Dimensionless Thermal Conductivity Research Articles

FRACTAL MODEL FOR EFFECTIVE THERMAL CONDUCTIVITY OF COMPOSITE MATERIALS EMBEDDED WITH A DAMAGED TREE-LIKE BIFURCATION NETWORK

Scientists worldwide have always been interested in the study of networks that resemble trees and porous materials. Therefore, based on fractal theory, this paper systematically studies the heat transfer problem of the damaged tree-like networks under different saturations in multi-medium composite materials and derives their dimensionless thermal conductivity (DTC). The research has shown that the dimensionless thermal conductivity (DTC) decreases with an increase in damaged channels. The saturation significantly impacts the heat transfer of the damaged tree networks, and it exhibits linear changes with the ratio of the gas phase, liquid phase, and solid phase. Additionally, it can be observed that the fractal dimension of length distribution and diameter distribution, bifurcation numbers, bifurcation series, porosity, and other parameters in the networks are related to thermal conductivity. The comparison between the data derived from this model and experimental data shows that the proposed model can effectively deepen our understanding of the heat transfer mechanism of the damaged networks in composite materials under different saturation levels. Additionally, the model in this paper does not have an empirical constant, which avoids the influence of potential factors.

THERMAL CONDUCTIVITY OF UNSATURATED FIBROUS MEDIA BY FRACTAL-MONTE CARLO SIMULATIONS

The effective thermal conductivity of unsaturated fibrous media is simulated by Fractal-Monte Carlo simulations in this work. Based on the probability model and thermal-electrical analogy method, the proposed model of the effective thermal conductivity can be expressed as an explicit function of fractal dimensions, [Formula: see text], [Formula: see text], intrinsic thermal conductivities, [Formula: see text], [Formula: see text], [Formula: see text], the straight capillary length of a unit cell [Formula: see text], liquid saturation [Formula: see text], pore size [Formula: see text] as well as porosity [Formula: see text]. Each parameter of the proposed model has physical meaning and the function has no empirical constant. The results calculated by Fractal-Monte Carlo simulations have been compared with the experimental data available, and an excellent agreement has been shown for an appropriate range of porosity except for the condition at a lower liquid saturation. It is found that, when [Formula: see text] and [Formula: see text], the dimensionless effective thermal conductivity of unsaturated fibrous media decreases with increment in fractal tortuosity dimension and porosity and decrement in liquid saturation. On the other hand, when [Formula: see text] and [Formula: see text], the correlations are opposite. The present Fractal-Monte Carlo technique may have its own advantages in predicting other transports in unsaturated fibrous media.

Improving the Sierpinski carpet model arithmetic to accurately predict the thermal conductivity of fatty acid/carbon composite phase-change materials

The effective thermal conductivity is an important physical parameter of composite phase-change materials (PCMs). In this study, a new analytic expression of the dimensionless thermal conductivity is proposed to predict the dimensionless thermal conductivity of fatty acid–carbon materials composite PCMs based on the Sierpinski carpet model and the thermal resistance network model of a fractal geometry combined with the principle of the weighted average. The new analytical expression can be used to overcome the issue of the predicted values deviating considerably from the experimental results when the fractal series n is too small. At the same time, an accurate solution for the dimensionless ratio (t+) of the width of the carbon materials to the side length of the Sierpinski carpet is presented. Compared with the general solution algorithm proposed by Song, the commonly used quadratic parallel (QP) prediction model, and the Maxwell–Eucken model, this algorithm is more reliable. The proposed prediction model can be used to predict the effective thermal conductivity of fatty acid–carbon materials.

