the equilibrium at the interface is known as
equilibrium relation or equation. For example, the equilibrium relation for cumeneisobutyric acid-water system is derived from the distribution of isobutyric acid (the
solute) between cumene and water data reported by Bailes et al. [37]. From this data
the equilibrium equation for the system is obtained which is
Cd = 0.135Cc1.85 ,
(2.24)
21
where Cd and Cc are concentrations of isobutyric acid in cumene and aqueous
respectively. Another example is the equilibrium equation of butanol-succinic acidwater system where the equation is
Cd = 1.086Cc − 0.849 × 10−3 Cc2 − 0.162 × 10−4 Cc3 .
2.5.2
(2.25)
The Penetration Theory
In Whitman two-film theory, we only consider the mass transfer across the
interface as a steady-state process of molecular diffusion whereby in this study
the process of mass transfer in RDC column is actually an unsteady-state process.
Therefore a theory proposed by Higbie (see Slater[38]) known as penetration theory is
introduced here.
The theory was about a mechanism of mass transfer involving the following
processes:
• The movement of eddies from the bulk of a fluid with concentration cb to the
interface at a distance z from its original position.
• At the interface the solute transfer takes place by unsteady state molecular
diffusion for a short exposure time t.
• This bulk of fluids is then being replaced by another bulk of fluids as a result of
eddy diffusion.
The equation governing the transfer process is given by
∂2c
∂c
=D 2
∂t
∂z
(2.26)
with the initial and boundary conditions
c = cb ,
z > 0,
t=0
(2.27)
c = ci ,
z = 0,
t>0
(2.28)
c = cb ,
z → ∞,
t>0
(2.29)
22
The solution of the above diffusion can be shown as
2
c − cb = (ci − cb )(1 − √
π
where
√2
π
√z
4Dt
0
√z
4Dt
exp(−u2 )du),
(2.30)
0
exp(−u2 )du is readily evaluated since it is actually a tabulated error
z
of function (erf( √4Dt
)). Therefore the expression (2.30) can be written as
c − cb = (ci − cb )(1 − erf ( √
z
)).
4Dt
(2.31)
By Fick’s first law [38], the rate of mass transfer per unit area across the interface
at any instant can be found by evaluating
∂c
= −D
∂t z=0
D
.
= (ci − cb )
πt
Jt
Averaging over time of exposure, te , gives
(ci − cb ) te D
J =
dt
te
πt
0
D
= 2(ci − cb )
.
πe
(2.32)
(2.33)
From (2.22) and (2.23), the film mass transfer coefficient of the continuous and dispersed
phase are kx = 2 πDe and ky = 2 πDe respectively.
2.5.3
Dispersed Phase Mass Transfer Coefficient
Several theoretical models have been proposed for the estimation of the
dispersed phase mass transfer coefficient (Godfrey and Slater[22]). They found that
the dispersed phase mass transfer coefficient, kd is time-dependent. In general, three
situations arise depending on the state of the drops which are
•
Molecular diffusion − Newman developed an equation for resistance in a solid
sphere that is
∞
d 6 1
−4n2 π 2 Dd t exp
kd = − ln 2
6t
π
n2
d2
1
23
Then, in 1953, Vermeulen[39] proposed a useful approximation to this equation as in
Talib[5], ie
kd ≈ −
•
−4n2 π 2 Dd t 1/2 d ln 1 − (1 − exp
)
6t
d2
Circulating of drop − A circulating motion inside drops is induced by drag forces
arising from relative velocity of motion between a drop and continuous phase. Kroglg
and Brink (see [7]) provided a general solution for the problem of heat transfer which can
be used for both phases . Then this idea was expanded by Calderbank and Korchincki
to obtain an equation for the mass transfer coefficient in drops with laminar internal
circulation (see Godfrey and Slater[22]). The equation is
∞
√
d 6 1
−4n2 π 2 Dd t kd = − 2.25 ln 2
exp
6t
π
n2
d2
1
•
Oscillating drops − As drops become larger their shape may change due to the
nature of drag force involved. At some critical size, drops can start to oscillate in shape
and drag force are such that terminal velocity decrease as the drop size increase further.
Many models have been proposed to predict mass transfer coefficient under oscillating
condition. Skelland et al. [40] suggested that
kd = 31.4
2.5.4
Dd 4Dd t −0.43 µd −0.125 V 2 ρc 0.37
.
d
d2
ρDd d
γ
Continuous Phase Mass Transfer Coefficient
There are three types of flow around the drops which influence the transfer of
solute from outside a stagnant drop. They are radial diffusion, natural convection and
forced convection. The continuous phase mass transfer coefficient for these types of
flow are correlated as a Sherwood number and are given respectively as
Shc = a1
where a1 is a constant,
Shc = a1 + a2
gρc ∆ρd3
n
Scc ,
2
µc
where Scc is Schmidt number defined as Scc = µc ρc Dc , a1 , a2 , n are constants and ∆ρ
is the difference in density and
Shc = a1 + a2 (Rf )n (Scc )l ,
24
where a1 , a2 , n, l are constants.
The transfer is by radial diffusion if the continuous phase is stagnant. Whilst the
transfer is by natural convection if the continuous phase around the drop is subjected
to convection. For the latter type of flow, the transfer is by forced convection if the
continuous phase around the drop is subject to an external force forcing the continuous
phase to flow past the drop with velocities up to those of complete turbulence.
2.5.5
Overall Mass Transfer Coefficient
If the equilibrium distribution of solute strongly favours one phase, then the
principle resistance to mass transfer lies in the other phase. A brief explanation about
this concept can be found in [7]. In this work, the required overall dispersed phase
mass transfer coefficient, Kodi for drops with size di in stage n, is defined as
Kodi =
∞
4Dd βn2 tr,i di 6L2
exp
−
ln
,
2
6ti
βn2 (βn2 + L(L − 1))
d
i
n=1
where, L known as Sherwood number, βn cot βn + L − 1 = 0. The first six values of βn
for specified value of L are given by Crank[11].
2.6
The Existing Forward Mathematical Models of the Processes in the
RDC column
In this section, the existing forward mathematical model of the processes
involved in the RDC column are reviewed.
This review only covers the most
recent researches on the models which were produced by Talib[5], Ghalehchian[6] and
Arshad[7] and Mohamed[41].
2.6.1
Talib’s work
Drop Distribution Model (Hydrodynamic Model)
Talib had modelled the break-up process for the drops moving up the RDC
column by assuming that each compartment has ten numbers of classes or cells of
equal widths which hold drops with sizes in the specified range.
25
Light phase drops entering the extraction column from the distributer has the
chance of breaking into smaller drops on hitting the first rotor disc. The drops then
moved into the first compartment. Depending on their sizes, the drops are placed in the
appropriate cell. In a given cell all drops are then treated as having the same average
diameter size when considering possible breakage as they moved past the next disc.
Continuing in this way up the column, the number of drops and their size distribution
for all the compartments in the column can be determined.
In Talib’s work, the distribution of the drops were determined by two methods
namely the Monte Carlo and the Expected Value methods. In the simulation drop
break-up using the Monte Carlo Method, drops are considered as entering and moving
up the column one at a time. Meanwhile, the simulation by the latter method considered
the break-up of a swarm of N drops. Beside that, the simulation of drop break-up using
this method uses the probability, p and beta distribution, φ(x, y) differently from the
first method which uses random number.
Even though the distribution of the drops are found to be similar for both
methods, Talib concluded that the latter method was more efficient due to the less
simulation time needed and data requirement. The Monte Carlo Method requires
detailed information including the number of daughter drops produced from the breakup of a drop, which depends on the size of the mother drop, the rotor speed and the
liquid used. However, the Expected Value Method requires only the average number
daughters produced.
Furthermore, Talib had introduced another model, Dynamic Expected Value
Method which was a modification of the EVM. The model was expected to give a more
realistic representation because it is based on different drop velocities for each class of
the drops whilst in the previous model, it was assumed that all drops have the same
velocity irrespective of their sizes.
Mass Transfer Model
In Talib’s early work, he had introduced two unsteady stage-wise models of
the mass transfer process in the RDC column namely the Initial Approach of mass
Transfer(IAMT) and the Boundary Approach of mass Transfer (BAMT). The first
model is based on the start up process of the drops entering a column with an
26
undisturbed continuous phase whilst the second model is based on the presence of
drops throughout the length of the RDC column.
Talib also introduced the concept of the diffusion in a sphere, the theory of
the film mass transfer coefficients and the two film-theory. Beside that at the early
development of the models, Talib used the linear driving force for both the drop and
the continuous phases. Since the interface of a liquid drop in a continuous phase is
spherical in RDC column, Vermuelen[39] stated that the driving force in a drop can
be considered as non-linear which is known as the quadratic driving force. Talib then
incorporated this idea into the IAMT mass transfer model.
2.6.2
Ghalehchian’s work
In Ghalehchian’s work, hydrodynamic and mass transfer experimental results
from a pilot RDC column of 23 stages were used. Then a new stage wise model with
back-flow was developed. The model took into account the influence of drop breakage
at each stage.
Generally, Ghalehchian had produced the new steady state model of mass
transfer which is also basically based on the mathematical equation discussed in
Talib[5]. Figure shows the stage wise mass transfer process, where e is the back-flow
coefficient which is equal to Fd /Fc . Fd and Fc here, are dispersed and continuous phase
flow rate. The new model was said to be more realistic by including the idea of axial
mixing. The model also considered the extraction of unclean solution.
2.6.3
Mohamed’s work
In Mohamed’s work, the mathematical modelling of simultaneous drop diffusing
in RDC column is developed. In this model, it is assumed that the distribution of the
drops in the column is in equilibrium and the mass transfer from the continuous phase
to every drop occur simultaneously.
The total concentration for every drop is obtained by the Simultaneous Discrete
Mass Transfer(S-DMT) model. The model is actually based on Discrete Mass Transfer
27
Fc , Cc , Nst 1
Fd , Cd , Nst
Nst
Nst-1
eFc , Cc ,n
n
(1 e) Fc , Cc ,n
eFc , Cc ,n1
2
1st stage
Fc , Cc ,1
Fd , Cd ,0
Figure 2.4: Stage wise back-flow for mass transfer process
Model as discussed in Talib[5]. From the model, Mohamed concludes that the two
drops will provide more cross-section area for the mass transfer compared to a drop for
same total volume.
2.6.4
Arshad’s work
In Arshad’s work, the hydrodynamic model is close to reality by following the
process from an undisturbed state into steady state. The model was found to reach
steady state quicker compared with Talib’s. The model was expected to update the
28
value of the hold-up and the velocities of the drops moving up the column before
they reach the final stage. Then Arshad used the mass transfer model developed by
Ghalehchian to combine with the new hydrodynamic model. Arshad considered four
different physical/chemical systems of two different sizes of the RDC column.
In addition, Arshad observed and analyzed the simulation data to examine the
effects of varying input variables on output values yield. The analysis was done by
Principle Component Analysis (PCA) method. Arshad also had provided a review on
Artificial Neural Network (ANN) and Fuzzy Logic (FL) modelling. At the final stage
of his work, Arshad had introduced these concept to the RDC system.
Besides producing a new mass transfer model which is based on the IBVP with
varied boundary condition, another aim of this research is to develop a model that
can determine the value of input parameters for a desired value of output parameters.
This type of modelling is called inverse modelling. Therefore the reviews on the inverse
problem including the terminologies, real world examples, classes of inverse problem
and the steps to get the solution of the inverse problem are provided in the following
section.
2.7
2.7.1
Inverse Modelling
Introduction
Inverse Problems or more precisely Inverse Modelling, are the most challenging
in computational and applied science and have been studied extensively. Although
there is no precise definition, the Inverse Problems refers to a wide range of problems
that are generally described by saying that they are concerned with the determination
of inputs or sources from observed outputs or responses. This terminology is contrary
with the forward problem. The opposition of terminologies can be illustrated in the
schematic diagrams as shown in Figures 2.5 and 2.6.
From the mathematical point of view, the distinguishing aspect of inverse
problem is that they are usually ill-posed problems. Ill-posed means that an inverse
problem will generally violate one or more of the properties of a well-posed problem as
defined by Hadamard in [42]. According to Hadamard, a problem is called well-posed
29
Input
Output
Process
Model
x
?
Effect
Cause
Figure 2.5: Forward problem
Input
?
Output
Process
Model
y
Effect
Cause
Figure 2.6: Inverse problem
if its solution satisfies the following three requirements:
• existence
• uniqueness
• stability.
The distinctions between the ill-posed and well-posed problem are tabulated in Table
2.1.
Table 2.1: The ill-posed and well-posed problems
Well-posed problem
i. a solution always exists
ii. there is only an unique solution
iii. a small change in the problem
leads to a small change in the solution
Ill-posed problem
i. a solution may not exist
ii. there may be more than one solution
iii. a small change in the problem
leads to a big change in the solution
For a long time it was thought that only well-posed problems have physical
meaning.
Nowadays the thought is different, ill-posed problems are no longer
discriminated. On the contrary, they have become synonyms for mathematically very
difficult but particularly interesting problems.
Inverse modelling can be defined as the process in obtaining the value of inputs
30
or sources from observed outputs or responses. In other words, it is the process of
solving the inverse problem. In order to solve an inverse problem or in order to develop
an inverse model, the following points have to be considered:
• the need to study and understand the process of a system (study the relationships
between the variables or parameters of the system).
This means study the
possibility of mathematical modelling of the process,
• the need to study the technique of solving this system (methods of solving the
forward problem),
• development of an algorithm for the numerical solution of the inverse problem.
Even though inverse problems have been enormously influential in many area of
industrial applications, but from the literature on the RDC column we found that the
inverse model of the process had never been studied. This motivated us to explore the
inverse problem area in general. Before we are able to construct the inverse modelling
of the process in the RDC column, a review on the inverse modelling itself including
some examples in other applications are rather necessary. Therefore as a start, in the
following section the typical examples of inverse problem in other areas will be given.
2.7.2
Inverse Problem in Sciences and Engineering
In the last twenty years, the field of inverse problems has undergone rapid
development. This is due to the fact that computing technology and the development
of powerful numerical methods have enormously increased which made it possible to
simulate complex problems. In addition, there exist many problems in sciences and
engineering which are ill-posed and in need of solution. This leads to a growing appetite
and stimulation of mathematical research particularly on the uniqueness solution and
developing stable and efficient numerical methods for solving such problems.
There are many examples of inverse problem in many fields of science and
engineering. A famous example is the X-ray CT (computed tomography) problem
[43]. X-ray CT is a medical imaging technique that produces images of a single plane
through the human body. In X-ray CT, a tomographic image is reconstructed from
31
X-ray shadow images taken from a set of different directions. The inverse problem
defined in this application is the determination of the mass density of the human body
using absorbtion X-ray.
Another interesting set of applications is related to optical or diffusion
tomography [44]. Optical or diffusion tomography refers to the use of low-energy
probes to obtain images of highly scattering media. It was recently discovered by
several experimental groups in the US that ultrafast laser pulses (about 10 picosecond
pulses) can propagate through so-called turbid media. More importantly, one can
detect differences in detector responses due to the presence of small inclusions with
different optical properties hidden in such media. The most obvious examples of turbid
media are coastal water and biological tissues. This discovery opened opportunities to
discover mines in coastal waters, to image early stage breast tumors etc.
Other examples such as inverse scattering for constructing the potential energy
from the phases of the scattered waves, identification of unknown heat conductivity
by means of temperature measurements on the boundary, estimation of structure and
properties of the earth etc [45, 46, 47, 48]. Alberg[45] introduced the concept of wave
splitting. In his works, the direct and the inverse scattering problems are solved for
a homogeneous semi-infinite by the use of time-domain technique. This technique has
been very successful in the solution of the inverse scattering problem. According to
Alberg[45], this method, which is based upon a wave-splitting concept in conjunction
with imbedding or Green function technique, has proved to be very efficient.
Abdullah and Louis[46] introduced an application of the method of approximate
inverse to a two-dimensional inverse scattering problem. They determined that the
refractive index from an inverse experiment is a nonlinear inverse problem and proposed
to split this nonlinear problem into an ill-posed linear problem and a well-posed
nonlinear part. This was done by first solving the data equation for the induced
sources, consisting of the product of the refractive index and the field inside the
object. This procedure retains the nonlinear relation between the two unknowns and
treats it implicitly. The linear problem is efficiently solved by applying the method of
approximate inverse. The nonlinear part is solved by treating the object equation. The
use of the method of approximate inverse makes it possible to determine the refractive
index and to locate inhomogeneities in the inverse medium problem. Further examples
32
of inverse problems can be found in [49].
The following subsection will provide the information of the general
classification for inverse problems.
2.7.3
Classes of Inverse Problem
Inverse problems may be classified in different ways. One way of classifying
inverse problems is by the type of information that is being sought in the solution
procedure. The classes are
• Backward inverse problem
In this class of problem, the initial conditions are to be found. For example,
consider the heat equation
ut = uxx ,
u(0, t) = u(π, t) = 0,
u(x, 0) = f (x).
(2.34)
(2.35)
(2.36)
The solution of the above heat equation is
u(x, t) =
∞
2
fn e−n t sin nx.
(2.37)
n=0
Then we formulate the backward problem as follows. Given g(x) = u(x, 1), can
we recover f (x) = u(x, 0)?. It is clear that, in the example above we are seeking
the solution of the initial condition which is u(x, 0).
• Coefficient Inverse Problem
Coefficient inverse problem is a parameter estimation problem where a constant
multiplier or parameter in a governing equation is to be found.
This type
of problem is very common in physics when the physical laws governing the
process are known but information about parameters is needed. Here, we wish to
determine the missing parameter values in order to get the mathematical model
which best describes the phenomena.
33
As an example, consider the heat conduction in a material occupying a domain
X whose temperature is kept at 0 at the boundary. The temperature u after
sufficiently long time can be modelled by:
ut = κuxx ,
(2.38)
u(0, t) = u(π, t) = 0,
(2.39)
u(x, 0) = 0,
(2.40)
where κ is characteristic of the material and is known as heat conductivity. The
inverse problem is to determine the heat conductivity, κ from the measurement
of the temperature (observed) and the flux κuxx on the boundary.
• Boundary Inverse Problem
Some missing information at the boundary of a domain is to be found. Note that
this can be a function estimation problem when this boundary condition changes
with time. It is also known as the control problem.
There is an example of a boundary inverse problem in the Inverse Heat
Conduction Problem
∂2T
∂T
=D 2,
∂t
∂z
where the unknown thermal action at the boundary of the object is to be found
based on observations (measurements) of the temperature on the interior of the
object.
2.7.4
Solution of Inverse Problem
Solution of inverse problem involves determining unknown causes (or inputs)
based on observation of their effects (or outputs).
This is in contrast to the
corresponding direct problem, whose solution involves finding effects based on a
complete description of their causes.
For example, methods of solving the inverse problem in mass transfer
require a rational combination of physics, mathematics and engineering including
experimentation knowledge. This combination actually forms a new research paradigm.
The same methodology can be applied to many research problems, in particular to
design and control system.
34
The characterization of new or unknown processes in solving the inverse
problems includes the following steps:
1. Construction of a mathematical model of the process from basic physical laws
(mathematical statement of the direct problem governing the process).
2. Development of inverse problem (IP) algorithms necessary to solve the
corresponding inverse problem.
The inverse problem naturally arises when
parameters or functions in the mathematical model need to be determined.
3. Mathematical validation of the IP algorithm by numerical experiments.
4. Design of experiments to gather information about the process in order to solve
the IP.
5. Experimental (data gathering).
6. Use of the IP algorithm to identify the unknown information about the process.
Analysis of the results, including statistical treatment is crucial to verify the
correctness of the results and ultimately, the adequacy of the description of the
process.
The following section will describe the concept of Fuzzy set to be used in solving
the problem mentioned above.
2.8
Fuzzy Logic Modelling
Fuzzy set techniques have been recognized as a powerful tool for the
development of models for systems that are not amenable to conventional modelling
approaches due to the lack of precise, formal knowledge about the system, strongly
nonlinear behavior or time varying characteristics [50]. Also at the computational
level, fuzzy models can be regarded as flexible mathematical structures, similar to
neural networks, that can approximate a large class of complex nonlinear systems to a
desired degree of accuracy [50].
Liquid-liquid extraction in an agitated RDC column is a complex system for
mathematical modelling. Even though satisfactory mathematical models have been
35
developed, the prediction of the values of the input parameters for desired values of
output parameters still could not be done successfully. Therefore, in this study a fuzzy
approach for modelling the system is considered.
The first subsection briefly reviews several basic concepts of fuzzy set theory
where it starts with an introduction to crisp set theory. In the subsequent subsection,
two systems which are based on fuzzy algorithm will be presented and then several
procedures for the building of fuzzy models are outlined.
2.8.1
The Basic Concepts of Fuzzy set Theory
The Crisp Set Theory
The ordinary crisp set theory is defined in such a way that individuals in a
universe are divided into two groups: members and non-members. The fundamental
review of crisp set theory and the crisp set operations can also be seen in many
literatures, for examples in [51, 52, 53].
The Fuzzy Set Theory
Fuzzy sets are not intended to replace crisp set theory but rather to bring
mathematics closer to reality. A fuzzy set may be defined as follows:
Definition 2.1. [51] Let X be the universal set with typical element denoted by x. A
fuzzy set F in X is characterized by a membership function µF : X → [0, 1], with the
value µF (x) representing the grade of membership of x in F .
Fuzzy sets are always mapping a universal set into [0, 1]. Conversely, every
function µ : X → [0, 1] may be considered as a fuzzy set ([52]). For example one can
define a set F1 = {x ∈ |x is about a2 } with triangular membership function as below

1

, x ∈ [a1 , a2 )
 ax−a

2 −a1



 1,
x = a2
(2.42)
µF1 (x) =
−x+a3


,
x
∈
(a
,
a
]
2
3

a3 −a2



 0,
otherwise
then the graphical description can be expressed as in Figure 2.7.
36
µF (x)
1
1
x
a1
a
2
a3
Figure 2.7: F1 = {x ∈ |x is about a2 }
Since a function can be represented by a set of ordered pairs, any fuzzy set F
can be written as
F = {(x, µF (x))|x ∈ X}
The Alpha Cut(α cut)
Definition 2.2. [51] An α-cut, Aα , is a crisp set which contains all the elements of
the universal set X that have a membership function at least to the degree of α and can
be expressed as
Aα = {x ∈ X|µA (x) ≥ α}
(2.43)
In addition, the set Aα = {x ∈ X|µA (x) > α} is called the strong α-cut. The
alpha cut is an important concept in procedures for creating the fuzzy environment as
well as assisting defuzzification.
The Fuzzy Numbers
A convex and normalized fuzzy set F , is called a fuzzy number. The definition
of convex and normalized fuzzy set are given below:
Definition 2.3. [51] Let F = {(x, µF (x))|x ∈ X} be a fuzzy set. F is called a convex
fuzzy set if
µF (λx1 + (1 − λ)x2 ≥ min[µF (x1 ), µF (x2 )], ∀x1 , x2 ∈ X and ∀ λ ∈ [0, 1].
Definition 2.4. A fuzzy set F is normal if there exists at least an element with
membership grade of 1.
Extension Principles
A principle for fuzzifying a crisp function is called an extension principle. It
was proposed by Zadeh [52] in 1965 to allow the extension of any point operations to
37
operations involving fuzzy sets. Klir[51] defined the extension principle as follows: Let
φ : X n → F (Y ) with φ(µ1 , µ2 , ..., µn )(y)) = sup{min{µ1 , µ2 , ..., µn }|(x1 , x2 , ..., xn ) ∈
X n and y = φ(x1 , x2 , ..., xn )}
In other words, a function φ : X n → F (Y ), which maps the tuples (x1 , x2 , ..., xn )
of X n to the crisp value φ(x1 , x2 , ..., xn ) of Y , can be extended in a proper way to a
function φ : (F (X))n → F (Y ). The extension principle extends the mapping in such a
way that it preserves the image of the elements of X.
2.8.2
Fuzzy System
Recently, logical models, which take a different form from mathematical ones,
have come into use, [53]. According to Terano[53], this type of models are divided into
two different types which are structural model and fuzzy logic model.
In structural model, the principle factors that make up the problem are
determined and the relationships among these factors are investigated and then
represented graphically. Although this type of model is effective for use with complex,
ambiguous problem but the representation of the principle factors is not clear. It is
because it uses a two-valued, “yes” or “no”, system to represent the relationships of
these factors.
The latter type of model is more suitable for expressing the ambiguity of
meaning found in natural language. The detailed explanation of how to develop fuzzy
modelling for a system will be found in the following subsection.
2.8.3
Fuzzy Modelling
The basic principles of fuzzy modelling were laid down by Zadeh [54] in 1973.
The area has grown rapidly since then, especially concerning the complexity of a system.
There are three principles in developing a fuzzy model, which are
• the use of linguistic variables in place of or in addition to numerical variables,
38
• the characterization of simple relations between variables by conditional fuzzy
statements,
• the characterization of complex relations by fuzzy algorithms.
These principles form the basis of two methods used for fuzzy modelling, namely
the direct approach and system identification. In the first method, the system is first
described linguistically using terms from natural language and is then translated into
the formal structure of a fuzzy system. The second method is developed from structure
identification to the parameter identification. Figure 2.8 shows a fuzzy modelling
approach compared to mathematical modelling.
INPUT CRISP
VALUES
Mathematical Modelling
OUTPUT CRISP
VALUES
Fuzzy Modelling
FUZZIFICATION
FUZZY
ENVIRONMENT
DEFUZZIFICZTION
Figure 2.8: The fuzzy logic modelling
2.8.4
Remarks
In this section the inverse problems by fuzzy approach on MISO and MIMO
systems, which were solved by Ahmad[8] and Ismail[10] are briefly presented.
39
Ahmad’s work
Ahmad developed an algorithm which was used in determining the optimized
electrical and geometrical parameters of microstrip lines. The problem was presented
as MISO system of some algebraic equations. The procedure involved in solving this
problem was taken in three phases which were:
• Fuzzification Phase
In this phase all the input and performance parameters required for the model
were fuzzified using the chosen membership function.
In Ahmad’s work,
triangular membership function was used.
• Fuzzy Environment Phase
The fuzzified values from the fuzzifaction phase were processed by Zadeh’s
extention principle to obtain the induced performance parameters.
• Defuzzification Phase
In this phase the optimal solution was determined.
In this work, Ahmad had also used a triangular induced performances parameter
and produced theorems of optimized defuzzified values which were used to determine
a most appropriate or optimized value of generated combination data.
Ismail’s work
Ismail produced a fuzzy algorithm for decision making in a MIMO system where
the system was modelled by a crisp state-space equation. Similar to Ahmad’s work, the
development of the algorithm is based on the three phases. In Ismail work, a triangular
induced performances parameter was used in order to get the optimized value of the
generated combination data.
2.9
Summary
In the beginning of this chapter, an overview of liquid-liquid extraction process
was provided. A review has been given, starting with the principle of the process and
40
it was then followed by the classification of the extraction equipment. To achieve the
aim of this research, the review on the RDC column was briefly given including the
important processes involved.
The mass transfer in the column are effected by the drop distribution and
breakage phenomena. The detailed description about these terms were provided leading
to the review on the mass transfer itself. The theoretical details on the mass transfer
coefficient were also included. A review on existing forward mathematical modelling by
the most recent researchers were also presented. These reviews are significantly used as
a background in developing a new mass transfer model; the works are detailed starting
in Chapter 3.
In section 2.7, the inverse problem in general, including the definition, the
examples of real world problems, the classes of the inverse problem and the steps
involved in solving the problems were reviewed. These reviews will serve as a motivation
to develop an inverse model of the mass transfer in the RDC column. The development
of the model is described starting in Chapter 5. Section 2.8 provides the review on
the Fuzzy Concepts. These concepts are applied in Chapters 5 and 6 to develop an
algorithm for solving the inverse problem.
CHAPTER 3
THE FORWARD MASS TRANSFER MODEL
3.1
Introduction
The existing mass transfer models mentioned in Chapter 2 were based on
a radial diffusion equation with constant boundary conditions.
However a mass
transfer model with varied boundary conditions has yet to be developed. Therefore,
in this chapter, a modified forward mass transfer model with time dependent function
boundary condition will be discussed. This function is derived from the experimental
data obtained in [5].
Following this derivation, Section 3.3.1 presents the details on how the solution
of the diffusion equation of the time varying boundary condition is obtained. This is
followed by the derivation of the new fractional approach to equilibrium. In Section
3.4, the simulations for different drop sizes are carried out to see their effects on the
new fractional approach to equilibrium. Then a comparison between the fractional
approach to equilibrium introduced in [11] and the new one is also carried out in the
last section of this chapter.
3.2
The Forward Mass Transfer Model
In the RDC column, the mechanism of mass transfer across an interface between
two liquid phases is based on penetration theory. This theory was proposed by Higbie
in 1935 (see Slater [38]), which assumed that a packet of fluid with bulk concentration
travel to the interface at a distance from its original position. At the interface, the fluid
42
packets undergo molecular diffusion for a short exposure of time, before being replaced
by another fluid packet.
The model discussed in this chapter is based on the model of the mass transfer
developed by Talib [5]. Talib [5] assumed that at each stage i, the drop has an initial
uniform concentration as well as the concentration of the medium phase. When a drop
enters a stage i, solutes from the uniform medium concentration surrounding the drop
are transferred to the drop or vice versa depending on the concentration difference
between the drop and the medium. In this study, only the transfer of solute from the
medium to the drop will be considered.
The study of the shape of the moving drops has been found useful in
understanding the dynamics of the moving drops since the drag on the drops depends
on their shapes during movement in another medium. The shape of liquid drops moving
in liquids is dependent on the balance between the hydrodynamic pressure exerted on
account of the relative velocities of the drop and field liquids, and the surface forces
which tend to make the drop a sphere. In this study we assume that all the drops
are spherical in shape, therefore the amount of solute transferred to the drops can be
obtained by using the concept of diffusion in a sphere.
3.2.1
Diffusion in a Sphere
Consider a sphere of radius a. The radial diffusion equation is
∂C
=D
∂t
∂2C
2 ∂C
+
2
∂r
r ∂r
(3.1)
where C = C(r, t) is the concentration at distance r from the center of the sphere at
time t and D is the diffusion constant.
If we make the substitution u = Cr, Equation (3.1) becomes
∂2u
∂u
=D 2
∂t
∂r
(3.2)
43
If the sphere of radius a has initial uniform concentration c1 and the surface of
the sphere is maintained at a constant concentration c0 , the diffusion equation of the
sphere with a constant diffusion coefficient D is given by the initial boundary value
problem (IBVP),
∂2u
∂u
= D 2 , 0 ≤ r < a,
∂t
∂r
u(0, t) = 0, t > 0
t≥0
(3.3)
(3.4)
u(a, t) = ac0 ,
t>0
(3.5)
u(r, 0) = rc1 ,
0≤r<a
(3.6)
These equations can be solved by the method of separation of variables.
Setting u(r, t) = R(r)T (t), we will get an ordinary differential equation of
T
R
=
= −λ2 ,
R
DT
where λ2 is a separation constant. Thus the general solutions for the space and time
variations are given as below,

 B1 + B2 r,
λ=0
R(r) =
 A cos λr + A sin λr, λ =
0
1
2

 B3 ,
λ=0
T (t) =
2
 Be−Dλ t , λ = 0
(3.8)
Therefore, the general solution for u(r, t) is

 (B1 + B2 r)B3 ,
λ=0
u(r, t) = R(r)T (t) =
2
 (A cos λr + A sin λr)Be−Dλ t , λ =
0
1
2
where A1 , A2 , A3 , B, B1 , B2 , B3 are the arbitrary constants.
Simplifying the above equation, we get

 (H + Ir),
λ=0
u(r, t) =
2
 (J cos λr + K sin λr)e−Dλ t , λ =
0
Using the superposition rule, we get
2
u(r, t) = (H + Ir) + (J cos λr + K sin λr)e−Dλ t ,
44
subject to boundary condition of Equations (3.4) and (3.5), we get
2
u(0, t) = 0 = H + Je−Dλ t ,
2t
Since the coefficient of H and e−Dλ
t>0
(3.12)
are linearly independent on the t interval,
it follows from (3.12) that we need H = 0 and J = 0, thus
2
u(r, t) = Ir + K sin λre−Dλ t ,
and
2
u(a, t) = ac0 = Ia + K sin λae−Dλ t ,
t>0
or
2
a(c0 − I) − K sin λae−Dλ t = 0.
t>0
(3.15)
2
Again invoking the linear independence of coefficient of a(c0 − I) and e−Dλ t , it follows
from (3.15) that a(c0 − I) = 0 and K sin λa = 0, which gives
I = c0
K = 0,
or
sin λa = 0,
(or both)
Here, the rule is to make the choice so as to maintain as robust a solution as possible,
then we take sin λa = 0 implies λn =
nπ
a
for n = 1, 2, 3, ......
Thus the solution becomes
u(r, t) = c0 r +
∞
Kn sin
n=1
nπr −Dn22 π2 t
e a
.
a
To find the Kn , and hence to complete the solution of the problem, we now set t = 0
in the expression on the right and replace u(r, 0) by the initial condition u(r, 0) = rc1 ,
then we obtain
u(r, 0) = rc1 = c0 r +
∞
Kn sin
n=1
or
r(c1 − c0 ) =
∞
Kn sin
n=1
nπr
,
a
nπr
.
a
This shows that the Kn are the coefficients in the half-range Fourier sine series
expansion of r(c1 − c0 ) over the interval 0 ≤ r ≤ a. Thus the Kn are given by
2
Kn =
a
a
0
r(c1 − c0 ) sin
nπr
dr,
a
45
and so
Kn =
−2a
(c1 − c0 )(−1)n .
nπ
Thus the solution of IBVP of (3.3)-(3.6) becomes
∞
(−1)n
nπr −Dn22 π2 t
2a
u(r, t) = c0 r + (c0 − c1 )
sin
e a
π
n
a
(3.21)
n=1
or
∞
C(r, t) = c0 +
(−1)n
nπr −Dn22 π2 t
2a
(c0 − c1 )
sin
e a
,
πr
n
a
(3.22)
n=1
where C(r, t) is the concentration of the drop at time t. In our work, we are interested
in the average concentration of the sphere, Cav , given by equation
Cav =
Ct
,
4πa3 /3
where the total concentration Ct of the sphere is obtained from
a
Ct =
C(r, t)4πr2 dr.
(3.23)
(3.24)
0
According to Crank [11], the total amount of diffusing substance entering or
leaving the drop which is denoted as fractional approach to equilibrium is used to relate
the analytical results to a mass transfer coefficient that is
F =
Cav − c1
c0 − c 1
(3.25)
where Cav is the average concentration of the drop at time t and c1 and c0 are the
initial and boundary concentrations respectively.
Using this concept, the fractional approach to equilibrium is derived for the
problem of equations (3.3) to (3.6) , which is
∞
6 1 Dn22π2 t
(e a ),
Fc (t) = 1 − 2
π
n2
(3.26)
1
where the subscript c in Fc indicates that this term is already derived by Crank [11].
3.3
The Modified Model
As mentioned in previous section, we are interested in developing an improved
model of mass transfer of which the boundary condition is a function of time. To
46
achieve this, we consider the normalized data obtained from the experimental work of
mass transfer in the RDC column with 152mm diameter and 23 stages of the iso-butyric
acid/cumene/watersystem (Talib [5]). In this system the iso-butyric acid in water is
acting as the feed(continuous phase) and the cumene is the solvent(dispersed phase).
The geometrical details of the RDC column used and the physical properties of this
system are given in Appendices A.1 and A.2.
Table 3.1: Normalized dispersed and continuous phase concentrations
Stage number
dispersed(drop)
continuous(medium)
0
0
0.912
7
0.118
0.947
11
0.162
0.960
15
0.232
0.981
19
0.269
0.992
23
0.285
0.997
24
0.294
1.00
Note: Stage 0 in Table 3.1 is the feed and exit for the dispersed and continuous phases
respectively and stage 24 is the exit and feed for the dispersed and continuous phases
respectively.
From the normalized data (see Table 3.1), we find that the concentration of the
continuous phase depends on the stage of the RDC column (the concentration is lesser
at the lower stage than the upper stage). This phenomenon has shown that there is
a mass transfer from the continuous to the dispersed phase. To relate the changes of
concentration with time, we consider 10 different classes of drops being formed from a
single mother drop as the mother drop hits the first rotor disc of the column. These 10
different sizes of daughter drops have different velocities depending on their sizes. The
velocity of the drops can be calculated using equation
v = vk (1 − h)m ,
(3.27)
where vk is the characteristic velocity of the drop. h is the hold up which is defined as
the ratio of the total volume of the freely moving drops present in the column to the
47
volume of the column and m is a constant.
0.07
velocity of the drops(m/s)
0.06
0.05
0.04
0.03
0.02
0.01
0
1
2
3
4
5
radius(m)
6
7
ï3
x 10
Figure 3.1: The velocity of 10 different sizes of drops in the RDC column.
The velocity of each drop is plotted against time as in Figure 3.1. From these
velocities the time spent by each drop in the compartment can be found. The residence
time for each size of the drop is tabulated in Table 3.2. The relationship between the
concentration of the continuous phase and the time taken for the 10 different sizes of
drops to reach a particular stage of the column can then be established. To achieve this,
we use the least squares method. In this method, we have to predict the relationship
between the two parameters by letting fˆ1 as the predicted function of the concentration
of the continuous phase where fˆ1 = aˆ1 i + bˆ1i t by a given value of t. The values of aˆ1 i
and bˆ1i can be obtained from
bˆ1i =
Siti f1
Siti ti
and
aˆ1 i = f¯1 − bˆ1i t¯i
where
Sitf1 =
and
ti f1 −
f1 ti
n
48
Siti ti =
t2i
( ti )2
−
n
where n = 7, t = tri stgj and tri is the resident time for drop of size i, meanwhile stgj
is the stage at j given data, for example stg2 = 7. This method chooses the prediction
bˆ1i that minimizes the sum of squared errors of prediction (f1 − fˆ1 )2 for all sample
points.
Table 3.2: The values of residence time and the slip velocity for each drop size
Drop Size(i)
0.0004
0.0011
0.0018
0.0025
0.0032
0.0039
0.0046
0.0053
0.0060
0.0067
Time(tri )
6.5225
2.9145
1.8427
1.5256
1.3760
1.2943
1.2487
1.3235
1.4257
1.5378
Velocity
0.0117
0.0261
0.0412
0.0498
0.0552
0.0587
0.0609
0.0574
0.0533
0.0494
Thus, by using this method, it is found that the concentration of the continuous
phase depends on the function of time t, that is f1i (t) = a1i + b1i t, where i corresponds
to the different sizes of the drops and a1 and b1 are constants. The values of a1 for the
ten different sizes of drops are the same but the values of b1 might differ according to
drop sizes (refer Table 3.3). To find the best function which represents all the functions
of t, the mean of the slopes of these linear functions is taken as the slope of the new
function. This new function represents the relationship between the concentration of
continuous phase and the resident time for all the 10 different drop sizes.
By assuming that the concentration on the surface of the drop is the same as
the concentration of the continuous phase (the medium), we can use the new function
f1 (t) as the boundary condition of equation (3.1). Thus, we get a modified model of
mass transfer of a single drop of which the boundary condition is a function of t, that
is
f1 (t) = a1 + b1 t,
(3.32)
49
Table 3.3: The values of a1 and b1
3.3.1
a1
b1
0.9671
0.0304 ×10− 3
0.9671
0.0681 ×10− 3
0.9671
0.1077 ×10− 3
0.9671
0.1301 ×10− 3
0.9671
0.1442 ×10− 3
0.9671
0.1533 ×10− 3
0.9671
0.1589 ×10− 3
0.9671
0.1499 ×10− 3
0.9671
0.1392 ×10− 3
0.9671
0.1290 ×10− 3
The Analytical Solution
The constant concentration ac0 in Equation (3.5) is now replaced by f1 (t) from
Equation (3.32), producing
u(a, t) = af1 (t),
t > 0,
(3.33)
while holding the other conditions unchanged. Rewrite Equations (3.3)-(3.6), where,
we get IBVP of time-varying function boundary condition of
∂u
∂2u
= D 2 , 0 ≤ r < a,
∂t
∂r
u(0, t) = 0, t > 0
u(a, t) = af1 (t) = f (t),
u(r, 0) = rc1 . 0 ≤ r < a
t>0
t≥0
(3.34)
(3.35)
(3.36)
(3.37)
The solution of this IBVP with varied boundary condition for conduction of
heat in solid is given by Carslaw and Jaeger in [55]. In this section, we show the
detailed steps in order to get the solution of the problem. The method of separation
of variables does not apply directly to the situation where time-varying boundary
50
condition arise. However, we show how by reformulating the problem it can be reduced
to a nonhomogeneous diffusion equation.
Now, let the solution of these equations be
u(r, t) = U (r, t) + V (r, t),
(3.38)
substitute this equation into (3.34) and rearrange the terms to obtain
Ut (r, t) − DUrr (r, t) = −[Vt (r, t) − DVrr (r, t)].
The appropriate boundary conditions are then
U (0, t) = −V (0, t),
t>0
U (a, t) = f (t) − V (a, t),
t>0
while the initial condition becomes
U (r, 0) = rc1 − V (r, 0),
0 ≤ r < a.
The idea now is to make the boundary conditions for U (r, t) homogeneous by
making a suitable choice for V (r, t). We will choose the simplest particular solution.
This is accomplished by setting
V (r, t) =
r
f (t).
a
(3.41)
This choice for V (r, t) converts the equation for U (r, t), which is a nonhomogeneous
diffusion equation, although now it is subject to the homogeneous boundary conditions.
Rewrite IBVP of U (r, t), we get
r
Ut − DUrr = − f (t) 0 ≤ r < a, t ≥ 0
a
U (0, t) = U (a, t) = 0, t > 0
r
U (r, 0) = g(r) − f (0), 0 ≤ r < a
a
(3.42)
(3.43)
(3.44)
where the related homogeneous problem is
vt − Dvrr = 0,
0 ≤ r < a,
t≥0
v(0, t) = v(a, t) = 0, t > 0
r
v(r, 0) = g(r) − f (0), 0 ≤ r < a
a
(3.45)
(3.46)
(3.47)
51
The solution of (3.45)-(3.47) by the method of separation of variables is
v(r, t) =
where An =
2
a
a
0
∞
An e
−Dn2 π 2 t
a2
ϕn (r),
(3.48)
n=1
nπr
[g(r) − ar f (0)] sin nπr
a dr and ϕn (r) = sin a .
The next step is to find a solution of the inhomogeneous problem of equations
(3.42)-(3.47) in the form of a series like (3.48), but in which the parameters An are
replaced by functions of t. The product An e
−Dn2 π 2 t
a2
will then become a function Tn (t)
so that the solution will be a series
∞
U (r, t) =
Tn (t)ϕn (r),
(3.49)
U (r, t)ϕn (r) dr.
(3.50)
n=1
where
2
Tn (t) =
a
a
0
We assume that Ut (r, t) is a continuous function in the region t > 0, 0 ≤ r ≤ a.
Under these circumstances, the integral in (3.50) has a derivative with respect to t
which can be calculated by differentiation under the integral sign. Referring to diffusion
equation of (3.42)-(3.44), we get,
Tn (t)
2 a
Ut (r, t)ϕn (r) dr.
a 0
r
2 a
[DUrr (r, t) − f (t)]ϕn (r) dr.
a 0
a
a
2 2D a
DUrr (r, t)ϕn (r) dr − 2 f (t)
rϕn (r) dr.
a 0
a
0
=
=
=
(3.51)
The last term of (3.51), denoted as
qn (t) = −
2 f (t)
a2
a
0
rϕn (r) dr,
is a known function since − a22 f (t) is given and using integration by parts we will get
2D
a
0
a
2D
Urr (r, t)ϕn (r) dr =
a
Further because ϕn = −λ2 ϕn and λ =
2D
a
a
0
DU (r, t)ϕn (r) dr.
nπ
a
a
0
Urr (r, t)ϕn (r) dr =
=
−2D a 2
λ U (r, t)ϕn (r) dr
a
0
−2Dn2 π 2 a
U (r, t)ϕn (r) dr,
a3
0
52
but from (3.50),
2D
a
a
0
Urr (r, t)ϕn (r) dr = −
Dn2 π 2
Tn (t)
a2
(3.54)
and substitute (3.54) into (3.51), we get
Tn (t) = −
Tn (t) +
Dn2 π 2
Tn (t) + q( t)
a2
Dn2 π 2
Tn (t) = qn (t)
a2
(3.55)
Equation (3.55) is a first-order linear differential equation where the integrating
factor is e
Dn2 π 2 t
a2
. Therefore the solution of (3.55) is
t
−Dn2 π 2 (t−τ )
−Dn2 π 2 t
2
a
a2
+
e
qn (τ ) dτ.
Tn (t) = Cn e
(3.56)
0
Setting t = 0 in (3.50), we get the general equation of Tn , which is
a
U (r, 0)ϕn (r) dr
Tn (0) = Cn =
0
nπr
r
2 a
dr
[g(r) − f (0)] sin
=
a 0
a
a
a
2
nπr
2
=
g(r) sin
dr +
(−1)n f (0),
a 0
a
nπ
(3.57)
and
qn (t) =
2 f (t)(−1)n .
nπ
(3.58)
The coefficient Tn (t) in (3.49) are completely known and hence problem (3.42)(3.44) has been solved. We have
∞
U (r, t) =
nπr
2 −Dn22 π2 t
e a
sin
a
a
n=1
a
0
nπr
dr +
g(r) sin
a
∞
nπr
(−1)n −Dn22 π2 t
e a
+
sin
n
a
n=1
t
∞
Dn2 π 2 τ
2 (−1)n −Dn22 π2 t
nπr
2
e a
sin
f (τ )e a
dτ .
π
n
a
0
2
f (0)
π
n=1
From (3.41) and u(r, t) = V (r, t) + U (r, t), the solution for diffusion equation of (3.34)(3.37) is
∞
u(r, t) =
2 −Dn22 π2 t
nπr
r
f (t) +
e a
sin
a
a
a
n=1
∞
a
0
nπr
dr +
g(r) sin
a
nπr
(−1)n −Dn22 π2 t
e a
+
sin
n
a
n=1
t
∞
Dn2 π 2 τ
2 (−1)n −Dn22 π2 t
nπr
2
e a
sin
f (τ )e a
dτ .
π
n
a
0
2
f (0)
π
n=1
53
By taking f (t) = a1 + b1 t which gives us
t
Dn2 π 2 τ
Dn2 π 2 t
a2
a2
2
2
a
a
f (τ )e
dτ = b1
e
−
Dn2 π 2
Dn2 π 2
0
and also we know that from the initial condition g(r) = c1 r which resulted in
r=t
g(r) sin
r=0
−a2
nπr
dr = c1
(−1)n ,
a
nπ
then
∞
r
2c1 a (−1)n+1 −Dn22 π2 t
nπr
sin
(a1 + b1 t) +
e a
+
a
π
n
a
u(r, t) =
n=1
∞
2a1 (−1)n −Dn22 π2 t
nπr
sin
e a
−
π
n
a
n=1
∞
a2 2b1
Dπ 3
(−1)n+1
nπr
+
sin
3
n
a
2b1
Dπ 3
(−1)n+1 −Dn22 π2 t
nπr
e a
sin
3
n
a
n=1
∞
2
a Knowing that
n=1
∞
(−1)n+1
n3
1
(3.60)
(r3 − a2 r)π 3
nπr
=−
,
a
12a3
sin
substituting this equation into (3.60) and rearranging them , give us
u(r, t) =
∞
r
b1
c1 a − a1 (−1)n+1 −Dn22 π2 t
nπr
(a1 + b1 t) +
(r3 − a2 r) + 2
e a
+
sin
a
6Da
π
n
a
∞
a2 2b1
Dπ 3
1
(−1)n+1
n3
1
e
−Dn2 π 2 t
a2
sin
nπr
.
a
(3.61)
Thus the solution of the diffusion equation of (3.1) with respect to the initialboundary condition of Equations (3.35), (3.36) and (3.37) is
C(r, t) =
∞
1
b1
c1 a − a1 (−1)n+1 −Dn22 π2 t
nπr
(a1 + b1 t) +
(r2 − a2 ) + 2
e a
+
sin
a
6Da
πr
n
a
∞
a2 2b1
Dπ 3 r
1
1
(−1)n+1
n3
e
−Dn2 π 2 t
a2
sin
nπr
.
a
(3.62)
As mentioned in the previous section, we are interested in the average
concentration of the sphere, Cav , given by Equation (3.23). Substituting (3.62) into
54
(3.24), we get
b1
1
(a1 + b1 t) +
(r2 − a2 ) 4πr2 dr +
a
6Da
r=0
r=a
∞
c 1 a − a1
nπr (−1)n+1 −Dn22 π2 t
2
sin
e a
4πr2 dr +
πr
n
a
r=0
1
r=a
∞
2
2b1 a
nπr (−1)n+1 −Dn22 π2 t
4πr2 dr
e a
sin
3
3
Dπ
r
n
a
r=0
Ct =
r=a (3.63)
1
= A+B+E
where
(3.64)
b1
1
(a1 + b1 t) +
(r2 − a2 ) 4πr2 dr
a
6Da
r=0
4 2
4πb1 a4
=
πa (a1 + b1 t) −
,
3
45D
r=a
∞
c1 a − a1 (−1)n+1 −Dn22 π2 t
nπr 2
B =
e a
4πr2 dr
sin
πr
n
a
r=0
A =
r=a ∞
1
8a2 (c1 a − a1 ) 1 −Dn22 π2 t
e a
,
=
π
n2
1
r=a
∞
2b1 a2 (−1)n+1 −Dn22 π2 t
nπr E =
e a
sin
4πr2 dr
3
3
Dπ
r
n
a
r=0
=
∞
a4 8b1
Dπ 3
1
1
1
e
n4
−Dn2 π 2 t
a2
.
Thus the average concentration Cav of the sphere is
∞
Cav =
b1 a
6(c1 a − a1 ) 1 −Dn22 π2 t
(a1 + b1 t)
−
+
e a
+
a
15D
π2a
n2
1
∞
6b1 a 1 −Dn22 π2 t
e a
.
Dπ 4
n4
(3.65)
1
Following with this result, we derive the new fractional approach to equilibrium
based on Equation (3.25), as
Fnew (t) =
Cav − c1
f (t)/a − c1
(3.66)
where f (t)/a and c1 are the boundary condition and initial condition of IBVP of
equation (3.1) respectively. By substituting (3.65) into (3.66), we get
Fnew =
b1 a
(a1 + b1 t)
−
+
a(a1 + b1 t − c1 ) 15D(a1 + b1 t − c1 )
∞
6(c1 a − a1 )
1 −Dn22 π2 t
e a
+
π 2 a(a1 + b1 t − c1 )
n2
6b1 a
4
Dπ (a1 + b1 t − c1 )
c1
.
(a1 + b1 t − c1 )
1
∞
1
1 −Dn22 π2 t
e a
−
n4
(3.67)
55
Avg conc of the drop when the boundary condition is f(t)
0.08
0.07
0.06
0.05
0.04
0.03
0.02
2
4
6
8
10
12
t
14
16
18
20
22
Figure 3.2: Sorption curve for sphere with surface concentration a1 + b1 t
3.4
Simulations for Different Drop Sizes
We consider the drop of size 0.00705m in diameter. The time taken for the
drop to move upwards in one compartment is 0.8868 seconds. From the least square
method, we found that f1 (t) is equal to 0.9187 + 0.0041t. Then let f (t) = a1 + b1 where
a1 = a(0.9187) and b1 = a(0.0041). Substitute this value and all the parameters into
(3.65) and we will get the relationship between the average concentration of the sphere
and the time, t. The relationship can easily be seen in Figure 3.2.
The comparison between this fractional approach to equilibrium, Fnew (t) and
the one obtained by Talib[5] is made based on the graph plotted in Figure 3.3. In
addition, simulations are also carried out to see the effect on fractional approach to
equilibrium with variations in drop sizes. A graphical representation of the effect of
variations in drop sizes is shown in Figure 3.4.
The concentration profiles of the curves shown in Figure 3.4 show that smaller
drops reach equilibrium concentration with the medium at a much faster rate than
larger drops. The profile of fractional approach to equilibrium of the modified model
compared with Crank’s[11] and Vermuelen’s[39] is similar.
56
0.08
Fractional Aprroach To Equilibrium
0.07
0.06
0.05
0.04
0.03
0.02
F(t)ïModified Model
F(t)ïTalib
0.01
0
5
10
15
20
25
t
Figure 3.3: Fractional approach to equilibrium vs. time
3.5
Discussion and Conclusion
In this chapter the modified mass transfer model was formulated based on
experimental data. By least square method and the assumption that the concentration
on the surface of the drop is the same as the concentration of the continuous phase, the
boundary condition of the IBVP was found to be a time dependent function, f1 (t) =
a1 + b1 (t). The analytical solution of the model was then detailed in Section 3.3.1. This
was followed by the derivation of the new fractional approach to equilibrium.
For comparison purposes, the new fractional approach to equilibrium profiles of
the time-dependent boundary condition and that of Talib[5] are shown in Figure 3.3.
Although the model considered here is a modification of the model proposed by Talib[5],
the new fractional approach to equilibrium profile agrees with the result obtained by
Talib. In conclusion, the new fractional approach to equilibrium gives a better theory
for further investigation of the mass transfer process in the RDC column. This is
because the new term was derived from the time-dependent boundary condition which
represents the real phenomena of the process in the column.
For further analysis, the simulations with variations in drop sizes were also
carried out to see their effect on the new fractional approach to equilibrium. From
Figure 3.4, the smaller the drop,the equilibrium concentration is more rapidly attained
57
1
0.9
Smaller Drop
Fractional Aprroach To Equilibrium
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
F(t)ïModified Model
F(t)ïTalib
Fv
Larger Drop
0
5
10
15
20
25
30
35
40
45
50
t
Figure 3.4: Fractional Approach to equilibrium vs. time for different drop sizes
. This is due to the fact that the smaller drop provides larger surface area to the volume
ratio, which causes a better absorption of the mass from the continuous phase.
The IBVP given by Equations (3.3), (3.4), (3.6) and (3.32) holds for a nonmoving drop in a stagnant medium. In a real RDC column drops are rising or falling in
the continuous phase, which induces internal circulation. The internal circulation has
the effect of distributing the solute uniformly in the drop to give the drop a uniform
concentration as it moves to the exit point. Therefore the following chapter will discuss
the mass transfer model of a moving drop in a non-stagnant medium.
CHAPTER 4
MASS TRANSFER IN THE MULTI-STAGE RDC COLUMN
4.1
Introduction
In the previous chapter we have shown the development of the modified mass
transfer model of time-dependent boundary value problem (BVP). The model involved
only the mass transfer of a single drop in a stagnant medium. The profile of the
fractional approach to equilibrium of the modified model agrees with the model
introduced by Talib[5] and Vermuelen[39].
Since in the RDC column the drops are moving in the continuous phase with
counter current direction, the mass transfer model of a moving drop in a non-stagnant
medium will be considered here. The new fractional approach to equilibrium is also
incorporated in developing the model.
According to the two-film theory, the concentrations of the two phases at
an interface where the equilibrium exist, is governed by the principles of physical
equilibrium. Due to this phenomenon at the interface, the boundary condition given
by Equation (3.36) has to be redefined. In the following section the IBVP with new
boundary condition will be given. In this section, the mass transfer model based on
modified quadratic driving force which is called Time-dependent Quadratic Driving
Force(TQDF) is described. Based on this model, we design a Mass Transfer of A Single
Drop Algorithm. This is then followed by a more realistic Mass Transfer of Multiple
Drops Algorithm. For comparison purposes, a normalization and the de-normalization
59
techniques are given in Section 4.5.
An alternative method of calculating the mass transfer for a Multi-Stage System
is also presented in the form of an algorithm named as the Mass Transfer Steady State
Algorithm. Finally, a discussion about the models is also presented.
4.2
The
Diffusion
Process
Based
On
The
Concept
Of
Interface
Concentration
Consider the previous IBVP. The diffusion process based on the concept of
interface concentration is obtained by replacing Equation (3.32) with the interface
condition. Then we get
∂2u
∂u
= D 2 , 0 ≤ r < a,
∂t
∂r
us (a, t) = f (cs ), t ≥ 0
u(r, 0) = rc1 ,
t≥0
0≤r<a
(4.1)
(4.2)
(4.3)
where us is the drop surface concentration, cs is the concentration of medium at the
drop surface and us (a, t) = f (cs ) is known as the equilibrium equation, which expresses
the concentration of the drop in equilibrium with the medium at the drop surface.
A drop with concentration cdin1 entering a column is subjected to the
concentration of the first compartment, cc1 . Solutes from the continuous phase are
transferred to the drop. Now the concentration of the drop is cdout1 . On reaching
the next compartment, the drop concentration, cdin2 , where cdin2 = cdout1 , is now
in a continuous phase concentration of the second compartment, cc2 , then the drop
concentration, cdout2 after leaving the second compartment can be obtained.
By
applying the same approach to the drop as it moves through every compartment, the
drop concentration cdoutn at the final compartment, ccn can be determined.
The mass transfer model based on the linear driving force is realistic if the
interface of the two liquids in contact is a simple plane. In 1953, Vermuelen[39] had
shown that if one of the interfaces of the two liquids is spherical, than the driving
60
force in a drop can be considered as non-linear. His expression known as quadratic
driving force was used successfully by previous researcher as can be found in [5]. In
this study we used the same concept and the new fractional approach to equilibrium is
also incorporated into the idea to get a new driving force named as Time Dependent
Quadratic Driving Force(TQDF).
The following section explains the rate of the mass transfer or flux across the
drop surface into the drop where the derivation of the time dependent quadratic driving
force is shown.
4.2.1
Flux Across The Drop Surface Into The Drop
The rate of mass transfer across the surface of the sphere given by flux J is
defined as
Jap =
dC
,
dt
(4.4)
where ap is considered as ratio of the surface area to the volume of a drop. From
Equation (3.66) the fractional approach to equilibrium of the new model is
Fnew (t) =
Cav − c1
f1 (t) − c1
and since the profile of fractional approach to equilibrium of the modified model is
similar to that of Crank[11] and Vermuelen[39] , we replace Fnew (t) with the one used
by Vermuelen that is
Fv = (1 − e−Dπ
2 t/a2
)0.5 .
(4.6)
Thus, Equation (3.66) becomes
Fv =
Cav − c1
f1 (t) − c1
(4.7)
Differentiating the above equation with respect to t, gives us
d
(Fv ) =
dt
1 Dπ 2 1 − Fv2 (t)
(
) =
)(
2 a2
Fv (t)
d Cav − c1
(
),
dt f1 (t) − c1
1
d
Cav − c1 d
(Cav ) −
f1 (t)
f1 (t) − c1 dt
(f1 (t) − c1 )2 dt
(4.8)
61
By taking Fv as (4.7) and rewriting, Equation (4.8) becomes,
(Cav − c1 ) d
1 Dπ 2 (f1 (t) − c1 )2 − (Cav − c1 )2
d
(Cav ) =
+
f1 (t)
2
dt
2 a
(Cav − c1 )
(f1 (t) − c1 ) dt
(4.9)
Substituting this equation into (4.4) and rearranging them, gives us
J
1 Dπ 2 (f1 (t) − c1 )2 − (Cav − c1 )2
1 Cav − c1 d
f1 (t)
(
)+
2
2ap a
Cav − c1
ap (f1 (t) − c1 ) dt
1 Cav − c1 d
4 Dπ 2 (f1 (t) − c1 )2 − (Cav − c1 )2
f1 (t)
(
)+
2
2ap d
Cav − c1
ap (f1 (t) − c1 ) dt
d Cav − c1 d
1 Dπ 2 (f1 (t) − c1 )2 − (Cav − c1 )2
(
f1 (t)
)+
3 d
Cav − c1
6 (f1 (t) − c1 ) dt
=
=
=
where ap =
6
d.
2
(4.10)
2
av −c1 )
The term ( (f1 (t)−cC1 )av−(C
) is known as the time-dependent
−c1
quadratic driving force. In this study, the quadratic driving force term is different
from the one used by Talib[5]. Here, f1 (t) is taken to be the surface concentration of
the drop instead of a constant, c0 which is used by Talib.
The rate of the mass transfer from the bulk concentration of the continuous
phase to the surface is given in the following section.
4.2.2
Flux in the Continuous Phase
The flux transfer in the continuous phase is given by
J = kc (cb − cs ),
(4.11)
where kc is the film mass transfer coefficient of the continuous phase. cb is the bulk
concentration in the continuous phase and cs is the concentration at the interface.
(cb −cs ) is the linear concentration driving force from the continuous bulk concentration
to the drop surface.
62
4.2.3
Process of Mass Transfer Based on Time-dependent Quadratic
Driving Force
The mass transfer across the surface of the sphere given by flux J defined in
2
2
av −c1 )
) is called the time-dependent quadratic
Equation (4.10) where ( (f1 (t)−cC1 )av−(C
−c1
driving force. Meanwhile the flux transfer in the continuous phase is given by (4.11).
As stated before, at the interface, (4.10) and (4.11) are equal, that is
d Cav − c1 d
1 Dy π 2 (f1 (t) − c1 )2 − (Cav − c1 )2
(
f1 (t) = kc (cb − cs ),
)+
3 d
Cav − c1
6 (f1 (t) − c1 ) dt
(4.12)
where Dy is the molecular diffusivity in the drop phase. By substituting (4.7) into
(4.12), we will get
Dy π 2
d
1 − Fv (t)2
1
(f1 (t) − c1 )(
) + Fv (t) f1 (t) = kc (cb − cs ).
3d
Fv (t)
6
dt
(4.13)
In this work, the concentration of both phases is in a normalized form that is,
the concentration is dimensionless which lies in the interval between zero and one. For
simplicity and to differentiate the latter terms from the original terms, we denote Cav ,
c1 , f1 (t), cb and cs as yav , y0 , ys , xb and xs respectively. Then (4.13) becomes
Dy π 2
d
1 − Fv (t)2
1
(ys − y0 )(
) + Fv (t) f1 (t) = kx (xb − xs ).
3d
Fv (t)
6
dt
(4.14)
By rearranging this equation, we get
ys =
3d
Fv
3d
Fv
d
1
kx (xb − xs )
−(
)(
)( )Fv (t) f1 (t) + y0
Dπ 2
1 − Fv2
Dy π 2 1 − Fv2 6
dt
(4.15)
Now, consider the situation at the drop surface. Equilibrium between the
medium and the concentration of drop is governed by equation
ys = f (xs ),
(4.16)
where for Cumene/Iso-butyric acid/Water system f (xs ) = x1.85
s .
The drop and medium concentrations, ys and xs , at the surface are found by
solving the non-linear equations of (4.15) and (4.16). In order to solve these equations,
we used bisection method. First, we substitute (4.16) into (4.15) which give us
63
x1.85
=
s
0 =
3d
Fv
3d
Fv
d
1
kx (xb − xs )
−(
)(
)( )Fv (t) f1 (t) + y0
Dπ 2
1 − Fv2
Dy π 2 1 − Fv2 6
dt
3d
Fv
3d
Fv
d
1
kx (xb − xs )
−(
)(
)( )Fv (t) f1 (t) + y0 − x1.85
s .
2
2
2
2
Dπ
1 − Fv
Dy π 1 − Fv 6
dt
Then let
g(xs ) =
3d
Fv
3d
Fv
d
1
kx (xb − xs )
−(
)(
)( )Fv (t) f1 (t) + y0 − x1.85
s . (4.17)
2
2
2
2
Dπ
1 − Fv
Dy π 1 − Fv 6
dt
With the assumption that the values of ys and xs lie between 0 and 1, the
root of Equation (4.17) must lie in the interval [0, 1]. Let the root be c, then by the
bisection method we get c =
0+1
2 .
This value is then substituted into Equation (4.17).
If g(c = 1/2) = 0 then the root of g is c = 1/2. Otherwise we have to repeat the
process of determining the value of c by checking the values of (g(0) × g(1/2)) and
(g(1) × g(1/2)). If (g(0) × g(1/2)) < 0 set a = 0 and b = 1/2, then c =
g(1) × g(1/2) < 0 set a = 1/2 and b = 1 then c =
1/2+1
2 .
0+1/2
2 .
If
With these values, repeat the
process until the root of g is obtained. These steps are well presented in the algorithm
below.
Bisection Algorithm
The following algorithm is used to calculate xs .
Step 1: Choose the initial solution of g, c, to lie in the interval [a, b] and initialize it
to
a+b
2 .
Set i = 1.
Step 2: Calculate the values of g(ai ), g(bi ) and g(ci ) using Equation (4.17).
Step 3: If g(ci ) ≤ 0.00001, set c = ci and stop.
else go to Step 4.
Step 4: If (g(ai ) × g(ci )) < 0 set i = i + 1, ai+1 = ai , bi+1 = ci and ci+1 =
ai +ci
2 ,
then
repeat Steps 2 to 3,
else if (g(ai ) × g(bi )) < 0 set ai+1 = ci , bi+1 = bi and ci+1 =
Steps 2 to 3.
ci +bi
2 ,
then repeat
64
The value of xs is substituted into Equation (4.15) or (4.16) to obtain ys . This
value is then used to calculate the average concentration of the drop using equation,
yav = Fnew (t)(ys − y0 ) + y0 ,
(4.18)
where Fnew (t) is the new fractional approach to equilibrium. Then the amount of mass
transfer of the drops can be obtained by applying mass balance equation, that is
Fx (xin − xout ) = Fy (yout − yin ),
(4.19)
where Fx and Fy are the flow rates of the continuous phase and the dispersed phase
respectively. The concentrations xin and yin are the uniform initial concentrations of
the continuous and drop phase. In this case xin and yin are xb and y0 respectively.
Meanwhile xout and yout are the exit concentration of the continuous and drop phase
respectively where we take yav as yout .
y
=y
out
ave
xin=xb
MASS BALANCE EQUATION
yin=y0
xout
Figure 4.1: Schematic diagram to explain the mass balance process
Equations (4.15)-(4.19) are used to calculate the amount of mass transfer from
the continuous phase to the drop. Based on various studies [21, 24, 31], the process
in the RDC column is very complicated because it involves not only mass transfer of
a single drop but infinitely many drops. These drops have different sizes and different
velocities. In this work we modelled the distribution of the drops along the column
exactly according to the model discussed by Arshad[7].
Before we construct a model that describes the mass transfer process as close
as the real process, the following section will explain the process of the mass transfer
of only a single drop with known size in the multi-stage RDC column.
65
4.3
Mass Transfer of a Single Drop
In this section, the process of the mass transfer of a single drop in the continuous
unsteady state medium of the 23 stages RDC column is considered. As explained
in the previous section, each compartment in the RDC column corresponds to the
stage number. The model concerns the mass transfer process of a single drop in every
compartment, where each compartment has its own medium concentration.
In this model, we assume that the continuous phase is continuously flowing in
the column with a unit concentration. Then a drop is injected into the column with
zero concentration. We also assume that the mass transfer takes place only when the
drop reaches the first compartment. Here the new fractional approach to equilibrium
is used, which is based on Equation (3.66) where the equation of average concentration
of the drop Cav , is given by (3.65) such that the new fractional approach to equilibrium
is (3.67).
The time t given in the Equation (3.67) is replaced with residence time tr,i of
a particular drop i in a compartment. Using this residence time and Equations (4.15)(4.19), the drop concentration in the first compartment is obtained. This concentration
is then taken as the initial concentration of the drop as the drop enters the second
compartment. This process is repeated through the final stage. The second drop is
then injected into the column with zero concentration but this time the concentration
of the medium, xout as calculated in the first batch of the process is used. We stop the
simulation when the steady state of the concentration of the drop at every compartment
is reached. In other words, the simulation is completed when the difference between
the concentration at iteration t and iteration t − 1 is very small. This condition must
be satisfied at each compartment.
The model described above is presented in Subsection 4.3.1.
66
4.3.1
Algorithm 4.1: Algorithm for Mass Transfer Process of a Single Drop
(MTASD Algorithm)
The process of mass transfer will continuously take place until the concentration
of the continuous phase is in equilibrium with the surface concentration of the drop.
The algorithm below describes the detail of the process of the mass transfer from stage
1 up to stage 23 for a single drop.
Algorithm to find the concentration of the liquids after the extraction
process of a single drop in 23 stages RDC Column.
This algorithm calculates the amount of the mass transfer from the continuous
phase to the drop.
Step 1: Input all the geometrical details and physical properties of the system. Set
iitr = 1, xin = 1 and yin = 0.
Step 2: Input initial values, that is xin and yin . Set j = 1 (stage 1)
Step 3: Calculate the value of fractional approach to equilibrium based on Varmulene
Equation (4.6) and the new Equation (3.67) which was based on the varied
boundary condition.
Step 4: Calculate the surface concentration of the medium and drop, xs and ys
respectively by solving the non-linear equations (4.15) and (4.16). Assume the
bulk concentration of the medium, xb is xin and the initial drop concentration,
y0 is yin .
Step 5: If ys > yin go to Step 6 else, set yout = yin , then go to Step 7.
Step 6: Determine the average concentration of the drops using Equation (4.18). This
value is taken to be the output concentration of the drop at jth stage, yout .
Step 7: Determine the concentration of the medium at the jth stage by using mass
balance equation of (4.19). This value is taken to be xout at the jth stage.
Step 8: If j > 23 go to Step 10,
else go to Step 9.
67
Step 9: Update the initial value for the next stage.
9a: If iitr = 1, set xin = 1, yin = yout (iitr , j), ∀j = 1, 2, ...n
else go to 9b.
9b: If j ≤ n − 2 set xin = xout (iitr − 1, j + 2) and yin = yout (iitr , j),
else (j = n − 1) set xin = 1, yin = yout (iitr , j).
Set j = j + 1. Repeat Steps 3 to 8.
Step 10: Take = 0.0001. If |yout (iitr , j) − yout (iitr − 1, j)| ≤ , stop, else go to Step
11.
Step 11: Update the initial value for the next iitr .
11a: Start with iitr = 1 set xin = xout (iitr , j = 2), yin = 0
11b: iitr = iitr + 1 set xin = xout (iitr , j = 2), yin = 0 ∀iitr = 2, 3, 4, .....
Set iitr = iitr + 1. Repeat Steps 2 to 10.
Figure 4.2 is the schematic representation of the mass transfer process of a
single drop in 23 stages RDC column in the form of a flow chart.
4.3.2
Simulation Results
Using Algorithm 4.1, we run the program to produce simulation results of
the mass transfer process for a single drop in a 23 stage RDC column. The profile
concentrations of continuous and dispersed phase along the column are shown in
Figure 4.3. For comparison purposes we also plot the concentrations of the continuous
and dispersed phase based on the new mass transfer model and Crank solution as
seen in Figure 4.4. Simulations were also carried out for different drop sizes. The
concentrations of the drop of different drop sizes are shown in Table 4.1.
4.4
Mass Transfer of Multiple Drops
In a real RDC column, the dispersed phase is injected into the column in the
form of drops. These drops will rise up the column if their density is less than that of
68
Start
Input Geometrical Details &
Physical Properties
Set itr = 1; xin =1, yin = 0
Input Initial Values (xin, yin)
Set j =1
Calculate equations (4.6 ) and (3.66)
Solving non-linear equations (4.15) & (4.16)
Yes
ys>yin?
j=j+1
Determine aveg
conc eqn (4.18)
iitr=iitr+1
No
Determine Medium conc
Mass Balance eqn (4.19)
Yes
j>23?
No
Update initial values
H d 0.0001
No
Update initial values
Yes
Stop
Figure 4.2: Flow chart of mass transfer process in the 23-stage RDC column for MTASD
Algorithm
69
Table 4.1: The concentration of the drops along the column
Drop size
Stage
No
d1
d2
d3
d4
d5
d6
d7
d8
d9
d10
1
0.043
0.0093
0.0039
0.0026
0.0021
0.0018
0.0017
0.0017
0.0017
0.0017
2
0.0831
0.0185
0.0078
0.0051
0.0041
0.0036
0.0034
0.0034
0.0034
0.0034
3
0.1206
0.0275
0.0117
0.0077
0.0062
0.0054
0.0051
0.005
0.0051
0.0051
4
0.156
0.0364
0.0155
0.0102
0.0082
0.0072
0.0067
0.0067
0.0068
0.0069
5
0.1894
0.0452
0.0194
0.0128
0.0102
0.009
0.0084
0.0083
0.0084
0.0086
6
0.221
0.0538
0.0232
0.0153
0.0123
0.0108
0.0101
0.01
0.0101
0.0103
7
0.251
0.0624
0.0269
0.0178
0.0143
0.0126
0.0117
0.0117
0.0118
0.0119
8
0.2795
0.0708
0.0307
0.0203
0.0163
0.0144
0.0134
0.0133
0.0134
0.0136
9
0.3066
0.0792
0.0344
0.0228
0.0183
0.0162
0.015
0.0149
0.0151
0.0153
10
0.3326
0.0874
0.0381
0.0253
0.0203
0.018
0.0167
0.0166
0.0167
0.017
11
0.3573
0.0955
0.0418
0.0278
0.0223
0.0197
0.0183
0.0182
0.0184
0.0187
12
0.3809
0.1035
0.0455
0.0302
0.0243
0.0215
0.02
0.0198
0.02
0.0203
13
0.4036
0.1115
0.0491
0.0327
0.0263
0.0232
0.0216
0.0215
0.0217
0.022
14
0.4252
0.1193
0.0528
0.0351
0.0283
0.025
0.0232
0.0231
0.0233
0.0237
15
0.446
0.1271
0.0564
0.0375
0.0302
0.0267
0.0249
0.0247
0.0249
0.0253
16
0.4659
0.1347
0.06
0.04
0.0322
0.0285
0.0265
0.0263
0.0266
0.027
17
0.485
0.1423
0.0635
0.0424
0.0341
0.0302
0.0281
0.0279
0.0282
0.0286
18
0.5033
0.1498
0.0671
0.0448
0.0361
0.0319
0.0297
0.0295
0.0298
0.0303
19
0.5209
0.1572
0.0706
0.0472
0.038
0.0337
0.0313
0.0311
0.0314
0.0319
20
0.5379
0.1645
0.0741
0.0496
0.04
0.0354
0.0329
0.0327
0.033
0.0335
21
0.5541
0.1717
0.0776
0.052
0.0419
0.0371
0.0345
0.0343
0.0346
0.0352
22
0.5698
0.1789
0.0811
0.0543
0.0438
0.0388
0.0361
0.0359
0.0362
0.0368
23
0.5848
0.1859
0.0845
0.0567
0.0458
0.0405
0.0377
0.0375
0.0378
0.0384
Note: Initial concentration of continuous phase is 1 at stage 24 and initial concentration
of dispersed phase, di = 0, i = 1, 2, 3, ..., 10 at stage 0.
70
1
0.9
0.8
Concentration
0.7
0.6
0.5
0.4
0.3
0.2
Drop conc of new b.condition
Medium conc of new b.condition
0.1
0
0
5
10
15
20
25
Stage No
Figure 4.3: The profile of the medium and drop concentration along the column with
respect to the new fractional approach to equilibrium
the continuous phase. In this mass transfer model, the process of solute transfer from
continuous phase to the drops is described as follows.
We assume that initially the continuous phase has a unit concentration, that
is in each stage j, for j = 1, 2, 3, ..., n = 23, the initial concentration of the continuous
phase, x(iitr , j) is one where iitr is the iteration number and j is the stage number.
Then the first batch of drops with the same size is injected into the column. Each drop
entering the first stage of the column has zero concentration.
This group of drops will move upward and break into smaller drops as they hit
the first rotor disc. As in [5], the daughter drops are modelled as such that they are
divided into ten different classes of size. It has to be noted that the mass transfer process
in the real RDC column occurs simultaneously. Here we define the concentration of a
certain group of drops with class size i, di in stage j as y (i) (iitr , j). As these drops with
i (i
initial concentration yin
itr = 1, j = 1) enter the first compartment, they are subjected
to the medium concentration of the first compartment, xin (iitr , j).
(i)
The drop surface concentration, ys (iitr = 1, j = 1) in equilibrium with the
(i)
continuous phase, ys (iitr = 1, j = 1) at the interface is then obtained by Equations
(4.15) and (4.16). In these equations, bulk concentration of the continuous phase, xb is
71
1
0.9
0.8
Concentration
0.7
0.6
0.5
0.4
0.3
Drop conc of new b.condition
Medium conc of new b.condition
Drop conc of constant b.condition
Medium conc of constant b.condition
0.2
0.1
0
0
5
10
15
20
25
Stage No
Figure 4.4: The profile of the medium and drop concentration along the column with
respect to the new fractional approach to equilibrium and Crank solution
(i)
replaced by xin (iitr , j) = 1 and y0 is replaced by ys (iitr = 1, j = 1). After obtaining
the drop surface concentration for each size, the next step is to determine the drop
(i)
average concentration, yav (iitr , j). This is obtained by using Equation (4.18). Then,
the total concentration of the drops in each cell can be obtained from
(i)
(i)
(i)
,
ytotal = N (i) (j) × Vdrop × yav
(4.20)
where N (i) (j) is the number of the drops in each cell i at stage j.
The next step is to calculate the average concentration of the drops in the first
compartment by using
N cl=23
Yav =
i=1
N cl=23
i=1
(i)
(i)
N (i) × Vdrop × yav
(i)
N (i) × Vdrop
.
(4.21)
The continuous phase concentration, xout (iitr , j) after some amount of solute was
transfered to the drops in the first compartment can be determined by using mass
balance of Equation (4.19).
The process continues to the second stage. Before we start the process, the
initial value of the drops and the medium have to be updated. The initial values of the
(i)
(i)
drops at second stage are equal to yin (iitr = 1, j = 2) = yav (iitr = 1, j = 1). Meanwhile
at this time, the continuous phase concentration remains the same, (x(iitr , j) = 1).
72
After the updating process is completed, the process of mass transfer as explained
above is repeated through the final stage.
Now, the process proceeds to the next iteration. Here, the updating process
for the initial value of the drops and the continuous phase concentration also need
to be done. At the second iteration the initial value of the drops is zero, whilst the
continuous phase concentration, xin (iitr = 2, j = 1) = xout (iitr = 1, j = 2). The mass
transfer process is said to achieve the steady state if there exist = 0.0001 such that
|yout (iitr , j) − yout (iitr − 1, j)| ≤ .
The procedure to calculate the amount of mass transfer as explained in the
above subsection is divided into two algorithms. The first is the Basic Mass Transfer
Algorithm. In this algorithm, the amount of mass transfer from the continuous phase
to the drops is calculated for given values of initial concentrations. The other is the
main algorithm which is denoted as the Mass Transfer Multiple Drops Algorithm.
4.4.1
Basic Mass Transfer(BMT) Algorithm
This algorithm calculates the amount of mass transfer from the continuous
phase to the drops.
Algorithm 4.2: Basic Mass Transfer(BMT) Algorithm
(i)
(i)
Input: xin and yin . Output: xout and yout .
Step 1: Read the input values. Calculate the value of the fractional approach to
equilibrium based on the Varmulene equation, (4.6), the Crank equation, (3.26)
and the new equation, (3.67) which is based on the varied boundary condition.
(i)
(i)
Step 2: Calculate the surface concentration of the medium and drops, xs and ys ,
for i = 1, 2, 3, ...10 respectively by solving the non-linear equations of (4.15) and
(4.16) using bi-section method. Set the bulk concentration of the medium, xb is
xin and the initial drop concentration, y0 is yin .
73
(i)
(i)
(i)
(i)
Step 3: If ys > yin for i = 1, 2, 3, ...10 go to Step 4, else set yout = yin , and go to
Step 6.
(i)
Step 4: Determine the average concentration of the drops, yav for i = 1, 2, 3, ...10
using Equation (4.18).
Step 5a: Calculate the total concentration of the drops in each cell (i):
(i)
(i)
(i)
ytotal = N (i) × Vdrop × yav
where N (i) is the number of drops in each cell(i) at stage j.
Step 5b. Calculate the average concentration of the drops in jth stage using Equation
(4.21).
Set Yav = yout at stage j.
Step 6: Determine the concentration of the medium at jth stage by using the mass
balance equation of (4.19).
(i)
Algorithm 4.2 is used in the Algorithm 4.3 to calculate xout and yout at every
stage.
4.4.2
Algorithm for the Mass Transfer Process of Multiple Drops in the
RDC Column (MTMD Algorithm)
In the RDC column, the mass transfer process involved a swarm of drops.
Therefore, to provide a more realistic mass transfer model in the RDC column, we will
discuss the algorithm for the mass transfer process of the multiple drops as described
in previous section. The mass transfer process is based on the drop distribution as
explained in [5].
Algorithm 4.3: MTMD Algorithm
The algorithm calculates the amount of mass transfer from the continuous phase
to the drops.
74
Step 1: Input all the geometrical details and physical properties of the system. Set
(i)
iitr = 1, xin = 1 and yin = 0, ∀i = 1, 2, 3, ..., 10.
(i)
Step 2: Initialize xin and yin , set j = 1.
(i)
Step 3: Apply BMT algorithm and calculate xout and yout . If j > 23 go to Step 5,
else go to Step 4.
Step 4: Update the initial value for the next stage.
(i)
(i)
4a: If iitr = 1 ∀j = 1, 2, 3, ...n set xin = 1, yin = yav (iitr , j)
else go to 4b.
(i)
(i)
4b: If (j <= n − 2) set xin = xout (iitr − 1, j + 2), yin = yav (iitr , j))
(i)
(i)
else (j = n − 1) xin = 1, yin = yav (iitr , j)
Set j = j + 1. Repeat Step 3.
Step 5: Set = 0.0001. If |yout (iitr , j) − yout (iitr − 1, j)| ≤ , stop, else go to Step 6.
Step 6: Update the initial value for the next iitr .
(i)
6a: Start with iitr = 1 set xin = xout (iitr , j = 2), yin = 0
(i)
6b: iitr = iitr + 1 set xin = xout (iitr , j = 2), yin = 0 ∀iitr = 2, 3, 4, .....
Set iitr = iitr + 1. Repeat Steps 2 to 5.
The algorithm is presented as a flow chart in Figure 4.5.
4.4.3
Simulation Results
The simulation of the mass transfer model based on MTMD Algorithm were
carried out. For comparison purposes, the fractional approach to equilibrium based on
the Crank solution is also used. To validate the algorithm, we use the experimental
data from the SPS report (see Talib[5]). These data were produced by experimental
work on the mass transfer process of an RDC column with the geometrical properties
and system physical properties as given in Appendices A.1 and A.2. The results of the
simulations can be found in Figure 4.6.
75
Start
Input Geometrical Details &
Physical Properties
Set iitr = 1; xin =1, y(i)in = 0
Read Initial Values (xin, yin)
Set j =1
BMT
Update initial
value
( iitr iitr 1 )
j
j>23r?
No
j 1
Update Initial Value
Yes
No
H 0.0001?
Yes
Stop
Figure 4.5: Flow chart for mass transfer process of MTMD Algorithm
Before the curve of the experimental data can be plotted (Figure 4.6), a few
steps of normalization have to be considered. The first and second experimental data
are given in Tables 4.2 and 4.3 respectively. Since the simulation of the forward
modelling program uses normalized data, we need to find a technique to normalize
the experimental data.
4.5
The Normalization Technique
To normalize the data, an equilibrium equation governing the mass transfer
process of the system needs to be known. In this study, the system used is the isobutyric acid/cumene/water system and the equilibrium equation of the system is
yO = 0.135x1.85
A ,
(4.22)
76
1
0.9
Medium−new model
Medium−Talib Model
Drop−New model
Drop− Talib Model
Medium− Exp
Drop− Exp
0.8
Concentration
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
5
10
15
20
25
Stage No
Figure 4.6: The concentration of continuous and dispersed phase of new model, Talib
model and experimental
where yO is the organic(cumene-drop) phase and xA is the aqueous(continuous) phase,
all measured in gram per litre(g/l).
The normalization technique used is explained in the following procedure:
4.5.1
Normalization Procedure
Procedure 1: (Normalization Procedure)
Step 1: Assume that xAF is the feed concentration of continuous phase (iso-butyric
acid in feed) and ysmax g/l is the iso-butyric acid in equilibrium with xAF , the
relation xAF and ysmax is given by
ysmax = 0.135x1.85
AF ,
(4.23)
where yO and xA are replaced by ysmax and xAF respectively.
Step 2: Dividing Equation (4.22) by (4.23), gives
y = x1.85
(4.24)
77
Table 4.2: Experiment 1-Continuous phase (aqueous) and dispersed phase (organic)
concentrations
Rotating disc contactor column with 152mm diameter and 23 stages
Mass transfer direction: continuous phase to drop phase
System : Cumene/Iso-butyric acid/Water
Rotor speed NR = 5rad/s
Flow ratio (dispersed)/continuous phase) : 0.3333
Continuous phase (aqueous)
Dispersed phase (organic)
Flow rate: 3.75 l/m
Flow rate: 1.25 l/m
Feed concentration: 36.02 g/l
Feed concentration: 28.66 g/l
Exit concentration: 23.97 g/l
Exit concentration: 63.10 g/l
Stage
1
5
9
13
17
21
Continuous
24.02
24.95
26.18
27.85
30.10
32.91
Stage
3
7
11
15
19
23
Dispersed
-
36.34
40.08
46.46
52.80
57.33
where
y=
yO
,
(4.25)
xA
.
xAF
(4.26)
ysmax
and
x=
Step 3: Determine the normalized values of the dispersed phase concentration using
Equation (4.25). In this study we assume that the feed concentration of the
dispersed phase is zero. In order to satisfy this assumption, we use yO = yexp −yF
where yexp is the experimental value of dispersed phase concentration at particular
stage and yF is the feed concentration of the dispersed phase.
Step 4: Calculate the normalized values of the continuous phase concentration for
each corresponding stage by mass balance equation, (4.19).
78
Table 4.3: Experiment 2-Continuous phase (aqueous) and dispersed phase (organic)
concentrations
Rotor speed NR = 4.12rad/s
Flow ratio (dispersed)/continuous phase) : 0.3333
Continuous phase (aqueous)
Dispersed phase (organic)
Flow rate: 5.0 l/m
Flow rate: 1.67 l/m
Feed concentration: 39.64 g/l
Feed concentration: 27.28 g/l
Exit concentration: 25.12 g/l
Exit concentration: 60.98 g/l
Stage
1
5
9
13
17
21
Continuous
27.32
28.69
30.10
31.94
34.10
36.83
Stage
3
7
11
15
19
23
Dispersed
-
39.42
44.88
53.42
57.95
59.85
The detailed calculation for the normalized values of the dispersed and
continuous phase are shown in the following example.
Example 1
In this example we use the experimental data 1 from Table 4.2.
Step 1: From Table 4.2, xAF = 36.02, Substitute this into (4.23), we get ysmax =
102.3151,
Step 2: Calculate the normalized value for dispersed concentration at stage zero (this
stage corresponds to the feed ) using Equation (4.25), that is y0 =
28.66−28.66
102.3151
= 0.
Step 3: Repeat step 2 for stage 7,11,15,19,23 and 24, where stage 24 corresponds to
the exit stage. At stage 7 we will get y7 =
36.34−28.66
102.3151
= 0.0751.
Step 4: With the assumption that the feed concentration of the continuous phase
is normalized so that its value is 1, Equation (4.19) is used to calculate the
79
normalized continuous phase for stage 0,7,11,15,19 and 23. In this case we have
to calculate the normalized concentration at stage 23 first, that is x23 = x24 −
Fy
Fx (y24
− y23 ) = 1.0 − 0.333(0.337 − 0.280) = 0.9810. Repeat this step for stage
19, 15, 11, 7 and 0.
The results for all stages can be found in Table 4.4. The normalized process can
also be done by first normalizing the continuous phase concentration followed by the
dispersed phase which uses the mass balance equation. The same procedure is applied
to the data in Table 4.3 which produced the normalized data in Table 4.4.
Due to the fact that the experimental data was not given for every stage, there
was no data for the continuous concentration at stage 3, 7, 11,15, 19 and 23. In these
circumstances, we have to construct a technique for de-normalization process to get the
values of the concentrations in g/l at this stages.
Table 4.4: Experiment 1-Normalized continuous and dispersed phase concentrations
Stage
Continuous(x)
Dispersed(y)
Normalized x
Normalized y
0
23.97
28.66
0.899
0
1
24.02
36.34
0.921
0.075
40.08
0.932
0.112
46.46
0.951
0.174
52.80
0.970
0.236
57.33
0.9810
0.280
63.10
1.0
0.337
3
5
24.95
7
9
26.18
11
13
27.85
15
17
30.10
19
21
32.91
23
24
36.02
80
38
36
Continuous Phase Concentration
34
32
30
28
26
24
22
Exp Data W/O Deïnorm Value
Exp Data With Deïnorm Values
0
5
10
15
20
25
Stage No
Figure 4.7: The continuous phase concentration along the column: Experiment Data 1
4.5.2
De-normalization Procedure
Procedure 2 (De-normalization Procedure)
Step 1: Assume that xj and yj are the normalized concentration of the continuous
and dispersed phase respectively at stage i . Xj and Yj are the experimental
concentration value of the continuous and dispersed phase respectively. Assume
also that the normalized concentration of the medium and its experimental value
has a linear relationship, that is, its gradient is m =
X24 −X0
x24 −x0 .
Step 2: With the assumption that the normalized concentration of the medium
and its experimental value has a linear relationship, calculate the experimental
concentration of the medium of its respective normalized value:
Xj = X24 − m(1 − xj ).
(4.27)
For example at stage 0, X0 = X24 − m(1 − x0 ).
Step 3: Repeat step 2 until all the approximated experimental values at the
corresponding stage are calculated.
For the Data of Experiment 1, the de-normalization process will produce the
approximated experimental data at corresponding stages as can be seen in Table 4.5. To
81
Table 4.5: Experiment 1-De-normalized continuous concentrations
Stage
Continuous(x)
Normalized x
0
23.97
0.899
1
24.02
3
5
24.95
7
26.5953
9
26.18
11
27.9036
13
27.85
15
30.1743
17
30.10
19
32.441
21
32.91
23
33.9919
0.9810
24
36.02
1.0
0.921
0.932
0.951
0.970
see the effect of this de-normalization process on the experimental data, points with and
without the de-normalization data are plotted against stage number. From the graph
(see Figure 4.7), we can see that the trace of points containing de-normalized values at
certain stage, oscillate about the trace of points of the actual experimental data. This
phenomenon is explained by the fact that the de-normalized values at that particular
stages are calculated from the normalized values which are calculated through Step 4
in Procedure 1. In other words, the normalized values are not directly calculated from
the actual experimental values. Due to this reason there are some errors which affect
the smoothness of the de-normalized data curve(trace of the points).
In this case we have to construct a better technique for the de-normalization
process so that the de-normalized value curve will follow the behaviour of the actual
experimental data curve. In order to do this we include the error factor in the denormalization process, that is, in Step 2 of Procedure 2, Equation (4.27) becomes
82
Xj = X24 − m(1 − xj ) ± ê,
(4.28)
where ê is the error factor.
1.4
1.2
Error (Concentration)
1
0.8
0.6
0.4
0.2
0
0
5
10
15
20
25
Stage No
Figure 4.8: The error between the continuous phase concentration of Experiment Data
1 with and without de-normalized values
The differences between the two graphs in Figure 4.7 are then calculated along
the column. The relationship between the differences and the stage number is then
shown in Figure 4.8. It is then observed from Figure 4.8 that the error curve with
respect to stage number of the column has quadratic-like curve. By using Matlab Basic
Curve Fitting Tool-box, we represented the error data as a quadratic curve (see Figure
4.9) where the quadratic equation is
ê(j) = −0.0074j 2 + 0.1889j + 0.0001,
(4.29)
and j is the stage number of the column. This error function is then applied to Equation
(4.28) which resulted in a corrected de-normalized continuous phase concentration data.
These values are tabulated in Table 4.8. For comparison purposes, the corrected data
is then plotted against stage number of the column in Figure 4.10.
In the next section, another algorithm for the mass transfer process of the
multiple drops will be presented. In this model, the time when the next swarm of
drops is injected into the column is taken into account.
83
1.4
1.2
Error(Concentration)
1
0.8
0.6
0.4
data 1
quadratic
0.2
0
0
5
10
15
20
25
Stage No
Figure 4.9: The error is fit to Quadratic-like curve
Table 4.6: The error by quadratic fitting
4.6
Stage
0
7
11
15
19
23
24
Error
0
0.9608
1.1848
1.1725
0.9240
0.4391
0
Forward Model Steady State Mass Transfer of Multiple Drops
In a real RDC column, the drops are continuously injected into the column
according to the dispersed phase flow rate. This means that in order to produce the
mass transfer model as close as possible to the real process, the time when the next
swarm of drops is injected into the column need to be taken into consideration.
In this model, the mass transfer is calculated via the distribution of the drops
which is assumed to be in a steady state. The model can be explained as follows.
As in MTMD algorithm, we assumed that initially the continuous phase has a unit
concentration, while the first batch of drops is injected into the column with zero
concentration. This first swarm of drops with the same size will break into smaller
drops as they hit the first rotor disc.
These daughter drops are distributed into the cells according to their sizes as
explained in Section 4.4 . At the same time the mass transfer process occurs. The
84
38
36
Continuous Phase Concentration
34
32
30
28
26
Exp Data W/O Deïnorm Value
Exp Data With Deïnorm Values
Exp Data With Deïnorm Corrected Values
24
22
0
5
10
15
20
25
Stage No
Figure 4.10: The continuous phase concentration along the column with corrected value
: Experiment Data 1
equations used are exactly the same as in MTMD algorithm. The steps of calculation
can easily be understood if we refer to the flow chart in Figure 4.11.
In the following algorithm, when the second swam of drops is injected into the
column, the drops will also move upward and break into smaller drops as they hit the
first rotor disc (in this algorithm we assumed that the number of iteration is equal to
the number of batches of drops injected into the column). Now, the second batch will
fill the first compartment whilst the first one moves to the second compartment. The
steps for calculating the mass transfer of the drops in the first compartment are exactly
the same as the first batch of the drops.
However, for the second compartment, we take the initial concentration of the
drops, yin as the output concentration of the drops when the iteration is equal to one,
that is yin (iitr = 2, j = 2) = yout (iitr = 1, j = 1). At this time the initial concentration
of the continuous phase remains the same, xin = 1. The complete procedure for
calculating the mass transfer at this stage is shown in Figure 4.12.
Now, when the third swam of drops enter the column, the same phenomenon
will occur, but this time the initial concentration of the continuous phase at the first
compartment is subjected to the output concentration of the continuous phase at the
85
Initial Value
xin = 1, yin = 0
...
...
...
...
...
..
.........
.....
j=1
...
...
...
...
...
..
..........
.....
Calculate
xout , yout
...
...
...
...
...
.........
.....
..............................................
.
.
.
.
.
.
.
.
.
.......
..
.....
.......
....
.....
...
...
..
.....
..
...
itr
..
.....
.....
.
......
.
.
.
.
..
.........
.........
..................
............................
Go to i
=1
Figure 4.11: Flow chart for mass transfer process at iitr = 1
second compartment when the iteration is equal to two, that is xin (iitr = 3, j = 1) =
xout (iitr = 2, j = 2). Meanwhile the initial concentration of the drops, yin (iitr =
3, j = 1) = 0. The initial concentrations for the mass transfer at the second and third
compartments can be determined by following the steps given in the flow chart in Figure
4.13.
The same steps apply to the 4th, 5th, 6th, ..., nth swarm of drops. The step that
explains the way to determine the initial concentrations at particular stage is shown
in Figure 4.14. The phenomenon explained above will continue until the first batch
or group reaches the 23rd compartment (stage). At this instance the column is full
of drops. The iteration will continue until the concentration of the drops is in steady
state. In other words, the difference of the concentration for both phases at time t and
t − 1 is very small or negligible. The schematic diagram in Figure 4.15 illustrates the
phenomenon explained above.
The steps to calculate the amount of mass transfer as explained in this section
is divided into three algorithms. The first one is the Basic Mass Transfer Algorithm.
86
Initial Value
xin = 1, yin = 0
...
...
...
...
...
..
.........
.....
j=1
...
...
...
.
......................................................................................................................................................
..
...
...
...........
....
...
...
....
..
...
...
...
out . out
...
...
...
...
...
...
...
...
...
...
...
.
...
.........
....................................
...
.....
...............
.........
.
.
..
.
.
.........
.
......
..... .............
.
.
......
.....
.
.
.
.
.......
.....
...
.
.....
.
.
.
.
.
.
.
.
.
.
.
.
...
.......
.
......
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.......
.............................................
.
.
.
.......
.
.
...
.
...
itr
itr
....
.......
.
.
.
.
...
.
.
.
.
.......
.....
....
.
.
....... .............
.
.
.
.
.
.
.......
...........
...........
........
..
.................................................
...
...
...
...
.....
...
...
...
...
..........
...
.....
...
...
...
...
...
...
..
......................................
Calculate
x ,y
j =j+1
j=i ?
Yes
Go to i
=3
No
Update Initial Value
xin = 1
yin = yout (iitr − 1, j)
Figure 4.12: Flow chart for mass transfer process at iitr = 2
In this algorithm, the amount of mass transfer from the continuous phase to the drops
is calculated for given values of initial concentrations. The next subsection is the main
algorithm which is denoted as the Mass Transfer Steady State Algorithm. It is then
followed by the Updating Mechanism Algorithm which is divided into two, these are
the Updating Initial Value for Next Iteration and the Next Stage Algorithms.
4.6.1
Algorithm 4.4:
Algorithm To Find The Drop Concentration of
a Steady State Distribution in 23 Stages RDC Column (MTSS
Algorithm)
The algorithm calculates the amount of the mass transfer from the continuous
phase to the drops.
Step 1: Input all the geometrical details and physical properties of the system. Set
87
x
in
y
in
Initial values
xout (i 1,2)
itr
0
j=1
Calculate
xout (i , j ), yout (i , j )
itr
itr
j=j+1
Yes
j=iitr?
Go to
iitr=4
No
Update initial value
(j=2)
Yes
j=1
x
in
y
in
1
yout (i 1, j 1)
itr
No
x
in
y
in
1
yout (i 1, j )
itr
Figure 4.13: Flow chart for the mass transfer process at iitr = 3
(i)
iitr = 1, xin = 1 and yin = 0, ∀i = 1, 2, 3, ..., 10.
(i)
Step 2: Read initial values, that is xin and yin , set j = 1.
Step 3: If iitr ≤ n, go to step 4, else go to Step 7.
(i)
Step 4: Apply BMT algorithm to calculate xout and yout .
Step 5: If j < iitr , go to Step 6, else update initial value for iitr = iitr + 1 go to Step
3,
Step 6: Update the initial value for the next stage. Set j = j + 1, repeat Steps 4 to
5.
Step 7: Now iitr = n + 1. Read the input values and set j = 1.
88
Initial values
x
in
y
in
x out ( i
itr
1, 2 )
0
j=1
Calculate
xout (i , j ), yout (i , j )
itr
itr
j=j+1
Go to
iitr=iitr+1
Yes
j=iitr?
No
Update initial value
j=1?
Yes
x
x out ( i
1, j 2 )
itr
y out ( i 1, j )
itr
in
y
in
No
j tn2
Yes
x
1
in
y
y (i 1, j)
in out itr
No
x
in
y
in
xout (i 1, j 2)
itr
yout (i 1, j 1)
itr
Figure 4.14: Flow chart describing the mass transfer process for iitr = 4, 56, ..., n
(i)
Step 8: Apply BMT algorithm to calculate xout and yout .
Step 9: If j < n. Update the initial values for the next stage. Set j = j + 1, repeat
Steps 8 to 9, else go to Step 10.
Step 10: Set = 0.0001. If |yout (iitr , j) − yout (iitr − 1, j)| ≤ , stop, else Update the
initial value for the next iitr . Set iitr = iitr + 1, repeat Steps 8 to 9.
To update the initial value for the next stage and for the next iteration, the
following algorithms are considered.
89
i itr = 1 ;j
1
iitr
4;j 4
iitr
3;j 3
iitr
4;j 3
iitr
2;j 2
iitr
3;j 2
iitr
4;j 2
iitr
2;j 1
iitr
3;j 1
iitr
4;j 1
Figure 4.15: Schematic diagram of the mass transfer process in the 23-stage RDC
column
90
Start
Input Geometrical Details &
Physical Properties
Set iitr = 1; xin =1, y(i)in = 0
Read Initial Values (xin, yin)
Set j =1
No
iitr d n ?
Yes
Update initial
value
( iitr iitr 1 )
BMT
No
j
j 1
j<iitr?
Yes
Update Initial Value
iitr=n+1. Read input value. Set j=1
Update initial
value
( iitr iitr 1 )
BMT
j<n?
Yes
Update
initial value
(j
j 1
No
No
H 0.0001
Yes
Stop
Figure 4.16: Flow chart for mass transfer process of MTSS Algorithm
91
4.6.2
Updating Mechanism Algorithm
Algorithm 4.5: Updating the Initial Value for Next Iteration (iitr ) (UIVI)
Algorithm
The Algorithm is used to update the initial values for the next iteration.
Step 1 Read the current position of iitr and j.
(i)
Step 2 If iitr = 1, the updating input values of next iteration is xin = 1, yin = 0
else (iitr > 1), the updating input values of next iteration is xin = xout (iitr −1, 2),
(i)
yin = 0
Algorithm 4.5: Updating the Initial Value for the Next Stage (j) (UIVS)
Algorithm
The Algorithm is used to update the initial values for the next stage (j)
Step 1: Read the current position of iitr and j.
Step 2: If iitr ≤ n go to Step 3 else go to Step 5.
Step 3: If j < iitr
(i)
(i)
if (1 < iitr ≤ 3) ⇒ xin = 1, yin = yav (iitr − 1, j)
else(4 ≤ iitr ≤ n)
(i)
if (j = 1) ⇒ xin = xout (iitr − 1, j + 2), yin = 0
(i)
(i)
elseif j ≥ iitr − 2 ⇒ xin = 1, yin = yav (iitr − 1, j)
else (2 ≤ j < iitr − 2) ⇒ xin = xout (iitr − 1, j + 2),
(i)
(i)
yin = yav (iitr − 1, j)
else (j < iitr ) go to Step 4
Step 4: Apply the UIVI Algorithm to update the initial values for the next iteration.
Step 5: (iitr > n)
(i)
If j = 1 ⇒ xin = xout (iitr − 1, j + 2), yin = 0
92
(i)
(i)
elseif j ≥ n − 2 ⇒ xin = 1 yin = yav (iitr − 1, j)
(i)
(i)
else(2 ≤ j < n − 2) ⇒ xin = xout (iitr − 1, j + 2), yin = yav (iitr − 1, j).
4.6.3
Simulation Results
The simulations of the mass transfer model based on the MTSS Algorithm were
carried out. For comparison purposes, the output concentrations of the continuous and
dispersed phase for both MTSS and MTMD algorithms are listed in Table 4.7. To
analyze the result graphically, the six curves from MTSS, MTMD and experimental
data are plotted in Figure 4.17.
4.7
Discussion and Conclusion
A detailed description of the development of the mass transfer models has been
presented in this chapter. It begins with the concept of the diffusion equation which
is based on the interface concentration. In these models, the new fractional approach
to equilibrium was used to get the flux across the drop surface of Equation (4.10).
From this derivation, the term referred to Time Dependent Quadratic Driving Force
was formulated.
The MTASD Algorithm was designed based on the concept explained above.
This algorithm calculates the amount of solute transfer from the continuous phase to a
single drop. The simulations of the algorithm were also carried out for different size of
drops. The range of the size is from 0.0004 to 0.0007 meter in diameter. The output
concentrations of the drops for each size were listed in Table 4.1. From the data, it can
be seen that, the concentration of a smaller drop is higher than the bigger one at every
stage. This is because the smaller drop provides larger surface area compared to the
other. In fact, the velocity of the smaller drop is less, meaning that the smaller drop
has a higher residence time in each compartment.
Besides the new fractional approach to equilibrium, we also run the MTASD
algorithm using the fractional approach to equilibrium based on the Crank solution.
93
1
0.9
0.8
Concentration
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
5
10
15
MediumïMTMD
DropïMTMD
MediumïMTSS
DropïMTSS
MediumïExp
DropïExp
20
25
Stage No
Figure 4.17: The concentration of continuous and dispersed phase of MTMD, MTSS
Algorithm and Experimental
The simulation data of the medium and drop concentrations were plotted in Figure 4.4.
The profile of the curve using the new fractional approach to equilibrium agrees with
the one based on the Crank solution.
The idea to provide a model which is close to the real process of the mass
transfer in the column has driven us to develop the MTMD algorithm. This algorithm
calculates the mass transfer of the multiple drops. The drop distribution as explained
in Talib[5] was considered. To validate the algorithm, we used the experimental data
in Tables 4.2 and 4.3. These data have to be normalized, before the comparison of
the data could be made. Figure 4.6 shows six curves of the continuous and dispersed
phase concentrations for the mass transfer model developed by Talib, the new model
of MTMD algorithm and from the experimental data.
Although the result of MTMD algorithm showed that the model agrees with
the profile of the experimental data, in Section 4.6 another algorithm named MTSS
algorithm for the multiple drops mass transfer process was presented. In MTMD
algorithm the calculation of the mass transfer for the first batch of the drops was
94
carried out up to the final stage without considering when the second batch of drops
was injected into the column. It seems that, in this algorithm the second batch of drops
was only injected when the first batch reached the top of the column. Similarly, the
third batch of drops would be injected when the calculation of mass transfer for the
second batch had been completed for all stages.
On the other hand, we considered the time when the next swarm of drops
was injected in MTSS algorithm. In this work, we assumed that the flow rate of the
dispersed phase was equal to the simulation time for each iteration. Therefore, in
the MTSS algorithm, the mass transfer for the second batch of the drops would be
calculated even as the first one just reached the second compartment. The process
of construction of the algorithm was illustrated in flow charts of Figures 4.11, 4.12,
4.13 and 4.14. The output concentrations for the continuous and dispersed phase from
MTSS and MTMD were listed in Table 4.7. From these data, we conclude that the
outputs from both algorithms do not give much difference. Figure 4.17 clearly is in
agreement with the above conclusion.
However, MTSS Algorithm has close trait to the real phenomenon of the mass
transfer process in the RDC column. It is because in the real RDC column the mass
transfer occurs simultaneously as explained in the MTSS Algorithm. Therefore, we
conclude that MTSS Algorithm gives a better representation of the real mass transfer
process and hence it is expected to produce better simulation results when compared
to the experimental data. The latter conclusion is in accordance with the dispersed
and continuous phase concentrations curves as shown in Figure 4.17.
The MTASD, MTMD and MTSS algorithms described in this chapter can be
used successfully to calculate the amount of mass transfer from the continuous phase
to dispersed phase. But in most industrial applications, the value of input parameters
that is needed to be identified instead of the value of the output parameters. The
identified input values will be used in the extraction process to produce the desired
output concentrations. This problem is known as the inverse problem. Traditionally,
such problem can only be solved by repeated simulation of the above algorithms. This
method consumes a lot of computer time and it will be costly if actual processes are
involved. Therefore, in the following chapter a new technique which is based on fuzzy
approach is introduced in order to overcome this problem.
95
Table 4.7: The concentration of the dispersed and continuous phase according MTMD
and MTSS Algorithm
Stage
Continuous
Dispersed
Continuous
Dispersed
(MTMD)
(MTMD)
(MTSS)
(MTSS)
1
0.959
0.003
0.8949
0.0027
2
0.9594
0.0068
0.8951
0.0033
3
0.96
0.0117
0.8966
0.0078
4
0.9608
0.0176
0.8985
0.0136
5
0.9619
0.0245
0.9009
0.0206
6
0.9631
0.0324
0.9036
0.0287
7
0.9645
0.0412
0.9067
0.038
8
0.966
0.051
0.9102
0.0485
9
0.9678
0.0616
0.914
0.06
10
0.9695
0.073
0.9181
0.0725
11
0.9715
0.0853
0.9226
0.086
12
0.9733
0.0982
0.9275
0.1005
13
0.9754
0.1119
0.9325
0.1157
14
0.9773
0.1262
0.9379
0.1319
15
0.9794
0.1411
0.9436
0.1488
16
0.9814
0.1564
0.9494
0.1663
17
0.9835
0.1722
0.9554
0.1843
18
0.9855
0.1883
0.9616
0.203
19
0.9875
0.2047
0.9679
0.2219
20
0.9895
0.2212
0.9743
0.241
21
0.9915
0.2378
0.9808
0.2606
22
0.9933
0.2544
0.9872
0.2799
23
0.9953
0.2694
0.993
0.2974
24
0.9971
0.2783
0.9966
0.3081
No
CHAPTER 5
THE INVERSE MODEL OF MASS TRANSFER: THEORETICAL
DETAILS AND CONCEPTS
5.1
Introduction
Basically this chapter introduces an Inverse Single Drop Single Stage-Fuzzy
(ISDSS-Fuzzy) model which represents the mass transfer process of a single drop in
a single stage RDC column. This model is a basis for the inverse model of the mass
transfer process in the real RDC column. It begins with a discussion of the formulation
of the inverse model for mass transfer process in the RDC column.
Section 5.2.1 presents the mappings of (5.1) and (5.2) which represent the
forward model involved. The mappings are in the form of functions of several variables.
Then Subsection 5.3.1 describes the three phases involved in developing the inverse
model. To validate the model, an example is used by implementing it to the problem
of mass transfer process for a single drop in a single stage RDC column.
The detail of the process is written in the form of an algorithm, which
is presented in Section 5.4.
We also used the Theorem of Optimized Defuzzified
and its corollary which are introduced by Ahmad in [8] in order to determine the
optimal combination of input parameters for the desired output parameters. Finally, a
discussion is presented and conclusions are drawn on the presented work.
97
5.2
Inverse Modelling in RDC Column
The MTASD, MTMD MTSS algorithms described in the previous chapter can
be used successfully to calculate the amount of mass transfer from continuous phase
to dispersed phase. But this type of modelling, which is known as forward modelling
is not efficient enough to determine the required input parameters in order to produce
certain values of output parameters. The determination of the input values by trial
