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The discrete nature of the dispersed phase (swarm of droplet) in stirred and pulsed liquid-liquid extraction columns makes its mathematical modelling of such complex system a tedious task. The dispersed phase is considered as a population of droplets distributed randomly with respect to their internal properties (such as: droplet size and solute concentration) at a specific location in space. Hence, the population balance equation has been emerged as a mathematical tool to model and describe such complex behaviour. However, the resulting model is too complicated. Accordingly, the analytical solution of such a mathematical model does not exist except for particular cases. Therefore, 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. However, modelling and simulation of liquid extraction columns is not an easy task because of the discrete nature of the dispersed phase, which consist of population of droplets. The natural frame work for taking this into account is the population balance approach.
In part of this doctoral thesis work, a rigours mathematical model based on the bivariate population balance frame work (the base of LLECMOD ‘‘Liquid-Liquid Extraction Column Module’’) for the steady state and dynamic simulation of pulsed (sieve plate and packed) liquid-liquid extraction columns is developed. The model simulates the coupled hydrodynamic and mass transfer for pulsed (packed and sieve plate) extraction columns. The model is programmed using visual digital FORTRAN and then integrated into the LLECMOD program. Within LLECMOD the user can simulate different types of extraction columns including stirred and pulsed ones. The basis of LLECMOD depends on stable robust numerical algorithms based on an extended version of a fixed pivot technique after Attarakih et al., 2003 (to take into account interphase solute transfer) and advanced computational fluid dynamics numerical methods. Experimental validated correlations are used for the estimation of the droplet terminal velocity in extraction columns based on single and swarm droplet experiments in laboratory scale devices. Additionally, recent published correlations for turbulent energy dissipation, droplet breakage and coalescence frequencies are discussed as been used in this version of LLECMOD. Moreover, coalescence model from literature derived from a stochastical description have been modified to fit the deterministic population model. As a case study, LLECMOD is used here to simulate the steady state performance of pulsed extraction columns under different operating conditions, which include pulsation intensity and volumetric flow rates are simulated. The effect of pulsation intensity (on the holdup, mean droplet diameter and solute concentration) is found to have more profound effect on systems of high interfacial tension. On the hand, the variation of volumetric flow rates have substantial effect on the holdup, mean droplet diameter and solute concentration profiles for chemical systems with low interfacial tension. Two chemical test systems recommended by the European Federation of Chemical Engineering (water-acetone (solute)-n-butyl acetate and water-acetone (solute)-toluene) and an industrial test system are used in the simulation. Model predictions are successfully validated against steady state and transient experimental data, where good agreements are achieved. The simulated results (holdup, mean droplet diameter and mass transfer profiles) compared to the experimental data show that LLECMOD is a powerful simulation tool, which can efficiently predict the dynamic and steady state performance of pulsed extraction columns.
In other part of this doctoral thesis work, the steady state performance of extraction columns is studied taking into account the effect of dispersed phase inlet condition (light or heavy phase is dispersed) and the direction of mass transfer (from continuous to dispersed phase and vice versa) using the population balance framework. LLECMOD, a program that uses multivariate population balance models, is extended to take into account the direction of mass transfer and the dispersed phase inlet. As a case study, LLECMOD is used to simulate pilot plant RDC columns where the steady state mean flow properties (dispersed phase hold up and droplet mean diameter) and the solute concentration profiles are compared to the available experimental data. Three chemical systems were used: sulpholane–benzene–n-heptane, water–acetone–toluene and water–acetone–n-butyl acetate. The dispersed phase inlet and the direction of mass transfer as well as the chemical system physical properties are found to have profound effect on the steady state performance of the RDC column. For example, the mean droplet diameter is found to persist invariant when the heavy phase is dispersed and the extractor efficiency is higher when the direction of mass transfer is from the continuous to the dispersed phase. For the purpose of experimental validation, it is found that LLECMOD predictions are in good agreement with the available experimental data concerning the dispersed phase hold up, mean droplet diameter and solute concentration profiles in both phases.
