Effective Radial Heat Conductivity Research Articles

Theoretical and Numerical Study of Electron Transport Through Magnetic Island

We investigate electron heat diffusion across a low mode magnetic island numerically in cylindrical geometry and compare with the theory in tokomak plasma physics. In order to keep the simulation results reliable, a modified simulation code, TM8P, was used in our simulation work. The correlation of the effective radial heat conductivity, χr, and several high ratio of parallel magnetic field heat diffusion coefficient to the perpendicular coefficient, \( {{\chi_{\parallel } } \mathord{\left/ {\vphantom {{\chi_{\parallel } } {\chi_{ \bot } }}} \right. \kern-0pt} {\chi_{ \bot } }} \), is studied in this article. It’s found that the theoretical prediction by Yu conform with the numerical results of TM8P when \( {{\chi_{\parallel } } \mathord{\left/ {\vphantom {{\chi_{\parallel } } {\chi_{ \bot } }}} \right. \kern-0pt} {\chi_{ \bot } }} \) is not bigger than 1012, i.e. the value of width of magnetic island to the width of diffusive layer, w/wc, is a small number.

Comparison of a Pseudocontinuous, Heterogeneous 2D Conductive Monolith Reactor Model to a 3D Computational Fluid Dynamics Model

Conductive catalytic honeycombs of a low-void fraction have gained renewed interest for use in the production of bulk chemicals because of favorable heat transfer properties and a low pressure drop compared to fixed-bed reactors. In this work, a pseudocontinuous, heterogeneous 2D conductive honeycomb reactor model is compared to a detailed 3D computational fluid dynamics model for the case of an irreversible, exothermic first order reaction with emphasis on the description of heat transfer. Excellent agreement in terms of maximum temperature and conversion is found for moderate conditions preferable for technical purposes when using the symmetric model for calculation of the effective radial heat conductivity. Deviations of maximum temperatures at harsher conditions are attributed to the use of global heat and mass transfer coefficients in the 1D channel model and the inherent assumption of a radially fully segregated flow in the continuum approach.

Radial Dispersion in Liquid Upflow through Solid SiC Foams

The radial dispersion process of an inert tracer in an upward liquid flow (superficial velocity from 3 × 10−3 to 13 × 10−3 m·s−1) through high-porosity (0.85 and 0.90 void fractions) solid foam samples of various lengths (6−15 cm) has been investigated. As in a standard packed bed the radial Peclet number based on the “cell diameter” is found to be of the order of 10 when the process is governed by mechanical dispersion. The results are in agreement with the flow contribution to the effective radial heat conductivity measured independently on the same type of foams.

Heat Diffusion in a Non-Local Tokomak Stochastic Magnetic Field

Heat transport across a non-local stochastic magnetic field was studied for the first time. Eleven incompact low m perturbed magnetic islands were used in our calculation. Parallel heat diffusion coefficient to the perpendicular coefficient was found still to be a key factor in influencing the effective radial heat conductivity and the results in this paper were compared with earlier studies in a local stochastic magnetic field.

The effective thermal properties of solid foam beds: Experimental and estimated temperature profiles

The effective radial heat conductivity of a solid foam packing and the wall heat transfer coefficient are determined under fluid flow conditions typical of catalytic reactors. A detailed 2-D heterogeneous model is phase-averaged in order to rigorously define lumped heat transfer parameters. The resulting pseudo-homogeneous model involves two fitting parameters only and it is successfully compared with experiments. First, experiments with packed extrudates validate the approach in comparison with known results. A second experiment with solid foams (polyurethane and SiC) allows correlating the radial heat conductivity to the nature of the solid, its morphology and fluid flow characteristics. The method is inspired from the correlations for particles and seems very promising. Conversely, determining the wall heat transfer coefficient yields only an average value (110 W m −2 K −1 ± 15%) and correlation with fluid velocity is impossible in the studied range 0.018–0.32 m s −1.

