Packed beds effective thermal conductivity

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.)... 

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... 

Resistance models are useful for making quick estimations of packed bed effective thermal conductivities, yet they tend to be highly subjective and, in practice, lack accuracy. Other types of models have been developed including numeric methods [17]. The most useful and accurate model was developed... 

Investigator Type of correlation Phases involved Mohamad et al. [36] Packed-bed effective thermal conductivity due to conduction Fluid-solid... 

The heat-transfer properties of the suspended-bed operations appear superior to those of the packed-bed operations. In particular, the former have a high effective thermal conductivity, and the development of uneven temperature distributions is therefore less likely. 

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 ... 

Illustration 12.7 indicates how to estimate an effective thermal conductivity for use with two-dimensional, pseudo homogeneous packed bed models. 

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

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

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)]... 

Effective thermal conductivities and heat transfer coefficients are given by De Wasch and Froment (1971) for the solid and gas phases in a heterogeneous packed bed model. Representative values for Peclet numbers in a packed bed reactor are given by Carberry (1976) and Mears (1976). Values for Peclet numbers from 0.5 to 200 were used throughout the simulations. 

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.

Relations for the Nusselt number and the effective thermal conductivity kc for the annular packed bed are... 

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. 

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). 

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. ... 

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]...

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 ... 

Figure 17.36. Effective thermal conductivity and wall heat transfer coefficient of packed beds. Re = r/ G/jjL, dp = SVpjAp, s = poros-...

Due to the Langrangian formulation applied to the solid phase, the use of an effective thermal conductivity as usually applied to porous media is not necessary. In a packed bed heat is transported between solid particles by radiation and conduction. For materials with low thermal conductivity, such as wood, conduction contributes only to a minor extent to the overall heat transport. Furthermore, heat transfer due to convection between the primary air flow through the porous bed and the solid has to be taken into account. Heal transfer due to radiation and conduction between the particles is modelled by the exchange of heat between a particle and its neighbours. The definition of the neighbours depends on the assembly of the particles on the flow field mesh. 

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. 

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. 

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