Abstract
Homotopy techniques in nonlinear problems are getting increasingly popular in engineering practice. The main reason is because the homotopy method deforms continuously a difficult problem under study into a simple problem, which then can be easy to solve. This study explores several homotopy approaches to obtain semi- or approximate analytical solutions for various cases involving mechanistic phenomena such as aggregation and breakage. The well-established approximate analytical methods namely, the Homotopy Perturbation Method (HPM), the Homotopy Analysis Method (HAM), and the more recent forms of homotopy approaches such as the Optimal Homotopy Asymptotic Method (OHAM) and the Homotopy Analysis Transform Method (HATM) have been used to solve using a general mathematical framework based on population balances. In this study, several test cases have been discussed such as conditions in which the aggregation kernel is not only constant, but also sum or product dependent. Furthermore cases involving pure breakage, pure aggregation and a combined aggregation-breakage have been studied to understand the sensitivity of these homotopy-based methods in solving PBM. In all these cases, the solutions have been analytically studied and compared with literature. Using symbolic computation and carefully chosen perturbation parameters, the approximate analytical solutions are compared with each other and with the available analytical solution. A convergence analysis of the solution methods is made in comparison to the available solution. The case studies indicate that OHAM performs slightly better than both HATM and HPM in solving nonlinear equations such as the PBEs.
Acknowledgements
It is important to note that certain parts of the study were presented in the 6th International Conference on Advanced Computational Methods in Engineering, ACOMEN 23-28th June 2014, Gent, Belgium. Zehra Pınar wishes to express her sincere thanks to the Scientific and Technological Research Council of Turkey (TUBITAK) for a scholarship to attend ACOMEN.
List of symbols
aggregation function, volume-based
number density function, in terms of particle volume
- S(v)
breakage frequency, volume based
- v
particle volume
boundary of domain ψ
total number fraction at time t
- Ψ
problem domain
stoichiometric kernel for breakage distribution function, volume-based
embedding parameter of homotopy-based methods
nucleation rate
auxiliary function of OHAM
expansion residual
initial approximation
growth rate
- Rv
maximum particle size
References
Dutta, A., D. Constales, R. Van Keer, and G. Heynderickx. 2013. “Implementation of Homotopy Perturbation Method to Solve a Population Balance Model in Fluidized Bed.” International Journal of Chemical Reactor Engineering 11: 1–12.10.1515/ijcre-2012-0047Search in Google Scholar
Hasseine, A., and H.-J. Bart. 2015. “Adomian Decomposition Method Solution of Population Balance Equations for Aggregation, Nucleation, Growth and Breakup Processes.” Applied Mathematical Modelling 39: 1975–1984.10.1016/j.apm.2014.09.027Search in Google Scholar
Hasseine, A., S. Senouci, M. Attarakih, and H.-J. Bart. 2015. “Two Analytical Approaches for Solution of Population Balance Equations: Particle Breakage Process.” Chemical Enginering & Technology 38: 1574–1584.10.1002/ceat.201400769Search in Google Scholar
He, J.-H. 1999. “Homotopy Perturbation Technique.” Computer Methods in Applied Mechanics and Engineering 178: 257–262.10.1016/S0045-7825(99)00018-3Search in Google Scholar
Kumar, S., J. Singh, D. Kumar, and S. Kapoor. 2014. “New Homotopy Analysis Transform Algorithm to Solve Volterra Integral Equation.” Ain Shams Engineering Journal 5: 243–246.10.1016/j.asej.2013.07.004Search in Google Scholar
Liao, S.J. 2003. Beyond Perturbation: Introduction to the Homotopy Analysis Method. Boca Raton: Chapman & Hall/CRC Press.10.1201/9780203491164Search in Google Scholar
Marinca, V., and N. Herişanu. 2008. “Application of Optimal Homotopy Asymptotic Method for Solving Nonlinear Equations Arising in Heat Transfer.” International Communications in Heat and Mass Transfer 35: 710–715.10.1016/j.icheatmasstransfer.2008.02.010Search in Google Scholar
McCoy, B.J., and G. Madras. 2003. “Analytical Solution for a Population Balance Equation with Aggregation and Fragmentation.” Chemical Engineering Science 58: 3049–3051.10.1016/S0009-2509(03)00159-3Search in Google Scholar
Melzak, Z. A. 1953. “The Effect of Coalescence in Certain Collision Processes, Quart.” Applications Mathematical 11: 231–234.Search in Google Scholar
Patil, D.P., and J.R.G. Andrews. 1998. “An Analytical Solution to Continuous Population Balance Model Describing Floc Coalescence and Breakage – A Special Case.” Chemical Engineering Science 53: 599–601.10.1016/S0009-2509(97)00314-XSearch in Google Scholar
Pınar, Z, A. Dutta, G. Bény, and T. Özis. 2015b. “Analytical Solution of Population Balance Equation Involving Growth, Nucleation and Aggregation in Terms of Auxiliary Equation Method.” Applied Mathematics & Information Sciences 9: 2467–2475.Search in Google Scholar
Pınar, Z., A. Dutta, G. Bény, and T. Öziş. 2015a. “Analytical Solution of Population Balance Equation Involving Aggregation and Breakage in Terms of Auxiliary Equation Method.” Pramana – Journal Physical 84: 9–21.10.1007/s12043-014-0838-ySearch in Google Scholar
Qamar, S., and G. Warnecke. 2007. “Solving Population Balance Equations for Two-Component Aggregation by a Finite Volume Scheme.” Chemical Engineering Science 62: 679–693.10.1016/j.ces.2006.10.001Search in Google Scholar
Ramkrishna, D. 2000. Population Balances: Theory and Applications to Particulate Systems in Engineering. San Diego, CA: Academic Press.Search in Google Scholar
Ríos-Morales, D., C. O. Castillo-Araiza, and M. G. Vizcarra-Mendoza. 2014. “Study of the Agglomeration Mechanism of a Natural Organic Solid in a Bench-Scale Wet Fluidized Bed Using Statistical Analysis and Discretized Population Balance.” Chemical Engineering Communications 201: 23–40.10.1080/00986445.2012.759562Search in Google Scholar
Scott, W. T. 1968. “Analytic Studies of Cloud Droplet Coalescence.” Journal of the Atmospheric Sciences 25: 54–65.10.1175/1520-0469(1968)025<0054:ASOCDC>2.0.CO;2Search in Google Scholar
Yu, M., J. Lin, J. Cao, and M. Seipenbusch. 2015. “An Analytical Solution for the Population Balance Equation Using a Moment Method.” Particuology 18: 194–200.10.1016/j.partic.2014.06.006Search in Google Scholar
Ziff, R.M., and E.D. McGrady. 1985. “The Kinetics of Cluster Fragmentation and Depolymerization.” Journal of Physics A: Mathematical and General 18: 3027–3037.10.1088/0305-4470/18/15/026Search in Google Scholar
© 2018 Walter de Gruyter GmbH, Berlin/Boston