and error consumes a lot of computer time and it will be costly if actual processes were
involved. These difficulties inspired us to develop an alternative method in order to
overcome the problems. Hence, a new technique which is based on fuzzy approach is
introduced here to determine the input concentration of both phases for a certain value
of output concentrations. This type of modelling is called inverse modelling.
Inverse Modelling is the process of obtaining the input parameters or
determining the causes for desired output parameters[42]. In other words, in inverse
modelling, the desired responses are given and a model is used to estimate the input
parameters. Before the inverse model can be developed, we have to consider the first
two steps below, these are Steps 1 and 2. After the inverse model has been developed,
we have to consider Step 3 in order to get the solution of the model.
1: The understanding and construction of a forward mathematical model of the system.
In this case, we already have the forward mathematical model of the system as
described in Chapters 3 and 4.
2: Studying the technique of solving this problem.
The forward algorithms of the mass transfer process in the RDC column have
been developed successfully as explained in Chapter 4.
3: Development of inverse problem algorithms necessary to solve the corresponding
inverse problem.
Traditionally, the determination of input parameters for desired value of output
parameters of mass transfer process has been addressed by repeated simulation
of forward problem.
The following subsection will give the formulation of inverse problem for the
mass transfer process in the RDC column.
98
5.2.1
Formulation of the Inverse Problem
Basically the forward model of the mass transfer process in the RDC column
consists of IBVP of diffusion equation (4.1)-(4.3) and nonlinear equations (4.15) and
(4.16), linear algebraic equations (4.18) and mass balance equation of (4.19). But the
IBVP of diffusion equation of (4.1)-(4.3) is actually embedded in (4.15) and (4.18).
Thus the multivariate system modelled by Equations (4.15), (4.16), (4.18) and (4.19)
can be simplified as the multiple input multiple output (MIMO) system of
h1 (xin , yin ) = yout ,
(5.1)
h2 (xin , yin , h1 ) = xout ,
(5.2)
where h1 is the mapping that represents equations (4.15), (4.16) and (4.18) to produce
the first output while h2 represents the mass balance equation to produce the second
output. The MIMO system can be represented as a block diagram as in Figure 5.1.
Equations (5.1) and (5.2) can be written in simplified form of
h(h1 , h2 ) = (yout , xout ),
(5.3)
or h : hi → 2 where h is a functional and hi is a space of functions and 2 is a two
dimensional Euclidean space. In other words, h is a functional from a space of functions
to a plane.
x
in
Average
Drop
Concentration
y
out
yout
yin
Mass Balance
Equation
xout
Figure 5.1: The MIMO system
Since the mass transfer process of the multiple drops in the multi-stage RDC
column is being considered, (5.3) is used repeatedly in order to accomplish the final
99
Output 2
Output 1
h1(xinn, yinn)
h2(xinn, yinn, h1)
h1(xin3, yin3)
h2(xin3, yin3, h1)
h1(xin2, yin2)
h2(xin2, yin2, h1)
h1(xin1, yin1)
h2(xin1, yin1, h1)
Input 1
Final Stage
2nd Stage
1st Stage
Input 2
Figure 5.2: Schematic diagram of the forward model in a Multi-stage RDC column
100
outputs. The multi-stage process in the RDC column is shown in Figure 5.2. In this
forward process, the values of the output parameters, yout and xout can be determined
if the values of the input parameters are given. Now, consider the inverse problem of
this system, which is to determine the input parameters, xin and yin for a desired values
of the output parameters, xout and yout . Since the exact solution of this problem is
hard to attain due to its ill-posed characteristics, an approximation method has to be
considered. The following section describes the method used in obtaining the solution
of the problem.
5.3
Inverse Modelling Method
The approach adopted in this work is based on the basic principles of fuzzy
modelling which was laid down by Zadeh[51]. He stated indirectly that fuzzy modelling
can provide an approximate and yet effective means of describing the behavior of
systems which are complex or ill-defined to admit use of precise mathematical analysis.
Therefore in this section, the description of the fuzzy approach is illustrated starting
with the statement of the multivariate equations of the system used.
The multivariate systems modelled by Equations (5.1) and (5.2) can be written
as
(n)
(n)
(n)
h1 (xin , yin ) = yout ,
(n)
(n)
(n)
(n)
(5.4)
(n)
h2 (xin , yin , h1 ) = xout ,
(5.5)
or
(n)
(n)
h(h1 , h2 ) = (yout , xout ),
(5.6)
where the superscript n refers to nth-stage of RDC column. When n = 1 which is
(1)
(1)
the first stage, the output parameters are yout and xout . These output parameters will
(2)
(1)
(2)
(1)
become input parameters for the second stage which are yin = yout and xin = xout .
This process continues up to the final stage.
In our study, these parameters are determined through experimental data,
simulation of forward model data or by suggestion of experts. These values are modelled
by the concept of fuzzy numbers. Fuzzy number defined in Chapter 2 is a fuzzy set
101
that is convex and normal. A fuzzy number of dimension one is considered in the early
phases of the development of the inverse model. Specifically, triangular fuzzy number
is employed through out the whole process of developing the model.
In formulating the inverse model, the approach introduced by Ahmad in [8] is
modified by considering the MIMO system of Equation (5.6).
5.3.1
Fuzzy Flow Chart
Ahmad introduced Fuzzy Flow chart to design a technique that optimizes
geometrical and electrical parameters of the microstrip lines in order to reduce the
crosstalk level[8]. Based on this idea we developed an inverse model to overcome
the difficulties explained in Section 5.2. The procedure in developing the model is
divided into three phases. The phases are the Fuzzification, Fuzzy Environment and
Defuzzification phase.
CRISP VALUE
FUZZIFICATION
FUZZY VALUE
FUZZY
ENVIRONMENT
FUZZY VALUE
DEFUZZIFICATION
CRISP VALUE
Figure 5.3: Fuzzy Algorithm
102
5.3.2
Fuzzification Phase
According to Klir et.al.[51], variables involved in an engineering design are
usually referred to as parameters. The parameters are classified as input, output and
performance parameters. In our problem, the input parameters are the geometrical
configurations and physical properties, and the input concentrations of continuous and
dispersed phase. The geometrical configurations are the diameters of the rotor disc,
the column, the rotor speed etc. Meanwhile the physical properties are the viscosity
and the density of the continuous and dispersed phase etc. The output parameters are
the output concentration of the continuous and dispersed phase and the performance
parameter is the hold-up. These parameters are specified in Table 5.1.
Table 5.1: Design parameters
Input Parameters
Output Parameters
Performance Parameters
Geometrical Properties
Continuous Phase Conc.
Holdup
Physical Properties
Dispersed Phase Conc.
Continuous Phase Conc.
Dispersed Phase Conc.
In our model, we assume that the input parameters of the geometrical
configurations and physical properties are fixed for certain values. These values are
taken from experimental data ( see in Appendix A). The performance parameter is
also assumed to be fixed. The actual input parameters of the model are the input
concentration of the continuous and dispersed phase. On the other hand the output
parameters are the output concentration of the continuous and dispersed phase.
In the fuzzification phase, the input and output parameters x(0) , y (0) and x(n) ,
y (n) respectively are fuzzified. For simplicity we denote the first input and output
parameters which are the input and output concentrations of the continuous phase,
(0)
(n)
(0)
(n)
x(0) , x(n) as p1 , q1
and the input and output concentrations of the dispersed phase,
y (0) , y (n) as p2 , q2 . These notations will be applicable throughout this thesis. The
descriptions of these notations are shown in Figure 5.4.
In this process the determined crisp values of the parameters are fuzzified by
the membership function. Here, the triangular membership function is used due to its
suitability to the nature of the problem. For example, if [a1 , a2 ] is the domain of the
103
(n)
(0)
q1
p1
Forward System
(0)
p2
(n)
q2
Figure 5.4: The view of the input and output parameters of the system
input parameter, then the end-points of this interval will be assigned zero fuzzy value.
If we refer to Figure 2.7 in Chapter 2, there is a value of input parameter, say, a2 ,
a2 ∈ [a1 , a2 ] which will give the close preferred output parameter. This input value
will be assigned to a fuzzy value of one. Therefore, any input value, p1 ∈ [a1 , a3 ] will
be assigned a fuzzy value according to it’s about a2 . Besides the suitability reasoning,
many industrial applications used triangular membership function due to its simplicity
and computational efficiency [51].
(0)
Now, assume an input pi
that takes value in the set Pi ∈ [ai , bi ] then the
(0)
preferences for the different values of pi
can be expressed as a fuzzy set FPi on Pi .
Similarly let QPi be the preferred output parameters which take all the input parameters
(0)
as its variables and is presented by fuzzy set FQPi . Each value of pi
(0)
corresponds to the membership value of pi
(0)
∈ Pi , FPi (pi )
in the set. The subscript i referred to the
different input or output parameter.
The following subsection describes the fuzzy environment phase where all the
fuzzified input values from fuzzification phase are used.
5.3.3
Fuzzy Environment Phase
The fuzzified input parameters from fuzzification phase are then used to
determine the induced output parameters. This process can be done by assuming
that all the fuzzy sets (taken from the previous phase), FPi , express preferences of all
(0)
input parameters pi
∈ Pi with Pi ⊂ R+ to be determined, normalised and convex. P
is a closed interval positive real number.
In this phase, the input, output and performance parameter must be
determined. We should also be able to specify the functions used which map the
input parameters to the output parameters. Let the function be h. Then, select the
104
value of α-cut, such that α1 , α2 , α3 , ..., αk ∈ (0, 1] which are equally spaced. After the
selection of the α-cut has been made, all the α-cut for every FPi must be determined.
According to Definition 2.2, the α-cut for every FPi is in the form of an interval.
The next step is to generate all 2m combinations of the endpoints of the intervals
which represent the αk -cut for every FPi . Since in this study we assume that the input
parameters are the input concentration of the continuous and dispersed phase, thus
m = 2. Therefore we will get 4 combination for every αk -cut. These combinations
are then used in the forward model as the values of its input parameters in order to
determine their corresponding output parameters, rk with respect to each value of the
α-cut. To get the induced performance parameter, Find , we have to determine the
minimum and maximum values of the output parameters, rk with respect to each value
of the α-cut.
After the induced performance parameter Find has been determined, plot these
points on the graph. The fuzzified output parameters, Fh(Pi ) must also be plotted on
the same axes for each i. Next the intersections between the fuzzified output parameters
Fh(Pi ) and the induced output parameters Find have to be found. This step is followed
by the determination of the largest fuzzy membership value for the intersection, say
(0)
f ∗ . Finally, the corresponding value of the output parameters, say h(pi )∗ has to be
determined.
The steps of the process of getting the optimal combination of the input
parameters values for the corresponding f ∗ will be given in the following subsection.
5.3.4
Defuzzification Phase
In this phase the optimal combination of the input parameters will be
determined. First we must determine the α-cut for FPi of the corresponding f ∗ . After
this value has been determined, all 4 combinations of the endpoints of the intervals
representing α = f ∗ -cut must be generated. These four combination of inputs values
are actually the possible solutions of the problem.
The next step is to determine the corresponding output parameter, rk∗ for each
105
of the combinations with respect to α = f ∗ . As stated in Chapter 2, one of the ill-posed
characteristics is that there may be more than one solution. Hence the inverse problem
in this study truly satisfies the ill-posed characteristics. Finally, since there are four
possible solutions, the optimal combination of input parameters must be determined.
The following example is used to describe the above method.
5.3.5
Numerical Example
In this section, we consider the problem of determining the values of the input
parameters for the desired values of the output parameters of the mass transfer process
of a single drop in a single stage RDC column. This is simply a Multiple Input Multiple
Output (MIMO) system. As stated in previous section, the first step is to determine
the preferred input and output parameters.
The purpose of this example is to see whether the method described above works
for the system. Further more, this example will also provide the detailed calculations
and as such a motivation to produce a better inverse model of the mass transfer
process in the RDC column. Since the system is a single drop single stage, there
is no experimental data available in the literature. Therefore to provide the preferred
input and output parameters, the simulation data of the forward model is used. The
specification of these values are given in Table 5.2 and 5.3.
Table 5.2: Preferred input values
Input Parameters
Domain
Suggested Value
Continuous Conc. (CCin )
[32.8
55.6]
45.48
Dispersed Conc.(CDin )
[11.2
14.5]
13.78
Table 5.3: Preferred output values
Output Parameters
Domain
Suggested Value
Continuous Conc.(CCout )
[28
40]
36
Dispersed Conc.(CDout )
[23
35]
30
106
1
x
1
x2
0.9
0.8
Membership Value
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
10
15
20
25
30
35
40
Concentration (g/l)
45
50
55
60
Figure 5.5: Triangular fuzzy number of the input parameters
These values are now ready to be fuzzified by the triangular membership
function. Figures 5.5 and 5.6 show the triangular fuzzy number of the preference
input and output parameters respectively. The two limits of the domain will have
fuzzy values of zeroes whereas the suggested value will be assigned a fuzzy value equal
to one. Now let’s choose the α-cut to be 0, 0.2, 0.4, 0.6, 0.8 and 1.0. Using the α-cut
definition, we calculate the α-cut of all the input and output parameters obtained from
Figure 5.5 and 5.6. For example, take α = 0.2, then the α = 0.2-cut for preferred input
is
Aα=0.2 = {pi ∈ Pi |µA (pi ) ≥ 0.2},
if i=1, the α = 0.2-cut is
Aα=0.2 = {p1 ∈ P1 |µA (p1 ) ≥ 0.2},
= [32.8 + 0.2(45.48 − 32.8), 55.6 − 0.2(55.6 − 45.48)],
= [35.34, 53.58].
The same procedure is used to calculate the α-cut for each value of α. These values
are then listed in Tables 5.4 and 5.5.
The next step is to generate all the possible combinations of the endpoints of
the interval representing each α-cut for the input parameters. We have 4 combination
107
1
y
1
y2
0.9
0.8
Membership Value
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
22
24
26
28
30
32
Concentration (g/l)
34
36
38
40
Figure 5.6: Triangular fuzzy number of the output parameters
Table 5.4: α-cuts values for input parameters
α-cuts Values
Input Parameters
0.2
0.4
Continuous Conc.
[35.34 53.58]
[37.87 51.55]
[40.41
Dispersed Conc.
[11.72
[12.23 14.21]
[12.75 14.07]
14.36]
0.6
49.53]
0.8
1.0
[42.94 47.50]
[45.48
45.48]
[13.26
[13.78
13.78]
13.92]
of 2-tuple for each α-cuts. For example, if α = 0.2, the alpha-cut for the first and
second input parameters are [35.34, 53.58] and [11.72, 14.36] respectively. Therefore the
four combinations of the end-points of these intervals are (35.34, 11.72), (35.34, 14.36),
(53.58, 11.72) and (53.58, 14.36). The first tuple is the value for the first input parameter
and the second tuple is the second input parameter. The same procedure is applied
to the rest of the alpha-cuts. The combinations of the end-points with respect to the
corresponding α-cuts can then be found in Table 5.6.
After all the possible combinations are identified, each of these combinations
will be mapped to the output value by the forward model. This forward model consists
of an IBVP of diffusion equation and a few of the algebraic equations as explained
in Chapter 3 and then simplified as the MIMO system of (5.3). For example, with
α = 0.2, the 1st input combination, (35.34,11.72) will map to output parameters of
108
Table 5.5: α-cuts values for output parameters
α-cuts Values
Output Parameters
0.2
0.4
0.6
0.8
1.0
Continuous Conc.(CCout )
[29.6
39.2]
[31.2 38.4]
[32.8 37.6]
[34.4
36.8]
[36.0 36.0]
Dispersed Conc.(CDout )
[24.4
34.0]
[25.8 33.0]
[27.2 32.0]
[28.6
31.0]
[30.0
30.0]
Table 5.6: The combination for each α-cuts values parameters
Combination of
α-cuts Values
input (CCin , CDin )
0.2
0.4
0.6
0.8
1.0
1st combination
(35.34,11.72)
(37.87,12.23)
(40.41,12.75)
(42.94,13.26)
(45.48,13.78)
2nd combination
(35.34,14.36)
(37.87,14.21)
(40.41,14.07)
(42.94,13.92)
(45.48,13.78)
3rd combination
(53.58,11.72)
(51.55,12.23)
(49.53,12.75)
(47.50,13.26)
(45.48,13.78)
4th combination
(53.58,14.36)
(51.55,14.21)
(49.53,14.07)
(47.50,13.92)
(45.48,13.78)
27.89 and 22.35 respectively. The complete set of the output parameter values for their
corresponding input values are listed in Table 5.7.
Table 5.7: The output of each combination of each α-cuts
Combination of
α-cuts Values
input (CCin , CDin )
0.2
0.4
0.6
0.8
1
CC
CD
CC
CD
CC
CD
CC
CD
CC
CD
1st combination
27.89
22.35
29.95
23.78
32.01
25.21
34.07
26.63
36.13
28.06
2nd combination
42.90
32.02
41.21
31.03
39.52
30.04
37.82
29.05
36.13
28.06
3rd combination
27.40
23.80
29.58
24.87
31.76
25.93
33.95
26.99
36.13
28.06
4th combination
43.37
30.57
41.57
29.94
39.76
29.31
37.94
28.68
36.13
28.06
The mapped output parameters are then used to determine the induced output
parameters, Find . The induced output parameters can be obtained by taking the
minimum and maximum value (endpoints of interval) for each element of output
parameters. As an example, in Table 5.7, the output concentration of the continuous
phase, q1 is equal to 27.89, 42.9, 27.4 and 43.37 for each input combination of α = 0.2.
In this procedure we will take the minimum value of 27.4 and the maximum of 43.37.
This indicates that α = 0.2-cut for the first induced output parameter is [27.4, 43.37].
109
The same process is repeated for different values of alpha to obtain the corresponding
α-cut. These values are listed in Table 5.8.
Table 5.8: The min and max of the combination for each α-cuts values
Combination of
α-cuts Values
input(CCin , CDin )
0.2
0.4
0.6
0.8
1.0
min
max
min
max
min
max
min
max
min
max
CCout
27.40
43.37
29.58
41.57
31.76
39.76
33.95
37.94
36.13
36.13
CDout
22.35
32.02
23.78
31.03
25.21
30.04
26.63
29.05
28.06
28.06
The induced output values are then used to plot the curves of induced output
parameters. The induced output parameters are actually triangular fuzzy numbers. The
intersection between the induced and the preferred output for both output parameters
are shown in Figure 5.7 and 5.8. From Figure 5.7, the intersection occurs at the
maximum side of the induced triangle meanwhile Figure 5.8 shows that the intersection
occurs at the minimum side of the induced triangle.
1
induced
preferred
0.9
0.8
0.7
f*=0.8377
Fuzzy Values
0.6
0.5
0.4
0.3
0.2
0.1
0
20
25
30
28.8636
Dispersed Phase Concentration
35
Figure 5.7: Intersection between induced and preferred output for dispersed phase
concentration
The intersection of the two curves in both figures will provide the f ∗ -values
110
1
induced
preferred
0.9
f* =0.9561
0.8
0.7
Fuzzy value
0.6
0.5
0.4
0.3
0.2
0.1
0
25
30
35
40
45
50
Continuous Phase Concentration
Figure 5.8: Intersection between induced and preferred output for continuous phase
concentration
which are z = 0.8377 and z = 0.9561 respectively. These f ∗ -values are then processed
in the defuzzification phase. In this phase, defuzzification is carried out to get the best
possible combination of the input parameters in order to produce the output parameters
which are close to the desired output values. Each of the four combinations of the
endpoints of the interval are determined and these values are then used to calculated
the output parameters. All the data are given in Table 5.9.
Table 5.9: Input combination with fuzzy value z = 0.8377
Combination of Input(CCin , CDin )
Input Value
Output Value
1st combination
(43.42, 13.36)
26.90
2nd combination
(43.42, 13.90)
27.20
3rd combination
(47.12, 13.36)
28.56
4th combination
(47.12, 13.90)
28.86
For the four combinations of the input parameters given in Table 5.9, we have to
choose only one combination which can provide the optimal solution. To achieve this,
Ahmad[8] introduced an optimization defuzzified theorem. By this theorem, the fourth
combination of input parameters i.e. (47.12, 13.90) of fuzzy value 0.8377 is chosen as
111
Table 5.10: Input combination with fuzzy value z = 0.9561
Combination of Input(CCin , CDin )
Input Value
Output Value
1st combination
(44.92, 13.67)
35.67
2nd combination
(44.92, 13.81)
35.65
3rd combination
(45.92, 13.67)
36.52
4th combination
45.92, 13.81)
36.50
the best solution. Meanwhile for the second output parameter we choose the second
combination of input parameter that is (44.92, 13.81) of z = 0.9561.
From both of the input combinations we choose the one with the higher
membership value. In this case we take fuzzy value of z = 0.9561 where the best
input combination of (44.92, 13.81) is chosen. These input values are then used in the
forward model producing the optimal solution of (35.65, 27.82).
Table 5.11: Optimized input parameters
Input Parameter
Calculated Input Values
Preferred Values
Error(%)
CCin
44.92
45.48
1.23
CDin
13.81
13.78
0.22
From the above example, the values of the input parameters for the desired
values of the output parameters are successfully determined. The values are shown in
Tables 5.11 and 5.12. The input values differ from the preferred values with an error of
1.23% and 0.22% respectively. The percentage error for each of the output parameters
of the system are 1.06% and 7.27% respectively.
Table 5.12: Calculated output parameters
Output Parameter
Calculated Output Values
Preferred Values
Error(%)
CCout
35.62
36
1.06
CDout
27.82
30
7.27
The complete process of the inverse model in determining the input parameters
112
for the desired output parameters is explained in detail by the following algorithm.
5.4
Inverse Modelling of the Mass Transfer Process of a Single Drop in a
Single Stage RDC Column Fuzzy-Based Algorithm(ISDSS-Fuzzy)
5.4.1
Algorithm 5.1: Inverse Fuzzy-Based Algorithm (ISDSS-Fuzzy)
The algorithm has the following steps:
Step 1: Let h1 : I1 × I2 −→ O2 , h2 : I1 × I2 × O2 −→ O1 where O2 ∈ Q2 , O1 ∈ Q1 ,
Q1 , Q2 ∈ R+ be the output parameters such that r1 = h1 (p1 , p2 ) and r2 =
h2 (p1 , p2 , h1 (p1 , p2 )).
Step 2: Select appropriate value for α-cut, such that α1 , α2 , α3 , ...αk ∈ (0, 1].
Step 3: For each Pi , determine the end points of all the αk -cuts, FIi (i = 1, 2).
Step 4: For each Qi , determine the end points of all the αk -cuts for preferred output
parameters, FQP (i = 1, 2).
Step 5: Generate all 2m combinations of all the endpoints of intervals representing
αk -cuts. Each combination is an m-tuple (in this problem m = 2).
Step 6: Determine r1 = h1 (p1 , p2 ) and r2 = h2 (p1 , p2 , h1 (p1 , p2 )) for each 2-tuple j ∈
1, 2, ...2m .
Step 7: For each α-cuts, determine the induced output parameters, Find by taking the
min value and max value of each element of i i.e let
Find = [min rj , max rj ]
for all j = 1, 2
Step 8: Set FQP ∧ Find and find the fuzzy number of f = sup(FQP ∧ Find )
Step 9: Find the α-cut of FIi for corresponding value of f .
Step 10: Repeat Steps 5 and 6 for α = f and denote the corresponding output
parameter as rj for each 2-tuple j ∈ 1, 2, ...2n .
113
Step 11: Determine the optimal combination of input parameter and stop.
The value determined in the final step of the algorithm is the approximate value
of the input parameter which will produce the desired value of the output parameter.
The value is determined by the Theorem of Optimized Defuzzified Value[8] and its
corollary as stated below.
Theorem 5.1. ([8]) If h∗i = rj∗ = max rj such that µ(ri∗j ) = f ∗ , for some (rj , f ∗ ) ∈
Find , then rj∗ = h∗i = max [hi (p∗1 , p∗2 )] where µ(p∗i ) = f ∗ .
Corollary 5.1. ([8]) If h∗i = rj∗ = min rj such that µ(ri∗j ) = f ∗ , for some (rj , f ∗ ) ∈
Find , then rj∗ = h∗i = min [hi (p∗1 , p∗2 )] where µ(p∗i ) = f ∗ .
Refer to [8] for the proof of Theorem 5.1 and Corollary 5.1. The theorem
indicates that if the preferred fuzzy intersects on the maximum side of the fuzzy
induced, then the set of optimized parameters is the set for the maximum of the induced
values. Furthermore, the corollary indicates that if the preferred fuzzy intersects on
the minimum side of the fuzzy induced, then the set of optimized parameters is the set
for the minimum of the induced values.
5.5
Simulation Results
The simulations of the mass transfer process for a single drop single stage system
by ISDSS-Fuzzy Algorithm are carried out. The input data for the simulations is based
on the data used in Section 5.3.5. Here, the domain of the input parameters is enlarged
while the domain of preferred outputs remains unchanged. For the first simulation,
the domains of CCin and Cdin are [30.8, 57.6] and [9.2, 17.5] respectively. The input
data for second simulation are [28.8, 59.6] and [7.2, 19.5]. In both simulations the same
suggested value is used for the input and the output parameters.
The result of the simulations are tabulated in Tables 5.13 and 5.14.
For
Simulation 1, the preferred output dispersed phase concentration intersected on the
maximum side of induced triangular, therefore by Theorem 5.1 the optimal solution
is (47.10, 14.29) with the maximum output of 28.35. Whilst, the intersection of the
114
Table 5.13: Simulation 1: The results of input domains [30.8, 57.6] and [9.2, 17.5]
Combination
Cd
CC
z = 0.866
z = 0.9752
Input
Output
Input
Output
1st combination
(43.51, 13.17)
26.83
(45.12, 13.67)
35.84
2nd combination
(43.51, 14.29)
27.45
(45.12, 13.87)
35.81
3rd combination
(47.10, 13.17)
27.73
(45.78, 13.67)
36.40
4th combination
(47.10, 14.29)
28.35
(45.78, 13.87)
36.37
Table 5.14: Simulation 2: The results of input domains [28.8, 59.6] and [7.2, 19.5]
Combination
Cd
CC
z = 0.8822
z = 0.9819
Input
Output
Input
Output
1st combination
(43.75, 13.24)
26.98
(45.21, 13.70)
35.91
2nd combination
(43.75, 14.22)
27.52
(45.21, 13.85)
35.89
3rd combination
(46.91, 13.24)
27.86
(45.70, 13.70)
36.33
4th combination
(46.91, 14.22)
28.40
(45.70, 13.85)
36.30
induced output continuous phase concentration and preferred output occurred on the
minimum side of the induced triangle. Therefore by Corollary 5.1 the optimal solution
is (45.12, 13.87) with the minimum output of 35.89.
For comparison purposes, we calculate the percentage errors between the
optimal solutions obtained from the algorithm and the suggested values.
The
comparison is also made between the errors calculated from the output solution and the
preferred output parameters. These values are listed in Table 5.15. On the other hand,
the errors between the output solutions and the preferred outputs for the different input
domains are also given in Table 5.16.
5.6
Discussion and Conclusion
The ISDSS-Fuzzy Algorithm was developed through the three phases of the
fuzzy system. Steps 1 to 4 described the fuzzification phase. In this phase, the preferred
115
Table 5.15: The errors between the calculated input values and preferred values for
different input domain
Simulation
Input Parameter
Numbers
CCin
Cdin
Eg 5.3.5
[32.8 55.6]
[11.2 14.5]
Sim 1
[30.8 57.6]
Sim 2
[28.8 59.6]
Calculated Input Values
CCin
Preferred Values
Errors (%)
Cdin
CCin
Cdin
CCin
Cdin
44.92
13.81
45.48
13.78
1.23
0.22
[9.2 17.5]
45.12
13.87
45.48
13.78
0.79
0.65
[7.2 19.5]
45.21
13.85
45.48
13.78
0.59
0.5
Table 5.16: The errors between the calculated output values and preferred values for
different input domain
Simulation
Numbers
Calculated Output Values
CCout
Preferred Values
Errors (%)
Cdout
CCout
Cdout
CCout
Cdout
Eg 5.3.5
35.62
27.82
36
30
0.92
7.53
Sim 1
35.81
27.94
36
30
0.53
6.87
Sim 2
35.89
27.97
36
30
0.32
6.77
input and output parameters were fuzzified. Steps 5 to 8 described the processing of the
fuzzified parameters in the fuzzy environment. The phase of defuzzification was then
described in Steps 9 to 11. The values determined in the final step of the algorithm are
the approximate optimal value of input parameters that will produce the desired value
of the output parameters. These values were determined by the Optimized Defuzzified
Value Theorem or its corollary.
In early stages of the development of the inverse model for the mass transfer
process in the RDC column, we implemented the ISDSS-Fuzzy Algorithm to the single
drop single stage system of the column. The system involved is the multiple input
multiple output (MIMO) system. Since MIMO system can always be separated into
a group of multiple input single output (MISO) [36, 56], we considered two MISO
systems to represent the problem involved.
The system consists of the IBVP of
the diffusion equation of (3.34)-(3.37) where from these equations, (3.65) was then
determined together with the mass balance equation of (4.19). For clear description,
we illustrate the separation of the model by a diagram in Figure 5.9.
116
x(0)
1
(n)
y2
(0)
x1
MISO
(0)
y(n)
2
x2
MIMO
(0)
(0)
x2
(n)
y1
x1
x(0)
MISO
y(n)
1
2
Figure 5.9: The MIMO system is separated into 2 MISO system
As shown in Figure 5.9, the MIMO system representing the mass transfer
process in the RDC column is now separated into two independent MISO systems. After
the completion of the fuzzification process , the fuzzified input parameters were then
processed to produce the induced output parameters. Since the MIMO system is now
separated into two independent MISO systems, we will get two different values from the
intersection points between two sets of two triangles. Table 5.9 presented the result of
intersection between the induced and preferred concentration of the continuous phase.
Meanwhile Table 5.10 presented the result of the intersection between the induced and
the preferred concentration of the dispersed phase.
A decision was made in order to determine the appropriate fuzzy value between
f ∗ = 0.8377 and f ∗ = 0.9561. Since the fuzzy value corresponds to the degree of
desirability, the largest value f ∗ = 0.9561 was chosen. With this value we arrived at
the stage where the determination of the optimal combination of the input parameters
from amongst the four combinations has to be resolved. The choice must produce
the least error when compared to the preferred values. This process was done in the
defuzzification phase.
The simulations of the mass transfer process for a single drop single stage
system by ISDSS-Fuzzy Algorithm were also carried out for different domains of input
parameters. The aim of the simulation is to see the effect of the input domain on the
solution. Tables 5.15 and 5.16 showed the percentage of errors between the optimal
solutions obtained from the algorithm and the suggested values and the errors between
the calculated and the preferred output parameters for the different simulations. From
the tables, we conclude that the change in the input domain does affect the output of
the algorithm. The tables showed that, if the domain is enlarged, the calculated input
and output values are closer to the preferred values.
From the numerical example shown in this chapter, it is clear that the ISDSS-
117
Fuzzy Algorithm is applicable to the Single Drop Single Stage RDC system. However,
the type of output parameters for MIMO system of the mass transfer process in the
RDC column is actually two dependent parameters. This situation is well illustrated
in the diagram shown in Figure 5.1. Therefore the inverse model of a Single Drop of a
Single Stage system which is in the form of the ISDSS-Fuzzy Algorithm is not sufficient
to represent the real situation. Hence in the next chapter, we will show a new approach
using fuzzy logic in developing the inverse model of the mass transfer process in such a
way that the norm of separating the MIMO system into a group of MISOs is no longer
necessary.
CHAPTER 6
INVERSE MODEL OF MASS TRANSFER IN THE MULTI-STAGE
RDC COLUMN
6.1
Introduction
Due to the inadequacy of the ISDSS-Fuzzy Algorithm described in Chapter 5
in representing the inverse model of the MIMO system, a new model is proposed. The
proposed model is an inverse model of the MIMO system without the separation of the
system into the MISOs.
To develop the model, fuzzy number of dimension two is used instead of the
one used in Chapter 5. Therefore in Section 6.2, we describe some theoretical details
involved. The details are about the relationship between two crisp sets and this is
followed by the relationship between two fuzzy sets. We also include some examples
which can explain the concept more clearly. From fuzzy relation we extend the concept
of fuzzy number of dimension one to dimension two. It uses the concept of the domain
of confidence in 2 which is actually a generalization of the interval in .
Section 6.4 discusses the development of the Inverse Model of Mass Transfer
Process of a Single Drop in a Single Stage RDC Column based on two dimensional
fuzzy number. The constructed algorithm referred to the Inverse Single Drop Single
Stage-2D Fuzzy Algorithm. To validate the algorithm we also used the same data
as in 5.3.5 so that the comparison of both algorithms can be carried out. We then
implement the latter algorithm to the mass transfer process in the multi-stage RDC
119
column. Some modification of the algorithm is considered before the implementation
can be done successfully.
6.2
Theoretical Details
Let U and V be two arbitrary classical crisp sets. The Cartesian product of U
and V , denoted by U × V , is the crisp set of all ordered pairs (u, v) such that u ∈ U
and v ∈ V ; that is,
U × V = {(u, v)| u ∈ U and v ∈ V }.
In general, the Cartesian product of arbitrary n crisp sets U1 , U2 , ...Un , denoted by
U1 × U2 × ... × Un , is the crisp set of all n-tuples (u1 , u2 , ...un ) such that ui ∈ Ui for
i ∈ 1, 2, ...n, that is
U1 × U2 × ... × Un = {(u1 , u2 , ...un )| ui ∈ Ui }.
6.2.1
Relation
A crisp relation among crisp sets U1 , U2 , ...Un , is a subset of the Cartesian
product U1 × U2 × ... × Un , that is, if we use Q(U1 , U2 , ...Un ) to denote a relation among
U1 , U2 , ...Un , then
Q(U1 , U2 , ...Un ) ⊂ U1 × U2 × ... × Un .
Example 6.1. Let U = {2, 3, 4} and V = {3, 4, 5}. Then the Cartesian product of U
and V is the set U × V = {(2, 3), (2, 4), (2, 5), (3, 3), (3, 4), (3, 5), (4, 3), (4, 4), (4, 5)}. A
relation between U and V is a subset of U × V . For example, let Q(U, V ) be a relation
named “the first element is smaller than the second element”, then
Q(U, V ) = {(2, 3), (2, 4), (2, 5), (3, 4), (3, 5), (4, 5)}
The following example shows the relation between two infinite sets.
120
Example 6.2. Let S = [24, 32] and T = [12, 18]. Then the Cartesian product of S and
T is the set S × T = {(s, t) : s ∈ S, t ∈ T } which is a rectangular surface and the four
vertices of the rectangle are (24, 12), (24, 18), (32, 12), (32, 18). Let P (S, T ) be a relation
named “the boundary points on the rectangular surface of S × T ”, then
P (S, T ) = {(s, t) : (24, t), (32, t) where t ∈ T and (s, 12), (s, 18) where s ∈ S}
(6.5)
Example 6.3. Let S = [a1 , an ] ∈ and T = [b1 , bn ] ∈ be two intervals of two input
parameters of a MIMO system. Let the Cartesian product of S and T be the relation
between the two input parameters in 2 and denoted as S × T = {(s, t) : s ∈ S, t ∈
T }. If there is a preference point in S × T say, (am , bm ), where a1 ≤ am ≤ an and
b1 ≤ bm ≤ bn , then this point will give the optimal solution for the system. If we use a
number in the interval [0, 1] to represent the degree of “the point in S × T that will give
the optimal solution”, then this concept may be represented by the following relational
matrix:
S
[a1 , an ] [a2 , an−1 ] · · ·
am

