Fluidization of Particles by Fluids, by Martin Rhodes

-- 7: Heat Transfer in Fluidized Beds --

7.1: Gas-Particle Heat Transfer

Gas to particle heat transfer is relevant where a hot fluidized bed is fluidized by cold gas. The fact that particle-to-gas heat transfer presents little resistance in bubbling fluidized beds can be demonstrated by the following example:

Consider a fluidized bed of solids held at a constant temperature T_s. Hot fluidizing gas at temperature T_g0 enters the bed. At what distance above the distributor is the difference between the inlet gas temperature and the bed solids temperature reduced to half its original value?

Consider an element of the bed of height L at a distance L above the distributor (Figure 11).

Figure 11: Analysis of gas-particle heat transfer in an element of a fluidized bed

Let the temperature of the gas entering this element be T_g and the change in gas temperature across the element be T_g. The particle temperature in the element is T_s.

The energy balance across the element gives:

where
            a = surface area of solids per unit volume of bed
            C_g = specific heat capacity of the gas
            _g = gas density
            h_gp = particle-to-gas heat transfer coefficient
            U = superficial gas velocity

Integrating with the boundary condition T_g = T_g0 at L = 0,

The distance over which the temperature distance is reduced to half its initial value, L_0.5 is then:

For a bed of spherical particles of diameter x, the surface area per unit volume of bed, a = 6 (1 - ) / x where is the bed voidage. Using the correlation for h_gp in Eq. 47, then

For a bed of spherical particles of diameter x, the surface area per unit volume of bed, a = 6 (1 - ) / x where is the bed voidage. Using the correlation for h_gp in Eq. 47 , then

As an example we will take a bed of particles of mean size 100 m, particle density 2500 kg/m³ fluidized by air of density 1.2 kg/m³, viscosity 1.84 x 10^-5 Pas, conductivity 0.0262 W/mK and specific heat capacity 1005 J/ (kg K).

Using the Baeyens equation for U_mf (Equation 11), U_mf = 9.3 x 10^-3 m/s. The relative velocity between particles and gas under fluidized conditions can be approximated as U_mf / under these conditions.

Hence, assuming a fluidized bed voidage of 0.47, U_rel = 0.02 m/s.

Substituting these values in Equation 51, we find L_0.5 = 0.95 mm. So, within 1 mm of entering the bed the difference in temperature between the gas and the bed will be reduced by half. Typically for particles less than 1 mm in diameter the temperature difference between hot bed and cold fluidizing gas would be reduced by half within the first 5 mm of the bed depth.

7.2: Bed-to-surface heat transfer

In a bubbling fluidized bed the coefficient of heat transfer between bed and immersed surfaces (vertical bed walls or tubes) can be considered to be made up of three components which are approximately additive (Botterill, 1975):
h = h_pc + h_gc + h_r

h_pc is the particle convective heat transfer coefficient and describes the heat transfer due to the motion of packets of solids carrying heat to and from the surface. h_gc is the gas convective heat transfer coefficient describing the transfer of heat by motion of the gas between the particles. h_r is the radiant heat transfer coefficient. Figure 12, after Botterill (1986) gives an indication of the range of bed-surface heat transfer coefficients and the effect of particle size on the dominant heat transfer mechanism.

Figure 12: Range of fluidized bed-to-surface heat transfer coefficients

Particle convective heat transfer: On a volumetric basis the solids in the fluidized bed have about one thousand times the heat capacity of the gas and so, since the solids are continuously circulating within the bed, they transport the heat around the bed. For heat transfer between the bed and a surface the limiting factor is the gas conductivity, since all the heat must be transferred through a gas film between the particles and the surface (Figure 13).

Figure 13: Heat transfer from bed particles to an immersed surface

The particle-to-surface contact area is too small to allow significant heat transfer. Factors affecting the gas film thickness or the gas conductivity will therefore influence the heat transfer under particle convective conditions. Decreasing particle size, for example, decreases the mean gas film thickness and so improves h_pc. However, reducing particle size into the Group C range will reduce particle mobility and so reduce particle convective heat transfer. Increasing gas temperature increases gas conductivity and so improves h_pc.

Particle convective heat transfer is dominant in Group A and B powders. Increasing gas velocity beyond minimum fluidization improves particle circulation and so increases particle convective heat transfer. The heat transfer coefficient increases with fluidizing velocity up to a broad maximum hmax and then declines as the heat transfer surface becomes blanketed by bubbles. This is shown in Figure 14 for powders in Groups A, B and D.

Figure 14: Effect of fluidizing gas velocity on bed-surface heat transfer coefficient in a fluidized bed

The maximum in h_pc occurs relatively closer to U_mf for Group B and D powders since these powders give rise to bubbles at U_mf and the size of these bubbles increase with increasing gas velocity. Group A powders exhibit a non-bubbling fluidization between U_mf and U_mb and achieve a maximum stable bubble size.

Botterill (1986) recommends the Zabrodsky (1966) correlation for h_max for Group B powders:

and the correlation of Khan et al. (1978) for Group A powders:

Gas convective heat transfer is not important in Group A and B powders where the flow of interstitial gas is laminar but becomes significant in Group D powders, which fluidize at higher velocities and give rise to transitional or turbulent flow of interstitial gas. Botterill suggests that the gas convective mechanism takes over from particle convective heat transfer as the dominant mechanism at Re_mf 12.5 (Re_mf is the Reynolds number at minimum fluidization and is equivalent to an Archimedes number Ar 26,000). In gas convective heat transfer the gas specific heat capacity is important as the gas transports the heat around. Gas specific heat capacity increases with increasing pressure and in conditions where gas convective heat transfer is dominant, increasing operating pressure gives rise to an improved heat transfer coefficient h_gc.

Botterill (1986) recommends the correlations of Baskakov and Suprun (1972) for h_gc:

where U_m is the superficial velocity corresponding to the maximum overall bed heat transfer coefficient.

For temperatures beyond 600^oC radiative heat transfer plays an increasing role and must be accounted for in calculations. The reader is referred to Botterill (1986) or Kunii and Levenspiel (1990) for treatment of radiative heat transfer or for a more detailed look at heat transfer in fluidized beds.