Internal Radiative Heat Transfer Research Articles

Fully Coupled Material Response and Internal Radiative Heat Transfer for Three-Dimensional Heat Shield Modeling

A material response code is strongly coupled with a radiative transfer equation (RTE) to evaluate the effect of a spectrally resolved heat flux on the thermal response of a heat shield. A P1 approximation model of RTE is used to account for radiation heat transfer within the material. First, the RTE model is verified by comparing the numerical results with the analytical solution. Next, the coupling scheme is verified by comparing the temperature histories computed by the pure conduction scheme with the ones computed by conduction coupled with radiative emission. The verification study is conducted using test cases from the literature (radiant heating, arc jet heating, and space shuttle entry) as well as on a 3D Block, a 2D IsoQ sample, and the Stardust Return Capsule. The verification results are satisfactory for all cases. Thus, the verification results indicate that the coupling approach can accurately simulate the thermal response of the material. The coupling scheme was then used to simulate a laser heating experiment that studied the impact of spectral radiative heat transfer on ablative material. The results from the laser ablation simulations exhibit a behavior analogous to the experimental observations, indicating the importance of spectral radiative flux on the material response.

Investigating Numerically the Effect of Wind on Fire Spread Between Two Informal Settlements Dwellings

AbstractPrevious full-scale fire studies revealed that the role of wind on fire spread between informal settlement dwellings was critical. However, the influence of wind conditions on informal settlement dwellings fire spread is currently understudied in the literature. This study aimed to investigate the effect of external wind conditions on fire spread between two informal settlement dwellings with a distance of 1 meter between them. A parametric numerical analysis was performed using the computational fluid dynamics code Fire Dynamics Simulator. The numerical models were benchmarked through laboratory experiments. The investigation included an analysis of the fire spread mechanism, flashover conditions, and heat transfer processes at the boundaries of the dwellings. Simulations were conducted with burning wood cribs as fuel and three wind speeds (6 m/s, 10 m/s, and 14 m/s) with four wind directions (East, West, South, and North). Results showed that wind speed and direction had a significant impact on the fire dynamics of the origin dwelling and its spread to neighboring dwellings. The wind direction also influenced the time to flashover in both dwellings, with a delay observed when the wind flowed through the alley between the two dwellings. The total heat transfer coefficient was found to be directly proportional to the wind speed for all directions. The internal radiative heat transfer coefficient of one wall was found to represent the total heat transfer coefficient in different scenarios. This study highlights the complexity of determining the role of wind in urban fire spread and underscores the need for further research in this area.

Conjugated heat transfer analysis of the Start-Up Monitoring Module (STUMM) for IFMIF-DONES including heat generation mapping

The Start-Up Monitoring Module (STUMM) is designed for the commissioning phase of the IFMIF-DONES (International Fusion Materials Irradiation Facility – Demo Oriented NEutron Source). The main aim is the detailed characterization of the irradiation field in space for the validation of the nuclear models. The thermo-hydraulic analysis of the STUMM was carried-out in the Ansys FLUENT software predicting the steady-state temperature distribution in the full range of the allowable inlet temperatures (15–60 °C), inlet pressures (3–3.5 bar), and the maximum allowable helium flow rate of 0.18 kg/s. The heat generation maps obtained from MCNP analyses were mapped on the CFD mesh via the in-build FLUENT functionality of “profiles”. To ensure the conservation of the global heat generation a correction coefficient αcorr was introduced to modify the MCNP profiles before the import to CFD: qCFD(x,y,z)=αcorr qMCNP(x,y,z). A dedicated test case of a cylinder subjected to a point irradiation source was studied, confirming the convergence of the global heat with a refinement of the MCNP and CFD discretization, where the MCNP model was discretized with cuboid cells and the CFD with a polyhedral. The global STUMM model was solved with 5 MCNP profiles with cell sizes from 5 mm to 1 mm (0.2–23.3 mln data points). Even though the smallest feature of the global CFD model was 0.5 mm (thickness of a detector wall) the MCNP profile with ∼4 mm cuboid cells gave almost identical results to the finest 1 mm profile, confirming the precision of the method and no need of solving very finely discretized MCNP models. With the direct transfer of results between MCNP and CFD model with the in-build FLUENT functionality of “profiles”, the simulation set-up was fast. A negligible impact of the internal radiation heat transfer on the temperatures was found (with the Discrete Ordinates (DO) model). Two possible flow directions of the helium were analysed for the worst-case scenarios. With the maximum pressure drop of 4.12 kPa, the maximum helium outlet temperature of 71.6 °C, and the max temperature of the detectors of 157.5 °C all the design requirements were satisfied, as values were far from the limits of 60–80 kPa, 150 °C and 250 °C respectively.

