Effective Thermal Conductivity Of Composites Research Articles

Qualitative Equivalence Between Electrical Percolation Threshold and Effective Thermal Conductivity in Polymer/Carbon Nanocomposites

In this study, a qualitative equivalence between the electrical percolation threshold and the effective thermal conductivity of composites filled with cylindrical nanofillers has been recognized. The two properties are qualitatively compared on a wide range of aspect ratios, from thin nanoplatelets to long nanotubes. Statistical continuum theory of strong-contrast is utilized to estimate the thermal conductivity of this type of heterogeneous medium, while the percolation threshold is simultaneously evaluated using the Monte Carlo simulations. Statistical two-point probability distribution functions are used as microstructure descriptors for implementing the statistical continuum approach. Monte Carlo simulations are carried out for calculating the two-point correlation functions of computer generated microstructures. Finally, the similarities between the effective conductivity properties and percolation threshold are discussed.

Limiting Mechanisms of Thermal Transport in Carbon Nanotube-Based Heterogeneous Media

The outstanding thermal properties of carbon nanotubes (CNTs) have generated considerable interest and research activity. A very promising application is the use of vertically aligned arrays of CNTs as thermal interface materials (TIMs) for electronic systems. TIMs require a high thermal conductivity, a low thermal interface resistance with the adjacent microprocessor and heat sink, and significant mechanical compliance to help minimize the impact of mismatched thermal expansion coefficients. However, due to high thermal interface resistances of CNT films, more detailed measurements and improved fabrication methods are needed. This review paper presents current patents and last experimental/ computational techniques investigating heat transfer and limiting mechanisms of CNTs, CNT-thermal interface structures, CNT-polymer nanocomposites (PNCs), and thermal boundary resistance (TBR) of the CNTs and different surrounding matrices (metals, nanofluids and polymers). Effects on directional thermal conductivities of aligned CNTs in PNCs/TIMs are predicted using a random walk simulation. The TBR of CNT-matrix and inter-CNT contact significantly affect the effective thermal transport properties including anisotropy ratios. When CNT-CNT contact is significant or CNT-CNT TBR is low (relative to the CNT-matrix TBR), then heat transport is dominated by CNT-CNT contact effects, rather than CNT-matrix interfacial effects. The effects of CNT orientation, type (single-versus multi-wall), inter-CNT contact, volume fraction and TBR on the effective thermal conductivities of CNT-PNCs/ TIMs are quantified. The simulation results agreed well with reported experimental data for randomly-oriented SWNT-epoxy and polymethyl methacrylate nanocomposites. These simulation results can be very useful for developing techniques to enhance the effective thermal conductivity of composites using conductive nanomaterials embedded in matrices, and assist experimentalists in interpreting heat conduction measurements. Keywords: Carbon nanotube, thermal conductivity, thermal interface material composite, thermal boundary resistance, random walk algorithm, CNT-matrix TBR, CNT-polymer nanocomposites, nanocomposites, nanofluids, CNT-matrix, integrated circuits, (graphenes, anisotropic heat conduction, thermal resistance, composite material, nanotube-polymer composites

Derivation of governing equation for predicting thermal conductivity of composites with spherical inclusions and its applications

A governing differential equation for predicting the effective thermal conductivity of composites with spherical inclusions is shown to be simply derived by using the result of the generalized self-consistent model. By applying the equation to composites including spherical inclusions such as graded spherical inclusions, microballoons, mutiply-coated spheres, and spherical inclusions with an interphase, their effective thermal conductivities are easily predicted. The results are compared with those in the literatures to be consistent. It can be stated from the investigations that the effective thermal conductivity of composites with spherical inclusions can be estimated as long as their conductivities are expressed as a function of their radius.

Prediction of effective thermal conductivities of woven fabric composites using unit cells at multiple length scales

Abstract

Effective Thermal Conductivity of Composites with Different Particle Geometries and Interfacial Thermal Resistance

A new micromechanical method, the weighted residual self-consistent method (WRSCM) is developed to study the effective thermal conductivity of two-phase composites with different particle geometries in the presence of a thermal barrier resistance at the interface between constituents. The imperfect interface involves the continuity of the normal flux but allow for a finite temperature differences across the interface. Within the framework of self-consistent scheme, the effective thermal conductivity of two-phase composite is obtained using numerical iterative method on the basis of a surface integral of temperature over the imperfect interfaces. Numerical results show that for the given composite system, due to the existence of an interfacial thermal resistance, the particle geometries have significant impact on the effective thermal conductivity of composites.

Bounds on the effective thermal conductivity of composites with imperfect interface

A new model is developed to bound the effective thermal conductivity of composites with thermal contact resistance between spherical inclusions and matrix. To construct the trial temperature and heat flux fields which satisfy the necessary interface conditions, the transition layer for each spherical inclusion is introduced. For the upper bound, the trial temperature field needs to satisfy the thermal contact resistance conditions between spherical inclusions and transition layers and the continuous interface conditions between transition layers and remnant matrix. For the lower bound, the trial heat flux field needs to satisfy the continuous interface conditions between different regions. It should be pointed out that the continuous interface conditions mentioned above are absolutely necessary for the application of variational principles, and the thermal contact resistance conditions between spherical inclusions and transition layers are suggested by the author. According to the principles of minimum potential energy and minimum complementary energy, the bounds of the effective thermal conductivity of composites with imperfect interfaces are rigorously derived. The effects of the size and distribution of spherical inclusions on the bounds of the effective thermal conductivity of composites are analyzed. It should be shown that the present method is simple and does not need to calculate the complex integrals of multi-point correlation functions. Meanwhile, the present method provides an entirely different way to bound the effective thermal conductivity of composites with imperfect interface, which can be developed to obtain a series of bounds by taking different trial temperature and heat flux fields. In addition, the present upper and lower bounds are finite when the thermal conductivity of spherical inclusions tends to ∞ and 0, respectively.

Nonsteady Effective Thermal Conductivity of Dense Fibrous Composites with Graded Interface

The multiple scattering of thermal waves by dense fibers with a functionally graded interface in composites is considered, and the analytical solution of the nonsteady effective thermal conductivity of composites is presented. The Fourier heat conduction law is applied to analyze the propagation of thermal waves in the fibrous composite. The scattering and refraction of thermal waves by cylindrical fibers with a nonhomogeneous interface layer in the matrix are expressed by using the wave function expansion method. The theory of quasi-crystalline approximation and the conditional probability density function are employed to treat the multiple scattering of thermal waves from the dense fibers. Through analysis, it is found that the nonsteady effective thermal conductivity under higher frequencies is quite different from the steady thermal conductivity. In different regions of wave frequency, the effects of the functional form of gradation and the properties and thickness of the functionally graded layers on the nonsteady effective thermal conductivity show great differences.

Rigorous bounds on the effective thermal conductivity of composites with ellipsoidal inclusions

A new method is developed to derive the bounds of the effective thermal conductivity of composites with ellipsoidal inclusions. The transition layer for each ellipsoidal inclusion is introduced to make the trial temperature field for the upper bound and the trial heat flux field for the lower bound satisfy the continuous interface conditions which are absolutely necessary for the application of variational principles. According to the principles of minimum potential energy and minimum complementary energy, the bounds of the effective thermal conductivity of composites with ellipsoidal inclusions are rigorously derived. The effects of the distribution and geometric parameters of ellipsoidal inclusions on the bounds of the effective thermal conductivity of composites are analyzed. It should be shown that the present method is simple and needs not calculate the complex integrals of multi-point correlation functions. Meanwhile, the present method provides a powerful way to bound the effective thermal conductivity of composites, which can be developed to obtain a series of bounds by taking different trial temperature and heat flux fields. In addition, the present upper and lower bounds still are finite when the thermal conductivity of ellipsoidal inclusions tends to ∞ and 0, respectively.

Computational Investigation of the Effective Thermal Conductivity of Interpenetrating Network Composites

In this article, the effective thermal conductivity of composites with a network structure as reinforcement was calculated using finite element method (FEM), and a tetrakaidecahedron configuration was employed to represent the network structure. A spectrum of effective thermal conductivity of these composites was simulated. An analytical relationship among the effective thermal conductivity Keff, intrinsic thermal conductivities of the matrix and network reinforcement phase, and the volume fraction of the network was developed by fitting the numerical results. The simulation results have been validated by comparing with previous published experimental ones, and excellent agreement has been achieved.

