Thiele Analytical Approximation

Thiele [4], having noticed the precision of the Percus-Yevick equation, postulated that an exact analjhical solution could be found. Starting with this equation and after considerable mathematical manipulations he arrived at the equation for pressure as 

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General analytical solution is not possible. A variety of numerical solutions have been developed, as well as analytical approximations, as described in the aforementioned reviews. The analytical solutions are generally applicable to either the thln film" regime - l.e., in which diffusion times are comparable to or less than reaction times, and minimal flux enhancement occurs, or the "thick film" regime in which perturbations from local reaction equilibrium are small (18). Frledlander and Keller (19) showed that there is a characteristic length scale, A, a function of the reaction and diffusion constants, such that L/X is a measure of the approach to local reaction equilibrium. It is thus similar to the Thiele modulus of porous catalysts. 

For hard spheres of diameter a, the PY approximation is equivalent to c(r) = 0 for r > o supplemented by the core condition g(r) = 0 for r < o. The analytic solution to the PY approximation for hard spheres was obtained independently by Wertheim [32] and Thiele [33]. Solutions for other potentials (e.g. Leimard-Jones) are... 

This approach is analytically correct for isothermal reactors and first-order rate laws, since concentration does not appear in the expression for the Thiele modulus. For other (nonlinear) rate laws, concentration changes along the reactor affect the Thiele modulus, and hence produce changes in the local effectiveness factor, even if the reaction is isothermal. Problem 21-15 uses an average effectiveness factor as an approximation. 

The concentration and temperature Tg will, for example, be conditions of reactant concentration and temperature in the bulk gas at some point within a catalytic reactor. Because both c g and Tg will vary with position in a reactor in which there is significant conversion, eqns. (1) and (15) have to be coupled with equations describing the reactor environment (see Sect. 6) for the purpose of commerical reactor design. Because of the nonlinearity of the equations, the problem can only be solved in this form by numerical techniques [5, 6]. However, an approximation may be made which gives an asymptotically exact solution [7] or, alternatively, the exponential function of temperature may be expanded to give equations which can be solved analytically [8, 9]. A convenient solution to the problem may be presented in the form of families of curves for the effectiveness factor as a function of the Thiele modulus. Figure 3 shows these curves for the case of a first-order irreversible reaction occurring in spherical catalyst particles. Two additional independent dimensionless paramters are introduced into the problem and these are defined as... 

IS a modified Damkohler number = A nhsCno ts the dimensionless NH3 adsoiption constant, D, is the molecular diffusivity of species 1 is the effective intraporous diffusivity of species i evaluated according to the Wakao-Smith random pore model [411. Equation (4) is taken from Ref. 39. Equations (6)-(8) provide an approximate analytical solution of the intraporous diffusion-reaction equations under the assumption of large Thiele moduli (i.e., the concentration of the limiting reactant is zero at the centerline of the catalytic wall) the same equations are solved numencally in Ref. 36. 

For hard spheres, the PY and mean-spherical approximations are identical. This identity does not extend to other potentials (charged hard spheres, for instance). The PY approximation is useful for molecules with short-ranged potentials, and the analytic solutions for hard spheres have been obtained by Wertheim (1963) and Thiele (1963). 

In solving (134) Di is usually considered constant, although Dj depends on the gas composition. This approximation seems reasonable considering the uncertainty in t. For simple reaction kinetics, the solution to (134) is given in terms of the Thiele modulus [102]. The more complicated kinetics for ammonia synthesis does not give a simple analytical solution. Generally, a numerical integration has to be carried out. 

In the case of internal diffusion, analytical solutions can be used direcdy only for simple kinetics, such as first or zero orders thus, approximations should be applied, since many reactions are not of zero or first order. Alternatively, utilization of the generalized diagrams of Aris [7] is possible (Fig. 10.19). Such diagrams relate for various kinetics effectiveness factors and the generafized Thiele modulus expressed by Eq. (10.121). 

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