[b1 , bn ]
T


[b2 , bn−1 ] 


..

.


bm

0
0
0
0.2
0
0
..
0
.
0




,



1
where the relation of the Cartesian product of the two intervals is the relation
as (6.5) defined in Example 6.2.
Example 6.3 shows that we need to generalize the concept of classical relation
in order to formulate more relationships in the real world. Therefore in the following
subsection, the concept of fuzzy relation is thus introduced.
121
6.2.2
Fuzzy Relation
Definition 6.1. [57]
A fuzzy relation R in UF1 × UF2 ×, ... × UFn is defined as the fuzzy set
R = {((u1 , u2 , ..., un ), µR (u1 , u2 , ..., un ))|(u1 , u2 , ..., un ) ∈ UF1 × UF2 ×, ... × UFn }, (6.6)
where µR : UF1 × UF2 ×, ... × UFn −→ [0, 1].
As a special case, a binary fuzzy relation is a fuzzy set defined in the Cartesian
product of two fuzzy sets.
Definition 6.2. (Cartesian Product of Two Fuzzy Sets)[59]
Let A and B be fuzzy sets in X and Y , respectively. The Cartesian product
of A and B, denoted by A × B, is a fuzzy set in the product space X × Y with the
membership function
µA×B (x, y) = min(µA (x), µB (y))
(6.7)
A × B is characterized by two-dimensional membership function.
The following discussion is based on the theories of fuzzy relation in 2 .
6.3
Fuzzy Number of Dimension Two
Definition 6.3. Let CFαi and DFαi be the sets of all triangular fuzzy numbers with
membership values of αi , where i ∈ [0, 1]. The fuzzy pyramidal number is the Cartesian
product of
CFαi × DFαi = {(x, y) : x ∈ CFαi , y ∈ DFαi }.
(6.8)
The fuzzy number of dimension two must satisfy the following properties[57]:
122
1. ∀x0 ∈ C: µR (x0 , y) ∈ [0, 1], is a convex membership function.
2. ∀y0 ∈ D: µR (x, y0 ) ∈ [0, 1], is a convex membership function.
3. ∀α ∈ [0, 1] and for all α-level, {R}α {R}α = {(x, y) : (x, y) ∈ C ×D, µR (x, y) ≥ α},
is a convex surface.
4. ∃(xn , yn ) ∈ C × D: µR (xn , yn ) = 1.
Theorem 6.1. The fuzzy pyramidal number is fuzzy number of dimension two.
Proof The pyramidal fuzzy number is a fuzzy number of dimension two if it satisfies
the four properties above. Let the pyramidal fuzzy number be defined as in Definition
6.3. Consider the triangular fuzzy numbers be defined as Equations

x−a2


+ 1 a1 ≤ x ≤ a2

 a2 −a1
x−a2
CF =
a2 −a3 + 1 a2 ≤ x ≤ a3



 0
otherwise





and
DF =
y−b2
b2 −b1
y−b2
b2 −b3



 0
(6.9)
+ 1 b1 ≤ y ≤ b2
+ 1 b2 ≤ y ≤ b3
(6.10)
otherwise.
Now, our next step is to show that the properties are satisfied by the pyramidal fuzzy
number.
1. Let C = [a1 , a3 ] and D = [b1 , b3 ]. Let x0 be any point in C where C ⊂ X. If we
fix the value of x (in this case we let this value equals x0 ) and let y vary from
b1 to b3 ∀α ∈ [0, 1], the function µR (x0 , y), which represents this condition is a
trapezoidal membership function for any value of x0 where x0 ∈ [a1 , a3 ] \ {a2 }
and triangular membership function when x0 = a2 . From [57], the trapezoidal
and triangular membership functions are convex.
2. The proof is the same as in (i).
3. Let {R}α=α0 be α-level for R which is defined from the Cartesian product of the
triangular fuzzy numbers C and D at α-cut = α0 , (In other words the Cartesian
product of two closed interval). From this definition, we get {R}α=α0 as a domain
of confidence for the rectangular domain (see Figure 6.2). Now we want to prove
123
that the rectangular domain is convex domain. The rectangular domain resulted
from the Cartesian product of two closed interval of [a1 , a3 ] and [b1 , b3 ], which
contains infinitely many pairs of elements in the form of (a, b) where a ∈ [a1 , a3 ]
and b ∈ [b1 , b3 ]. The boundary of the rectangular domain is four line segments.
Each line segment divide the plane into two half planes. Thus the rectangular
domain is the intersection of four half planes. Half plane is convex. From Theorem
9 in [58], the intersection of the convex sets is also a convex set. Hence the
rectangular domain, {R}α=α0 is convex.
4. Let xn ∈ C where µC (xn ) = 1 and yn ∈ D where µD (xn ) = 1. The Cartesian
product of xn and yn will give the point (xn , yn ) in R2 such that µ((xn , yn )) = 1
µ(x,y)
1
CF
α= 0.2
DF
x
C ×D
F
F
y
Figure 6.1: Pyramidal fuzzy number
6.3.1
Alpha-level
Consider fuzzy sets CF and DF with their triangular membership functions
defined as Equations 6.9 and 6.10.
Let R be the fuzzy relation on R2 with its
corresponding pyramidal membership function. Then R can be written as a fuzzy
124
µ(x)
µ(y)
1
1
a
1
X
a3
a
2
b1
(a)
b2 b
3
(b)
Y
Y
b
3
α=1
b2
αïlevel curve
•
b1
a
1
a2
a3
X
(c)
Figure 6.2: Pyramidal fuzzy number from Cartesian product of two triangular fuzzy
numbers
set
R = {((x, y), µ(x, y))|(x, y) ∈ CFαi × DFαj , µ(x, y) ∈ [0, 1]}.
(6.11)
The α-level set of R denoted by {R}α , is defined as
{R}α =



