Journal of Applied and Computational Mechanics
Analysis of Particulate Systems with Numerical Inverse Laplace Transform
Document Type : Research Paper
Authors
- Clauderino da Silva Batista 1
- Helder Kiyoshi Miyagawa 1
- Emanuel Negrão Macêdo 1
- João Nazareno Nonato Quaresma 1
- Helcio Rangel Barreto Orlande 2
1 School of Chemical Engineering, Federal University of Pará, UFPA, Campus do Guamá, Rua Augusto Corrêa, 01, Belém, PA, 66075-110, Brazil
2 Laboratory of Transmission and Technology of Heat, Mechanical Engineering Department, Federal University of Rio de Janeiro, Rio de Janeiro, RJ, 21945-970, Brazil
Abstract
The technique of numerical inversion of the Laplace transform is applied to solve the population balance equation (PBE). The model considers the dispersed phase systems in which nucleation and heterogeneous condensation are present. The studied phenomena model corresponds to a nonlinear integro-partial-differential equation. Test cases are solved considering two different collision mechanisms, the first-order removal mechanism and the effect of simultaneous coagulation and growth. Numerical results are compared with the analytical solution and with the literature. Based on these results, the technique applied in this work demonstrates to be a tool to solve problems in particulate systems, particularly for aerosol modeling where coagulation is the most important inter-particle mechanism affecting the size distribution.
Keywords
Main Subjects
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[1] Amini, H., He, X., Tseng, Y.-C., Kucuk, G., Schwabe, R., Schultz, L., Maus, M., Schroder, D., Rajniak, P., Bilgili, E., A semi-theoretical model for simulating the temporal evolution of moisturetemperature during industrial fluidized bed granulation, European Journal of Pharmaceutics and Biopharmaceutics, 151, 2020, 137–52.
[2] Kaur G., Singh M., Kumar J., De Beer T., Nopens I., Mathematical modelling and simulation of a spray fluidized bed granulator, Processes, 6, 2018, 195.
[3] Gong, H., Yang, Y., Yu, B., Luo, X., Peng, Y., Jiang, Y., Coalescence characteristics of droplets dispersed in oil subjected to electric and centrifugal fields, Colloids and Surfaces A: Physicochemical and Engineering Aspects, 648, 2022, 129398.
[4] Costa, C.B.B., Maciel, M.R.W., Filho, R.M., Considerations on the crystallization modeling: population balance solution, Computers & Chemical Engineering, 31, 2007, 206–18.
[5] Farias, L.F., de Souza, J.A., Braatz, R.D., da Rosa, C.A., Coupling of the population balance equation into a two-phase model for the simulation of combined cooling and antisolvent crystallization using openfoam Comput, Chemical Engineering, 123, 2019, 246–56.
[6] Vasquez, C.C.R., Lebaz, N., Ramière, I., Lalleman, S., Mangin, D., Bertrand, M., Fixed point convergence and acceleration for steady state population balance modelling of precipitation processes: Application to neodymium oxalate, Chemical Engineering Research and Design, 177, 2022, 767–777.
[7] Maluta, F., Buffo, A., Marchisio, D., Montante, G., Paglianti, A., Vanni, M., Numerical and Experimental Analysis of the Daughter Distribution in Liquid-Liquid Stirred Tank, Chemical Engineering & Technology, 44(11), 2021, 1994–2001.
[8] Rajamani, K., Pate, W.T., Kinneberg, D.J., Time-driven and event-driven Monte Carlo simulations of liquid–liquid dispersions: a comparison, Industrial & Engineering Chemistry Fundamentals, 25, 1986, 746.
[9] Kiparissides, C., Challenges in particulate polymerization reactor modeling and optimization: a population balance perspective, Journal of Process Control, 16, 2006, 205–24.
[10] Kirse, C., Briesen, H., Numerical solution of mixed continuous-discrete population balance models for depolymerization of branched polymers, Computers & Chemical Engineering, 73, 2015, 154–71.
[11] Kotoulas, C., Kiparissides, C., A generalized population balance model for the prediction of particle size distribution in suspension polymerization reactors, Chemical Engineering Science, 61, 2006, 332–46.
[12] Barrasso, D., Eppinger, T., Pereira, F.E., Aglave, R., Debus, K., Bermingham, S.K., Ramachandran, R., A multi-scale, mechanistic model of a wet granulation process using a novel bi-directional PBM-DEM coupling algorithm, Chemical Engineering Science, 123, 2015, 500–13.
[13] Barrasso, D., Walia, S., Ramachandran, R., Multi-component population balance modeling of continuous granulation processes: a parametric study and comparison with experimental trends, Powder Technology, 241, 2013, 85–97.
