Analytical solution of the advection–diffusion transport equation using a change-of-variable and integral transform technique

This paper presents a formal exact solution of the linear advection–diffusion transport equation with constant coefficients for both transient and steady-state regimes. A classical mathematical substitution transforms the original advection–diffusion equation into an exclusively diffusive equation. The new diffusive problem is solved analytically using the classic version of Generalized Integral Transform Technique (GITT), resulting in an explicit formal solution. The new solution is shown to converge faster than a hybrid analytical–numerical solution previously obtained by applying the GITT directly to the advection–diffusion transport equation.     

Semianalytical method of solution for solid phase diffusion in lithium ion battery electrodes: Variable diffusion coefficient

A semianalytical methodology based on the integral transform technique is proposed to solve the diffusion equation with concentration dependent diffusion coefficient in a spherical intercalation electrode particle. The method makes use of an integral transform pair to transform the nonlinear partial differential equation into a set of ordinary differential equations, which is solved with less computational efforts. A general solution procedure is presented and two illustrative examples are used to demonstrate the usefulness of this method for modeling of diffusion process in lithium ion battery electrode. The solutions obtained using the method presented in this study are compared to the numerical solutions.     

Determination of moisture diffusion coefficient in transformer paper using thermogravimetric analysis

This paper presents the methodology used and the results obtained in the determination of moisture diffusion coefficient of non-impregnated transformer insulating paper. In order to establish the diffusion coefficient, drying curves of paper samples were obtained by means of thermogravimetric experiments. The diffusion coefficient parameters were found by applying an optimization process based on genetic algorithms. The error function between measured and simulated curves was determined, and the parameters achieving the best correspondence between measured and estimated values were obtained. As a result, a new equation for the diffusion coefficient of non-impregnated insulating paper is proposed, depending on average moisture concentration, temperature and insulation thickness. The proposed coefficient was validated through experimental cases finding a good agreement between the experimental drying curves and those obtained by simulation using the diffusion coefficient. The proposed diffusion coefficient can be used for the determination of the time required to dry power transformers in factory.     

Effect of sweep gas on hydrogen permeation of supported Pd membranes: Experimental and modeling

Membrane reactor processes are being increasingly proposed as an attractive solution for pure hydrogen production due to the possibility to integrate production and separation inside a single reactor vessel. High hydrogen purity can be obtained through dense metallic membranes, especially palladium and its alloys, which are highly selective to hydrogen. The use of thin membranes seems to be a good industrial solution in order to increase the hydrogen flux while reducing the cost of materials. Typically, the diffusion through the membrane layer is the rate-limiting step and the hydrogen permeation through the membrane can be described by the Sieverts’ law but, when the membrane becomes thinner, the diffusion through the membrane bulk becomes less determinant and other mass transfer limitations might limit the permeation rate. Another way to increase the hydrogen flux at a given feed pressure, is to increase the driving force of the process by feeding a sweep gas in the permeate side. This effect can however be significantly reduced if mass transfer limitations in the permeate side exist. The aim of this work is to study the mass transfer limitation that occurs in the permeate side in presence of sweep gas. A complete model for the hydrogen permeation through PdAg membranes has been developed, adding the effects of concentration polarization in retentate and permeate side and the presence of the porous support using the dusty gas model equation, which combines Knudsen diffusion, viscous flow and binary diffusion. By studying the influence of the sweep gas it has been observed that the reduction of the driving force is due to the stagnant sweep gas in the support pores while the concentration polarization in the permeate is negligible.     

Unidimensional formulation of fick's second law for prismatic bodies

Using the concept of similarity numbers, the analytical solution of Fick's second law for diffusion in sphere was conveniently modified to describe a three-dimensional diffusion process. A method to calculate the similarity numbers for prismatic bodies was developed. Concentration values calculated by means of the proposed equation were compared with the corresponding to the analytical solution for prism with satisfactory results.     

On the self-similar,early-time,anomalous diffusion in random networks — Approach by fractional calculus

In a recent work, Zhang and Padrino (2017) derived an equation for diffusion in random networks consisting of junction pockets and connecting channels by applying the ensemble average method to the mass conservation principle. The resulting integro-differential equation was solved numerically using finite volumes for the test case of one-dimensional diffusion in the half-line. For early time, they found that the numerical predictions of pocket mass density depend on the similarity variable xt ^−1/4, representing sub-diffusion. They argue that the sub-diffusive behavior is a consequence of the time needed to establish a linear concentration profile inside a channel. By theoretical analysis of the diffusion equation for small time, they confirmed this finding. Nevertheless, they did not present an exact solution for the small time limit to compare with. Here, starting with their small-time leading order diffusion equation in (x,t) space, we use elements of fractional calculus to cast it into a form for which an analytical solution has been obtained in the literature for the same boundary and initial conditions in terms of the Fox H-function (Schneider and Wyss, 1989). For ease of computation, we express the solution in terms of the Meijer G-function. We compare the exact solution with Zhang and Padrino's numerical predictions, resulting in excellent agreement, thereby validating their numerical approach.     

