Glaser and Thodos [Am. Inst. Chem. Eng. J., 4, 63 (1958)] give a correlation involving individual particle shape and bed porosity. Kunii and Suzuki [Int. J. Heat Mass Transfer, 10, 845 (1967)] discuss heat and mass transfer in packed beds of fine particles.

Particle-to-fluid heat-transfer coefficients in gas fluidized beds are predicted by the relation (Zenz and Othmer, op. cit.)

where Gis the superficial mass velocity at incipient fluidization.

A more general equation is given by Frantz [Chem. Eng., 69(20), 89 (1962)]:

where h is based on true gas temperature.

Bed-to-wall coefficients in dilute-phase transport generally can be predicted by an equation of the form of Eq. (5-50). For example, Bonilla et al. (American Institute of Chemical Engineers Heat Transfer Symp., Atlantic City, N.J., December 1951) found for 1- to 2-| m chalk particles in water up to 8 percent by volume that the coefficient on Eq. (5-50) is 0.029 where k, p, and c were arithmetic weighted averages and the viscosity was taken equal to the coefficient of rigidity. Farber and Morley [Ind. Eng. Chem., 49, 1143 (1957)] found the coefficient on Eq. (5-50) to be 0.025 for the upward flow of air transporting silica-alumina catalyst particles at rates less than 2 kg solids kg air (2 lb solids/lb air). Physical properties used were those of the transporting gas. See Zenz and Othmer (op. cit.) for additional details covering wider porosity ranges.

The thermal performance of cylindrical rotating shell units is based upon a volumetric heat-transfer coefficient

where Vr = volume. This term indirectly includes an area factor so that

thermal performance is governed by a cross-sectional area rather than by a heated area. Use of the heated area is possible, however: QQ

For heat transfer directly to solids, predictive equations give directly the volume V or the heat-transfer area A, as determined by heat balance and airflow rate. For devices with gas flow normal to a fluidized-solids bed,

where Atp = A3t4 as explained above, cp = volumetric specific heat, and Fg = gas flow rate. For air, cp at normal temperature and pressure is about 1100 J/(m3-K) [0.0167 Btu/(ft3°F)]; so

(A3t4)Fg where h = 0.0009 (SI) or 60 (U.S. customary). Another such equation, for stationary vertical-shell and some horizontal rotary-shell and pneumatic-transport devices in which the gas flow is parallel with and directionally concurrent with the fluidized bed, is the same as Eq. (11-62) with A 314 replaced by A 3t2. If the operation involves drying or chemical reaction, the heat load Q is much greater than for sensible-heat transfer only. Also, the gas flow rate to provide moisture carry-off and stoichiometric requirements must be considered and simultaneously provided. A good treatise on the latter is given by Pinkey and Plint (Miner. Process., June 1968, p. 17).

Evaporative cooling is a special patented technique that often can be advantageously employed in cooling solids by contactive heat transfer. The drying operation is terminated hefore the desired final moisture content is reached, and solids temperature is at a moderate value. The cooling operation involves contacting the burden (preferably fluidized) with air at normal temperature and pressure. The air adiabatically absorbs and carries off a large part of the moisture and, in doing so, picks up heat from the warm (or hot) solids particles to supply the latent heat demand of evaporation. For entering solids at temperatures of 180° C (350° F) and less with normal heat-capacity values of 0.85 to 1.0 kJ/(kgK) [0.2 to 0.25 Btu/(lb° F)], the effect can be calculated by:

1. Using 285 m3 (1000 ft3) of airflow at normal temperature and pressure at 40 percent relative humidity to carry off 0.45 kg (1 lb) of water [latent heat 2326 kJ/kg (1000 Btu/lb)] and to lower temperature by 22 to 28°C (40 to 50°F).

2. Using the lowered solids temperature as t3 and calculating the remainder of the heat to be removed in the regular manner by Eq. (11-62). The required air quantity for (2) must be equal to or greater than that for (1).

When the solids heat capacity is higher (as is the case for most organic materials), the temperature reduction is inversely proportional to the heat capacity.

A nominal result of this technique is that the required airflow rate and equipment size is about two-thirds of that when evaporative cooling is not used. See Sec. 20 for equipment available.

Convective Heat Transfer Equipment using the true convec-tive mechanism when the heated particles are mixed with (and remain with) the cold particles is used so infrequently that performance and sizing equations are not available. Such a device is the pebble heater as described by Norton (Chem. Metall. Eng., July 1946). For operation data, see Sec. 9.

Convective heat transfer is often used as an adjunct to other modes, particularly to the conductive mode. It is often more convenient to consider the agitative effect a performance-improvement influence on the thermal diffusivity factor a, modifying it to ae, the effective value.

A pseudo-convective heat-transfer operation is one in which the heating gas (generally air) is passed over a bed of solids. Its use is almost exclusively limited to drying operations (see Sec. 12, tray and shelf dryers). The operation, sometimes termed direct, is more akin to the conductive mechanism. For this operation, Tsao and Wheelock [Chem. Eng., 74(13), 201 (1967)] predict the heat-transfer coefficient when radiative and conductive effects are absent by h = bG08 (11-63)

where b = 14.31 (SI) or 0.0128 (U.S. customary), h = convective heat transfer, and G = gas flow rate.

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