[14] Kostoglou, M., Extended cell average technique for the solution of coagulation equation, Journal of Colloid and Interface Science, 306, 2007, 72–81.
[15] Tandon, P., Rosner, D.E., Monte Carlo simulation of particle aggregation and simultaneous restructuring, Journal of Colloid and Interface Science, 213, 1999, 273–86.
[16] Ramabhadran, T.E., Peterson, T.W., Seinfeld, J.H., Dynamics of Aerosol Coagulation and Condensation, AIChE Journal, 22, 1976, 840-851.
[17] Gerstlauer, A., Gahn, C., Zhou, H., Rauls, M., Schreiber, M., Application of population balances in the chemical industry - current status and future needs, Chemical Engineering Science, 61, 2006, 205–217.
[18] Litster, J.D., Smit, D.J., Hounslow, M.J., Adjustable Discretized Population Balance for Growth and Aggregation, AIChE Journal, 41, 1995, 591-603.
[19] Yua, M., Lin, J., Cao, J., Seipenbuschc, M., An analytical solution for the population balance equation using a moment method, Particuology, 18, 2015, 194–200.
[20] Bertin, D., Cotabarren, I., Piña, J., Bucalá, V., Population balance discretization for growth, attrition, aggregation, breakage and nucleation, Computers and Chemical Engineering, 84, 2016, 132–150.
[21] Marchisio, D.L., Pikturna, J.T., Fox, R.O., Vigil, R.D., Barresi, A.A., Quadrature Method of Moments for Population-Balance Equations, AIChE Journal, 49, 2003, 1266-1276.
[22] Omar, H.M., Rohani, S., Crystal Population Balance Formulation and Solution Methods: A Review, Crystal Growth & Design, 17, 2017, 4028−4041.
[23] Handwerk, D.R., Particle Size Distributions via Mechanism-Enabled Population Balance Modeling, The Journal of Physical Chemistry C, 124(8), 2020, 4852-4880.
[24] Ghadipasha, N., Romagnoli, J.A., Tronci, S., Baratti, R., On-line control of crystal properties in nonisothermal antisolvent crystallization, AIChE Journal, 61, 2015, 2188–2201.
[25] Wu, B., Li, J., Li, C., He, J., Luo, P., Antisolvent crystallization intensified by a jet crystallizer and a method for investigating crystallization kinetics, Chemical Engineering Science, 211, 2020, 115259.
[26] Frederix, E.M.A., Stanic, M., Kuczaj, A.K., Nordlund, Geurts, M.B.J., Characteristics-based sectional modeling of aerosol nucleation and condensation, Journal of Computational Physics, 326, 2016, 499–515.
[27] Nguyena, T.T., Laurenta, F., Foxa, R.O., Massot, M., Solution of population balance equations in applications with fine particles, Mathematical Modeling and Numerical Schemes, 325, 2016, 129-156.
[28] Peterson, T.W., Gelbard, F., Seinfeld, J.H., Dynamics of Source-Reinforced, Coagulating, and Condensing Aerosols, Journal of Colloid and Interface Science, 63, 1978, 426-445.
[29] Gelbard, F., Seinfeld, J.H., Numerical Solution of the Dynamic Equation for Particulate Systems, Journal of Computational Physics, 28, 1978, 357-375.
[30] Drake, R.L., Topics in Current Aerosol Research, Hidy, G. and Brock, J., Eds., Pergamon, Oxford, 1972.
[31] Seinfeld, J.H., Pandis, S.N., Atmospheric chemistry and physics: from air pollution to climate change, Wiley-Interscience, New York, 2006.
[32] Abramowitz, M., Stegun, I.A., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover Publications, New York, 1972.
[33] Hussanan, A., Khan, I., Shafie, S., An Exact Analysis of Heat and Mass Transfer Past a Vertical Plate with Newtonian Heating, Journal of Applied Mathematics, 2013, 2013, 1-9
[34] Abate, J., Valkó, P.P., Multi-precision Laplace transform inversion, International Journal for Numerical Methods in Engineering, 60(5), 2004, 979-993
[35] Krylov, V.I., Skoblya, N.S., Handbook of Numerical Inversion of Laplace Transforms, Israel Program for Scientific Translation, Jerusalem, 1969.
[36] Pilatti, C., Rodriguez, B.D., do, A., Prolo Filho, J.F., Performance Analysis of Stehfest and Power Series Expansion Methods for Solution to Diffusive and Advective Transport Problems, Defect and Diffusion Forum, 396, 2019, 99-108.
[37] IMSL Library, Math/Lib, Houston, TX, 1991.
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