Pellet center concentration

A property of the effectiveness factor is that the product of the Thiele modulus and effectiveness factor approaches unity as the value of the Thiele modulus approaches infinity. In this asymptotic region of strong diffusion effects, the pellet center concentration approaches zero. Therefore, a generalized Thiele modulus can be defined as ... 

The pellet center concentration Q is still an unknown quantity. It can be obtained by integrating Eq. 4.71 ... 

The reactor point effectiveness of Eq. 4.83 is in a form suitable for direct inclusion in reactor conservation equations. However, the pellet center concentration Q has to be specified. As discussed earlier, C can be set to zero when G 3. For G less than 3, Q can be calculated in principle using Eq. 4.74, but this involves trial and error procedures, which are cumbersome. Instead, an approximate solution for Q can be based on the fact that the value of the integral / in Eq. 4.84 is rather insensitive to errors in the estimated value of Q. If we let AQ be the error in C, the corresponding maximum relative error in the integral I for a second-order reaction, for instance, is 3 AQC /Cg. A 30% error in Q will result in a maximum error of less than 10% for = 1. 

The basic approximations made in arriving at the reactor point effectiveness are (1) isothermal pellet, (2) negligible external mass transfer resistance, and (3) estimation of the pellet center concentration by a simple relationship when the reaction is not severely diffusion-limited. The first two approximations are quite adequate in view of the fact that the mass Biot number is of the order of hundreds under realistic reaction conditions. Both theoretical and experimental justifications for these approximations have been given in Chapter 4. The first approximation will be relaxed when reactions affected by pore-mouth poisoning are considered since a definite temperature gradient then exists within the pellet. An additional approximation is the representation of the difference between the Arrhenius exponentials evaluated at the pellet surface and the bulk-fluid temperatures by a linear rela-... 

Here, 7]w is given by Eq. 12.28, and Tjd is simply tj given by Eq. 12.18 with Cas replaced by Cass/H. The pellet center concentrations necessary for rju, and y]d can be obtained from the following approximate relations ... 

Q pellet center concentration coke concentration (mole/pellet volume) ... 

Derive the first-order correction of the effectiveness factor 171 for > 3 (Eq. 7.57). Also obtain 171 when is less than 3, i.e., when the pellet center concentration is not close to zero. 

The approximation for the pellet center concentration when 0.3 < 4>c < 3 is based on the fact that the value of the integral in Eq. 4.73 is quite insensitive to an error in the estimated value of Q. Because of this tolerance, the approximation is adequate in the region of moderate diffusion effect (0.3 < < >c < 3). On the other hand, the error in the calculated value of rji can be significant as <(>g approaches zero. This should not pose any problem since the reaction can be considered diffusion-free for all practical purposes when the value of a is small, say less than 0.3. 

In view of the fact that G at the reactor inlet is 5.6, the pellet center concentration was set equal to zero. Eq. (J) in Table 10.1 was used to evaluate the value of bulk temperature was used for the evaluation of the equilibrium constant The relationship between conversion and residence time (equivalently the reactor... 

In Eq. 10.54, the pellet center concentration has been set to zero for the diffusion-limited case being considered. As pointed out in Chapter 5, the global rates are usually valid for y less than 0.5 (refer to Sections 4-8 and 5-6 for details). 

This means that Eq. 10.90 is solved until (Ao)j reaches zero for use in Eq. 10.93. The grid index j corresponding to the value of (Ao)c calculated from Eq. 10.92 can be located for the values of ( 40 corresponding to thisy, which are the approximate pellet center concentrations. 

A simple search can also be used in which various values of t are calculated for assumed values of Tin until the minimum is found. Here again, the pellet center concentration, Q. is obtained from ... 

Obtain an expression that can be solved for the optimal Tin which maximizes Cout using Eq. 13.1, given Qn and t. Assume that the pellet center concentration Cc is negligible. 

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