FRACTAL ANALYSIS FOR THERMAL CONDUCTIVITY OF DUAL POROUS MEDIA EMBEDDED WITH ASYMMETRIC TREE-LIKE BIFURCATION NETWORKS

Heat transport in tree-like bifurcation networks has been widely studied in various fields. In this work, we investigate heat conduction in the dual porous media embedded with asymmetric tree-like bifurcation networks. In addition, considering the effects of nonuniform tube shape, we assume that the bifurcated tube shows sinusoidal fluctuations. Based on the fractal distribution of pore size and bifurcation structure, we established a dimensionless effective thermal conductivity (ETC) model of the dual porous media. The dimensionless ETC ([Formula: see text] obtained is related to the porosity ([Formula: see text], the fluid–solid thermal conductivity ratio ([Formula: see text], the pore area fractal dimension [Formula: see text] and the structural parameters of the bifurcation network (bifurcation level [Formula: see text], length ratio [Formula: see text], radius ratio [Formula: see text], fluctuation amplitude factor [Formula: see text], bifurcation angle [Formula: see text]. To verify the validity of this model, a comparison of the present dimensionless ETC model with available experimental data was carried out and the results were in good agreement. We have discussed the effects of each parameter on the dimensionless thermal conductivity in detail and constructed parametric planes to evaluate the structural parameters more directly. The model has positive implications for revealing the heat transport mechanism in asymmetric tree-like bifurcation dual porous media.

INVESTIGATIONS ON STRESS-DEPENDENT THERMAL CONDUCTIVITY OF FRACTURED ROCK BY FRACTAL THEORY

Stress field is a key external factor affecting the thermodynamic properties of rock in many scientific and engineering fields ranging from geoscience to practical problems in geothermal resources, oil and gas exploitation, mineral exploitation, etc. In this work, the stress-dependent models for the effective thermal conductivity of fractured rock are proposed by combining the fractal geometry theory and the two-part Hooke’s model. The proposed model for the effective thermal conductivity is a function of the effective stress and structure-parameters and physical parameters of fractured rock, such as porosity ([Formula: see text], fracture orientations ([Formula: see text] and [Formula: see text] as well as [Formula: see text] and [Formula: see text]. The results show that thermal conductivity increases nonlinearly with increasing stress at low stress range due to the closure of fractures, then, it exhibits an almost linear behavior of pressure dependence at high stress range for most fractures closing completely. The proposed thermal conductivity model coupled with stress is evaluated by comparison with the available experimental data. The model predictions show a good agreement with the available experimental data. A more detailed investigation of the essential influence of structures parameters of a set of fractures on the dimensionless stress-dependent thermal conductivity is performed. The proposed model may provide a good quantitative understanding of the stress effect on thermal conductivity of fractured rock.

ASSESSMENT OF TEMPERATURE STRESSES IN THE LITHOSPHERE OF THE EARLY MOON

In the early stages of the Moon’s formation, its growing lithosphere experienced complex time-varying temperature and gravitational stresses. Despite the subsequent intense impact transformation of the surface, during the gravimetric study of the GRAIL space mission, the presence of ancient deep intrusions was detected. The analysis of linear gravitational anomalies shows the expansion of the outer rigid layer of the planet at a certain early stage of the Moon’s evolution due to the excess of temperature stresses over gravitational compression. Obtaining the dependence of the time interval of the lithosphere expansion on a number of dimensionless thermal conductivity parameters will make it possible to refine existing models of the thermal and geochemical evolution of the early Moon.

Influence of solder condition on effective thermal conductivity of two-directional random fibres: Pore-scale simulation

It is indicated that the solder joint of the metal fibrous materials is a critical factor impacting the heat conduction. To reveal the mechanism by which solder joint sizes, solder joint skips, solder flux materials, and filling media affect the thermal conductivity of fibres, pore-scale numerical simulation is employed to study the thermal transport in two-directional (2-D) random fibres. Satisfactory agreement with existing data validates the numerical model. The dimensionless effective thermal conductivity (ETC) of the porous fibres increases with the solder joint sizes. As the solder joint size (i.e., solder joint ratio) increases by 3.06%, the in-plane (ke-in) and out-of-plane (ke-out) dimensionless ETC increase by 9.0% and 437.2%, respectively. However, the solder joint skips will weaken the thermal conductivity of the fibres. For the same fibre, the ETC of the fibre increases as the thermal conductivity of solders increases. Further, when the dissimilarity in thermal conductivity between the filling medium and the fibre is reduced, the fibre is less affected by the solder joint skips. Finally, it should be supplemented that the in-plane and out-of-plane ETC (ke-in and ke-out) of the fibre without any solder joint are reduced by an average of 14.3% and 98.8%, respectively.