Development of temperature-responsive transmission switch film (TRTSF) using phase change material for self-adaptive radiative cooling

To overcome the problem of unable automatic turn on and off response to the ambient temperature of current static radiative cooling systems, we prepared a series of temperature-responsive transmission switch film (TRTSF) samples using n-octadecane as phase change material (PCM) and polydimethylsiloxane (PDMS) as carrier material. The phase change cycle permeability analysis showed that the TRTSF samples with PCM content above 30.0 w.t.% exhibit a serious leakage problem due to a large number of holes on surface. The TRTSF samples with 10.0–30.0 w.t.% PCM content were further analysed by Differential scanning calorimetry (DSC) and Thermogravimetry (TG), and compared with the theoretical PCM contents, which verified the reliability of the curing results of PCM in TRTSF from various aspects. The ultraviolet–visible light-near-infrared (UV–VIS-NIR) and mid-infrared (MIR) transmittance characterization of TRTSF revealed that the switching effect was obvious during solid–liquid phase change process. TRTSF microstructure analysis revealed that the particle size distribution of PCM microspheres was between 1 μm and 6 μm. The UV–VIS-NIR band (wavelength at 0.25–1.1 μm) transmittance of PCM decreased to less than 3.7% after solidification due to the interfacial reflectance of solid PCM particles increased to more than 66.8% even in monolayer, thus the solid PCM particles would produce Mie scattering and isolate the internal and external radiation heat transfer. However, the TRTSF can be used as the radiative cooling emission layer since its optical properties were similar to the PDMS when the PCM is in liquid phase. Overall, the TRTSF film prepared in this study successfully adapted the temperature switching properties, also, is expected to be further used for the development of self-adaptive radiative cooling systems.

Experimental and Theoretical Study on the Internal Convective and Radiative Heat Transfer Coefficients for a Vertical Wall in a Residential Building

Experimental studies on internal convective (CHTC) and radiative (RHTC) heat transfer coefficients are very rarely conducted in real conditions during the normal use of buildings. This study presents the results of measurements of CHTC and RHTC for a vertical wall, taken in a selected room of a single-family building during its everyday use. Measurements were performed using HFP01 heat flux plates, Pt1000 sensors for internal air and wall surface temperatures and a globe thermometer for mean radiant temperature measured in 10 min intervals. Measured average CHTC and RHTC amounted to 1.15 W/m2K and 5.45 W/m2K, compared to the 2.50 W/m2K and 5.42 W/m2K recommended by the EN ISO 6946, respectively. To compare with calculated CHTC, 14 correlations based on the temperature difference were applied. Obtained values were from 1.31 W/m2K (given by Min et al.) to 3.33 W/m2K (Wilkes and Peterson), and in all cases were greater than the 1.15 W/m2K from measurements. The average value from all models amounted to 2.02 W/m2K, and was greater than measurements by 75.6%. The quality of models was also estimated using average absolute error (AAE), average biased error (ABE), mean absolute error (MAE) and mean bias error (MBE). Based on these techniques, the model of Fohanno and Polidori was identified as the best with AAE = 68%, ABE = 52%, MAE = 0.41 W/m2K and MBE = 0.12 W/m2K.

Heat transfer analysis of rectangular porous fins in local thermal non-equilibrium model

