Intraparticulate difusion. Extension of the concepts of thiele modulus and efficiency, страница 6

Consider two parallel first-order reactions:

As we have seen in Chapter 1, for a batch reactor the selectivity is defined by:

but

Hence the selectivity       without diffusion limitation can be written:

(without diffusion limitation)

The denominator does not need to be clarified. Its value depends on what is considered as an ideal case. This could be, for example:

If intraparticulate diffusion becomes extensive, since each of the reactions can develop independently of the other, we have to take account of two independent efficiency factors h₁ and h₂ so that:

 (with diffusion limitation) =

and

If these diffusion limitations are severe, both for    and for   , the    will be inversely proportional to   , and we can write:

 (10.30)

can be higher than        depending on the respective values of          ,          and         .

10.5.2. Twin first-order reactions

Consider the system of twin reactions represented by:

For a batch reactor, the instantaneous selectivity     is written by definition

because         represents the maximum number of moles of A₂ which could be obtained per mole of A₁ if reaction 2 did not occur:

If intraparticulate diffusion intervenes significantly to limit the penetration rate of a), we can define a Thiele modulus:

                               and an efficiency

so that

(10.31)

As foreseeable, the diffusion limitation has no effect on the selectivity of twin reactions.

10.5.3. Consecutive reactions

Consider the reactions:

By definition, the instantaneous selectivity for a batch reactor is:

because          represents the maximum number of moles of A₂ which could be obtained per mole of A₁ if reaction 2 did not occur.

Selectivity without diffusion limitation is written:

Hence if we define

 (10.32)

The integrated selectivity:

is obtained by integrating (10.32). To do this, we can write:

The result of this integration, using the assumption w^o = 0> i-s- "o intermediate compound in the reactor feed, leads to:

or

and since

In these conditions:

 (10.33)

and

If the reactions are both clearly affected by intraparticulate diffusion, we can show that the instantaneous selectivity is written [2]:

(10.34)

where

   = molar flux of A₂ leaving the grain,

= molar flux of A₁ entering the grain,

= ratio of the two fluxes in an ideal situation.

Knowing also that:

we can calculate the integrated selectivity

by writing

Integrating, and using the assumption         , this gives:

and consequently

(10.35)

and

If m is larger than 1, which frequently occurs, selectivity with diffusion limitation is lower. For values of m of around 10, a 50 per cent loss of selectivity can be observed. Decreasing the particle size helps to offset this loss of selectivity, at least partially.

Note a particular feature of expression (10.34). At the very start of the reaction, we normally expect to observe only the appearance of A₂, without the occurrence of the subsequent reaction, and consequently an initial selectivity of 1.

In the case of a pronounced diffusion limitation, it is observed that the initial selectivity is equal to              , which means that, from the outset of the reaction, product A₃ appears resulting from the disappearance of A₂. This clearly indicates that A₂, trapped in the pores, cannot escape from them without damage and is subject to the subsequent reaction, so that even the first observation concerning the interior of the fluid phase reveals the product resulting from the disappearance of A₂. This process can be exploited to identify a possible diffusion limitation.