Numerical analysis for finite volume schemes for population balance equations [Elektronische Ressource] / von Rajesh Kumar

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• Publié par: otto-von-guericke-universitat_magdeburg
• Publié le: 01 janvier 2011
• Nombre de lectures: 14
• Langue: English
• Poids de l'ouvrage: 1 Mo

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Numerical analysis of finite volume schemes
for population balance equations
Dissertation
zur Erlangung des akademischen Grades
doctor rerum naturalium
(Dr. rer. nat.)
vonM.Sc., Rajesh Kumar
geb. am 08.07.1981 in Jhajha, India
genehmigt durch die Fakult¨at fur¨ Mathematik
der Otto-von-Guericke-Universit¨at Magdeburg
Gutachter:
Prof. Dr. rer. nat. habil. Gerald Warnecke
Prof. Dr. Ansgar Jungel¨
Eingereicht am 12.11.2010
Verteidigung am: 18.03.2011Acknowledgements
Asachild,Iwasalwayscuriousaboutthetitle’Dr.’ Innotebooks,Irewrotemynamealongwith
the title ’Dr.’ It was a childhood fantasy to be called ’Dr. Rajesh Kumar’. However, when I
grew up and decided to pursue Mathematics more specifically, this fantasy was shaping into a
stronger dream. In my IIT days, I was keen on getting my dream fulfilled. And now I am here,
with all the blessings and good wishes of friends and families, with the title I always dreamt of.
Foremost, I express my deep gratitude to my supervisor Prof. Dr. Gerald Warnecke. Besides
imparting in me the basic fundamentals of Mathematics, he gave me remarkable suggestions
and supervision, which improved my scientific writing. My work grew better with his advices
and criticism during our group seminars and discussions. His patience and support helped me
to finish my thesis successfully. Under his tutelage, I grew into an able researcher.
I am also grateful to the DFG-Graduiertenkolleg-828, “Micro-Macro-Interactions in Structured
Media and Particle Systems”, Otto-von-Guericke Universit¨at Magdeburg for the financial sup-
port all through this PhD program.
I would like to express my sincere and deep gratitude to Dr. Jitendra Kumar. His advice and
encouragement during the course of my research has been great help. The numerical discussions
with him helped me to implement the matlab codes efficiently.
At the department of Chemical Process Engineering, I am thankful to Prof. Jurgen¨ Tomas for
giving me an opportunity to work in an interdisciplinary project.
I am thankful to the assistance of all members of the Institute of Analysis and Numerics,
especially Dr. Walfred Grambow and Stephanie Wernicke. I deeply appreciate the fruitful
discussions with my close friends Ankik and Vincent.
To all my friends in Magdeburg and in India who provided support and encouragement to me
during my stay here. I am overwhelmingly grateful to all of them: Yashodhan, Ashwini, Ankik,
Mini, Vincent, James, Vikrant, Bala, Thiru, Sashi, Sangeeta, Anita, Prabhat, Pankaj and Ravi.
I would also like to thank Maxim at TU Eindhoven for the same.
To my very special friend Ankik and Yashodhan, I will always remember the evening walk, the
time we spent together after dinner near the library. Our long evenings playing table tennis
will always be memorable. I am also obliged to Chetna bhabhi and her delicious food and
encouragement. The sheer joy I had while I spent my time with Babu Mahika will always
remain with me.
At Indian Institute of Technology Roorkee, I am indebted to Prof. Ramesh Chand Mittal who
helped me to get the position in Eramus Mundus program for pursuing my Master studies. I
also thank Mr. Rajesh Jha, my school teacher to whom I owe my knowledge of and love for
Mathematics.I am deeply grateful to my wife, Smita for her patience and understanding especially during
the last period of my work. I also must thank my in-laws for their warmth and affection.
Most importantly, I am eternally obliged to my family Maa-Papaji, brother Rajeev, all sisters,
brother-in-laws and their kids in India for their moral support and unconditional love without
which this work could not be completed.
Last but not the least, my heartly thanks to my Khatti Meethi for her care, affection and love.
Her inspiration and belief in me drove me this far and made me realize who I am. You are so
precious and lovable to me and will always be close to my heart.i
Abstract
Thisthesisdescribesthenumericalanalysisoffinitevolumeschemesforpopulationbalanceequa-
tions in particulate processes, incorporating aggregation, breakage, growth and source terms.
These equations are a type of partial integro-differential equations. Such equations can be
solved analytically only for some specific aggregation and breakage kernels. This motivates us
to study numerical schemes and the numerical analysis for these equations.
