Donald L. Koch

Dispersion in fixed beds

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 Ficks 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 Peclet number ℙ = Ua / D f , where U is the average velocity through the bed. a is the particle radius and D f 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 Peclet 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 Peclet 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 ℙ 2 . Even if the bed particles are impermeable, non-mechanical contributions that grow as ℙ ln ℙ and ℙ 2 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 Peclet number are summarized in tabular form in §6. Because the same physical mechanisms promote dispersion in dilute and dense fixed beds, the predicted Peclet-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.

Moderate-Reynolds-number flows in ordered and random arrays of spheres

Lattice-Boltzmann simulations are used to examine the effects of fluid inertia, at moderate Reynolds numbers, on flows in simple cubic, face-centred cubic and random arrays of spheres. The drag force on the spheres, and hence the permeability of the arrays, is calculated as a function of the Reynolds number at solid volume fractions up to the close-packed limits of the arrays. At Reynolds numbers up to O (10 2 ), the non-dimensional drag force has a more complex dependence on the Reynolds number and the solid volume fraction than suggested by the well-known Ergun correlation, particularly at solid volume fractions smaller than those that can be achieved in physical experiments. However, good agreement is found between the simulations and Erguns correlation at solid volume fractions approaching the close-packed limit. For ordered arrays, the drag force is further complicated by its dependence on the direction of the flow relative to the axes of the arrays, even though in the absence of fluid inertia the permeability is isotropic. Visualizations of the flows are used to help interpret the numerical results. For random arrays, the transition to unsteady flow and the effect of moderate Reynolds numbers on hydrodynamic dispersion are discussed.

The first effects of fluid inertia on flows in ordered and random arrays of spheres

Theory and lattice-Boltzmann simulations are used to examine the effects of fluid inertia, at small Reynolds numbers, on flows in simple cubic, face-centred cubic and random arrays of spheres. The drag force on the spheres, and hence the permeability of the arrays, is determined at small but finite Reynolds numbers, at solid volume fractions up to the close-packed limits of the arrays. For small solid volume fraction, the simulations are compared to theory, showing that the first inertial contribution to the drag force, when scaled with the Stokes drag force on a single sphere in an unbounded fluid, is proportional to the square of the Reynolds number. The simulations show that this scaling persists at solid volume fractions up to the close-packed limits of the arrays, and that the first inertial contribution to the drag force relative to the Stokes-flow drag force decreases with increasing solid volume fraction. The temporal evolution of the spatially averaged velocity and the drag force is examined when the fluid is accelerated from rest by a constant average pressure gradient toward a steady Stokes flow. Theory for the short- and long-time behaviour is in good agreement with simulations, showing that the unsteady force is dominated by quasi-steady drag and added-mass forces. The short- and long-time added-mass coefficients are obtained from potential-flow and quasi-steady viscous-flow approximations, respectively.

Moderate Reynolds number flows through periodic and random arrays of aligned cylinders

The effects of fluid inertia on the pressure drop required to drive fluid flow through periodic and random arrays of aligned cylinders is investigated. Numerical simulations using a lattice-Boltzmann formulation are performed for Reynolds numbers up to about 180.The magnitude of the drag per unit length on cylinders in a square array at moderate Reynolds number is strongly dependent on the orientation of the drag (or pressure gradient) with respect to the axes of the array; this contrasts with Stokes flow through a square array, which is characterized by an isotropic permeability. Transitions to time-oscillatory and chaotically varying flows are observed at critical Reynolds numbers that depend on the orientation of the pressure gradient and the volume fraction.In the limit Re[Lt ]1, the mean drag per unit length, F, in both periodic and random arrays, is given by F/(μU) =k1+k2Re2, where μ is the fluid viscosity, U is the mean velocity in the bed, and k1 and k2 are functions of the solid volume fraction ϕ. Theoretical analyses based on point-particle and lubrication approximations are used to determine these coefficients in the limits of small and large concentration, respectively.In random arrays, the drag makes a transition from a quadratic to a linear Re-dependence at Reynolds numbers of between 2 and 5. Thus, the empirical Ergun formula, F/(μU) =c1+c2Re, is applicable for Re>5. We determine the constants c1 and c2 over a wide range of ϕ. The relative importance of inertia becomes smaller as the volume fraction approaches close packing, because the largest contribution to the dissipation in this limit comes from the viscous lubrication flow in the small gaps between the cylinders.

Particle pressure and marginal stability limits for a homogeneous monodisperse gas-fluidized bed : kinetic theory and numerical simulations

A linear stability analysis is performed for the homogeneous state of a monodisperse gas-fluidized bed of spherical particles undergoing hydrodynamic interactions and solid-body collisions at small particle Reynolds number and finite Stokes number. A prerequisite for the stability analysis is the determination of the particle velocity variance which controls the particle-phase pressure. In the absence of an imposed shear, this velocity variance arises solely due to the hydrodynamic interactions among the particles. Since the uniform state of these suspensions is unstable over a wide range of values of particle volume fraction φ and Stokes number St , full dynamic simulations cannot be used in general to characterize the properties of the homogeneous state. Instead, we use an asymptotic analysis for large Stokes numbers together with numerical simulations of the hydrodynamic interactions among particles with specified velocities to determine the hydrodynamic sources and sinks of particle-phase energy. In this limit, the velocity distribution to leading order is Maxwellian and therefore standard kinetic theories for granular/hard-sphere molecular systems can be used to predict the particle-phase pressure and rheology of the bed once the velocity variance of the particles is determined. The analysis is then extended to moderately large Stokes numbers for which the anisotropy of the velocity distribution is considerable by using a kinetic theory which combines the theoretical analysis of Koch (1990) for dilute suspensions (φ [Lt ] 1) with numerical simulation results for non-dilute suspensions at large Stokes numbers. A linear stability analysis of the resulting equations of motion provides the first a priori predictions of the marginal stability limits for the homogeneous state of a gas-fluidized bed. Dynamical simulations following the detailed motions of the particles in small periodic unit cells confirm the theoretical predictions for the particle velocity variance. Simulations using larger unit cells exhibit an inhomogeneous structure consistent with the predicted instability of the homogeneous gas–solid suspension.

Kinetic theory for a monodisperse gas–solid suspension

The fluid‐dynamic and solid‐body interactions among a suspension of perfectly elastic particles settling in a viscous gas are studied. The Reynolds number of the particles, Re≡ρfUa/μ, is small but their Stokes number St≡mŪ/(6πμa2) is large, indicating that particle inertia and viscous forces in the fluid are important. Here, ρf is the density of the fluid, m is the mass of a particle, U is the average velocity of the particles, a is their radius, and μ is the fluid viscosity. Equations for the particle velocity distribution and averages of the fluid and particle velocities are derived. For very large Stokes numbers, St≫φ−3/2, where φ is the particle volume fraction, solid‐body collisions lead to a nearly Maxwellian velocity distribution. On the other hand, at smaller Stokes numbers, St≪φ−3/2, fluid‐dynamic interactions play a more important role in determining the particle velocity distribution and the distribution is not Maxwellian. The amount of energy contained in the particle velocity fluctuations is...

Observations of fibre orientation in simple shear flow of semi-dilute suspensions

The orientations of fibres in a semi-dilute, index-of-refraction-matched suspension in a Newtonian fluid were observed in a cylindrical Couette device. Even at the highest concentration ( nL 3 = 45), the particles rotated around the vorticity axis, spending most of their time nearly aligned in the flow direction as they would do in a Jeffery orbit. The measured orbit-constant distributions were quite different from the dilute orbit-constant distributions measured by Anczurowski & Mason (1967 b ) and were described well by an anisotropic, weak rotary diffusion. The measured ϕ-distributions were found to be similar to Jefferys solution. Here, ϕ is the meridian angle in the flow-gradient plane. The shear viscosities measured by Bibbo (1987) compared well with the values predicted by Shaqfeh & Fredricksons theory (1990) using moments of the orientation distribution measured here.

Anomalous diffusion in heterogeneous porous media

The dispersion of a passive tracer resulting from flow through heterogeneous porous media is studied. When the correlation length for the permeability fluctuations is finite, a normal diffusive process, with the mean-square displacement of the tracer growing linearly with time, is obtained at long times. However, it is shown that, when the correlation length diverges, anomalous diffusion occurs in which the mean-square displacement grows faster than linearly with time. The space–time evolution of the tracers concentration is calculated and shown to be universal—uniquely related to the covariance of the permeability field—in the anomalous regime.

Clustering of aerosol particles in isotropic turbulence

It has been recognized that particle inertia throws dense particles out of regions of high vorticity and leads to an accumulation of particles in the straining-flow regions of a turbulent flow field. However, recent direct numerical simulations (DNS) indicate that the tendency to cluster is evident even at particle separations smaller than the size of the smallest eddy. Indeed, the particle radial distribution function (RDF), an important measure of clustering, increases as an inverse power of the interparticle separation for separations much smaller than the Kolmogorov length scale. Motivated by this observation, we have developed an analytical theory to predict the RDF in a turbulent flow for particles with a small, but non-zero Stokes number. Here, the Stokes number (St) is the ratio of the particles viscous relaxation time to the Kolmogorov time. The theory approximates the turbulent flow in a reference frame following an aerosol particle as a local linear flow field with a velocity gradient tensor and acceleration that vary stochastically in time

A non-local description of advection-diffusion with application to dispersion in porous media

When the lengthscales and timescales on which a transport process occur are not much larger than the scales of variations in the velocity field experienced by a tracer particle, a description of the transport in terms of a local, averaged macroscale version of Ficks law is not applicable. Here, a non-local transport theory is developed in which the average mass flux is not simply proportional to the average local concentration gradient, but is given by a convolution integral over space and time of the average concentration gradient times a spatial- and temporal-wavelength-dependent diffusivity. The non-local theory is applied to the transport of a passive tracer in the advective field that arises in the bulk fluid of a porous medium, and the complete residence-time distribution - space-time response to a unit source input - of the tracer is determined. It is also shown how the method of moments may be simply recovered as a special case of the non-local theory. While developed in the context of and applied to tracer dispersion in porous media, the non-local theory presented here is applicable to the general problem of determining the average transport behaviour in advection-diffusion-type systems in which spatial and temporal variations are occurring on scales comparable with the scale of interest.