E STRUCTURA DE GEMPRO

a. Prediction by Bubble Dynamics Method

A design engineer estimates the entrainment rate on the basis of experience or from knowledge of the data generated through research. There are several mecha-nisms by which elutriation or entrainment occurs in a gas fluidized bed. This topic was reviewed by Kunii and Levenspiel,⁵⁸ Leva and Wen,³²³ Geldart,³²⁴ and Wen et al.³⁰⁴ Some salient features are briefly presented here for the benefit of designers.

The solids carried by clouds and wakes of the bubbles are exchanged³²⁵ between the emulsion and dispersed phases in a gas fluidized bed. When the gas bubbles reach the fluidized bed surface, they erupt, throwing the solids present in the wake into the freeboard. From knowledge of the number of such bubbles erupting and the

fraction of wake occupying the bubble, the entrainment rate has been computed in the following manner.

If N_b is the number of bubbles and f_w is the fraction of wake volume, the solids entrained per unit area of the fluidizing column of cross-sectional area A is given by:

(1.199)

where d_b can be computed from the data presented in Table 1.16. The term N_b /A can be evaluated on the basis of the assumption that the total bubble volume is equal to A(U – U_mf). Thus,

(1.200)

Using Equations 1.200 and 1.199, the entrainment rate can be expressed as:

Ws = fw(1 –
mf ) ρ_p (U – Umf ) (1.201) Use of Equation 1.201 requires knowledge of fw, which can be determined along the lines suggested by Rowe and Partridge.³²⁵ According to them, fw is 30%. It can be seen that Ws is influenced by U, Umf, and fw. In addition to these parameters, the bed diameter, the distributor, and the bed internals, which affect the hydrodynamics of fluidization, also affect entrainment. These factors do not explicitly appear in Equa-tion 1.201. However, with an appropriate choice of db and fw, the effects of these parameters can be taken into account.

The radial variation of the gas velocity is disturbed by the presence of elutriated solids above the bed surface. The difference between the average velocity and the

Figure 1.29 Entrainment process and related terminology. Above TDH only fines are carried out, below TDH fines and coarse particles are present, and at TDH coarse particles are disengaged and returned back to the bed.

W N

A

d f

s

b b

w mf p

= ⎛⎝⎜ ⎞

⎠⎟^π₆³

(

)

N A U U

b d

mf

b

=

( )

( )

– π ³ 6

maximum velocity across the cross-section of a fluidizing column is rather small near the surface of the bed where bubble eruption is continuous. This would cause the reduction in elutriation to be less than what would otherwise be expected.

b. Prediction by Mass Balance Method

Prediction of particle entrainment on the basis of bubble eruption and the asso-ciated particles in the bubble wake requires thorough understanding of bubble dynamics and reliable and pertinent data. Another method for predicting elutriation is by applying mass balance considerations to a continuous or batch system. Let us now evaluate algebraically the elutriation rate (E_i) for the ith component in a mul-ticomponent system. In this connection, reference may be made to Figure 1.30.

The mass balance for a continuous system with the feed rate (F), the discharge rate (B) at the bottom, and the elutriation rate (E) is

F = E + B (kg/s) (1.202)

For a component i with its mass fraction X_iF, X_iE, and X_iB, respectively, in F, B, and E, FX_iF = EX_iE + BX_iB (kg/s) (1.203) Equations 1.202 and 1.203 can be used to calculate E, if all the other parameters (i.e., the mass fraction of component i and the feed or discharge rate) are known.

Details of the algebraic method of evaluating the elutriation rate can be found elsewhere in the literature. Only the gist will be presented here so as to understand the basics of the method.

The elutriation rate of the ith component from mass M is given by:

(1.204)

Figure 1.30 Illustration of parameters used in mass balance method.

E d

dt MX K AX

i =

(

)

where K is the elutriation rate constant and A is the area of cross-section. From Equation 1.204, X_iB,t at any time t in the bed is

(1.205)

In other words, the mass of the ith component at time t is

(1.206)

where M_iB,o is equal to MX_iB,o. The total entrainment (E) at the exit is

(1.207)

and the solids concentration (C_s,E) at the exit is

(1.208)

Thus, knowledge of E is useful in predicting the solids concentration at the exit and also for the design of post-reactor equipment such as cyclones or settling chambers. The other important parameters required to predict the conversion in the freeboard are particle velocity and its residence time. These parameters can be estimated from knowledge of the pressure drop across a section (∆z) in the freeboard.

Thus, for a section ∆z if ∆P is known, the solid holdup density (ρ_sz) is given by:

(1.209)

The particle residence time (t_z) can be expressed as:

(1.210)

U_PZ = C_sZ U/ρ_s (1.212) Equation 1.212 is useful in predicting the particle slip velocity (Usl), which is given by:

(1.213)

c. Kunii and Levenspiel Method

The mechanism of entrainment discussed above is based on bubble dynamics.

An entirely different method of evaluating entrainment, as proposed by Kunii and Levenspiel,³²⁶ can also be used. This model was developed based on the mass balance of three distinct phases: (1) gas stream with dispersed solids (phase 1) moving upward and receiving solids from descending as well as ascending agglomerates, (2) ascending agglomerates (phase 2) losing solids by transfer to an upward-moving gas stream (phase 1) and also to downward-moving agglomerates, and (3) descending agglomerates (phase 3) gaining solids from phase 2 and losing solids to phase 1.

According to this model, under normal entrainment, solids are elutriated from the bed and also returned to it. The entrainment rate is

E = Eoe^–aH (1.214)

where E_o is the entrainment at H = 0 (i.e., just above the bed surface) and the constant a is a function of the rate coefficients for the transfer of solids among the three phases and the associated phase velocities. Both E_o and H can be computed for a specific system from a plot of ln E versus H.

Kunii and Levenspiel³²⁷ later analyzed their complex model and evaluated the decay constant a by using data from the literature. They also attempted to explain the fast fluidization phenomenon by using their predictive model and presented several sample calculations of their model for practical applications.

d. Complexity of Parameter Determination

In order to evaluate the elutriation rate from Equation 1.204, the elutriation rate constant (K_E) should be known. Geldart³²⁴ presented a list of correlations in this regard. A simple correlation in a dimensionless form, as proposed by Geldart,³²⁴ is

(1.215)

Correlation 1.215 is picked up from the literature only to demonstrate the simplicity of its use. However, the limited range of applicability of this correlation makes designers aware of their restricted choice. The reason why numerous correlations have been proposed to date lies in the complex nature of the parameters that influence

U U

sl = Upz

–

K U

U U

E g

t

ρ = ⎡

⎣⎢

⎤ 23 7. exp – .5 4 ⎦⎥

entrainment or elutriation. The elutriation rate is affected by the gas flow rate, the size distribution of the solids, the column diameter, and the freeboard height. The effects of several other parameters, such as particle shape, fluid viscosity, and particle surface morphology, have not yet been systematically studied. The effect of the bed diameter also has yet to be clearly established, although it is known that the quality of fluidization is affected by the bed diameter. A small bed diameter can give rise to slugs and can thus increase the elutriation rate. As far as gas dispersion is concerned, elutriation with a dispersed gas is expected to be high. With good quality fluidization, where the gas distribution is appropriate and there are several small gas bubbles, elutriation rates were found to be low.³²⁸ The use of mechanical stirrers and inserts inside the fluidized bed was found to reduce the elutriation rate.³²⁹