In a further part of this doctoral thesis, a mathematical model is developed for liquid extraction columns based on the multivariate population balance equation (PBE) and the primary secondary particle method (PSPM) introduced by Attarakih, 2010 (US Patent Application: 0100106467). It is extended to include the momentum balance for the dispersed phase. The advantage of momentum balance is to eliminate the need for often conflicting correlations used in estimating the terminal velocity of single and swarm of droplets. The resulting mathematical model is complex due to the integral nature of the population balance equation. To reduce the complexity of this model, while maintaining most of the information drawn from the continuous population balance equation, the concept of the PSPM is used. Based on the multivariate population balance equation and the PSPM a mathematical model is developed for any liquid extraction column. The secondary particle could be envisaged as a fluid particle carrying information about the distribution as it is evolved in space and time, in the meanwhile the primary particles carry the mean properties of the population such as total droplet concentration; mean droplet diameter dispersed phase hold up and so on. This information reflects the particle-particle interactions (breakage and coalescence) and transport (convection and diffusion). The developed model is discretized in space using a first-order upwind method, while semi-implicit first-order scheme in time is used to simulate a pilot plant RDC extraction column. Here the effect of the number of primary particles (classes) on the final predicted solution is investigated. Numerical results show that the solution converge fast even as the number of primary particle is increased. The terminal droplet velocity of the individual primary particle is found the most sensitive to the number of primary particles. Other mean population properties like the droplet mean diameter, mean hold up and the concentration profiles are also found to converge along the column height by increasing the number of primary particles. The predicted steady state profiles (droplet diameter, holdup and the concentration profiles) along a pilot RDC extraction column are compared to the experimental data where good agreement is achieved.
In addition to this a robust rigorous mathematical model based on the bivariate population balance equation is developed to predict the steady state and dynamic behaviour of the interacting hydrodynamics and mass transfer in Kühni extraction columns. The developed model is extended to include the momentum balance for the calculation of the droplet velocity. The effects of step changes in the important input variables (such as volumetric flow rates, rotational speed, inlet solute concentrations etc.) on the output variables (dispersed phase holdup, mean droplet diameter and the concentration profiles) are investigated.
The last topic of this doctoral thesis is developed to transient problems. The unsteady state analysis reveals the fact that the largest time constant (slowest response) is due to the mass transfer. On the contrary, the hydrodynamic response of the dispersed phase holdup is very fast when compared to the mass transfer due to the relative fast motion of the dispersed droplets with respect to the continuous phase. The dynamic behaviour of the dispersed and continuous phases shows a lag time that increases away from the feed points of both phases. Moreover, the solute concentration response shows a highly nonlinear behaviour due to both positive and negative step changes in the input variables. The simulation results are in good agreement with the experimental ones and show the usefulness of the model.

An isogeometric Reissner-Mindlin shell derived from the continuum theory is presented. The geometry is described by NURBS surfaces. The kinematic description of the employed shell theory requires the interpolation of the director vector and of a local basis system. Hence, the definition of nodal basis systems at the control points is necessary for the proposed formulation. The control points are in general not located on the shell reference surface and thus, several choices for the nodal values are possible. The proposed new method uses the higher continuity of the geometrical description to calculate nodal basis system and director vectors which lead to geometrical exact interpolated values thereof. Thus, the initial director vector coincides with the normal vector even for the coarsest mesh. In addition to that a more accurate interpolation of the current director and its variation is proposed. Instead of the interpolation of nodal director vectors the new approach interpolates nodal rotations. Account is taken for the discrepancy between interpolated basis systems and the individual nodal basis systems with an additional transformation. The exact evaluation of the initial director vector along with the interpolation of the nodal rotations lead to a shell formulation which yields precise results even for coarse meshes. The convergence behavior is shown to be correct for k-refinement allowing the use of coarse meshes with high orders of NURBS basis functions. This is potentially advantageous for applications with high numerical effort per integration point. The geometrically nonlinear formulation accounts for large rotations. The consistent tangent matrix is derived. Various standard benchmark examples show the superior accuracy of the presented shell formulation. A new benchmark designed to test the convergence behavior for free form surfaces is presented. Despite the higher numerical effort per integration point the improved accuracy yields considerable savings in computation cost for a predefined error bound.

Unidirectional (UD) composites are the most competitive materials for the production
of high-end structures. Their field of application spreads from the aerospace up to
automotive and general industry sector. Typical examples of components made of
unidirectional reinforced composite materials are rocket motor cases, drive shafts or
pressure vessels for hydrogen storage. The filament winding technology, the pultrusion
process and the tape placement are processes suitable for the manufacturing
using UD semi-finished products. The demand for parts made of UD composites is
constantly increasing over the last years. A key feature for the success of this technology
is the improvement of the manufacturing procedure.
Impregnation is one of the most important steps in the manufacturing process. During
this step the dry continuous fibers are combined with the liquid matrix in order to create
a fully impregnated semi-finished product. The properties of the impregnated roving
have a major effect on the laminate quality, and the efficient processing of the
liquid matrix has a big influence on the manufacturing costs.
