The Thiele modulus was developed by Ernest Thiele in his paper 'Relation between catalytic activity and size of particle' in 1939. Thiele reasoned that with a large enoughparticle, the reaction rate is so rapid that diffusion forces are only able to carry product away from the surface of the catalyst particle. Therefore, only the surface of the catalyst would be experiencing any reaction. The Thiele Modulus was then developed to describe the relationship between diffusion and reaction rate in porous catalyst pellets with no mass transfer limitations. This value is generally used in determining the effectiveness factor for catalyst pellets. The Thiele modulus is represented by different symbols in different texts, but is defined in Hill as hT.
Overview
The derivation of the Thiele Modulus begins with a material balance on the catalyst pore. For a first-order irreversible reaction in a straight cylindrical pore at steady state: where is a diffusivity constant, and is the rate constant. Then, turning the equation into a differential by dividing by and taking the limit as approaches 0, This differential equation with the following boundary conditions: and where the first boundary condition indicates a constant external concentration on one end of the pore and the second boundary condition indicates that there is no flow out of the other end of the pore. Plugging in these boundary conditions, we have The term on the right sidemultiplied by C represents the square of the Thiele Modulus, which we now see rises naturally out of the materialbalance. Then the Thiele modulus for a first order reaction is: From this relation it is evident that with large values of, the rate term dominates, and the reaction is fast while slow diffusion limits the overall rate. Smaller values of the Thiele modulus represent slow reactions with fast diffusion.
Other Forms
Other order reactions may be solved in a similar manner as above. The results are listed below for irreversible reactions in straight cylindrical pores.
Second Order Reaction
Zeroth Order Reaction
Effectiveness Factor
The effectiveness factor η relates the diffusive reaction rate with the rate of reaction in the bulk stream. For a first order reaction in a slab geometry, this is:
