Thermal conductivity in packed beds

Argo and Smith (106, 107) have presented a detailed discussion of heat transfer in packed beds and have proposed the following relation for the effective thermal conductivity in packed beds ... 

Yagi, S., and Kunii, K., Studies on Effective Thermal Conductivities in Packed Beds, AIChE J., 3 373 (1957)... 

S. Yagi, D. Kunii, and N. Wakao, Studies on Axial Effective Thermal Conductivities in Packed Beds, AlChE J, (6) 543-546,1960. 

A. Matsuura, Y. Hitcka, T. Akehata, and T. Shirai, Effective Radial Thermal Conductivity in Packed Beds with Gas-Liquid Down Flow, Heat Transfer—Japanese Research, (8) 44-52,1979. 

Yagi S, Kunii D (1957) Studies on effective thermal conductivities in packed beds. AIChE J 3(3) 373-380... 

Matsuura A, Hitaka Y, Akehata T, Shirai T. Effective radial thermal conductivity in packed beds with downward cocurrent gas-liquid flow. Heat Transf. Jpn Res. 1979 8 44. 

H. Hofmann Industrial process kinetics and parameter estimation, Adv.Chem.Ser. 109(1972)519-534 /2/ P. Trambouze, H. van Landeghem and J.P. Wauquier Les reac-teurs chimiques. Edition Technip, Paris 1984 /3/ H. Hofmann, G. Emig and W. Rdder The use of an integral reactor with sidestream-analysis for the investigation of complex reactions, EFCE Publ.Ser. 37(1984)4 19-426 /4/ S. Yagi and D. Kunii Studies on the effective thermal conductivities in packed beds, AIChE J. (1957)373-381 /5/ D. Kunii and J.M. Smith Heat transfer characteristics of porous rocks, AIChE J. (1960)71-78 /6/ N. Wakao and J.M. Smith Diffusion in catalyst pellets, Chem.Eng.Sci. 17(1962)825-834... 

Specchia, V, and S. Sicardi, "Modified Correlation for the Conductive Contribution of Thermal Conductivity in Packed Bed... 

There is even more uncertainty in estimating the heat-transfer coefficient at the wall of the tube than in estimating the effective thermal conductivity in the bed of catalyst. The measurement is essentially a difficult one, depending either on an extrapolation of a temperature profile to the wall or on determining the resistance at the wall as the difference between a measured over-all resistance and a calculated resistance within the packed bed. The proper exponent to use on the flow rate to get the variation of the coefficient has been reported as 0.33 (C4), 0.47 (C2), 0.5 and 0.77 (HI), 0.75 (A2), and 1.00 (Ql). 

Heat Transfer in a Packed Bed (Effective Thermal Conductivity) In a bed of solid particles through which a reacting fluid is passing, heat can be transferred in the radial direction by a number of mechanisms. However, it is customary to consider that the bed of particles and the gas may be replaced by a hypothetical solid in which conduction is the only mechanism for heat transfer. The thermal conductivity of this solid has been termed the effective thermal conductivity k. With this scheme the temperature T of any point in the bed may be related to and the position parameters r and z by the differential equation... 

Investigator Type of correlation Phases involved Model associated Model equation Kunii and Smith [29] Effective thermal conductivity of packed bed Fluid-solid One-dimensional heat transfer model Spheres in cubic array = <°- W>(K K) (in - ) .21 ... 

Smirnov El, Kuzmin VA, Zolotarskii lA. Radial thermal conductivity in cylindrical beds packed by shaped particles. Chemical Engineering Research and Design 2004 82 293-296. 

Heat transfer in the bed of a rotary kiln is similar to heat transfer in packed beds except that in addition to the heat flow in the particle assemblage of the static structure (Figure 8.3), there is an additional contribution of energy transfer as a result of advection of the bed material itself. The effective thermal conductivity of packed beds can be modeled in terms of thermal resistances or conductance within the particle ensemble. As shown in Figure 8.3 almost all the modes of heat transfer occurs within the ensemble, that is, particle-to-particle conduction and radiation heat transfer as well as convection through the interstitial gas depending upon the size distribution of the material and process temperature. Several models are available in the literature for estimating the effective thermal conductivity of packed beds. 

A. J. Slavin, F. A. Londry, and J. Harrison. "A new model for the effective thermal conductivity of packed beds of solid spheroids Alumina in helium between 100 and 500°C," Int J. Heat Mass Transfer, 43, 2059-2073, 2000. 

