Letters to the editor

Abstract

Recently Chang (1982) reported a numerical method for the estimation of the effectiveness factor for catalyst particles of high Thiele modulus with the Michaelis Menten type kinetics. The report by Lakshmanan (1983) suggested a collocation method with a trial function for the concentration gradient near the catalyst surface. Both methods require a trial and error procedure. In this letter, a simple method using the general modulus (Bischoff, 1965) is suggested and shown to be useful in obtaining the effectiveness for catalyst particles of high Thiele moduli.According to Bischoff (1965), the general modulus, M , with Michaelis Menten kinetics, is where St is the concentration of substrate on the catalyst surface and L is the distance measured from the surface (at L , dS/dr = 0).In the case of high Thiele moduli, the curvature effect on the mass transfer rate in a spherical catalyst particle can be assumed negligible. This is a valid assumption when the thickness of "effective reaction zone" (Paterson and Cresswell, 1971) is very small relative to the particle radius, R, due to a high Thiele modulus. When the above conditions are met, the overall reaction rate per catalyst particle, +overall, can be written (Bischoff, 1965; Moo Young and Kobayashi, 1972) asWe can express the concentration of substrate on the catalyst surface (unmeasurable) asInserting Equation (4) into (3) and using the dimensionless variables defined previously (Chang, 1982; Lakshmanan, 1983), we can derive the implicit function with respect to 7 as follows Equation ( 5 ) is the working equation for the TABLE 1. VALUES 7 values (a) Chang (1982) (b) Nine point collocation (c) This work % difference (Lakshmanan (1983)) 1 (@$-@ x 100, nine point collocation % difference 1(v x 100, this work determination of 7 with given p,@ and Sh.The values of 7 calculated with Equation (5) compare very well with the values reported for the case of 4 = 100. (Chang, 1982; Lakshmanan, 1983). These are shown in Table 1. As can be seen the 7 values calculated fromEquation ( 5 ) agree very well with the values of numerical calculations (Chang, 1983) and with those by nine point collocation (Lakshmanan, 1983). The calculation of 7 with Equation (5) is straightforward and does not require trial and error. In conclusion, Equation (5) can be used successfully in calculating the effectiveness factor for catalyst particles of high Thiele modulus with the Michaelis Menten type kinetics (or the Langmuir type, or the Monod type). NOTATION D = diffusion coefficient k K , = Michaelis Menten constant L to the inside of the particle M = general modulus T = radial coordinate 9 = reaction rate R = radius of a catalyst particle St = the concentration of substrate on the catalyst surface So = the concentration of substrate in bulk stream Sh = Sherwood number V,,, = maximum reaction rate = external mass transfer coefficient = distance measured from the surface Greek Letters 0 = saturation constant 17 = effectiveness factor 4 = Thiele modulus O F~F O R~= 100 Sh = 100; 4 = 100 p = 0.01 ...