{(x, y)|x ∈ [(α − 1)(a2 − a1 ) + a2 , (α − 1)(a2 − a3 ) + a2 ],


y ∈ [(α − 1)(b2 − b1 ) + b2 , (α − 1)(b2 − b3 ) + b2 ], µR (x, y) ≥ α} 0 < α ≤ 1



 cl(supp R)
α=0
(6.12)
The α-level set of a pyramidal fuzzy number is a closed and bounded surface.
The Cartesian product of two close intervals will result in a domain of confidence
for the rectangular domain. From the definition of Cartesian product of two closed
intervals [60] if [a1 , a3 ] and [b1 , b3 ] are closed intervals, their product,
[a1 , a3 ] × [b1 , b3 ] = {(x, y) : a1 ≤ x ≤ a3 , b1 ≤ y ≤ b3 }
(6.13)
is called a closed rectangle in R2 .
In order to show that {R}α is a bounded domain, here we use the Euclidean
distance. We know that,
a21 + b21 ≤ a23 + b23 .
125
Pick any point in {R}α , say (a, b), we get
because a ≤ a3 and b ≤ b3 . Since
a2
+
b2
≤ a23 + b23 ,
a23 + b23 > 0, by the theorem of positive real number,
[61] ∃ M ∈ Z+ , such that
2
2
a +b ≤
a23 + b23 ,
< M
(6.16)
Hence, it is shown that the α-level set of a pyramidal fuzzy number is a closed and
bounded domain.
This section provides the theoretical details which become the basis for
accomplishing the aim of this chapter. Thus, in the following section, the development
of the inverse model of the mass transfer based on two dimensional fuzzy number will
be presented.
6.4
Inverse Modelling of the Mass Transfer Based on Two Dimensional
Fuzzy Number
In this section, the modification of the ISDSS-Fuzzy Model is shown. The
modification use the two dimensional fuzzy number concept as discussed in Section 6.3.
In this model the suggested input parameters are processed in three phases in
order to produce the optimal solution as described in the ISDSS-Fuzzy Algorithm. In
the first phase, the input parameters are fuzzified by a triangular membership function,
while the preferred output parameters are fuzzified by pyramidal membership function.
Figure 6.2 shows how the Cartesian product of two triangular fuzzy numbers produce
a pyramidal fuzzy number. All the α-cuts for every fuzzified input, FPi , must be
determined as explained in Subsection 5.3.3. The same goes for the α-cuts for every
fuzzified output, FQi , which is determined by Equation (6.12). In the next phase,
the fuzzifed input values are employed for calculating the associated fuzzy sets for the
output parameter.
126
The fuzzified input and output parameters are then processed by Zadeh’s
extension principle [51] to produce the most appropriate output data.
The steps
involved are exactly the same as in Subsection 5.3.3. However, in this model, the
MIMO system as described in Figure 5.1 is considered. Therefore, for each α-cut, the
fuzzified input values FPi are mapped onto two dimensional Euclidean space by the
functional
 h of Equation (5.3). Let the corresponding output parameters be r, where
r1
rj =   and j ∈ 1, 2, ...2m .
r2
j
The subsequent step is the determination of the induced output parameters,
Find . This is done by taking the minimum and maximum values of each element of
r i.e let u1 = min(r1 )j , u2 = min(r2 )j and let v1 = max(r1 )j , v2 = max(r2 )j , for all
j ∈ 1, 2, ...2m and we define
Rαl = {(r1 , r2 , µ(r1 , r2 )) : αl ≤ µ(r1 , r2 ) ≤ αl+1 , u1 ≤ r1 ≤ v1 , u2 ≤ r2 ≤ v2 },
therefore Find =
(6.17)
Rαl . The α-cut of Find is defined as
l∈k




 u1 + (v1 − u1 )t


= r : 0 ≤ t ≤ 1, µ(r) ≥ α .
[r]α =

 u + (v − u )t
2
2
2
(6.18)
The intersection points between the pyramidal preferred output and the
triangular induced plane are determined by plotting the two curves on the same axes.
The optimal fuzzy value is then used to determined the possible combinations of input
parameters. These combinations of input parameters are actually the output data of
the inverse model. The output data is then defuzzified in the fuzzification phase in
order to obtain the best possible combination of the input parameters.
The steps to determine the outputs of the above model may conveniently be
performed using the following algorithm:
6.4.1
The ISDSS-2D Fuzzy Algorithm
All the fuzzy sets FIi expressing preferences of all input parameters Pi ∈
[ai , bi ] ⊂ + (i = 1, 2) are determined, normalized and convex. Let QP be preferred
127
performances parameter which takes all the input parameters as its variables and is
presented by fuzzy set FQP . Here, QP refers to Equation (5.3) in order to get the
values of yout and xout .
Algorithm 6.1: The ISDSS-2D Fuzzy Algorithm
Step 1: Let h1 : I1 × I2 −→ O2 , h2 : I1 × I2 × O2 −→ O1 and h(h1 , h2 ) = (O2 , O1 )
where O2 ∈ Q2 , O1 ∈ Q1 . Let h : h1 × h2 −→ 2 is the performance parameter
such that r = h(h1 , h2 ).
Step 2: Select appropriate value for α-cut, such that α1 , α2 , α3 , ...αk ∈ (0, 1].
Step 3: For each Pi , determine the end points of all the αk -cuts, FIi (i = 1, 2).
Step 4: Generate all 2m combinations of all the endpoints of intervals representing
αk -cuts. Each combination is an m-tuple (in this problem m = 2).

 

r1
O2
 for each 2-tuple j ∈ 1, 2, ...2m .
Step 5: Determine rj =   = h(h1 , h2 ) = 
r2
O1
j
Step 6: For each α-cuts, determine the induced output parameters, Find by taking the
min value and max value of each element of r i.e let u1 = min(r1 )j , u2 = min(r2 )j
and let v1 = max(r1 )j , v2 = max(r2 )j , for all j ∈ 1, 2, ...2m . Apply equations
(6.17) and (6.18) to obtain the α-cuts of Find .
Step 7: Set FhP ∧ Find , where Find is the induced output parameter and determine
the fuzzy number of f = sup(FhP ∧ Find ).
Step 8: Find the α-cut of FIi for the corresponding value of f .
Step 9: Repeat step 4 and 5 for α = f and denote the corresponding performance
parameter as rj for each 2-tuple j ∈ 1, 2, ...2m .
Step 10: Determine the optimal combination of input parameters and stop.
The value determined in the final step of the algorithm is the approximate value
of the input parameters which is hoped to produce the desired value of the output
parameter. The value is determined by Theorem 6.4, which is an extension of Theorem
of Optimized Defuzzified Value [8]. However, we have to show that the induced solution
for RDC column, Find , is convex and normal before Theorems 6.4 works. The definition
of the convexity of a fuzzy set is given in order to show that Find is a convex fuzzy set.
128
Definition 6.4. [51] A fuzzy set A ⊂ F(P ) is convex if and only if
µA (λp1 + (1 − λ)p2 ) ≥ min[µA (p1 ), µA (p2 )],
∀λ ∈ [0, 1] and ∀p1 , p2 ∈ F(P ),
where min denotes the minimum operator.
This definition is then followed by the formulation of the following lemma.
Lemma 6.1. If A ⊆ F(P ) and µA (λp1 + (1 − λ)p2 ) ≥ min[µA (p1 ), µA (p2 )] ∀λ ∈
[0, 1] and ∀p1 , p2 ∈ F(P ) then [a]α2 ⊆ [a]α1 for all α2 ≥ α1 .
Proof Assume A ⊆ F(P ) and µA (λp1 + (1 − λ)p2 ) ≥ min[µA (p1 ), µA (p2 )] ∀λ ∈
[0, 1] and ∀p1 , p2 ∈ F(P ). Let α2 = µA (λp1 + (1 − λ)p2 ) and min[µA (p1 ), µA (p2 )] = α1 .
By the properties of alpha-cut([51]), if there exist α2 ≥ α1 , then [a]α2 ⊆ [a]α1 .
What follows is the definition of the normality of a fuzzy set.
Definition 6.5. [62] A fuzzy set A ⊂ F(P ) is normal if and only if sup µA (p) = 1.
p∈A
We state the following theorem which then is proven by construction.
Theorem 6.2. Let h(h1 , h2 ) = (O2 , O1 ) where O2 ∈ Q2 , O1 ∈ Q1 . Let h : h1 × h2 −→
2 be the performance parameter such that r = h(h1 , h2 ). Then if all the fuzzy set FIi
expressing preferences of all input parameter Pi ∈ Ii ⊂ + (i = 1, 2) is convex, it follows
that the induced solution for RDC column, Find is also a convex fuzzy set.
Proof (by construction) Given that I1 , I2 is convex, i.e if [Ii ]α is a closed interval
for each α and αk+1 ≥ αk ⇒ [Ii ]αk+1 ⊆ [Ii ]αk ∀i = 1, 2 and αk ∈ [0, 1] for k = 1, 2, ..., n.
Find all end points of [Ii ]αk and denote as {Iimin , Iimax }αk . Now, determine all the
combination of end points for every [Ii ] of each αk and write as {< I1 , I2 >}αk .
Generate h1 (I1 , I2 )αk , h2 (I1 , I2 , O2 )αk and h(h1 , h2 )αk .


Determine rj = 


r1
O2
 = Y(h1 , h2 ) = 
 for each 2-tuple j ∈ 1, 2, ...2m .
r2
O1
j
Then for each α-cuts, determine the induced performance parameters, Find by taking
the min value and max value of each element of r i.e if u1 = min(r1 )j , u2 = min(r2 )j
129
and let v1 = max(r1 )j , v2 = max(r2 )j , for all j ∈ 1, 2, ...2m and we define Rαl =
{(r1 , r2 , µ(r1 , r2 )) : αl ≤ µ(r1 , r2 ) ≤ αl+1 , u1 ≤ r1 ≤ v1 , u2 ≤ r2 ≤ v2 }, therefore
Rα l .
Find =
l∈k
Next, we prove that if αk+1 ≥ αk and all α-cut of Find is a closed domain ⇒ [r]αk+1 ⊆
α-cut of Find which is defined
[r]αk ∀i = 1,
2and αk ∈ [0, 1] for
 k = 1, 2, ..., n. Obviously

 u1 + (v1 − u1 )t
 = r : 0 ≤ t ≤ 1, µ(r) ≥ α is a closed domain ∀ α ∈
as [r]α = 

 u + (v − u )t
2
2
2
Rαl and r = (r1 , r2 ) = Y(h1 , h2 )αk+1
[0, 1]. Take r ∈ [r]αk+1 , therefore r ∈
l=k+1,...n
where h1 (I1 , I2 )αk+1 = r2 and h2 (I1 , I2 , r2 )αk+1 = r1 for some (I1 , I2 ) where Ii ∈
[Ii ]αk+1 ∀i = 1, 2.
Since Ii is convex i.e Ii ∈ [Ii ]αk+1 ⊆ [Ii ]αk ∀i = 1, 2., which implies that r ∈
Rαl i.e r ∈ [r]αk .
l=k,k+1,...n
Therefore [r]αk+1 ⊆ [r]αk . Hence the induced solution for RDC column, Find , is a
convex fuzzy set.
We then state the following theorem which is used in proving Theorem 6.4.
Theorem 6.3. Let h(h1 , h2 ) = (O2 , O1 ) where O2 ∈ Q2 , O1 ∈ Q1 . Let h : h1 × h2 −→
2 be the performance parameter such that r = h(h1 , h2 ). If all the fuzzy set FIi
expressing the preferences of all input parameters Pi ∈ Ii ⊂ + (i = 1, 2) is normal,then
it follows that the induced solution for RDC column, Find , is also normal.
Proof Since all the fuzzy sets FIi , expressing preferences of all input parameters Pi ∈
Ii ⊂ + (i = 1, 2) are normal, then there exist p1 ∈ I1 and p2 ∈ I2 , such that µ(p1 ) =
µ(P2 ) = 1. Look, h1 (p1 , p2 ) = O2 , h2 (p1 , p2 , h1 ) = O1 , and h(h1 , h2 ) = (O2 , O1 ) = r
such that µ(r) = 1. Hence Find is normal.
We use the following Theorem 6.4 to determine the optimal combination of the
input parameters.
Theorem 6.4. Let P = {(rj , µ(rj )), a ≤ rj ≤ c : µ(a) = 0, µ(c) = 0, µ(b) =
1, a ≤ b ≤ c, j ∈ Z + } where (rj , µ(rj )) ∈ Find and let S = {(rl , µ(rl )),
b ≤ rl ≤ c : µ(a) = 0, µ(b) = 1} be the max side of the induced plane, where
130
(rl , µ(rl )) ∈ Find and S ⊂ P . If there exist QP = rl such that µ(rl ) = f and
(rl , f ) ∈ S where f = sup(FQP ∩ Find ) for some (rl , µ(rl )) ∈ Find , then rl = QP =
max QP (P1 , P2 ) where µ(Pi ) = f .
Proof Suppose QP = rl ∈ S such that µ(rl ) = f where f = sup(FQP ∩ Find )
for some (rl , µ(rl )) ∈ Find . Determine all the f -cuts of all FIi to create all 2-tuples
of (P1 , P2 ) such that µ(Pi ) = f and (Pi , f ) ∈ FIi . Set rl = max QP (P1 , P2 ),
therefore (rl , f ) ∈ Find . However, since Find is normal and convex, this imply that
rl = rl .
The theorem above is used if the intersections lie on the maximum side of the
induced plane. However if the intersections lie on the minimum side of the induced
plane, the following Theorem is used.
Theorem 6.5. Let P = {(rj , µ(rj )), a ≤ rj ≤ c : µ(a) = 0, µ(c) = 0, a ≤
b ≤ c, j ∈ Z + } where (rj , µ(rj )) ∈ Find and let S = {(rl , µ(rl )),
a ≤ rl ≤ b : µ(a) = 0, µ(b) = 1} be the min side of the induced plane, where
(rl , µ(rl )) ∈ Find and S ⊂ P . If there exist QP = rl such that µ(rl ) = f and
(rl , f ) ∈ S where f = sup(FQP ∩ Find ) for some (rl , µ(rl )) ∈ Find , then rl = QP =
min QP (P1 , P2 ) where µ(P )i ) = f .
Proof Suppose QP = rl ∈ S such that µ(rl ) = f where f = sup(FQP ∩ Find )
for some (rl , µ(rl )) ∈ Find . Determine all the f -cuts of all FIi to create all 2-tuples
of (P1 , P2 ) such that µ(Pi ) = f and (Pi , f ) ∈ FIi . Set rl = min QP (P1 , P2 ),
therefore (rl , f ) ∈ Find . However, since Find is normal and convex, this imply that
rl = rl .
Theorems 6.4 and 6.5 indicate that if the preferred fuzzy intersects on the
maximum or minimum side of the fuzzy induced plane, then the set of optimized
parameters is the set of the maximum or minimum norm of the induced values. The
theorems enable the decision maker to identify the best optimized value from the
predicted results in the final phase of the algorithm.
131
6.4.2
Numerical Example
To study the capability of the proposed method, a numerical example is given
by considering a system with two input and two output parameters for the mass transfer
of a single drop in a single stage RDC column. The input parameters of the numerical
example in Subsection 5.3.5 is used. The domains of the preferred input and output
parameters are given in Tables 5.2 and 5.3.
These figures are simply fuzzy numbers where the preferred and the domain
are set to have the highest and lowest membership values respectively. The α-cuts of
all the input parameters with the increment of 0.2 are calculated as listed in Table
5.4.
Then these values are used to calculate the fuzzified induced plane, Find as
mentioned in Step 6 of Algorithm 6.1. For example when α = 0.2, the values of
the input parameters, p1 and p2 are in the interval [35.34, 53.58] and [11.72, 14.36].
After all the possible combinations of the end points of these two input parameters
are identified, each of these combinations is then used in the forward model to get the
respective r. Since
we will get four different values
 wehave
 fourinput
 combinations,
 

27.89
42.90
27.40
43.37
,
,
,
.
of r which are 
22.35
32.02
23.80
30.57
Now, let u1 = min(r1 )j = 27.40, u2 = min(r2 )j = 22.35 and let
 v1 =max(r1 )j =
27.40
 and v =
43.37, v2 = max(r2 )j = 32.02, where j = 1, 2, 3, 4, then u = 
22.35


43.37
. Then the values for Rα=0.2 becomes R0.2 = {(r1 , r2 , µ(r1 , r2 )) = 0.2 : αl ≤

32.02
0.2 ≤ αl+1 , u1 ≤ r1 ≤ v1 , u2 ≤ r2 ≤ v2 }. The calculation must be repeated for different
values of α. After the value of R has been calculated for every α ∈ [0, 1], we will get the
Rαl . The induced output generated from this
induced output parameters, Find =
l∈k
derivation is actually a triangular plane as can be seen in Figure 6.3. The α = 0.2-level
set of Find is
{R}0.2


 

 u1 + (v1 − u1 )t

27.4 + 15.97t
=
 = r : 0 ≤ t ≤ 1, µ(r) ≥ 0.2 .
= 
 u + (v − u )t

22.35 + 9.67t
2
2
2
Similarly the output parameters given in Table 5.3 are also ready to be fuzzified
by triangular membership function. Now, let CF and DF be the fuzzy sets for fuzzified
132
1
0.9
0.8
Membership value
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
35
30
25
20
CDout
25
30
35
40
45
50
CCout
Figure 6.3: The Induced Plane
CCout and CDout respectively. By Definition 6.2, the Cartesian product of two fuzzy
sets CF and DF , denoted by CF × DF is also a fuzzy set and by Definition 6.3 the
Cartesian product of the triangular fuzzy numbers will generate a pyramidal fuzzy
number of dimension two. Then with these definitions we calculate the alpha-level set
of the fuzzy relation of the preferred output parameter according to equation (6.12).
For example when α = 0.2, we obtain
{R}0.2 = {(q1 , q2 )|(q1 , q2 ) ∈ [29.6, 39.2] × [24.4, 34.0]}
= {(q1 , q2 )|(q1 , q2 ) ∈ Rec(a1 , a2 , a3 , a4 )},
where
Rec((a
domain and ((a1 , a2 , a3 , a4 )) =
2 , a
3 , a4 )) is the rectangular


  1 , a

29.6
29.6
39.2
39.2

,
,
,
 corresponds to the four points of the
24.4
34.0
24.4
34.0
rectangular domain resulting from the Cartesian product of the two intervals. The
values of alpha-level set with increment of 0.2 are tabulated in Table 6.1. These
values are then used to generate the pyramidal fuzzy number of the preferred output
parameters as shown in Figure 6.4.
The next step is to determine the intersection points between the preferred
and the induced plane. To determine the intersection points we first have to find
the equation of the induced plane and the plane on the pyramidal surface where the
133
Table 6.1: The values of alpha-level set
α-level Set( A set of points in the rectangle surface)
Fuzzy Value
0.2
0.4
0.6
0.8
1.0
The Four Points of the Rectangle Vertices

 
 
 

29.6
29.6
39.2
39.2

 
 
 


,
,
,

24.4
34.0
24.4
34.0

 
 
 

31.2 31.2 38.4 38.4

,
,
,

25.8
33.0
25.8
33.0

 
 
 

32.8
32.8
37.6
37.6

 
 
 


,
,
,

27.2
32.0
27.2
32.0

 
 
 

34.4
34.4
36.8
36.8

 
 
 


,
,
,

28.6
31.0
28.6
31.0

 
 
 

36.0
36.0
36.0
36.0

 
 
 


,
,
,

30.0
30.0
30.0
30.0
intersection occurred. To do so, the following steps are considered:
Step 1 Determination of the equation of the induced plane.
Identify the three points of the vertices of the triangular plane. In this example
the three points are P (25.22, 20.93, 0), Q(45.20, 33.01, 0) and P (36.13, 28.06, 1),
which determine the vectors PQ and PR where PQ = (19.98, 12.08, 0) and PR =
(10.91, 7.13, 1). The cross product of these two vectors is
"
"
"
"
" i
j
k "
"
"
"
"
N1 : PQ × PR = " 19.98 12.08 0 " = 12.08i − 19.98j + 10.66k.
"
"
"
"
" 10.91 7.13 1 "
Thus the equation of induced plane is
12.08(q1 − 25.22) − 19.98(q2 − 20.93) + 10.66z = 0,
12.08q1 − 19.98q2 + 10.66z = −113.489
Step 2 Observation of which planes of the pyramidal surface intersect with
the induced plane.
In this example, the intersection lies on two planes of the pyramidal, say, plane 1
and plane 2. The next step is to identify the equations of these two planes.
134
1
Membership value
0.8
0.6
0.4
0.2
0
35
40
30
38
36
34
25
32
30
CDout
20
28
CCout
Figure 6.4: The preference output
Step 3 Determination of the equation of the preferred plane.
Plane 1: The three points are P (28, 23, 0), Q (40, 23, 0) and R (36, 30, 1). Then,
P Q = (12, 0, 0) and P R = (18, 7, 1). The
"
"
" i j
"
"
N2 : P Q × P R = " 12 0
"
"
" 8 7
cross product of these two vectors is
"
"
k "
"
"
0 " = 0i − 12j + 84k.
"
"
1 "
Therefore the equation of plane 1 is
0(q1 − 28) − 12(q2 − 23) + 84z = 0,
Plane 1: − 12q2 + 84z = −276
Plane 2: The three points are Q (40, 23, 0), P (40, 35, 0) and R (36, 30, 1). Then,
QP = (0, 12, 0) and Q R = (−4, 7, 1). The cross product of these two vectors
is
"
"
" i
j k
"
"
N3 : QP × Q R = " 0 12 0
"
"
" −4 7 1
"
"
"
"
"
" = 12i + 0j + 48k.
"
"
"
Therefore the equation of plane 2 is
12(q1 − 40) + 48z = 0,
Plane 2: 12q1 + 48z = 480
135
Membership value
1
0.5
0
35
30
25
20
CDout
35
30
25
45
40
50
CCout
(a)
32
CDout
30
28
26
24
22
26
28
30
32
34
36
38
40
42
44
CCout
(b)
Figure 6.5: (a)The intersection between preferred and induced Output. (b) Level curve
of (a)
Step 4 Identification of the line segment from the intersection between the
induced plane and the two surfaces of the pyramidal preferred output.
Identify the line segments of the intersection. To achieve this, first solve the
simultaneous equations of the induced plane and plane 1.
12.08q1 − 19.98q2 + 10.66z = −113.489,
−12q2 + 84z = −276
For simplicity, the end point of the line segment lying on the Q1 Q2 -plane is
considered. That point is the point of intersection when the fuzzy value is equal
to zero, i.e z = 0. Therefore
 we get
 q2 = 23 and q1 = 28.69. Thus, one of the
28.69




point of the intersection is  23 . We know that N1 and N2 are the normal


0
vector of the induced and plane 1 respectively. Therefore, the cross product of
136
these two vectors is
"
"
"
"
" i
j
k "
"
"
"
"
N1 × N2 = " 12.08 −19.98 10.66 " = −1550.36i − 1014.72j − 144.96k.
"
"
"
"
" 0
−12
84 "


28.69




The equation of the line segment that contains the point  23  and parallel


0
to the above vector is
q2 − 23
z
q1 − 28.69
=
=
= t1
−1550.36
−1014.72
−144.96
(6.19)
The same procedure is applied in order to determine the intersection points
between the induced plane and plane 2. The two simultaneous equations are
12.08q1 − 19.98q2 + 10.66z = −113.489,
12q1 + 0q2 + 48z = −276
If z = 0, we will get q1 = 40 and q2 = 29.84. Next find the vector which is parallel
to the line segment of intersection between induced plane and plane 2
"
"
"
"
" i
j
k "
"
"
"
"
N1 × N3 = " 12.08 −19.98 10.66 " = −959.04i − 451.88j + 239.76k.
"
"
"
"
" 12
0
48 "
Thus the equation of the line when the two planes intersect is
q2 − 29.84
z
q1 − 40
=
=
= t2
−959.04
−451.88
239.76
(6.20)
Step 5 Determination of the supremum point.
This point is the point of intersection which has the highest fuzzy value. To
indentify the point, we must solve (6.19) and (6.20) simultaneously. Then we get
t1 = −5.31 × 10−3
= 3.21 ×10−3 . Substitute these values into (6.19) or
 t2 
 and
q
36.92
 1 

  

(6.20), to obtain q2  =  28.39 .
  

z
0.7697
Thus f ∗ = 0.7697. Now we want to determine the α-cut of the fuzzified input
parameters, FIi for f ∗ = 0.7697. In this case, we get CCin = [42.48, 47.87] and CCout =
137
[13.17, 13.95]. Again we have to generate all 4 combinations of all the endpoints of the
interval representing α = f ∗ -cut. Repeat Step 5 in the IDSS-2D Fuzzy Algorithm.
The output values for the corresponding input parameters are tabulated in
Table 6.2. Since f ∗ lies on the maximum size of the induced plane, by Theorem 6.4, the
combination of the input parameters with the maximum norm of their respective output
parameter is chosen. The result of the implementation of the ISDSS-2D Algorithm
shows that the 4th combination, (47.87, 13.95) of the input parameters is taken as the
optimal solution in order to produce the desired output. The percentage of errors of
each of the input and output parameters are tabulated in Table 6.3.
Table 6.2: Input combination with fuzzy value z = 0.7697
Input Value
Output Value
output 1st combination
(42.48, 13.17)
(33.69, 26.37)
42.78
2nd combination
(42.48, 13.95)
(33.55, 26.80)
42.94
3rd combination
(47.87, 13.17)
(38.27, 28.80)
47.90
4th combination
(47.87, 13.95)
(38.13, 29.23)
48.04
Combination of Input(CCin , CDin )
Table 6.3: Error of input and output parameters
Calculated
Preferred
Input
Calculated
Preferred
Output
Input Values
Input Values
Error (%)
Output Values
Output Values
Error (%)
(47.87, 13.95)
(45.48, 13.78)
(5.26,1.23)
(38.13, 29.23)
(36.0, 30.0)
(5.92, 2.57)
6.4.3
Simulation Results
The simulations for different input domains of ISDSS-2D-Fuzzy Algorithm are
carried out. The output data is tabulated in Tables 6.4 and 6.5.
For comparison purposes, we tabulate the percentage errors of the calculated
values of the input and output parameters with respect to the suggested values
respectively for the ISDSS-Fuzzy and ISDSS-2D-Fuzzy Algorithms as can be seen in
Table 6.6.
138
Table 6.4: The errors between the calculated input values and the preferred values for
the different Input Domain
Simulation
Input Parameter
Calculated Input Values
Preferred Values
Errors (%,%)
Numbers
CCin
Cdin
(CCin, Cdin)
(CCin, Cdin)
(CCin, Cdin)
Eg of 6.4.2
[32.8 55.6]
[11.2 14.5]
(47.87, 13.95)
(45.48, 13.78)
(5.26, 1.23)
Sim 1
[30.8 57.6]
[9.2 17.5]
(48.06, 14.57)
(45.48, 13.78)
(5.68, 5.76)
Sim 2
[28.8 59.6]
[7.2 19.5]
(48.43, 14.97)
(45.48, 13.78)
(6.48, 8.66)
Table 6.5: The errors between the calculated output values and the preferred values
for the different input domain
Simulation
Calculated Output Values
Preferred Values
Errors (%,%)
(CCout, Cdout)
(CCout, Cdout)
Numbers
f∗
Eg of 6.4.2
0.7697
(38.13, 29.23)
(36.0, 30.0)
(5.92, 2.57)
Sim 1
0.7868
(38.18, 29.66)
(36.0, 30.0)
(6.05, 1.15)
Sim 2
0.7914
(38.41, 30.04)
(36.0, 30.0)
(6.70, 0.13)
(CCout, Cdout)
The following section will describe the development of the inverse model of the
mass transfer of a single drop in a multi-stage RDC column. The model is constructed
by implementing the ISDSS-2D Fuzzy Algorithm with some modification.
6.5
Inverse Model of the Mass Transfer of a Single Drop in a Multi-Stage
RDC Column Based on Two Dimensional Fuzzy Number
The procedure in developing the model is also taken in three phases as described
in Section 5.3.1.
As in ISDSS-2D Fuzzy Model, the preferred input and output
parameters are fuzzified using the triangular and the two dimensional pyramidal
membership functions respectively.
(0)
Let p1
(0)
∈ P1 and p2
∈ P2 be the initial
concentrations of the continuous and dispersed phase where P1 and P2 are two bounded
intervals. The fuzzified values of these parameters are FP (0) and FP (0) respectively. Let
(n)
q1
(n)
∈ Q1 and q2
1
2
∈ Q2 be the preferred output concentrations of the continuous and
139
Table 6.6: The errors of the output solution of ISDSS and ISDSS-2D-Fuzzy Algorithms
ISDSS-Fuzzy
Simulation
ISDSS-2D-Fuzzy
Input Error
Output Error
Input Error
Output Error
Numbers
CCin
Cdin
CCin
Cdin
CCin
Cdin
CCin
Cdin
Eg of 6.4.2
1.23
0.22
0.92
7.53
5.26
1.23
5.92
2.57
Sim 1
0.79
0.65
0.53
6.87
5.68
5.76
6.05
1.15
Sim 2
0.59
0.5
0.32
6.77
6.48
8.66
6.70
0.13
dispersed phase where Q1 and Q2 are two bounded intervals. Then the fuzzified values
of the output concentrations are FQ1 ×Q2 .
The fuzzified input data is then processed in the fuzzy environment phase to
produce the output data. Steps 2 to 5 in ISDSS-2D Fuzzy Algorithm are applied to
(i)
the fuzzified input parameters to produce the corresponding output values, rj where


(i)
r
(i)
1 
and j = 1, 2, ...2m . The superscript (i) corresponds to the ith stage
rj = 
(i)
r2
j
of the column. Since there are four combinations of the fuzzified input values, we will
(1)
obtain four different values of r(i) for each α-cut. If i = 1, rj corresponds to the output




(1)
(1)
r
q
(1)
1 
1 
=
of the column which are
concentrations of the first stage, rj = 
(1)
(1)
r2
q2
j



j
(2)
(1)
q1
p1
 =
 . The
assumed to be the input concentrations of the second stage, 
(2)
(1)
p2
q2
j
j
process in obtaining the output parameters for the successive stage is repeated through
the final stage of the column. Then, the value of the output concentrations at the final
stage is

(n)
rj
(n)


r1
(n)

q1
 =

=
(n)
(n)
r2
q2
j
j
The subsequent step is the determination of the induced output parameters,
Find . This determination will be done by taking the minimum and maximum values
(n)
(n)
of each of the element of r(n) i.e let u1 = min(r1 )j , u2 = min(r2 )j
(n)
and let v1 =
(n)
max(r1 )j , v2 = max(r2 )j , for all j ∈ 1, 2, ...2m . Then Equations (6.17) and (6.18)
are used to calculate the α-cuts of Find . The intersection points between Find and FQP
will be determined by plotting the curve of Find and the preferred output parameters
140
q1
q2
CRISP VALUE
DEFUZZIFICATION
FUZZY VALUE
h23(h1,h2)
Fuzzy Value
FUZZY
ENVIRONMENT
hn(h1,h2)
Fuzzy Value
h2(h1,h2)
Fuzzy Value
h1(h1,h2)
FUZZY VALUE
FUZZIFICATION
CRISP VALUE
p1
p2
Figure 6.6: ISDMS-2D Fuzzy model
141
on the same axes. This step is followed by choosing the optimal fuzzy values, f ∗ , of all
the intersection points.
Subsequently, Steps 8 to 11 of ISDSS-2D Fuzzy Algorithm are applied in order
to determine the optimal combination of the input parameters. These steps are taken in
the defuzzification phase of the algorithm. Figure 6.6 represents the Inverse Single Drop
in a Multi-Stage Model (ISDMS-2DFuzzy) of the mass transfer in the RDC Column.
The following algorithm describes in detail the steps involved in determining
the input parameters for the desired output parameters for the mass transfer of a single
drop in the multi-stage RDC column.
6.5.1
The Inverse of Single Drop Multi-stage-2D Fuzzy(ISDMS-2D Fuzzy)
Algorithm
(1)
All the convex and normalized fuzzy sets FIi expressing the preferences of all
(1)
the input parameters Pi
(n)
∈ [ai , bi ] ⊂ + (i = 1, 2) are determined. Let QP
be the
preferred output parameters which take all the input parameters as its variables and
are presented by the fuzzy set FQ(n) . The algorithm can be represented as follows.
P
Algorithm 6.2: ISDMS-2D Fuzzy Algorithm
(1)
(1)
(1)
(1)
Step 1: Let h1 : I1 ×I2 −→ O2 , h2 : I1 ×I2 ×O2 −→ O1 and h(h1 , h2 ) = (O2 , O1 )
(1)
(1)
(1)
(1)
(1)
(1)
where O2 ∈ Q2 , O1 ∈ Q1 . Let h : h1 × h2 −→ 2 is the output parameter
(1)
(1)
such that r(1) = h(h1 , h2 ).
Step 2: Select the appropriate value for the α-cut, such that α1 , α2 , α3 , ...αk ∈ (0, 1].
(1)
Step 3: For each Pi , determine the end points of all the αk -cuts, FI (1) (i = 1, 2).
i
Step 4: Generate all 2m combinations of all the endpoints of the intervals representing
the αk -cuts. Each combination is an m-tuple (in this problem m = 2).