A study of dual phage lag model of heat transfer for solidification of binary eutectic alloy

In this paper, we developed a generalized non-Fourier model of energy transfer for solidification of a binary eutectic system (Al-Cu alloy). The mathematical model is divided into three stages: stage I (liquid region), II (liquid and mushy region), and III (liquid, mushy and solid region). The analysis of the model has been done in dimensionless form. This model is a boundary value problem of the system of second-order hyperbolic partial differential equations for Stage I, the moving boundary problem of the System of second-order hyperbolic partial differential equations for Stage II and III. For the solution, we applied Laplace transform technique in Stage I and Legendre wavelet spectral Galerkin method in Stage II and III. The effect of unit-less parameters: Relaxation time, Predvoditelev numbers, dimensionless thermal conductivity, Fourier numbers, Stefan number, solid fraction on temperature, and position of moving interface are discussed in detail.

Thermal conductivity prediction of pure refrigerants and mixtures based on entropy-scaling concept

The residual entropy scaling (RES) concept builds up connection between the transport properties and thermodynamic equation of state directly. However, the low-density divergence issue of the reduced dimensionless thermal conductivity might not be fixed properly through current literature models. This study proposes a new RES model coupled with the original perturbed-chain statistical associating fluid theory (PC-SAFT) equation of state (EoS) for the thermal conductivities of 11 typical refrigerants (including hydrocarbons, hydrofluorocarbons, hydrofluoroolefins and carbon dioxide) and their multicomponent mixtures. Model parameters are adjusted to saturated vapor and liquid data obtained from REFPROP. The combination of a modified reference thermal conductivity term as an extension of the original work by Rosenfeld et al. and the new formulation of the reduced term could provide reasonable predictive results for the investigated pure and mixture systems with AAD% of 3.95% and 5.76%, respectively. As expected, quasi-linear behaviors in both dilute and dense fluid regions could be observed for the refrigerant systems under study. The extended corresponding states (ECS) and the other two RES approaches reported in the literatures are used as reliable tools for assessing the predictive capability of the proposed model. However, the critical enhancement of thermal conductivity and the correction of the reference EoS especially near the critical point will be the priority in the future.

ANALYSIS OF THERMAL CONDUCTIVITY OF DAMAGED TREE-LIKE BIFURCATION NETWORK WITH FRACTAL ROUGHENED SURFACES

Tree bifurcation network has always been a hot issue concerning scientists all over the world. However, the research on the heat conduction properties of damaged tree-like bifurcation networks is insufficient, and many issues remain unsolved. Therefore, this paper systematically studies the heat conduction phenomenon of a new damaged tree-like bifurcation network with fractal roughened surfaces (FRS), and derives the equivalent thermal conductivity of damaged network. According to the study, the dimensionless thermal conductivity (DTC) shows a downward trend with an increase in roughness. In addition, it can be seen that an increase in the number of damaged channels leads to a downward trend in the optimal thermal conductivity, but a slow upward trend in the critical diameter. Another interesting phenomenon is that the factors leading to the change in the optimal thermal conductivity do not include the total number of bifurcation levels, bifurcation number or the fractal dimension of length distribution. The model established in this paper does not contain any empirical constants to ensure that each parameter has its physical significance, revealing the heat transfer mechanism of tree-like bifurcation network more profoundly.

