Bubble swarms in power-law liquids at moderate Reynolds numbers: Drag and mass transfer

Abstract The motion of and mass transfer from swarms of mono-size non-circulating spherical bubbles in clean power-law liquids have been numerically investigated at moderate Reynolds and Peclet numbers. A simple sphere-in-a-sphere cell model has been used to account for hydrodynamic interactions between neighbouring bubbles. A wide range of values of the gas hold-up has been considered so as to evaluate the detailed drag and mass transfer characteristics of dense swarms of bubbles and of almost single bubbles rising in power-law liquids. Extensive numerical results have been obtained to elucidate the effects of the Reynolds number ( Re ), Schmidt number ( Sc ), power-law index ( n ) and gas hold-up ( Φ ) on the drag coefficient and Sherwood number over the ranges of conditions: 0.6 ≤  Φ  ≤ 10 −6 , 1 ≤  Re  ≤ 200, 1 ≤  Sc  ≤ 1000 and 0.6 ≤  n  ≤ 1.6. The drag coefficient ( C d ) decreases, whereas, the average Sherwood number ( Sh avg ) increases with the decreasing values of n . However, the effect of the power-law index is found to be more significant on the rising velocity of swarms of bubbles than that on the rate of mass transfer. Based on the present numerical results, predictive correlations have been proposed for drag coefficient and Sherwood number which can be used to estimate the free rising velocity and mass transfer coefficients respectively in a new gas–liquid application involving power-law continuous phase.

[1]  Chris D. Rielly,et al.  Chemical Engineering for the Food Industry , 1991 .

[2]  G. Marrucci.  Communication. Rising Velocity of Swarm of Spherical Bubbles , 1965 .

[3]  John Happel,et al.  Viscous flow in multiparticle systems: Slow motion of fluids relative to beds of spherical particles , 1958 .

[4]  R. Chhabra,et al.  Mass transfer from a single fluid sphere to power-law liquids at moderate Reynolds numbers , 2007 .

[5]  A. Jarzebski,et al.  Drag and mass transfer in multiple drop slow motion in a power law fluid , 1986 .

[6]  A. Margaritis,et al.  Comments on the free rise of gas bubbles in non-Newtonian liquids , 2005 .

[7]  Further remarks on the drag of a swarm of bubbles , 1995 .

[8]  J. Khinast,et al.  Prediction of mass transfer coefficients in non-Newtonian fermentation media using first-principles methods. , 2007, Biotechnology and bioengineering.

[9]  Jyeshtharaj B. Joshi,et al.  Bubble Formation and Bubble Rise Velocity in Gas−Liquid Systems: A Review , 2005 .

[10]  E. Ruckenstein.  On mass transfer in the continuous phase from spherical bubbles or drops , 1964 .

[11]  Drag and mass transfer in slow non-newtonian flows over an ensemble of newtonian spherical drops or bubbles , 1987 .

[12]  Efstathios E. Michaelides,et al.  Mass and heat transfer from fluid spheres at low Reynolds numbers , 2000 .

[13]  A. Jarzebski,et al.  Drag and mass transfer in a creeping flow of a carreau fluid over drops or bubbles , 1987 .

[14]  V. Kelessidis.  An explicit equation for the terminal velocity of solid spheres falling in pseudoplastic liquids , 2004 .

[15]  R. Chhabra.  Rising velocity of a swarm of spherical bubbles in power law fluids at high Reynolds numbers , 1998 .

[16]  D. Kee,et al.  A note on the drag coefficient of a single gas bubble in a power‐law fluid , 1999 .

[17]  S. Kuwabara,et al.  The Forces experienced by Randomly Distributed Parallel Circular Cylinders or Spheres in a Viscous Flow at Small Reynolds Numbers , 1959 .

[18]  P. Roache.  Perspective: A Method for Uniform Reporting of Grid Refinement Studies , 1994 .

[19]  Numerical simulation of the drag on a swarm of bubbles , 1994 .

[20]  M. Moo-Young,et al.  Bubble drag and mass transfer in non‐newtonian fluids: Creeping flow with power‐law fluids , 1969 .

[21]  G. Juncu.  Conjugate mass transfer to a spherical drop accompanied by a second-order chemical reaction inside the drop , 2002 .

[22]  Robert W. Field,et al.  Bubble Column Reactors , 1991 .

[23]  R. D. Felice,et al.  The particle-in-a-tube analogy for a multiparticle suspension , 1996 .

