Conductivity Of Heterogeneous Media Research Articles

Heat diffusion at the boundary of stratified media: Homogenized temperature field and thermal constriction

Heat diffusion in stratified materials with the layers running parallel to the main heat flux direction is analyzed with special emphasis on the temperature field near the boundaries. In a previous work, a semi-analytical general solution was proposed as an extension of the thermal quadrupole method for heat conduction in heterogeneous media. In this paper, the steady solution is separated into an homogenized transfer in series with a constriction term, and a conductive boundary layer is defined. The same decomposition method is implemented for the semi-infinite transient case, and some simplified models are obtained from asymptotic expansions. For long times, the transient averaged signal is found to be superposed to the steady constriction matrix effect. The main application is to better envision experimental temperature field analysis for thermal non-destructive evaluation methods.

Accuracy of the renormalization method for computing effective conductivities of heterogeneous media

The accuracy of the renormalization method for upscaling two-dimensional hydraulic conductivity fields is investigated, using two canonical 2 × 2 blocks: a checkerboard geometry and a geometry in which three of the cells have conductivity K1 and the other has conductivity K2. The predictions of the renormalization algorithm are compared to the arithmetic, harmonic and geometric means, as well as to theoretical predictions and finite element calculations. For the latter geometry renormalization works well over the entire range of the conductivity ratio K2/K1, but for the checkerboard geometry the error becomes unbounded as the conductivity ratio grows.

Scale Dependency of Hydraulic Conductivity in Heterogeneous Media

AbstractVarious types of sediments and rocks were analyzed for the relationship between hydraulic conductivity (K) and scale of measurement No variations of K with scale were observed for homogeneous media such as quartz‐arenites (quartz sandstones). However, hydraulic conductivity increased with scale of measurement in heterogeneous media. The scaling behavior can be described with the equation K = c (V)m, where c is a parameter characteristic of the geological medium that relates to geological variables such as average pore size and pore interconnectivity in porous media, and probably fracture opening and fracture interconnectivity in fractured media. V is the volume of tested material (used as scale measure), and m is the exponent of the relationship (slope of the line on a log‐log plot). The value of the exponent depends on the type or types of flow present. Porous flow media have an exponent of 0.5, multiple flow media an exponent between 0.5 and 1.0, and fracture and conduit flow controlled media an exponent of about 1.0. The more dominant fracture/conduit flow is relative to porous flow, the closer the exponent is to 1.0. K increases with scale up to a rock volume after which the aquifer approaches the properties of an equivalent homogeneous medium and K remains constant with scale. This volume (upper bound of the relationship) is related to the degree of heterogeneity in a medium. It is at a much larger scale in karstic media (if encountered at all) than in nonkarstic and more homogeneous media. Both confined and unconfined aquifers exhibit a similar scale dependence.

Numerical Green’s Function for Steady-State Heat Conduction in Heterogeneous Media

Numerical Green's function for steady-state heat conduction problems is derived in a finite-sized medium that may contain inclusions (fibers) in the matrix phase. Green's function is approximated by employing the Galerkin method that uses permissible functions which satisfy the homogeneous boundary condition for the given geometry. The present approach allows physical fields in a medium that contain multiple inclusions to be expressed through isolated integrals semi-analytically while retaining all the relevant material parameters.

Generalized Boundary Integral Equation for Transient Heat Conduction in Heterogeneous Media

Using the generalized boundary integral method developed by the authors for steady heat conduction in heterogeneous media as a point of departure, a generalized dual reciprocity boundary element method (BEM) is presented for the solution of transient heat conduction problems in heterogeneous media. In the process, new interpolating radial basis functions are dee ned. This method retains the boundary-only discretization feature of the BEM. Two- and three-dimensional numerical examples provide validation of the proposed method. Excellent agreement is found between analytical and BEM-computed results.