(1)
(1)
r
O
(1)
(1) (1)
1 
2 
= h(h1 , h2 ) = 
Step 5: Determine rj = 
for each 2-tuple j ∈
(1)
(1)
r2
O1
j
1, 2, ...2m by applying MTASD Algorithm.
142
Step 6: Repeat Step 6.5.1 by taking the output rj (1) in the first stage as the input
(2)
Pi
for the next stage. These process continues through the final stage.
Step 7: For each α-cuts, determine the induced output parameters, Find by taking the
(n)
minimum and maximum values of each element of r(n) i.e let u1 = min(r1 )j ,
(n)
(n)
(n)
and let v1 = max(r1 )j , v2 = max(r2 )j , for all j ∈ 1, 2, ...2m
u2 = min(r2 )j
and apply Equations (6.17) and (6.18) to obtain Find .
Step 8: Set FQP ∧ Find , where Find is the induced output parameter at the final stage
and find the fuzzy values of f = sup(FQP ∧ Find ).
Step 9: Find the α-cut of FI (1) for the corresponding value of f .
i
Step 10: Repeat step 4 and 5 for α = f and denote the corresponding output
(n)
parameter as r j
for each 2-tuple j ∈ 1, 2, ...2m .
Step 11: Determine the optimal combination of input parameters and stop.
In Step 6.5.1, the optimal solution of the algorithm is determined by applying
Theorem 6.4 or 6.5. The three phases of the algorithm are illustrated as a flow chart
in Figure 6.7.
For the multi-stage RDC column, Theorem 6.2 is extended to the following
corollary for verifying the convexity of the induced solution.
Corollary 6.1. If all the fuzzy set FI (i) where superscript (i) corresponds to ith stage,
i
(i)
expressing the preferences of all the input parameters Pi
(i)
∈ Ii
⊂ + (i = 1, 2) are
convex at any stage, then the induced solution for the respective stage of the RDC
column, Find is also convex.
Proof The proof is repetitive of Theorem 6.2 for any stage of RDC column.
As a special case for our 23-stage RDC column, we have the following corollary.
Corollary 6.2. If all the fuzzy set FI (i) expressing the preferences of all the input
parameters
(i)
Pi
∈
(i)
Ii
i
⊂
+ (i
= 1, 2) are convex, then the induced solution for 23-stage
RDC column, Find is also convex.
143
Start
Specify the value of input
and performance
parameters
Fuzzify these values by
triangular membership
function.
Fuzzified Performance
Parameter
Fuzzified Input Parameter
Forward Mathematical
equations
Fuzzy
Environment
No
Final
Stage?
Fuzzy values
Yes
Determine the fuzzy
number of the optimal
intersection point
Fuzzy values
Fuzzy values
Defuzzification
Crisp values
Stop
Figure 6.7: The flow chart representing the three phases
144
Again, for the multi-stage RDC column, Theorem 6.3 is extended to the
following corollary for verifying the normality of the induced solution.
Corollary 6.3. If all the fuzzy set FI (i) expressing the preferences of all the input
(i)
parameter Pi
(i)
∈ Ii
i
⊂ + (i = 1, 2) is normal at any stage , then the induced solution
for the respective stage RDC column, Find is also normal.
Proof The proof is repetitive of Theorem 6.3 for any stage of RDC column.
In view of the fact the 23-stage RDC column is considered, we have the following
corollary for verifying the normality of the induced solution.
Corollary 6.4. If all the fuzzy set FI (i) expressing the preferences of all the input
parameter
(i)
Pi
∈
(i)
Ii
i
⊂
+ (i
= 1, 2) is normal, then the induced solution for the
23-stage RDC column, Find is also normal.
The results of the simulations of this algorithm are presented in the following
subsection.
6.5.2
Simulation Results
The simulations of ISDMS-2D-Fuzzy Algorithm are carried out for the different
input domains. The input data used in the simulations are tabulated in Table 6.7.
These are increments of two units for both sides of the input domains of each input
parameter in the successive simulation without changing the preferred values. However
the domains and the preferred values of the output parameters remain unchanged. The
results of the simulations are tabulated in Tables 6.8, 6.9 and 6.10.
Table 6.8 shows the four combinations of the input parameters which are
mapped onto four of two tuples of output parameters for each simulation. The fuzzy
value, f ∗ of each simulation is also given. The percentage errors of the calculated input
parameters with respect to their preferred values are listed in Table 6.9. Whilst Table
6.10 shows the errors between the calculated output parameters and their respective
preferred values for each simulation.
145
Table 6.7: The input data for simulations of ISDMS-2D-Fuzzy Algorithm
Simulation
Input Domain
Preferred Input
Output Domain
Preferred Output
Numbers
CCin
Cdin
CCin
Cdin
CCin
Cdin
CCin
Cdin
Sim 1
[32.8 42.6]
[22.5 34.5]
39.64
27.28
[26.99 35.05]
[72.0 116.0]
32.62
98.71
Sim 2
[30.8 44.6]
[20.5 36.5]
39.64
27.28
[26.99 35.05]
[72.0 116.0]
32.62
98.71
Sim 3
[28.8 46.6]
[18.5 38.5]
39.64
27.28
[26.99 35.05]
[72.0 116.0]
32.62
98.71
Table 6.8: The results of ISDMS-2D-Fuzzy simulations
Comb
Sim 1
Sim 2
Sim 3
f ∗ = 0.9859
f ∗ = 0.9864
f ∗ = 0.9864
No
Input
Output
Input
Output
Input
Output
1st
(39.54, 27.21)
(32.54, 98.32)
(39.52, 27.19)
(32.52, 98.22)
(39.49, 27.16)
(32.49, 98.10)
2nd
(39.54, 27.38)
(32.54, 98.49)
(39.52, 27.41)
(32.52, 98.44)
(39.49, 27.43)
(32.49, 98.37)
3rd
(39.68, 27.21)
(32.65, 98.78)
(39.71, 27.19)
(32.67, 98.84)
(39.73, 27.16)
(32.69, 98.91)
4th
(39.68, 27.38)
(32.65, 98.95)
(39.71, 27.41)
(32.67, 99.06)
(39.73, 27.43)
(32.69, 99.18)
6.6
Implementation of ISDMS-2D-Fuzzy Algorithm on the Mass Transfer
of Multiple Drops in Multi-stage System
The implementation of ISDSS-2D-Fuzzy and ISDMS-2D-Fuzzy algorithms were
discussed in Sections 6.4 and 6.5. However the implemented models involves only the
mass transfer of a single drop. Therefore, in this section, the mass transfer of the
multiple drops is considered.
The forward model of the mass transfer of the multiple drops has been discussed
in Chapter 4. Based on this model, we implement the concept of ISDMS-2D-Fuzzy
Model to produce an inverse model which is capable of determining the value of the
input parameters of the mass transfer for the multiple drops. The aim of developing
this model is to handle real world problems which arise in the chemical industry in order
to determine the best input for the desire values of output. In a real RDC column, the
process of the mass transfer involves multiple drops.
146
Table 6.9: ISDMS-2D-Fuzzy: The errors between the calculated input values and
preferred values for different input domain
Simulation
Input Parameter
Calculated Input Values
Preferred Values
Errors (%)
Numbers
CCin
Cdin
CCin
Cdin
CCin
Cdin
CCin
Cdin
Sim 1
[32.8 42.6]
[22.5 34.5]
39.54
27.21
39.64
27.28
0.25
0.26
Sim 2
[30.8 44.6]
[20.5 36.5]
39.52
27.19
39.64
27.28
0.30
0.33
Sim 3
[28.8 46.6]
[18.5 38.5]
39.49
27.16
39.64
27.28
0.38
0.44
Table 6.10: ISDMS-2D-Fuzzy: The errors between the calculated output values and
preferred values for different input domain
Simulation
Calculated Output Values
Numbers
f∗
Sim 1
Preferred Values
Errors (%)
CCout
Cdout
CCout
Cdout
CCout
Cdout
0.9859
32.54
98.32
32.62
98.71
0.26
0.39
Sim 2
0.9864
32.52
98.22
32.62
98.71
0.32
0.50
Sim 3
0.9864
32.49
98.10
32.62
98.71
0.39
0.62
The procedure in developing the model has three phases as described in Section
6.5. Basically the steps in ISDMS-2D-Fuzzy Algorithm are applied in order to get the
final output of the model. The difference is only on the forward model used in Step 6.5.1
of ISDMS-2D-Fuzzy Algorithm. In order to determine the induced output parameters,
Find , the MTMD forward model is employed instead of the MTASD in Section 6.5. The
successive steps of ISDMS-2D-Fuzzy Algorithm is then followed to produce the output
of the model. The results of the simulations of this model are presented in the following
subsection.
6.6.1
Simulation Results
The simulations of IMDMS-2D-Fuzzy Algorithm are carried out on two different
sets of input data. For the first set, the preferred input values are the feed continuous
and dispersed phase concentrations of the Experimental Data 1 as in Table 4.2. While
147
for the second set, the feed continuous and dispersed phase concentrations from the
Experimental Data 2 of Table 4.2 are taken as the preferred input values. These input
data are listed in Tables 6.11 and 6.14 respectively.
Table 6.11: Set Data 1: The input data for simulations of IMDMS-2D-Fuzzy Algorithm
Simulation
Input Domain
Preferred Input
Output Domain
Preferred Output
Numbers
CCin
Cdin
CCin
Cdin
CCin
Cdin
CCin
Cdin
Sim 1
[32.80 44]
[15.0 30.0]
36.02
28.66
[21.25 32.70]
[41.51 75.65]
24.50
60.18
Sim 2
[30.80 46]
[13.0 32.0]
36.02
28.66
[21.25 32.70]
[41.51 75.65]
24.50
60.18
Sim 3
[28.80 48]
[11.0 34.0]
36.02
28.66
[21.25 32.70]
[41.51 75.65]
24.50
60.18
Table 6.12: IMDMS-2D-Fuzzy: The errors between the calculated input values and
preferred values
Simulation
∗
Calculated Input Values
Preferred Values
Errors (%,%)
(CCin, Cdin)
(CCin, Cdin)
(CCin, Cdin)
Numbers
f
Sim 1
0.9507
(35.86, 27.99)
(36.02, 28.66)
(0.44, 2.34)
Sim 2
0.9530
(35.77, 27.92)
(36.02, 28.66)
(0.69, 2.58)
Sim 3
0.9555
(35.70, 27.87)
(36.02, 28.66)
(0.89, 2.76)
As in previous algorithms, the simulations are also carried out for the different
input domains. The errors of the solutions are tabulated in Tables 6.12 and 6.13 for
the first set of data and Tables 6.15 and 6.16 for the second set of data. Error 1 in
Tables 6.13 and 6.16 indicates the errors between the calculated and preferred output.
Error 2 indicates the error between the calculated output and the experimental values.
6.7
Discussion and Conclusion
In this chapter, we first presented the theoretical details involved in developing
the inverse model of the mass transfer in the RDC column which is based on two
dimensional fuzzy number. The details start with the crisp relation which was then
followed by the fuzzy relation. Based on the fuzzy relation concept, the pyramidal
148
Table 6.13: IMDMS-2D-Fuzzy: Errors of calculated output against and preferred values
and Experimental Data 1
Simulation
Calculated Output
Preferred Values
Exp Value
Errors 1(%,%)
Errors 2(%,%)
Numbers
Values
(CCin, Cdin)
(CCin, Cdin)
(CCin, Cdin)
(CCin, Cdin)
(CCin, Cdin)
Sim 1
(24.34, 59.25)
(24.50, 60.18)
(23.97, 63.1)
(0.64, 1.54)
(1.54,6.10)
Sim 2
(24.26, 59.05)
(24.50, 60.18)
(23.97, 63.1)
(0.99, 1.87)
(1.21,6.42)
Sim 3
( 24.18, 58.88)
(24.50, 60.18)
(23.97, 63.1)
(1.31, 2.16)
(0.88,6.69)
Table 6.14: Set Data 2: The input data for simulations of IMDMS-2D-Fuzzy Algorithm
Simulation
Input Domain
Preferred Input
Output Domain
Preferred Output
Numbers
CCin
Cdin
CCin
Cdin
CCin
Cdin
CCin
Cdin
Sim 1
[32.8 44]
[15.0 30.0]
39.64
27.28
[18.89 30.09]
[41.52 75.65]
25.73
64.91
Sim 2
[30.8 46]
[13.0 32.0]
39.64
27.28
[18.89 30.09]
[41.52 75.65]
25.73
64.91
Sim 3
[28.8 48]
[11.0 34.0]
39.64
27.28
[18.89 30.09]
[41.52 75.65]
25.73
64.91
fuzzy number of dimension two was derived. This derivation was followed by the
verification that the properties of the two dimensional fuzzy number has been satisfied.
The definition of the alpha-level of the pyramidal fuzzy number was also given. It
was then proved that the alpha-level is closed and bounded. The boundedness and
closedness of the alpha-level are necessary to ensure the existence of the solution of the
inverse model.
The development of the model starts with the mass transfer of the single drop
single stage, ISDSS-2D-Fuzzy system. In this model, the three phases of the fuzzy
algorithm were used as in ISDSS-Fuzzy Model described in Chapter 5. However, in
ISDSS-2D-Fuzzy model, the two dimensional fuzzy number concept was employed.
Therefore, for each α-level, the fuzzified input parameters were mapped by Equation
(5.3). This process was taken in the fuzzy environment phase.
In addition, a triangular plane was used as the induced output parameter
149
Table 6.15: IMDMS-2D-Fuzzy: The errors between the calculated input values and
preferred values
Simulation
Calculated Input Values
Preferred Values
Errors (%,%)
Numbers
f∗
(CCin, Cdin)
(CCin, Cdin)
(CCin, Cdin)
Sim 1
0.9811
(39.72, 27.33)
(39.64, 27.28)
(0.20, 0.18 )
Sim 2
0.9806
(39.76, 27.37)
(39.64, 27.28)
(0.30, 0.33)
Sim 3
0.9807
(39.80, 27.41)
(39.64, 27.28)
(0.40, 0.48)
Table 6.16: IMDMS-2D-Fuzzy: Errors of the calculated output against preferred values
and Experimental Data 2
Simulation
Calculated Output
Preferred Values
Exp Value
Errors 1(%,%)
Errors 2(%,%)
Numbers
Values
(CCin, Cdin)
(CCin, Cdin)
(CCin, Cdin)
(CCin, Cdin)
(CCin, Cdin)
Sim 1
(25.81, 65.11)
(25.73, 64.91)
(27.28, 60.98)
(0.33, 0.31)
(5.38, 6.77)
Sim 2
(25.86, 65.22)
(25.73, 64.91)
(27.28, 60.98)
(0.49, 0.48)
(5.21, 6.95)
Sim 3
(25.89, 65.33)
(25.73, 64.91)
(27.28, 60.98)
(0.64, 0.64)
(5.10, 7.13)
instead of the one used by Ahmad[8] and Ismail et al.[10]. In their studies, they
considered only the points on the boundary of the triangular plane as the induced
output parameter. The idea of using the triangular plane was inspired by the result of
the nonexistent intersection points between the induced triangular and the preferred
output parameters.
The optimal solution of the algorithm was then determined in the defuzzification
phase by either Theorem 6.4 or Theorem 6.5. These theorems respectively indicated
that if the preferred fuzzy intersects on the maximum or minimum side of the fuzzy
induced plane, then the set of the optimized parameters is the set of the maximum
or minimum norm of the induced values. Prior to the proof of the theorems, we have
shown that the induced solution, Find is normal and convex. To show the convexity of
the induced solution, Lemma 6.1 was used.
A numerical example was also presented to study the capability of the method
150
used in developing the algorithm. Since, the system discussed involves the single drop
single stage system, the same input data of the Numerical Example in Subsection
5.3.5 was used. Thus, from the ISDSS-2D-Fuzzy Algorithm, the optimal solution is
(47, 87, 13.95) with an error of 5.26 and 1.23 percents respectively. The optimal solution
would be mapped to the output parameters of values (38.13, 29.23). These values
differ from the suggested values by 5.92 and 2.57 percents. From our comparison, it
has been empirically found that the error of the optimal solution from the example
in Subsection 5.3.5 is lesser than the error in Subsection 6.4.2. So is the error of the
output parameters. Even though the error in ISDSS-2D-Fuzzy Algorithm is greater,
the model is a appropriate representation of the dependent behavior of the two output
parameters of the system.
Besides the example in Subsection 6.4.2, the simulations of ISDSS-2D-Fuzzy
were also carried out for the different input domains. The input data used was exactly
the same as in the simulations of the ISDSS-Fuzzy algorithm. The errors of the optimal
solutions and the calculated output were tabulated in Tables 6.4 and 6.5. Table 6.4
showed that the larger the domain the bigger the error of the optimal solution for both
parameters. On the other hand, Table 6.4 showed that the error of calculated CCout
contradicted with the error of calculated Cdout . In other words as the domain became
larger the error of CCout increased while the error of Cdout decreased.
With the new approach, we then implemented the ISDSS-2D-Fuzzy Algorithm
to the multi-stage system. The system considered in this work is divided into two.
The first one is the single drop multi-stage system which is named ISDMS-2D-Fuzzy.
The inverse model of the system was developed by implementing the ISDSS-2DFuzzy Algorithm with some modification. The modification is on the repeated uses
of Equation (5.3) in order to get r for every stage. At the final stage this value was
processed to produce the induced output parameters. The details of the process were
described in the ISDMS-2D-Fuzzy Algorithm.
Again, the optimal solution is determined by either Theorem 6.4 or 6.5 in the
defuzzification phase. Subsequently, based on the normality and convexity theorems of
the induced solution for the single stage system, we constructed Corollaries 6.1 and 6.3
for multi-stage system. Specifically, since we discussed the mass transfer process in the
23-stage RDC column, Corollaries 6.1 and 6.3 were followed by Corollaries 6.2 and 6.4
151
for the convexity and normality of the induced solution.
The simulations of the ISDMS-2D-Fuzzy Algorithm were also carried out for
the different input domains. In the successive simulation, there were increments of
two units for both sides of each input domain. The errors of the calculated input and
output parameters were listed in Tables 6.9 and 6.10 respectively. These tables showed
that as the input domain became larger the error of the calculated input and output
parameters increased.
The other multi-stage system is the multi-stage for multiple drops system. The
procedure involved in developing this type of model is the same as in ISDMS-2D-Fuzzy
system. The difference is only on the forward model used in order to determined the
induced solution. The forward model of MTSS was used in IMDMS-2D-Fuzzy instead
of MTASD in the ISDMS-2D-Fuzzy Algorithm.
To validate the model, the Experimental Input Data 1 and 2 were used in the
simulations of the algorithm. The preferred output values were chosen by applying
the forward MTSS Algorithm. Subsequently, a comparison was made between the
calculated output of the IMDMS-2D-Fuzzy Algorithm and the experimental values.
The percentage error, Error 2 was calculated and tabulated in Tables 6.13 and 6.16
for both sets of data. The aim of this comparison is to point out the difference of the
output between the inverse modelling by fuzzy concept and the experimental values.
It is observed that both errors are less than 10%.
To summarize, the formulation of an inverse mass transfer for multiple drops
in a multi-stage RDC column was presented based on two dimensional fuzzy number
concept. In general, we conclude that this new technique for the determination of
the optimal input parameters gives useful information and provides a faster tool for
decision-makers.
CHAPTER 7
CONCLUSIONS AND FURTHER RESEARCH
7.1
Introduction
This chapter provides a summary and an overall conclusion of the findings
presented in this work and also gives an outline of some further research which are
worthwhile investigating in the future.
7.2
Summary of the Findings and Conclusion
The initial task of the work was to formulate an equation that will be used as
the boundary condition of the IBVP. This equation was expected to be a time varying
function. This was achieved by using the experimental data from [5]. From the data,
it was found that the concentration of the continuous phase depends on the stage of
the RDC column. In this work, the following assumptions are adopted:
• there are ten different classes of drops with different velocities depending on their
sizes,
• mass transfer of a single solute from continuous phase to a single drop,
• the drop is spherical and there is no coalescence of drops,
• the concentration of the drop along the radius r is assumed to be uniform
153
• drop contact time for mass transfer coefficient estimation is residence time in the
compartment.
With these assumptions and by the least square method, it was found that the boundary
condition is a function of t, that is f1 (t) = a1 +b1 t. The analytical solution of the IBVP
with the new boundary condition was detailed in Subsection 3.3.1.
The derivation of the new fractional approach to equilibrium was then
considered based on the analytical solution of the varied boundary condition IBVP. The
comparison of the new and existing fractional approach to equilibrium was carried out
by plotting the curves with respect to time on the same axes as in Figure 3.3. The curve
of the new fractional approach to equilibrium profile agrees with the result obtained by
Talib[5]. Therefore, from this initial task we conclude that the new fractional approach
to equilibrium, Fnew , represents the real phenomena of the mass transfer and hence
gives better tool for further development of the improved mass transfer model in the
column.
The development of the improved mass transfer model is one of the main aims
in this research. Therefore the IBVP which is based on the interface concentration was
considered. With this consideration and the new fractional approach to equilibrium,
a new driving force named Time-dependent Quadratic Driving Force (TQDF),
2
2
av −c1 )
), was derived. The process of mass transfer of a single drop based
( (f1 (t)−cC1 )av−(C
−c1
on TQDF is governed by:
1. The equilibrium equations
ys = f (xs ),
2. The interface equation
ys =
3d
k (x
Dπ 2 x b
Fv
Fv
3d
d
1
− xs ) 1−F
2 − ( D π 2 )( 1−F 2 )( 6 )Fv (t) dt f1 (t) + y0 ,
y
v
3. The average concentration of the drop
yav = Fnew (t)(ys − y0 ) + y0 ,
4. The mass balance equation
Fx (xin − xout ) = Fy (yout − yin ).
v
154
Based on these equations, the MTASD Algorithm was designed. This algorithm
calculates the amount of mass of a solute transfer from the continuous phase to a single
drop in the column. The process of mass transfer is said to be in a steady state, if there
exist ε = 0.0001 where the difference of the concentration at t = n and t = n − 1 is
less or equal to ε at every stage. At this point, the concentration of the drop interface
is in equilibrium with the medium. The complete description of the algorithm is well
illustrated as a flow chart in Figure 4.2.
The simulation of the MTASD Algorithm was also carried out using the Crank
solution of fractional approach to equilibrium, Fc . The validation of the MTASD
Algorithm was done empirically by plotting the curves of the simulation results from
both Fnew and Fc . It was found from Figure 4.3 that the curves of the dispersed and
continuous phase concentrations from the improved model agree with the curves from
the existing model in [5].
Based on various studies, the mass transfer process in the RDC column is very
complicated because it involves not only the mass transfer of a single drop but infinitely
many drops. These drops have different sizes and different velocities. Therefore, a more
realistic MTMD Algorithm is constructed which was later refined as another algorithm,
MTSS Algorithm. Both of the algorithms calculate the mass transfer of multiple drops
in 23-stages RDC column.
In this model, the total concentration of the drops in each cell is obtained by
applying Equation (4.20). Then using Equation (4.21), the average concentration of
the drop in each compartments is calculated. Finally the mass balance equation is
applied in order to obtain the amount of solute transfer from the continuous to the
dispersed phase. In the MTSS Algorithm the calculation of the mass transfer was done
simultaneously with respect to iteration time. In other word, as an example, the mass
transfer for iitr = 2 is calculated at stage two for the first swam of drops and at stage
one for the second swam of drops. This is contrary to MTMD Algorithm. In the
MTMD algorithm, the mass transfer at iitr = 1 is calculated at every stage without
considering the second swam of drops. The simulation data of both algorithms and the
experimental data are then plotted in Figure 4.17.
From this figure, it was empirically found that the output from both algorithms
do not give significant difference. However, MTSS Algorithm is more realistic due to
155
the fact that mass transfer in the real RDC column occurs simultaneously as explained
in the MTSS Algorithm. In conclusion, MTSS Algorithm gives a better representation
of the real mass transfer process and hence it is expected to produce better simulation
results when compared to experimental data. The dispersed and continuous phase
concentrations curves in Figure 4.17 clearly show the agreement of the above conclusion.
For more definitive conclusion, the improved mass transfer model, where by
the output can be simulated MTSS Algorithm, gives a useful information and provides
better simulation results and hence better control system for the RDC column.
This research has established a technique for solving the inverse problem of
determining the values of input parameters for the desired values of output parameters.
In achieving the task, the multivariate equations involved in modelling the forward
mass transfer process as explained in Chapters 3 and 4 are simplified as MIMO system
of Equation (5.6). The development of the inverse algorithm necessary to solve the
corresponding inverse problem is as described in Section 5.2. The essential feature
of the method used was the fuzzy algorithm. This algorithm requires three phases
of a structure-based fuzzy system, these are fuzzification of the input variables, fuzzy
environment and defuzzification. In the early stage of the development, the ISDSSFuzzy Algorithm was constructed by considering the separation of the MIMO into
