Application of Gaussian cubature to model two-dimensional population balances

Open access

Abstract

In many systems of engineering interest the moment transformation of population balance is applied. One of the methods to solve the transformed population balance equations is the quadrature method of moments. It is based on the approximation of the density function in the source term by the Gaussian quadrature so that it preserves the moments of the original distribution. In this work we propose another method to be applied to the multivariate population problem in chemical engineering, namely a Gaussian cubature (GC) technique that applies linear programming for the approximation of the multivariate distribution. Examples of the application of the Gaussian cubature (GC) are presented for four processes typical for chemical engineering applications. The first and second ones are devoted to crystallization modeling with direction-dependent two-dimensional and three-dimensional growth rates, the third one represents drop dispersion accompanied by mass transfer in liquid-liquid dispersions and finally the fourth case regards the aggregation and sintering of particle populations.

Bałdyga J., Bourne J.R., 1993. Drop breakup and intermittent turbulence. J. Chem. Eng. Jpn., 26, 738-741. DOI: 10.1252/jcej.26.738.

Bałdyga J., Bourne J.R., 1995. Interpretation of turbulent mixing using fractals and multifractals. Chem. Eng. Sci., 50, 381-400. DOI: 10.1016/0009-2509(94)00217-F.

Bałdyga J., Podgórska W., 1998. Drop break-up in intermittent turbulence: maximum stable and transient sizes of drops. Can. J. Chem. Eng., 76, 456-470. DOI: 10.1002/cjce.5450760316.

Batchelor G.K., 1980. Mass transfer from small particles suspended in turbulent fluid. J. Fluid Mech., 98, 609-623. DOI: 10.1017/S0022112080000304.

Borchert C., Nere N., Ramkrishna D., Voigt A., Sundmacher K., 2009. On the prediction of crystal shape distributions in a steady-state continuous crystallizer. Chem. Eng. Sci., 64, 686-696. DOI: 10.1016/j.ces.2008.05.009.

Brändström A., 1966. On the existence of acid salts of monocarboxylic acids in water solutions. Acta Chem. Scand., 20, 1335-1343. DOI: 10.3891/acta.chem.scand.20-1335.

DeVuyst E.A., Preckel P.V, 2007. Gaussian cubature: A practitioner’s guide. Math. Comput. Modell., 45, 787–794. DOI: 10.1016/j.mcm.2006.07.021.

Golub G.H., Welsch J.H., 1969. Calculation of Gauss quadrature rules. Math. Comput. 23, 221-230. DOI: 10.1090/S0025-5718-69-99647-1.

Gordon R.G., 1968. Error bounds in equilibrium statistical mechanics. J. Math. Phys. 9, 655-663. DOI: 10.1063/1.1664624.

Gunawan R., Fusman I., Braatz R.D., 2004. High resolution algorithms for multidimensional population balance equations. AIChE J., 50, 2738-2749. DOI: 10.1002/aic.10228.

Hill P.J., Ng, K.M., 1995. New discretization procedure for the breakage equation. AIChE J., 41, 1204-1216. DOI: 10.1002/aic.690410516.

Hulburt H.M., Katz S., 1964. Some problems in particle technology. A statistical mechanical formulation. Chem. Eng. Sci., 19, 555-574. DOI: 10.1016/0009-2509(64)85047-8.

Marchisio D.L., Fox R.O., 2005. Solution of population balance equations using direct quadrature method of moments. J. Aerosol Sci., 36, 43-73. DOI: 10.1016/j.jaerosci.2004.07.009.

McGraw R., 1997. Description of aerosol dynamics by the quadrature method of moments. Aerosol Sci. Technol., 27, 255-265. DOI: 10.1080/02786829708965471.

Okamoto Y., Nishikawa M., Hashimoto K., 1981. Energy dissipation rate distribution in mixing vessels and its effects on liquid-liquid dispersion and solid-liquid mass transfer. International Chemical Engineering, 21, 88-94.

Sack R.A., Donovan A.F., 1972. An algorithm for Gaussian quadrature given modified moments. Numer. Math. 18, 465-478. DOI: 10.1007/BF01406683.

Sen M., Chaudhury A., Singh R., Ramachandran R., 2014. Two-dimensional population balance model development and validation of a pharmaceutical crystallization process. Am. J. Mod. Chem. Eng., 1, 13-29.

Silva J.E., Paiva A.P., Soares D., Labrincha A., Castro F., 2007. Crystallization from solution, In Trends in Hazardous Materials Research. Nova Science Publishers, New York.

Sorgato, I., 1983. Statistical approach to kinetics. Università Degli Studi di Padova-Istituto di Impianti Chimici, Antoniana S.p.A, Padova, Italy.

Wright D.L., McGraw R., Rosner D.E., 2001. Bivariate extension of the quadrature method of moments for modeling simultaneous coagulation and sintering of particle populations. J. Colloid Interface Sci., 236, 242-251. DOI: 10.1006/jcis.2000.7409.

Chemical and Process Engineering

The Journal of Committee of Chemical and Process of Polish Academy of Sciences

Journal Information


IMPACT FACTOR 2016: 0.971

CiteScore 2016: 1.03

SCImago Journal Rank (SJR) 2016: 0.395
Source Normalized Impact per Paper (SNIP) 2016: 0.873

Cited By

Metrics

All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 199 195 7
PDF Downloads 97 95 6