ARTICLE pubs.acs.org/IECR
Flow Past a Porous Permeable Sphere: Hydrodynamics and Heat-Transfer Studies Ashok K. Jain and Suddhasatwa Basu* Department of Chemical Engineering, Indian Institute of Technology Delhi, New Delhi 110016, India ABSTRACT: Flow past a porous permeable sphere has been studied using standard computational fluid dynamics (CFD) software (FLUENT) for a wide range of Reynolds numbers. Brinkman’s extension of Darcy’s law in the inner region and NavierStokes equations in the outer region of the porous permeable sphere were employed to solve the fluid flow phenomena. The results are presented in terms of three dimensionless parameters, e.g., the particle Reynolds number, the permeability ratio, and the drag ratio. The computed drag ratio asymptotically approaches zero at high permeability ratios and unity at low permeability ratios. A peak in the drag ratio versus permeability ratio plot is observed at high particle Reynolds number and in the intermediate range of permeability ratio. The drag ratios obtained from CFD simulation show good agreement with the experimental data available in the literature at different Reynolds numbers and permeability ratios. The results of heat transfer from porous permeable sphere are presented in terms of four dimensionless parameters: the particle Reynolds number (Re), permeability ratio, the Prandtl number (Pr), and the Nusselt number (Nu). The correlation obtained from the CFD simulation data for heat transfer from a porous permeable sphere is useful in predicting Nu for a wide range of Re and Pr at different permeability ratios.
’ INTRODUCTION Flow through and around permeable bodies is a topic that manifests itself in several practical situations, such as sedimentation of sludge flocs,1 motion of clusters in gassolid fluidized-bed reactors,2 settling of flocs in liquidsolid reactors,3 and sinking plankton and snow aggregates at the bottom of the ocean.4,5 Prediction of the settling velocity of a porous permeable body and heat transfer from a porous permeable body is essential for the overall assessment of the above processes. In view of this, flow through and around a porous permeable sphere and heat transfer from it were studied. Although the shape of a floc is not exactly like that of a sphere, a porous permeable sphere is considered in the present study, because of the simplicity in analyzing a spherical geometry and the availability of experimental data, and, for the purpose of comparing the results of a porous permeable sphere with those for an impermeable sphere. The initial theoretical studies on sedimentation of a porous sphere in a laminar flow regime were carried out by Sutherland and Tan6 and Neale et al.7 Neale et al.7 provided the analytical solution for creeping flow, relative to a single permeable sphere and clusters of permeable spheres. Matsumoto and Suganumo8 carried out experiments in the low Reynolds number regime (Re ≈ 0.1) with a model aggregate made of steel wool. They proposed a correlation between the drag ratio and the permeability ratio. The terminal velocity and drag at different Reynolds numbers were experimentally measured and estimated, respectively, by Masliyah and Polikar.3 They buttressed their experimental results in the laminar flow region with the theoretical prediction of Neale et al.7 Johnson et al.9 performed extensive experimental and theoretical analyses on sedimentation of aggregated latex microspheres in the low-Reynolds-number regime and proposed an empirical relationship for the drag coefficient. The aggregated structure was highly porous in nature and was considered as fractals. Noymer et al.10 studied the steady r 2011 American Chemical Society
flow past a permeable cylinder using CFD simulation software (PHOENICS). Drag was computed for flow past a single, permeable cylinder for Re = 10, 100, and 1000, and for permeability ratios from 1.7 105 to 10. The computed drag agreed with the measured drag for a porous permeable cylinder in the wind tunnel experiment. Noymer et al.10 found that (i) the drag on a porous permeable cylinder becomes greater than that for an impermeable cylinder in the intermediate range of permeability ratio and (ii) no clear reason for this observation is known (hence, it requires further investigation). Wu and Lee11 studied the flow profile of a falling porous sphere in a cylindrical column. They computed the hydrodynamic drag force exerted on a permeable sphere for Reynolds numbers ranging from 0.1 to 40, using CFD simulation software (FIDAP 7.5). They have studied, in great detail, the effect of the wall on a falling porous and nonporous sphere. Recently, Bhattacharya and Sekhar12 produced an analytical solution for viscous flow past a permeable sphere with an impermeable solid core, using the jumped stress boundary condition. All of these studies were restricted to hydrodynamic study on a permeable sphere. Only a few studies were carried out on heat transfer from a permeable sphere. Furthermore, most of the studies on hydrodynamics were restricted to the low-Reynolds-number regime, except the work of Wu and Lee,11 which is not experimentally verified, and Noymer et al.,10 which is on porous cylinder. A generalized correlation for drag coefficient that is valid for a wide range of
Special Issue: Nigam Issue Received: July 28, 2011 Accepted: October 21, 2011 Revised: October 19, 2011 Published: October 21, 2011 2170
dx.doi.org/10.1021/ie201647p | Ind. Eng. Chem. Res. 2012, 51, 2170–2178
Industrial & Engineering Chemistry Research
ARTICLE
Reynolds numbers and permeability ratios is not available in the literature. Acrivos and Taylor13 studied the problem of forced convection from an isothermal impermeable sphere for small and large Peclet number (Pe) values. [Pe = Re Pr, where Pr is the Prandtl number.] Their analysis is valid when Re and Pe are both