a group of MISOs system. In this algorithm, fuzzy number of dimension one and
triangular membership function were employed.
As described in Chapter 5, the output parameters for MIMO system of the
mass transfer in the RDC column are actually two dependent parameters. Due to this
reason a new approach is introduced based on two dimensional fuzzy number. In this
work, we deduced the following results:
• the derivation of pyramidal fuzzy number from the Cartesian product of two
triangular fuzzy numbers,
• the verification of the properties of the two dimensional fuzzy number,
• the definition of the alpha-level set of the pyramidal fuzzy number,
• this alpha-level set is closed and bounded.
156
The boundedness and closedness of the alpha-level are necessary to ensure the existence
of the solution of the inverse model.
These deductions brought the study to the development of a series of algorithms
for solving the inverse problem corresponding to the improved forward models. The
respective algorithms are:
• ISDSS-2D-Fuzzy for Single Drop Single Stage system,
• ISDMS-2D-Fuzzy for Single Drop Multi-stage system
• Implementation of ISDMS-2D-Fuzzy Algorithm to Multiple Drops Multi-stage
system with some modifications.
In these algorithms, all the input parameters were fuzzified to create fuzzy environment.
This is then processed to produce the induced output parameters. The best output
parameters were extracted through defuzzification phase. This optimal solution was
determined by either Theorem 6.4 or Theorem 6.5. This study has also led us to the
establishment of:
• Lemma 6.1. This lemma is used for the proof of Theorem 6.2,
• Theorem 6.1. This theorem states that the fuzzy pyramidal number is fuzzy
number of dimension two.
• Theorem 6.2. This theorem states about the convexity of the induced solution,
• Theorem 6.3. This theorem states about the normality of the induced solution,
• Theorem 6.4. This theorem indicates that if the preferred fuzzy intersects on the
maximum of the fuzzy induced plane, then the set of optimized parameters is the
set of the maximum norm of the induced values.
• Theorem 6.5. This theorem indicates that if the preferred fuzzy intersects on the
minimum of the fuzzy induced plane, then the set of optimized parameters is the
set of the minimum norm of the induced values,
• Corollary 6.1. This corollary states about the convexity of the induced solution
for multi-stage RDC column,
157
• Corollary 6.2. This corollary states about the convexity of the induced solution
for 23 stages RDC column,
• Corollary 6.3. This corollary states about the normality of the induced solution
for multi-stage RDC column,
• Corollary 6.4. This corollary states about the normality of the induced solution
for 23 stages RDC column.
(i)
This study proved that if all fuzzy set FIi expressing the preferences of all
(i)
the input parameter Pi
(i)
∈ Ii
⊂ + (i = 1, 2) are convex and normal, then the
induced solution for the 23-stage RDC column, Find are also convex and normal. It
also showed that the presented method is able to solve the inverse problem of MIMO
system and capable of determining the optimal value of the output parameters. Their
corresponding input parameters are then chosen to be the best suggested values.
In addition, the percentage of relative error between the actual outputs obtained
from the approximate solution of the presented algorithm and the target output was
found to be less than 10%. The inverse models have successfully eliminated the trial
and error aspect of the forward process in determining the correct inputs for the desired
outputs. Therefore, we conclude that this new technique gives a useful information and
provides a faster tool for decision-makers.
7.3
Further Research
This research presents the improved mathematical forward mass transfer model
for simulation of RDC Column and also established a technique for assessing the inverse
models of the corresponding improved forward mass transfer models. All the objectives
of the research are achieved successfully. However, the following research suggestions,
in our opinion are worthwhile investigations:
• Development of Inverse Model of the hydrodynamic process.
The parameters involved in the hydrodynamic process in the RDC column
are complexly interrelated.
Therefore only certain parameter values can be
158
controlled and adjusted such as that of rotor speed (Nr ), dispersed phase flow
rate (Fd ) and interfacial tension (γ). Although interfacial tension could not be
controlled directly but at least by varying this value will provide us with some
useful information. These three parameters are determined or fixed outside the
RDC column, but once they are applied to the modelling, it will give whatever
calculated value for the holdup. This is an inverse problem of type coefficient
inverse problem.
• Development of the intra-stage control system for the RDC column.
In this study the inverse problem in determining the value of the input parameter
for the desired value of output of 23-stage RDC column has been successfully
solved. Intra-stage control system is the control system inside the RDC column.
The inverse algorithm developed in this study only need the information of the
input and output parameters outside the RDC column. Whilst for the intrastage control, more information is needed in particular the information on the
concentrations of both liquids at certain stage or if possible at every stage in the
RDC column.
• Further investigation and development on the theory of two dimensional fuzzy
number in multi-stage systems.
• Development of the integrated model of the hydrodynamic and mass transfer
processes.
Parallel processing is suggested to be introduced in order to develop the integrated
model of the hydrodynamic and mass transfer processes. This integrated model
is hoped to give better simulation and better control system for the RDC column.
159
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APPENDIX A
Geometrical and Physical Properties of RDC Column
Geometrical properties of RDC column
Number of stages
23
Height of a compartment(m)
0.076
Diameter of rotor disc(m)
0.1015
Diameter of column(m)
0.1520
Diameter of stator ring(m)
0.1110
Rotor speed(rev/s)
4.2
Table A.1: The geometrical properties of the rotating disc contactor (RDC) column.
165
166
Physical properties of the system(cumene/isobutyric acid/water)
Continuous phase:isobutyric acid in water
Dispersed phase:isobutyric acid in cumene
Viscosity of continuous phase (kg/ms)
0.100E-2
Viscosity of Dispersed phase (kg/ms)
0.710E-3
Density of continuous phase (kg/ms3 )
0.100E+4
Density of dispersed phase (kg/ms3 )
0.862E+3
Molecular diffusivity in the continuous phase (m2 /s)
0.850E-9
Molecular diffusivity in the dispersed phase (m2 /s)
0.118E-8
Table A.2: The physical properties of the system used.
APPENDIX B
GLOSSARY
A glossary of the acronyms used in the thesis is provided below.
The acronyms
represents some useful mathematical concepts or terms and the names for some
algorithms.
NAME
MEANING
ANN
Artificial Neural Network
BAMT
Boundary Approach of Mass Transfer
BMT
Basic Mass Transfer
EVM
Expected Value Mathod
FL
Fuzzy Logic
IAMT
Initial Approach of Mass Transfer
IBVP
Initial Boundary Value Problem
IP
Inverse Problem
IMDMS
Inverse Multiple Drops MUlti-stage
ISDSS
Inverse Single Drop Single Stage
MIMO
Multiple Input Multiple Output
MISO
Multiple Input Single Output
MTASD
Mass Transfer of a Single Drop
MTMD
Mass Transfer of Multiple Drops
MTSS
Mass Transfer steady State
PCA
Principle Component Analysis
167
168
RDC
Rotating Disc Contactor
TQDF
Time-dependent Quadratic Driving Force
S-DMT
Simultaneous Discrete Mass Transfer
UIVI
Updating Initial Value for Next Iteration
UIVS
Updating Initial Value for Next Stage
X-ray CT
X-ray Computed Tomography
APPENDIX C
PAPERS PUBLISHED DURING THE AUTHOR’S CANDIDATURE
From the material in this thesis there are, at the time of submission, papers which have
been published, presented or submitted for publication or presentation as following:
Books
P1. Maan, N., Talib, J., Arshad, K.A. and Ahmad, T. Inverse Modeling Of Mass
Transfer Process in the RDC Column by Fuzzy Approach.
In: Mastorakis,
N.E., Manikopoulos, C., Antoniou, G.E., Mladenov, V.M. and Gonos, I.F. Recent
Advances in Intelligent Systems and Signal Processing. Greece: WSEAS Press.
2003: 348–353
Papers published in journals
P2. Normah Maan, Jamalludin Talib and Khairil Anuar Arshad Use of Neural Network
for Modeling of Liquid-liquid Extraction Process in the RDC Column. Matematika.
19(1).2003. 15–27
P3. Maan, N., Talib, J., Arshad, K.A. and Ahmad, T. On Determination of Input
Parameters of The Mass Transfer Process in the Multiple Stages RDC Column by
Fuzzy Approach: Inverse Problem Matematika. Vol 20(2): 2004
170
Papers published in proceedings
P4. Normah Maan, Jamalludin Talib and Khairil Anuar Arshad Mass Transfer Model
of a Single Drop in the RDC Column. Prosiding Simposium Kebangsaan Sains
Matematik Ke 10 2002; 217–227.
P5. Maan, N., Talib, J., Arshad, K.A. and Ahmad, T. Inverse Modeling Of Mass
Transfer Process in the RDC Column by Fuzzy Approach. Proceedings of the 7th
WSEAS International Multiconference on Circuits, systems, Communications and
Computers. July 7-10. Corfu, Greece: WSEAS, Paper No. 457-281.
P6. Maan, N., Talib, J., Arshad, K.A. and Ahmad, T. On Inverse Problem of the Mass
Transfer Process in the Multiple Stages RDC Column: Fuzzy Approach. Abstract
of the International Conference on Inverse Problem: Modeling and Simulation.
2004.
P7. Maan, N., Talib, J., Arshad, K.A. and Ahmad, T. On Mass Transfer Process
In Multiple Stages RDC Column: Inverse Problem. Presented in the Simposium
Kebangsaan Sains Matematik Ke 11, Promenade Hotel, Kota Kinabalu, Sabah Dis
22-24, 2003.
APPENDIX D
PROGRAM MAT-LAB: INVERSE ALGORITHM
%Final Revision- successfully run
% ===========================================================
% Filename : ISDMS-2D Fuzzy
%===========================================================
%Chapter6
=============================================================
% A Fuzzy Algorithm For Inverse Modeling Of Mass Transfer Process in The
Rotating Disc Contactor Column.
=============================================================
% Reference: Normah Maan,
Department of Mathematics, Faculty of Science, University
%
Technology Malaysia, 81310 Skudai Johor Bharu Johor,
=============================================================
% n-the number of stages
n=23
a=0.00705;
Dy=0.118*10^-8;
%t=100
Fy=1.25;
Fx=3.75;
tic;
I=[28.8 39.64 46.6;18.5 27.28 38.5];%Sim3
O=[31.5 39.20 41.6;25.8 31.97 39.86]
% Hit any key to print the input matrix and the output matrix
pause
fprintf('Input matrix is' ),I
fprintf('Output matrix is' ),O
% Hit any key to calculate the end-points for input matrix
pause
k=5
h = 1/k;
m = 2;
for i = 1:m
y=0;
for j = 1:k+1
p(2*i-1,j)= y * (I(i,2)-I(i,1))+I(i,1);
p(2*i,j)= I(i,3) - y * (I(i,3)-I(i,2));
y = y+h;
end
end
171
172
disp('the end-points for input matrix: '),p
CCin=zeros(k+1,m);
for i=1:m
for j=1:k+1
CCin(1,1)=p(1,1);
CCin(j,i)=p(i,j);
end
end
CCin
Cdin=zeros(k+1,m);
for i=1:2
for j=1:k+1
Cdin(j,i)=p(i+m,j);
end
end
Cdin
% Hit any key to calculate the end-points for output matrix
pause
%prefered output
for i = 1:m
y=0;
for j = 1:k+1
po(2*i-1,j)= y * (O(i,2)-O(i,1))+O(i,1);
po(2*i,j)= O(i,3) - y * (O(i,3)-O(i,2));
y = y+h;
end
end
disp('the end-points for output matrix: '),po
CCout=zeros(k+1,m);
for i=1:m
for j=1:k+1
CCout(1,1)=po(1,1);
CCout(j,i)=po(i,j);
end
end
CCout
Cdout=zeros(k+1,m);
for i=1:2
for j=1:k+1
Cdout(j,i)=po(i+m,j);
end
end
Cdout
% Hit any key to create the x-points, y-points and z-points
173
pause
dCC=zeros(1,k)
H=10
for j=1:k
dCC(j)=(CCout(j,2)-CCout(j,1))/H;
end
dCC
for j=1:k
dCd(j)=(Cdout(j,2)-Cdout(j,1))/H;
end
dCd
Xaxis=zeros(k+1,H+1) ;
for j=1:k
for i=1:H+1
Xaxis(j,1)=CCout(j,1);
Xaxis(j,i)=CCout(j,1)+ (i-1)*dCC(1,j);
Xaxis(k+1,i)=CCout(k+1,2);
end
end
Xaxis
Yaxis=zeros(k+1,H+1)
for j=1:k
for i=1:H+1
Yaxis(j,1)=Cdout(j,1);
Yaxis(j,i)=Cdout(j,1)+ (i-1)*dCd(1,j);
Yaxis(k+1,i)=Cdout(k+1,2);
end
end
Yaxis
%Generate the points to plot the graph of preferred output
y1=Yaxis(:,1:1);
y1_1=y1(:,[1 1 1 1 1 1 1 1 1 1 1]);
y11=Yaxis(:,11:11);
y11_11=y11(:,[1 1 1 1 1 1 1 1 1 1 1]);
x1=Xaxis(:,1:1);
x1_1=x1(:,[1 1 1 1 1 1 1 1 1 1 1]);
x11=Xaxis(:,11:11);
x11_11=x11(:,[1 1 1 1 1 1 1 1 1 1 1]);
Zaxis=zeros(k+1,H+1);
for j=1:k+1
for i=1:H+1
Zaxis(1,:)=0 ;
Zaxis(j,:)=(j-1)*h;
end
end
174
Zaxis
fnX=[Xaxis ;Xaxis ;(x1_1) ;(x11_11) ];
fnY=[y1_1;y11_11;Yaxis;Yaxis];
fnZ=[Zaxis;Zaxis;Zaxis;Zaxis];
% Hit any key to plot preferred output
pause
figure('name','Membership function for preferred output')
mesh(fnX,fnY,fnZ),%drawnow;
xlabel('CCout');
ylabel('CDout');
zlabel('Membership function');
% Hit any key to process the input values to get the induced performance parameter
%Generate all 2^n combinations of all the endpoints of intervals
%representing alpha-k cuts
pause
fuzzy_val=zeros(m,4,k+1);
for i=1:k+1
for j=1:2
for l=1:m
fuzzy_val(1,2*j-1,i)=CCin(i,1);
fuzzy_val(1,2*j,i)=CCin(i,2);
fuzzy_val(2,3*j-2,i)=Cdin(i,1);
fuzzy_val(2,j+1,i)=Cdin(i,2);
end
end
end
fuzzy_val
%Determine rj=(r1,r2)=Y(h1,h2)#####Y1pt0=[min(r1) max(r1)];
%Y2pt0=[min(r2) max(r2)];
%For each alpha cuts, determine the induced performance parameters,
Y1pt=zeros(1,2,k+1);
Y2pt=zeros(1,2,k+1);
y1pt=zeros(1,2,k+1);
y2pt=zeros(1,2,k+1);
for i=1:k+1
[Y1pt(:,:,i),Y2pt(:,:,i)]=fw1n2((fuzzy_val( 1:1,1:4,i))',(fuzzy_val( 2:2,1:4,i))',I)
end
Y1pt
Y2pt
Fxy_ind=zeros(k+1,2^m);
for i=1:k+1
for j=1:m
%Fxy_ind(i,j)=Y1pt(:,j,n,i);
Fxy_ind(i,j)=Y1pt(:,j,i);
Fxy_ind(i,2+j)=Y2pt(:,j,i);
175
end
end
Fxy_ind
F_ind=zeros(1,k+1)
L=0
for i=1:k+1
F_ind(:,i)=L;
L=L+h
end
F_ind
Out_minx=[Fxy_ind(:,1:1) F_ind'];
Out_maxx=[Fxy_ind(:,2:2) F_ind'];
Out_miny=[Fxy_ind(:,3:3) F_ind'];
Out_maxy=[Fxy_ind(:,4:4) F_ind'];
minmax_x=[Out_minx;Out_maxx];
minmax_y=[Out_miny;Out_maxy];
%induced output
dx=zeros(1,k)
H=10
for j=1:k
dx(j)=(Fxy_ind(j,2)-Fxy_ind(j,1))/H;
end
dx
dy=zeros(1,k) ;
for j=1:k
dy(j)=(Fxy_ind(j,4)-Fxy_ind(j,3))/H;
end
dy
% Hit any key to Generate the points to plot the graph of induced output
pause
inx_axis=zeros(k+1,H+1) ;
for j=1:k
for i=1:H+1
inx_axis(j,1)=Fxy_ind(j,1);
inx_axis(j,i)=Fxy_ind(j,1)+ (i-1)*dx(1,j);
inx_axis(k+1,i)=Fxy_ind(k+1,2);
end
end
inx_axis
iny_axis=zeros(k+1,H+1) ;
for j=1:k
for i=1:H+1
iny_axis(j,1)=Fxy_ind(j,3);
iny_axis(j,i)=Fxy_ind(j,3)+ (i-1)*dy(1,j);
iny_axis(k+1,i)=Fxy_ind(k+1,4);
176
end
end
iny_axis
zz=(F_ind)'
inz_axis=zz(:,[1 1 1 1 1 1 1 1 1 1 1])
figure('name','The interction between the preferred and the induced output.')
subplot(2,1,1);
mesh(inx_axis,iny_axis,inz_axis),hold on;
mesh(fnX,fnY,fnZ),drawnow;
xlabel('CCout');
ylabel('CDout');
zlabel('Membership function');
subplot(2,1,2)
contour(inx_axis,iny_axis,inz_axis,12),hold on;
%surf(inxx,inyy,inzz),hold on;
%prefered output
contour(fnX,fnY,fnZ,12),drawnow;
%mesh(fnX,fnY,fnZ)%,drawnow
%view([10,45]);
grid on;
xlabel('CCout');
ylabel('CDout');
zlabel('Membership function');
%The equation of induced plane
P=[Y1pt(1,1,1) Y2pt(1,1,1) 0]
Q=[Y1pt(1,2,1) Y2pt(1,2,1) 0]
R=[Y1pt(1,1,k) Y2pt(1,1,k) 1]
PQ=-P+Q%vector PQ
PR=-P+R%vector PR
%To find the cross product of vector PQ and PR i.e vector normal to the induced
plane
PQPR=[(PQ(1,2)*PR(1,3))-(PQ(1,3)*PR(1,2))
-1*((PQ(1,1)*PR(1,3))(PQ(1,3)*PR(1,2))) (PQ(1,1)*PR(1,2))-(PQ(1,2)*PR(1,1))]
%Therefore the equation of induced plane is ax+by+cz+d=0
InPlane=[PQPR(1,1) PQPR(1,2) PQPR(1,3) (PQPR(1,1)*-1*P(1,1))+(PQPR(1,2)*1*P(1,2))+(PQPR(1,3)*-1*P(1,3))]
%to find the equation of one of the plane of the prefered surface
PP=[O(1,3) O(2,1) 0];
QQ=[O(1,1) O(2,1) 0];
RR=[O(1,2) O(2,2) 1];
PPQQ=-PP+QQ;
PPRR=-PP+RR;
[Plane_pref1,x_1,n1n2]=Pt_inter(PP,PPQQ,PPRR,PQPR,InPlane)
%linein1 is the line intersection between the induced plane and preferred surface
linein1=[n1n2(1,1) x_1(1,1) ; n1n2(1,2) x_1(2,1); n1n2(1,3) x_1(3,1)];
% i.e x=t1*n1n2(1,1)+x(1,1),y=t1*n1n2(1,2)+x(2,1),z=t1*n1n2(1,3)+x(3,1),
%###################################################################
%equation of the adjacent lines of this plane1 line PPRR,
%i.e x=t2*PPRR(1,1) +PP(1,1),y=t2*PPRR(1,2) +PP(1,2),z=t2*PPRR(1,3) +PP(1,3)
177
line1=[PPRR(1,1) PP(1,1); PPRR(1,2) PP(1,2) ;PPRR(1,3) PP(1,3)];
%solve to get x, y, z i.e the common point
[X1,Y1,Z1]=Optimal_pt1(linein1,line1,PPRR,PP,n1n2,x_1)
%$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
QQ=[O(1,1) O(2,1) 0];
QQRR=[RR(1,1)-QQ(1,1) RR(1,2)-QQ(1,2) RR(1,3)-QQ(1,3)];
line2=[QQRR(1,1) QQ(1,1); QQRR(1,2) QQ(1,2) ;QQRR(1,3) QQ(1,3)];
[X2,Y2,Z2]=Optimal_pt1(linein1,line2,QQRR,QQ,n1n2,x_1)
%to find the equation of 2nd plane of the prefered surface
SS=[O(1,1) O(2,3) 0];
SSQQ=-SS+QQ;
SSRR=[RR(1,1)-SS(1,1) RR(1,2)-SS(1,2) RR(1,3)-SS(1,3)];
[Plane_pref2,x_2,n1n3]=Pt_inter(SS,SSQQ,SSRR,PQPR,InPlane)
%linein2 is the line intersection between the induced plane and preferred surface
linein2=[n1n3(1,1) x_2(1,1) ; n1n3(1,2) x_2(2,1); n1n3(1,3) x_2(3,1)];
% i.e x=t1*n1n3(1,1)+x(1,1),y=t1*n1n3(1,2)+x(2,1),z=t1*n1n3(1,3)+x(3,1),
%###################################################################
%equation of the adjacent lines of this plane1 line PPRR,
%i.e x=t2*SSRR(1,1) +SS(1,1),y=t2*SSRR(1,2) +SS(1,2),z=t2*SSRR(1,3) +SS(1,3)
line3=[SSRR(1,1) SS(1,1); SSRR(1,2) SS(1,2) ;SSRR(1,3) SS(1,3)];
%solve to get x, y, z i.e the common point
[X3,Y3,Z3]=Optimal_pt1(linein2,line3,SSRR,SS,n1n3,x_2)
%$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
QQRR=[O(1,2)-O(1,1) O(2,2)-O(2,1) RR(1,3)-QQ(1,3)];
line4=[QQRR(1,1) QQ(1,1); QQRR(1,2) QQ(1,2) ;QQRR(1,3) QQ(1,3)];
[X4,Y4,Z4]=Optimal_pt1(linein2,line4,QQRR,QQ,n1n3,x_2)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%to find the equation of 3rd plane of the prefered surface
TT=[O(1,3) O(2,3) 0];
SS=[O(1,1) O(2,3) 0];
RR=[O(1,2) O(2,2) 1];
TTSS=-TT+SS;
TTRR=-TT+RR;
%TTRR=[O(1,2)-O(1,1) O(2,2)-O(2,1) RR(1,3)-TT(1,3)];
[Plane_pref3,x_3,n1n4]=Pt_inter(TT,TTSS,TTRR,PQPR,InPlane)
%linein3is the line intersection between the induced plane and preferred surface
linein3=[n1n4(1,1) x_3(1,1) ; n1n4(1,2) x_3(2,1); n1n4(1,3) x_3(3,1)];
% i.e x=t1*n1n2(1,1)+x(1,1),y=t1*n1n2(1,2)+x(2,1),z=t1*n1n2(1,3)+x(3,1),
%###################################################################
%equation of the adjacent lines of this plane1 line PPRR,
%i.e x=t2*PPRR(1,1) +PP(1,1),y=t2*PPRR(1,2) +PP(1,2),z=t2*PPRR(1,3) +PP(1,3)
line5=[TTRR(1,1) TT(1,1); TTRR(1,2) TT(1,2) ;TTRR(1,3) TT(1,3)];
%solve to get x, y, z i.e the common point
[X5,Y5,Z5]=Optimal_pt1(linein3,line5,TTRR,TT,n1n4,x_3)
%$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
%SS=[O(1,1) O(2,3) 0];
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SSRR=[O(1,2)-O(1,1) RR(1,2)-SS(1,2) RR(1,3)-SS(1,3)];
line6=[SSRR(1,1) SS(1,1); SSRR(1,2) SS(1,2) ;SSRR(1,3) SS(1,3)];
[X6,Y6,Z6]=Optimal_pt1(linein3,line6,SSRR,SS,n1n4,x_3)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%to find the equation of THE 4TH plane of the prefered surface
PP=[O(1,3) O(2,1) 0];
TT=[O(1,3) O(2,3) 0];
RR=[O(1,2) O(2,2) 1];
PPTT=-PP+TT;
PPRR=-PP+RR;
[Plane_pref4,x_4,n1n5]=Pt_inter(PP,PPTT,PPRR,PQPR,InPlane)
%linein1 is the line intersection between the induced plane and preferred surface
linein4=[n1n5(1,1) x_4(1,1) ; n1n5(1,2) x_4(2,1); n1n5(1,3) x_4(3,1)];
% i.e x=t1*n1n2(1,1)+x(1,1),y=t1*n1n2(1,2)+x(2,1),z=t1*n1n2(1,3)+x(3,1),
%###################################################################
%equation of the adjacent lines of this plane1 line PPRR,
%i.e x=t2*PPRR(1,1) +PP(1,1),y=t2*PPRR(1,2) +PP(1,2),z=t2*PPRR(1,3) +PP(1,3)
line7=[PPRR(1,1) PP(1,1); PPRR(1,2) PP(1,2) ;PPRR(1,3) PP(1,3)];
%solve to get x, y, z i.e the common point
[X7,Y7,Z7]=Optimal_pt1(linein4,line7,PPRR,PP,n1n5,x_4)
%$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
TT=[O(1,3) O(2,3) 0];
%TTRR=[O(1,2)-O(1,1) O(2,2)-O(2,1) RR(1,3)-TT(1,3)];
line8=[TTRR(1,1) TT(1,1); TTRR(1,2) TT(1,2) ;TTRR(1,3) TT(1,3)];
[X8,Y8,Z8]=Optimal_pt1(linein4,line8,TTRR,TT,n1n5,x_4)
%Proses of Defuzzification
Ans=zeros(8,3);
Ans=[X1 Y1 Z1;X2 Y2 Z2;X3 Y3 Z3;X4 Y4 Z4;X5 Y5 Z5;X6 Y6 Z6;X7 Y7 Z7;X8
Y8 Z8]
count=0
for ii=1:1:8
if (Ans(ii,3)>=0) & (Ans(ii,3)<=1)
count=count+1
for jj=1:1:3
AnsZ(count,jj)=Ans(ii,jj)
end
else
count=count
end
end
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%AnsSup=zeros(count,3)
for ii=1:1:count;
Sup_Z=max(AnsZ(:,3))
if Sup_Z==max(AnsZ(ii,3))
II=ii
Sup_X=AnsZ(II,1)
Sup_Y=AnsZ(II,2)
end
end
Sup_X
Sup_Y
Sup_Z
%Determine all alpha cuts of the fuzzified input parameter by using
%the highest fuzzy membership value of the intersection
endpt_Sup_Z=[((I(1,2)-I(1,1))*Sup_Z+I(1,1)) ((I(1,2)-I(1,3))*Sup_Z+I(1,3));((I(2,2)I(2,1))*Sup_Z+I(2,1)) ((I(2,2)-I(2,3))*Sup_Z+I(2,3))]
%Defuzzified
%Generate all 2^n combinations of all the endpoints of intervals representing alpha-k
%cuts
nn=2
Com=zeros(2,1,2^nn);
for j=1:2^nn
DCom(:,:,1)=[endpt_Sup_Z(1,1);endpt_Sup_Z(2,1)];
DCom(:,:,2)=[endpt_Sup_Z(1,1);endpt_Sup_Z(2,2)];
DCom(:,:,3)=[endpt_Sup_Z(1,2);endpt_Sup_Z(2,1)];
DCom(:,:,4)=[endpt_Sup_Z(1,2);endpt_Sup_Z(2,2)];
end
fprintf( '2^n combination of all the endpoints of intervals representing alpha-0.0 cuts
'),ComSup_Z=[DCom(:,:,1) DCom(:,:,2) DCom(:,:,3) DCom(:,:,4)]
%Determine rj=(r1,r2)=Y(h1,h2)#####Y1pt0=[min(r1) max(r1)];
%Y2pt0=[min(r2) max(r2)];
toc;
[x_out,y_out]=Defw1n2((ComSup_Z( 1:1,1:4))',(ComSup_Z( 2:2,1:4))',I)
%x_out(:,:,:)=(xx_out.*(I(1,3)-I(2,1)))+I(2,1);
%y_out(:,:,:)=(yy_out.*(I(1,3)-I(2,1)))+I(2,1);
[Output,Out_para]=minnorm(x_out,y_out,n)
%[Output,Out_para]=maxnorm(x_out,y_out,n)
for i=1:2
Error(i)=abs(O(i,2)-Out_para(1,i))*100/O(i,2);
end
Error
180
function [Fxind,Fyind]=fw1n2(x_in,y_in,I)
n=23
time=20
KL=4
[xoout(:,:),yoout(:,:)]=fw1_2n2(n,I)
xoout
yoout
X_out=zeros(KL,1);
Y_out=zeros(KL,1);
XX_out=zeros(KL,1);
YY_out=zeros(KL,1);
ysmax=zeros(KL,1)
for L=1:1:KL
ysmax(L,1)=0.135*(x_in(L,:))^1.85
for j=1:n
X_out(L,:)=xoout;% the continuous phase is flowing out at the first stage
Y_out(L,:)=yoout;
YY_out(L,:)=(Y_out(L,:)*ysmax(L,1))+y_in(L,1);
XX_out(L,:)=X_out(L,:)*x_in(L,:);
end
end
X_out
Y_out
Fxind=zeros(1,2);Fyind=zeros(1,2);
Fxind(:,:)=[min(XX_out(:,:)) max(XX_out(:,:))];
Fyind(:,:)=[min(YY_out(:,:)) max(YY_out(:,:))];
Fxind
Fyind
function [XX_out,YY_out]=Defw1n2(x_in,y_in,In)
n=23;
time=20;
KL=4;
[xoout(:,:),yoout(:,:)]=fw1_2n2(n,In);
X_out=zeros(KL,1);
Y_out=zeros(KL,1);
XX_out=zeros(KL,1);
YY_out=zeros(KL,1);
ysmax=zeros(KL,1)
for L=1:1:KL
ysmax(L,1)=0.135*(x_in(L,:))^1.85
X_out(L,:)=xoout;% the continuous phase is flowing out at the first stage
Y_out(L,:)=yoout;
YY_out(L,:)=(Y_out(L,:)*ysmax(L,1))+y_in(L,1);
XX_out(L,:)=X_out(L,:)*x_in(L,:);
end
XX_out
YY_out
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%solve to get x, y, z i.e the common point
function [X,Y,Z]=Optimal_pt1(linein,line1,PPRR,PP,n1n2,x)
AA=[linein(1,1) -1*line1(1,1) ;linein(2,1) -1*line1(2,1) ];
bb=[-1*(linein(1,2)-line1(1,2));-1*(linein(2,2)-line1(2,2))];
t=inv(AA)*bb;
x1=t(2,1)*PPRR(1,1) +PP(1,1)
x2=t(1,1)*n1n2(1,1)+x(1,1)
if (x1-x2)<=0.000001;
X=x1
else
X=fprintf( 'x1 not equal to x2:No intersection point ')
end
if X<0
X=fprintf( 'no intersection point ')
else
X==X;
end
X
y1=t(2,1)*PPRR(1,2) +PP(1,2)
y2=t(1,1)*n1n2(1,2)+x(2,1)
if (y1-y2)<=0.000001;
Y=y1;
else
Y=fprintf( 'y1 not equal to y2 ');
end
if Y<0
Y=fprintf( 'no intersection point ');
else
Y=Y;
end
Y
z1=t(2,1)*PPRR(1,3) +PP(1,3);
z2=t(1,1)*n1n2(1,3)+x(3,1);
if (z1-z2)<=0.000001
Z=z1;
if 0<=Z<=1
Z=Z
else
Z=fprintf( 'no intersection point ');
end
else
Z=fprintf( 'z1 not equal to z2 :No intersection point.');
end
function [Plane_pref,x,n1n2]=Pt_inter(Pt_P,PPQQ,PPRR,PQPR,InPlane)
PPQQPPRR=[(PPQQ(1,2)*PPRR(1,3))-(PPQQ(1,3)*PPRR(1,2)) 1*((PPQQ(1,1)*PPRR(1,3))-(PPQQ(1,3)*PPRR(1,2))) (PPQQ(1,1)*PPRR(1,2))(PPQQ(1,2)*PPRR(1,1))]
%Therefore the equation of this plane is ax+by+cz+d=0
Plane_pref=[PPQQPPRR(1,1) PPQQPPRR(1,2) PPQQPPRR(1,3)
(PPQQPPRR(1,1)*-1*Pt_P(1,1))+(PPQQPPRR(1,2)*1*Pt_P(1,2))+(PPQQPPRR(1,3)*-1*Pt_P(1,3))]
Plane1=Plane_pref
A=[InPlane(1,1) InPlane(1,2) InPlane(1,3);Plane1(1,1) Plane1(1,2) Plane1(1,3);0 0 1]
b=[-1*InPlane(1,4);-1*Plane1(1,4);0]
x=inv(A)*b%i.e x is one of the points on the intersection between these two planes
n1n2=[(PQPR(1,2)*PPQQPPRR(1,3))-(PQPR(1,3)*PPQQPPRR(1,2)) 1*((PQPR(1,1)*PPQQPPRR(1,3))-(PQPR(1,3)*PPQQPPRR(1,1)))
(PQPR(1,1)*PPQQPPRR(1,2))-(PQPR(1,2)*PPQQPPRR(1,1))]
182
function [Output,Out_para]=maxnorm(xout,yout,n)
for i=1:4
norma(i)=sqrt((xout(i,1))^2+(yout(i,1))^2)
end
Output=max(norma)
for i=1:4
if Output==norma(i)
Out_para=[xout(i,1) yout(i,1)]
end
end
function [Output,Out_para]=minnorm(xout,yout,n)
for i=1:4
norma(i)=sqrt((xout(i,1))^2+(yout(i,1))^2)
end
Output=min(norma)
for i=1:4
if Output==norma(i)
Out_para=[xout(i,1) yout(i,1)]
end
end