Abstract
Objective: In this work, we obtained the analytical and approximate solutions of the population balance equations (PBEs) involving the breakup process in batch and continuous flow by applying the Adomian decomposition method and piecewise continuous basis functions, respectively. Methods: The key to the advanced numerical method is to represent the number distribution function of the dispersed phase through the orthogonal Chebyshev basis polynomials. It is a straightforward and effective method that has the advantage of simultaneously giving the distribution and the different required moments. Therefore, it does not require the construction of the distribution from moments computations obtained by the transformation of the initial problem and the lost information. Results: The performance of this numerical approach is evaluated by solving breakup equation and comparison against analytical solutions obtained from the Adomian decomposition method, which generally allows the analysis of this approach. Conclusion: The numerical results obtained by the present numerical method were compared with the new analytical solutions of the PBE. It was found that both piecewise continuous basis functions and analytical solutions have comparable results.
Highlights
Population balance models of dispersed phases find many scientific and engineering applications including liquid-liquid, liquid-vapor and solid-liquid dispersions, nanoparticle physics, pharmaceutical industries, polymerization and bioreactors [1 - 11].The complex structure of the population balance equations (PBEs) allows for an analytical solution only for some simple breakup kernels [8 - 10]
Population Balance Equation for the Breakup Process in Continuous Flow: The PBE that describes the evolution of the droplet number density, n(v,t) in a continuous well-stirred vessel, with droplet formation and loss terms due to breakup, can be expressed as [1]
The agreement is excellent by the finite element method with expansion coefficients based on Chebyshev polynomials and the analytical solution
Summary
The key to the advanced numerical method is to represent the number distribution function of the dispersed phase through the orthogonal Chebyshev basis polynomials. It is a straightforward and effective method that has the advantage of simultaneously giving the distribution and the different required moments. It does not require the construction of the distribution from moments computations obtained by the transformation of the initial problem and the lost information
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