11. External Diffusion Effects on Heterogeneous Reactions^*

Topics

Seven Steps in a Catalytic Reaction
External Diffusion Across a Stagnant Film
Relative Rates of Diffusion and Reaction
Mass Transfer in a Packed Bed of Catalyst Particles
Shrinking Core Model

7 Steps in a Catalytic Reaction

1. Mass transfer (diffusion) of the reactant(s) from the bulk fluid to the external surface of the catalyst pellet

2. Diffusion of the reactant from the pore mouth through the catalyst pores to the immediate vicinity of the internal catalytic surface

3. Adsorption of reactant A onto the catalytic surface

4. Reaction on the surface of the catalyst

5. Desorption of the products from the surface

6. Diffusion of the products from the interior of the pellet to the pore mouth at the external surface

7. Mass transfer of the products from the external pellet surface to the bulk fluid

We shall now focus on steps 1, 2, 6, and 7. Because the reaction below does not occur in the bulk phase (only at the surface, at z = delta), we shall first consider steps 1 and 7.

Binary Diffusion

Diffusion is the spontaneous intermingling or mixing of atoms or molecules by random thermal motion. Mass transfer is any process in which diffusion plays a role.

The molar flux is just the molar flow rate, F_A, divided by the cross sectional area, A_C, normal to the flow. W_A = F_A/A_C

Molar flux of A W_A (moles/time/area) with respect to fixed coordinate system

W_A = J_A + B_A

J_A = diffusional flux of A with respect to bulk motion, i.e. molar average velocity

B_A = flux of A resulting from bulk flow

One dimension for constant total concentration- Ficks First Law

Adolph Fick (1829-1901)

Gases: D_Ab~10^-5m²/s

Liquids: D_Ab~10^-9m²/s

Prediction of Gas Phase Diffusivities

External Diffusion Across a Stagnant Film

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Species A diffuses from the bulk (z=0) to a catalytic surface (z=d) where it reacts instantaneously to form B.

In-Out+Generation=Accumulation

Types of Boundary Conditions

1. Specify a concentration a boundary

2. Specify a flux at a boundary

a) No mass transfer across a boundary

[E.g., at pipe wall]

therefore

b) Reaction at a boundary

c) Diffusional flux to a boundary is equal to the convective flux away from the boundary.

e.g.,

3. Planes of Symmetry

[E.g., cylinder]

4 Common Cases of the Constitutive Equation

1) Dilute concentrations (liquids)

Constant total concentration

2) Equal Molar Counter Diffusion (EMCD)

3) Diffusion through a Stagnant Film

4) Negligable Diffusion (Plug Flow)

W_A=? for 2A<=>B

Relative Rates of Diffusional Transport

Comparing J_A and B_A for diffusion through a stagnant film

Measurement of Gas Phase Diffusivities

Mole Balance with Diffusion and Reaction in 3-D

Measurement of Liquid Phase Diffusivities

Relaitve Rates of Diffusion and Reaction

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1.	Mole Balance on Species A at steady state


	Integrating:
	W_A=K'
2.	Rate Law / Constitutive Equation
	Constitutive Equation


	Rate Law on Surface

3.	Boundary Conditions
	Z=0 C_A=C_A0
	Z=d C_A=C_A0

	The rate of arrival of molecules on the surface equals the rate of reaction on the surface.

	k_C is the mass transfer transfer coefficient. It can be found from a correlation for the Sherwood number:

	which in turn is a function of the Reynolds Number

	and the Schmidt Number

	For packed beds:



	We see if we increased the velocity by a factor of 4, then the mass transfer coefficient, and hence the rate would increase by a factor of 2.
	The flux to the surface is equal to the rate of reaction on the surface:




	Let's look at the effect of increasing the velocity. We know that k_c increases with increasing velocity, while k_r is independent of velocity. At low velocities, the reaction is diffusion limited with k_c >>k_r and -r_A=k_c C_AO
	CASE 1 when k_c >>> k_r, then reaction is diffusion limited
	, rapid reaction on the surface, meaning that the overall reaction rate () is a function of velocity

	CASE 2 when k_c >>> k_r, then reaction is reaction rate limited
	, slow surface reaction, meaning that the overall reaction rate (W_A=k_rC_AO) is independent of velocity


	At high velocities, k_c >> k_r and -r_A is independent of velocity

Mass Transfer in a Packed Bed of Catalyst Particles

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Mole Balance


Rate Law / Constitutive Equation

		for single pellets
		for packed beds

	If
		k_r >>> k_c
	Then


	We want to know how the mass transfer coefficient varies with the physical properties (e.g., D_AB) and the system operating variables.





For isothermal operation, taking the rates for case 1 and case 2, the product of the mass transfer coefficient and the area a_cis

This equation tells us how the product of our mass transfer coefficient and surface area would change, if we were to change our operating conditions. In other words, it will help us answer "What if..." questions about our system.
For example: What effect does pressure drop have when all other variables remain the same?

	For:

	Substituting:


	If

	Then