Thermalhydraulic comparison of supercritical fluids in minichannel heat sink to assess the suitability of macroscopic scaling rules

Suitability of supercritical fluids as heat transport media in mini-/microscale has not been explored much, despite showing enormous potential in macrochannels. One major hindrance thereof is the absence of scaling rules devised specifically for such scales. Present numerical study explores the thermalhydraulic characteristics of five different supercritical fluids, identified based on the diversity in their respective critical-point parameters, for a particular minichannel employing a conventional set of scaling relations. Supercritical fluids exhibit consistently-superior heat transfer coefficient compared to single-phase, and even boiling, channels. Substantial level of similarity has been observed in the visual nature of the dimensionless momentum and thermal fields and local recirculation patterns across the fluids, as a consequence of the imposed scaling. Noticeable difference, however, exists in characterizing dimensionless groups, such as local Reynolds number and local and area-averaged Nusselt number, pointing toward the deficiency of the scaling model. R134a can be earmarked as the optimum fluid among the considered ones, both in terms of Nusselt number and dimensionless pressure drop, though it experiences the largest entropy generation as well. It is suggested to consider the profiles of Prandtl number and dimensionless thermal conductivity with dimensionless enthalpy for improving the used scaling rules.

Numerical simulation of the melting and solidification processes of two organic phase change materials in spherical enclosures for cold thermal energy storage applications

The encapsulation of phase change materials (PCMs) is a convenient alternative for latent heat thermal energy storage systems (LHTESSs) because of the excellent relationship between their storage volume and the heat transfer surface. The goal is to establish a unified heat transfer behavior of encapsulated PCM. Computational fluid dynamics (CFD) numerical simulations are performed with ANSYS/Fluent of the melting and solidification processes of the commercial organic PCMs RT5HC and RT10HC. The numerical study involve a parametric study of the sphere diameter (40, 60, and 80 mm), thermal gradient (difference between wall temperature and phase change temperature) for melting (1–5 °C), and thermal gradient for solidification (1–3 °C). The numerical simulations implemented the experimentally obtained and temperature-dependent thermophysical properties of thermal conductivity (k), phase change enthalpy (L), specific heat (cp), and viscosity (μ) of the PCMs. The computational simulations allowed us to determine the behavior of the heat transfer coefficient (h) as a function of the liquid fraction (φ) and solid fraction (φso) of the PCM. Furthermore, nonlinear regression adjustments to the Nusselt number (Nu) curves allowed the fitting of power-law curves with suitable correlation parameters. Finally, the coefficients (a, b, and c) of the proposed fitting power-law curves were plotted and adjusted via linear regression through the origin as a function of the dimensionless numbers of Archimedes (Ar), Stefan (Ste), Grashof (Gr), Prandtl (Pr), and dimensionless thermal conductivity (κ), unifying the behavior of the melting and solidification processes through two correlations of the Nu number. These correlations will be helpful in the design of LHTESS tanks for their application in heat-pump-type air-conditioning systems in buildings.

Theoretical Analysis of Two Collinear Cracks in an Orthotropic Solid under Linear Thermal Flux and Linear Mechanical Load

This paper studies the problem of a cracked orthotropic solid subject to linear thermal flux and linear mechanical load. The proposed extended partially insulated crack model is employed to simulate two collinear cracks. Taking advantage of Fourier transform technique and superposition theory, the closed form of some physical quantities and fracture parameters is obtained. Some simple examples are employed to demonstrate dimensionless thermal conductivity ( R c ) between the upper and below crack regions, and the proposed coefficient ( ε ) has great effects on some physical quantities and fracture parameters.