The present work is related to the development of a new method for the impregnation
of carbon fiber rovings with thermoset resin. The developed impregnation unit consists
of a sinusoidal cavity without any moving parts. The unit in combination with an
automated resin mixing-dosing system allows complete wet-out of the fibers, precise
calibration of the resin fraction, and stable processing conditions.
The thesis focuses on the modeling of the impregnation process. Mathematical expressions
for the fiber compaction, the gradual increase of the roving tension, the
static pressure, the capillarity inside the filaments of the roving, and the fiber permeation
are presented, discussed, and experimentally verified. These expressions were
implemented in a modeling algorithm. The model takes into account all the relevant
material and process parameters. An experimental set-up based on the filament
winding process was used for the validation of the model. Trials under different conditions
have been performed. The results proved that the model can accurately simulate
the impregnation process. The good impregnation degree of the wound samples
confirmed the efficiency of the developed impregnation unit. A techno economical
analysis has proved that the developed system will result to the reduction of the
manufacturing costs and to the increase of the productivity.

Induction welding is a technique for joining of thermoplastic composites. An alternating
electromagnetic field is used for contact-free and fast heating of the parts to be
welded. In case of a suitable reinforcement structure heat generation occurs directly
in the laminate with complete heating in thickness direction in the vicinity of the coil.
The resulting temperature field is influenced by the distance to the induction coil with
decreasing temperature for increasing distance. Consequently, the surface facing the
inductor yields the highest, the opposite surface the lowest temperature.
The temperature field described significantly complicates the welding process. Due to
complete heating the laminate has to be loaded with pressure in order to prevent delamination,
which requires the usage of complex and expensive welding tools. Additionally,
the temperature difference between the inductor and the opposite side may
be greater than the processing window, which is determined by the properties of the
matrix polymer.
The induction welding process is influenced by numerous parameters. Due to complexity
process development is mainly based on experimental studies. The investigation
of parameter influences and interactions is cumbersome and the measurement
of quality relevant parameters, especially in the bondline, is difficult. Process simulation
can reduce the effort of parameter studies and contribute to further analysis of
the induction welding process.
The objective of this work is the development of a process variant of induction welding
preventing complete heating of the laminate in thickness direction. For optimal
welding the bondline has to reach the welding temperature whereas the other domains
should remain below the melting temperature of the matrix polymer.
For control of the temperature distribution localized cooling by an impinging jet of
compressed air was implemented. The effect was assessed by static heating experiments
with carbon fiber reinforced polyetheretherketone (CF/PEEK) and polyphenylenesulfide
(CF/PPS).
The application of localized cooling could influence the temperature distribution in
thickness direction of the laminate, according to the specifications of the welding
process. The temperature maximum was shifted from the inductor to the opposite side. This enables heating of the laminate to welding temperature in the bondline and
concurrently preventing melting and effects connected to this on the outer surface.
Inductive heating and the process variant with localized cooling were implemented in
three-dimensional finite-element process models. For that purpose, the finiteelement-
software Comsol Multiphysics 4.1 was used for the development of fully
coupled electromagnetic-thermal models which have been validated experimentally.
A sensitivity analysis for determination of different processing parameters of inductive
heating was conducted. The coil current, field frequency, and heat capacity were
identified as significant parameters. The cooling effect of the impinging jets was estimated
by appropriate convection coefficients.
For transfer of the developed process variant to the continuous induction welding
process, a process model was created. It represents a single overlap joint with continuous
feed. With the help of process modeling a parameter set for welding of
CF/PEEK was determined and used for joining of specimens. In doing so, the desired
temperature field was achieved and melting of the outer layers could be prevented.

Thermoplastic polymer-polymer composites consist of a polymeric matrix and a
polymeric reinforcement. The combination of these materials offers outstanding
mechanical properties at lower weight than standard fiber reinforced materials.
Furthermore, when both polymeric components originate from the same family or,
ideally, from the same polymer, their sustainability degree is higher than standard
fiber reinforced composites.
A challenge of polymer-polymer composites is the subsequent processing of their
semi-finished materials by heating techniques. Since the fibers are made of meltable
thermoplastic, the reinforcing fiber structure might be lost during the heating process.
