Solution methodologies for the population balance equations describing the hydrodynamics of liquid-liquid extraction contactors [Elektronische Ressource] / vorgelegt von Menwer Attarakih

Solution Methodologies for the Population Balance Equations
Describing the Hydrodynamics of Liquid-Liquid Extraction
Contactors





Vom Fachbereich für Maschinenbau und Verfahrenstechnik
der Technischen Universität Kaiserslautern
zur Erlangung des akademischen Grades

Doktor – Ingenieur (Dr.-Ing.)

genehmigte Dissertation




Vorgelegt von

M. Sc. Chem. Eng. Menwer Attarakih

aus Amman - Jordanien





Eingereicht am: 19.05.2004
Mündliche Prüfung am: 01.07.2004


Promotionskommission:

Vorsitzender: Prof. Dr.-Ing. P. Steinmann
Referenten: Prof. Dipl.-Ing. Dr. techn. Hans-Jörg Bart
Associated Prof. Naim M. Faqir

Dekan: Prof. Dr.-Ing. P. Steinmann



D 386
2004

ACKNOWLEDGMENT
This work is performed during my stay as a Ph.D. student at the University of Kaiserslautern / Germany
at the Institute of Process Engineering chaired by Prof. Dipl.-Ing. Dr. techn. Hans-Jörg Bart.
I would like first to thank my advisor Prof. Hans-Jörg Bart for his permanent and extensive support to
accomplish this work. I highly appreciate his excellent advice and guidance during all the stages of this
long term project, and in particular for his contributions to my publications.

I would like also to express my deep thanks to Associated Prof.. Naim Faqir from the University of
Jordan/ Chem. Engng. Dept. for his valuable remarks, patience, and his contributions for preparing my
publications during his many research visits to the Institute of Thermal Process Engineering here at the
University of Kaiserslautern.

Second I am very grateful to Prof. Dr.-Ing. P. Steinmann and the Associated Prof. Naim M. Faqir for
serving as Committee Members.

This work is partially supported by the German Academic Exchange Service (DAAD), the precious
scholarship from my home University: Al-Balaqa Applied University as well as the financial support
granted by the chairman of the Institute of Process Engineering: Prof. Hans-Jörg Bart, to all those I
gratefully acknowledge this support.

I am deeply indebted to all my colleagues at the Institute of Process Engineering and especially the
Extraction Group (Martin Simon and Stephan Schmidt) whose continuous help greatly facilitated my
research.

I would like to thank Mr Dennis Bosse for translating the abstract, and Mrs Schneider for her help in
checking the language of some parts of this thesis. The help of Dr. Krätz is also greatly acknowledged.

To my mother that she has been suffering for more than three years without any chance to see me; to the
whole members of my family who are still waiting for me; I say to all of them ″PLEASE FOREGIVE
ME″.

Finely, I am thankful to my wife: Feda′ for her endless patience and continuous support, to my little
children: Balqees, Maya and Kariem for their ever shining smiles of kindness and love.


Kaiserslautern, in July 2004


Menwer Attarakih




































To my mother, wife and the little children: Balqees, Maya and Kariem



























ABSTRACT

Solution Methodologies for the Population Balance Equations
Describing the Hydrodynamics of Liquid-Liquid Extraction
Contactors



The polydispersive nature of the turbulent droplet swarm in agitated liquid-liquid contacting equipment
makes its mathematical modelling and the solution methodologies a rather sophisticated process. This
polydispersion could be modelled as a population of droplets randomly distributed with respect to some
internal properties at a specific location in space using the population balance equation as a mathematical
tool. However, the analytical solution of such a mathematical model is hardly to obtain except for
particular idealized cases, and hence numerical solutions are resorted to in general. This is due to the
inherent nonlinearities in the convective and diffusive terms as well as the appearance of many integrals
in the source term.
In this work two conservative discretization methodologies for both internal (droplet state) and external
(spatial) coordinates are extended and efficiently implemented to solve the population balance equation
(PBE) describing the hydrodynamics of liquid-liquid contacting equipment. The internal coordinate
conservative discretization techniques of Kumar and Ramkrishna (1996a, b) originally developed for the
solution of PBE in simple batch systems are extended to continuous flow systems and validated against
analytical solutions as well as published experimental droplet interaction functions and hydrodynamic
data. In addition to these methodologies, we presented a conservative discretization approach for droplet
breakage in batch and continuous flow systems, where it is found to have identical convergence
characteristics when compared to the method of Kumar and Ramkrishna (1996a).
Apart from the specific discretization schemes, the numerical solution of droplet population balance
equations by discretization is known to suffer from inherent finite domain errors (FDE). Two approaches
that minimize the total FDE during the solution of the discrete PBEs using an approximate optimal
moving (for batch) and fixed (for continuous systems) grids are introduced (Attarakih, Bart & Faqir,
2003a). As a result, significant improvements are achieved in predicting the number densities, zero and
first moments of the population.
For spatially distributed populations (such as extraction columns) the resulting system of partial
differential equations is spatially discretized in conservative form using a simplified first order upwind
scheme as well as first and second order nonoscillatory central differencing schemes (Kurganov &
Tadmor, 2000). This spatial discretization avoids the characteristic decomposition of the convective flux
based on the approximate Riemann Solvers and the operator splitting technique required by classical
upwind schemes (Karlsen et al., 2001).
The time variable is discretized using an implicit strongly stable approach that is formulated by careful
lagging of the nonlinear parts of the convective and source terms.
The present algorithms are tested against analytical solutions of the simplified PBE through many case
studies. In all these case studies the discrete models converges successfully to the available analytical
solutions and to solutions on relatively fine grids when the analytical solution is not available. This is
accomplished by deriving five analytical solutions of the PBE in continuous stirred tank and liquid-liquid
extraction column for especial cases of breakage and coalescence functions.
As an especial case, these algorithms are implemented via a windows computer code called LLECMOD
(Liquid-Liquid Extraction Column Module) to simulate the hydrodynamics of general liquid-liquid
extraction columns (LLEC). The user input dialog makes the LLECMOD a user-friendly program that
enables the user to select grids, column dimensions, flow rates, velocity models, simulation parameters,
dispersed and continuous phases chemical components, and droplet phase space-time solvers. The
graphical output within the windows environment adds to the program a distinctive feature and makes it
very easy to examine and interpret the results very quickly. Moreover, the dynamic model of the
dispersed phase is carefully treated to correctly predict the oscillatory behavior of the LLEC hold up. In
this context, a continuous velocity model corresponding to the manipulation of the inlet continuous flow
rate through the control of the dispersed phase level is derived to get rid of this behavior.

Key words: Liquid-liquid dispersion; Hydrodynamics; Population balance; Droplet breakage; Droplet
coalescence; Conservation laws; Numerical Solution.

KURZFASSUNG
Lösungsansätze zur Beschreibung der Hydrodynamik in der
Flüssig-flüssig Extraktion auf Basis von Populationsbilanzen
Der poyldisperse Charakter von turbulenten Tropfenschwärmen in gerührten Extraktionsapparaten
erschwert deren mathematische Modellierung sowie das Finden von geeigneten Lösungsstrategien. Mit
Hilfe von Populationsbilanzen (PBE) können solche Systeme als eine Verteilung von Tropfen mit
unterschiedlichen internen Eigenschaften, z.B. Konzentration oder Temperatur, zeit- und ortsaufgelöst
mathematisch beschrieben werden. Aufgrund der mathematischen Komplexität können nur für wenige
Spezialfälle die PBE analytisch gelöst werden und es müssen numerische Lösungstrategien entwickelt
werden. Di