Dispersion in fixed beds

1985 ◽  
Vol 154 ◽  
pp. 399-427 ◽  
Author(s):  
Donald L. Koch ◽  
John F. Brady
Keyword(s):  
Peclet Number ◽  
Molecular Diffusion ◽  
Volume Fraction ◽  
Effective Diffusivity ◽  
Ensemble Averaging ◽  
Péclet Number ◽  
Physical Mechanisms ◽  
Molecular Diffusivity ◽  
Fixed Beds ◽  
Effective Diffusivities

A macroscopic equation of mass conservation is obtained by ensemble-averaging the basic conservation laws in a porous medium. In the long-time limit this ‘macro-transport’ equation takes the form of a macroscopic Fick's law with a constant effective diffusivity tensor. An asymptotic analysis in low volume fraction of the effective diffusivity in a bed of fixed spheres is carried out for all values of the Péclet number ℙ = Ua/Df, where U is the average velocity through the bed. a is the particle radius and Df is the molecular diffusivity of the solute in the fluid. Several physical mechanisms causing dispersion are revealed by this analysis. The stochastic velocity fluctuations induced in the fluid by the randomly positioned bed particles give rise to a convectively driven contribution to dispersion. At high Péclet numbers, this convective dispersion mechanism is purely mechanical, and the resulting effective diffusivities are independent of molecular diffusion and grow linearly with ℙ. The region of zero velocity in and near the bed particles gives rise to non-mechanical dispersion mechanisms that dominate the longitudinal diffusivity at very high Péclet numbers. One such mechanism involves the retention of the diffusing species in permeable particles, from which it can escape only by molecular diffusion, leading to a diffusion coefficient that grows as ℙ2. Even if the bed particles are impermeable, non-mechanical contributions that grow as ℙ ln ℙ and ℙ2 at high ℙ arise from a diffusive boundary layer near the solid surfaces and from regions of closed streamlines respectively. The results for the longitudinal and transverse effective diffusivities as functions of the Péclet number are summarized in tabular form in §6. Because the same physical mechanisms promote dispersion in dilute and dense fixed beds, the predicted Péclet-number dependences of the effective diffusivities are applicable to all porous media. The theoretical predictions are compared with experiments in densely packed beds of impermeable particles, and the agreement is shown to be remarkably good.

The effective diffusivity of ordered and freely evolving bubbly suspensions

2018 ◽  
Vol 840 ◽  
pp. 215-237 ◽  
Author(s):  
Aurore Loisy ◽  
Aurore Naso ◽  
Peter D. M. Spelt
Keyword(s):  
Numerical Simulations ◽  
Peclet Number ◽  
Random Media ◽  
Chemical Species ◽  
Effective Diffusivity ◽  
Passive Scalar ◽  
Solid Particles ◽  
Péclet Number ◽  
Mechanical Dispersion ◽  
Fixed Beds

We investigate the dispersion of a passive scalar such as the concentration of a chemical species, or temperature, in homogeneous bubbly suspensions, by determining an effective diffusivity tensor. Defining the longitudinal and transverse components of this tensor with respect to the direction of averaged bubble rise velocity in a zero mixture velocity frame of reference, we focus on the convective contribution thereof, this being expected to be dominant in commonly encountered bubbly flows. We first extend the theory of Kochet al.(J. Fluid Mech., vol. 200, 1989, pp. 173–188) (which is for dispersion in fixed beds of solid particles under Stokes flow) to account for weak inertial effects in the case of ordered suspensions. In the limits of low and of high Péclet number, including the inertial effect of the flow does not affect the scaling of the effective diffusivity with respect to the Péclet number. These results are confirmed by direct numerical simulations performed in different flow regimes, for spherical or very deformed bubbles and from vanishingly small to moderate values of the Reynolds number. Scalar transport in arrays of freely rising bubbles is considered by us subsequently, using numerical simulations. In this case, the dispersion is found to be convectively enhanced at low Péclet number, like in ordered arrays. At high Péclet number, the Taylor dispersion scaling obtained for ordered configurations is replaced by one characterizing a purely mechanical dispersion, as in random media, even if the level of disorder is very low.

