Particle diameter ratio, heat transfer

A. Grab, U. Nowak, M. Schreier, R. Adler, Radial heat transfer in fixed-bed packing with small tube/particle diameter ratios. Heat Mass Transfer, 45, 417-425 (2009). 

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 ... 

Dixon A, Di Costanzo M, Soucy B (1984) Fluid-phase radial transport in packed beds of low tube-to-particle diameter ratio. Int J Heat Mass Transfer 27(10) 1701-1713... 

Mears [68] derived criteria for negligible axial mass and heat transfer effects in nonisothermal reactors with uniform wall temperature (so that the rate deviates <5% from the one observed from a plug-flow model), extending the results from Young and Finlayson [139]. In terms of the bed length to particle diameter ratio, it writes as... 

Similar results have also been obtained for different commercial and development cylindrical type catalysts with one or more holes in a 70 mm tube, where the tube-to-particle diameter ratio ranged from 3.4 to 5, but at identical operating conditions [120]. The average heat flux was as large as 160 kW/m. It should be added that the pressure drop varied with more than a factor 2 between the individual experiments. The small overall particle size dependence on the heat transfer coefficient in Equation (3.25) where the dp is raised to the power a-1 appears thus to be correct. 

However, in the region of Reynolds numbers of interest, heat transfer through the solid packing is significant, as already shown, and therefore we prefer a continuum description, in spite of reservations on its suitability to beds of low tube to particle diameter ratio. 

The recent experimental study by Melanson and Dixon [13] provides estimates of k g and the solid phase Biot number Big for spheres, cylinders and rings in beds of low tube to particle diameter ratio. It also throws up important questions about the mechanisms of solid conduction and wall heat transfer in such beds which seem to defy current theories. 

Several workers have inferred the existence of a radial velocity profile in a packed bed of low tube-to-particle diameter ratio from measurements of the fluid velocity at the bed exit [30]. However, their results are in considerable disagreement. A semi-theoretical study, using a modified Brinkmann model [31], indicates the existence of a steep maximum in the velocity next to the wall, but this remains unsubstantiated. Non-intrusive measurements of gas velocity within the packed bed are needed before a proper evaluation of the interactions of radial velocity, radial heat transfer, conversion and reaction selectivity are forthcoming. 

Dixon A G., Cresswell D L, Paterson W R., "Heat transfer in packed beds of low tube/particle diameter ratio", ACS Symp. Series 65, 238 (1978). 

Heat Transfer in Packed Beds of Low Tube/Particle Diameter Ratio... 

In spite of much research (1-7), identification of the relevant heat transfer parameters in packed beds and their subsequent estimation continue to provide challenging problems,especially so for beds having a small tube to particle diameter ratio, where so few experimental data are reported. 

An experimental evaluation of homogeneous continum models of steady state heat transfer in packed beds of low tube/particle diameter ratio has been carried out. It was found that both axial and radial conduction effects were important in such beds for N j 500, which covers the flow range in many industrial reactors. Heat transfer resistance at the wall was significant, but of secondary importance. 

Cp a = specific heat of air at constant pressure AT jj = temperature rise for stoichiometric combustion D = surface average particle diameter Pa = air density Pf = fuel density

equivalence ratio B = mass transfer number... 

Wall-to-bed heat-transfer coefficients were also measured by Viswanathan et al. (V6). The bed diameter was 2 in. and the media used were air, water, and quartz particles of 0.649- and 0.928-mm mean diameter. All experiments were carried out with constant bed height, whereas the amount of solid particles as well as the gas and liquid flow rates were varied. The results are presented in that paper as plots of heat-transfer coefficient versus the ratio between mass flow rate of gas and mass flow rate of liquid. The heat-transfer coefficient increased sharply to a maximum value, which was reached for relatively low gas-liquid ratios, and further increase of the ratio led to a reduction of the heat-transfer coefficient. It was also observed that the maximum value of the heat-transfer coefficient depends on the amount of solid particles in the column. Thus, for 0.928-mm particles, the maximum value of the heat-transfer coefficient obtained in experiments with 750-gm solids was approximately 40% higher than those obtained in experiments with 250- and 1250-gm solids. 

The Nusselt number with respect to the tube Nu(= hdt/k) is expressed as a function of four dimensionless groups the ratio of tube diameter to length, the ratio of tube to particle diameter, the ratio of the heat capacity per unit volume of the solid to that of the fluid, and the tube Reynolds number, Rec = (ucdtp/p,). However, equation 6.59 and other equations quoted in the literature should be used with extreme caution, as the value of the heat transfer coefficient will be highly dependent on the flow patterns of gas and solid and the precise geometry of the system. 

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)]... 

With the measurements subject to fluctuations of 20 or 30%, no accurate description of the profile is possible. All that can be said is that with moderate ratios of tube to particle diameter, the maximum velocity is about twice the minimum, and that when the particles are relatively small, the profile is relatively flat near the axis. It is fairly well established that the ratio of the velocity at a given radial position to the average velocity is independent of the average velocity over a wide range. Another observation that is not so easy to understand is that the velocity reaches a maximum one or two particle diameters from the wall. Since the wall does not contribute any more than the packing to the surface per unit volume in the region within one-half particle diameter from the wall, there is no obvious reason for the velocity to drop off farther than some small fraction of a particle diameter from the wall. In any case, all the variations that affect heat transfer close to the wall can be lumped together and accounted for by an effective heat-transfer coefficient. Material transport close to the wall is not very important, because the diffusion barrier at the wall makes the radial variation of concentration small. 

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