Hence, the mechanical properties of an overheated polymer-polymer composite
would decline, and finally, they would be even lower than the neat matrix. A decrease
of process temperature to manage the heating challenge is not reasonable since the
cycle time would be increased at the same time. Therefore, this work pursues the
adaption of a fast and selective heating method on the use with polymer-polymer
composites. Inductively activatable particles, so-called susceptors, were distributed in
the matrix to evoke a local heating in the matrix when being exposed to an
alternating magnetic field. In this way, the energy input to the fibers is limited.
The experimental series revealed the induction particle heating effect to be mainly
related to susceptor material, susceptor fraction, susceptor distribution as well as
magnetic field strength, coupling distance, and heating time. A proper heating was
achieved with ferromagnetic particles at a filler content of only 5 wt-% in HDPE as
well as with its respective polymer fiber reinforced composites. The study included
the analysis of susceptor impact on mechanical and thermal matrix properties as well
as a degradation evaluation. The susceptors were identified to have only a marginal
impact on matrix properties. Furthermore, a semi-empiric simulation of the particle
induction heating was applied, which served for the investigation of intrinsic melting
processes.
The achieved results, the experimental as well as the analytic study, were
successfully adapted to a thermoforming process with a polymer-polymer material,
which had been preheated by means of particle induction.

Nanoparticle-Filled Thermoplastics and Thermoplastic Elastomer: Structure-Property Relationships
(2012)

The present work focuses on the structure-property relationships of
particulate-filled thermoplastics and thermoplastic elastomer (TPE). In this work
two thermoplastics and one TPE were used as polymer matrices, i.e. amorphous
bisphenol-A polycarbonate (PC), semi-crystalline isotactic polypropylene (iPP),
and a block copolymer poly(butylene terephthalate)-block-poly(tetramethylene
glycol) TPE(PBT-PTMG). For PC, a selected type of various Aerosil® nano-SiO2
types was used as filler to improve the thermal and mechanical properties by
maintaining the transparency of PC matrix. Different types of SiO2 and TiO2
nanoparticles with different surface polarity were used for iPP. The goal was to
examine the influence of surface polarity and chemical nature of nanoparticles on
the thermal, mechanical and morphological properties of iPP composites. For
TPE(PBT-PTMG), three TiO2 particles were used, i.e. one grade with hydroxyl
groups on the particle surface and the other two grades are surface-modified with
metal and metal oxides, respectively. The influence of primary size and dispersion
quality of TiO2 particles on the properties of TPE(PBT-PTMG)/TiO2 composites
were determined and discussed.
All polymer composites were produced by direct melt blending in a twin-screw
extruder via masterbatch technique. The dispersion of particles was examined by
using scanning electron microscopy (SEM) and micro-computerized tomography
(μCT). The thermal and crystalline properties of polymer composites were characterized by using thermogravimetric analysis (TGA) and differential
scanning calorimetry (DSC). The mechanical and thermomechanical properties
were determined by using mechanical tensile testing, compact tension and
Charpy impact as well as dynamic-mechanical thermal analysis (DMTA).
The SEM results show that the unpolar-surface modified nanoparticles are better
dispersed in polymer matrices as iPP than polar-surface nanoparticles, especially
in case of using Aeroxide® TiO2 nanoparticles. The Aeroxide® TiO2 nanoparticles
with a polar surface due to Ti-OH groups result in a very high degree of
agglomeration in both iPP and TPE matrices because of strong van der Waals
interactions among particles (hydrogen bonding). Compared to unmodified
Aeroxide® TiO2 nanoparticles, the other grades of surface modified TiO2 particles
are very homogenously dispersed in used iPP and TPE(PBT-PTMG). The
incorporation of SiO2 nanoparticles into bisphenol-A PC significantly increases
the mechanical properties of PC/SiO2 nanocomposites, particularly the resistance
against environmental stress crazing (ESC). However, the transparency of
PC/SiO2 nanocomposites decreases with increasing nanoparticle content and
size due to a mismatch of infractive indices of PC and SiO2 particles. The different
surface polarity of nanoparticles in iPP shows evident influence on properties of
iPP composites. Among iPP/SiO2 nanocomposites, the nanocomposite
containing SiO2 nanoparticles with a higher degree of hydrophobicity shows
improved fracture and impact toughness compared to the other iPP/SiO2
composites. The TPE(PBT-PTMG)/TiO2 composites show much better thermal and mechanical properties than neat TPE(PBT-PTMG) due to strong chemical
interactions between polymer matrix and TiO2 particles. In addition, better
dispersion quality of TiO2 particles in used TPE(PBT-PTMG) leads to dramatically
improved mechanical properties of TPE(PBT-PTMG)/TiO2 composites.