Analytical Solution For Diffusion Research Articles

Nodal integral method for convection-diffusion transport using linear and higher order quadrilateral elements

Generic nodal integral method (NIM) based scheme, utilizing nine noded 2D quadratic elements along with four noded 2D linear elements, is developed to solve the fluid flow and heat transfer equations in complex geometries. Non-linear (quadratic) quadrilateral elements are used for discretization of boundary region and linear quadrilateral elements are used for interior region. Lagrange interpolation functions are used to map both type of elements to corresponding square computational elements. The scheme for Neumann and mixed type of boundary conditions are also developed for quadratic elements. C1 type continuity condition is imposed at the interfaces of adjacent elements. Numerical results are compared with analytical solutions for diffusion and advection-diffusion equations. Results for Navier–Stokes equations in curved domain are compared with previously reported experimental as well as numerical results. The comparative study has also been done between presently developed scheme using quadratic and linear elements (referred as scheme-1) and scheme based on complete discretization with linear elements (referred as scheme-2). The comparison shows that both the schemes are of nearly second order accurate while the scheme-1 is more accurate than scheme-2. The results show that the efficient mapping of curved surface with quadratic elements improves the accuracy of NIM schemes.

Analytical solutions for diffusion of organic contaminant through composite liner considering degradation in leachate and soil liner

Analytical solutions for analysing organic contaminant diffusion through a composite liner are presented in this paper. The degradation of organic contaminants in leachate and soil liner are both taken into account. The validity and accuracy of the proposed analytical solutions are verified by comparing with a numerical method. The mass flux and concentration of organic contaminants are calculated by the proposed solutions for zeroconcentration and zero-mass flux boundary condition at the base of composite liner, respectively. The calculated mass flux and concentration at the base of the composite liner decrease with the decrease of contaminant half-life in leachate and soil liner. It can be concluded from parametric studies that increasing the thickness or sorption capacity of the soil liner is effective to improve the performance of composite liner. The solutions can be used for preliminary design of composite liners, and verification of complicated numerical methods.

An analytical solution for diffusion and nonlinear uptake of oxygen in a spherical cell

We develop a new analytical solution for a reactive transport model that describes the steady-state distribution of oxygen subject to diffusive transport and nonlinear uptake in a sphere. This model was originally reported by Lin [S.H. Lin, Oxygen diffusion in a spherical cell with nonlinear oxygen uptake kinetics, J. Theor. Biol. 60 (1976) 449–457] to represent the distribution of oxygen inside a cell and has since been studied extensively by both the numerical analysis and formal analysis communities. Here we extend these previous studies by deriving an analytical solution to a generalized reaction–diffusion equation that encompasses Lin’s model as a particular case. We evaluate the solution for the parameter combinations presented by Lin and show that the new solutions are identical to a grid-independent numerical approximation.

Surface exchange reactions and diffusion in composite materials: A finite element approach

A two-dimensional square grain model has been applied to model oxygen exchange processes between a gas phase and a ceramic composite consisting of two randomly distributed phases of equal grain size (side length of squares). Both average diffusion profiles for thin films and the time dependence of the total amount of exchanged oxygen (relaxation curves) have been calculated numerically by means of the finite element method. The boundary conditions refer to an instantaneous change of the oxygen partial pressure in the surrounding gas phase, which gives rise to surface exchange reactions as well as to diffusion in the composite. Both local equilibrium at the interface between different phases (host phase and inclusions) and blocking heterophase boundaries have been taken into account. The numerical results are compared with the analytical solution for diffusion in a homogeneous medium introducing effective diffusion and surface exchange coefficients. When the relaxation time for effective medium diffusion is considerably shorter than that for the transport process from the host phase into the inclusions, relaxation curves with two separate time constants are predicted. Based on analytical approximations, relaxation times for various limiting cases are given.

