Diffusion effects reactant concentration

A hydrocarbon is cracked using a silica-alumina catalyst in the form of spherical pellets of mean diameter 2.0 mm. When the reactant concentration is 0.011 kmol/m3, the reaction rate is 8.2 x 10"2 kmol/(m3 catalyst) s. If the reaction is of first-order and the effective diffusivity De is 7.5 x 10 s m2/s, calculate the value of the effectiveness factor r). It may be assumed that the effect of mass transfer resistance in the. fluid external Lo the particles may be neglected. 

Table 1.6 Characteristic quantities to be considered for micro-reactor dimensioning and layout. Steps 1, 2, and 3 correspond to the dimensioning of the channel diameter, channel length and channel walls, respectively. Symbols appearing in these expressions not previously defined are the effective axial diffusion coefficient D, the density thermal conductivity specific heat Cp and total cross-sectional area S, of the wall material, the total process gas mass flow m, and the reactant concentration Cg [114].

If diffusion of reactants to the active sites in pores is slower than the chemical reaction, internal mass transfer is at least partly limiting and the reactant concentration decreases along the pores. This reduces the reaction rate compared to the rate at external surface conditions. A measure of the reaction rate decrease is the effectiveness factor, r, which has been defined as ... 

Now consider the other extreme condition where diffusion is rapid relative to chemical reaction [i.e., hT( 1 — a) is small]. In this situation the effectiveness factor will approach unity for both the poisoned and unpoisoned reactions, and we must retain the hyperbolic tangent terms in equation 12.3.124 to properly evaluate Curve C in Figure 12.11 is calculated for a value of hT = 5. It is apparent that in this instance the activity decline is not nearly as sharp at low values of a as it was at the other extreme, but it is obviously more than a linear effect. The reason for this result is that the regions of the catalyst pore exposed to the highest reactant concentrations do not contribute proportionately to the overall reaction rate because they have suffered a disproportionate loss of activity when pore-mouth poisoning takes place. 

Schematic representation of reactant concentration profiles in various global rate regimes. I External mass transfer limits rate. II Pore diffusion limits rate. Ill Both mass transfer effects are present. IV Mass transfer has no influence on rate.

The study of the intra-phase mass transfer in SCR reactors has been addressed by combining the equations for the external field with the differential equations for diffusion and reaction of NO and N H 3 in the intra-porous region and by adopting the Wakao-Smith random pore model to describe the diffusion of NO and NH3 inside the pores [30, 44]. The solution of the model equations confirmed that steep reactant concentration gradients are present near the external catalyst surface under typical industrial conditions so that the internal catalyst effectiveness factor is low [27]. 

Thus we can say that if the Thiele modulus is much less than unity, there is no pore diffusion effect on the reaction rate because the reactant concentration remains at Cas down the pore, but if the Thiele modulus is much greater than unity, then the rate is proportional to l/[Pg.287]

This sieve effect cannot be considered statically as a factor that only determines the amount of accessible acid groups in the resin in such a way that the boundary between the accessible and non-accessible groups would be sharp. It must be treated dynamically, i.e. the rates of the diffusion of reactants into the polymer mass must be taken into account. With the use of the Thiele s concept about the diffusion into catalyst pores, the effectiveness factors, Thiele moduli and effective diffusion coefficients can be determined from the effect of the catalyst particle size. The apparent rates of the methyl and ethyl acetate hydrolysis [490] were corrected for the effect of diffusion in the resin by the use of the effectiveness factors, the difference in ester concentration between swollen resin phase and bulk solution being taken into account. The intrinsic rate coefficients, kintly... 

The burning velocity is controlled by the rates of chemical reactions and transport processes in the reaction zone, and chemical reaction rates vary exponentially with temperature and depend on the partial pressures of the reactants (concentrations). These arc strongly influenced by diffusion (molecular and/or eddy). Any treatment, therefore, which ignores these effects would not be complete. 

Under the ribs, the species diffusion in the direction orthogonal to the cell plane, i.e. cross-plane diffusion, is obviously impossible. Hence, the reactant concentrations on the electrode/electrolyte site under the ribs are driven only by in-plane diffusion. Due to the strong diffusion property of H2, in-plane diffusion allows H2 to penetrate under the ribs, while in the case of O2, a relevant concentration reduction is noticed. The effect of the ribs is significant at all operating conditions, but it becomes predominant at high fuel utilization (not shown in the figures). 

Figure 13 shows the potential and concentration distributions for different values of dimensionless potential under conditions when internal pore diffusion (s = 0.1) and local mass transport (y = 10) are a factor. As expected the concentration and relative overpotential decrease further away from the free electrolyte (or membrane) due to the combined effect of diffusion mass transport and the poor penetration of current into the electrode due to ionic conductivity limitations. The major difference in the data is with respect to the variation in reactant concentrations. In the case when an internal mass transport resistance occurs (y = 10) the fall in concentration, at a fixed value of electrode overpotential, is not as great as the case when no internal mass transport resistance occurs. This is due to the resistance causing a reduction in the consumption of reactant locally, and thereby increasing available reactant concentration the effect of which is more significant at higher electrode overpotentials. 

