Governing Diffusion Equation Research Articles

Divergence of Velocity Fields in Electrochemical Systems

The passage of current through electrochemical systems results in the development of concentration gradients in the electrolytic phase that can be modeled using concentrated solution theory. Application of this theory requires knowledge of three concentration-dependent transport coefficients, which are often taken to be conductivity, diffusion coefficient, and the cation transference number with respect to the solvent velocity. The governing diffusion equation for molar concentration contains two additional terms—the thermodynamic factor which is related to activity of the electrolytic species and the solvent velocity. The main advance in this paper is the derivation of an expression for the divergence of the solvent velocity. Solving this equation requires knowledge of the partial molar volume of the electrolyte. Analogous expressions are derived for the mass average and molar average velocities. These velocities occur naturally in the diffusion equation if concentration is expressed as weight fraction and mole fraction of the electrolytic species, respectively.

Early-Time 1D Analysis of Shale-Oil and -Gas Flow

Summary We present a new semianalytic method to solve the nonlinear pressure-diffusion equation at early time, before reservoir boundaries are encountered, and under constant bottomhole pressure (BHP), applicable to the analysis of unconventional reservoirs. We assume that the flow rate is inversely proportional to the square root of time since the beginning of production. The method is an extension of the semianalytic solution proposed by Schmid et al. (2011) for spontaneous imbibition; we replace the solution for saturation with one for pressure, while extending the functional form of the governing diffusion equation. The solution can accommodate arbitrary pressure-dependent nonlinear rock and fluid properties as well as production caused by desorption. The mathematical formulation is presented for a general nonlinear case and tested by use of synthetic data. Field production from the Barnett Shale is then used to estimate effective matrix permeability. The model can be used to predict production if the rock and fluid properties are known, or can be used to constrain reservoir properties from production data. It is a complement to traditional pressure- or rate-transient analysis; if the response of a well for constant-pressure production can be determined, our method can be used to determine reservoir properties, without any approximations inherent in linearizing the flow equations.

Analytical model for 1-D contaminant diffusion through clay barriers

Studies on diffusion transport of contaminants through clays are vitally important for performance assessment of waste disposal facilities. Laboratory through-diffusion (double-reservoir) experiment represents a well-established and routinely used technique for the determination of design mass transport parameters in a single test. Availability of accurate and simple theoretical models is valuable for simulating diffusion transport of contaminants through clays for the design and evaluation processes of many waste disposal facilities. The theoretical solutions for this problem are commonly obtained by numerically solving the governing diffusion equation in combination with finite mass boundary conditions. The lack of analytical solutions to this problem encouraged the development of numerical solutions. However, a high computational cost is associated with such time-marching numerical solutions. Closed-form analytical solutions are proposed, in this work, for one-dimensional (1D) transient through diffusion transport of sorptive contaminants through homogeneous clays. Proposed analytical solutions are compared with the solution obtained using existing methods on experimental data. The proposed analytical solutions are simple and accurate for theoretically estimating transient diffusion transport of sorptive contaminants through clays. The proposed model has ready application for analysing routine laboratory through-diffusion experimental data.

Transport Processes in Vaginal Films that Release Anti-HIV Microbicide Molecules

A promising method for blocking sexual transmission of HIV is application of topical microbicide molecules to mucosal surfaces and the fluids contacting them. There is widespread agreement that more effective and diverse drug delivery vehicles, as well as better active ingredients, must be developed to increase microbicide efficacy. There is now great interest in developing a variety of delivery vehicles to suit the preferences of a diverse group of users. These include intravaginal rings and dissolving films, as well as the original microbicide gels. Here, we develop a mechanistic mathematical model of the dissolution of a microbicide-bearing polymer film, and subsequent distribution of its active drug throughout the vaginal lumen. The film dissolves by first imbibing (or taking up) solvent (vaginal fluid), whereupon its material structure changes in a way that frees individual polymer molecules in the film to move. In the model, the polymer structural relaxation via water uptake forms a two-phase, glassy-rubbery system in which interfacial movement follows Fickian (i.e. diffusion dominated) dynamics. However, at some intermediate time the interfacial motion ceases to follow Fickian dynamics, due to viscoelastic stresses in the polymer. As the glassy polymer transforms into a rubbery polymer, drugs carried within the film are freed to move throughout the relaxed network, and begin to diffuse through the vaginal lumen in all directions, especially laterally between the apposed vaginal walls. Therefore, as the initially fully glassy polymer matrix relaxes to a two-phase, glassy-rubbery system, the governing diffusion equation is simultaneously solved to track the motion of released drug molecules in the vaginal lumen and model the drug delivery. This is the first mechanistic model of how a vaginal film releases microbicidal molecules to neutralize HIV virions and inhibit transmission. [Supported by NIH AI077289]

