Numerical solution of the bivariate population balance equation for the interacting hydrodynamics and mass transfer in liquid-liquid extraction columns

A comprehensive model for predicting the interacting hydrodynamics and mass transfer is formulated on the basis of a spatially distributed population balance equation in terms of the bivariate number density function with respect to droplet diameter and solute concentration. The two macro- (droplet breakage and coalescence) and micro- (interphase mass transfer) droplet phenomena are allowed to interact through the dispersion interfacial tension. The resulting model equations are composed of a system of partial and algebraic equations that are dominated by convection, and hence it calls for a specialized discretization approach. The model equations are applied to a laboratory segment of an RDC column using an experimentally validated droplet transport and interaction functions. Aside from the model spatial discretization, two methods for the discretization of the droplet diameter are extended to include the droplet solute concentration. These methods are the generalized fixed-pivot technique (GFP) and the quadrature method of moments (QMOM). The numerical results obtained from the two extended methods are almost identical, and the CPU time of both methods is found acceptable so that the two methods are being extended to simulate a full-scale liquid-liquid extraction column.     

AGGLOMERATION, BREAKAGE, POPULATION BALANCE, AND CRYSTALLIZATION KINETICS OF REACTIVE PRECIPITATION PROCESS

The agglomeration and breakage of particles play a significant role in determining final particle size distribution (PSD) and other qualities such as filtering characteristics and impurity content. In reactive precipitation processes, especially during the precipitation of fine particles, the agglomeration and breakage of particles normally cannot be neglected. In this study, the agglomeration and breakage of particles during the reactive precipitation process of procaine penicillin has been investigated experimentally through a continuous steady MSMPR crystallizer. Based on the population balance theory, a crystallization kinetics model including agglomeration and breakage is established, in which the breakage of particles is expressed by a two-body equal-volume birth function and a two-body power-law death function. The crystallization kinetics model is shown to be more suitable than size-dependent growth models as ASL and MJ2.     

Agglomeration and breakage of nanoparticles in stirred media mills—a comparison of different methods and models

The increasing industrial demand for nanoparticles challenges the application of stirred media mills to grind in the sub-micron size range. It was shown recently [Mende et al., 2003. Mechanical production and stabilization of submicron particles in stirred media mills. Powder Technology 132, 64-73] that the grinding behavior of particles in the sub-micron size range in stirred media mills and the minimum achievable particle size is strongly influenced by the suspension stability and thus the agglomeration behavior of the suspension. Therefore, an appropriate modeling of the process must include a superposition of the two opposing processes in the mill i.e., breakage and agglomeration which can be done by means of population balance models. Modeling must now include the influence of colloidal surface forces and hydrodynamic forces on particle aggregation and breakup. The superposition of the population balance models for agglomeration and grinding with the appropriate kernels leads to a system of partial differential equations, which can be solved in various ways numerically. Here a modified h-p Galerkin algorithm which is implemented in the commercially available software package PARSIVAL developed by CiT (CiT GmbH, Rastede, Germany) and the moment methodology according to [Diemer and Olsen, 2002a. A moment methodology for coagulation and breakage problems: Part I—analytical solution of the steady-state population balance. Chemical Engineering Science 57 (12), 2193-2209; Diemer and Olsen, 2002b. A moment methodology for coagulation and breakage problems: Part II—moment models and distribution reconstruction. Chemical Engineering Science 57 (12), 2211-2288] are used and compared to explicit data on alumina. This includes a comparison of the derived particle size distributions, moments and its accuracy depending on the starting particle size distribution and the used agglomeration and breakage kernels. Finally, the computational effort of both methods in comparison to the prior mentioned parameters is evaluated in terms of practical application.     

An extended cell-average technique for a multi-dimensional population balance of granulation describing aggregation and breakage

This paper focuses on obtaining the numerical solution to a three-dimensional population balance model (PBM) of granulation using the cell-average technique first proposed by [22]. Conventionally, linear grids are used for the solution of PBMs, but the ability to incorporate non-linear grids would be more advantageous given that a larger size range can be covered using fewer number of grids, thus reducing computational overhead. Furthermore, the use of linear representation of grids in PBMs to represent industrial granulation processes that span a wide granule size range is computationally prohibitive and results show that a non-linear grid representation is computationally more efficient with comparable accuracy. Parallelization of the PBM via a multi-core strategy has also been incorporated in order to reduce the simulation time of the model. Incorporating the cell average technique along with parallelization of the overall model lends credence to the overall use of the model for effective granulation process design and analysis.     

Is the adjustment of the impeller speed a reliable attempt to influence granule size in continuous dry granulation?

