Calculation of the effective diffusivity of heterogeneous media using the lattice-Boltzmann method

Identifying the dominant transport mechanism in single nanoscale pores and 3D nanoporous media

Gas transport mechanisms can be categorized into viscous flow and mass diffusion, both of which may coexist in a porous media with multiscale pore sizes. To determine the dominant transport mechanism and its contribution to gas transport capacity, the gas viscous flow and mass diffusion processes are analyzed in single nanoscale pores via a theoretical method, and are simulated in 3D nanoporous media via pore-scale lattice Boltzmann methods. The apparent permeability from the viscous flow and apparent diffusivity from the mass diffusion are estimated. A dimensionless parameter, i.e., the diffusion-flow ratio, is proposed to evaluate the dominant transport mechanism, which is a function of the apparent permeability, apparent diffusivity, bulk dynamic viscosity, and working pressure. The results show that the apparent permeability increases by approximately two orders of magnitude when the average Knudsen number (Kn avg) of the nanoporous media or Knudsen number (Kn) of single nanoscale pores increases from 0.1 to 10. Under the same conditions, the increment in the apparent diffusivity is only approximately one order of magnitude. When Kn < 0.01, the apparent permeability has a lower bound (i.e., absolute permeability). When Kn > 10, the apparent diffusivity has an upper bound (i.e., Knudsen diffusivity). The dominant transport mechanism in single nanoscale pores is the viscous flow for 0.01 < Kn < 100, where the maximum diffusion-flow ratio is less than one. In nanoporous media, the dominant transport relies heavily on Kn avg and the structural parameters. For nanoporous media with the pore throat diameter of 3 nm, Kn avg = 0.2 is the critical point, above which the mass diffusion is dominant; otherwise, the viscous flow is dominant. As Kn avg increases to 3.4, the mass diffusion is overwhelming, with the maximum diffusion-flow ratio reaching ∼4.

Prediction of the moments in advection-diffusion lattice Boltzmann method. II. Attenuation of the boundary layers via double-Λ bounce-back flux scheme

Impact of the unphysical tangential advective-diffusion constraint of the bounce-back (BB) reflection on the impermeable solid surface is examined for the first four moments of concentration. Despite the number of recent improvements for the Neumann condition in the lattice Boltzmann method-advection-diffusion equation, the BB rule remains the only known local mass-conserving no-flux condition suitable for staircase porous geometry. We examine the closure relation of the BB rule in straight channel and cylindrical capillary analytically, and show that it excites the Knudsen-type boundary layers in the nonequilibrium solution for full-weight equilibrium stencil. Although the d2Q5 and d3Q7 coordinate schemes are sufficient for the modeling of isotropic diffusion, the full-weight stencils are appealing for their advanced stability, isotropy, anisotropy and anti-numerical-diffusion ability. The boundary layers are not covered by the Chapman-Enskog expansion around the expected equilibrium, but they accommodate the Chapman-Enskog expansion in the bulk with the closure relation of the bounce-back rule. We show that the induced boundary layers introduce first-order errors in two primary transport properties, namely, mean velocity (first moment) and molecular diffusion coefficient (second moment). As a side effect, the Taylor-dispersion coefficient (second moment), skewness (third moment), and kurtosis (fourth moment) deviate from their physical values and predictions of the fourth-order Chapman-Enskog analysis, even though the kurtosis error in pure diffusion does not depend on grid resolution. In two- and three-dimensional grid-aligned channels and open-tubular conduits, the errors of velocity and diffusion are proportional to the diagonal weight values of the corresponding equilibrium terms. The d2Q5 and d3Q7 schemes do not suffer from this deficiency in grid-aligned geometries but they cannot avoid it if the boundaries are not parallel to the coordinate lines. In order to vanish or attenuate the disparity of the modeled transport coefficients with the equilibrium weights without any modification of the BB rule, we propose to use the two-relaxation-times collision operator with free-tunable product of two eigenfunctions Λ. Two different values Λ_{v} and Λ_{b} are assigned for bulk and boundary nodes, respectively. The rationale behind this is that Λ_{v} is adjustable for stability, accuracy, or other purposes, while the corresponding Λ_{b}(Λ_{v}) controls the primary accommodation effects. Two distinguished but similar functional relations Λ_{b}(Λ_{v}) are constructed analytically: they preserve advection velocity in parabolic profile, exactly in the two-dimensional channel and very accurately in a three-dimensional cylindrical capillary. For any velocity-weight stencil, the (local) double-Λ BB scheme produces quasi-identical solutions with the (nonlocal) specular-forward reflection for first four moments in a channel. In a capillary, this strategy allows for the accurate modeling of the Taylor-dispersion and non-Gaussian effects. As illustrative example, it is shown that in the flow around a circular obstacle, the double-Λ scheme may also vanish the dependency of mean velocity on the velocity weight; the required value for Λ_{b}(Λ_{v}) can be identified in a few bisection iterations in given geometry. A positive solution for Λ_{b}(Λ_{v}) may not exist in pure diffusion, but a sufficiently small value of Λ_{b} significantly reduces the disparity in diffusion coefficient with the mass weight in ducts and in the presence of rectangular obstacles. Although Λ_{b} also controls the effective position of straight or curved boundaries, the double-Λ scheme deals with the lower-order effects. Its idea and construction may help understanding and amelioration of the anomalous, zero- and first-order behavior of the macroscopic solution in the presence of the bulk and boundary or interface discontinuities, commonly found in multiphase flow and heterogeneous transport.

Effects of viscosity, surface tension, and evaporation rate of solvent on dry colloidal structures: a lattice Boltzmann study

Understanding the mechanisms of how colloidal solution properties and drying processes result in dry colloidal structures is essential for industrial applications such as paint, ceramics, and electrodes. In this study, we develop a computational method to simulate the drying process of colloidal suspensions containing solid particles and polymers. The method consists of a solvent evaporation model, a fluid particle dynamics method, and a two-phase lattice Boltzmann method. We determine that a high-viscosity solvent, small surface tension, and a high evaporation rate of the solvent lead to a structure with dispersed particles and interconnected pores. When these conditions are not present, the particles agglomerate and the pores are disconnected.

Continuum versus discrete model: a comparison for multicellular tumour spheroids

We study multicellular tumour spheroids with a continuum model based on partial differential equations (PDEs). The model includes viable and necrotic cell densities, as well as oxygen and glucose concentrations. Viable cells consume nutrients and become necrotic below critical nutrient concentrations. Proliferation of viable cells is contact-inhibited if the total cellular density locally exceeds volume carrying capacity. The model is discussed under the assumption of spherical symmetry. Unknown model parameters are determined by simultaneously fitting the cell number to several experimental growth curves for different nutrient concentrations. The outcome of the PDE model is compared with an analogous off-lattice agent-based model for tumour growth. It turns out that the numerically more efficient PDE model suffices to explain the macroscopic growth data. As in the agent-based model, we find that the experimental growth curves are only reproduced when a necrotic core develops. However, evaluation of morphometric properties yields differences between the models and the experiment.