Magnetohydrodynamics and Soret Effects on Bioconvection in a Porous Medium Saturated With a Nanofluid Containing Gyrotactic Microorganisms

2014 ◽  
Vol 136 (5) ◽  
Author(s):  
S. Shaw ◽  
P. Sibanda ◽  
A. Sutradhar ◽  
P. V. S. N. Murthy
Keyword(s):  
Porous Medium ◽  
Differential Equations ◽  
Sherwood Number ◽  
Peclet Number ◽  
Volume Fraction ◽  
Soret Effect ◽  
Magnetic Field Effect ◽  
Lewis Number ◽  
Solute Concentration ◽  
Péclet Number

We investigate the bioconvection of gyrotactic microorganism near the boundary layer region of an inclined semi infinite permeable plate embedded in a porous medium filled with a water-based nanofluid containing motile microorganisms. The model for the nanofluid incorporates Brownian motion, thermophoresis, also Soret effect and magnetic field effect are considered in the study. The governing partial differential equations for momentum, heat, solute concentration, nanoparticle volume fraction, and microorganism conservation are reduced to a set of nonlinear ordinary differential equations using similarity transformations and solved numerically. The effects of the bioconvection parameters on the thermal, solutal, nanoparticle concentration, and the density of the micro-organisms are analyzed. A comparative analysis of our results with previously reported results in the literature is given. Some interesting phenomena are observed for the local Nusselt and Sherwood number. It is shown that the Péclet number and the bioconvection Rayleigh number highly influence the local Nusselt and Sherwood numbers. For Péclet numbers less than 1, the local Nusselt and Sherwood number increase with the bioconvection Lewis number. However, both the heat and mass transfer rates decrease with bioconvection Lewis number for higher values of the Péclet number.

scholarly journals Dispersion in the large-deviation regime. Part 2. Cellular flow at large Péclet number

2014 ◽  
Vol 745 ◽  
pp. 351-377 ◽  
Author(s):  
P. H. Haynes ◽  
J. Vanneste
Keyword(s):  
Peclet Number ◽  
Eigenvalue Problems ◽  
Large Deviation ◽  
Effective Diffusivity ◽  
Rate Function ◽  
Homogenization Theory ◽  
Péclet Number ◽  
Stagnation Points ◽  
Scalar Dispersion ◽  
Scalar Concentration

AbstractA standard model for the study of scalar dispersion through the combined effect of advection and molecular diffusion is a two-dimensional periodic flow with closed streamlines inside periodic cells. Over long time scales, the dispersion of a scalar released in this flow can be characterized by an effective diffusivity that is a factor$\mathit{Pe}^{1/2}$larger than molecular diffusivity when the Péclet number$\mathit{Pe}$is large. Here we provide a more complete description of dispersion in this regime by applying the large-deviation theory developed in Part 1 of this paper. Specifically, we derive approximations to the rate function governing the scalar concentration at large time$t$by carrying out an asymptotic analysis of the relevant family of eigenvalue problems. We identify two asymptotic regimes and, for each, make predictions for the rate function and spatial structure of the scalar. Regime I applies to distances$|\boldsymbol {x}|$from the scalar release point that satisfy$|\boldsymbol {x}|= O(\mathit{Pe}^{1/4} t)$. The concentration in this regime is isotropic at large scales, is uniform along streamlines within each cell, and varies rapidly in boundary layers surrounding the separatrices between adjacent cells. The results of homogenization theory, yielding the$O(\mathit{Pe}^{1/2})$effective diffusivity, are recovered from our analysis in the limit$|\boldsymbol {x}|\ll \mathit{Pe}^{1/4} t$. Regime II applies when$|\boldsymbol {x}|=O(\mathit{Pe}\, t/{\rm log}\, \mathit{Pe})$and is characterized by an anisotropic concentration distribution that is localized around the separatrices. A novel feature of this regime is the crucial role played by the dynamics near the hyperbolic stagnation points. A consequence is that in part of the regime the dispersion can be interpreted as resulting from a random walk on the lattice of stagnation points. The two regimes overlap so that our asymptotic results describe the scalar concentration over a large range of distances$|\boldsymbol {x}|$. They are verified against numerical solutions of the family of eigenvalue problems yielding the rate function.