A Comparative Study of Electron Heat Transport in a Stochastic and a Non‐Stochastic Magnetic Field

AbstractWe studied the electron heat transport across a non‐local stochastic magnetic field and a non‐local non‐stochastic magnetic field in tokamak plasmas. Analytical results and numerical simulation results were compared with conditions of a stochastic and a non‐stochastic magnetic field respectively in this article. Parameter of stochasticity in perturbed magnetic field was found not tobe a key factor in influencing effective radial heat conductivity in radial direction of tokomak when Chirikov parameter is close to one (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Study of Heat Transport Across a Quasi-Stochastic Magnetic Field in Magnetic Confined Plasma Physics

Heat diffusion across a non-local quasi-stochastic magnetic field in tokamak plasma is numerically studied. The perturbed magnetic field is found to be a key factor in influencing effective radial heat conductivity whether the magnetic field is stochastic or not. Being different from previous work, a non-local perturbed magnetic field is used. Analytical results and numerical simulation results are compared between the conditions with a full and a quasi-perturbed stochastic field. The analytical results are found to be still consistent with numerical simulation results when the perturbed field is quasi-stochastic.

Heat Diffusion across a Strong Stochastic Magnetic Field in Tokamak Plasmas

We investigate heat diffusion across a local strong stochastic magnetic field by using eleven low-m perturbed magnetic islands. A maximum stochasticity of 38.82 between two neighboring rational surfaces is attained. The correlation between the effective radial heat conductivity χr and the ratio of the parallel heat diffusion coefficient to the perpendicular coefficient, χ|/χ⊥, numerically studied and compared with earlier work.

The Influence of Periodic Operation on the Effective Radial Thermal Conductivity in Trickle Bed Reactors

AbstractThe effect of a periodic liquid feed on the effective radial heat conductivity, Λr, in a trickle bed was investigated. The mean Λr under periodic operation was found to be well estimated by time‐weighed averaging of steady state values. The actual effect of periodic operation can therefore be predicted by the shape of the Λr, uL — curve in steady state operation. The effect of periodic operation is attenuated with decreasing gas flow rates and with decreasing liquid flow rate amplitudes in a bed of spherical particles. Practically no effect was observed with cylindrical particles.

Significance of the radial porosity profile for the description of heat transport in wall-cooled packed beds

The influence of a radial porosity and velocity profile on the predicted temperature and concentration profiles in wall-cooled packed beds is studied, with and without an exothermic first-order chemical reaction, on the basis of literature correlations for the effective transport coefficients. Furthermore, values for the effective heat transport coefficients are obtained from “cold-flow” experiments by means of model fitting, with and without taking the radial velocity profile into account. The radial porosity and velocity profiles are approximately by step-function, which is referred to as the “two-region model”. It is shown that the effective radial heat conductivity can be taken constant over the radius, despite the wall effect. Nevertheless, the influence of a radial superficial velocity profile can be significant through the convective term in the heat balance, especially for low tube-to-particle diameter ratios. The predicted NTU can increase the order of 20% for high values of the Reynolds number and up to 100% for low values. This is confirmed by the results obtained from the model fitting. In case of a first-order exothermic reaction, significantly higher values for the hot-spot temperatures are predicted, if a radial porosity and velocity profile is incorporated in the heat and mass balances. This is found to be mainly caused by the non-uniform distribution of active catalyst over the radius, due to the porosity profile.

Gas pressure drop and heat dispersion in a layer of fibrous material

The pressure drop and effective radial heat conductivity have been measured in fibrous layers of insulation materials exhibiting unisotropic stratified structure. Both quantities have been found to vary with the two principal orientations of strata in the fibrous material with respect to the direction of the flow. Pressure losses have been successfully interpreted in terms of the Ergun equation, valid for beds of particulate solids, applying the concept of the equivalent diameter to the fibrous material. The equivalent diameters were correlated with the void fraction of the material, allowing direct use of the Ergun equation for calculation of the energy losses. The heat dispersion has been found detectably unisotropic but, for technical purposes, the problem may be solved on the basis of an isotropic model with an error of about 20%. A correlation of the effective thermal conductivity of the material for this case also has been presented. The effective heat conductivities were correlated as Peclet numbers, using the effective ‘particle’ diameter as the scale length both in the Peclet and the Reynolds number. The effective ‘particle’ diameters for the fibrous materials were obtained by fitting the Ergun equation to pressure drop vs. gas velocity data. A joint plot of the Peclet vs. Reynolds number, interpreting the effective thermal conductivities of layers of fibrous material as well as particulate beds, suggests that the adopted approach is also physically meaningful. The results presented represent a contribution to the process of thermally self-sustained curing of adhesive in the manufacture of fibrous insulating materials.

