5.5.2 Flow Through a Packed Bed

The majority of gas-phase reactions are catalyzed by passing the reactant through a packed bed of catalyst particles.

image

The equation used most often to calculate pressure drop in a packed porous bed is the Ergun equation4,5:

Ergun equation

5-22

image

Term 1 is dominant for laminar flow, and Term 2 is dominant for turbulent flow, where

image

In calculating the pressure drop using the Ergun equation, the only parameter that varies with pressure on the right-hand side of Equation (5-22) is the gas density, ρ. We are now going to calculate the pressure drop through a packed bed reactor.

Because the PBR is operated at steady state, the mass flow rate at any point down the reactor, image (kg/s), is equal to the entering mass flow rate, image (i.e., equation of continuity),

image

Recalling Equation (4-16), we have

4-16

image

5-23

image

Combining Equations (5-22) and (5-23) gives

image

Simplifying yields

5-24

image

where β0 is a constant that depends only on the properties of the packed bed (φ, DP) and the fluid properties at the entrance conditions (i.e., μ, G, ρ0, T0, P0).

5-25

image

For tubular packed-bed reactors, we are more interested in catalyst weight, rather than the distance z down the reactor. The catalyst weight up to a distance of z down the reactor is

5-26

image

where Ac is the cross-sectional area. The bulk density of the catalyst, ρb (mass of catalyst per volume of reactor bed), is just the product of the density of the solid catalyst particles, ρc, and the fraction of solids, (1 – φ):

ρb = ρc(1 – φ)

Bulk density

Using the relationship between z and W [Equation (5-26)] we can change our variables to express the Ergun equation in terms of catalyst weight:

image

Use this form for multiple reactions and membrane reactors.

Further simplification yields

5-27

image

Let y = (P / P0), then

Used for multiple reactions.

5-28

image

where

5-29

image

We will use Equation (5-28) when multiple reactions are occurring or when there is pressure drop in a membrane reactor. However, for single reactions in packed-bed reactors, it is more convenient to express the Ergun equation in terms of the conversion X. Recalling Equation (4-20) for FT,

4-20

image

Differential form of Ergun equation for the pressure drop in packed beds.

where, as before,

4-22

image

[Nomenclature note: y with the subscript A0—i.e., yA0—is the inlet mole fraction of species A, while y without any subscript is the pressure ratio—i.e., y = (P/P0)].

Substituting for the ratio (FT/FT0), Equation (5-28) can now be written as

5-30

image

Use for single reactions.

We note that when ε is negative, the pressure drop ΔP will be less (i.e., higher pressure) than that for ε = 0. When ε is positive, the pressure drop ΔP will be greater than when ε = 0.

For isothermal operation, Equation (5-30) is only a function of conversion and pressure:

5-31

image

Recalling Equation (5-21), for the combined mole balance, rate law, and stoichiometry,

5-21

image

Two coupled equations to be solved numerically

we see that we have two coupled first-order differential equations, (5-31) and (5-21), that must be solved simultaneously. A variety of software packages (e.g., Polymath) and numerical integration schemes are available for this purpose.

Analytical Solution. If ε = 0, or if we can neglect (εX) with respect to 1.0 (i.e., image), we can obtain an analytical solution to Equation (5-30) for isothermal operation (i.e., T = T0). For isothermal operation with ε = 0, Equation (5-30) becomes

5-32

image

Isothermal with ε = 0

Rearranging gives us

image

Taking y inside the derivative, we have

image

Integrating with y = 1 (P = P0) at W = 0 yields

(y)2 = 1 – αW

Taking the square root of both sides gives

5-33

image

Pressure ratio only for ε = 0

image

Be sure not to use this equation if ε ≠ 0 or if the reaction is not carried out isothermally, where again

5-29

image

Equation (5-33) can be used to substitute for the pressure in the rate law, in which case the mole balance can then be written solely as a function of conversion and catalyst weight. The resulting equation can readily be solved either analytically or numerically.

If we wish to express the pressure in terms of reactor length z, we can use Equation (5-26) to substitute for W in Equation (5-33). Then

5-34

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