scholarly journals Dispersion due to molecular diffusion and macroscopic mixing in flow through a network of capillaries

1960 ◽  
Vol 7 (2) ◽  
pp. 194-208 ◽  
Author(s):  
P. G. Saffman
Keyword(s):  
Porous Medium ◽  
Correlation Function ◽  
Viscous Fluid ◽  
Molecular Diffusion ◽  
Random Network ◽  
Mean Velocity ◽  
Molecular Diffusivity ◽  
Flow Through ◽  
The Mean ◽  
Effective Diffusivities

This paper is concerned with the dispersion of a material quantity in the steady flow of a viscous fluid through a random network of capillaries (which is a useful model of a porous medium), for the case in which molecular diffusion and macroscopic mixing, due to the randomness of the streamlines, are both important. A Lagrangian correlation function is introduced and the longitudinal and lateral effective diffusivities are thereby calculated for all values ofUl/κ less than some large value. Here,ldenotes the length of a capillary,Uthe mean velocity of the fluid, and κ the molecular diffusivity of the material quantity. The theory is compared with experimental observations of dispersion in flow through granular beds.

On Simplified Models for the Rate- and Time-Dependent Performance of Stratified Thermal Storage

2007 ◽  
Vol 129 (3) ◽  
pp. 214-222 ◽  
Author(s):  
Ying Ji ◽  
K. O. Homan
Keyword(s):  
Entropy Generation ◽  
Peclet Number ◽  
Volume Fraction ◽  
Thermal Storage ◽  
Time Dependent ◽  
Simplified Models ◽  
Péclet Number ◽  
Thermal Mixing ◽  
Discharge Volume ◽  
Cumulative Entropy

In direct sensible thermal storage systems, both the energy discharging and charging processes are inherently time-dependent as well as rate-dependent. Simplified models which depict the characteristics of this transient process are therefore crucial to the sizing and rating of the storage devices. In this paper, existing models which represent three distinct classes of models for thermal storage behavior are recast into a common formulation and used to predict the variations of discharge volume fraction, thermal mixing factor, and entropy generation. For each of the models considered, the parametric dependence of key performance measures is shown to be expressible in terms of a Peclet number and a Froude number or temperature difference ratio. The thermal mixing factor for each of the models is reasonably well described by a power law fit with Fr2Pe for the convection-dominated portion of the operating range. For the uniform and nonuniform diffusivity models examined, there is shown to be a Peclet number which maximizes the discharge volume fraction. In addition, the cumulative entropy generation from the simplified models is compared with the ideally-stratified and the fully-mixed limits. Of the models considered, only the nonuniform diffusivity model exhibits an optimal Peclet number at which the cumulative entropy generation is minimized. For each of the other models examined, the cumulative entropy generation varies monotonically with Peclet number.

Mixing of the fluid phase in slowly sheared particle suspensions of cylinders

2017 ◽  
Vol 818 ◽  
pp. 807-837 ◽  
Author(s):  
Kjetil Thøgersen ◽  
Marcin Dabrowski
Keyword(s):  
Diffusion Coefficient ◽  
Fractal Dimension ◽  
Peclet Number ◽  
Volume Fraction ◽  
Fluid Phase ◽  
Particle Suspensions ◽  
Péclet Number ◽  
Self Diffusion ◽  
Wide Range ◽  
Self Diffusion Coefficient