Scattering of Thermal Waves and Unsteady Effective Thermal Conductivity of Particular Composites with Functionally Graded Interface

In this study, thermal wave method is applied to investigate the unsteady effective thermal conductivity of particular composites with a functionally graded interface, and the analytical solution of the problem is obtained. The Fourier heat conduction law is applied to analyze the propagation of thermal waves in the particular composite. The scattering and refraction of thermal waves by a spherical particle with an inhomogeneous interface layer in the matrix are analyzed, and the results of the single scattering problem are applied to the composite medium. The wave fields in different material layers are expressed by employing spherical wave function expansion method, and the expanded mode coefficients are determined by satisfying the boundary conditions of the layers. The theory of Waterman and Truell is applied to obtain the effective propagating wave number and the unsteady effective thermal conductivity of composites. Through the numerical examples, it is found that in different region of wave frequency, the effect of the thickness of the interface on the unsteady effective thermal conductivity of composites shows great difference. The effect of the thermal conductivity ratio of the particles and matrix on the unsteady effective thermal conductivity is closely related to the incident frequency of thermal waves. The variation of the interface properties between the particles and matrix also expresses great effect on the unsteady effective thermal conductivity of composites. Finally, comparison with the results obtained from other methods is made.

The specific heat and effective thermal conductivity of composites containing single-walland multi-wall carbon nanotubes

We present a study of the specific heat and effective thermal conductivity in anisotropicand randomly oriented multi-wall carbon nanotube (MWCNT) and randomly orientedsingle-wall carbon nanotube (SWCNT) composites from 300 to 400 K. Measurements onrandomly oriented MWCNTs and SWCNTs were made by depositing a thin film of CNTswithin a calorimetric cell. Anisotropic measurements were made on MWCNTs grown insidethe highly ordered, densely packed nanochannels of anodic aluminum oxide. The specificheat of randomly oriented MWCNTs and SWCNTs showed similar behavior to the specificheat of bulk graphite powder. However, the specific heat of aligned MWCNTs issmaller and has weaker temperature dependence than that of the bulk above roomtemperature. The effective thermal conductivity of randomly oriented MWCNTs andSWCNTs is similar to that of powder graphite, exhibiting a maximum value near364 K indicating the onset of phonon–phonon scattering. The effective thermalconductivity of the anisotropic MWCNTs increased smoothly with increasingtemperature and is indicative of the one-dimensional nature of the heat flow.

Evaluation of the effective thermal conductivity of composite polymers by considering the filler size distribution law

We present an empirical model for the effective thermal conductivity (ETC) of a polymer composite that includes dependency on the filler size distribution—chosen as the Rosin-Rammler distribution. The ETC is determined based on certain hypotheses that connect the behavior of a real composite material A, to that of a model composite material B, filled with mono-dimensional filler. The application of these hypotheses to the Maxwell model for ETC is presented. The validation of the new model and its characteristic equation was carried out using experimental data from the reference. The comparison showed that by using the size distribution law a very good fit between the equation of the new model (the size distribution model for the ETC) and the reference experimental results is obtained, even for high volume fractions, up to about 50%.

Multiple scattering of thermal waves and non-steady effective thermal conductivity of composites with dense coated fibers

In this study, the multiple scattering of thermal waves by dense coated fibers in composites is considered, and the analytical solution of the non-steady effective thermal conductivity of composites is presented. The Fourier heat conduction law is applied to analyze the propagation of thermal waves in the fibrous composite. The scattered and refracted fields in different material zones are expressed by using wave functions expanded method. The addition theorem for Bessel functions is used to accomplish the translation between different coordinate systems. The theory of quasicrystalline approximation and the conditional probability density function are employed to treat the multiple scattering of thermal waves from the dense fibers in matrix. The effective propagating wave number and non-steady effective thermal conductivity of composites are obtained. As an example, the effects of the material properties of the coating on the effective thermal conductivity of composites are graphically illustrated and analyzed. Analysis shows that the non-steady effective thermal conductivity under higher frequencies is quite different from the steady thermal conductivity. With the increase of the volume fraction of fibers, the effect of the thickness of the coating on the non-steady effective thermal conductivity increases greatly. In different region of wave frequency, the effects of the thickness and properties of the coating and the volume fraction of fibers on the effective thermal conductivity show great difference. Comparisons with the steady thermal conductivity obtained from other methods are also presented.

