Effectiveness factor nonisothermal pellet

FIGURE 10.3 Nonisothermal effectiveness factors for first-order reactions in spherical pellets. (Adapted from Weisz, P. B. and Hicks, J. S., Chem. Eng. Sci., 17, 265 (1962).)... 

ILLUSTRATION 12.4 EFFECTIVENESS FACTOR DETERMINATION FOR A NONISOTHERMAL CATALYST PELLET-EXOTHERMIC REACTION... 

The above suggests that the discussion of the Aris numbers for simple reactions also holds for nonisothermal pellets. For example, effectiveness factors larger than one are found if the number An, becomes negative. According to Equation 7.14 this is the case if... 

To evaluate the effectiveness factor for a first-order, isobaric, nonisothermal, flat plate catalyst pellet, the material and energy balances must be solved simultaneously. As shown previously, the mole balance in a slab is given by ... 

The above discussion of effectiveness factors is valid only for isothermal conditions. When a reaction is exothermic and nonisothermal, the effectiveness factor can be significantly greater than 1 as shown in Figiue 12-7. Values of t greater than 1 occur because the external smface temperature of the pellet is less than the temperature inside the pellet where the exothermic reaction is taking place. Therefore, the rate of reaction inside the pellet is greater than the rate at the surface. Thus, because the effectiveness factor is the ratio of the actual reaction rate to the rate at smface conditions, the effectiveness factor... 

P12-16c Determine the effectiveness factor for a nonisothermal spherical catalyst pellet in which a first-order isomerization is taking place. 

Even when is low, the center and surface temperatures may differ appreciably, because catalyst pellets have low thermal conductivities (Sec. 11-5). The combined effect of mass and heat transfer on can still be represented by the general definition of the effectiveness factor, according to Eq. (11-41). Hence Eq. (11-42) may be used to find r, provided rj is the nonisothermal effectiveness factor. The nonisothermal 17 may be evaluated in the same way as the isothermal 77, except that an energy balance must be combined with the mass balance. 

Let us return to the nonisothermal effectiveness factor. Weisz and Hicks solved Eqs. (11-46) and (11-72) numerically to obtain the concentration profile within the pellet. Then r was obtained from Eq. (11-51), which is not limited to isothermal conditions, provided is evaluated at the surface temperature. The results expressed 17 as a function of three dimensionless parameters ... 

Fig. 11-10 Nonisothermal effectiveness factors for first-order reactions in spherical catalyst pellets...

In the previous examples, we have exploited the idea of an effectiveness factor to reduce fixed-bed reactor models to the same form as plug-flow reactor models. This approach is useful and solves several important cases, but this approach is also limited and can take us only So far. In the general case, we must contend with multiple reactions that are not first order, nonconstant thermochemical properties, and nonisothermal behavior in the pellet and the fluid. For these cases, we have no alternative but to solve numerically for the temperature and species concentrations profiles in both the pellet and the bed. As a final example, we compute the numerical solution to a problem of this type. 

For the single-reaction, nonisothermal problem, we solved the so-called Weisz-Hicks problem, and determined the temperature and concentration profiles within the pellet. We showed the effectiveness factor can be greater than unity for this case. Multiple steady-state solutions also are possible for this problem, but for realistic values of the... 

Figure 3.7.a-I Effectiveness factor with first-order reaction in a spherical nonisothermal catalyst pellet from Weisz and Hicks [112]). 

ILLUSTRATION 12.4 Effectiveness Factor Determination for a Nonisothermal Catalyst Pellet Employed to Effect an Exothermic Reaction... 

Derive the energy balance for a flat-plate catalyst pellet operating under nonisothermal conditions (first-order exothermic reaction). Give a plausible argument why the effectiveness factor can in this case exceed unity. 

Heat Effects in a Catalyst Pellet The Nonisothermal Effectiveness Factor... 

The two ODEs, which are coupled by the two state variables Q and T, generally have to be solved numerically. The resulting concentration profile Q(x) can then be integrated over the pellet volume Vj, as was done in the isothermal case to obtain ffie nonisothermal effectiveness factor E , ... 

A typical, unsealed plot of versus the nonisothermal Thiele modulus is shown in Figure 9.10. Two additional parameters that contain the thermal factors make their appearance here the Arrhenius number EJRT which contains the important activation energy E and the dimensionless parameter P, which reflects the effect due to the heat of reaction and the transport resistances. For p = 0 (i.e., for a vanishing heat of reaction or infinite thermal conductivity), the effectiveness factor reduces to that of the isothermal case. P > 0 denotes an exothermic reaction, and here the rise in temperature in the interior of the pellet is seen to have a significant impact on E which may rise above unity and reach values as high as 100. This means that the overall reaction rate in the pellet is up to 100 times faster than would be the case at the prevailing surface conditions. This is due to the strong exponential dependence of reaction rate on temperature, as expressed by the Arrhenius relation... 

For the nonisothermal pellet being considered, /c, in Eq. 5.38 has been replaced by ka, i.e., the rate constant evaluated at the conditions at the boundary. With the assumption of negligible external mass transfer resistance, C, has also been replaced by the bulk fluid concentration Ct. The internal effectiveness factor for the fresh inner core is rewritten as ... 

The existence of internal resistances complicates the analysis of transport effects for trickle-beds since the pellet cannot necessarily be assumed isothermal. Reactions in which the heat effect is negligible are considered first, and the case of a nonisothermal pellet will be treated in the following section. For arbitrary kinetics kfiC), the internal, isothermal effectiveness factor (Chapter 4) is ... 

Figure 4.5 Nonisothermal effectiveness factors for irreversible first-order reactions in spherical pellets. (Weisz and Hicks 1962. Reprinted with permission from Chemical Engineering Science. Copyright by Pergamon Press, Inc.)...

Figure 2, (a) Geometry of partially deactivated pellet (b) effective activity factor vs. dtffusUmal modulus for aeactivation of nonisothermal pellet (21)... 

Figure 6.5 Effectiveness factor vs. Thiele modulus for nonisothermal first-order chemical reaction within a spherical catalyst pellet.

There are several factors that may be invoked to explain the discrepancy between predicted and measured results, but the discrepancy highlights the necessity for good pilot plant scale data to properly design these types of reactors. Obviously, the reaction does not involve simple first-order kinetics or equimolal counterdiffusion. The fact that the catalyst activity varies significantly with time on-stream and some carbon deposition is observed indicates that perhaps the coke residues within the catalyst may have effects like those to be discussed in Section 12.3.3. Consult the original article for further discussion of the nonisothermal catalyst pellet problem. 

Relative rates of CF and GAS are determined by a complex combination of both thermodynamic and kinetic factors [1]. The pellet deformation due to these processes can be effectively studied in cyclic, nonisothermal experiments. 

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