Approximate Solution Methods for Solid-State Diffusion in Phase-Change Electrodes

Approximate solution methods have been developed for solid-state diffusion in phase-change electrodes, which, when incorporated into a porous electrode battery model, would provide a computational advantage over the exact rigorous solution without sacrificing accuracy. A shrinking core approach has been used to describe the phase boundary movement in a spherical electrode particle, where a shell of one phase covers a core of the other phase. Two approximations have been developed in this work, which account for diffusion in the shell while taking into account the moving phase interface. It is assumed that there are no concentration gradients in the core at any time. The validity of the approximations has been tested by solving the governing diffusion equation with the moving boundary using the arbitrary Lagrangian–Eulerian application mode in COMSOL Multiphysics. One of the approximations (order-2) is seen to perform very well over a wide range of discharge currents in predicting the positions of the moving interface, surface concentrations, concentration profiles, etc. The order-2 approximation is also seen to capture scenarios where diffusion continues in a single-phase particle after the core has been completely consumed.

Stable numerical evaluation of Grünwald–Letnikov fractional derivatives applied to a fractional IHCP

The computation of Grünwald–Letnikov fractional derivatives from noisy data is considered as an ill-posed problem and treated by mollification techniques. It is shown that, with the appropriate choice of the radius of mollification, the method is a regularizing algorithm. Next, the recovery of the boundary temperature and heat flux functions from one measured transient temperature data at some interior point of a one-dimensional semi-infinite conductor when the governing diffusion equation is of fractional type is discussed. A simple algorithm based on space marching mollification techniques and Grünwald–Letnikov fractional derivatives is introduced for the numerical solution of this inverse ill-posed problem. In all cases, stability and error estimates are included together with numerical examples of interest.

Degenerate Dirichlet Problems Related to the Invariant Measure of Elasto-Plastic Oscillators

A stochastic variational inequality is proposed to model a white noise excited elasto-plastic oscillator. The solution of this inequality is essentially a continuous diffusion process for which a governing diffusion equation is obtained to study the evolution in time of its probability distribution. The diffusion equation is degenerate, but using the fact that the degeneracy occurs on a bounded region we are able to show the existence of a unique solution satisfying the desired properties. We prove the ergodic properties of the process and characterize the invariant measure. Our approach relies on extending Khasminskii’s method (Stochastic Stability of Differential Equations, Sijthoff and Noordhoff, 1980), which in the present context leads to the study of degenerate Dirichlet problems with nonlocal boundary conditions.

Finite Element Analysis of Cell Coarsening in Plastic Foaming

In this study, cell coarsening in plastic foaming is investigated through numerical simulation. Cell coarsening occurring in two adjacent bubbles of different sizes in a finite volume of polymer melt is considered to be representative of the whole foaming system. A quadratic triangle-based finite element analysis with an implicit scheme for time evolution is utilized to solve the governing diffusion equation in the axisymmetric coordinate system. The effects of the bulk gas concentration, the intercellular distance, and the initial bubble sizes on cell coarsening are estimated. Efforts are made to improve a fundamental understanding of cell coarsening in plastic foaming.

Parallel simulation of anisotropic diffusion with human brain DT-MRI Data

We conduct simulations for the 3D unsteady state anisotropic diffusion process with DT-MRI data in the human brain by discretizing the governing diffusion equation on Cartesian grid and adopting a high performance differential–algebraic equation (DAE) solver, the parallel version of implicit differential–algebraic (IDA) solver, to tackle the resulting large scale system of DAEs. Parallel preconditioning techniques including sparse approximate inverse and banded-block-diagonal preconditioners are used with the GMRES method to accelerate the convergence rate of the iterative solution. We then investigate and compare the efficiency and effectiveness of the two parallel preconditioners. The experimental results of the diffusion simulations on a parallel supercomputer show that the sparse approximate inverse preconditioning strategy, which is robust and efficient with good scalability, gives a much better overall performance than the banded-block-diagonal preconditioner.

Irreversible adsorption of particles at random-site surfaces.

