Expression For Effective Thermal Conductivity Research Articles

Research on Effective Thermal Conductivity in Porous Media Embedded with Randomly Distributed Damaged Tree-like Bifurcation Networks

Due to the complexity of the microstructure of porous media, it is of great significance to explore the heat transport mechanism in porous media in many engineering applications. In this study, an expression for effective thermal conductivity (ETC) of porous media embedded with randomly distributed damaged tree-like bifurcation networks is derived based on the theory of thermodynamics and fractal features of tree-like bifurcation networks. We investigate the effect of heat conduction and heat convection in porous media embedded with randomly distributed damaged tree-like bifurcation networks on the ETC of the porous media. It is found that our fractal model has good consistency with the existing available experimental data. In addition, the influence of the microstructural parameters of the model on heat transfer in the porous media is analyzed in detail. The research results can provide significant theoretical guidance for the development and design of heat transfer systems.

Estimation of Out-of-Plane Effective Thermal Conductivity of Wire Mesh Using 3D Unit-Cell Model Incorporating Secondary Effects

AbstractThis study proposes a modified collocated parameter model to estimate the effective thermal conductivity (keff) of two-phase wire mesh materials. The study focused on the influence of parameters, primary as well as secondary, on the effective thermal conductivity of a wire mesh, namely, thermal conductivity ratio (α), concentration (γa), mesh number (M), and thermal contact conductance at the wire-to-wire interfacial area were all investigated in detail over a different temperature range in the normal to the wire mesh. The analytical expressions for effective thermal conductivity in the form of algebraic equations were derived by adopting the solid square cylinder unit-cell-based thermal resistance method for a three-dimensional spatially in-line touching periodic medium. The effective thermal conductivities of different wire mesh materials with solid–fluid combinations, possessing conductivity ratios of 1–10,000 and concentrations of 0–0.8, are predicted for various temperatures at constant pressure regimes using the developed model. Furthermore, the developed model was applied to estimate the effective thermal conductivities of brass and stainless-steel wire mesh, which are compared with values measured using the modified transient plane source (MTPS) instrument over the temperature range 283–460 K. For the brass and stainless-steel mesh—water, silicone oil, and air combinations, the estimated values of effective thermal conductivity obtained from the present model show an average deviation of fluids ± 2.6%, with air ± 4.4% in the z-direction, respectively, concerning the ones obtained from the experiments.

A theoretical model for effective thermal conductivity of open‐cell‐coated metal foams saturated with fluid/phase change material

The current research presents geometrical effective thermal conductivity (ETC) models for externally coated open-cell metal foams (MFs) saturated with fluid/phase change material (PCM) by considering three-dimensional configuration adopted from the tetrakaidecahedron structure. The three-dimension (3D) models consider different geometries, such as hexagonal and square, involving different shapes of ligaments (cylindrical, concave triprism) and nodes (cubic, pyramidal) for the analysis. The ETC of MF-composite increases with the coating thickness, the thermal conductivity of coating material, and infiltrating medium. The percentage enhancement in ETC is found to be significant with the increase in coating thickness compared with the increase in thermal conductivity of coating material and infiltrating medium. In addition, the ETC value increases with the increase in the porosity value, which can be explored at the manufacturing process during the design of thermal management systems (TMS). The present model involving hexagonal geometry with concave triprism ligaments and pyramidal nodes is found to be in excellent agreement with test data for all the infiltration cases with an average deviation of less than 3%. The present study reports the mathematical expressions for ETC as a function of various modeling parameters, which can be useful during the design of the TMS.

A FRACTAL MODEL FOR EFFECTIVE THERMAL CONDUCTIVITY OF DUAL-POROSITY MEDIA WITH RANDOMLY DISTRIBUTED TREE-LIKE NETWORKS

It is very difficult to characterize the transport properties of porous media due to the disorder of pores distribution in porous media. In this paper, we study heat conduction through porous media with randomly fractures which are considered as tree-like networks. An expression for effective thermal conductivity (ETC) of saturated dual-porosity media is derived based on the fractal characteristics of pores diameter and fractures size. It is shown that the ETC is a function of structure parameters of the fractal dimension ([Formula: see text]), the porosity ([Formula: see text]), the tortuous fractal dimension ([Formula: see text]) and the maximum diameter ([Formula: see text]) for the porous matrix; the fractal dimension ([Formula: see text]), the porosity ([Formula: see text]), the maximum diameter ([Formula: see text]), the diameter ratio ([Formula: see text]), the length ratio ([Formula: see text]) and the branching levels (n) for the fracture networks. The dependence of structure parameters on the ETC is studied in detail. The results of our model are compared with the available experiments, which show good agreement. The proposed model for the ETC does not contain empirical parameters. The model is useful for predicting the heat conduction of materials.