Fractional Steps methods for transient problems on commodity computer architectures

Fractional Steps methods are suitable for modeling transient processes that are central to many geological applications. Low memory requirements and modest computational complexity facilitates calculations on high-resolution three-dimensional models. An efficient implementation of Alternating Direction Implicit/Locally One-Dimensional schemes for an Opteron-based shared memory system is presented. The memory bandwidth usage, the main bottleneck on modern computer architectures, is specially addressed. High efficiency of above 2 GFlops per CPU is sustained for problems of 1 billion degrees of freedom. The optimized sequential implementation of all 1D sweeps is comparable in execution time to copying the used data in the memory. Scalability of the parallel implementation on up to 8 CPUs is close to perfect. Performing one timestep of the Locally One-Dimensional scheme on a system of 1000 3 unknowns on 8 CPUs takes only 11 s. We validate the LOD scheme using a computational model of an isolated inclusion subject to a constant far field flux. Next, we study numerically the evolution of a diffusion front and the effective thermal conductivity of composites consisting of multiple inclusions and compare the results with predictions based on the differential effective medium approach. Finally, application of the developed parabolic solver is suggested for a real-world problem of fluid transport and reactions inside a reservoir.

Multiple scattering of thermal waves and non-steady effective thermal conductivity of composites with dense coated particles

In this study, the multiple scattering of thermal waves from dense coated particles in composites is considered, and the analytical solution of the non-steady effective thermal conductivity of composites is presented. The Fourier heat conduction law is applied to analyze the propagation of thermal waves in the particular composite. The scattered and refracted temperature fields in different material zones are expressed by using wave functions expanded method. The addition theorem for spherical Bessel functions is used to accomplish the translation between different coordinate systems. The theory of quasicrystalline approximation and conditional probability density function are employed to treat the multiple scattering of thermal waves from the dense particles. The effective propagating wave number and non-steady effective thermal conductivity of composites are obtained. As an example, the effects of the material properties of the coating on the effective thermal conductivity of composites are graphically illustrated and analyzed. Analysis shows that the non-steady effective thermal conductivity under higher frequencies is quite different from the steady thermal conductivity. In the region of lower frequency, the effect of the properties of the coating on the effective thermal conductivity is greater. Comparisons with the steady effective thermal conductivity obtained from other methods are also presented.

Scattering of Thermal Waves and Non-steady Effective Thermal Conductivity of Unidirectional Fibrous Composites with Functionally Graded Interface

In this paper, a thermal wave method is applied to investigate the non-steady effective thermal conductivity of unidirectional fibrous composites with a functionally graded interface, and the analytical solution of the problem is obtained. The Fourier heat conduction law is applied to analyze the propagation of thermal waves in the fibrous composite. The scattering and refraction of thermal waves by a cylindrical fiber with an inhomogeneous interface layer in the matrix are analyzed, and the results of the single scattering problem are applied to the composite medium. The wave fields in different material layers are expressed by using the wave function expansion method, and the expanded mode coefficients are determined by satisfying the boundary conditions of the layers. The theory of Waterman and Truell is employed to obtain the effective propagating wave number and the non-steady effective thermal conductivity of composites. As an example, the effects of a graded interface on the effective thermal conductivity of composites are graphically illustrated and analyzed. Analysis shows that the non-steady effective thermal conductivity under higher frequencies is quite different from the steady thermal conductivity. In the region of intermediate and high frequencies, the effect of the properties of the interface on the effective thermal conductivity is greater. Comparisons with the steady thermal conductivity obtained from other methods are also presented.