• Analytical solutions on convective-radiative porous fin in LTNE is accomplished. • Darcy model and Boussinesq approximation are used to simulate the porous medium. • The two energy equations are solved using the Adomian Decomposition Method. • Results are given as temperature profiles and total Nusselt numbers. • Criteria to compare LTNE and LTE hypothesis are given for both scale analysis and ADM. In this study, an analytical solution of a porous fin with natural convection and radiation heat transfer is carried out. For the first time, the analysis is accomplished in Local Thermal Non-Equilibrium (LTNE) model. The investigation is carried out on a porous fin with finite length and adiabatic tip. The Darcy model and Boussinesq approximation for buoyancy effects are used to evaluate the infiltration velocity in the porous medium. Two energy equations are solved using the Adomian Decomposition Method (ADM). The solution is validated with the numerical solution of the finite difference method, and with the asymptotic solution for the Local Thermal Equilibrium (LTE) model. The results are presented in terms of temperature profiles and total average Nusselt numbers. They pointed out the effects of internal and external radiation and convection heat transfer, as well as thermal conductivity ratio and dimensionless thickness. It was found that solid phase temperature profiles decrease as the Biot, Bi , and Rayleigh, Ra *, numbers decrease, whereas the difference between the solid phase and fluid phase temperatures, for assigned Bi , decreases for lower Ra . Thermal conductivity ratio and dimensionless thickness increase engender higher solid phase temperature for assigned Bi and Ra *. The total Nusselt number increases as Ra *, Bi and external radiation increase, whereas it decreases with the thermal conductivity ratio. Criteria to compare LTNE and LTE assumptions are proposed, and they highlight the fact that the minimum Biot number to accept the LTE assumption becomes lower as the Rayleigh number decreases.

Regimes of filtrational gas combustion in a cylindrical annular burner with radiation heat transfer

A problem of premixed filtrational gas combustion in a cylindrical annular burner has been solved in the frame of one-dimensional diffusion-thermal model taking into account internal radiative heat transfer in porous burner and radiative heat flux emitted by internal layers of porous material. Two different stationary combustion regimes are detected under the same values of gas flow rate. It was shown that stationary flame radius depends on the initial ignition conditions. Solid and gas temperature characteristics depending on flame radius and gas flow rate are studied. Estimation of the burner radiation efficiency is carried out. Significant difference between temperature characteristics of gas and solid phases and burner radiation efficiency for different combustion regimes are shown.

Heat Transfer Model of a 50 kW Solar Receiver–Reactor for Thermochemical Redox Cycling Using Cerium Dioxide

This work reports on the development of a transient heat transfer model of a solar receiver–reactor designed for thermochemical redox cycling by temperature and pressure swing of pure cerium dioxide in the form of a reticulated porous ceramic (RPC). In the first, endothermal step, the cerium dioxide RPC is directly heated with concentrated solar radiation to 1500 °C while under vacuum pressure of less than 10 mbar, thereby releasing oxygen from its crystal lattice. In the subsequent, exothermic step, the reactor is repressurized with carbon dioxide as it cools, and at temperatures below 1000 °C, the partially reduced cerium dioxide is re-oxidized with a flow of carbon dioxide. To analyze the performance of the solar reactor and to gain insight into improved design and operational conditions, a transient heat transfer model of the solar reactor for a solar radiative input power of 50 kW during the reduction step was developed and implemented in ANSYS cfx. The numerical model couples the incoming concentrated solar radiation using Monte Carlo ray tracing, incorporates the reduction chemistry by assuming thermodynamic equilibrium, and accounts for internal radiation heat transfer inside the porous ceria by applying effective heat transfer properties. The model was experimentally validated using data acquired in a high-flux solar simulator (HFSS), where temperature evolution and oxygen production results from model and experiment agreed well. The numerical results indicate the prominent influence of solar radiative input power, where increasing it substantially reduces reduction time of the cerium dioxide structure. Consequently, the model predicts a solar-to-fuel energy conversion efficiency of >6% at a solar radiative power input of 50 kW; efficiency >10% can be obtained provided the RPC macroporosity is substantially increased, and better volumetric absorption and uniform heating is achieved. Managing the ceria surface temperature during reduction to avoid sublimation is a critical design consideration for direct absorption solar receiver–reactors.

Numerical Analysis of Thermal Stress in Semi‐Transparent Oxide Crystals Grown by Czochralski and EFG Methods