An integral equation approach to diffusion

A method of solution of transient diffusion, e.g. heat conduction, problems in homogeneous and isotropic media with internal sources and arbitrary (including nonlinear) boundary conditions and initial conditions is proposed. The method is based on the reduction of the problem to one only involving surface values of temperature and/or heat flux in the form of an integral equation through the introduction of fundamental solutions and the use of Green's theorem. The integral equation is solved numerically for a specific example.     

Liquid diffusion model for drying a stack of rough rice

Drying of a stack of rough rice was experimentally and theoretically investigated. Stacks of rough rice having different heights were dried with forced convection of warm air. A theoretical model was developed for predicting the bulk drying kinetics of a stack of rough rice using the analytical solution of liquid diffusion equation based on Fick's law. Effective diffusion coefficients were obtained minimizing the sum of squared differences between the theoretical results and experimental data obtained for various drying conditions. Drying air temperature is the most effective factor on the total rate of moisture removal from the stack. The bulk drying rate of a stack of rough rice was considerably reduced as compared to that of a single layer as the height of the stack increased. Agreement between the theoretical and experimental result is very good. Copyright © 2003 John Wiley & Sons, Ltd.     

Solar powered air conditioning system using rotary honeycomb desiccant wheel

A solar powered air conditioning system using liquid desiccant is proposed. A solar air heater containing a porous material is used for regeneration purpose in the proposed system. The honeycomb desiccant rotary wheel is constructed from iron wire and clothes layer impregnated with calcium chloride solution, in honeycomb form, is utilized for the regeneration and absorption processes. The effect of airflow rate and solar radiation intensity on the system regeneration and absorption processes are studied. The obtained results show that the system is highly effective in the regeneration process. An empirical equation to calculate the removed moisture as a function of air flow rate at solar noon is obtained. Also empirical equation for wheel effectiveness as a function of air flow rate for regeneration and absorption process was obtained.     

A nonlinear reaction–diffusion process using the Adomian decomposition method

A nonlinear ordinary differential equation with homogeneous Dirichlet boundary conditions describes a steady-state reaction–diffusion process with a nonlinear power-law heat source. In this paper, the solution is obtained using the Adomian decomposition method. The convergence of the method is investigated for several nonlinear sources involving integer positive, negative, decimal, and radical powers of the temperature.     

Mathematical description of the diffusion in a temperature field and measuring the heat of transport

A critical review of the existing analytical solutions of the thermodiffusion equation and methods for measuring heat of transport was performed. Particular emphasis is placed upon the fact that of the reported solutions yielding reasonable accuracy, there is no one solution that describes the diffusion profile behavior while making possible to measure the diffusion substance's heat of transport. Simple thermodiffusion equation solutions were obtained by taking into account the coordinate dependence of the diffusion coefficient and thermodiffusion factor in a temperature field. Such an approach has allowed the proposal of a new method for measuring the heat of transport similar to the conventional method for measuring a diffusing impurity's activation energy.     

Estimation of front surface temperature and heat flux of a locally heated plate from distributed sensor data on the back surface

We present a new method of solving the three-dimensional inverse heat conduction (3D IHC) problem with the special geometry of a thin sheet. The 3D heat equation is first simplified to a 1D equation through modal expansions. Through a Laplace transform, algebraic relationships are obtained that express the front surface temperature and heat flux in terms of those same thermal quantities on the back surface. We expand the transfer functions as infinite products of simple polynomials using the Hadamard Factorization Theorem. The straightforward inverse Laplace transforms of these simple polynomials lead to relationships for each mode in the time domain. The time domain operations are implemented through iterative procedures to calculate the front surface quantities from the data on the back surface. The iterative procedures require numerical differentiation of noisy sensor data, which is accomplished by the Savitzky–Golay method. To handle the case when part of the back surface is not accessible to sensors, we used the least squares fit to obtain the modal temperature from the sensor data. The results from the proposed method are compared with an analytical solution and with the numerical solution of a 3D heat conduction problem with a constant net heat flux distribution on the front surface.     