AN ANALYTICAL MODEL FOR EFFECTIVE THERMAL CONDUCTIVITY OF THE MEDIA EMBEDDED WITH FRACTURE NETWORKS OF POWER LAW LENGTH DISTRIBUTIONS

The effective thermal conductivity of porous media is of steady interest in the design of new materials. In this study, an analytical model for dimensionless effective thermal conductivity of media embedded with fracture networks of the power law length distributions is proposed. It is found that the proposed dimensionless effective thermal conductivity is a function of micro-structural parameters of media, such as the porosity [Formula: see text] and fracture orientation (dip [Formula: see text] and azimuth [Formula: see text]). The present results show that the dimensionless effective thermal conductivity of the media increases with the increase in the ratio [Formula: see text] and the power law exponent [Formula: see text] at [Formula: see text]. Inversely, when [Formula: see text], it decreases with the increase in the power law exponent [Formula: see text]. In addition, the dimensionless effective thermal conductivity is gradually independent of the orientation as the ratio [Formula: see text]. The present model may provide a significant insight into the mechanism of heat transfer in the media embedded with fracture networks of power law length distributions.

A NOVEL MODEL FOR EFFECTIVE THERMAL CONDUCTIVITY OF TREE-LIKE BRANCHING NETWORK WITH FRACTAL ROUGHENED SURFACES

The tree-like branching network shows prominent advantages in heat and mass transfer, which has been studied for a long time. In this paper, a novel model for the effective thermal conductivity is proposed for the tree-like branching network with fractal roughened surfaces. The proposed model is a function of structure parameters of the tree-like branching network with fractal roughened surfaces, including the relative roughness ([Formula: see text], the length ratio ([Formula: see text], the diameter ratio ([Formula: see text], the total number of branching levels ([Formula: see text] and the branching number ([Formula: see text]. The influence of structure parameters on the effective thermal conductivity is discussed individually based on the systematic data analysis. It is found that the relative roughness has a clear effect on the thermal conductivity. Furthermore, the effective thermal conductivity decreases with an increase in the relative roughness. It is also seen that the effective thermal conductivity decreases with an increase in the length ratio. In addition, there is an interesting phenomenon that the effective thermal conductivity increases initially and decreases afterwards with an increase in the diameter ratio. The determined dimensionless effective thermal conductivity of the whole tree-like branching network with roughened surfaces is in good agreement with existing model reported in the literature. The proposed model does not include any empirical constant and every parameter has physical meaning, which can better reveal the mechanism of heat transfer in the tree-like branching network.

The Analysis of a Cracked Material under Combined Unsymmetric Thermal Flux and Symmetric Linear Mechanical Loading

An improved partially permeable crack model is put forward to deal with the problem of a single crack embedded in an orthotropic or isotropic material under combined unsymmetric thermal flux and symmetric linear mechanical loading. With the application of the Fourier transform technique (FTT), the thermoelastic field is given in a closed form. Numerical results show combined unsymmetric linear thermal flux, symmetric linear mechanical loading, and dimensionless thermal conductivity, and the coefficient has influences on fracture parameters. For the improved partially permeable crack, the mode II stress intensity factor and the energy release rate might be zero or positive under combined unsymmetric thermal flux and symmetric linear mechanical loading. Therefore, closure of the crack tip region need not be considered under combined unsymmetric thermal flux and symmetric linear mechanical loading when making use of fracture parameters as a criterion.

FRACTAL STUDY ON THE HEAT TRANSFER CHARACTERISTICS IN THE ROUGH MICROCHANNELS

In this paper, considering the structure of the shape factor and the rough element, combined with fractal geometry theory, the shape factor, the relative increase in the heat transfer between the rough surface and the smooth surface in a fractal unit, and the fractal analytical expression of the dimensionless equivalent thermal conductivity caused by convection thermal conductivity are derived. In addition, the relationship between the microstructure parameters of the rough surface and the shape factor, the relative increase in convective heat transfer and the dimensionless equivalent thermal conductivity are studied. It is found that the relative increment of the convective heat transfer and the shape factor are proportional to the ratio of the minimum height and maximum height of the rough element and the fractal dimension, and the dimensionless equivalent thermal conductivity model deduced in this paper is compared with the existing experimental data to verify the validity of the model.

