Spheres, effectiveness factors

The solution of this equation is in the form of a Bessel function 32. Again, the characteristic length of the cylinder may be defined as the ratio of its volume to its surface area in this case, L = rcJ2. It may be seen in Figure 10.13 that, when the effectiveness factor rj is plotted against the normalised Thiele modulus, the curve for the cylinder lies between the curves for the slab and the sphere. Furthermore, for these three particles, the effectiveness factor is not critically dependent on shape. 

From Example 3.2 in Volume 3, for a sphere, the effectiveness factor is given by ... 

Hollow cylindrical catalyst pellets are sometimes employed in commercial chemical reactors in order to avoid excessive pressure drops across a packed bed of catalyst. A more complex expression for the effectiveness factor is obtained for such geometry. This case was first discussed by Gunn [4]. Figure 2 illustrates the effectiveness factor curves obtained for the slab, sphere and cylinder. 

Thus we have expressions for the effectiveness factor for different catalyst geometries. The Thiele modulus can be computed from catalyst geometry and surface area parameters. The characteristic size is 21 for a porous slab and 2R for a cylinder or sphere. While the expressions for r)(0) appear quite different, they are in fact very similar when scaled appropriately, and they have the same asymptotic behavior,... 

In Figure 7.4 the effectiveness factor is plotted against the Thiele modulus for spherical catalyst particles. For low values of 0, Ef is almost equal to unity, with reactant transfer within the catalyst particles having little effect on the apparent reaction rate. On the other hand, Ef decreases in inverse proportion to 0 for higher values of 0, with reactant diffusion rates limiting the apparent reaction rate. Thus, decreases with increasing reaction rates and the radius of catalyst spheres, and with decreasing effective diffusion coefficients of reactants within the catalyst spheres. 

The result of solving these equations is often expressed as an effectiveness factor, the ratio of the total rate of reaction to the rate that would obtain if the composition and temperature were everywhere equal to the boundary composition and temperature. For the sphere, this would be... 

Fig. 3.3. Effectiveness factors for flat plate, cylinder and sphere...

An enzyme which hydrolyzes the cellobiose to glucose, /3-glucosidase is immobilized in a sodium alginate gel sphere (2.5 mm in diameter). Assume that the zero-order reaction occurs at every point within the sphere with k0 = 0.0795 mol/sm3, and cellobiose moves through the sphere by molecular diffusion with Ds = 0.6 x 10 5 cm2 /s (cellobiose in gel). Calculate the effectiveness factor of the immobilized enzyme when the cellobiose concentration in bulk solution is 10 mol/m3. 

Figure 5.8 Influence of geometry on the link between the Thiele modulus if) and the effectiveness factor rj (1 slab 2 cylinder 3 sphere) (Aris, 1969).

Figure 5.9 Effectiveness factors rj for a slab (P), cylinder (C), and sphere (S) as functions of the Thiele modulus if) (Aris, 1965).

The shape of the catalysts does not appreciably affect the effectiveness factor. Emig and Holfman (5) have shown that the greatest difference between the effectiveness factors of such diverse shapes as sphere and infinite plate remain within 10%. Therefore, if effectiveness factor is known for one catalyst shape, it can be used for other forms with slight error. 

Comparing the effectiveness factors in Figure 5 and Figure 6, it can be seen, at the same value of Thiele modulus , the effectiveness factors for a sphere (a, = 3) is greater than that for a plane (ag = 1), (down curve in Figure 6b). It can be concluded that effectiveness factors for a cylinder shape (a, = 2) will be in... 

FIG. 19-15 Effectiveness factors versus Thiele modulus for a first-order reaction in spheres under adiabatic conditions. [Weisz and Hicks, Chem. Eng. Sci., 17 265 (1962).]... 

Figure 6 shows the effectiveness factor for any of the three different pellet shapes as a function of the generalized Thiele modulus p. It is obvious that for larger Thiele moduli (i.e. p > 3) all curves can be described with acceptable accuracy by a common asymptote t] — 1 / p. The largest deviation between the solutions for the individual shapes occurs around p x 1. However, even for the extremely different geometries of the flat plate and the sphere, the deviation of the efficiency... 

Figure 9. Effectiveness factor ij as a function of the Weisz modulus iji. Combined influence of intraparticle and interphase mass transfer on the effective reaction rate (isothermal, first order, irreversible reaction in a sphere, Biot number Bim as a parameter).

Figure 18. Effectiveness factor rj of a first-order reversible reaction versus the Weisz modulus ip (related to the forward rate constant k+). Influence of intraparticle diffusion on the effective reaction rate (isothermal reaction in a sphere, equal diffusivitics i,e = Die, equilibrium constant as a parameter).

Fig. 7.11. Effectiveness factor for slab (-), cylinder (-----) and sphere (-----) geometry as...

Figure 6.2. Effectiveness factor i) versus the Thiele modulus

Table 6.1 Effectiveness factor 17 as a function of the zeroth Aris number An, for first-order kinetics an infinitely long slab, an infinitely long cylinder and a sphere...

Figure 6.11 holds for a slab. Similar figures can be obtained for other catalyst geometries. This is illustrated in Figure 6.13 where the effectiveness factor is plotted versus 8 for zeroth-order kinetics in an infinite slab, infinite cylinder and a sphere. Figure 6.13 has been constructed on the basis of the formulae given in Table 6.6. Hence, the discussion that follows is not restricted to a slab, but holds for any arbitrary catalyst geometry. 

Table 6.6 Effectiveness factor tj as a function of the Thiele modulus 8 (given by Equation 6.48) for zeroth-order kinetics an infinite slab, an infinite cylinder and sphere...

Effectiveness factor for a first-order reaction in a sphere as a function of the Thiele modulus. 

A low value of the Thiele modulus results from a small diffusional resistance. For this case, the effectiveness factor is approximately 1 (values of (f) typically less than 1). Large values of the Thiele modulus are characteristic of a diffusion-limited reaction with an effectiveness factor less than 1. For (f> 1, the value of the effectiveness factor in a sphere approaches 3/[Pg.201]

Effectiveness factors for sphere, infinite cylinder, and finite cylinder pellet geometries where the Thiele modulus is based on equal VpjSp. Individual points correspond to the numerical solutions of the material balance on a finite cylinder. 

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