Thermal conduction properties of microcracked media: Accounting for the unilateral effect

Studies dedicated to the homogenization approach of microcracked media are largely focused on the determination of effective elastic properties. Some works investigate other properties, but most of them consider only open cracks. This paper intends to provide effective thermal properties related to the conduction problem taking into account the unilateral effect (opening/closure of cracks). Such analysis considers steady-state heat transfer within an initially isotropic media weakened by randomly oriented cracks. According to the boundary conditions, estimates and bounds based on Eshelby-like formalism are developed to derive closed-form expressions for effective thermal conductivity and resistivity in a fixed microcracking state.

Estimation of effective thermal conductivity of packed beds incorporating effects of primary and secondary parameters

This paper presents a detailed description and experimental validation of a modified collocated unit cell model for the estimation of effective thermal conductivity of packed beds. The effect of primary and secondary parameters influencing the effective thermal conductivity viz. conductivity ratio (α) and concentration (ν) of the spherical particles (fraction of total volume occupied by the particles) of the packed bed, inter-particle gap, thermal contact conductance between the surfaces in contact, temperature, pressure and Knudsen number respectively are investigated in detail. Analytical expressions for effective thermal conductivity are derived for spatially periodic 2D square cylinder and in-line touching geometries based on the unit cell based thermal resistance approach. Using the developed model, the effective thermal conductivity of various packed materials in the conductivity ratio (ks/kf) range between 1–1000 and concentration (ν) 0 to 1 are predicted for varied temperature and pressure regimes. Predictions obtained from the developed model are compared against existing models available in literature in the pressure range 10−2 to 103 kPa at constant temperature. Furthermore, effective thermal conductivities of packed beds comprising of glass and ceramic beads are predicted using the developed model and compared with measured values obtained using the standard square guarded hot plate (SGHP) apparatus in the temperature range of 323–673 K. Both values are found to agree within ±12.24% and ±14.54% for glass and ceramic beads respectively.

Expression for ETC of the solid phase of randomly packed granular materials

Effective thermal conductivity (ETC) is an important parameter describing the thermal behavior of particulate materials, and has been extensively examined in the past decades. In this paper, a theoretical model called parallel-column model is proposed and based on this model, an analytical expression is developed to predict the ETC of the solid phase of granular materials. The ETC is expressed in terms of particle size distribution, the compressive strain and thermal conductivity of individual particle. Different types of contact force models and grain size distribution is incorporated into the parallel-column model and the expression for ETC is derived for different types of contact force models and grain size distribution.

Expression for effective thermal conductivity of randomly packed granular material

This paper proposes an effective analytical expression for the accurate evaluation of the effective thermal conductivity (ETC) of randomly packed granular material with high solid-to-fluid conductivity ratio. The procedure is a novel combination of two key techniques: the parallel-column model and the pipe-network model. The ETC can be predicted using the proposed expression without microscopic analysis of the granular sample. Several practical issues associated with the accuracy and efficiency of the analytical expression are discussed, and test cases are provided to illustrate the applicability of the expression.

Prediction of Effective Thermal Conductivity of Cu/Solder System by Using Interfacial Layer in Two Phase System

A theoretical model is presented to predict effective thermal conductivity of real two phase materials from the values of thermal conductivity of the constituent phases and their volume fraction. Here we develop a numerical expression for effective thermal conductivity of Cu-solder system in two phase system is being comprised of contributions from both the phases with interfacial layer. A new correlation term F is introduced for highly conducting phase, non-linear fl ow of heat fl ux lines and random distribution of the phases. Here, we get best fi tting relation for F. Comparison of the proposed relation with different models has also been made. The values predicted by the proposed model are in close and good agreement with experimental values and more accurate than those obtained from existing models.

A new model for heat transfer through the contact network of randomly packed granular material

A new model termed as the parallel-column model for heat transfer through the contact network of the granular materials with random packing structure is presented. In the framework of the proposed model, the expression for effective thermal conductivity (ETC) of granular materials is derived. The numerical tests are performed on the granular samples with different structural sizes and particle size distributions. Initial arrangements of particles in the granular samples are generated using the inwards packing methods. The validity of the proposed model is verified by the good agreement between theoretical prediction given by the proposed expression and numerical results over a wide range of structural sizes and particle size distributions. It is demonstrated that the parallel-column model is a simplified and physically acceptable model, and the proposed expression is competent to accurately predict the effective thermal conductivity of granular materials while with no need of particle scale analysis of granular materials.