Irreversible adsorption of negatively charged polystyrene latex particles (averaged diameter 0.9 microm) at heterogeneous surfaces was studied experimentally. The substrate bearing a controlled number of adsorption sites was produced by precovering mica sheets by positively charged polystyrene latex (averaged diameter of 0.45 microm). Positive latex (site) deposition was carried out under diffusion-controlled transport conditions and its coverage was determined by direct particle counting using the optical microscopy. Deposition kinetics of larger latex particles (averaged diameter 0.9 microm) at heterogeneous surfaces produced in this way was studied by direct optical microscope observations in the diffusion cell (under no-convection transport conditions). It was demonstrated that the structure of larger particle monolayers, characterized in terms of the pair correlation function, showed much more short-range ordering than it was predicted for homogeneous surface monolayers at the same coverage. This was found in agreement with theoretical predictions derived from the Monte Carlo simulations. On the other hand, particle adsorption kinetics was quantitatively interpreted in terms of numerical solutions of the governing diffusion equation with the nonlinear boundary condition derived from Monte Carlo simulations. From these kinetic measurements maximum (jamming) coverage of particles was determined in an accurate way by extrapolation. It was concluded that both the monolayer structure and jamming coverage were strongly influenced by the site multiplicity (coordination) effect.

One‐dimensional fluid diffusion induced by constant‐rate flow injection: Theoretical analysis and application to the determination of fluid permeability and specific storage of a cored rock sample

We describe the conceptual design and first application of a new method for simultaneously measuring the fluid permeability and specific storage of a rock sample. In our laboratory tests, a constant flow rate single‐stroke piston pump injects fluid into a cored rock specimen placed between two reservoirs in which fluid pressure is recorded. For this geometry we have derived a new analytic solution of the governing diffusion equation describing the one‐dimensional fluid flow. This new analytic solution in the time domain consists of two parts: an asymptotic linear function of time, and a transient part which decays to zero as time increases. The model predicts that the fluid pressures of the upstream and downstream reservoirs both increase linearly with time after the initial transient vanishes. The slope of the linear pressure variation depends on the specific storage of the rock sample for a given test condition, and the differential pressure between the two reservoirs is related to the permeability. If the downstream pressure is not recorded, the permeability can be calculated from the zero intercept of the linear upstream fluid pressure variation. This calculation is quite straightforward and no tedious history curve matching is required. We applied our new method to measure fluid permeability and specific storage of Westerly granite. The measured values of the permeability are consistent with those published in the literature. The main advantages of our method are the reliability of the testing method, its economy of time, and the flexibility in adapting the system parameters to tests at different conditions.

Performance of ILU preconditioning techniques in simulating anisotropic diffusion in the human brain

We conduct simulations for the unsteady state anisotropic diffusion process in the human brain by discretizing the governing diffusion equation on a face-centered cubic grid and adopting a high performance differential-algebraic equation solver, IDA, to deal with the resulting large-scale system of DAEs. Incomplete LU preconditioning techniques are used with the GMRES method to accelerate the convergence rate of the iterative solution. We then investigate and compare the efficiency and effectiveness of a number of ILU preconditioners, and find out that the ILUT with a dual dropping strategy gives the best overall performance when it is provided with the optimum choices of the fill-in parameter and the threshold dropping tolerance.

First-passage time for randomly excited non-linear oscillators

An approximate method of computing first-passage probabilities for non-linear oscillators subjected to stationary wide-band random excitation is discussed. The method is based upon an approximation of the energy envelope of the oscillator response as a one-dimensional Markov process, governed by an appropriate diffusion equation. The adoption of suitable absorbing and reflecting boundary conditions enables the evolution of the first-passage probability to be computed. An efficient and simple algorithm for the numerical solution of the governing diffusion equation is outlined which is based upon an implicit, finite-difference approximation. Numerical results for the mean first-passage time are compared with the results of a finite element numerical solution of the exact two-dimensional Pontriagin-Vitt equation for this statistic.

Interfacial geometry and d-variation effects in two-phase systems

Numerical solutions of the governing diffusion equation for two-phase concentration dependent diffusion coefficients are examined. Solutions were also calculated for planar, cylindrical, and spherical geometries to compare the effect of interface geometries with those caused by concentration-dependent diffusion coefficients, and two methods of averaging D were considered to determine the best averaging method for different types of D-variations. The effects of interface-location criteria on mass conservation and convergence of interface location, diffusion coefficient variation in the alpha and beta-phases of a two-phase binary alloy system, effect of D(alpha) variation in a cylindrical couple on beta-phase thickness, and geometry and D-variation effects on the degree of homogenization were determined. It is concluded that typical D(alpha)-variations can have a greater influence on the kinetics of interdiffusion than the geometry.