In "External Mass Transfer" and "Thiele Modulus" external mass transfer and internal mass transfer have been examined, but seperately. In many systems we have porous catalysis but some external mass transfer effects. The video below examines the global reaction rate where the rate of reaction is taken from the bulk concentration with both external and internal mass transfer effects,
Amadeo (1995) examine the reaction rate of the low-temperature water-gas sift reaction. To do this they use a small amount of
a powdered catalyst and product a full reaction model with surface adsorption kinetics as,
$$-r=k\frac{\displaystyle p_{\text{CO}}p_{\text{H}_2\text{O}}(1-\beta)}{\displaystyle 1+\sum_iK_ip_i}$$
where
$$\beta=\frac{p_{CO_2}p_{H_2}}{p_{CO}p_{H_2O}K_e}$$
Which is for the equlibrium of the reaction.
This provides a good model for the reaction rate. However, it does not take into account the effects of using real catalyst
pellets with mass transfer. This can be taken into account by using the Overall Effectiveness Factor, $\Omega$.
Moe (1963) examine the reaction rate on an industrial system with an approximately 2 cm diameter catalyst. The equation they produced
is not from first principles, but just empirically correlates the data.
The graph below shows the variation of the reaction rate with catalyst size. The limitation is the internal mass transfer in the pores
as the external mass transfer rate is higher than the reaction rate.
Now that we have expressions for the rate of reactions with mass transfer effects we can design catalytic reactors by inserting these rates into the key design equations. The video below gives a simple integratable example for a single first order reaction,
References:
N. E. Amadeo and M. A. Laborde (1995), "Hydrogen Production from the Low-Temperature Water-Gas Shift Reaction: Kinetics and Simulation
of the Industrial Reactor", International Journal of Hydrogen Energy, 20:949-965.
J. M. Moe (1963), "Low-temperature CO conversion", Symposium on Production of Hydrogen, Am. Chem. Sot. Div. Petr. Chem. B29.