Determination of a diffusion coefficient in a membrane by electronic speckle pattern interferometry: a new method and a temperature sensitivity study

In this work, a method has been developed to easily determine the effective diffusion coefficient (De) of a solute in a permeable membrane using electronic speckle pattern interferometry. Fringes are introduced parallel to the direction of diffusion during the diffusion process and De can be calculated by simple measurements on the interference pattern. For a fast and convenient determination of De, a mathematical expression has been derived from the analytical solution of diffusion in two media separated by a resistance. The De obtained when fringes are introduced is in agreement with that obtained when fringes are not introduced. The effect of temperature variation on the optical path of the reference and the object beams has also been investigated. The error introduced into the calculation of De, when the temperature oscillation is not taken into account, has been compared for the case when fringes are not introduced during the diffusion experiment and the case when fringes are introduced. In the first case, the relative error can be greater than 100%. Interestingly, in the latter case, the error caused by temperature oscillation is considerably reduced, and no error is introduced if the temperature changes homogeneously over the whole diffusion cell used for the diffusion experiment.

Review of the use of Péclet numbers to determine the relative importance of advection and diffusion in low permeability environments

In low permeability environments, transport by advection is often neglected based on a Péclet number criterion. Such a criterion usually states that if the Péclet number (Pe) is much smaller than 1, diffusion dominates over advection and transport may be modeled considering diffusion only. Unfortunately, up to 10 different Péclet number definitions exist and for a particular case these different definitions lead to very diverse Péclet number values, differing several orders of magnitude from each other. In this paper, the different Péclet number definitions are therefore evaluated on their ability to determine the relative importance of transport by advection and by diffusion in low permeability environments. This is done by comparing the results of the analytical solution for pure diffusion with the analytical solution for diffusion, advection and dispersion for a large number of different input parameter values. The relation between the different Péclet numbers and the difference between the calculated concentration considering diffusion only and the calculated concentration considering both diffusion and advection is studied. These calculations show that some Péclet number definitions are not well suited to decide whether advection may be neglected in low permeability media.

Transit-Time Design for Diffusion through Composite Liners

Transit-time design methods are presented in this paper for determining the design thickness for composite liners consisting of a geomembrane and a compacted soil liner or geosynthetic clay liner. The design methods are based on a closed-form analytical solution for transient solute diffusion of volatile organic compounds in a composite liner and results from a numerical model. An analytical solution for diffusion in a two-layer soil profile, which is useful for transit-time design of composite liners, is also presented. The analytical solutions are used to develop graphical solution charts that can be used to design composite liners for which the effluent concentration and contaminant flux are less than a specified value. Design examples are included for a composite liner having a compacted soil liner and a composite liner having a geosynthetic clay liner. The method is relatively simple to apply and can be used for preliminary design of composite liners, evaluating experimental results, and verifying more complex numerical models.

VOLUME-OF-FLUID CALCULATION OF HEAT OR MASS TRANSFER ACROSS DEFORMING INTERFACES IN TWO-FLUID FLOW

A new volume-of-fluid (VOF)-based numerical method for calculating heat transfer or mass transfer of a species within and between fluids with deforming interfaces is described. The algorithm is tested first against an analytical solution for diffusion from a sphere, and good agreement between theory and calculation is shown. The method is then demonstrated by predicting (a) heat transfer from a rising bubble when the bubble forms a toroidal shape, and (b) mass transfer from a rising drop when the drop phase controls diffusion. The method is shown to be a viable approach for complex interfacial heat/mass transfer.

Diffusion of Moisture in Adhesively Bonded Joints

In this article, diffusion of moisture in adhesively bonded composite joints is discussed and analysed experimentally, analytically and numerically. The experimental studies concentrate on moisture diffusion in adhesive films and in unidirectional and multidirectional composite substrates exposed to two different conditioning environments, namely 45°C/85% RH and 90°C/97% RH for the absorption studies and 90°C/ambient for the desorption studies. The coefficients of diffusion are determined from the water uptake plots. The analytical solutions for diffusion in joints with impermeable adherends are based on the classical theory of diffusion and are used to derive equations in two-dimensions for different adhesive fillet shapes, namely radiused fillet, triangular fillet and rectangular fillet. In the finite element analysis, the diffusion of moisture from the composite substrates into lap-strap joints is also taken into account. Both unidirectional and multidirectional composites are considered, as well as two different fillet shapes, i.e., rectangular and triangular fillet. A comparison between the results obtained using FEA and those obtained using the analytical solution is made. Finally, fatigue test data for lap-strap joints aged and tested in different environments is presented and a tentative link between fatigue threshold and water concentration at the site of failure initiation is made, indicating a semi-empirical method of predicting the strength of joints subjected to moisture-induced degradation.

