1. CHEE 32322.1 Internal Mass Transfer in Porous Catalysts We have examined the potential influence of external mass transfer on the rate of heterogeneous reactions. However, where active sites are accessible within the particle, internal mass transfer (molecular diffusion) has a tremendous influence on the rate of reaction within the catalyst. Numerous examples exist:  Encapsulated or entrapped enzymes  Microporous catalysts for catalytic cracking (zeolites) The diffusion rate of reactants and products within the particle often determines the rate at which a microporous catalyst functions.

2. CHEE 32322.2 Concentration Gradients of Diffusing Reactants and Products In a Uniform Catalyst Particle Figure 4-19 Spherical catalyst particle with diffusing reactant. C s is the concentration of reactant at the particle periphery.

3. CHEE 32322.3 Thiele Model for Intraparticle Mass Transport A simple but conceptually useful treatment of intraparticle mass transfer describes the diffusion of a reactant within a uniform catalyst particle. Assumptions/Simplifications  Uniform catalytic activity throughout the spherical particle  Uniform properties of solid  Irreversible, first-order kinetics; rate = k C A Given that mass transfer occurs by molecular diffusion, an analytical expression for the transport and consumption of our reactant can be written. A material balance on a shell between radii of r and r+dr is: Rate of A into - Rate of A out of = rate of consumption of A surface at r+dr surface at r within the shell volume

4. CHEE 32322.4 Thiele Model for Intraparticle Mass Transport Rate of A into - Rate of A out of = rate of consumption of A surface at r+dr surface at r within the shell volume divide by (4  r) and taking the limit as  r goes to zero yields: the molar flux, N A is given by Fick’s Law of diffusion: after substituting in for N A yields: with boundary conditions:

5. CHEE 32322.5 Thiele Model: Reactant Concentration Profile The reactant concentration profile for this simple case is: Note that at a Thiele modulus of 15, the reactant is consumed before penetrating further than r/R = 0.6. The reaction becomes increasingly that of a surface process, with the internal region remaining inert. Diffusion with reaction in a spherical porous catalyst particle: conc’n profiles of the reactant for a first-order isothermal reaction

6. CHEE 32322.6 Overall Reaction Rate in a Catalyst Particle Although knowledge of our reactant concentration as a function of radial position is interesting, what we require for process design is an expression for the reaction rate.  Local rate of reaction is rate = k C(r), so we could integrate over r=0 to r=R A simpler method follows from the fact that at steady state, the rate of reaction will equal the rate of mass transport of reactant into the particle.

7. CHEE 32322.7 Effectiveness Factor An experimentally accessible quantity that describes the extent of diffusion resistance in a porous catalyst is called the effectiveness factor,  = reaction rate / maximum rate. In our simple example, the maximum rate of reaction for a particle is the particle volume times the maximum rate per unit volume: Dividing our reaction rate by the maximum yields:

8. CHEE 32322.8 Effectiveness Factor Note that internal diffusion resistance decreases with decreasing . Therefore, the influence of diffusion on the reaction rate supported by a particle is reduced when particle radius is reduced, D AB is high and the rate constant is relatively small.

9. CHEE 32322.9 Effectiveness Factor: Example Suppose an encapsulated enzyme reaction (irreversible, first-order heterogeneous) is carried out without an external mass-transfer limitation, but it suffers from a diffusion limitation. Two different bead sizes were examined under identical conditions. Measured RateBead Radius Bead 13.0*10 -5 mole/g bead sec0.01 m Bead 2 15.0*10 -5 mole/g bead sec0.001 m Estimate the Thiele Modulus and effectiveness factor for each bead, and determine what size bead is needed to virtually eliminate an internal diffusion resistance.