We introduce a finite element model for neutrally buoyant particle suspensions of cylinders at zero Reynolds number and infinite Péclet number in the purely hydrodynamic limit, which allows us to access a high-accuracy fluid velocity field at any time during the simulation. We use the diffusive strip method to characterize the development of the concentration field in the fluid phase of sheared suspensions from initial thin filaments, and characterize the structures that form with their fractal dimension. We find that the growth of the fractal dimension of the filaments scales with the increase of mean square displacement in the fluid phase. Further, we measure the concentration distribution of tracers in the fluid phase, as well as the shear-induced self-diffusion coefficient in both the solid phase and the fluid phase. We demonstrate that the shear-induced self-diffusion coefficient is slightly larger in the fluid phase at infinite Péclet number. Finally, we investigate enhanced mass diffusivity in the fluid phase by systematic measurements of the shear-induced self-diffusion coefficient in the fluid phase for a wide range of fluid tracer Péclet numbers. We find that the functional dependence $D_{s}/D=1+\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D719}^{\unicode[STIX]{x1D6FC}}Pe^{\unicode[STIX]{x1D701}}$ (where $D_{s}$ is the shear-induced self-diffusion coefficient, $D$ is the molecular diffusivity and $\unicode[STIX]{x1D719}$ is the particle volume fraction) fits the observations fairly well. We measure the constants $\unicode[STIX]{x1D6FD}=2.98\pm 0.39$, $\unicode[STIX]{x1D6FC}=1.61\pm 0.26$ and $\unicode[STIX]{x1D701}=0.900\pm 0.031$.

On Laminar Dispersion for Flow Through Round Tubes

1979 ◽  
Vol 46 (4) ◽  
pp. 750-756 ◽  
Author(s):  
J. S. Yu
Keyword(s):  
Peclet Number ◽  
Molecular Diffusion ◽  
Initial Conditions ◽  
Local Concentration ◽  
Axial Distance ◽  
Dimensionless Time ◽  
Axially Symmetric ◽  
Péclet Number ◽  
Average Flow Velocity ◽  
Diffusion Convection

A general method for the solution of the axially symmetric transient diffusion-convection equation for laminar dispersion in round tubes subject to arbitrary square-integrable initial conditions is analytically developed. The solution representing the local concentration is expressed by a series in terms of the zeroth-order Bessel function, and the order of approximation (equal to the number of terms in the series) required at a given value of the dimensionless time τ for flow with a specified Peclet number Pe is clearly established. It is shown that the approximation used by Gill, et al. [5–8], is a special case of the present analysis under certain conditional assumptions. For the case of fundamental interest with an initial input concentrated at a section of the tube, the mean concentration as a function of the axial distance measured from the origin of a coordinate moving with the average flow velocity determined by the present method at given values of the Peclet number and the dimensionless time is compared with those by Taylor [1], Lighthill [4], Chatwin [9], Gill, et al. [7], and Hunt [23]. The comparison of the concentration profiles shows that Lighthill’s solution is perhaps valid as τ → 0, Hunt’s solution obtained by first-order perturbation approximation yields too large a dispersion by molecular diffusion even at small times, and the other solutions are asymptotically correct at large values of time for flow with high Peclet numbers.

The structure of turbulent density interfaces

1974 ◽  
Vol 65 (1) ◽  
pp. 45-63 ◽  
Author(s):  
P. F. Crapper ◽  
P. F. Linden
Keyword(s):  
Peclet Number ◽  
Molecular Diffusion ◽  
Interface Structure ◽  
Mixing Process ◽  
Temporal Response ◽  
Péclet Number ◽  
A Value ◽  
Density Interfaces ◽  
Diffusive Processes

The structure of density interfaces upon which turbulent motions have been imposed is investigated in the laboratory. Particular attention is paid to the profiles of density across such interfaces, and the temporal response of the interface to the imposed turbulence. Interfaces over a range of Péclet numbers and Richardson numbers are examined. It is found that the interface thickness h/l (non-dimensionalized with respect to the length scale l of the turbulence) is a function of the Péclet number but independent of the Richardson number. At low Péclet numbers (Pe [lsim ] 200) molecular diffusion is important in the determination of the interface structure and a diffusive core (across which all transport occurs by molecular diffusion) is formed in the centre of the interface. At higher Péclet numbers the interface structure appears to be determined by non-diffusive processes and h/l becomes approximately constant at a value of about 1·5. Some information concerning the intermittent nature of the mixing process at high Péclet number is obtained from records of salinity fluctuations measured at a fixed point. Finally, the implications of these data concerning the interpretations of the measurements of entrainment across an interface made by Turner (1968) are discussed.

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