Conductivity Of Porous Materials Research Articles

Inverse Problem for a Two-Dimensional Anomalous Diffusion Equation with a Fractional Derivative of the Riemann–Liouville Type

The article presents a method for solving the inverse problem of a two-dimensional anomalous diffusion equation with a Riemann–Liouville fractional-order derivative. In the first part of the present study, the authors present a numerical solution of the direct problem. For this purpose, a differential scheme was developed based on the alternating direction implicit method. The presented method was accompanied by examples illustrating its accuracy. The second part of the study concerned the inverse problem of recreating the model parameters, including the orders of the fractional derivative, in the anomalous diffusion equation. Equations of this type can be used to describe, inter alia, the heat conductivity in porous materials. The ant colony optimization algorithm was used to solve this problem. The authors investigated the impact of the distribution of measurement points, the use of different mesh sizes, and the input data errors on the obtained results.

Thermophysical properties of porous materials

The study of the porosity of thermal insulation made of refractory materials is an important task for the power industry, since the thermal conductivity of porous materials depends on the shape and especially the location of the pores. An analytical review of existing technologies shows that research in this area is not enough to simulate the process of heat and mass transfer in porous alumina material. Experimental determination of the characteristics of heat and mass transfer in porous materials during the formation of a porous structure is a pressing scientific problem. This article analyzes the influence of the composition of materials on the formation of pores, as well as the effect of various impurities and temperature on the thermal conductivity of the material.

A Two-Parametric Model for Gas Flow in Low-Permeable Porous Materials

Capillary flow at low pressure gradients is determined by the pore structure of the matrix and the viscosity of the fluid. Traditionally, the conductivity of porous materials is characterized by a single parameter—the coefficient of permeability—which includes all geometric and structural resistances of the capillary structure. The fluid is described by its dynamic viscosity, which is independent of pressure in liquids. Gases, on the other hand, are highly compressible which means that their coefficient of permeability is becoming pressure dependent. The Klinkenberg effect takes account of this behaviour by introducing the notion of intrinsic permeability; nevertheless, it remains applicable only to a limited pressure range. At low- and high-pressure gradients, the Klinkenberg linearization deviates considerably from the real flow behaviour. This paper demonstrates that microporous capillary resistance to gas flow cannot be adequately described by a single material parameter. On the basis of ideal gas law conditions, the mass flow is derived by a two-parametric power law. Pore pressure decline is characterized by a permeability exponent n and a reference velocity v1, replacing both dynamic viscosity of the fluid and permeability. The actual pressure distribution over the flow path is expressed by means of an equation of motion which leads to a logarithmic linearization of the material parameters. Velocity and acceleration vectors are derived and discussed in relation to the actual pressure distribution. The two parameters n and v1 remain constant throughout and independent of pore pressure and are thus used to describe the flow rates over the entire pressure range. This model is then validated by means of flow analyses based on data given in the literature and compared to the classical theory of permeability.

Size- and shape-dependent effective conductivity of porous media with spheroidal gas-filled inclusions

Porous media containing gas-filled inclusions embedded in a solid phase constitute an important class of natural or artificial materials of both theoretical and practical interest. In these materials, thermal conductivity is one of the most important properties. In a variety of situations of practical interest, when the characteristic size of gas-filled inclusions is comparable with the mean free path of gas molecules and when the slip flow regime is considered, the behavior of gas near solid surfaces cannot be described by classical thermal conductivity equations. In fact, the boundary conditions at the solid surfaces must be modified by considering that the temperature and normal heat flux simultaneously suffer a discontinuity. The first purpose of the present work is to develop an efficient and accurate micromechanical model capable of estimating the effective conductivity of porous materials while taking into account the discontinuities of the temperature and normal heat flux across solid surfaces and the non-spherical form of gas-filled inclusions. The second purpose of the present work is to study the dependencies of the effective conductivity on the size and shape of gas-filled inclusions. By applying the micromechanical model based on the differential scheme and by using the solution results obtained for auxiliary dilute problem accounting for modified boundary conditions on surface solids, the closed-form expression for the effective conductivity is obtained. Numerical results are provided to illustrate the dependence of the effective conductivity on the size and shape of gas-filled inclusions in the case of randomly oriented inclusions.

Heat transfer and shear-induced migration in dense non-Brownian suspension flows: modelling and simulation

Modelling and simulation are developed, generalized and validated for both heat transfer and shear-induced particle migration in dense non-colloidal laminar suspension flows. Past theory and measurements for the effective thermal conductivity in porous materials at zero shear rate are coupled to more recent effective thermal diffusivity measurements of sheared suspensions. The suspension effective heat transfer affected by the local shear rate ($\dot{\unicode[STIX]{x1D6FE}}$), the phenomenon of shear-induced particle migration (SIM), the buoyancy effects ($\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}$) and the thermal Péclet number ($Pe_{d_{p}}=\dot{\unicode[STIX]{x1D6FE}}d_{p}^{2}/\unicode[STIX]{x1D6FC}_{f}$, where $d_{p}$ is the diameter of rigid particles and $\unicode[STIX]{x1D6FC}_{f}$ is the fluid phase thermal diffusivity) at the particle scale are all considered in the present constitutive three-dimensional modelling. Moreover, the influence of the temperature, the shear rate and the particle volume fraction ($\unicode[STIX]{x1D719}$) on the suspension effective viscosity ($\unicode[STIX]{x1D702}_{S}$), the suspension effective thermal properties and the fluid density ($\unicode[STIX]{x1D70C}_{f}$) are taken also into account. The present contribution represents an emerging field of heat transfer applications of complex fluid flows and is very beneficial for many future applications where concentrated suspension laminar flows with conjugate heat transfer may be present (e.g. for designing more innovative and compact heat exchangers).

A fractional approach to wicking fluid flow in nonwoven porous fabric

Purpose – The purpose of this paper is to employ a fractional approach to predict the permeability of nonwoven fabrics by simulating diffusion process. Design/methodology/approach – The method described here follows a similar approach to anomalous diffusion process. The relationship between viscous hydraulic permeability and electrical conductivity of porous material is applied in the derivation of fractional power law of permeability. Findings – The presented power law predicted by fractional method is validated by the results obtained from simulation of fluid flow around a 3D nonwoven porous material by using the lattice-Boltzmann approach. A relation between the fluid permeability and the fluid content (filling fraction), namely, following the power law of the form, was derived via a scaling argument. The exponent n is predominantly a function of pore-size distribution dimension and random walk dimension of the fluid. Originality/value – The fractional scheme by simulating diffusion process presented in this paper is a new method to predict wicking fluid flow through nonwoven fabrics. The forecast approach can be applied to the prediction of the permeability of other porous materials.

Numerical Simulation on the Effective Thermal Conductivity of Porous Material

The thermal conductivity of porous material is an important basic parameter, but it is not easy to study, due to the complexity of the structure of porous material. In the present work, we show a numerical simulation method to study the thermal conductivity of the porous material. We generate 200 material models with random distribution of solid skeleton and air for a fixed porosity, then we get the effective thermal conductivity of the porous material by Monte Carlo statistical analysis. The results are in good agreement with the previous empirical formula. The numerical results show that the effective thermal conductivity of porous material depends on the thermophysical properties of solid skeleton and air, the pore distribution and pore structure, the numerical error decreases with the increase in the number of grids, this finite element method can be used to estimate the effective thermal conductivity of composites and maybe has broad application prospects in terms of computing the effective thermal conductivity and other physical properties of composite material with known components.

Prediction of effective stagnant thermal conductivities of porous materials at high temperature by the generalized self-consistent method

The generalized self-consistent method is developed to deal with porous materials at high temperature, accounting for thermal radiation. An exact closed form formula of the local effective thermal conductivity is obtained by solving Laplace's equation, and a good approximate formula with uncoupled conductive and radiative effects is given. A comparison with available experimental data and theoretical predictions demonstrates the accuracy and efficiency of the present formula. Numerical examples provide a better understanding of interesting interaction phenomena of pores in heat transfer. It is found that the local effective thermal conductivity divides into two parts. One, attributed to conduction, is independent of pore radius for a fixed porosity and, furthermore, is independent of temperature (actually, it is approximately independent of the temperature) if it is non-dimensionalized by the thermal conductivity of the matrix. The other is due to thermal radiation in pores and strongly depends on the temperature and pore radius. The radiation effect can not be neglected at high temperature and in the case of relatively large pores.