Size of polymeric particles forming hemodialysis membranes determined from water and solute permeabilities

Regarding hemodialysis membranes as layers packed with uniform polymeric particles, the size of the particles is determined using the Kozeny–Carman equation. Diameter of the spheres forming cellulosic membranes is the same order as the size of primary polymeric particles determined by electron microscopy in a previous article. Pore radii of the membranes calculated by the Kozeny–Carman equation are in agreement with those determined by the tortuous capillary pore model. The result suggests that an estimate of a pore radius of a membrane is feasible by the Kozeny–Carman equation only with water permeability of the membrane. Intramembrane diffusion coefficients of vitamin B12 calculated from an equation derived from the analogy of heat conduction in heterogeneous media consisting of a continuous phase and particles are larger than the experimental values. The result suggests the failure of the analogy between heat conduction and diffusion of vitamin B12 in a heterogeneous medium. © 1998 John Wiley & Sons, Inc. J Appl Polym Sci 67:833–840, 1998

Diagrammatic theory of effective hydraulic conductivity

This work presents a stochastic diagrammatic theory for the calculation of the effective hydraulic conductivity of heterogeneous media. The theory is based on the mean-flux series expansion of a log-normal hydraulic conductivity medium in terms of diagrammatic representations and leads to certain general results for the effective hydraulic conductivity of three-dimensional media. A selective summation technique is used to improve low-order perturbation analysis by evaluating an infinite set of diagrammatic terms with a specific topological structure that dominates the perturbation series. For stochastically isotropic media the selective summation yeilds the anticipated exponential expression for the effective hydraulic conductivity. This expression is extended to stochastically anisotropic media. It is also shown that in the case of non homogeneous media the uniform effective hydraulic conductivity is replaced by a non-local tensor kernel, for which general diagrammatic expressions are obtained. The non-local kernel leads to the standard exponential behavior for the effective hydraulic conductivity at the homogeneous limit.

Transient Diffusion and Conduction in Heterogeneous Media: Beyond the Classical Effective-Medium Approximation

In this paper steady-state and transient diffusion and conduction in heterogeneous media, such as composite solids, are studied. The heterogeneities are of percolation type in the sense that a randomly-selected fraction of the system does not allow transport to take place. We present several new extensions of the classical effective-medium approximation (EMA) for diffusion and conduction in regular network models of heterogeneous media, including an EMA for topologically-random networks, such as the Voronoi network, for which we introduce a novel stochastic Green function, and a cluster (higher order) EMA for transient diffusion and other dynamical properties of heterogeneous media. In all cases the predictions are compared with the results of Monte Carlo simulations.

Nonlinear electrical conductivity in heterogeneous media

A new method is proposed for estimating the effective behaviour of strongly nonlinear composite materials. The central idea is to write down second–order Taylor expansions for the constituent phase energy functions, about appropriately chosen reference fields. This allows the restatement of the problem for the nonlinear composite in terms of a corresponding problem for a fictitious linear composite with anisotropic constituents, subjected to a given distribution of polarization fields. The procedure thus permits the straightforward conversion of well–known estimates for linear composites, such as the Maxwell–Garnett and effective–medium approximations, into corresponding estimates for nonlinear composites. The resulting nonlinear Maxwell–Garnett and effective-medium estimates are exact to second order in the contrast.

A mixture theory for heat conduction in heterogeneous media

A mixture theory based on the ensemble-averaging technique is proposed for studying the thermal behaviour of heterogeneous media. Balance equations with partial heat fluxes and interaction terms are introduced for a two-component medium. Relations between these quantities and the ensemble-averaged (mean) temperatures of separate components are derived. The limits of application of widely used postulates of proportionality of the interaction term of difference in mean temperature of components is discussed. The effective conductivities of components in the mixture and the interaction coefficient are introduced and related to the so-called microstructure functions, which are functions of the microgeometry of the medium and the thermal properties of its components. Examples of the application of the theory to the calculation of the mean properties of a heterogeneous medium are presented.

