Catalyst de-activation and poisoning

Catalysts can become de-activated in a number of ways during the course of the operation. A common cause of de-activation is the sintering of the particles to form a continuous matte as a result of the development of local hot spots within the body of the reactor. Another is the deposition of carbon on the surface of the particles, as in the case of the fluidised cracking of hydrocarbons in the oil industry, a process described in Volume 2, Chapter 6. In this case the spent catalyst is transported to a second fluidised bed in which the carbon coating is burned off and the regenerated catalyst is then recycled. 

When homogeneous poisoning occurs, since no reaction will be possible on the poisoned fraction ( , say, as shown in Fig. 3.11) of active surface it is reasonable to suppose that the intrinsic activity of the catalyst is in proportion to the fraction of active surface remaining unpoisoned. To find the ratio of activity of the poisoned catalyst to the activity of an unpoisoned catalyst one would compare the stationary flux of reactant to the particle surface in each case. For a first-order reaction 

A fraction ( of the active surface of some porous slab-shaped catalyst pellets becomes poisoned. The pellets are used to catalyse a first-order isothermal chemical reaction. Find an expression for the ratio of the activity of the poisoned catalyst to the original activity of the unpoisoned catalyst when (a) homogeneous poisoning occurs, (b) selective poisoning occurs. 

Now from equation 3.13 the concentration of reactant is a function of the distance the reactant has penetrated the slab. Thus  

If the slab were poisoned the activity would be 4(1 - ) rather than k and then  

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