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Hydrodynamic diffusion and mass transfer across a sheared suspension of neutrally buoyant spheres
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10.1063/1.3098446
/content/aip/journal/pof2/21/3/10.1063/1.3098446
http://aip.metastore.ingenta.com/content/aip/journal/pof2/21/3/10.1063/1.3098446

Figures

Image of FIG. 1.
FIG. 1.

The Sherwood number for mass transfer across a stationary suspension is plotted as a function of the particle volume fraction . The triangles and diamonds are experimental measurements with and 22, respectively. The line is the Maxwell theoretical prediction, Eq. (6).

Image of FIG. 2.
FIG. 2.

Experimentally measured Sherwood number for mass transfer across sheared suspensions with is plotted as a function of the Peclet number. Circles are measurements in the absence of particles. The diamonds, squares, and triangles are measurements in particle suspensions with , 0.17, and 0.25, respectively. The error bars are standard deviations of five experimental measurements, each made with a fresh suspension.

Image of FIG. 3.
FIG. 3.

Mass transfer rate as a function of Peclet number in suspensions with . The filled and open squares are experimental measurements with and 22, respectively. The upper and lower solid lines are the theory, Eqs. (9), (11), and (12), for and 22, respectively. The dashed lines are the theoretical predictions neglecting the effects of fluid inertia, i.e., setting Re to zero in Eqs. (12) and (11).

Image of FIG. 4.
FIG. 4.

The rate of mass transfer measured as a function of the Peclet number in sheared suspensions of polystyrene spheres with and in the present work (triangles) is compared with the mass transfer rate in a suspension of red-blood cells with and by Wang and Keller (Ref. 15) (diamonds) and the heat transfer rate in suspensions of polystyrene spheres in a viscous oil with and by Sohn and Chen (Ref. 16) (squares).

Image of FIG. 5.
FIG. 5.

The mass transfer rate obtained from the numerical simulations as a function of the Peclet number for and several values of the particle volume fraction. The symbols indicate the transport rate obtained from the simulated mass flux to the wall. The lines are the theoretical model, Eq. (9), with the parameters listed in Table I which are obtained by fitting the mean concentration profile. The diamonds and bottom line are for , squares and middle line for , and triangles and top line for .

Image of FIG. 6.
FIG. 6.

The mass transfer rate obtained from the numerical simulations as a function of the Peclet number for and several values of the Couette-gap-to-particle-radius ratio. The symbols are obtained from the simulated mass flux to the wall. The lines are the theoretical model, Eq. (9), with the parameters listed in Table I which are obtained by fitting the mean concentration profile. The diamonds and bottom line are for , triangles and middle line for , and squares and top line for .

Image of FIG. 7.
FIG. 7.

A typical profile of the average concentration in the fluid phase obtained from the numerical simulations as a function of the distance from one of the planar walls. The concentration is normalized by the concentration in equilibrium with the wall at . The solid line is the fluid-phase concentration averaged over the plane and over time (after a statistical steady state is established). The dashed line is the best fit to the concentration profile in the middle third of the channel. The slope and intercept of this line are and , which are used to determine the hydrodynamic diffusivity and boundary layer thickness. This profile corresponds to , , , and and exhibits a concentration slip .

Image of FIG. 8.
FIG. 8.

Sketch of the profile of the hydrodynamic diffusivity as a function of distance from the planar wall normalized by the particle radius. The scaling of the wall-normal fluid velocity fluctuations inferred from the no-slip boundary condition and continuity imply that the diffusivity is proportional to as . A profile that satisfies this constraint and approaches a constant as is likely to have a very steep slope for a region where so that there will exist a broad range of intermediate Pe for which the boundary layer thickness obtained by equating the hydrodynamic and molecular diffusivities is weakly dependent on Pe.

Image of FIG. 9.
FIG. 9.

Inertial migration of a single sphere in planar Couette flow. The wall-normal component of the velocity of a spherical particle normalized by is plotted as a function of the distance from the wall. The line is the asymptotic theory of Vasseur and Cox (Ref. 39) and the symbols are results of lattice-Boltzmann simulations with (diamonds), 0.25 (squares), and 0.1 (triangles).

Image of FIG. 10.
FIG. 10.

The mass transfer rate obtained from numerical simulations is plotted as a function of particle Reynolds number at a fixed value of the Peclet number , , and .

Image of FIG. 11.
FIG. 11.

The simulated mass transfer rate is plotted as a function of the Peclet number for two values of the particle Reynolds number: for the lower line and triangles and for the upper line and diamonds . The symbols are obtained from the simulated mass flux to the wall. The lines are the theoretical model, Eq. (9), with the parameters listed in Table I which are obtained by fitting the mean concentration profile. The solid particle volume fraction is 0.1 and .

Image of FIG. 12.
FIG. 12.

The mass transfer rate is plotted as a function of the Peclet number for . The symbols are experimental measurements and the lines are the theoretical model, Eqs. (9), (11), and (12), with the parameters determined solely from the numerical simulations. The diamonds and lowest line are for , squares and middle line for , and triangles and highest line for .

Image of FIG. 13.
FIG. 13.

The mass transfer rate is plotted as a function of the Peclet number for . The symbols are experimental measurements and the lines are the theoretical model, Eqs. (9), (11), and (12), with the parameters determined solely from the numerical simulations. The diamonds and lowest lines are for , squares and middle line for , and triangles and highest line for .

Tables

Generic image for table
Table I.

Numerical simulation results for suspension structure and mass-transfer-model parameters. The simulation conditions are characterized by the particle Reynolds number Re, the Couette-gap-to-particle-radius ratio , and the nominal solid volume fraction . The mass transfer results are obtained from the mass flux and average concentration profiles and are averaged over four Peclet numbers in the range of . The suspension structure is characterized by the bulk volume fraction averaged over the middle third of the channel, , and the thickness of a wall depletion layer normalized by the sphere radius, . The parameters appearing in the mass transfer model, Eq. (9), are the mass transfer boundary layer thickness normalized by the sphere radius and the hydrodynamic diffusivity , which is normalized by .

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/content/aip/journal/pof2/21/3/10.1063/1.3098446
2009-03-27
2014-04-18
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Hydrodynamic diffusion and mass transfer across a sheared suspension of neutrally buoyant spheres
http://aip.metastore.ingenta.com/content/aip/journal/pof2/21/3/10.1063/1.3098446
10.1063/1.3098446
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