Stagnant Thermal Conductivity Research Articles

A three-energy equation model and estimation of effective thermal properties for transient analysis of bi-disperse packed bed thermocline storage system

For the first time, a complete set of the volume averaged governing equations based on a three-energy equation model has been proposed to attack the transient thermal response of a bi-disperse packed bed thermocline storage system, in which its void is filled by smaller particles (i.e. solid filler). Effective permeability, stagnant thermal conductivity and effective thermal conductivities for three individual phases, namely, the fluid phase, large particle phase and small particle phase of the bi-disperse packed bed storage system, have been evaluated taking full account of tortuosity and appealing to a volume averaging theory. Subsequently, the three individual energy equations in the three-energy model were coupled and analytically solved to reveal a transient behavior of the temperature fields of the individual phases. The effective discharging efficiency of the bi-disperse packing is found to decrease substantially as compared with that of the mono-disperse packing under equal pumping power. Though one can save a substantial initial cost by filling the void by solid fillers, one must pay a penalty of a higher cost in running the bi-disperse packed bed thermocline storage system. The analysis presented in this study serves to estimate both initial and running costs of a bi-disperse packed bed thermocline storage system.

Investigation of Effective Thermal Conductivity of Spherical Packed Beds Using In-Line Contacting Three-Dimensional Cubic Unit Cell Model

This paper estimates the effective stagnant thermal conductivity of a spherical packed bed using an experimentally validated modified collocated unit cell model. The study focused on the effect of primary and secondary parameters on effective thermal conductivity. The effects of concentration of the spherical packed bed, thermal conductivity ratio, contact conductance resulting from particle-to-particle contact, and moderate temperature are studied in detail. Analytical expressions are derived for the in-line contacting three-dimensional solid cubic arrays by adopting the unit-cell-based thermal resistance method for determining the effective stagnant thermal conductivity. The effective thermal conductivity of different heterogeneous packed beds with solid-fluid materials possessing conductivity ratio in the range of 1–1000 and concentrations in the range of 0–1 are predicted for various temperatures at constant pressure regimes using the developed model. Furthermore, the developed model is used to estimate the effective stagnant thermal conductivities of packed beds filled with glass and ceramic bead materials, which are compared with values measured using the steady-state square guarded hot plate apparatus over the moderate temperature range of 323–673 K. The values for both uniform-sized glass and ceramic beads are found to be within ±11.26% and ±13.35% of each other.

A Novel Slug Heat Test Theoretical and Indoor Model Research for Determining Thermal Property Parameters of Aquifers and Rock-Soil Skeletons

As important parameters for characterizing heat transfer, thermal property parameters of aquifers and rock-soil skeletons have important research significance in the development and utilization of geothermal resources. The slug heat test is inspired by the slug test, and the heat is instantaneously excited in the test well so as to change the temperature of test section in the test well instantaneously. Based on the thermal radial convection-dispersion theory and the principle of heat conservation, the theoretical model of the slug heat test is established, and the model is solved by Laplace transform and inverse transform to obtain multiple sets of standard curves under different conditions. The slug heat tests were conducted in the indoor model, the slug heat test data under different hydrodynamic conditions were fitted with the standard curves and the thermal property parameters, including effective thermal conductivity, stagnant thermal conductivity, thermal mechanical dispersion coefficient, thermal dispersive degree, thermal diffusivity, heat capacity of aquifer, heat capacity and thermal conductivity of rock-soil skeletons, were accurately obtained. The test results are in good agreement with the empirical values. Meanwhile, the effective thermal conductivity of the aquifer also clearly increases with the increase of flow rate. The excitation temperature difference had little effect on the effective thermal conductivity of the aquifer. At the same time, numerical simulation methods are used to establish a numerical model consistent with the indoor test model, and the numerical model is assigned with the thermal property parameters obtained from the indoor slug heat test, and the measured values of temperature changes in the test well during the slug heat test under different hydrodynamic and excitation strength conditions are compared with the simulated values for verification. The research results show that the slug heat test has the characteristics of high applicability, simple operation and rapid testing, and can effectively determine the thermal properties parameters of aquifers and rock-soil skeletons.

Predicting thermal and mechanical performance of stochastic and architected foams

This study conducted pore-scale simulations to estimate the stagnant thermal conductivity and mechanical performance of stochastic metal foams and architected metal foams. The stagnant thermal conductivity of 5-, 10-, 20- and 40-ppi commercial aluminum foams were estimated to be 6.74, 6.49, 5.97, and 7.16 W/m/K, respectively. Thermal conductivity depended mainly on porosity and was not a strong function of pore size. The pore-scale models simulated thermal conductivities of 40-ppi Al foam along three orthogonal directions are 7.16, 7.92, and 7.56 W/m/K, respectively, indicating a strong isotropic nature. Compared with stochastic foams, the architected foam with a triply periodic minimal surface showed 103% higher thermal conductivity, 488% higher stiffness, and 265% more energy absorption capability. The findings reported here not only provide a fundamental understanding of the interplay between performance and architecture but also open a new avenue to create multifunctional foams.

An Analytical and Numerical Estimation of the Effective Thermal Conductivity of Complex Metal Frame Core Structures

An analytical and numerical study was conducted for estimation of the effective thermal conductivities of curved metal frame core structures, which can replace metal foams, in views of their advantages over the metal foams for both load bearing and heat dissipation. The trajectory of the frame ligament and its cross-sectional area were allowed to vary arbitrarily in the three-dimensional (3D) space. The analytical formula obtained by extending the formula previously proposed by Bai et al. (2017, “A General Expression for the Stagnant Thermal Conductivity of Stochastic and Periodic Structures,” ASME J. Heat Transfer, 140(5), p. 052001) was examined by comparing it with the numerical results directly obtained from full 3D numerical computations. An air layer partially filled with a collection of coiled circular rods was treated both analytically and numerically. Furthermore, the effect of lattice nodes on the effective thermal conductivity was investigated by introducing an analytical model with the lattice ligaments merging together at one nodal point. The analytical expressions thus derived for the lattice structures with nodes were applied to tetrahedral structure and octet-truss structure to find their effective thermal conductivities, which are found to agree closely with the 3D numerical results. Thus, the present analytical expressions can be used to customize the structure to meet its desired thermal performance.

A General Expression for the Stagnant Thermal Conductivity of Stochastic and Periodic Structures

A general expression has been obtained to estimate thermal conductivities of both stochastic and periodic structures with high-solid thermal conductivity. An air layer partially occupied by slanted circular rods of high-thermal conductivity was considered to derive the general expression. The thermal conductivity based on this general expression was compared against that obtained from detailed three-dimensional numerical calculations. A good agreement between two sets of results substantiates the validity of the general expression for evaluating the stagnant thermal conductivity of the periodic structures. Subsequently, this expression was averaged over a hemispherical solid angle to estimate the stagnant thermal conductivity for stochastic structures such as a metal foam. The resulting expression was found identical to the one obtained by Hsu et al., Krishnan et al., and Yang and Nakayama. Thus, the general expression can be used for both stochastic and periodic structures.

The stagnant thermal conductivity of porous media predicted by the random walk theory