AbstractCrystal growth of oxides is generally difficult since large curvatures of the growth interface in these systems generate high thermal stress, dislocations and crystal cracking. Three‐dimensional numerical modeling is applied to investigate thermal stress distribution in sapphire and langatate La3Ta0.5Ga5.5O14 (LGT) semi‐transparent crystals grown by Czochralski (Cz) and Edge‐defined Film‐fed Growth (EFG) techniques. The analysis of thermal stress distribution in a sapphire ingot grown in a Czochralski furnace shows high von Mises stresses distributed almost symmetrically on large areas in the crystal. Thermal stress computations for piezoelectric langatate crystals grown in a Czochralski configuration show non‐symmetrical von Mises distribution with higher stress on one side of the ingot. These numerical results are in agreement with experimental results showing non‐symmetrical cracking at the outer surface of the crystal. 3D modeling of multi‐die EFG growth of white sapphire ribbons shows that the von Mises stress is almost constant when the number of ribbons is increased from two to ten. Two models are applied to simulate the internal radiative heat transfer in the sapphire crystals: P1 approximation and the Rosseland radiation model. Numerical results show that applying Rosseland formula introduces significant errors in temperature field calculations especially in the case of the EFG configuration.

Energy efficiency comparison of Stirling engine types (α, β, and γ) using detailed CFD modeling

In this work, a comparison of the power output and the energy efficiency of three types of Stirling engine (α, β and γ) is conducted numerically. The three Stirling engine types do not include a regenerator. The Stirling engine is set to work between temperature limits of 300 K and 800 K, where air is adopted as the working fluid. To make the model more realistic, the internal radiation heat transfer inside the Stirling engine is considered. Also, the thermal properties of air are taken to be depending on temperature. The model is validated against an experimental data set from the literature. The results show that γ-type Stirling engine is being able to produce the highest power output and the highest thermal efficiency (9.223 W and 9.8%). It is followed by β-type Stirling engine (8.634 W and 7.5%). In contrast, the lowest performance is achieved by α-type Stirling engine (0.908 W and 1.8%). Therefore, the α-type Stirling engine is modified to have an annular connecting pipe instead of circular connecting pipe. Consequently, its performance improved dramatically to produce 9.856 W with a thermal efficiency of 10.5%.

Two-dimensional heat and mass transfer and thermodynamic analyses of porous microreactors with Soret and thermal radiation effects—An analytical approach

Transport of heat and mass and the thermodynamics of porous microreactors with thermal diffusion and radiation effects are investigated analytically. The examined configuration includes an axisymmetric, thick-wall microchannel with an iso-flux thermal boundary condition imposed on the external surfaces. The microchannel is filled with porous materials and accommodates a zeroth order homogenous chemical reaction. Internal radiative heat transfer is modelled in addition to heat convection and conduction, while the local thermal non-equilibrium approach is taken within the porous section of the system. The transport of species is coupled with that of heat via the inclusion of thermodiffusion or Soret effect. Two-dimensional heat and mass transfer differential equations are solved analytically. The results are subsequently used to predict the thermodynamic irreversibilities inside the reactor and a thorough analysis of local and total entropy generation rates is performed. Also, the changes in Nusselt number, calculated on the internal walls of the microreactor, versus various parameters are reported. It is shown that the radiation effects can impact the temperature of the solid phase of the porous medium and lead to alteration of Nusselt number. It is further observed that the transfer of mass is the main source of irreversibility in the system. The findings are of particular use for the design and analysis of the microreactors with homogenous chemical reactions and can be also used for the validation of computational models.

Thermal modelling and experimental validation of a semi-transparent water wall system for Sydney climate

A transient heat balance model (THBM) based on energy conservation is developed for predicting the thermal performance of a semi-transparent water wall system. In order to validate the THBM against field measurements, real climate conditions in Sydney, Australia are adopted and the time variations of both internal and external convective and radiative heat transfer coefficients are calculated using empirical correlations from the literature. An alternative concrete wall model is also developed for further validation of the THBM. Good agreements between the THBM simulation and field data are achieved for both the water wall and concrete wall systems. The validated THBM is then adopted in a sensitivity analysis in order to assess the thermal performance of the semi-transparent water wall system under different configurations. It is found that the daily maximum water and air temperatures decrease significantly with the decrease of the transmissivity of the semi-transparent panels, whereas an increase of the attenuation coefficient of solar radiation in water has an insignificant impact on the water and air temperatures. The present results also indicated that increasing the thickness of the water column can reduce the fluctuations of the water temperature and the diurnal peak air temperature. From the practical application point of view, reducing the transmissivity of the external panel provides the most effective way to mitigate over-heating in the semi-transparent water wall system.