As the field of continuous manufacturing of solid pharmaceutics is developing, the interest in implementing continuous granulation methods is growing. Process analytical technology tools should be integrated to ensure the monitoring of the product quality and therefore enforce control strategies.Three single materials which are often used in dry granulation and additionally two formulations, one containing ibuprofen and the other acetaminophen were processed at various process parameters. They all differed in their compaction and fracture behavior. A statistical analysis of the influence of process parameters was executed, to work out which parameters could be used for a granule size control approach in continuous dry granulation. Thereby, the specific compaction force and the impeller speed were found to be significant factors in each design of experiment. However, the impeller speed was evaluated as the only suitable parameter to control granule size, as an impact on granule density is unlikely. Nevertheless, some restrictions such as an upper impeller speed limitation to avoid excessive fines and a lower speed limitation to impede a downturn of the throughput, have to be considered. Furthermore, a decreasing median granule size was observed at higher throughputs for plastically deforming materials and formulations.     

Breakage behavior of two different coal types in a grinding process and importance for thermal power stations

This study investigates the effect of changes in coal properties in thermal power plants (TPPs) due to the breakage behavior of coal in the grinding stage. A grinding process is carried out in a stirred mill for different periods of time to determine the specific breakage rate. Two different types of coal with different properties and similar calorific values are used to compare their breakage rates. The results of the stirred mill grinding are presented in a comparative manner. Furthermore, washing tests are performed on the coal to determine the solid fossil fuel structure. It is shown that the solid fossil fuel structure affects linear and non-linear movement that results in different breakage behaviors.     

Influence of mechanical properties on impact fracture: Prediction of the milling behaviour of pharmaceutical powders by nanoindentation

The impact grinding behaviour of materials can be characterized by the two breakage parameters f_Mat and xW_m,min [Vogel and Peukert, Powder Technol. 129 (2003) 101-110]. These parameters are usually determined by single particle milling tests. The parameters are useful for predicting the selection function and the breakage function and thus enable modelling of impact milling processes. So far, no detailed correlations have been established between the breakage parameters f_Mat and xW_m,min and intrinsic material properties. In this work, we study the correlation between the breakage parameters of pharmaceutical powders and their mechanical properties (hardness, Young's modulus and fracture toughness) that are determined from indentation experiments. It will be shown that f_Mat and xW_m,min can be expressed in terms of the brittleness index (defined as the ratio of hardness to fracture toughness H/K_c). This correlation allows the prediction of the breakage probability of a material by using only a small number of crystals.     

Effect of debris on the silicon wafering for solar cells

The wafers used by the photovoltaic industry are mostly produced by multi-wire slurry sawing. One of the key factors determining the wafer quality (presence of saw marks and chips, increased roughness, wafer thickness variations and wafer strength) is the abrasive slurry. For cost reduction, the slurry is regularly exchanged and the debris it contains is removed in a recycling operation. To optimise the slurry usage, it is of utmost importance to understand the effects of the silicon debris concentration in the slurry. This was studied by sawing several bricks one after the other with the same slurry. It was found that when the amount of debris is too high (more than 4% of the slurry volume), saw marks appear on the wafers and they become more fragile. Finally, a first qualitative model explaining the apparition of the saw marks and the reduction of wafer strength is proposed.     

A population balance model for the chocolate roller refining process

A study was performed to analyze the evolution of particle size distribution in the chocolate roller refining process. A mathematical model based on the continuity and population balance equations was developed to interpret industrial data. The industrial data were provided for chocolate pastes with powder/crystalline sugar particles at low/high shear rates. The parameters of the breakage frequency and fragment distribution function were estimated using industrial data. After parameter fitting, the model shows good agreement with the experimental results for varying conditions with a single consistently chosen set of parameters. This provides confidence that the general model structure is suitable for process evaluation. The population balance model was used to analyze the influence of changing some process variables on the efficiency of grinding. The results show that there might exist optimum values of the roll’s diameter and rotational velocity for a specific configuration of the equipment.     

Dispersion of oil droplets in rotor–stator mixers: Experimental investigations and modeling

Forming emulsions by mixing of immiscible liquids, water and silicone oils was carried out by applying the in-line high-shear rotor–stator mixers. In experimental part investigations were carried out using experimental rig consisting of two tanks and an in-line Silverson rotor-stator (150/250) MS; the system was operated in a multiple pass (MP) mode, which can be compared with a single pass (SP) mode experiments. Emulsification of 1 wt.% silicone oils (Dow Corning 200 fluid) with viscosities of 9.4, 48 and 339 mPa s was investigated. Emulsions were stabilized by adding 0.5 wt.% of sodium laureth sulphate. Effects of rotor speed, number of passes and the drop viscosity on the drop size were investigated.Numerical simulations were carried out using the k–ɛ model of turbulence and the multiple reference frame method (MRF) linked to the population balance equation. The population balance was expressed and solved using the quadrature method of moments (QMOM). The breakage kernel for drops whose diameter falls within the inertial subrange of turbulence was defined based on the multifractal model of intermittent turbulence.