Strong pore diffusion

Figure 8.10(b) shows a plot of if/ = cAlcAs as a function of z, the fractional distance into the particle, with the Thiele modulus cj) as parameter. For = 0, characteristic of a very porous particle, the concentration of A remains the same throughout the particle. For (f> = 0.5, characteristic of a relatively porous particle with almost negligible pore-diffusion resistance, cA decreases slightly as z —> 1. At the other extreme, for = 10, characteristic of relatively strong pore-diffusion resistance, cA drops rapidly as z increases, indicating that reaction takes place mostly in the outer part (on the side of the permeable face) of the particle, and the inner part is relatively ineffective. 

Strong pore-diffusion resistance (beyond point H) ... 

The asymptotic solution ( - large) for tj is [2/(n + l)]1/2/, of which the result given by 8.5-14c is a special case for a first-order reaction. The general result can thus be used to normalize the Thiele modulus for order so that the results for strong pore-diffusion resistance all fall on the same limiting straight line of slope - 1 in Figure 8.11. The normalized Thiele modulus for this purpose is... 

The rate is inversely proportional to particle size. This is an indication of strong pore-diffusion resistance, in which t) - llcp" as " - large. Since (f>" a Le for fixed other conditions (surface kinetics, De, and c ), if we compare measured rates for two particle sizes (denoted by subscripts 1 and 2), for strong pore-diffusion resistance,... 

Strong Pore-Diffusion Resistance Some Consequences... 

Here, we consider the consequences of being in the region of strong pore-diffusion resistance (77 - 11(f)" as apparent activation energy (f)" is given by equation 8.5-20b. 

Figure 18.7 Shows the limits for negligible and for strong pore diffusion resistance.

Thus, in the regime of strong pore diffusion, an nth-order reaction behaves like a reaction of order (n + l)/2 or... 

To find how pore resistance influences the rate evaluate Mj or then find from the above equations or figures, and insert < into the rate equation. Desirable processing range Fine solids are free of pore diffusion resistance but are difficult to use (imagine the pressure drop of a packed bed of face powder). On the other hand a bed of large particles have a small Ap but are liable to be in the regime of strong pore diffusion where much of the pellets interior is unused. 

Here we examine how strong pore diffusion modifies the true instantaneous fractional yield for various types of reactions however, we leave to Chapter 7... 

Now for strong pore diffusion in one size of pore we already know that the observed order of reaction, activation energy, and k ratio for multiple reactions will differ from the true value. Thus from Eqs. 30 and 32... 

A packed bed reactor converts A to R by a first-order catalytic reaction, A R. With 9-mm pellets the reactor operates in the strong pore diffusion resistance regime and gives 63.2% conversion. If these pellets were replaced by 18-mm pellets (to reduce pressure drop) how would this affect the conversion ... 

These expressions show that in the regime of strong pore diffusion resistance a decreases (see Eq. 39), causing to also decrease. This means that S rises with time, as shown in Fig. 21.6. However a decreases faster than (Arises so the reaction rate decreases with time. 

Find the kinetics of reaction and deactivation, both in the diffusion-free and in the strong pore diffusion resistance regime. 

Note In the strong pore diffusion regime the rate is lower but the catalyst deactivates more slowly. Actually, for the catalyst used here if we could have been free of diffusional resistances reaction rates would have been 360 times as fast as those measured. 

This represents fairly strong pore diffusion resistance. 

Our reaction A R proceeds isothermally in a packed bed of large, slowly deactivating catalyst particles and is performing well in the strong pore diffusion regime. With fresh pellets conversion is 88% however, after 250 days conversion drops to 64%. How long can we run the reactor before conversion drops to... 

Under conditions of strong pore diffusion the reaction A R proceeds at 700°C on a slowly deactivating catalyst by a first-order rate... 

From the following data, reported by Krishnaswamy and Kitterell, AIChE 28, 273 (1982), develop rate expressions to represent this decomposition, both in the diffusion free regime and in the strong pore diffusion regime, for the catalyst at hand. Note that the conversion decreases with time in all runs showing that the catalyst deactivates with use. 

If (j)< 0.5, tj 1 results. If 0 > 5, then tj = 1/0 and the substrate concentration decreases very rapidly towards zero in the pore this case is termed strong pore diffusion. 

In the region of strong pore diffusion effects, the rate varies inversely with particle size, as shown by Eq. (4.34). Experimental tests with different particle sizes are often used to check for pore diffusion limitations. If the rate varies with 7 , the effectiveness factor is low, but the value of r] cannot be determined. If the rate increases less than twofold when R is halved, the data can perhaps be fitted to the appropriate curve in Figure 4.8 to determine 0 and T] for both sizes and thus obtain the true kinetic constant. However, more accurate values of k, , and r] are obtained when crushed catalyst is tested and the particle size is reduced until there is no further... 

In the region of strong pore diffusion limitations, the reaction rate will increase less rapidly with increasing temperature than if diffusion effects are absent. If the rate constant follows the Arrhenius relationship and Knudsen diffusion dominates, the apparent activation energy is about half... 

The true allowable temperature difference is somewhat higher than the value from Eq. (5.61), because Cg decreases as Tg increases. The stability depends on the order of the reaction and the relative concentration change. Another stabilizing effect is the decrease in apparent activation energy with temperature if the reaction is in the region of moderate to strong pore diffusion limitations. 

Some early studies of trickle-bed reactors used /le-in. or Vs-in. catalyst pellets, but the trend has been to smaller sizes because of higher reaction rates. Extrudates with diameters of Vl6-in. or V32-in. are often used. Shah and Paraskos [40] studied desulfurization and demetallization of crude oil in a small reactor at 400°C and 200 psia. They used V32-in. extrudates (790 /am X 2-3 mm) or the same catalyst crushed and sieved to give an average size of 550 /am. The apparent rate constants for the crushed catalyst were greater than for extrudates by a factor of about 1.3 for sulfur removal and 2-3 for vanadium and nickel removal. The higher factor for metal removal indicates a strong pore diffusion limitation. The metals are present... 

Thus, with strong pore diffusion limitations, the observed activation energy is one-half the true one. This provides one possible experimental test for the presence of pore diffusion problems, since if the observed is = 5-10 kcal/mol (21-42 kJ/mol), it is probably one-half of a true chemical activation energy value. However, if the observed is 2 20 kcal/mol (84 kJ/mol) it could be the true one or one-half of 40 kcal/mol (168 kJ/mol), and so the test is inconclusive in this case. 

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