Thermophysical characteristics of a wickless heat pipe in microgravity – Constrained vapor bubble experiment

Abstract Wickless heat pipes are being studied for use in cooling critical components of spacecraft. The wickless design is thought to produce a simpler and lighter heat transfer system than heat pipes containing wicks or mechanically driven systems. The constrained vapor bubble experiment (CVB) is one such system tested on the International Space Station where the Bond Number (ratio of gravitational force to surface force) is small maximizing the affects of capillarity. The CVB is essentially a square, fused silica spectrophotometer cuvette evacuated and then partially filled with pentane as the working fluid. Along with temperature and pressure measurements, the two-dimensional thickness profile of the menisci formed at the corners of the quartz cuvette was determined using an interferometry based system contained with the station’s Light Microscopy Module (LMM). The CVB can be viewed as a hollow fin and its behavior analyzed using a simple, one-dimensional heat transfer model. That model, coupled with the visual observation of the vapor–liquid distribution inside the fin, provides an enhanced understanding of what the measured temperature and pressure profiles represent and the heat transfer mechanisms controlling the operation of the device. The internal heat transfer processes were found to be very complicated, multi-dimensional, and greatly dependent on internal and external radiative heat transfer. Internal radiative exchange was found to be more significant than originally anticipated as was the effect Marangoni forces on internal convective heat transfer. An analysis of the temperature profiles in conjunction with vapor–liquid interface mapping showed that the system could be separated into a number of discrete operation zones depending on the dominant mode of heat transfer.

Numerical Characterization of the Thermal Performance of Static Porous Insulation Layers on Steel Substrates in Furnace Tests

The thermal performance of insulated steel plates in standard furnace tests is analyzed numerically. The effect of internal radiation heat transfer across the pores of the insulation layer is also estimated and included in the analysis. It transpires that for a fixed heating regime and substrate, the thermal performance of an insulation layer depends on three dimensionless heat transfer parameters corresponding to the furnace Biot number, the ratio of the thermal diffusion timescale in the insulation to the test duration, and the ratio of radiation heat flux to conduction heat flux across a pore. Furthermore, a method for characterizing the heat transfer environment within the furnace is also discussed.

Numerical modeling of radiative heat transfer in Bridgman solidification of semi-transparent BaF 2 crystals

The radiative heat transfer during Bridgman solidification of semi-transparent barium fluoride (BaF 2) crystals is numerically investigated by using the commercial software FIDAP. This code uses the P-1 approximation for the participating media modeling. The thermal field and the solid–liquid interface shape are computed for an opaque melt-crystal sample, a semi-transparent grey sample and a semi-transparent non-grey sample. The transient numerical analysis of the latent heat influence on the interface deflection shows a significant effect on the interface shape. In the case of an opaque sample, this effect is huge because of the small thermal conductivity of the BaF 2 melt. The interface curvature is drastically reduced when the latent heat is taken into account, and the growth front, which has a convex shape, becomes flat when the growth rate increases. The latent heat effect is reduced in the case of the participating BaF 2 sample because the effective thermal conductivity of the melt is augmented by the internal radiative heat transfer. The internal radiative effect is small at low solidified fractions but becomes significant when the crystal length increases, leading to a more curved interface. When the growth rates are greater than a critical value, the interface becomes concave and a destabilization of the growth process can occur. These results are in agreement with previous experimental measurements of the interface curvature and analytical investigations of the factors affecting the interface deflection.

A global analysis of heat transfer in the CZ crystal growth of oxide: Recent developments in the model

In this paper, first, we reviewed our previous studies of the global analysis of heat transfer in an inductively heated Czochralski (CZ) furnace to grow oxide single crystals, in which it was assumed that the system is in a steady state and is axially symmetric. Particularly, it was emphasized that internal radiative heat transfer within oxide single crystals is important because oxide crystals are often semitransparent to infrared radiation. The results of the global analysis revealed that the melt/crystal interface shape is strongly affected by the absorption coefficient of such crystals as well as melt convection. Next, a new simple global model, in which unsteady, three-dimensional melt convection is taken into account in addition to radiative heat transfer in such crystals, is introduced. Here, it was shown numerically that when the crystal rotational Reynolds number is relatively large, melt convection becomes unsteady and three dimensional.