Severalmathematicalresultsareavailableontheexistenceofweaksolutionsfortheaggregation-
breakage equations with different classes of aggregation and breakage kernels. Recently, Bour-
gade and Filbet [7] have investigated the convergence of finite volume approximated solutions
towards weak solutions of the continuous binary aggregation-breakage equations under the as-
sumptions of local boundedness of the kernels. Furthermore, they have shown a first order error
estimate only on uniform meshes with more restricted kernels. However, the case of multiple
fragmentation and error analysis on general meshes were not discussed. A similar approach is
also suitable to show the convergence of the finite volume discretized solutions towards weak
solution of the continuous equations when multiple breakage is taken into account. This is the
first aim of our work.
The second aim is to study the convergence analysis of a finite volume method for the aggrega-
tionandmultiplebreakageequationsonfivedifferenttypesofuniformandnon-uniformmeshes.
We observe that the scheme is second order convergent independently of the grids for the pure
breakage problem. Moreover, for pure aggregation as well as for combined equations the tech-
niqueshowssecondorderconvergenceonlyonuniform,non-uniformsmoothandlocallyuniform
meshes. In addition, we find only first order convergence on oscillatory and random grids.
Anumericalschemeissaidtobemomentpreservingifitcorrectlyreproducesthetimebehaviour
of a given moment. Some authors have proposed different numerical methods which show
momentpreservationnumericallywithrespecttothetotalnumberortotalmassforanindividual
processofaggregation,breakage,growthandsourceterms. However,couplingofalltheprocesses
causes no preservation for any moments. Up to now, there was no mathematical proof which
gives the conditions under which a numerical scheme is moment preserving or not. The third
aim of this work is to study the criteria for the preservation of different moments. Based on this
criteriawedeterminezerothandfirstmomentspreservingconditionsforeachprocessseparately.
Further, we propose one moment and two moment preserving finite volume schemes for all the
coupled processes. We analytically and numerically verify the moment preserving results. The
numerical verifications are made for several coupled processes for which analytical solutions are
available for the moments.
The fixed pivot (FP) method and the cell average technique (CAT) for solving two-dimensional
aggregation equations using a rectangular grid were implemented in J. Kumar et al. [44]. Re-
cently, Chakraborty and Kumar [9] have studied the FP scheme for the same problem on two
different types of triangular grids. They found that the method shows better results for num-
ber density on grids as compared to rectangular grids. However, the discussion of
higher moments was ignored. In our work we compare different moments calculated by theii
FP technique on rectangular and triangular meshes with the analytical moments. Numerical
simulations show that the method does not improve the results for the higher moments. Fur-
ther we introduce a new mathematical formulation of the CAT for the two different types of
triangular grids as considered by Chakraborty and Kumar [9]. The new formulation is simple to
implement and gives better accuracy as compared to the rectangular grids. Three different test
problems are considered to analyze the accuracy of both schemes by comparing the analytical
andnumericalsolutions. Thenewformulationshowsgoodagreementwiththeanalyticalresults
both for number density and higher moments.
Finally we state some applications of aggregation-breakage equations in nano-technology. We
solve the equations using the cell average technique and compare the simulation results with
the experimental data by using a shear aggregation kernel together with two different breakage
kernels.iii
Zusammenfassung
DieseDoktorarbeitbeschreibtdienumerischeAnalysisvonFinite-Volumen-Methodenfur¨ Popu-
lationsbilanzgleichungen in Partikelprozessen, die Aggregation, Bruch, Wachstum und Quell-
termeeinbeziehen. DieseGleichungensindeineArtvonpartiellenIntegro-Differentialgleichungen.
Solche Gleichungen k¨onnen nur fur¨ einige spezielle Aggregations- und Bruchkerne analytisch
gel¨ost werden. Dies motiviert uns, numerische Verfahren und die numerische Analysis fur¨ diese
Gleichungen zu studieren.
EsgibtmehreremathematischeErgebnissezurExistenzvonschwachenL¨osungenfur¨ dieAggre-
gations-Bruch-GleichungenmitverschiedenenKlassenvonAggregations-undBruchkernen. Vor
kurzemuntersuchtenBourgadeundFilbet[7]dieKonvergenzvonFinite-Volumen-approximierten
L¨osungen gegen schwache L¨osungen der kontinuierlichen bin¨aren Aggregations-Bruch-Gleich-
ungen unter der Annahme der lokalen Beschr¨anktheit der Kerne. Weiterhin haben sie nur
Fehlerabsch¨atzungen erster Ordnung auf gleichm¨assigen Gittern mit eingeschr¨ankteren Kernen
gezeigt. Allerdings wurden der Fall multipler