Effectiveness factor nonisothermal case

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... 

Therefore, Equations 8.48 and 8.49 can be combined into one equation for concentration only. The effectiveness factor for the case considered can be calculated with the technique described in Chapter 7. It is important to stress that the effectiveness factor changes along the reactor because parameters of the reaction rate expression, Equation 6.18, e and a depend on the surface concentration and temperature. The calculated modified effectiveness factors for nonisothermal first-order reaction at different conversions = (1- CAJCa) are shown in Figure 8.9 versus the ratio of inner and outer diameters of the hollow cylinder. The parameters chosen for the calculation are ... 

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... 

Since the major thermal resistances in nonisothermal reaction systems are encountered in the boundary layer, while the major mass transfer resistances occur within the particle, we can entertain some simplification of the overall effectiveness factor problem we have been considering. This simplified model envisions interphase temperature gradients and intraphase concentration gradients only. For this case... 

Figure 11.9.a-l shows the relation between the effectiveness factors rj and tjo and the modulus [104, 108]. This relation can only be obtained by numerical integration of the system, Eqs. 11.9.a-l to 11.9.a-8, except for the cases already mentioned. With isothermal situations ri tends to a limit of 1 as 0 increases, with nonisothermal situations, however, r/ or tic, may exceed 1. Curve 1 corresponds to the t] concept, curves 2, 3, and 4 to r/o. The dotted part of curve 4 corresponds to a region of conditions within which multiple steady states inside the catalyst are... 

Theoretical work has been done on the effectiveness factor for nonisothermal particles. Fig. 3.13.1-1 shows the results of the computations by Weisz and Hicks [1962] for y = EIRT = 20. For > 0.1, that is, for sufficiently exothermic reactions, the effectiveness factor can exceed the value of 1. In such a case the temperature rise, which increases the value of the rate constant, would more than offset the decrease in reactant concentration Cas, so that fA averaged over the particle exceeds that at surface conditions. The converse is true for endothermic reactions. 

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. 

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... 

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 ... 

Use of the CSTR sequence as a model for nonideal reactors has been criticized on the basis that it lacks certain aspects of physical reality, such as the absence of backward communication between the individual mixing cell units. Such may be the case nonetheless the mathematical simplicity of the approach makes it very attractive, particularly for systems with complex kinetics, nonisothermal effects, or other complicating factors. 

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