Overall effectiveness factors

The particle effectiveness factor 17 defined by equation 8.5-5 takes into account concentration and temperature gradients within the particle, but neglects any gradients from bulk fluid to the exterior surface of the particle. The overall effectiveness factor y)0 takes both into account, and is defined by reference to bulk gas conditions (cA, Tg) rather than conditions at the exterior of the particle (cAj, Ts)  

as in Section 8.5.4, we treat the isothermal case for ijo, and relate tj0 to 17. may then be interpreted as the ratio of the (observed) rate of reaction with pore diffusion and external mass transfer resistance to the rate with neither of these present. 

and Hicks, J.S., The behaviour of porous catalyst particles in view of internal mass and heat difiusion effects, pp. 

Furthermore, at steady-state, (— rA) is also the rate of mass transfer of A across the exterior film, such mass transfer being in series with the combined intraparticle processes of diffusion and reaction hence, from the definition of kAg, 

On eliminating ( — rA) and cA from the three equations 8.5-46 to -48, for example, by first obtaining an expression for cAs from 8.5-47 and -48, and then substituting for Cp back in equation 8.5-47 and comparing the resulting equation with 8.5-46, we obtain 

both internal and external diffusion are important 

The molar rate of mass transfer from the bulk fluid to the external surface is 

This molar rate of mass transfer to the surface, is equal to the net (total) rate of reaction on and within the pellet  

Ma = r%- (external area I internal area) external area 

internal area mass of catalyst volume of catalyst,, ,  

---

The overall effectiveness factor for the first-order reaction is defined using the bulk gas concentration a. 

We consider the effects of cA and Tseparately, deferring the latter to Section 8.5.5. In focusing on the particle effectiveness factor, we also ignore the effect of any difference in concentration between bulk gas and exterior surface (cAg and cAs.) in Section 8.5.6, we introduce the overall effectiveness factor to take this into account. 

The starting points for the continuity and energy equations are again 21.5-1 and 21.5-6 (adiabatic operation), respectively, but the rate quantity7 (—rA) must be properly interpreted. In 21.5-1 and 21.5-6, the implication is that the rate is the intrinsic surface reaction rate, ( rA)int. For a heterogeneous model, we interpret it as an overall observed rate, (—rA)obs, incorporating the transport effects responsible for the gradients in concentration and temperature. As developed in Section 8.5, these effects are lumped into a particle effectiveness factor, 77, or an overall effectiveness factor, r]0. Thus, equations 21.5-1 and 21.5-6 are rewritten as... 

Recall than both A and B must diffuse into the catalyst for a bimolecular reaction to occur. This can create fairly complex concentration profiles of the two reactants within the catalyst, and the overall effectiveness factor is more complex than with the assumptions above. 

This equation cannot be integrated immediately as f is a function of the substrate concentration C. A possible procedure is calculation of the overall effectiveness factor... 

Also in case of diffusion-limited reactions where the overall effectiveness factor is used to describe the effect of diffusion on the rate of biocatalysis, the mathematics are the same as in the case of the batch reactor. Substitution of Equation (11.56) in Equation (11.23) thus yields ... 

This equation, again, cannot be integrated immediately as t is a function of the substrate concentration C., which changes when going from the entrace to the exit of the plug-flow reactor. The procedure to solve this equation is the calculation of the overall effectiveness factor at n substrate concentration in the interval C to Through this n (... 

It is possible to combine the resistances of internal and external mass transfer through an overall effectiveness factor, for isothermal particles and first-order reaction. Two approaches can be applied. The general idea is that the catalyst can be divided into two parts its exterior surface and its interior surface. Therefore, the global reaction rates used here are per unit surface area of catalyst. 

In the case that the mass transfer effects are not negligible, the required height of the packed bed is greater than that without mass transfer effects. In some practices, the ratio of the packed-bed heights without and with the mass transfer effects is defined as the overall effectiveness factor (-), the maximum value of which is unity. However, if the right-hand side of Equation 7.47 is multiplied it cannot be integrated simply, since is a function of C, U, and other factors. If the reaction is extremely rapid and the liquid phase mass transfer on the particle surface controls the overall rate, then the rate can be estimated by... 

In order to describe the hydrogenation in slurry reactors, and as the reaction is always controlled by mass transfer [1], we must consider both the mass-transfer and the reaction steps. Thus, we must introduce the overall effectiveness factor in the expression of reaction rate [3] ... 

The value of the overall effectiveness factor for a zero order reaction can be calculated with the following relation [4] ... 

By comparing this relationship with the solution for the effectiveness factor in absence of interphase concentration gradients (cq 51), it becomes obvious that the overall effectiveness factor t] can be expressed as the product of separate pore and external (film) effectiveness factors ... 

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 ... 

By substituting from eq 72 into eq 74, can be eliminated. The overall effectiveness factor can then be expressed in a more illustrative form ... 

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. 

Any of the curves in Fig. 10, which refer to different values of the modified Prater number fi, tend to approach a certain limiting value of the Weisz modulus for which the overall effectiveness factor obviously becomes infinitely small. This limit can be easily determined, bearing in mind that the effective reaction rate can never exceed the maximum interphase mass transfer rate (the maximum rate of reactant supply) which is obtained when the surface concentration approaches zero. To show this, we formulate the following simple mass balance, analogous to eq 62 ... 

Hence, as an alternative to Fig. 10, the overall effectiveness factor can also be plotted against the product rjDan, which again contains only measurable quantities, and which, as already stated, can never exceed unity. This leads to the representation shown in Figs. 11 and 12 for two different values of the Arrhenius number. 

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... 

The overall effectiveness factor H [which is not equal to ] is given by... 

---