Educ. Reso. for Part. Techn. 012Q-Rhodes
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Copyright © 2001 Martin Rhodes, Licensed to ERPT

Fluidization of Particles by Fluids, by Martin Rhodes

-- 5: Expansion of a Fluidized Bed --

5.1: Non-bubbling fluidization

In a non-bubbling fluidized bed beyond Umf the particle separation increases with increasing fluid superficial velocity whilst the pressure loss across the bed remains constant. This increase in bed voidage with fluidizing velocity is referred to as bed expansion (see Figure 4). The relationship between fluid velocity and bed voidage may be determined by recalling the analysis of multiple particle systems (see Rhodes, 1998, Chapter 2). For a particle suspension settling in a fluid under force balance conditions the relative velocity between particles and fluid, Urel is given by:

where Up and Uf are the actual downward vertical velocities of the particles and the fluid, and UT is the single particle terminal velocity in the fluid. In the case of a fluidized bed the time-averaged actual vertical particle velocity is zero (Up = 0) and so:

where Ufs is the downward volumetric fluid flux. In common with fluidization practice, we will use the term superficial velocity (U) rather than volumetric fluid flux. Since the upward superficial fluid velocity (U) is equal to the upward volumetric fluid flux (-Ufs), and Ufs = Uf , then:

Richardson and Zaki (1954) found the function f() which applied to both hindered settling and to non-bubbling fluidization. They found that in general, f() = n where the exponent n was independent of particle Reynolds number at very low Reynolds numbers, when the drag force is independent of fluid density, and at high Reynolds number, when the drag force is independent of fluid viscosity. ie.

Khan and Richardson (1989) suggested the correlation given below, which permits the determination of the exponent n at intermediate values of Reynolds number (although it is expressed in terms of the Archimedes number Ar there is a direct relationship between Rep and Ar). This correlation also incorporates the effect of the vessel diameter on the exponent.

Thus Equations, 21, 22 and 23 in conjunction with the correlation of Khan and Richardson above, permit calculation of the variation in bed voidage with fluid velocity beyond Umf. Knowledge of the bed voidage allows calculation of the fluidized bed height as illustrated below:

mass of particles in the bed = MB = (1 - ) p A H . . . . . (Eq.24)

If packed bed depth (H1) and voidage ( 1) are known then if the mass remains constant the bed depth at any voidage can be determined:

5.2: Bubbling Fluidization

The simplest description of the expansion of a bubbling fluidized bed is derived from the Two-Phase Theory of fluidization of Toomey and Johnstone (1952). This theory considers the bubbling fluidized bed to be composed of two phases; the bubbling phase (the gas bubbles) and the particulate phase (the fluidized solids around the bubbles). The particulate phase is also referred to as the emulsion phase. The theory states that any gas in excess of that required at incipient fluidization will pass through the bed as bubbles. Thus, in bubbling fluidization, bed expansion at velocities beyond minimum bubbling velocity is due to the presence of bubbles. Thus, referring to Figure 9,


Figure 9: Gas flows in a fluidized bed according to the Two-Phase Theory

Q is the actual gas flow rate to the fluid bed and Qmf is the gas flow rate at incipient fluidization, then

      gas passing through the bed as bubbles
           = Q - Qmf = (U - Umf) A . . . (Eq. 26)

      gas passing through the emulsion phase
           = Qmf = Umf A . . . . . (Eq.27)

Expressing the bed expansion in terms of the fraction of the bed occupied by bubbles, B:

where H is the bed height at U and Hmf is the bed height at Umf and UB is the mean rise velocity of a bubble in the bed (obtained from correlations -- see below). The voidage of the emulsion phase is taken to be that at minimum fluidization mf. The mean bed voidage is then given by:

In practice, the elegant Two-Phase Theory overestimates the volume of gas passing through the bed as bubbles (the visible bubble flow rate) and better estimates of bed expansion may be obtained by replacing (Q - Qmf) in Eq.28 with

visible bubble flow rate, QB = Y A (U - Umf) . . . . . (Eq.30)

where for Group A powders 0.8 < Y < 1.0
            for Group B powders 0.6 < Y < 0.8
            for Group D powders 0.25 < Y < 0.6

The above analysis requires a knowledge of the bubble rise velocity UB, which depends on the bubble size dBv and bed diameter D. The bubble diameter at a given height above the distributor depends on the orifice density in the distributor N, the distance above the distributor L and the excess gas velocity (U - Umf).

For Group B powders:

For Group A powders: bubbles reach a maximum stable size which may be estimated from:

where UT2.7 is the terminal free fall velocity for particles of diameter 2.7 times the actual mean particle diameter.

Bubble velocity for Group A powders is again given by Werther (1983):

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