Department of Mathematics

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Author: search.filters.author.Kumar, RajeshSubject: search.filters.subject.Population balance

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    Disintegration process of surface stabilized sol-gel TiO2 nanoparticles by population balances
    (Elsevier, 2009-12) Kumar, Rajesh
    Titanium oxide () is one of the most useful oxide materials, because of its widespread applications in photocatalysis, solar energy conversion, sensors and optoelectronics. The control of particle size and monodispersity of nanoparticles is a challenging task in processing. The control and prediction of these dynamics are based on the process conditions and the nature of chemicals. In this work, we investigate the effect on the surface stabilization with different surfactants and temperature. The steric stabilization of the polymer and various functional groups of dispersants are also considered. Narrow distributed spherical titania particles in the size range 10–100 nm are produced in a sol–gel synthesis from titanium tetra-isopropoxide. The influence of various precursor concentrations and different surfactants on the particle size distribution is investigated. The population balance model for disintegration leads to a system of integro-partial differential equations which is numerically solved by the cell average technique. The experimental results are also compared with the simulation using two different disintegration kernels.
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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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    Numerical simulation and convergence analysis of a finite volume scheme for solving general breakage population balance equations
    (Elsevier, 2013-01) Kumar, Rajesh
    This paper presents a finite volume scheme (FVS) for solving general breakage population balance equations (BPBEs). In particular, the number density based BPBE is transformed to the form of a mass conservation law. Then it becomes easy to apply the well known FVSs that have an important property of mass conservation. Following Kumar and Warnecke for fixed pivot (FP) method [16] and cell average technique (CAT) [15], the stability and the convergence analysis of the semi-discretized FVS are studied. Unlike the CAT and the FP method, the FVS is second order consistent independently of the type of meshes. We also observe that FVS gives second order convergence rate on four different types of uniform and non-uniform meshes with non-decreasing behavior of mesh width. Nevertheless, one order better accuracy than the FP method is achieved on locally uniform meshes. It is also noticed that on non-uniform random meshes the FVS shows one order and two orders higher accuracy than the CAT and the FP method, respectively. The mathematical results of convergence analysis are validated numerically by taking three test problems. The numerical simulations are also compared with the results obtained by the CAT and the FP method
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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.