Simulation of the dynamic (frequency-dependent) permeability and electrical conductivity in porous materials based on the concept of an ensemble of pores

The class of models of porous media based on the concept of an ensemble of pores with a certain distribution of the main geometrical parameters (e.g., pore size) is studied. The case of the saturation of the pore space with a single-phase multicomponent fluid mixture is studied with and without taking into account the transfer of electric charges. Transfer laws are derived from the condition of decreasing free energy. The hydrodynamic connectivity of pores (and electrical conductivity) is described by two kernels: one kernel describes the connectivity of pores in space, and the other describes the connectivity of pores in the elementary macrovolume. The frequency dependences of the dynamic permeability determined in laboratory experiments and the electrical conductivity of the porous medium were determined using the concept of an ensemble of pores. The relationship between the models considered and relaxation filtration models is established.

ESTIMATION OF THERMAL CONDUCTIVITY OF POROUS MATERIAL WITH FEM AND FRACTAL GEOMETRY

The relationship between thermal conductivity of porous material and fractal dimension is numerically simulated by using the finite element method. The solid matrix and pore space are generated randomly according to material porosity. Material parameters and element properties are changed by using ANSYS parameter design language. The effective thermal conductivity is derived according to thermal flux through some sections computed by FEM and Fourier heat transform law. The investigation shows that the effective thermal conductivity decreases with increasing porosity. The effective thermal conductivity will decrease exponentially with increasing fractal dimension of porosity space and increase exponentially with increasing fractal dimension of solid matrix.

Conductivity of Sm2TiO5 and Sm2Ti2O7

The oxygen-ion conductivity of porous materials, the coarse-grained pyrochlore-like Sm2Ti2O7 and fine-grained Sm2TiO5 compounds, produced by mechanical activation of initial oxides is studied at 400–1000 °C. The Sm2TiO5 samples contain ∼15 wt % of the nanosized pyrochlore-like Sm2TiO5 phase in addition to the rhombic phase. As determined by impedance spectroscopy, the ionic conductivities of Sm2TiO5 and Sm2Ti2O7 at 1000°C are 1.3 × 10−3 and 1.8 × 10−4 S cm−1, and the activation energies of the bulk and grainboundary conductivities of the materials are 1.04 and 1.24 eV for Sm2TiO5 and 1.69 and 1.80 eV for Sm2Ti2O7.

Thermal performance analysis of silica tiles

The thermal response of nonablating ceramic tiles is studied by the finite-element method. A continuum model is used to determine the thermal conductivity of porous materials. The temperature distribution for the one-dimensional model is compared with the available arc-jet result. Also presented are the 2D and 3D temperature contours and the heat-flux distributions for silica tiles. The expression for the pressure distribution in a silica tile is derived.

Electrical conductivity of sintered powder compacts

Some equations for calculating the electrical conductivity of porous materials are reviewed and their applicability to sintered powder compacts is discussed. A previous equation proposed by the authors, in which the effective conductivity of a sintered compact is composed of a function of the fully dense material conductivity, the compact porosity degree and tap porosity of the starting powder, is reviewed. The primary aim of the present work is to establish a connection between such equation and the percolation conduction theory.

