1. Heterogeneous Catalysis: Kinetics in Porous Catalyst Particles
Assoc. Prof. Dr. Nesrin E. Machin
Ref: S. Fogler (Ch 11)

2. Content Introduction Diffusion Mass Transfer
External Resistance to Mass Transfer
Mass Transfer-Limited Reactions in Packed Beds
Diffusion through a Spherical Catalyst Pellets
Thiele Modulus
Effectiveness Factor

3. Introduction 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.

4. Steps in a Catalytic Reaction

5. The Complications of the Rate Equation
Since more than one phase is present, the movement of material from phase to phase must be considered in the rate equation.
The rate expression in general will incorporate mass transfer terms in addition to the usual chemical kinetics term.
These mass transfer terms are different in type and numbers in the different kinds of heterogeneous systems; hence, no single rate expression has general application
In addition to an equation describing the rate at which the chemical reaction proceeds, one must also provide a relationship or algorithm
to account for the various physical processes which occur.

6. 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, FA, divided by the cross sectional area, AC, normal to the flow. WA = FA/AC
Molar flux of A, WA (moles/time/area) with respect to fixed coordinate system
WA = JA + BA
JA = diffusional flux of A with respect to bulk motion, i.e. molar average velocity
BA = flux of A resulting from bulk flow

7. WA= VA.CA Where VA is the particle velocity of A which is the vector average of millions of A molecules at a point. VA is the velocity wrt to a stationary coordinate, e.g., the lab bench. BA= V. CA= V. yA. CT where V is the molar average velocity of all species. V= VA.yA + VB.yB multiplying by the total concentration CT.V = CT (VA.yA + VB.yB )= VA.CA + VB.CB = WA + WB BA= yA (WA + WB) WA = JA + BA JA tells us of the flux of A with respect to the molar average velocity. WA = JA + yA (WA + WB)

8. The mass transfer flux by molecular diffusion is given according to Ficks law by:
Mass transfer flux due to bulk flow:

9. Z= 0
Z= 
Z= 0
z/ 
Z= 

13. External Resistance to Mass Transfer
For dilute solutions or equimolar diffusion:
Assuming film thickness
Is much smaller than the solid

14. Relative Rates of Diffusion and Reaction
Mole Balance on Species A at steady state (slab geometry)
Integrate
WA=K'

15. Mass Transfer Rate:
Boundary Conditions:
Z=0 CA=CA0
Z=d CA=CA0
Reaction Rate:

16. The rate of arrival of molecules on the surface equals the rate of reaction on the surface
kC is the mass transfer coefficient. It can be found from a correlation for the Sherwood number:
For Packed Beds

17. For Packed Beds
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:

18. Lets look at the effect of increasing the velocity
Lets look at the effect of increasing the velocity. We know that kc increases with increasing velocity, while kr is independent of velocity.
CASE 1
At low velocities, the reaction is diffusion limited with kr >> kc
and -rA= kc CAo
rapid reaction on the surface,
meaning that the overall reaction rate ( ) is a function of velocity

19. CASE 2
At high velocities, the reaction is reaction rate limited, kc >>> kr
slow surface reaction, meaning that the overall reaction rate (WA=kr CAo) is independent of velocity

20. At high velocities, kc >> kr and -rA is independent of velocity

21. Example:

23. And the surface concentration of reactant approaches zero,

25. Diffusion from the Bulk to Surface (external transport)
kC : Mass transfer coefficient,
U: Fluid velocity,
Dp: Particle diameter

26. Reactant Concentration Profiles
Reactant concentration profiles around a catalyst pellet for reaction control and for external mass transfer control

27. Definitions

29. Mass Transfer Limited Reactions in Packed Beds

35. Internal Diffusion Diffusion Through Spherical Catalyst Pellets
We now focus on steps 2 and 6 of our catalytic reaction . We shall carry a mole balance on species A as it diffuses and reacts in a catalyst pellet.
The pores in the pellet are not straight and cylindrical;
They are a series of tortuous, interconnecting paths of pore bodies and pore throats with varying cross-sectional areas.
Effective diffusion coefficient is used to describe the average diffusion taking place at any position r in the pellet

36. The pores in the pellet are not straight and cylindrical; rather, they are a series of tortuous, interconnecting paths of pore bodies and pore throats with varying cross-sectional areas.
( b )
( a ) accounts for the variations in the cross sectional area, normal to diffusion
(a)
(b)

37. Effect of Pore size on Diffusivity of Gas Molecules

38. Derivation of the Differential Equation Describing Diffusion and Reaction
Irreversible isomerization reaction
We now proceed to perform shell balance on A. The area that appears in the balance equation is the total area (voids and solids) normal to the direction of the molar flux.

39. where rm is some mean radius between r and r +delta r that is used to approximate the volume V of the shell and c is the density of the pellet.
The mole balance over the shell thickness delta r is:
After dividing by ( - 4r) and taking the limit as r , we obtain the following differential equation:

40. 1 mol of A reacts under conditions of constant temperature and pressure to form 1 mol of B, we have Equimolar Counter Diffusion (EMCD) at constant total molar concentration:
Substitute in differential equation
We need to substitute rate law now

41. At high temperatures, the denominator of the catalytic rate law approaches 1.
It is reasonable to assume that the surface reaction is of n th order in the gas-phase concentration of A within the pellet:
After substitution

42. By differentiating the first term and dividing through by ( - r2 De ) :

43. Writing the Equation in Dimensionless Form
Boundary conditions:

44. Rewrite the differential equation for the molar flux in terms of dimensionless variables:

45. The overall rate of reaction, MA , can be obtained by multiplying the molar flux into the pellet at the outer surface by the external surface area of the pellet, 4R2 ;
After dividing by , can be written as:
where

46. Thiele Modulus,  :
When the Thiele modulus is large, internal diffusion usually limits the overall rate of reaction; when n is small, the surface reaction is usually rate-limiting.
If for the first order irreversible reaction at high temperature,
A B ,

48. Solution to the Differential Equation for a First-Order Reaction

49. The arbitrary constants A1 and B1 can easily be evaluated with the aid of the boundary conditions:
Because the second boundary condition requires  to be finite at the center, ( =0 ), A1= 0
The constant B1 is evaluated from B.C.1,
Concentration Profile
Large values of the Thiele modulus indicate that the surface reaction is rapid and that the reactant is consumed very close to the external pellet surface and very little penetrates into the interior of the pellet.
If the porous pellet is to be plated with a precious metal catalyst (e.g., Pt), it should only be plated in the immediate vicinity of the external surface when large values characterize the diffusion and reaction, so that precious metal in the center is not wasted.

50. Internal Effectiveness Factor
The magnitude of the effectiveness factor (ranging from 0 to 1) indicates the relative importance of diffusion and reaction limitations. The internal effectiveness
factor is defined as;
The overall rate, -rA , is also referred to as the observed rate of reaction [rA(obs)]

51. For a first-order reaction in a
spherical catalyst pellet