Transport Limitations and the Thiele Diffusion Modulus

The effect of transport limitations can conveniently be evaluated by considering the spherical catalyst particle shown in Fig. 5.32. We will introduce a dimensionless quantity called the Thiele diffusion modulus (Og) [W. Thiele Ind. Eng. Chem. 31 

We start with an ideal, porous, spherical catalyst particle of radius R. The catalyst is isothermal and we consider a reaction involving a single reactant. Diffusion is described macroscopically by the first and second laws of Pick, stating that 

Under steady state conditions, the rate of diffusion inwards at the outside of the shell (r + dr) minus the rate of diffusion inwards at the inside of the shell (r) corresponds to the rate of reaction in the shell. In other words what came in and did not go out has reacted. The inward flux at position r is given by Pick s first law and hence the rate is the area of the shell A(r) = 4jtr multiplied by the flux  

The rate of reaction is given by the volume of the shell 4jrr dr [m -at] multiplied by the pore area per volume of catalyst S [m surfcat/iti cat] multiplied by the rate constant k [m /(m surfcat (itiol) s)] multiplied by the concentration of the reactant C (r). Thus the rate of reaction in the shell is 

By eliminating higher orders in dr this equation becomes 

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