A Discontinuous Finite-Element Formulation for Radiative Transfer in Axisymmetric Finite Cylindrical Enclosures and Coupling with Other Mode Heat Transfer

ABSTRACT This article presents a discontinuous finite-element formulation for numerical solution of internal thermal radiative transfer problems in axisymmetric cylindrical enclosures. While the computation of axisymmetric radiative transfer is essentially three-dimensional, an appropriate mapping procedure entailing the use of the axisymmetric and periodic conditions associated with the enclosures may be used to make it possible to compute the problem over a 2-D mesh only. This will result in a significant reduction in both computing time and memory storage requirement. Mathematical formulation and numerical implementation using the discontinuous Galerkin method for axisymmetric internal radiation heat transfer calculations are given. The procedures for incorporating the mapping in the discontinuous formulation to convert an essentially 3-D calculation into a 2-D calculation are discussed in detail. The computed results are given and compare well with the solutions reported in the literature using other methods. Examples include both nonscattering and scattering cases. The effects of both solid-angular and spatial discretization are discussed, and both the theory and results show that an even discretization of solid angle is important to ensure adequate numerical accuracy. The coupling of the discontinuous and conventional finite-element methods for mixed-mode heat transfer calculations, including swirling flow, conduction, natural convection, and thermal radiation in participating media, is illustrated in the last example.

A DISCONTINUOUS FINITE-ELEMENT FORMULATION FOR MULTIDIMENSIONAL RADIATIVE TRANSFER IN ABSORBING, EMITTING, AND SCATTERING MEDIA

This article presents a discontinuous finite-element formulation for the numerical solution of internal thermal radiation problems in multidimensional domains. In contrast with the conventional finite-element formulation, the discontinuous finite element is based on the idea of allowing the discontinuity of field variables across the internal interelement boundaries and thus is particularly useful for the integral-differential equations governing the thermal radiative transfer phenomena in emitting, absorbing, and scattering media. Mathematical formulation and numerical details using the discontinuous finite element method for internal radiation heat transfer calculations are given. Computational procedures are presented. A parallel computing algorithm exploiting the advantages of the localized discontinuous finite-element formulation and space/solid-angle discretization characteristics is presented. The computed results are given and compared with analytical solutions whenever available as reported in references. Examples include both nonscattering and scattering cases. Parallel computing performance is also evaluated for both 2-D and 3-D cases and results show that the parallel algorithm can be easily implemented within the framework of the discontinuous finite-element method. For the cases tested, the parallel scheme based on the solid-angle discretization gives a reduction in CPU time approaching to almost an idealized parallel CPU scenario.

Effect of internal radiative heat transfer on spoke pattern on oxide melt surface in Czochralski crystal growth

The effect of internal radiative heat transfer on the spoke pattern observed on an oxide melt surface was investigated by means of a three-dimensional numerical simulation of LiNbO 3 melt flow in an open Czochralski crucible. In the mathematical model, the assumptions that the melt was an incompressible Newtonian Boussinesq fluid, the flow was laminar and the melt free surface was flat were adopted. The crucible wall was assumed to be heated by a constant heat flux. The Marangoni effect on the melt free surface was taken into account. When the internal radiative heat transfer was ignored, i.e., the melt was assumed to be opaque, a spoke pattern was generated and the bulk melt flow was oscillatory; otherwise, the internal radiative heat transfer decreased the temperature gradient and hence suppressed the Marangoni instability, and consequently, the spoke pattern disappeared and the melt flow became steady and axisymmetric. The investigation provided one theoretical explanation for the experimental phenomenon observed by Okano et al. (J. Mater. Process. Manuf. Sci. 4 (1995) 41).

Effective thermal conductivity and internal thermal radiation in burning black liquor particles

The effective thermal conductivity of burning black liquor particles is studied. It is one of the most important parameters affecting the rate of particle heating and, consequently, processes controlled by heat transfer rate, such as the devolatilization studied here. A numerical combustion simulation sensitivity analysis, combined with experimental verification, showed that in order to have reasonable agreement between experimental and modeled carbon release rates and swelling during pyrolysis, a significant addition of internal thermal radiation heat transfer must be included in effective thermal conductivity. In terms of the Rosseland mean absorption coefficient a R in diffusion approximation for radiative thermal conductivity, the value ∼850 m m 1 gave the best correlation with experiments. This corresponds to a mean penetration length of 1.2 mm, which is very close to the largest experimentally observed pore size. Other available effective thermal conductivity models were also compared. For the 1-mm pores observed experimentally, these models predict values for thermal conductivity that are too small.