Department of Mathematics

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Author: search.filters.author.Kumar, RajeshSubject: search.filters.subject.Aggregation

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Now showing 1 - 4 of 4
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    Comparison of variational iteration and Adomian decomposition methods to solve growth, aggregation and aggregation-breakage equations
    (Elsevier, 2023-03) Kumar, Rajesh
    In this work, semi-analytical approaches such as the Adomian decomposition method (ADM), and variational iteration method (VIM) are examined to solve the aggregation, aggregation-breakage and pure growth equations in series forms. The analytical and truncated series solutions are compared for the number density and various moments. The solutions produced using ADM and VIM are mathematically equal in the pure growth case and provide closed-form solutions for constant growth rate. Additionally, Optimal variational iteration method (OVIM) is implemented to solve the growth and aggregation equations, which reduces the error compared to ADM and VIM to some extent but increases the computational cost. Furthermore, in this work, we provide the ADM and VIM formulations for the coupled aggregation-breakage model. Various test cases of each problem are taken to justify the efficiency and accuracy of the series approximated methods. These observations are shown numerically by comparing the finite term series solutions with the exact solutions of number density and moments.
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    Numerical methods for solving two-dimensional aggregation population balance equations
    (Elsevier, 2011-06) Kumar, Rajesh
    The cell average technique (CAT) and the fixed pivot (FP) method for solving two-dimensional aggregation population balance equations using a rectangular grid were implemented in Kumar et al. (2008). Recently, Chakraborty and Kumar (2007) have studied the FP scheme for the same problem on two different types of triangular grids and found that the method shows better results for number density as compared to the rectangular grids. However, they did not discuss the results for higher moments. Therefore, our first aim in this work is to compare different moments calculated by the FP technique on rectangular and triangular meshes with the analytical moments. Further we introduce a new mathematical formulation of the CAT for the two different types of triangular grids as considered by Chakraborty and Kumar (2007). 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 and numerical solutions. The new formulation shows good agreement with the analytical results for number density and higher moments.
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    A population balance model with simultaneous aggregation and breakage for the synthesis of Titanium dioxide nanoparticles
    (Indianjournals, 2013) Kumar, Rajesh
    In this work we discuss the applications of aggregation and multiple breakage equations in nano-technology. The kinetics of the aggregation and breakage processes during titanium dioxide (TiO2) nano-particle sol-gel synthesis is presented. Nano-particle precipitation in the batch reactor is discussed briefly and the particle size distributions (PSDs) are verified numerically by solving population balance equations (PBEs). The cell average technique (CAT) is used to solve the equations. The experimental data for the PSDs and the simulation results are compared by taking a shear flow aggregation kernel and two different breakage kernels, e.g. Austin and Diemer kernels. The comparisons are good enough to see the novelty of this work..
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    Moment preserving finite volume schemes for solving population balance equations incorporating aggregation, breakage, growth and source terms
    (World Scientific, 2013) Kumar, Rajesh
    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.