Numerical approximations of population balance equations in particulate systems [Elektronische Ressource] / von Jitendra Kumar

Sujets

• Mathematics

Informations

• Publié par: otto-von-guericke-universitat_magdeburg
• Publié le: 01 janvier 2007
• Nombre de lectures: 31
• Langue: English
• Poids de l'ouvrage: 2 Mo

Extrait

Numerical approximations of population
balance equations in particulate systems
Dissertation
zur Erlangung des akademischen Grades
doctor rerum naturalium
(Dr. rer. nat.)
von M.Sc. Jitendra Kumar
geb. am 01.04.1978 in Bijnor, India
genehmigt durch die Fakult at fur Mathematik
der Otto-von-Guericke-Universit at Magdeburg
Gutachter:
Prof. Dr. rer. nat. habil. Gerald Warnecke
Jun.-Prof. Dr.-Ing. Stefan Heinrich
Eingereicht am: 29.08.2006
Verteidigung am: 04.10.2006Acknowledgements
The life of a PhD student has never been easy but I have certainly relished this experience.
During this period I derived my inspiration from several sources and now I want to express my
deepest gratitude to all those sources.
Foremost, I owe special thanks to my supervisor Prof. Dr. Gerald Warnecke who has not only
encouraged me but has given his remarkable suggestions and invaluable supervision throughout
my thesis. His advices and constructive criticism have always been the driving force towards
the successful completion of my thesis.
I also deeply appreciate the help of all members of the Institute for Analysis and Numerics. Now
I would like to thank Mr. Narni Nageswara Rao for his assistance with numerous aspects of the
preparation of this thesis and Dr. Mathias Kunik for his valuable advice and discussion.
I must thank the nancial support I received from the DFG-Graduiertenkolleg-828, "Micro-
Macro-Interactions in Structured Media and Particle Systems", Otto-von-Guericke-Universit at
Magdeburg for this PhD program.
At the Institute of Process Equipment and Environmental Technology, I am indebted to Prof.
Dr.-Ing. Lothar M orl and Jun.-Prof. Dr.-Ing. Stefan Heinrich for their willingness to involve
me in this project. During past three years they have given me lot of nancial support and
provided me with the facilities to write this thesis.
I am very grateful to Dr.-Ing. Mirko Peglow, who spared lot of his precious time in advising and
helping me throughout the research work. This work would be unthinkable without his guidance
and persistent help.
At the University of She eld U.K., I would like to express my special thank to Prof. Mike
Hounslow for his hospitality treatment during my two weeks visit to She eld. His valuable
remarks and suggestions enabled me to complete this thesis e cien tly.
I am overwhelmingly grateful to all my friends and colleagues for their cooperation and guid-
ance. I want to express my deepest gratitude to my very special friends Mr. Ayan Kumar
Bandopadyaya and Mr. Amit Kumar Tyagi who have ever been encouraging and optimistic.
Now I would like to express my deep obligation to my parents and relatives back home in India.
Their consistent mental supports have always been the strong push for me during this period.
Finally, an especially profound acknowledgement is due to my wife Chetna whose forbearance
and supports have been invaluable.Contents
Nomenclature iii
1 General Introduction 1
1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Problem and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3 New Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.4 Outline of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2 Population Balances 14
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2 Aggregation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2.1 Population Balance Equation . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2.2 Existing Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3 Breakage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.3.1 Population Balance Equation . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.3.2 Existing Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.4 Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.4.1 Population Balance Equation . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.4.2 Existing Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.5 Combined Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3 New Numerical Methods: One-Dimensional 35
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.2 The Cell Average Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.2.1 Pure Breakage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.2.2 Pure Aggregation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.2.3 Pure Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
3.2.4 Simultaneous Aggregation and Breakage . . . . . . . . . . . . . . . . . . . 103
3.2.5 Sim and Nucleation . . . . . . . . . . . . . . . . . . 106
3.2.6 Simultaneous Growth and Aggregation . . . . . . . . . . . . . . . . . . . . 113
3.2.7 Sim Growth and Nucleation . . . . . . . . . . . . . . . . . . . . 118
3.2.8 Choice of Representative Sizes . . . . . . . . . . . . . . . . . . . . . . . . 119
3.3 Mathematical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
3.3.1 The Fixed Pivot Technique . . . . . . . . . . . . . . . . . . . . . . . . . . 123
3.3.2 Direct Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
3.3.3 The Cell Average Technique . . . . . . . . . . . . . . . . . . . . . . . . . . 132
iCONTENTS
3.4 The Finite Volume Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
3.4.1 Pure Breakage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
3.4.2 Simultaneous Aggregation and Breakage . . . . . . . . . . . . . . . . . . . 152
4 New Numerical Methods: Multi-Dimensional 158
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
4.2 Reduced Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
4.2.1 Mathematical Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
4.2.2 Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
4.2.3 Test Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
4.3 Complete Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
4.3.1 Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
4.3.2 Test Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
5 Conclusions 205
Appendices 208
A Analytical Solutions 208
A.1 Pure breakage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
A.2 Pure aggregation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
A.3 Pure growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
A.4 Simultaneous aggregation and breakage . . . . . . . . . . . . . . . . . . . . . . . 213
A.5 Sim and nucleation . . . . . . . . . . . . . . . . . . . . . . . 215
A.6 Simultaneous growth and aggregation . . . . . . . . . . . . . . . . . . . . . . . . 216
A.7 Sim growth and nucleation . . . . . . . . . . . . . . . . . . . . . . . . . 217
A.8 Analytical solutions of a two component PBE . . . . . . . . . . . . . . . . . . . . 217
A.9 of tracer weighted mean particle volume . . . . . . . . . . . 218
B Mathematical Derivations 219
B.1 Birth and death rates for aggregation in the xed pivot technique . . . . . . . . . 219
B.2 A di eren t form of the xed pivot technique . . . . . . . . . . . . . . . . . . . . . 221
B.3 Comparison of accuracy of local moments . . . . . . . . . . . . . . . . . . . . . . 222
B.4 Discrete birth and death rates for breakage . . . . . . . . . . . . . . . . . . . . . 223
B.5 Second moment of the Gaussian-like distribution . . . . . . . . . . . . . . . . . . 223
B.6 Discrete birth rate for aggregation . . . . . . . . . . . . . . . . . . . . . . . . . . 224
B.7 Stability condition for the combined problem . . . . . . . . . . . . . . . . . . . . 226
B.8 Mass conservation of modi ed DTPBE . . . . . . . . . . . . . . . . . . . . . . . . 227
B.9 Discrete birth and death terms of TPBE . . . . . . . . . . . . . . . . . . . . . . . 228
B.10 Two-dimensional discrete PBE . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
B.11 Consistency with the rst cross moment . . . . . . . . . . . . . . . . . . . . . . . 231
Bibliography 232
Curriculum Vitae 241
iiNomenclature
Latin Symbol
a ;a ;a ;a Fraction of particles1 2 3 4
3b Breakage function m
6 1B Birth rate m s
3 1~B Binary breakage function m s
3c Tracer volume m
C Constant
6 1D Death rate m s
E Error 1Q
3f Multi-dimensional number density function m [i ]jj
1F Mass ux s
g Volume density function
G Growth rate along internal coordinate j [i ]=sj j
h Enthalpy J
H Heaviside function
I Total number of cells
I Degree of aggregationagg
J Numerical mass ux 1=s
K Constant
l Mass of liquid kg
3m Tracer volume density function m
M Tracer v
6n Number density function m
3N Number m
3N Initial number of particles m0
i 3p Limit of integr