A heterogeneous model for heat transfer in packed beds

If transient heat transfer occurs in a packed bed or a reaction is carried out on the pellets, the heterogeneity of the bed is essential because of the heat flow between pellets and gas. Global heat parameters for the packed bed, such as λeff and αw, are usually derived from homogeneous models. Therefore, one must choose whether to attribute the values of λeff and αw to the gas phase, solid phase or to both the phases. To describe changing radial and axial temperature profiles in a packed bed we assumed the heat transport in the solid and gas phase to occur in series by three mechanisms: heat transfer from the solid to the gas, described by the pellet heat transfer coefficient, αp heat transfer through the gas to the outer wall, described by the effective radial heat conductivity, λeff; and heat transfer at the wall through the gas, given by the coefficient αw. The values of λeff, αw and αp were determined by fitting the measured temperature profiles with a dynamic heterogeneous model. Two different-diameter beds, packed with industrial, ring-shaped pellets, were employed. The superficial gas velocity, based on the empty tube, was varied up to 3 m/s. Air was used as the gas. The values obtained for λeff, αw and αp are in good agreement with the values reported in the literature. Also, the agreement between the measured temperature profiles and the profiles obtained by fitting is very good. This confirms that the series model is a reasonable assumption and is close to physical reality. Therefore, it is concluded that, if packed-bed heterogeneity is significant, the above series model can be used to describe the temperature profiles.

The statistical character of packed-bed heat transport properties

Packed beds are essentially heterogeneous on a pellet scale. For random packed beds this heterogeneity causes a statistical character both on a pellet and bed scale. We discuss experimental results which deal with bed-scale statistics. For packed beds on a laboratory scale, the results indicate that the bed-scale statistical behaviour introduces a spread up to a factor of three in correlations for the effective radial heat conductivity, λ eff, and the heat transfer coefficient at the wall, α w. This spread of a factor of three roughly equals the spread between correlations available in the literature. For cooled tubular reactors the statistical spread in the behaviour of the individual tubes will have a large influence on reactor operation.

Radial heat transport in packed beds at elevated pressures

Values were measured for the effective radial heat conductivity λ eff, r and the heat transfer coefficient at the wall α w in a packed bed. This was done for superficial velocities of 5 – 70 cm s −1 and at pressures from 1 – 10 bar. Values for λ eff, r and α w were obtained by simultaneous fitting of measured axial and radial temperature profiles. The bed diameter was 5 cm; it was filled with 6.1 mm Raschig rings. Nitrogen gas was used in all cases. Values could be determined with reasonable accuracy. The agreement with correlations presented in the literature is good for λ eff, r and less so for α w. The results obtained indicate that λ eff, r and α w are a function of the product ‘velocity times pressure’ only. The correlations found can be represented by λ eff, r/λ g = 21 + 0.23Pe with an average relative error of 4% and Bi = 2.9Pe −0.40 with an average relative error of 5%. The experiments covered the range 25 < PE < 350. These correlations were obtained for one specific gas and one specific set-up.

Determinations of effective heat transport coefficients for wall-cooled packed beds

The influence is studied of several assumptions, often made in literature, on the values for the effective radial heat conductivity, wall heat transfer coefficient and overall heat transfer coefficient, as obtained from experiments in wall-cooled packed beds without a chemical reaction. Especially the choice of the inlet boundary condition can have a large impact on the values obtained. The influence of the presence of a radial velocity profile and also the cross-correlation of the parameters obtained is studied.

Influence of tube and particle diameter on heat transport in packed beds

AbstractInfluence of the tube and particle diameter and shape, as well as their ratio, on the radial heat transport in packed beds has been studied. Heat transport experiments were performed with four different packings in three wall‐cooled tubes, which differed in inner diameter only. Experimental values for the effective radial heat conductivity and wall heat‐transfer coefficient for the pseudo‐homogeneous two‐dimensional model and the overall heat‐transfer coefficient for the one‐dimensional model are presented. Values were obtained for glass spheres, alumina cylinders, and alumina Raschig rings. The effective radial heat conductivity and wall heat‐transfer coefficient can both be correlated as a linear function of the gas flow rate. The Bodenstein number for heat at fully developed turbulent flow is influenced strongly by the shape of the packing: 10.9 for glass spheres, 7.6 for alumina cylinders, and 4.2 for alumina Raschig rings. For the same packing, no significant influence is found of the tube diameter on the effective radial heat conductivity or on the wall heat‐transfer coefficient. The overall heat‐transfer coefficient can be described very well by the so‐called “lump equation,” which gives the relations among the overall heat‐transfer coefficient, effective radial heat conductivity, and wall heat‐transfer coefficient. The “lump factor,” as used in the lump equation, has a best‐fit experimental value of 7.4.