Diffusion equations based on generalized continuum mechanics

We propose that an increasing value in entropy inequality is not a local value of entropy but an average value in a mesodomain. From this standpoint, new balance equations are obtained by applying the concept of generalized continuum mechanics to mass transfer. In the present paper, it is clarified that the influence of a mesoscopic gradient of mass concentration should be included in Fick's first law and Fourier's law. Even if thermal effects on diffusion are neglected, the diffusion equations obtained here are simultaneous differential equations with two undetermined values, which include the conventional diffusion equation as a special case. The conventional diffusion equation describing infinite velocity of propagation associated with diffusion entails a contradiction. The velocity of propagation defined here is shown in the results of a numerical analysis for an extreme initial state. Consequently, it is indicated that the present theory gives an acceptable solution to the above problem in the conventional theory. © 1998 Scripta Technica, Inc. Heat Trans Jpn Res, 26(2): 84–96, 1997     

Electrical cross effects in porous media with ice inclusions. 1. Diffusion mechanism

This paper is the first part of the theoretical work devoted to electrical cross effects in biporous media with ice inclusions. The fine-pored part of the medium is saturated by the electrolytic solution.A heat and mass transfer problem is formulated for an elementary cell of the medium on basis of diffusion mechanism. An equation set is proposed for finding the temperature, pressure, concentration, and electric potential fields. All the heat and mass flows through the cell depend on the cross thermodynamical forces.In the closed system the electric polarization induced by temperature gradient depends on the hydroconductivity of the fine-pored medium, the solution concentration and the ice content. At the low solution concentration and the high hydroconductivity the thermoelectric potential difference may be as great as 150 mV K^?1.In the open system the ice presence has no effect on the diffusion potential difference induced by concentration gradient.     

A simplified model for bi-component droplet heating and evaporation

A simplified model for bi-component droplet heating and evaporation is developed and applied for the analysis of the observed average droplet temperatures in a monodisperse spray. The model takes into account all key processes, which take place during this heating and evaporation, including the distribution of temperature and diffusion of liquid species inside the droplet and the effects of the non-unity activity coefficient (ideal and non-ideal models). The effects of recirculation in the moving droplets on heat and mass diffusion within them are taken into account using the effective thermal conductivity and the effective diffusivity models. The previously obtained analytical solution of the transient heat conduction equation inside droplets is incorporated in the numerical code alongside the original analytical solution of the species diffusion equation inside droplets. The predicted time evolution of the average temperatures is shown to be reasonably close to the measured one, especially in the case of pure acetone and acetone-rich mixture droplets. It is shown that the temperatures predicted by the simplified model and the earlier reported vortex model are reasonably close. Also, the temperatures predicted by the ideal and non-ideal models differ by not more than several degrees. This can justify the application of the simplified model with the activity coefficient equal to 1 for the interpretation of the time evolution of temperatures measured with errors more than several degrees.     

Heat and mass transfer influence on potential variation in a PEMFC membrane

Water transport in the membrane of a PEM fuel cell is provided essentially by a convective force, resulting from a pressure gradient, an osmotic force, due to a concentration gradient and an electric force caused by the protons migration from the anode to the cathode. Through these three types of force the two-dimensional behavior of electric potential has been studied in this paper. The adopted model in this work is based on the assumption of single phase and multi-species flow, supposed two-dimensional and transient in a porous medium. The species conservation equation is coupled with the energy equation through the diffusion coefficient of water and the heat convective flux. The set of governing equations in the form of convection–diffusion problem has been solved numerically using the finite volume method. The obtained results show the transient two-dimensional effect of heat and mass transfer on the voltage variation within the membrane.     

Thermal propagation in solids due to surface laser pulsation and oscillation

Most studies of heat transfer pertaining to a planar medium subjected to time-varying and spatially-decaying laser incidence along with external surface cooling are based on the diffusion theory (parabolic heat conduction equation), an approximation that implies a non-physical infinite speed of energy transport. In this study, temperature distributions within one-dimensional plates subjected to the aforementioned heating and cooling conditions, but with thermal energy propagation at a finite speed, are presented. Incident energy that is partially absorbed at the external surface is transferred through the plate by conduction, while the remaining energy is convectively cooled to the environment. The present investigation will examine three different time characteristics of the incident heat sources which include a continuously operating, a pulsed and an oscillatory laser source. The temperature results were obtained by using a simple and concise finite-difference algorithm based on the Godunov scheme, a method developed for the solution of resulting characteristic equations that govern thermal wave propagations within the solid.     

The hydrogen transport through the metal alloy membranes with a spatial variation of the alloy composition: Potential diffusion and enhanced permeation

It is feasible to make hydrogen separation membranes of an alloy whose elemental or phase composition controllably varies at least in one direction. However the problem of hydrogen transport through such membranes is found not to be solved with the standard equation of diffusion in heterogeneous media. That is because the effect of heterogeneity on diffusion phenomena is considered to be caused by only spatial variations of diffusion coefficient while the spatial difference in the potential energy of diffusing particles due to their interactions with the inhomogeneous medium is not taken into consideration. The corrected equation shows the existence of an additional driving force making possible the diffusion in the direction opposite to that prescribed by Fick's law even in the isothermal medium when there is no any external field of force. The solution for the hydrogen transport through the membrane made of the alloy of variable composition indicates the possibility of great increase in the hydrogen permeation flux due to the optimization of spatial distribution of the alloy composition.