Fluidization of Particles by Fluids, by Martin Rhodes

-- 9: A Simple Model for the Bubbling Fluidized Bed Reactor --

• the division of gas between the bubble phase and particulate phase
• the degree of mixing in the particulate phase
• the transfer of gas between the phases

The approach assumes the following:

• original Two-Phase Theory applies
• perfect mixing in the particulate phase
• no reaction in the bubble phase

Figure 17: Schematic of the Orcutt fluidized bed reactor model

The following is the nomenclature used
          C₀ = concentration of reactant at distributor
          C_p = concentration of reactant in the particulate phase
          C_B = concentration of the reactant in the bubble phase at height h above the distributor
          C_BH = concentration of reactant leaving the bubble phase
          C_H = concentration of reactant leaving the reactor

In steady state, the concentration of reactant in the particulate phase is constant throughout the particulate phase because of the assumption of perfect mixing in the particulate phase. Throughout the bed gaseous reactant is assumed to pass between particulate phase and bubble phase.

The overall mass balance on the reactant is:

Term (1) = U A C₀

Term (2) changes with height L above the distributor as gas is exchanged with the particulate phase. Consider an element of bed of thickness L at a height L above the distributor. In this element:

Where K_C is the mass transfer coefficient per unit bubble volume and _B is the bubble fraction. Integrating with the boundary condition that C_B = C₀ at L = 0:

At the surface of the bed, L = H and so the reactant concentration in the bubble phase at the bed surface is given by

and so, Term (2) = C_BH (U - U_mf) A

Term (3) = U_mf A C_p

Term (4): For a reaction which is jth order in the reactant under consideration the molar rate of conversion per unit volume of solids = k C_p^j, where k = reaction rate per unit volume of solids.

Therefore, [molar rate of conversion in the bed] = [molar rate of conversion per unit volume of solids] x [volume of solids per unit volume of particulate phase] x [volume of particulate phase per unit volume of bed] x [volume of bed]

Hence, Term (4),
[molar rate of conversion in the bed] = k C_p^j (1 - _p) (1 - _B) A H . . . . (Eq. 61)

where _p = particulate phase voidage

substituting these expressions for the term 1 - 4, the mass balance becomes Eq. 62:

From this mass balance C_p may be found. The reactant concentration leaving the reactor C_H is then calculated from the reactant concentrations and gas flows through the bubble and particulate phases:

In the case of a first order reaction (j = 1), solving the mass balance for Cp gives:

where = K_CH / U_B, equivalent to a number of mass transfer units for gas exchange between the phases. is related to bubble size and correlations are available. Generally decreases as bubble size increases and so small bubbles are preferred.

Thus from Equations 63 and 64, we obtain an expression for the conversion in the reactor:

where = (U - U_mf) / U, the fraction of gas passing through the bed as bubbles. It is interesting to note that although the Two-Phase Theory does not always hold, the Equation 65 often holds with still the fraction of gas passing through the bed as bubbles, but not equal to (U - U_mf) / U.

Readers interested in reactors of order different from unity, solids reactions, and more complex reactor models for the fluidized bed, are referred to Kunii and Levenspiel (1990).

Although the Orcutt model is simple, it does allow us to explore the effects of operating conditions, reaction rate and degree of interphase mass transfer on performance of a fluidized bed as a gas-phase catalytic reactor. Figure 18 shows the variation of conversion with reaction rate (expressed as k H_mf (1 - _p) / U) with excess gas velocity (expressed as ) calculated using Equation 65 for a first order reaction.

Figure 18: Conversion as a function of reaction rate and interphase mass transfer for = 0.75 for a first order gas phase catalytic reaction (based on Equation 65)

Noting that the value of is dictated mainly by the bed hydrodynamics, we see that:

• for slow reactions overall conversion is insensitive to bed hydrodynamics and so reaction rate k is the rate controlling factor
• for intermediate reactions both reaction rate and bed hydrodynamics affect the conversion.
• for fast reactions the conversion is determined by the bed hydrodynamics

  These results are typical for a gas-phase catalytic reaction in a fluidized bed.