Non-destructive Thermal Diagnostics of Porous Materials

Developed mathematical models of apparent thermal conductivity of porous materials are applied to non-destructive methods of thermal diagnostics. The non-destructive thermal diagnostics of porous materials can be used to estimate the size of pores and cracks in the range 10−9 to 10−3 m. A fractal model of porous structure and dependences of thermal conductivity/diffusivity on (experimental) gas pressure are used as a basis for structure parameter calculations. The measuring element (sensor) in this method is the mean free path of gas molecules in pores and cracks (Knudsen number) that is very sensitive to changes in gas pressure. Possible applications of the developed methods include non-destructive thermal diagnostics (NDTD) of nano- and micro-crack sizes; opening, closing and size changes of the cracks at high temperatures in a wide temperature range; evaluation of interfacial and contact heat barrier resistance for coatings; remote laser thermal diagnostics of the cracks; as well as obtaining data on strength, thermal shock behavior, failure and fatigue behavior of coatings and other structures. Examples of several applications of the NDTD method are presented.

Effects of Bubbles on the Hydraulic Conductivity of Porous Materials – Theoretical Results

In a porous material, both the pressure drop across a bubble and its speed are nonlinear functions of the fluid velocity. Nonlinear dynamics of bubbles in turn affect the macroscopic hydraulic conductivity, and thus the fluid velocity. We treat a porous medium as a network of tubes and combine critical path analysis with pore-scale results to predict the effects of bubble dynamics on the macroscopic hydraulic conductivity and bubble density. Critical path analysis uses percolation theory to find the dominant (approximately) one-dimensional flow paths. We find that in steady state, along percolating pathways, bubble density decreases with increasing fluid velocity, and bubble density is thus smallest in the smallest (critical) tubes. We find that the hydraulic conductivity increases monotonically with increasing capillary number up to Ca ∼ 10−2, but may decrease for larger capillary numbers due to the relative decrease of bubble density in the critical pores. We also identify processes that can provide a positive feedback between bubble density and fluid flow along the critical paths. The feedback amplifies statistical fluctuations in the density of bubbles, producing fluctuations in the hydraulic conductivity.

Lectrical Conductivity of Permeable Sintered Materials Composed of Discrete Randomly Oriented Fibers

A theory of the electrical properties of individual contacts is used to calculate the electrical conductivity of porous materials composed of discrete fibers with imperfect interparticle contacts. It is shown that the conductivity of these materials may be substantial even if the contacts are small. The materials' conductivity is significantly affected by a change in the relative length of the fibers.

Characteristic of Effective Thermal Conductivity in Porous Materials.

The effective thermal conductivities of porous materials are calculated with a finite element method by the fundamental equation of heat conduction in porous materials in the case of void fraction, pore shapes and thermal conductivity ratio of two phases.Effective thermal conductivities in the porous materials with spherical pores are larger than those with cylindrical and parallelepipedic pores, and agree closely with the Maxwell-Eucken equation. Since effective thermal conductivities in the case of cylindrical and parallelepipedic pores are almost the same, a correlative equation is obtained by using the numerical results for these pore shapes.To check these calculated results, the temperature distributions and effective thermal conductivities were measured experimentally under the same operating conditions as those of the theoretical analysis.

Computation of hydraulic conductivity of porous materials using computer-assisted tomography

The spatial distribution of porosity and continuity of pore space in glass bead and soil aggregate systems has been characterised by application of computer-assisted tomography to gamma-ray attenuation measurements. It has been demonstrated that a useful empirical relationship between hydraulic conductivity and pore size and continuity for the glass bead system as measured by CAT scanning can be derived.

Some Scaling Rules for the Overall Thermal Conductivity in Porous Materials

The overall thermal conductivity in an isotropic porous medium containing cavities of various shapes is studied in this work. The cavity shapes under consideration include spherical and cylindrical pores, prelate and oblate spheroids, and cylindrical cavities with rectangular cross sections. The unique feature lies in the derivation of the scaling rules governing the overall thermal conductivity in porous media containing various shapes of internal cavities. The temperature distribution around the representative cavity surface, the central quantity in the present differential formulation, is obtained by the boundary element method employing the exterior formulation.

A simple thermo-electric probe for the measurement of low water flow.

A probe for the measurement of thermal conductivity in porous materials (e.g. soils) was modified and used as a sensor to measure the flow rate of water in the range of 0.5 to 5 mm/s. A repeatability of 5% of the measured flow rate in water was obtained. (Abstract retrieved from CAB Abstracts by CABI’s permission)