Solids in divided form, such as powders, pellets, and lumps, are heated and/or cooled in chemical processing for a variety of objectives such as solidification or fusing (Sec. ii), drying and water removal (Sec. 20), solvent recovery (Secs. i3 and 20), sublimation (Sec. i7), chemical reactions (Sec. 20), and oxidation. For process and mechanical-design considerations, see the referenced sections.

Thermal design concerns itself with sizing the equipment to effect the heat transfer necessary to carry on the process. The design equation is the familiar one basic to all modes of heat transfer, namely,

where A = effective heat-transfer surface, Q = quantity of heat required to be transferred, At = temperature difference of the process, and U = overall heat-transfer coefficient. It is helpful to define the modes of heat transfer and the corresponding overall coefficient as Uco = overall heat-transfer coefficient for (indirect through-a-wall) conduction, Um = overall heat-transfer coefficient for the little-used convection mechanism, Uct = heat-transfer coefficient for the contactive mechanism in which the gaseous-phase heat carrier passes directly through the solids bed, and Ura = heat-transfer coefficient for radiation.

There are two general methods for determining numerical values for Uco, Ucv, Uct, and Ura. One is by analysis of actual operating data. Values so obtained are used on geometrically similar systems of a size not too different from the equipment from which the data were obtained. The second method is predictive and is based on the material properties and certain operating parameters. Relative values of the coefficients for the various modes of heat transfer at temperatures up to 980° C (i800° F) are as follows (Holt, Paper ii, Fourth National Heat Transfer Conference, Buffalo, i960): Convective i

Radiant 2

Conductive 20

Because heat-transfer equipment for solids is generally an adaptation of a primarily material-handling device, the area of heat transfer is often small in relation to the overall size of the equipment. Also peculiar to solids heat transfer is that the At varies for the different heat-transfer mechanisms. With a knowledge of these mechanisms, the At term generally is readily estimated from temperature limitations imposed by the burden characteristics and/or the construction.

Conductive Heat Transfer Heat-transfer equipment in which heat is transferred by conduction is so constructed that the solids load (burden) is separated from the heating medium by a wall.

For a high proportion of applications, At is the log-mean temperature difference. Values of Uco are reported in Secs. ii, i5, i7, and i9. A predictive equation for Uco is

Mdsdfc) awffl where h = wall film coefficient, c = volumetric heat capacity, dm = depth of the burden, and a = thermal diffusivity. Relevant thermal properties of various materials are given in Table ii-9. For details of terminology, equation development, numerical values of terms in typical equipment and use, see Holt [Chem. Eng., 69, i07 (Jan. 8, i962)].

Equation (ii-48) is applicable to burdens in the solid, liquid, or gaseous phase, either static or in laminar motion; it is applicable to solidification equipment and to divided-solids equipment such as metal belts, moving trays, stationary vertical tubes, and stationary-shell fluidizers.

Fixed (or packed) bed operation occurs when the fluid velocity is low or the particle size is large so that fluidization does not occur. For such operation, Jakob (Heat Transfer, vol. 2, Wiley, New York, i957) gives hDt /k = b1bD017(DpG/|it)0'83(c|it/k) (ii-49a)

where bi = i.22 (SI) or i.0 (U.S. customary), h = Um = overall coefficient between the inner container surface and the fluid stream, b = .2366 + .0092 [^) - 4.0672 [^

Dp = particle diameter, Dt = vessel diameter, (note that Dp/Dt has units of foot per foot in the equation), G = superficial mass velocity, k = fluid thermal conductivity, | = fluid viscosity, and c = fluid specific heat. Other correlations are those of Leva [Ind. Eng. Chem., 42, 2498 (i950)]:

h = 0.813 — e- 6D'm lDpG Dt h = 0.125 — Dt for — < 0.35 (11-50«) Dt for 0.35 < — < 0.60 (11-50k) Dt and Calderbank and Pogerski [Trans. Inst. Chem. Eng. (London), 35, 195 (1957)]:

A technique for calculating radial temperature gradients in a packed bed is given by Smith (Chemical Engineering Kinetics, McGraw-Hill, New York, 1956).

TABLE 11-3 Typical Overall Heat-Transfer Coefficients in Tubular Heat Exchangers

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