Permeability and thermal conductivity models of shale matrix with a bundle of tortuous fractal tree-like branching micropore networks

The complex pore structures are of great significance to gas transport and heat transfer in shale gas reservoirs, but an accurate prediction of permeability and thermal conductivity is a challenge to a porous medium with the complex geometry and distribution of pore networks. This study investigates the permeability and thermal conductivity of shale matrix with a bundle of tortuous fractal tree-like micropore networks. First, a new permeability model and a new dimensionless effective thermal conductivity model of shale matrix are developed to include the branching and tortuous characteristics of the fractal tree-like network. Then, the permeability model is validated by experimental data, and the dimensionless effective thermal conductivity model is compared with simulation results in literature. Last, sensitivity analysis is conducted to identify key parameters in these two new models. It is found that the two new models can well describe the gas transport and heat transfer in shale matrix with a bundle of tortuous fractal tree-like pore networks. The permeability model has much higher accuracy than traditional models. Tortuosity fractal dimension and radius fractal dimension of shale matrix with the bundle of tortuous fractal tree-like networks have negative influences on both permeability and thermal conductivity. Length ratio and radius ratio significantly alter the gas flow resistance and thermal resistance, and thus lead to the change of permeability and thermal conductivity. Branching angle has insignificant effect on the permeability and thermal conductivity and thus is ignorable.

Computational Evaluation of Thermal Response of Open-Cell Foam With Circular Pore

Abstract The evaluation of effective material properties in heterogeneous materials (e.g., composites or multicomponent structures) is critically relevant to a wide spectrum of applications, including nuclear power, electronic packaging, flame retardants, hypersonics, and gas turbine power. The work described in this paper is centered around the numerical assessment of the thermal behavior of porous materials obtained from finite element thermal modeling and simulation. Two-dimensional, steady-state analyses were performed on unit cells with centered, circular pores using a second-order accurate Galerkin finite element method (FEM). The effective thermal conductivities of the porous systems were examined, encompassing a range of porosities from 4.9% to 60.1%. The geometries of the models were generated based on ordered circular pores for each modeled porosity level. The system response quantity (SRQ) under investigation was the dimensionless effective thermal conductivity across the unit cell. The dimensionless effective thermal conductivity was compared across all simulated cases, producing a trend between porosity and effective thermal conductivity. In the presented investigation, the method of manufactured solutions (MMS) was used to perform code verification, and the grid convergence index (GCI) was employed to estimate discretization uncertainty as solution verification. Code verification concluded an approximately second order accurate Galerkin FEM solver. It was found that the introduction of porosity to the unit cell material structure reduces effective thermal conductivity, as anticipated. Numerical results obtained in this study are compared to an analytical solution and to a sample of empirical data. This approach can be readily generalized to study a wide variety of porous solids from ranging from structures at the nanoscale—such as nanocarbon tubes—to structures at macrolevel scales—such as geological features.

Thermal Boundary Evolution of Molten Pool During Wire and Arc Additive Manufacturing of Single Walls of 5A06 Aluminum Alloy

Molten pool uniformity is a prerequisite for wire and arc additive manufacture to achieve a uniform surface appearance. The thermal boundary is a key indicator to evaluate the thermodynamic state of the molten pool. This paper focuses on thermal analysis through finite element simulation and characterizes the thermal evolution of a molten pool during bottom-up deposition. The peak temperature of the substrate plate increases from 375.7 °C to 623.1 °C when peak current increases from 120 A to 180 A. The temperature gradient decreases from 40 °C/mm to 30 °C/mm. Weld speed increases from 0.15 m/min to 0.25 m/min; the temperature gradient is kept at about 30 °C/mm. Dimensionless thermal conductivity, Ge number, is proposed and defined as the criteria to estimate layer size variation. For Ip = 140 A, the average value of the Ge number is 87.7, and it shows the best numerical stability. Dimensionless thermal conductivity, Ge, is proposed and defined as the criteria to estimate the occurrence of layer size variation. Prolonging heating time and increasing cycle frequency, Ge number shows favorable stability during bottom-up deposition. The experimental results are consistent with analytical conclusions, which proves the validity of the Ge number as an indicator of deposition stability.