^{1}, Donald L. Koch

^{1}, Xiaolong Yin

^{1}and Claude Cohen

^{1}

### Abstract

We present experimental, theoretical, and numerical simulation studies of the transport of fluid-phase tracer molecules from one wall to the opposite wall bounding a sheared suspension of neutrally buoyant solid particles. The experiments use a standard electrochemical method in which the mass transfer rate is determined from the current resulting from a dilute concentration of ions undergoing redox reactions at the walls in a solution of excess nonreacting ions that screen the electric field in the suspension. The simulations use a lattice-Boltzmann method to determine the fluid velocity and solid particle motion and a Brownian tracer algorithm to determine the chemical tracer mass transfer. The mass transport across the bulk of the suspension is driven by hydrodynamicdiffusion, an apparent diffusive motion of tracers caused by the chaotic fluid velocity disturbances induced by suspended particles. As a result the dimensionless rate of mass transfer (or Sherwood number) is a nearly linear function of the dimensionless shear rate (Peclet number) at moderate values of the Peclet number. At higher Peclet numbers, the Sherwood number grows more slowly due to the mass transport resistance caused by a molecular-diffusion boundary layer near the solid walls. Fluid inertia enhances the rate of mass transfer in suspensions with particle Reynolds numbers in the range of 0.5–7.

This work was supported by the Department of Energy Office of Basic Energy Sciences Grant No. DE-FG02-03-ER46073. We thank Rebecca Gauthier, an undergraduate student, for her assistance with the experimental work.

I. INTRODUCTION

II. EXPERIMENTAL METHODS

III. SIMULATION METHODS

IV. RESULTS AND DISCUSSION

A. Mass transfer across a stationary suspension

B. Experimental measurements of mass transfer in a sheared suspension

C. A simple theoretical model

D. Numerical simulations of viscous suspensions

E. Effects of inertia

F. Comparison of model predictions with experiments

V. CONCLUSION

### Key Topics

- Mass diffusion
- 111.0
- Suspensions
- 94.0
- Hydrodynamics
- 70.0
- Reynolds stress modeling
- 33.0
- Diffusion
- 30.0

## Figures

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).

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).

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.

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.

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).

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).

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).

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).

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 .

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 .

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 .

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 .

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 .

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 .

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.

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.

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).

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).

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 .

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 .

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 .

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 .

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 .

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 .

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 .

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

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 .

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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