Zehner P, Schliinder EU Thermal conductivity of packed beds (in German), Chem Ing Tech 42 933-941, 1970. 

Heat transfer in packed beds. Effective thermal conductivity as a function of Reynolds number. Curve 1 Coberly and Marshall. Curve 2 Campbell and Huntington. Curve 3 Calderbank and Pogorski. Curve 4 Kwong and Smith. Curve 5 Kunii and Smith. (From G. F. Froment, Chemical Reaction Engineering, Adv. Chem. Ser., 109, 1970.)... 

The equations describing the concentration and temperature within the catalyst particles and the reactor are usually non-linear coupled ordinary differential equations and have to be solved numerically. However, it is unusual for experimental data to be of sufficient precision and extent to justify the application of such sophisticated reactor models. Uncertainties in the knowledge of effective thermal conductivities and heat transfer between gas and solid make the calculation of temperature distribution in the catalyst bed susceptible to inaccuracies, particularly in view of the pronounced effect of temperature on reaction rate. A useful approach to the preliminary design of a non-isothermal fixed bed catalytic reactor is to assume that all the resistance to heat transfer is in a thin layer of gas near the tube wall. This is a fair approximation because radial temperature profiles in packed beds are parabolic with most of the resistance to heat transfer near the tube wall. With this assumption, a one-dimensional model, which becomes quite accurate for small diameter tubes, is satisfactory for the preliminary design of reactors. Provided the ratio of the catlayst particle radius to tube length is small, dispersion of mass in the longitudinal direction may also be neglected. Finally, if heat transfer between solid cmd gas phases is accounted for implicitly by the catalyst effectiveness factor, the mass and heat conservation equations for the reactor reduce to [eqn. (62)]... 

Figure 1736. Effective thermal conductivity and wall heat transfer coefficient of packed beds. Re = dpG/fi, dp = 6Vp/Ap, s -porosity, (a) Effective thermal conductivity in terms of particle Reynolds number. Most of the investigations were with air of approx. kf = 0.026, so that in general k elk f = 38.5k [Froment, Adv. Chem. Ser. 109, (1970)]. (b) Heat transfer coefficient at the wall. Recommendations for L/dp above 50 by Doraiswamy and Sharma are line H for cylinders, line J for spheres, (c) Correlation of Gnielinski (cited by Schlilnder, 1978) of coefficient of heat transfer between particle and fluid. The wall coefficient may be taken as hw = 0.8hp.

Extensive experimental determinations of overall heat transfer coefficients over packed reactor tubes suitable for selective oxidation are presented. The scope of the experiments covers the effects of tube diameter, coolant temperature, air mass velocity, packing size, shape and thermal conductivity. Various predictive models of heat transfer in packed beds are tested with the data. The best results (to within 10%) are obtained from a recently developed two-phase continuum model, incorporating combined conduction, convection and radiation, the latter being found to be significant under commercial operating conditions. 

Specific heats of metals and hydrides are easily determined and typically fall in the range of 0.1-0.2 cal/g°C. Thermal conductivity is a little more difficult to determine. The conductivity of the metal or hydride phase is not sufficient the effective conductivity of the bed must be determined. This depends on alloy, particle size, packing, void space, etc. Relatively little data of an engineering nature is now available and must be generated for container optimization. Techniques to improve thermal conductivity of hydride beds are needed. As pointed out earlier, good heat exchange is the most important factor in rapid cycling. 

The effect of molecular conduction is negligible when the Reynolds number is greater than 40. These considerations lead to the following proposed working expression for the effective thermal conductivity in a packed bed. 

The choice of a model to describe heat transfer in packed beds is one which has often been dictated by the requirement that the resulting model equations should be relatively easy to solve for the bed temperature profile. This consideration has led to the widespread use of the pseudo-homogeneous two-dimensional model, in which the tubular bed is modelled as though it consisted of one phase only. This phase is assumed to move in plug-flow, with superimposed axial and radial effective thermal conductivities, which are usually taken to be independent of the axial and radial spatial coordinates. In non-adiabatic beds, heat transfer from the wall is governed by an apparent wall heat transfer coefficient. ... 

TABLE 17.15. Data for the Effective Thermal Conductivity, K, (kcal/mh°C), and the Tube Wall Film Coefficient, (kcal/m h C), in Packed Beds ... 

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