Cylindrical pore model Effectiveness factor

The equations for effectiveness factors that we have developed in this subsection are strictly applicable only to reactions that are first-order in the fluid phase concentration of a reactant whose stoichiometric coefficient is unity. They further require that no change in the number of moles take place on reaction and that the pellet be isothermal. The following illustration indicates how this idealized cylindrical pore model is used to obtain catalyst effectiveness factors. 

The Effectiveness Factor for a Straight Cylindrical Pore Second- and Zero-Order Reactions. This section indicates the predictions of the straight cylindrical pore model for isothermal reactions that are zero- and second-... 

Plots of effectiveness factors versus corresponding Thiele moduli for zero-, first-, and second-order kinetics based on straight cylindrical pore model. For large hr, values of r are as follows ... 

The measured value of k Sg is 0.716 cm3/(sec-g catalyst) and the ratio of this value to k ltTueSg should be equal to our assumed value for the effectiveness factor, if our assumption was correct. The actual ratio is 0.175, which is at variance with the assumed value. Hence we pick a new value of rj and repeat the procedure until agreement is obtained. This iterative approach produces an effectiveness factor of 0.238, which corresponds to a differs from the experimental value (0.17) and that calculated by the cylindrical pore model (0.61). In the above calculations, an experimental value of eff was not available and this circumstance is largely responsible for the discrepancy. If the combined diffusivity determined in Illustration 12.1 is converted to an effective diffusivity using equation 12.2.9, the value used above corresponds to a tortuosity factor of 2.6. If we had employed Q)c from Illustration 12.1 and a tortuosity factor of unity to calculate eff, we would have determined that rj = 0.65, which is consistent with the value obtained from the straight cylindrical pore model in Illustration 12.2. 

Illustration 12.2 indicates how this idealized cylindrical pore model is used to obtain catalyst effectiveness factors. 

WTien the hydrogen pressure is 1 atm and the temperature is 77 K, the experimentally observed (apparent) rate constant is 0.159 cm /(s- g catalyst). Determine the mean pore radius, the combined diffusivity for hydrogen in the pores of this catalyst, and the catalyst effectiveness factor based on the straight cylindrical pore model. 

Since theoretical calcination of effectiveness is based on a hardly realistic model of a system of equal-sized cylindrical pores and a shalq assumption for the tortuosity factor, in some industrially important cases the effectiveness has been measured directly. For ammonia synthesis by Dyson and Simon (Ind. Eng. Chem. Fundam., 7, 605 [1968]) and for SO9 oxidation by Kadlec et aJ. Coll. Czech. Chem. Commun., 33, 2388, 2526 [1968]). 

Cuprophan membranes made from regenerated cellulose are frequently used in hemodialysis. Model this membrane as one consisting of cylindrical capillaries of radius 18 A. Determine the separation factors of the dialysis membrane for two solutes, urea and vitamin B12. The characteristic radii of urea and vitamin B12 are 2.8 A and 8.5 A, respectively. The diffusion coefficients of urea and vitamin B12 at infinite dilution in isotonic saline at 37 °C are 1.81 x 10 and 0.38 x 10" cm /s, respectively. Compare the result for the given pore size estimate with that obtained from the data of Colton et al (1971), namely 16 (based on effective diffusion coefficients without any consideration of equilibrium partition coefficients in their Figure 6). (Ans. 23.5.)... 

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