COM MUN I CAT1ON
A NORMALIZATION FOR T H E T H I E L E MODULUS Following Petersen’s asymptotic method for a general case it is shown how the Thiele modulus, A, can be normalized so that the effectiveness factor, E, is asymptotically -1-l for large A, This provides a universal criterion for definite diffusion limitation and brings together the effectiveness factor curves for other parameters of the system. The method is illustrated by reference to the recent work of W a k a o and Smith. A parametric representation of E(A) is given in terms of indefinite integrals.
value of the effectiveness factor as a means of expressing the interaction of diffusion and reaction in porous pellets has been well proved since Thiele suggested it in 1939 (4). The dimensionless ratio of the reaction and diffusion rates, on which the effectiveness factor, E: chiefly depends, is often known as the Thiele modulus. In simple cases-e.g., nth order irreversible, isothermal reactions-this is the only parameter that is required, but in more complex cases a number of other dimensionless groups must be formed from the naturally occurring constants. In all cases, however, it seems ‘V1 desirable so to define the Thiele modulus. A. that E for large A, and we wish to show that this may always be done. This has two advantages: First: it allows a universal criterion for the region of pronounced diffusion limitation, and secondly, it often brings together whole families of curves for various other parameters and shapes. In such a case a single mean curve, with some appreciation of the variability, .may be more useful than a large number of precise curves, for the data with which the Thiele modulus is calculated will not always be known with great accuracy. It has been known for some time ( 7 ) that if the characteristic dimension of the particle is taken to be v p l s Z r the ratio of its volume to its external surface area. and this is used in the Thiele modulus, the curves of E us. -1 lie very much together for different shapes. The reason is that E is always approximately 1 for small I,and this choice of characteristic dimension makes E .i-’. It is clear on physical grounds that this must be so, for under extreme diffusion limitation only a thin ldyer of catalyst within the particle is in use and hence the local curvature is not important. It follows that in discussion of the asymptotic behavior of E we need only consider the geometry of the flat plate, of which u p Is, = d is the half thickness. Petersen’s asymptotic method, which he has used for a number of cases (2, 3 ) , deserves to be more Midely used in conveniently defining the Thiele modulus and can indeed be applied very generally. In many cases a variable, E , may be defined having the properties of extent or concentration. but made dimensionless in such a way that E = 1 a t the surface of the pellet and = 0 under equilibrium conditions. An example of this is to be found in a recent paper by M’akao and Smith (5), where E could be taken as - ,ye),’(,yo - ye) = Z;‘a THE
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in their notation. A similar variable can be defined almost trivially for the irreversible, isothermal reaction and after a little reduction for the nonisothermal reaction. If x is the distance from the central plane of the slab, 0 x d. the steady-state equation of diffusion and reaction can be written
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3 is certainly a safe criterion. Applying this to the macro-micropore structure of Wakao and Smith (5),we set
Hence by Equations 9 and 12
= Z / a , z = t/X’1’2,
Equations 12 and 14 provide a parametric representation of ( 1, being the paramthe function E(X). the variable {, 0 eter. This pair of formulas affords the easiest way of computing the effectiveness factor as a function of the Thiele modulus. hloreover, in the limit of strong diffusion limitation, equilibrium conditions will prevail at the center of the particle and hence j- = 0. Thus
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