Effective Stagnant Thermal Conductivity Research Articles

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.

Numerical and experimental study of stagnant effective thermal conductivity of a graphite pebble bed with high solid to fluid thermal conductivity ratios

In this work, a numerical method combining conduction and radiation is used to study the stagnant effective thermal conductivity of a simple cubic packed bed with high solid to fluid thermal conductivity ratios. The experiment is designed to verify the numerical model and the numerical results agree well with the experimental results. It is found that the effective thermal conductivity accounting for conduction and radiation declines first until 690 K as the temperature rises and then increases. This trend is a result of the interactions between different heat transfer components. Compared with the numerical values and experimental values, the ZBS correlation with a proper value for the empirical parameter φ selected is capable of predicting the stagnant effective thermal conductivity in the bulk region. Lastly the different heat transfer components caused by the solid/fluid conduction, contact conduction and radiation are extracted from the total effective thermal conductivity and their individual contributions are analyzed. The contact conduction makes a major contribution to the effective thermal conductivity but gets weaker as temperature rises. The radiation overtakes contact conduction after 920 K. The solid/fluid conduction has little influence on the effective thermal conductivity.

A supplementary analytical model for the stagnant effective thermal conductivity of low porosity granular geomaterials

A cylindrical unit cell model for predicting stagnant effective thermal conductivity of two-phase or three-phase low porosity granular geomaterials was proposed in this paper. In the present model, the cylindrical unit cell is made of a quasi-hemispherical/cylindrical solid particle, and the solid grain is surrounded by water or air or the mixtures of two phases. In addition to the thermal conductivity of each phase, the effects of porosity and saturation degree on the stagnant effective thermal conductivity of geomaterials were investigated by introducing the coefficients α and β that control the porosity and saturation degree, respectively. The stagnant effective thermal conductivity of cylindrical unit cell was obtained via lumped parameter method and a further spatial correction was then conducted. Comparisons with some empirical models and experimental data for three types of selected geomaterials indicate that this theoretical model successfully captures the stagnant effective thermal conductivity with variation of saturation degree and thus provides the stagnant effective thermal conductivity of granular geomaterials with acceptable accuracy in the porosity range [0, 0.33] at most degrees of saturation.

Forced convection through open cell foams based on homogenization approach: Transient analysis

Thermal transportation in case of open cell foams is a complex process that comprises of all the modes of heat transfer. In macroscopic numerical models the key parameters that govern these modes of heat transfer are stagnant effective thermal conductivity, ke, dispersion conductivity, kd, effective radiative conductivity, kr and volumetric heat transfer coefficient, hv. The purpose of this study is to determine these key parameters for the forced convection of air through open cell alumina foams. It is achieved by performing two different types of experiments on alumina foams having different PPI number (10, 20 and 30) and porosity (0.79–0.87). The first sets of experiments are based on the transient heat transfer and the aim of performing these experiments is to determine the volumetric heat transfer coefficient, hv. The second sets of experiments are based on the steady state heat transfer and they are the outcome of our previous study. The aim of implementing steady state experiments is to determine dispersion conductivity, kd by supplying the values of hv as an input parameter from the transient analysis. Further, to supply effective thermal conductivity data to both transient and steady state analyses, the Kelvin cell model is implemented. The findings of this study exhibit that the convection heat transfer in case of transient analysis and the thermal dispersion in case of steady state analysis are the dominant modes of heat transfer. Further, this study reveals that by employing the proposed combined analysis, it is possible to predict the values of key parameters rather accurately. But still, the accuracy of the results predicted through the macroscopic models largely depends on the accuracy of the experiments.

Forced convection through open cell foams based on homogenization approach: Steady state analysis

To investigate thermal transportation through open cell foams there are various microscopic and macroscopic numerical models along with their limitations available in the literature. The purpose of this study is to investigate the limitations of macroscopic models and to propose some reliable ideas as conclusion that can be used to overcome the existing limitations. Therefore, a combined experimental and numerical study is presented. The experimental study comprises of steady state forced convection experiments which involve three different regimes of heat transfer. Further, in macroscopic models these three regimes of heat transfer namely conduction, thermal dispersion and interstitial convection are governed by stagnant effective thermal conductivity, ke, dispersion conductivity, kd and volumetric heat transfer coefficient, hv respectively. Moreover, the complex structure of the open cell foams is simplified into a rather realistic Kelvin cell model for the determination of ke. The influence of the geometrical parameters such as pore diameter, d and foam porosity, ε is investigated by examining 10, 20 and 30 PPI (pores per inch) alumina foams for a porosity range of 0.79–0.87. The findings of this study reveal that it is important to consider both local thermal non-equilibrium (LTNE) and thermal dispersion together for improved analysis. Further, it is revealed that although with the above consideration, it is possible to exhibit the effect of geometrical parameters on each regime of heat transfer but the accuracy of the results predicted through macroscopic models remain under question as there is no basis to validate the results.

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.

Stagnant Effective Thermal Conductivity of Agro-Industrial Residues for Solid-State Fermentation

The stagnant effective thermal conductivities (K0) of sugar cane bagasse (SCB), wheat bran (WB), orange pulp and peel (OPP) and their combination (weight proportion 1:2:2 SCB/OPP/WB) were obtained using the line heat source method. These solid materials were applied to pectinase production via solid-state fermentation. The moisture content ranged from 4 to 80% (w.b.). A conduction mechanism through the porous media was observed, along with conduction through a liquid film and contact thermal resistance between the samples and the probe. K0 was low for intermediate moisture contents and approached the molecular conductivity of water for high moisture contents.

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.

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.

Experimental and numerical study of convection heat transfer of CO 2 at supercritical pressures in vertical porous tubes

Convection heat transfer of CO 2 at supercritical pressures in vertical sintered porous tubes with particle diameters of 0.1–0.12 mm and 0.2–0.28 mm was investigated experimentally and numerically. The study investigated the influence of the inlet fluid temperature, mass flow rate, pressure, particle diameter, heat flux, flow direction and buoyancy on convection heat transfer in porous tubes. The results show that the inlet temperature, pressure, mass flow rate, particle diameter and heat flux all strongly influence the convection heat transfer at supercritical pressures. When the inlet temperature is much larger than the pseudocritical temperature, T pc, the local heat transfer coefficients in porous tubes are much less than when the inlet temperature is less than T pc. For T 0 < T pc and wall temperatures not much larger than T pc, the local heat transfer coefficients have a maximum for both upward flow and downward flow along the porous tube when the fluid bulk temperatures are near T pc. Buoyancy caused the different variations in the local heat transfer coefficients along the porous tube for upward and downward flows. The results also show that the heat transfer coefficients increase as the particle diameter decrease. The numerical simulations were performed using the local thermal equilibrium model with consideration of the effects of variable porosity, thermal dispersion and area-of-contact stagnant effective thermal conductivity. The experimental and numerical results for the friction factor of CO 2 at supercritical pressures flowing in sintered porous tubes at constant temperature (without heating) corresponded very well with the known correlation. However, the predicted results for the friction factor of CO 2 at supercritical pressures flowing in the heated sintered porous tube differ from those measured in the experiments both for upward and downward flows. The calculated wall temperatures corresponded well with the measured wall temperatures.

A Closure Model for Transient Heat Conduction in Porous Media

Equations governing the transient heat conduction in porous materials consisting of solids and fluids of different thermal properties were derived with a volumetric average scheme under the assumption of nonthermal equilibrium. The derivation leads to a macroscopic two-equation system which requires the closure modeling of new unknown terms due to interfacial transport, namely, the tortuosity term and the interfacial heat transfer term. Closure relations were obtained from the microscopic equations for temperature fluctuation under quasi-steady assumption. The closure coefficients appeared in the closure relations then depend on the media geometry as well as thermal properties. To demonstrate these dependencies, the closure coefficient for the thermal tortuosity is evaluated based on the effective stagnant thermal conductivity model proposed by Hsu et al. (1995) for periodically packed cubes, and the coefficient for interfacial heat transfer based on a quasi-steady heat conduction of dispersed spheres immersed in fluids. The salient features as well as the applicability and limitation of the newly proposed transient heat conduction model were discussed.

The Effective Stagnant Thermal Conductivity of Porous Media with Periodic Structures

The Effective Stagnant Thermal Conductivity of Porous Media with Periodic Structures

放射流を伴う多孔質物体の熱伝導率の研究

A method for determining lateral effective thermal conductivity in a radial flowsystem is presented. The analytical and experimental studies were carried by using a radial flowmodel. The experimental apparatus used is shown in Fig. 3 and Fig. 4. It was packed withglass-beases. The apparatus was designed as a wedge-shaped vessel (see Fig. 4). Lateral effectivethermal conductivity may be calculated by Eq. (3) which is called Yagi and Kunii's equation.The relations between temperature, distance, flowrate and stagnant effective thermal conductivityetc. in a radial flow condition are expressed in Eq. (5), Eq. (6) and Eq. (10), respectively.Then, stagnant effective thermal conductivity λ0 and lateral effective thermal conductivityλe may be calculated from the measured value obtained in the experiments. The distribution oftemperature in the radial flow model was estimated Eq. (8) by substituting the value of λe deter-mined in this study. As the results, close agreement between measured and calculated values onthe distribution of temperature was obtained.

Thermal conductivity of some heat pipe wicks

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.

Stagnant Effective Thermal Conductivities of Soldered Packed-hed

Stagnant effective thermal conductivities of soldered packed-bed (ko0) for sphere packing were measured bythe method which is ordinarily used for metal rod. Values of 15-28kcal/m·hr·°C were obtained for solderedpacked-bed made by soldering iron and brass sphere with pewter.The ko0 of soldered packed-bed has been measured to be independent of the orientation of heat flux, since thedistribution of bridge-parts which connect the spherical packings of soldered packed-bed to each other, wasuniform throughout the whole surface of the sphere.A theoretical estimation of the stagnant effective thermal conductivity was tried as follows.At first, the resistance of theimal conduction was estimated on two spherical packings connected to eachother with solder and then a correction was introduced. In soldered packed-bed heat will be conducted along along zig-zag roundabout way. The ratio of the length of zig-zag path to the straight distance was named effectivepath length coefficient (α). The resistance of thermal condution should be multiplied α-fold of that of thefirst model.Three regular types of sphere packing values of α were calculated, and it was found that a has a simplelinear functional relation with the void fraction of soldered packed-bed.α= 0.85+1.18εThus ko0 may be estimated by the following equation, ko0= (1-ε) 3/αΩkpThe agreement between calculated and obserbed ko0 was very good.