Thiele modulus overall effectiveness factor

To derive an equation for determining the overall effectiveness factor, we first introduce a Thiele modulus which is related to the unknown surface temperature Ts ... 

Again, eq 75 cannot be used immediately to calculate the overall effectiveness factor, since the modulus fi, which is related to the unknown catalyst temperature, can only be determined when the overall efficiency has been specified (see eqs 71 and 72). Therefore, both sides of eq 74 arc multiplied by 2, resulting in an expression which relates the Weisz modulus ift to the modulus . Then, for a given value of fi, the corresponding value of ij/ is calculated, and from ij/ the unknown catalyst temperature 0S (eq 71). This temperature is substituted into eq 72 to obtain the corresponding value of the Thiele modulus <)>. Dividing ij/ by fi finally yields the overall effectiveness factor which is then plotted against i/f. 

However, whereas effectiveness factors above unity under nonisothcrmal conditions can be explained quite easily, the observation of multiple steady states is a new and unexpected feature. These arise at small values of provided the reaction is substantially exothermic and, additionally, has a high activation energy. This means that, for a single value of the Thiele modulus, several possible solutions for the steady state overall effectiveness factor may exist (operating points), usually up to three. The middle operating point is normally unstable. Whenever the temperature and/or the... 

Closure After completing this chapter, the reader should be able to derive differential equations describing diffusion and reaction, discuss the meaning of the effectiveness factor and its relationship to the Thiele modulus, and identify the regions of mass transfer control and reaction rate control. The reader should be able to apply the Weisz-Prater and Mears criteria to identify gradients and diffusion limitations. These principles should be able to be applied to catalyst particles as well as biomaierial tissue engineering. The reader should be able to apply the overall effectiveness factor to a packed bed reactor to calculate the conversion at the exit of the reactor. The reader should be able to describe the reaction and transport steps in slurry reactors, trickle bed reactors, fluidized-besd reactors, and CVD boat reactors and to make calculations for each reactor. 

These equations were nondimensionalized in terms of physically relevant parameters like the Thiele modulus and the Biot number for mass transfer, and an expression for overall effectiveness factor derived. An apparent rate constant for the organic reaction was derived as a function of both internal and external mass-transfer resistances... 

Equations for the overall effectiveness factor and the modified Thiele modulus are shown in Table 8.1 for several types of rate equations (but all dependent upon A alone). Figure 8.10 gives some idea as to the nature of the overall effectiveness factor for the first-order case treated above. 

Using Bischoflf s approximation (1965), implicit expressions can then be derived for the generalized Thiele modulus and overall effectiveness factor E. Based on these, plots can be prepared of E versus (the familiar Thiele modulus for the catalyst) for different values of cta = a[(1 + aM1 )/ w) and Kp [A. A few representative plots are shown in Figure 17.3. 

For a catalyst in the form of a flat plate the effectiveness factor is given by i]p = tanh cpijcpi (Equation 2.174) and the overall effectiveness factor can be expressed as a function of the Thiele modulus and the Biot number. 

The relationship shown in Equation 2.207 suffers from the fact that the Thiele modulus must be specified to estimate the catalyst efficiency. This is, in general, not possible as the intrinsic kinetics is not known. It is, therefore, more convenient to relate the overall effectiveness factor to the Weisz modulus, which is based only on observable parameters. 

The methods used for modeling-supported PTC systems are all based on the standard equations developed for porous catalysts in heterogeneous catalysis (Chapter 6). These are expressed in terms of an overall effectiveness factor that accounts both for the mass transfer resistances outside the supported catalyst particles (film diffusion resistance, expressed as a Biot number) and within them (intraparticle diffusional resistance, expressed in terms of a Thiele modnlns). Then, for any given solid shape, the catalytic effectiveness factor can be derived as a function of the Thiele modulus A. Thus, for a spherical support solid, we have... 

Here the attention is focused on the behavior of the overall effectiveness factor as a function of an overall Thiele modulus. For a given particle size,... 

If intraparticle diffusion controls the overall reaction rate, the Thiele modulus will be large (0 > 2) and then the effectiveness factor 77 is approximately 0. From eqn. (10) defining the Thiele modulus, it follows that, for a given reaction, the effectiveness factor will be... 

When the pore-diffusion is limiting, the effectiveness factor is rj = Thiele modulus, (p = rtJ3 (k/DP) 12, m being the radius of the catalyst particle, and DP the coefficient of pore-diffusion. The overall rate of the process depends then on the reciprocal modulus,

[Pg.77]

A measure of the absence of internal (pore diffusion) mass transfer limitations is provided by the internal effectiveness factor, t, which is defined as the ratio of the actual overall rate of reaction to the rate that would be observed if the entire interior surface were exposed to the reactant concentration and temperature existing at the exterior of the catalyst pellet. A value of 1 for rj implies that all of the sites are being utilized to their potential, while a value below, say, 0.5, signals that mass transfer is limiting performance. The value of rj can be related to that of the Thiele modulus, 4>, which is an important dimensionless parameter that roughly expresses a ratio of surface reaction rate to diffusion rate. For the specific case of an nth order irreversible reaction occurring in a porous sphere,... 

An effectiveness factor is then introduced, defined as the ratio between the overall reaction rate and the rate one would obtain if substrate concentration within the gel were uniform at the feed level. An overall substrate mass balance relates this factor to the Thiele modulus, 35... 

Compute the effectiveness factor versus Thiele modulus, plot this result and compare to Figure 7.17. Comment p.n the effect of pellet geometry on the overall reaction rale. 

For Pshooting method is equal to 1) such interval of the values of Thiele modulus exists in which the effectiveness factor T] = ro /rs exceeds unity. Consequently, the presence of an internal resistance to mass transport may lead to serious increase in the overall rate of the isothermal and non-isothermal, heterogeneous autocatalytic reactions compared to the values obtained for the vanishing or very large resistance. 

The same cannot be said under mass transfer control, where the surface distribution of concentration and reaction rates included in the Thiele modulus also depend on the mass transfer problem in the channel. This only happens for nonlinear kinetics and gives origin the distinction between an overall regime (with low wall concentration but still allowing for strong gradients to develop in the coating) and a purely interphase resistance (where channel concentration is so low that the effectiveness factor may even approach 1). 

Effectiveness factor approach (rj-approach) accounts for diffusion limitations in the washcoat. rj-approach is based on the assumption that one target species determines overall reactivity (Deutschmann, 2008). An effectiveness factor for a first-order reaction is calculated for the chosen species based on the dimensionless Thiele modulus (< ) (Hayes et al., 2007, 2012), and all reaction rates are multipfied by this factor at the species governing equation at the gas-surface interface. is calculated as... 

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... 

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