Analytical expressions for effective thermal conductivity of composite materials with inclusions of square cross-section

Thermal conductivity of composite materials with periodically distributed cylindrical inclusions of square cross-section is investigated. Analytical interpolation formula for the effective thermal conductivity in transversal direction valid in the entire range of volume fraction of inclusions and value of their thermal conductivity is derived. The methods of asymptotic homogenization, boundary shape perturbation and asymptotically equivalent functions are used. Asymptotic expansions of the effective thermal conductivity in limiting cases of size of inclusions and their thermal conductivity are obtained. The analytical expressions are evaluated numerically and compared to other models. It is shown that the discrepancy between these results is very low in all cases.

Effect of Geometry of Filler Particles on the Effective Thermal Conductivity of Two-Phase Systems

The present paper deals with the effect of geometry of filler particles on the effective thermal conductivity for polymer composites. In the earlier models, less emphasis has been given on the shape of filler particles. In this paper, expressions for effective thermal conductivity has been derived using the law of minimal thermal resistance and equal law of the specific equivalent thermal conductivity for three different shapes i.e. spherical, elliptical and hexagonal of filler particles respectively. Calculated values of effective thermal conductivity for various samples using the derived expressions then compared with experimental data available and other models developed in the literature. The results calculated are in good agreement with the earlier experimental data and the deviation, is least in our expressions showing the success of the model.

Thermal conductivity of mesoporous material MCM-41

In this paper, the mesoporous structural unit of MCM-41 is established first and the equilibrium molecular dynamics (EMD) numerical simulation is performed to obtain the shell thermal conductivity of MCM-41. Then, based on one-dimensional heat transfer analysis, the analytical expression for effective thermal conductivity of MCM-41 is obtained by coupling heat conduction in air-filled nano-pores and that in the shell. The effects of wall thickness, pore size and porosity on the thermal conductivities of the MCM-41 are further analyzed. It turns out that MCM-41 possesses good thermal insulation and the decrease of effective thermal conductivity is closely linear as porosity increases. Furthermore, the thermal conductivity shows that it is obviously anisotropic and its largest value is along the length of the pores, and that it has the quasi-one-dimensional characteristic.

Fractal analysis of effective thermal conductivity for three-phase (unsaturated) porous media

A fractal analysis of effective thermal conductivity for unsaturated fractal porous media is presented based on the thermal-electrical analogy and statistical self-similarity of porous media. Here, we derive a dimensionless expression of effective thermal conductivity without any empirical constant. The effects of the parameters of fractal porous media on the dimensionless effective thermal conductivity are discussed. From this study, it is shown that, when the thermal conductivity of solid phase and wet phase are greater than that of the gas phase (viz., ks∕kg>1, kw∕kg>1), the dimensionless effective thermal conductivity of unsaturated fractal porous media decreases with decreasing degree of saturation (Sw) and increasing fractal dimension for pore area (Df), fractal dimension for tortuosity (Dt), and porosity (ϕ); when the thermal conductivities of solid phase and wet phase are lower than that of the gas phase (viz., ks∕kg<1, kw∕kg<1), the trends were just opposite. Our model was validated by comparing the model prediction with existing experimental data. Excellent agreement was found except for the cases at very low level of saturation.

Effective thermal conductivity of real two-phase systems using resistor model with ellipsoidal inclusions

A theoretical model has been developed for real two-phase system assuming linear flow of heat flux lines having ellipsoidal particles arranged in a three-dimensional cubic array. The arrangement has been divided into unit cells, each of which contains an ellipsoid. The resistor model has been applied to determine the effective thermal conductivity (ETC) of the unit cell. To take account of random packing of the phases, non-uniform shape of the particles and non-linear flow of heat flux lines in real systems, incorporating an empirical correction factor in place of physical porosity modifies an expression for ETC. An effort is made to correlate it in terms of the ratio of thermal conductivities of the constituents and the physical porosity. Theoretical expression so obtained has been tested on a large number of samples cited in the literature and found that the values predicted are quite close to the experimental results. Comparison of our model with different models cited in the literature has also been made.