This study developed a predictive model based on the random walk theory to calculate the stagnant thermal conductivity of porous materials considering detailed pore geometries. This model can take the connectedness of solid matrix and filling fluid into consideration and simulate porous media with a wide range of porosities and geometries. The model was calibrated by comparing simulated thermal conductivities of simplified geometries proposed in literature with the corresponding experimental data. Results indicate that the stagnant thermal conductivities of porous media are very sensitive to the connectedness (or contact resistance) of the solid matrix, especially when the conductivity of the solid matrix (e.g. Al) is much higher than the conductivity of filling fluid (air). This study also imported reconstructed three-dimensional geometries (measured by micro computed tomography) of Al foams into the model and simulated their stagnant thermal conductivities. The measured Al foams have porosities around 0.9 and different pore sizes (5, 10, 20, and 40 parts per inch). The simulated stagnant thermal conductivities of Al foams slightly increased with decreased pore size at the porosity around 0.9.The model developed in this study is a powerful tool to predict stagnant thermal conductivity of porous media widely applied in a broad range of science and engineering disciplines. The low computational time of this model makes it a unique tool to provide real-time updates of material properties for on-site applications or other computational models. Furthermore, the solid matrix and the filling fluid in the porous medium could be multi-phase mixtures or composite materials due to the statistical nature of the random walk model.

Determination of thermal conductivity of baled agricultural biomass

Modeling of transport phenomena in materials with porous bed features requires thermophysical properties, such as porosity, permeability and thermal conductivity be known.The paper presents experimental data obtained in the course of investigation focused on analysis of stagnant thermal conductivity of biomass. Experiments were conducted using a custom designed and constructed experimental setup and implementing original experimental procedure. Results obtained enabled stagnant thermal conductivity of biomass to be determined for different biomass porosity values.Analysis of the experimental data enabled functional dependence of stagnant thermal conductivity on porosity to be determined, with porosity varied in a range 0.50–0.85. Results obtained indicate that reduced bed porosity, down to the value of 0.65, causes stagnant thermal conductivity of the bed to be reduced. Further porosity reduction, below the value of 0.65, results in increased stagnant thermal conductivity of the bed.Experimental procedure developed shall be useful for gathering experimental data on different materials planned to be analyzed when examining transport phenomena occurring during combustion of biomass bales. In addition, data to be acquired shall be helpful in modeling the transport phenomena associated with combustion in porous beds.

Modeling of Effective Stagnant Thermal Conductivity of Porous Media

Based on one-dimensional analysis of heat conduction, a general formula for the effective stagnant thermal conductivity of spatially periodic porous media is derived without assuming local thermal equilibrium. Furthermore, we discuss the contribution of the contact area between particles to the effective stagnant thermal conductivity in detail, and the modification of the formula is proposed to predict the actual effective stagnant thermal conductivity for the porous media. The present results are in good agreement with experimental results of Nozad et al. (1985, “Heat Conduction in Multi-Phase Systems I: Theory and Experiments for Two-Phase Systems,” Chem. Eng. Sci., 40(5), pp. 843–855) for a packed-sphere bed.

A theoretical model on the effective stagnant thermal conductivity of an adsorbent embedded with a highly thermal conductive material

The effective stagnant thermal conductivity of an adsorbent bed plays an important role in the system performance of adsorption cooling systems and other applications. It is proposed that embedding the adsorbent with a highly thermal conductive material by forming a coating layer on the adsorbent molecules could increase the overall heat transfer rate resulting in a higher COP and SCP. A theoretical model based on the area-contact approach was developed in this study to predict the effective stagnant thermal conductivity of a surface coated porous material. This modified area-contact model can also be employed in different applications where surface coated porous materials are used. It was found that the coating layer thickness, c, the solid/fluid thermal conductivity ratio, λ, and the porosity, ϕ, are factors that can affect the effective stagnant thermal conductivity. When λ=103, the effective stagnant thermal conductivity was improved by about 4.3 times for ϕ=0.4 and 9.7 times for ϕ=0.8 with a deformed factor, α=0.002 and a coating layer thickness, c=0.05. An FEM simulation model was built showing that the modified area contact model has better agreement with the simulation result than the area contact model. It is suggested that this modified area contact model can be applied to predict the effective stagnant thermal conductivity of a surface coated porous material.

CFD testing of the pointwise use of the Zehner–Schlünder formulas for fixed-bed stagnant thermal conductivity

For the conduction mode of heat transfer in a fixed bed, we have carried out an evaluation of whether the most widely-used equations for the overall stagnant bed effective thermal conductivity ke0, the Zehner–Schlünder cell model, can be used on a pointwise basis to obtain local values of ke0(r) using the local bed voidage, ε(r). A three-dimensional computational fluid dynamics (CFD) model of an annular test bed of spheres with tube-to-particle diameter ratio equal to four was constructed to obtain the local void fraction, and solved for the radial temperature profiles. Comparisons to a one-dimensional pseudo-continuum model with ke0(r) gave good agreement for a range of values of solid thermal conductivity, establishing that the Zehner–Schlünder cell model can be used in local models of fixed-bed heat transfer.

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.

Experimental determination thermo physical characteristics of balled biomass

The paper presents the results of an experimental investigation conducted to determine the permeability coefficients and the stagnant thermal conductivity i.e. the thermo physical properties deemed to be the most important features of the bed material when considering combustion in pusher-type furnaces (i.e. combustion of biomass bales in cigar burners).The appropriate experimental methods have been developed and suitable experimental apparatus designed and constructed in order to determine permeability and stagnant thermal conductivity coefficient of the soybean straw bales. The experimental investigation conducted was aimed at examining the effects of relevant biomass bale parameters on the aforementioned coefficients.Based on the numerous measurements performed, correlations were obtained that are deemed highly important for optimization of the biomass bale combustion, as well as for modeling the transport phenomena occurring in the porous bed formed during biomass bales combustion in cigar burners.Data collected during the course of research investigation were used to develop a detailed CFD model of straw bales combustion in cigar burners.

EFFECT OF POROSITY VARIATION IN DETERMINATION OF STAGNANT AND DISPERSION THERMAL CONDUCTIVITIES IN THE RESIN TRANSFER MOLDING PROCESS

A numerical analysis for determination of the stagnant and dispersion thermal conductivities in the Resin Transfer Molding (RTM) process is presented. A solution was obtained for the momentum and energy equations, which are used in this study and which represent the Brinkman−Forchheimer-extended Darcy model, and one-temperature energy equation. Subject to the appropriate boundary conditions, the governing equations are solved by the finite volume method on a collocated grid. In order to satisfy the pressure−velocity coupling in the solution, the SIMPLE algorithm was used and for determining the mass flux flowing past the cell faces, the Rhie−Chow interpolation was applied. Both constant and variable porosities near the channel wall were considered, so that the exact velocity distribution in a channel can be found. The numerical results for the temperature profile along with the centerline of the channel are fitted against available experimental data, and the effective thermal conductivity is obtained. Considering the quadratic relationship between the dispersion conductivity and Peclet number, the Chang model is proposed to predict the stagnant thermal conductivity as well as a correlation for the dispersion thermal conductivity.

A Local Thermal Non-Equilibrium Analysis of Fully Developed Forced Convective Flow in a Tube Filled with a Porous Medium

A local thermal non-equilibrium model has been considered for the case of thermally fully developed flow within a constant heat flux tube filled with a porous medium. Exact temperature profiles for the fluid and solid phases are found after combining the two individual energy equations and then transforming them into a single ordinary differential equation with respect to the temperature difference between the solid phase and the wall subject to constant heat flux. The exact solutions for the case of metal-foam and air combination reveal that the local thermal equilibrium assumption may fail for the case of constant heat flux wall. The Nusselt number is presented as a function of the Peclet number, which shows a significant increase due to both high stagnant thermal conductivity and thermal dispersion resulting from the presence of the metal-foam.

A synthesis of tortuosity and dispersion in effective thermal conductivity of porous media

Effects of tortuosity and dispersion on the effective thermal conductivity of fluid-saturated porous media are investigated analytically with help of a volume averaging theory. Firstly, a general expression for the effective stagnant thermal conductivity has been derived using a unit cell model, which consists of rectangular solids with connecting arms in an in-line arrangement. The validity of the expression for the stagnant thermal conductivity has been confirmed comparing the present results with available experimental and theoretical data for packed beds, porous foams and wire screens. Secondly, an general expression for the thermal dispersion conductivity has been sought with help of the two energy equations for solid and fluid phases, derived on the basis of a volume averaging theory. It has been shown that the interstitial heat transfer between the solid and fluid phases is closely associated with the thermal dispersion. The resulting expressions for the longitudinal and transverse thermal dispersion conductivities agree well with available experimental data and empirical correlations.

A NUMERICAL ANALYSIS OF THERMAL CONDUCTIVITY, THERMAL DISPERSION, AND STRUCTURAL EFFECTS IN THE INJECTION PART OF THE RESIN TRANSFER MOLDING PROCESS

Thermal conductivity, thermal dispersion, and structural effects in resin transfer molding (RTM) process are studied numerically. The injection part of the RTM process is modeled as a transport of resin flow through a fibrous porous medium in a long rectangular channel. The fluid flow is modeled using the Darcy-Brinkman-Forchheimer model and the heat transfer process using the energy equation based on local thermal equilibrium assumption. Both isotropic and anisotropic heat transfer in porous media are investigated. The governing equations are solved numerically for the isotropic heat transfer case and analytically for the anisotropic case. The numerical results are fitted to the available experimental data with at most 3% discrepancy, and the effective thermal conductivity of the fibrous porous medium is estimated. Taking into account the quadratic dependency of dispersion conductivity on Peclet number, the Chang model is recommended for the prediction of stagnant thermal conductivity as well as a correlation for dispersion conductivity. Finally, the effects of structural parameters on temperature distribution and velocity profiles are investigated.

Effects of variable viscosity on non-Darcy MHD free convection along a non-isothermal vertical surface in a thermally stratified porous medium

An analysis is performed for non-Darcy free convection flow of an electrically conducting fluid over an impermeable vertical plate embedded in a thermally stratified, fluid saturated porous medium for the case of power-law surface temperature. The present work examines the effects of non-Darcian flow phenomena, variable viscosity, Hartmann–Darcy number and thermal stratification on free convective transport and demonstrates the variation in heat transfer prediction based on three different flow models. The wall effect on porosity variation is approximated by an exponential function. The effects of thermal dispersion and variable stagnant thermal conductivity are taken into consideration in the energy equation. The resulting non-similar system of equations is solved using a finite difference method. Results are presented for velocity, temperature profiles and local Nusselt number for representative values of different controlling parameters.

Forced Convective Heat Transfer in Metallic Fibrous Materials

A numerical study is reported for high Reynolds number forced convection in a channel filled with rigid metallic fibrous materials of high porosity. The effects of convective and form inertia, viscous shear, and thermal dispersion are all considered together. Inertia and thermal dispersion are modeled. The numerical results suggest that heat transfer rate increases with increasing Reynolds number within a range, but not significantly beyond that range. The heat transfer rate also increases with stagnant thermal conductivity, and decreases with Darcy number. The fiber thickness was found to have significant influence on thermal dispersion. The range of applicability of the local volume averaging in terms of the significant parameters is discussed.

A theoretical and experimental study on stagnant thermal conductivity of bi-dispersed porous media

Based on the unit cell model, stagnant thermal conductivities of bi-dispersed porous media with different micro- and macro- porosities were predicted. Measurements of stagnant thermal conductivities of two samples of bi-dispersed porous media (with micro-pore diameter of 80 μm and macro-pore diameters of 200 μm and 400 μm) saturated with three different fluids (air, FC-72 and water) respectively were conducted. For comparison, the effective thermal conductivity of a mono-dispersed medium with pore diameter of 800 μm saturated with the same fluid was measured. The measured data are shown to be in agreement with those based on theoretical predictions. The results indicated that the effective thermal conductivity of a bi-dispersed porous medium is smaller than that of the mono-dispersed porous medium saturated with the same fluid because of the contact resistance at micro-scale and higher porosity for bi-dispersed porous media in comparison with the mono-dispersed porous media.