Intraparticulate difusion. Extension of the concepts of thiele modulus and efficiency, страница 3

We can conclude from this that the effect of the variation in the number of moles during the reaction is rarely important, especially considering the inaccuracy in the numerical value of the effective diffusivity coefficient.

10.3.4. Non-isothermal reactions

It is easy to imagine that an exothermic reaction occurring in a porous medium will liberate heat that will not necessarily escape from the grain. This will lead to an increase in temperature and hence an increase in reaction rate, etc. Given the capacity of the grain to transfer this heat, a thermal gradient will be established in the grain, with a mean temperature higher than the surface temperature for exothermic reactions, and a mean temperature lower than the surface temperature for endothermic reactions.

The important parameters to be considered in analyzing this thermal mechanism are obviously:

(a) The heat of reaction       (J/mol).

(b) Activation energy of the reaction      (J/mol).

(c) Reactant concentration at the surface C_1s (mol/m³), which we have called C_1R for spherical grains.

(d)  Effective thermal conductivity of the porous grain    (W/m . K).

(e)   Surface temperature      (K).

(f)   Diffusivity of the reactant in the porous medium         (m²|s).

It is logical to group these different parameters into dimensionless numbers. Thus the following have been introduced:

Prater number                (10.20)

Arrhenius number            (10.21)

where R  is the ideal gas constant (8.31 J/mol . K).

It should be recalled that the efficiency is the ratio of the apparent reaction rate, with respect to the grain, to the theoretical reaction rate, corresponding to a temperature Ts existing at every point of the grain, and to a concentration C_1s existing at every point of the grain. Hence, according to this definition, it is perfectly feasible to obtain efficiencies largely greater than 1.

For b₁< 0, the efficiency is always less than 1. This corresponds to endothermic reactions, doubly penalized by a mean temperature in the granule that is lower than T_s and a mean concentration in the grain lower than C_1s. For b₁> 0 (exothermic reactions), two effects act in opposition. The mean temperature of the grain may be substantially higher than T_s, but, by contrast, the mean concentration may be much lower than C_1s. Depending on the values of Ф, b₁ and b₂, this means that r\ may be lower or higher than 1.

As for b₂, it can be shown that the increase in efficiency, for the same surface temperature T_s and the same value of b₁, is greater with higher activation energy E. Table 10.2 shows the ranges of the usual variations of b₁ and b₂.

Table 10.2

Variation ranges of b₁and b₂

Note on the significance of parameter b₁

If a heat balance is carried out around a spherical layer of thickness dz, in steady-state conditions, we can write:

(10.22)

If  ,    and     can be considered constant, this expression can be integrated to obtain:

(10.23)

Knowing that the minimum value of C_1z is the value C_1z == 0, this means that:

maximum      in the grain =

and

(10.24)

Hence b₁ i represents the maximum relative increase (or decrease) in temperature in the grain.

If:

does not exceed the value 5 K, the effect of temperature on efficiency can be ignored.

For Ts = 500 K, this corresponds to b₁ = 0.01.

The main difficulty in estimating b₁ resides in the estimation of the values of    and especially    . Figure 10.6a illustrates the variation in efficiency with the Thiele modulus and the Prater number b₁, for a b₂of 20 corresponding to an average case. This figure calls for several comments: