Longitudinal Dispersion Coefficient: Effects of Particle-Size Distribution

Abstract

The effects of particle-size distribution on the longitudinal dispersion coefficient ($D_{\mathrm{L}})$ in packed beds of spherical particles are studied by simulating a tracer column experiment. The packed-bed models consist of uniform and different-sized spherical particles with a ratio of maximum to minimum particle diameter in the range of 1–4. The modified version of Euclidian Voronoi diagrams is used to discretize the system of particles into cells that each contains one sphere. The local flow distribution is derived with the use of Laurent series. The flow pattern at low particle Reynolds number is then obtained by minimization of dissipation rate of energy for the dual stream function. The value of $D_{\mathrm{L}}$ is obtained by comparing the effluent curve from large discrete systems of spherical particles to the solution of the one-dimensional advection–dispersion equation. Main results are that at Peclet numbers above 1, increasing the width of the particle-size distribution increases the values of $D_{\mathrm{L}}$ in the packed bed. At Peclet numbers below 1, increasing the width of the particle-size distribution slightly lowers $D_{\mathrm{L}}$.

Appendix A: Permeability

A fluid can only generate tangential viscous stresses on a solid surface with a non-slip boundary condition. The viscous force exerted on the particle is an integral of vorticity $\varvec{\upomega }^{{sph}}$ at the surface:

$$\begin{aligned} f_{\mathrm{viscous}} =\mu \int \int \limits _{{sph}} {{\varvec{\upomega }}^{{sph}}} \times {\mathbf{n}}dS, \end{aligned}$$

(14)

where n is the normal direction related to the particle surface, and index sph denotes the sphere. Then, normal force, in turn, depends on the pressure P distribution around the particle. The force, for example, in the $z$-direction is:

$$\begin{aligned} f_{z,{\mathrm{normal}}}&= \int \int \limits _{sph} {Pn_z } dS \nonumber \\&= R^{2}\int \limits _0^{2\pi } {\int \limits _0^\pi P } (\theta ,\phi )\sin (\theta )\cos (\theta )d\theta d\phi =-\frac{R^{2}}{2}\int \limits _0^{2\pi } {\int \limits _0^\pi {\frac{\partial P}{\partial \theta }} } \sin ^{2}\theta d\theta d\phi ,\qquad \end{aligned}$$

(15)

where axis $\theta$ = 0 coincides with $z$-axis, and inner integral has been integrated by parts.

According to the momentum equation, the change of pressure at the surface of the particle is

$$\begin{aligned} \frac{1}{R}\frac{\partial P}{\partial \theta }=\mu \frac{\partial \omega _\phi ^{sph} }{\partial n}. \end{aligned}$$

(16)

where $n$ is coordinate in normal direction from the surface of particle. Combining viscous and normal forces results in the following force on the particle

$$\begin{aligned} f_z =\mu R^{2}\int \limits _0^{2\pi } {\int \limits _0^\pi {\left( {{\omega } _\phi ^{sph} -\frac{R}{2}\frac{\partial {\omega } _\phi ^{sph} }{\partial n}} \right) } } \sin ^{2}\theta d\theta d\phi . \end{aligned}$$

(17)

The permeability tensor K can be obtained by summing drag forces to all the particles:

$$\begin{aligned} \mathbf{K }\sum _i {\mathbf{f }_i } =\mu V\left\langle \mathbf{v} \right\rangle , \end{aligned}$$

(18)

where $i$ is the sum over all particles, $V$ is the volume of the system, and $\left\langle \mathbf{v} \right\rangle$ denotes the average velocity of the system. Figure 8 shows the relationship between the obtained permeabilities and porosities in the packed-bed model for uniform-sized spheres that are compared to Kozeny–Carman formula.

Fig. 8

Symbols (circle) show calculated permeability versus porosity. Line represents Kozeny–Carman formula with multiplier 1.4: $\frac{K}{\bar{{d}}^{2}}=\frac{1.4\varepsilon ^{3}}{180(1-\varepsilon )^{2}}$; ${\bar{d}}$ is the average diameter of a sphere