An NMR pulsed field gradient study of the electrical and conventional heating of carrot

SummaryExperiments have been carried out using pulsed field gradient NMR to characterize the self‐diffusion of water within samples of carrot, both uncooked and cooked by either volumetric electrical or conventional heating. The method detected the presence of structure within uncooked carrot, and the disappearance of the structure on cooking. The NMR data were fitted to two models: (i) an analytical solution for diffusion between impermeable barriers and (ii) a numerical solution to the more physically realistic situation of diffusion between permeable barriers. The barrier spacings obtained with raw carrot data using both methods (i) and (ii) were consistent with cell sizes found in carrots. The effect of cooking on the model fit was either an increase in barrier spacing (method i), or an increase in barrier permeability (method ii). There was a statistically significant difference between samples cooked electrically and conventionally, but this was much less than the difference between cooked and uncooked material.

An approximate analytical solution for the nonlinear inverse axial dispersion model

An analytical solution for diffusion with homogenous n the order nonlinear chemical reaction in laminar flow of a Newtonian fluid through tubular reactor with unknown boundary conditions is presented. The results of the new solution have been compared with previously published numerical solutions and shown to be accurate.

An analytical solution for diffusion and reaction in a laminar flow tubular reactor

An exact analytical solution in terms of Laguerre hypergeometric function to problem of diffusion with first order chemical reaction in laminar flow through a tubular reactor is presented. A method of separation of variables followed by change of variables and Laplace transformation has been efficiently applied to the governing partial differential equation. The analytical solution obtained in this work has been compared with previously published numerical solution and shown to be an accurate and reliable solution.

Analytical solutions for diffusion on fractal objects.

A generalization of the Euclidean diffusion equation is proposed for diffusion on fractals on the basis of a scaling argument, a renormalization-group theory for the Green's function, and numerical tests. Conjectures on the applicability to natural fractals (such as porous media) are presented.

An analytical solution for diffusion in a shrinking body

AbstractAn analytical solution for the equation that describes the mass balance for a solvent diffusing out of a shrinking body gives the variation of the half‐thickness of an infinite slab and its diffusant content as functions of time. This solution is compared with numerical results obtained previously for a slab of finite thickness.

An approximate solution for diffusion and reaction in a non-ideal tubular reactor

An analytical solution for diffusion with homogeneous first order chemical reaction in laminar flow of a Newtonian fluid through tubular reactor is presented within the framework of Taylor's dispersion theory. The novelty about the work is that Galerkin's technique has been used for the first time to obtain the expression for the effective dispersion coefficient.

An approach in modeling the tubular polymer reactor

AbstractAn analytical solution for diffusion with a homogeneous first‐order reaction in the bulk and a heterogeneous reaction at the reactor wall in a non‐Newtonian laminar flow tubular reactor is presented by using the Galerkin technique. The effect of reaction rate constants on dispersion is studied under isothermal conditions. It is found that, for the same mean velocity of the flow, the effective dispersion coefficient decreases with increase in the chemical reaction rate constants.

Mass transport of solutes in dual‐porosity media

In many aquifers, solute mass transport is controlled by molecular diffusion between mobile and static water. This is especially the case in fissured aquifers in which the rock matrix has a high porosity but low permeability. In order to model mass transport in such a dual‐porosity medium, an analytical solution for diffusion from a constant concentration boundary has been incorporated into a two‐dimensional mass transport model. The model uses Galerkin's method of weighted residuals and isoparametric finite elements to solve the flow and transport equations. The method has been successfully applied to a largescale pollution event. Historical data on the event were available over a long period of time and permitted calibration of the model. In situations in which such historical data are not available, calibration is very difficult, particularly the determination of dispersivities. This is because the molecular diffusion between mobile and static water so greatly retards the transport of solutes that field tests at the appropriate scale are essentially preempted.