[24]  M. Dziubiński,et al.  Comments on bubble rising velocity in non-Newtonian liquids , 2003 .

[25]  H. Blanch,et al.  Bubble motion and mass transfer in non‐Newtonian fluids: Part I. Single bubble in power law and Bingham fluids , 1978 .

[26]  R. Chhabra,et al.  Mass Transfer from Ensembles of Newtonian Fluid Spheres at Moderate Reynolds and Peclet Numbers , 2007 .

[27]  Analytical Study of Drag and Mass Transfer in Creeping Power Law Flow across Tube Banks , 2004 .

[28]  G. Juncu.  Unsteady heat and/or mass transfer from a fluid sphere in creeping flow , 2001 .

[29]  R. Chhabra,et al.  Drag on a single fluid sphere translating in power-law liquids at moderate Reynolds numbers , 2007 .

[30]  D. Rodrigue.  A General Correlation for the Rise Velocity of Single Gas Bubbles , 2008 .

[31]  Vivek V. Ranade,et al.  Computational study of a single‐phase flow in packed beds of spheres , 2005 .

[32]  Z. Feng,et al.  Drag Coefficients of Viscous Spheres at Intermediate and High Reynolds Numbers , 2001 .

[33]  Free rise velocity of a swarm of spherical gas bubbles through a quiescent power law liquid , 1992 .

[34]  Efstathios E. Michaelides,et al.  Heat and mass transfer coefficients of viscous spheres , 2001 .

[35]  J. Hsu,et al.  Electrophoresis in a Carreau fluid at arbitrary zeta potentials. , 2004, Langmuir : the ACS journal of surfaces and colloids.

[36]  R. Chhabra,et al.  Rise velocity of a swarm of spherical bubbles through a non-Newtonian fluid: Effect of zero shear viscosity , 1988 .

[37]  Jianting Zhu.  A note on slow non-Newtonian flows over an ensemble of spherical bubbles , 2001 .

[38]  F. Harlow,et al.  Numerical Calculation of Time‐Dependent Viscous Incompressible Flow of Fluid with Free Surface , 1965 .

[39]  B. P. Leonard,et al.  A stable and accurate convective modelling procedure based on quadratic upstream interpolation , 1990 .

[40]  R. Chhabra,et al.  Bubbles, Drops, and Particles in Non-Newtonian Fluids , 2006 .

[41]  E. Michaelides.  Particles, Bubbles And Drops: Their Motion, Heat And Mass Transfer , 2006 .

[42]  Effect of the viscosity ratio on mass transfer from a fluid sphere at low to very high Peclet numbers , 2007 .

[43]  R. Chhabra,et al.  Sedimentation in Emulsions of Mono-size Droplets at Moderate Reynolds Numbers , 2006 .

[44]  B. Gal-or,et al.  Hydrodynamics of an ensemble of drops (or bubbles) in the presence or absence of surfactants , 1968 .

[45]  R. Chhabra,et al.  Mass transfer from a spherical bubble rising in power-law fluids at intermediate Reynolds numbers , 2007 .

[46]  R. Clift,et al.  Bubbles, Drops, and Particles , 1978 .

[47]  Vinayak Eswaran,et al.  Power Law Fluid Flow Through Beds of Spheres at Intermediate Reynolds Numbers: Pressure in Fixed and Distended Beds , 2004 .

[48]  D. G. Karamanev,et al.  Rise of gas bubbles in quiescent liquids , 1994 .

[49]  A. Saboni,et al.  Numerical study of the drag on a fluid sphere , 2002 .

[50]  Jianting Zhu.  Drag and mass transfer for flow of a Carreau fluid past a swarm of Newtonian drops , 1995 .

[51]  Denis Rodrigue,et al.  A simple correlation for gas bubbles rising in power‐law fluids , 2002 .

[52]  R. P. Chhabra,et al.  Rising velocity of a swarm of spherical bubbles in a power law non-Newtonian liquid , 1987 .

[53]  R. Chhabra,et al.  Drag of a Spherical Bubble Rising in Power Law Fluids at Intermediate Reynolds Numbers , 2007 .

[54]  T. Sundararajan,et al.  Hydrodynamics of newtonian fluid flow through assemblages of rigid spherical particles in intermediate reynolds number regime , 1991 .

[55]  R. Chhabra,et al.  Convective heat transfer for power law fluids in packed and fluidised beds of spheres , 2004 .