Numerical modelling of the effective conductivities of composites with arbitrarily shaped inclusions and highly conducting interface

The particulate composite under consideration consists of arbitrarily shaped micro-inclusions embedded in a matrix. The interface between the matrix and each inclusion is imperfect and highly conducting, and corresponds to the case of inclusions coated with a highly conducting material. More precisely, the temperature across the interface is continuous while the normal heat flux across the interface suffers a jump related to the surface flux. The purpose of the present work is to elaborate an efficient numerical tool for computing the effective conductivities of the composite by combining the level-set method and the extended finite element method (XFEM). The elaborated numerical procedure has the advantage of avoiding curvilinear coordinates and surface elements in treating imperfect interfaces. It is first validated with the help of some analytical exact and approximate results as benchmarks. It is then applied to determine the inclusion size and shape effects on the effective thermal conductivities of composites with highly conducting coated inclusions and arbitrary geometries. The elaborated numerical procedure is directly applicable to other physically analogous phenomena, such as electrical conduction, dielectrics and diffusion, and to the mathematically identical phenomenon of anti-plane elasticity.

Scattering of thermal waves and non-steady effective thermal conductivity of composites with coated particles

In this study, thermal wave method is proposed to predict the non-steady effective thermal conductivity of composites with coated particles, and the analytical solution of this problem is obtained. The Fourier heat conduction law is introduced to analyze the propagation of thermal waves in the particular composite. The scattering and refraction of thermal waves by a coated particle in the matrix are analyzed, and the results of the single scattering problem are applied to the composite medium. The wave fields in different material zones are expanded by using the spherical wave functions and Legendre polynomial, and the expanded mode coefficients are determined by satisfying the boundary conditions of the coating layer. The theory of Waterman and Truell is employed to obtain the effective propagating wave number and the non-steady effective thermal conductivity of composites. As an example, the effects of the material properties of the particles, coating and matrix on the effective thermal conductivity of composites under different wave frequencies are graphically illustrated and analyzed. Analysis shows that the non-steady effective thermal conductivity under higher frequencies is quite different from the effective thermal conductivity under lower frequencies. In the region of lower frequency, the effect of the properties of the coating on the effective thermal conductivity is greater. Comparisons with the steady effective thermal conductivity obtained from other methods are also presented.

Scattering of thermal waves and non-steady effective thermal conductivity of composites with coated fibers

In this study, thermal wave method is applied to predict the non-steady effective thermal conductivity of composites with coated fibers, and the analytical solution of the problem is obtained. The Fourier heat conduction law is applied to analyze the propagation of thermal waves in the fibrous composite. The scattering and refraction of thermal waves by a cylindrical fiber with coating in the matrix are analyzed, and the results of the single scattering problem are applied to the composite medium. The wave fields in different material layers are expressed by using the wave function expansion method, and the expanded mode coefficients are determined by satisfying the boundary conditions of the layer. The theory of Waterman and Truell is employed to obtain the effective propagating wave number and non-steady effective thermal conductivity of composites. As an example, the effects of the material properties of the coating on the effective thermal conductivity of composites are graphically illustrated and analyzed. Analysis shows that the non-steady effective thermal conductivity under higher frequencies is quite different from the steady thermal conductivity. In the region of lower frequency, the effect of the properties of the coating on the effective thermal conductivity is greater. Comparisons with the steady thermal conductivity obtained from other methods are also presented.

06/01521 Study on paraffin/expanded graphite composite phase change thermal energy storage material: Zhang, Z. and Fang, X. Energy Conversion and Management, 2006, 47, (3), 303–310

Effective thermal conductivity of composites of graphite foam infiltrated with phase change materials (PCM) was investigated numerically and experimentally. Graphite foam, as a highly-conductive, highly-porous structure, is an excellent candidate for infiltrating PCM into its pores and forming thermal energy storage composites with improved effective thermal conductivity. For numerical simulation, the graphite structure was modeled as a three-dimensional body-centered cube arrangement of uniform spherical pores, saturated with PCM thus forming a cubic representative elementary volume (REV). Thermal analysis of the developed REV was conducted for unidirectional heat transfer and the total heat flux was determined, which leads to the effective thermal conductivity evaluation. For experimental verification, a sample of graphite foam was infiltrated with PCM. The effective thermal conductivity was evaluated using the direct method of measuring temperature within the sample under fixed heat flux in unidirectional heat transfer. The results indicate a noticeable improvement in the effective thermal conductivity of composites compared to the PCM. Our numerical and experimental results are in agreement and are also consistent with reported experimental results on graphite foam. Moreover, the role of natural convection within the pores is found to be negligible.