A least squares method for the solution of population balance problems

Dorao, Carlos Alberto; Jakobsen, Hugo A.

doi:10.1016/j.compchemeng.2005.10.012

Cited by 71 publications

(48 citation statements)

References 28 publications

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“…A detailed explanation of the application of LSM to the direct PBE was reported by Dorao and Jakobsen. 22 The basic idea in the LSM is to minimize the integral of the square of the residual over the computational domain. For the cases in which the exact solution is sufficiently smooth, the convergence rate is exponential by increasing the polynomial degree of the approximation.…”

Section: Methodsmentioning

confidence: 99%

“…The least squares spectral method will be used to solve the resulting integrodifferential equation. The numerical model will be validated with the experimental data obtained by George Interpolation of the experimental data set at 1640 s -1 reported by George et al 22 et al 19 for capillary flow of short pineapple fiber reinforced low-density polyethylene composites. The main goal of this work is to show the evolution of a density function predicted with different breakage kernels and to compare the modeling results with experimental data.…”

Section: Introductionmentioning

confidence: 99%

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Novel Approach for Modeling the Dynamics of Fiber Breakage in Polymer Matrix Composites during Capillary Extrusion

et al. 2015

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ABSTRACT:In this work, a population balance equation (PBE) model is used to predict the evolution of the fiber breakage in thermoplastic polymer/short-fiber composites during capillary extrusion. The least squares spectral method was used to solve the resulting integrodifferential equation. The differences found between the experimental and simulated data were attributed to the selected breakage and redistribution functions. An experimental setup is proposed to find accurate breakage functions for this problem. The results indicate that the application of the PBE for such a breakage process could be a powerful tool for the design of injection molding molds. C 2015 Wiley Periodicals, Inc. Adv Polym Technol 2015, 0, 21635; View this article online at wileyonlinelibrary.com.

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Section: Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Novel Approach for Modeling the Dynamics of Fiber Breakage in Polymer Matrix Composites during Capillary Extrusion

et al. 2015

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“…The following algorithm achieves this for the model kernel Eq.(3). The presentation follows the method outlined in the supplementary material of [30] and is similar to the work in [31]. It is included here for completeness.…”

Section: B Stationary Casementioning

confidence: 99%

Nonlinear least-squares method for the inverse droplet coagulation problem

2013

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If the rates, K(x, y), at which particles of size x coalesce with particles of size y is known, then the mean-field evolution of the particle-size distribution of an ensemble of irreversibly coalescing particles is described by the Smoluchowski equation. We study the corresponding inverse problem which aims to determine the coalescence rates, K(x, y) from measurements of the particle size distribution. We assume that K(x, y) is a homogeneous function of its arguments, a case which occurs commonly in practice. The problem of determining, K(x, y), a function to two variables, then reduces to a simpler problem of determining a function of a single variable plus two exponents, µ and ν, which characterise the scaling properties of K(x, y). The price of this simplification is that the resulting least squares problem is nonlinear in the exponents µ and ν. We demonstrate the effectiveness of the method on a selection of coalescence problems arising in polymer physics, cloud science and astrophysics. The applications include examples in which the particle size distribution is stationary owing to the presence of sources and sinks of particles and examples in which the particle size distribution is undergoing self-similar relaxation in time.

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“…The method was implemented in the software package Parsival. Recently, Dorao and Jakobsen [4,5] showed the applicability of the least squares method [12,1,23,21] for solving PBEs. The least squares method (LSM) consists in finding the solution which minimizes the integral of the square of the residual over the computational domain.…”

Section: Introductionmentioning

confidence: 99%