Solving population balance equations for two-component aggregation by a finite volume scheme

2007 ◽  
Vol 62 (3) ◽  
pp. 679-693 ◽  
Author(s):  
Shamsul Qamar ◽  
Gerald Warnecke
Keyword(s):  
Finite Volume ◽  
Population Balance ◽  
Finite Volume Scheme ◽  
Balance Equations ◽  
Population Balance Equations ◽  
Volume Scheme ◽  
Two Component

MOMENT PRESERVING FINITE VOLUME SCHEMES FOR SOLVING POPULATION BALANCE EQUATIONS INCORPORATING AGGREGATION, BREAKAGE, GROWTH AND SOURCE TERMS

2013 ◽  
Vol 23 (07) ◽  
pp. 1235-1273 ◽  
Author(s):  
RAJESH KUMAR ◽  
JITENDRA KUMAR ◽  
GERALD WARNECKE
Keyword(s):  
Finite Volume ◽  
Population Balance ◽  
Coupled Processes ◽  
Source Terms ◽  
Balance Equations ◽  
Finite Volume Schemes ◽  
Average Technique ◽  
Population Balance Equations ◽  
Moment Preservation ◽  
First Moment

In this work we present some moment preserving finite volume schemes (FVS) for solving population balance equations. We are considering unified numerical methods to simultaneous aggregation, breakage, growth and source terms, e.g. for nucleation. The criteria for the preservation of different moments are given. The property of conservation is a special case of preservation. First we present a FVS which shows the preservation with respect to one-moment depending upon the processes under consideration. In case of the aggregation and breakage it satisfies first-moment preservation whereas for the growth and nucleation we observe zeroth-moment preservation. This is due to the well-known property of conservativity of FVS. However, coupling of all the processes shows no preservation for any moments. To overcome this issue, we reformulate the cell average technique into a conservative formulation which is coupled together with a modified upwind scheme to give moment preservation with respect to the first two-moments for all four processes under consideration. This allows for easy coupling of these processes. The preservation is proven mathematically and verified numerically. The numerical results for the first two-moments are verified for various coupled processes where analytical solutions are available.

scholarly journals Characterization of Simultaneous Evolution of Size and Composition Distributions Using Generalized Aggregation Population Balance Equation

Pharmaceutics ◽  
2020 ◽  
Vol 12 (12) ◽  
pp. 1152
Author(s):  
Mehakpreet Singh ◽  
Ashish Kumar ◽  
Saeed Shirazian ◽  
Vivek Ranade ◽  
Gavin Walker
Keyword(s):  
Finite Volume ◽  
Balance Equation ◽  
Numerical Scheme ◽  
Population Balance ◽  
Population Balance Equation ◽  
Finite Volume Scheme ◽  
Average Technique ◽  
Volume Scheme ◽  
Density Prediction ◽  
Cell Average

The application of multi-dimensional population balance equations (PBEs) for the simulation of granulation processes is recommended due to the multi-component system. Irrespective of the application area, numerical scheme selection for solving multi-dimensional PBEs is driven by the accuracy in (size) number density prediction alone. However, mixing the components, i.e., the particles (excipients and API) and the binding liquid, plays a crucial role in predicting the granule compositional distribution during the pharmaceutical granulation. A numerical scheme should, therefore, be able to predict this accurately. Here, we compare the cell average technique (CAT) and finite volume scheme (FVS) in terms of their accuracy and applicability in predicting the mixing state. To quantify the degree of mixing in the system, the sum-square χ2 parameter is studied to observe the deviation in the amount binder from its average. It has been illustrated that the accurate prediction of integral moments computed by the FVS leads to an inaccurate prediction of the χ2 parameter for a bicomponent population balance equation. Moreover, the cell average technique (CAT) predicts the moments with moderate accuracy; however, it computes the mixing of components χ2 parameter with higher precision than the finite volume scheme. The numerical testing is performed for some benchmarking kernels corresponding to which the analytical solutions are available in the literature. It will be also shown that both numerical methods equally well predict the average size of the particles formed in the system; however, the finite volume scheme takes less time to compute these results.

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