Effective Thermal Conductivity of Packed Beds

We saw in Chapter 7 that the heat flow between a temperature difference AT across a planar gap of Ax and area A is Q = kAAT/Ax = UAAT = G AT where G is the conductance. If we consider the unit 

Chapter 8 Heat Transfer Processes in the Rotary Kiln Bed 

It is worth mentioning that in a vacuum, Gq and Gj are zero and since Gs Gc, Equation (8.19) becomes G = Gr + Gc (Slavin et al, 2000). The use of this equation is for all practical purposes sufficient. 

The radiant heat tranter coefficient can be expressed in terms of the average temperature, T (Wakao, 1973) 

For solid particles larger than 0.5 mm, that is, nonradiating gas systems, the effective thermal conductivity at atmospheric pressure is little affected by solid-solid conductivity. The Nusselt number for 

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Figure 9.2 shows existing data for the effective thermal conductivity of packed beds. These data include both ceramic and metallic packings. More accurate results can be obtained from the semitheoretical predictions of Dixon and Cresswell (1979). Once Kr is known, the wall heat transfer coefficient can be calculated from... 

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 ... 

Figure 5.16 Effective thermal conductivity of packed beds.

Elsari M, Hughes R. Axial effective thermal conductivities of packed beds. Applied Thermal Engineering 2002 22 1969-1980. 

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. 

N. Wakao. "Effect of radiating gas on effective thermal conductivity of packed beds," Chem. Eng. Sci., 28, 1117-1118, 1973. 

K = effective thermal conductivity of packed bed Tk = temperature at inside radius of the vessel... 

Cheng GJ Structural evaluation of the effective thermal conductivity of packed beds, Sydney, Australia, 2003, The University of New South Wales. 

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 ... 

Flow through the porous bed enhances the radial effective or apparent thermal conductivity of packed beds [10, 26]. Winterberg andTsotsas [26] developed models and heat transfer coefficients for packed spherical particle reactors that are invariant with the bed-to-particle diameter ratio. The radial effective thermal conductivity is defined as the summation of the thermal transport of the packed bed and the thermal dispersion caused by fluid flow, or ... 

A. B. Duncan, G. P. Peterson, and L. S. Fletcher, Effective Thermal Conductivity With Packed Beds of Spherical Particles, J. Heat Tr. (Ill) 830,1989. 

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... 

ILLUSTRATION 12.7 DETERMINATION OF THE EFFECTIVE THERMAL CONDUCTIVITY OF A PACKED BED OF CATALYST PELLETS... 

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. 

Here, ae is the effective thermal diffusivity of the bed and Th the bulk fluid temperature. We assume that the plug flow conditions (v = vav) and essentially radially flat superficial velocity profiles prevail through the cross-section of the packed flow passage, and the axial thermal conduction is negligible. The uniform heat fluxes at each of the two surfaces provide the necessary boundary conditions with positive heat fluxes when the heat flows into the fluid... 

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). 

Figure 10.5 Temperature and conversion profiles in an adiabaiic packed-bed membrane reactor with a dimensionless effective thermal conductivity of the membrane as a parameter [Itoh and Govind, 1989b]...

Heat transfer of packed bed has been the subject of numerous studies. For cylindrical packed columns, a solution for determining temperature distributions was given using Bessel functions. Here, it is important to find out exact effective thermal conductivity of bed because of flowing gas and relatively high temperatures. Radial temperature distributions are more important than that of axial direction because the latter can be measured and controlled during the operation. 

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... 

Models have been constructed describing each of these heat transfer mechanisms. Yagi and Kunii [10] developed generalized resistance models for packed beds which others adapted for application to metal hydride beds [9,11-13]. For lower and moderate temperature applications of these models, radiation heat transfer can be neglected [9, 11, 12, 14, 15]. In general, the resistance model of effective thermal conductivity of a packed metal hydride bed can be described as ... 

Calculated effective thermal conductivity of a packed particle bed as a function of void fraction, hydrogen pressure, and particle characteristics. 

The calculations presented here are consistent with many models and measurements described in the literature [11-14, 17, 20, 21], Models and measurements indicate that the effective thermal conductivity of particles loaded in a packed bed is generally limited to values below 5 W/m K, even with significant increases in the particle thermal conductivity (Fig. 4.4(b)). More clever methods must be employed to enhance thermal conductivity to levels above 5 W/m K. Additionally, the models discussed above have been developed for distinct particles typical of classic/interstitial hydride materials. These classic/interstitial beds are generally characterized as unsintered powders while complex hydrides, such as sodium alanates, can become porous sintered solids as seen in Fig. 4.5. Application of packed particle models have not been directly applied to sintered solid materials. 

The effective thermal conductivity of the packed bed expressed in Eq. (126) includes two terms, the conductivity term with no gas flow, k°, and the convective term. The factor 0.1 was recommended by Yagi and Kunii (1957) for spherical particles. The conductivity term with no gas flow can be calculated below, following that suggested by Swift (1966) for orthorhombic particle packing with a voidage of 0.395. 

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