The intrapellet resistances to mass and heat transfer can be computed by integrating the diffusion/conduction balances using known surface composition and temperature.
A positive feature of this method
This method has an advantage compared to the effectiveness factor methods because it can readily handle complex kinetics and multiple reactions.
Problem 1: Effective transport parameters
The diffusion and conduction equations require effective diffusivity and conductivity parameters. Computation of these parameters is not recommended for the following reasons.
Diffusivity may be computed from the pellet porosity and tortuosity. The porosity may be readily determined by Hg and gas penetration, but the tortuosity is usually predicted via an ideal model. Another difficulty is the distribution of pore sizes from Knudsen size to molecular diffusion size. Because of this distribution, computing the overall diffusion parameter is a challenge and the result may not be accurate.
Carberry (2001) recommends measurement of the effective diffusivity, and he describes the measurement method, pg. 493. In that method, a cylindrical pellet is used and the diffusivity is measured in the axial direction. Since the measurement uses a single pellet, multiple pellets need to be tested to arrive at a statistical mean.
The effective thermal conductivity is even more difficult to compute from a model because of the complex parallel paths through gas and solids. Usually an order-of-magnitude estimate is used. Direct measurement is not usually attempted.
Problem 2: Non-spherical catalyst shape
There are several common shapes for commercial catalysts...spherical (agglomerates), short cylinders (tablets), long cylinders(extrudates). The diffusion/conduction equations usually assume a spherical shape. Thus, an effective radius is often used based on the external surface area and the catalyst volume.
Problem 3: Non-uniform catalyst activity
Some catalysts are made by impregnating the commercial size catalyst pellets with a solution of the active material and then drying the catalyst. This may lead to a distribution of activity from the exterior to the interior. This distribution will be unknown and therefore a model based on a uniform distribution will not be accurate.
Problem 4: A split boundary value problem
The boundary conditions are given at the catalyst surface and the "interior" (see Problem 5 also). The surface conditions are the values of the surface composition and temperature. The interior conditions are the derivatives (=0) of the dependent variables. Thus, a shooting method or collocation integration method are required. This problem can be solved, but it requires an extra effort and possibly extra computation time.
Problem 5: Unknown location of interior boundary
If a reaction rate is sufficiently high, the limiting reactant may disappear well before reaching the center of the particle. In actual practice, I have noticed that finite difference methods and orthogonal collocation methods may not reach solutions if the inner boundary is set at the zero radius. Thus, a trial and error was required to set the location of the inner boundary in order to achieve a solution. In addition, results depended upon the chosen location so the results are not reliable. Finally, for multiple reactions with different limiting reactants and reaction rates, one limiting reactant may disappear before the center whereas another may still be present at the center. This would require multiple boundary locations.
Problem 6: Iterations required to obtain surface conditions
The model assumes the surface conditions are known. In the overall reactor model, the surface conditions cannot be computed until the pellet production rates are known. Thus, iterations are required between the interphase model and the intraphase (film) model in order to converge on the surface conditions.
Problem 7: Computation time
Solving a set of split boundary ODEs, iteratively, at every location in a reactor surely will increase the computation time for the model. I don't know the extent of the computation time penalty because I haven't used this method. With today's fast computers, this may not be a problem for many applications, but it could be significant for studying catalyst decay or other dynamic scenarios.
Next: Effectiveness factor methods
Carberry, J.J., Chemical and Catalytic Reaction Engineering, Dover (2001)