Hostname: page-component-7c8c6479df-5xszh Total loading time: 0 Render date: 2024-03-28T11:29:45.229Z Has data issue: false hasContentIssue false

Population balance equation for turbulent polydispersed inertial droplets and particles

Published online by Cambridge University Press:  17 October 2017

F. Salehi*
Affiliation:
School of Aerospace, Mechanical and Mechatronic Engineering, The University of Sydney, Sydney 2006, NSW, Australia
M. J. Cleary
Affiliation:
School of Aerospace, Mechanical and Mechatronic Engineering, The University of Sydney, Sydney 2006, NSW, Australia
A. R. Masri
Affiliation:
School of Aerospace, Mechanical and Mechatronic Engineering, The University of Sydney, Sydney 2006, NSW, Australia
*
Email address for correspondence: fatemeh.salehi@sydney.edu.au

Abstract

This paper presents a probability density function (PDF) form of the population balance equation (PBE) for polysized and polyshaped droplets and solid particles in turbulent flows. A key contribution of this paper lies in the inclusion of an explicit consideration of the inertial effects and the shape of particles in the PDF-PBE formulation. The number density is taken as a function of droplet or particle size (volume) and shape as well as space and time. Potentially, other particle properties could also be included in the formulation. Inertial effects are quantified through the Stokes number, leading to accurate modelling of the different trajectories that are followed by droplets and/or particles with different sizes and shapes. To treat these effects, a new affordable approach is proposed and referred to as the method of Stokes binning. Here, the inertial dispersed elements are accelerated due to fluid dynamic forces associated with an averaged Stokes number in each bin. The model is validated against two data sets. The first data set includes a series of numerical test cases involving the injection of polyshaped droplets ranging in size from 1 to 50 $\unicode[STIX]{x03BC}\text{m}$ into a turbulent jet resulting in inlet Stokes numbers ranging from 0.03 to 75.2. The second data set consists of an experimental case focusing on the dispersion of 60 and 90 $\unicode[STIX]{x03BC}\text{m}$ spherical droplets in a turbulent round jet, resulting in inlet Stokes numbers of 53 and 122, respectively. The results confirm the ability of the approach to accurately model the polysized and polyshaped droplet dispersion using as few as eight Stokes bins. This approach has the potential to greatly reduce the computational cost of modelling the evolution of inertial droplets and particles in turbulent flows.

Type
Papers
Copyright
© 2017 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Balachandar, S. & Eaton, J. K. 2010 Turbulent dispersed multiphase flow. Annu. Rev. Fluid Mech. 42, 111133.CrossRefGoogle Scholar
Bini, M. & Jones, W. P. 2008 Large-eddy simulation of particle-laden turbulent flows. J. Fluid Mech. 614, 207252.Google Scholar
Chesnel, J., Reveillon, J., Menard, T. & Demoulin, F. 2011 Large eddy simulation of liquid jet atomization. Atomiz. Sprays 21 (9), 711736.Google Scholar
Colucci, P. J., Jaberi, F. A., Givi, P. & Pope, S. B. 1998 Filtered density function for large eddy simulation of turbulent reacting flows. Phys. Fluids 10 (2), 499515.Google Scholar
Druzhinin, O. A. & Elghobashi, S. 1998 Direct numerical simulations of bubble-laden turbulent flows using the two-fluid formulation. Phys. Fluids 10 (3), 685697.CrossRefGoogle Scholar
Elghobashi, S. & Truesdell, G. C. 1993 On the two-way interaction between homogeneous turbulence and dispersed solid particles. I: Turbulence modification. Phys. Fluids A 5 (7), 17901801.Google Scholar
Fox, R. O. 2012 Large-eddy-simulation tools for multiphase flows. Annu. Rev. Fluid Mech. 44, 4776.CrossRefGoogle Scholar
Ganser, G. H. 1993 A rational approach to drag prediction of spherical and nonspherical particles. Powder Technol. 77 (2), 143152.CrossRefGoogle Scholar
Jenny, P., Roekaerts, D. & Beishuizen, N. 2012 Modeling of turbulent dilute spray combustion. Prog. Energy Combust. Sci. 38 (6), 846887.Google Scholar
Jiang, X., Siamas, G. A., Jagus, K. & Karayiannis, T. G. 2010 Physical modelling and advanced simulations of gas–liquid two-phase jet flows in atomization and sprays. Prog. Energy Combust. Sci. 36 (2), 131167.Google Scholar
Jones, W. P. & Lettieri, C. 2010 Large eddy simulation of spray atomization with stochastic modeling of breakup. Phys. Fluids 22 (11), 115106.Google Scholar
Kennedy, I. M. & Moody, M. H. 1998 Particle dispersion in a turbulent round jet. Exp. Therm. Fluid Sci. 18 (1), 1126.Google Scholar
Kourmatzis, A. & Masri, A. R. 2015 Air-assisted atomization of liquid jets in varying levels of turbulence. J. Fluid Mech. 764, 95132.CrossRefGoogle Scholar
Kourmatzis, A., Pham, P. X. & Masri, A. R. 2015 Characterization of atomization and combustion in moderately dense turbulent spray flames. Combust. Flame 162 (4), 978996.Google Scholar
Kuerten, J. G. M. 2006 Subgrid modeling in particle-laden channel flow. Phys. Fluids 18 (2), 025108.CrossRefGoogle Scholar
Lasheras, J. C., Eastwood, C., Martınez-Bazán, C. & Montanes, J. L. 2002 A review of statistical models for the break-up of an immiscible fluid immersed into a fully developed turbulent flow. Intl J. Multiphase Flow 28 (2), 247278.Google Scholar
Liao, Y. & Lucas, D. 2009 A literature review of theoretical models for drop and bubble breakup in turbulent dispersions. Chem. Engng Sci. 64 (15), 33893406.CrossRefGoogle Scholar
Loth, E. 2008 Drag of non-spherical solid particles of regular and irregular shape. Powder Technol. 182 (3), 342353.Google Scholar
Masri, A. R. 2016 Turbulent combustion of sprays: from dilute to dense. Combust. Sci. Tech. 188 (10), 16191639.Google Scholar
Maxey, M. R. & Riley, J. J. 1983 Equation of motion for a small rigid sphere in a nonuniform flow. Phys. Fluids 26 (4), 883889.Google Scholar
Minier, J. & Peirano, E. 2001 The PDF approach to turbulent polydispersed two-phase flows. Phys. Rep. 352 (1), 1214.Google Scholar
Neuber, G., Kronenburg, A., Stein, O. T. & Cleary, M. J. 2017 MMC-LES modelling of droplet nucleation and growth in turbulent jets. Chem. Engng Sci 167, 204218.Google Scholar
Pan, T. W., Joseph, D. D., Bai, R., Glowinski, R. & Sarin, V. 2002 Fluidization of 1204 spheres: simulation and experiment. J. Fluid Mech. 451, 169191.CrossRefGoogle Scholar
Peirano, E., Chibbaro, S., Pozorski, J. & Minier, J. 2006 Mean-field/PDF numerical approach for polydispersed turbulent two-phase flows. Prog. Energy Combust. Sci. 32 (3), 315371.Google Scholar
Pope, S. B. 1985 PDF methods for turbulent reactive flows. Prog. Energy Combust. Sci. 11 (2), 119192.CrossRefGoogle Scholar
Pozorski, J. & Apte, S. V. 2009 Filtered particle tracking in isotropic turbulence and stochastic modeling of subgrid-scale dispersion. Intl J. Multiphase Flow 35 (2), 118128.Google Scholar
Prakash, A., Bapat, A. P. & Zachariah, M. R. 2003 A simple numerical algorithm and software for solution of nucleation, surface growth, and coagulation problems. Aerosol Sci. Technol. 37 (11), 892898.Google Scholar
Ramkrishna, D. 1985 The status of population balances. Rev. Chem. Engng 3 (1), 4995.Google Scholar
Rigopoulos, S. 2007 PDF method for population balance in turbulent reactive flow. Chem. Engng Sci. 62 (23), 68656878.CrossRefGoogle Scholar
Rigopoulos, S. 2010 Population balance modelling of polydispersed particles in reactive flows. Prog. Energy Combust. Sci. 36 (4), 412443.Google Scholar
Sporleder, F., Borka, Z., Solsvik, J. & Jakobsen, H. A. 2012 On the population balance equation. Rev. Chem. Engng 28, 149169.Google Scholar
Srinivasan, V., Salazar, A. J. & Saito, K. 2008 Numerical investigation on the disintegration of round turbulent liquid jets using LES/VOF techniques. Atomiz. Sprays 18 (7), 571617.CrossRefGoogle Scholar
Subramaniam, S. 2013 Lagrangian–Eulerian methods for multiphase flows. Prog. Energy Combust. Sci. 39 (2), 215245.Google Scholar
Sun, G., Hewson, J. C. & Lignell, D. O. 2017 Evaluation of stochastic particle dispersion modeling in turbulent round jets. Intl J. Multiphase Flow 89, 108122.CrossRefGoogle Scholar
Vo, S., Kronenburg, A., Stein, O. T. & Cleary, M. J. 2017 Multiple mapping conditioning for silica nanoparticle nucleation in turbulent flows. Proc. Combust. Inst. 36 (1), 10891097.Google Scholar
Wang, B., Kronenburg, A., Dietzel, D. & Stein, O. T. 2017 Assessment of scaling laws for mixing fields in inter-droplet space. Proc. Combust. Inst. 36 (2), 24512458.Google Scholar
Wang, Q. & Squires, K. D. 1996 Large eddy simulation of particle deposition in a vertical turbulent channel flow. Intl J. Multiphase Flow 22 (4), 667683.Google Scholar