Two-temperature discrete model for nonlocal heat conduction

The two-temperature discrete model for heat conduction in heterogeneous media is proposed. It is shown that the discrete model contains as limiting cases both hyperbolic and parabolic heat conduction equations for propagative and diffusive regimes, respectively. To obtain these limiting cases two different laws of continuum limit have been introduced. The evolution of the two-temperature system comprises three stages with distinct time scales: fast relaxation of each subsystem to local equilibrium, energy exchange between the subsystems and classical hydrodynamics

Transient conduction in heterogeneous media

A physical model of transient conduction in heterogeneous media is presented in this study. The thermal conductivity of the media is assumed to vary randomly. Temperature distributions in the media are generated using both random variable and constant average thermal conductivities. The results are employed to determine the percent differences in the predicted average temperature between these two cases and to assess the adequacy of the transient conduction model using average thermal conductivity to represent that with variable random thermal conductivity.

Heat conductivity of concentrated suspensions

According to the representation formulated in [i, 2], ordered rotation of suspension particles under the action of an external field results in intensification of the heat (mass, charge) transfer. Since the suspension as awhole remains fixed, the observer perceives this as a change in its effective heat conductivity which meanwhile becomes tensorial in nature. Intensification of heat transfer in magnetic suspensions subjected to a rotating field was observed in experiments [3, 4]. A unique interpretation of these observations as a result of internal rotation meets difficulties associated with the possibility of the origination of macroscopic suspension motion in the rotating field, although all possible measures for its exclusion where taken in [4]. According to the theoretical result [5], more than a tenfold increment in the effective heat conductivity is possible because of internal rotation. However, the theoretical model [5], being cellular, is based on replacement of the real system by some analog so that the degree of approximation to reality is unknown in the exact sense. This can be judged by examining the resulting particular result for the heat conductivity of a suspension of spherical particles at rest. We deal here with the classical object of the theory of generalized conductivity of heterogeneous media [6].

Application of Volterra–Wiener Series for Bounding the Overall Conductivity of Heterogeneous Media. I. General Procedure

A general method of placing optimal bounds on the overall conductivity of random heterogeneous media is proposed. It utilizes truncated Volterra-Wiener functional series, generated by the random conductivity field K(x), as classes of trial functions for the classical variational principles. The method simplifies, unifies and/or generalizes the earlier proposed variational techniques of Beran (3), Dederichs and Zeller (10), Hori (15), Kr6ner (17) and Prager (25). The general procedure is displayed in detail in the simplest case of interest, the construction of the optimal third-order bounds, that requires knowledge of the two- and three-point correlation functions for K(x). The evaluation of these bounds is reduced to the solution of an integrodifferential equation whose coefficients and kernels are expressed through the said correlation functions. A perturbation solution to this equation is given. For Miller's cell materials the equation is explicitly solved and the obtained optimal bounds are shown to coincide with those of Beran-Miller.

Application of Volterra–Wiener Series for Bounding the Overall Conductivity of Heterogeneous Media. II. Suspensions of Equi-sized Spheres

The variational procedure of Part I of this paper [this Journal, 47 (1987), pp. 831–849] is employed for bounding the overall conductivity $\kappa^*$ of a suspension of equi-sized spheres. The procedure uses classes of trial functions in the form of truncated Volterra–Wiener series generated here by the random density function of the system of sphere centers. Integrodifferential equations are derived whose solutions yield bounds, upper and lower, on $\kappa^*$ for given two- and three-point distribution functions $f_2 $, $f_3 $ of the system of spheres. The bounds are explicitly calculated to the order $c^2 $, where c is the volume fraction of the spheres; they include a statistical parameter $m_2 $ that can be simply evaluated by means of the two-point distribution function $f_2 $ and that reflects the tendency of the spheres to form clusters at higher c. The predictions of various approximate models of composite mechanics are checked against the $c^2 $-bounds. The check yields a number of curious results concerning the random constitution of suspensions of impenetrable spheres that can be described by means of these models.