Reactions in biphasic systems can take place either at the interface or in the bulk of one of the phases. The reaction at the interface depends on the reactants meeting at the interface boundary. This means, the interface area as well as the diffusion rate across the bulk of the phase plays an important role. On the contrary, in reactions that take place in the bulk phase, the reactants have to be transferred first through the interface before the reactions take place. In this case, the rate of diffusion across the interface is an important factor. Diffusion across the interface is more complicated than the diffusion across a phase, as the mass transfer of the reactant across the interface must be taken into account. Hence, the solubility of the reactants in each phase has to be considered, as this has an effect on diffusion across the interface. In a system where the solubility of a reactant is the same in both phases, the reactant diffuses from the concentrated phase to the less concentrated phase across the interface. This takes into account the mass transfer of the reactant from one phase into the other through the interface. The rate of diffusion J in such systems is described in Equation 4.1, where D is the diffusion coefficient, x is the diffusion distance and l is the interface thickness (Figure 4.9). 

In relation 8.6 - rA is the reaction rate in kmol/(kg catalyst s), pc density of solid catalyst, R particle radius, I ) i diffusion coefficient in liquid and CAs reactant concentration at the catalyst surface. If CWp 1 there are no diffusion limitations, but if CWP 1 the catalyst effectiveness is severely affected. 

From this figure, it can be concluded that the reduction of the effectiveness factor at large values of becomes more pronounced as the Biot number is decreased. This arises from the fact that the reactant concentration at the external pellet surface drops significantly at low Biot numbers. However, a clear effect of interphase diffusion is seen only at Biot numbers below 100. In practice, Bim typically ranges from 100 to 200. Hence, the difference between the overall and pore effectiveness factor is usually small. In other words, the influence of intraparticle diffusion is normally by far more crucial than the influence of interphase diffusion. Thus, in many practical situations the overall catalyst efficiency may be replaced by the pore efficiency, as a good approximation. 

Provided the interphase mass transfer resistance (1 /k() is sufficiently large, the reactant concentration at the external pellet surface will drop almost to zero. Thus, we may neglect the surface concentration cs compared to the bulk concentration q>. With cs — 0 in eq 115, it is obvious that in this case the reaction will effectively follow a first-order rate law. Moreover, it is also clear that the temperature dependence of the effective reaction rate is controlled by the mass transfer coefficient k(. This exhibits basically the same temperature dependence as the bulk diffusivity Dm, since the boundary layer thickness 5 is virtually not affected by temperature (kf = Dm/<5). Thus, we have the rule of thumb that the effective activation energy of an isothermal, simple, nth order, irreversible reaction will be less than 5-lOkJmor1 when the overall reaction rate is controlled by interphase diffusion. 

In the second case, when n n2, the point selectivity will be a function of the reactant concentration (see cq 131). Therefore, the apparent selectivity may be influenced by intraparticle diffusion, since a decrease of the reactant concentration towards the pellet center will have a different effect on the observable rates of... 

To keep the mathematics as simple as possible, we treat the catalyst pellet as an infinitely flat plate (b = 0 in eq 139). The solution of eq 139 depends on whether the reactant concentration will drop to zero at some point Xo inside the pellet, in the case that the reaction rate is strongly influenced by diffusion, or will be finite everywhere in the pellet interior, if there is only a moderate effect of diffusion. This is a general feature of zero-order reactions which arises from the assumption that the reaction will proceed at a constant rate until the reactant is completely exhausted. 

Most textbook discussions of effectiveness factors in porous, heterogeneous catalysts are limited to the reaction A - Products where the effective diffusivity of A is independent of reactant concentration. On the other hand, it is widely recognized by researchers in the field that multicomponent single reaction systems can be handled in a near rigorous fashion with little added complexity, and recently methods have been developed for application to multiple reactions. Accordingly, it is the intent of the present communication to help promote the transfer of these methods from the realm of the chemical engineering scientist to that of the practitioner. This is not, however, intended to be a comprehensive review of the subject. The serious reader will want to consult the works of Jackson, et al. 

For exothermic reactions (fi > 0) a sufficient temperature rise due to heat transfer limitations may increase the rate constant Ay. and this increase may offset the diffusion limitation on the rate of reaction (the decrease in reactant concentrations CA), leading to a larger internal rate of reaction than at surface conditions CAs. This, eventually, leads to 17 > 1. As the heat of reaction is a strong function of temperature, Eq. (9.24) may lead to multiple solutions and three possible values of the effectiveness factor may be obtained for very large values of /I and a narrow range of catalytic reactions, (3 is usually <0.1, and therefore, we do not observe multiple values of the effectiveness factor. The criterion... 

The effective diffusion coefficient may become a function of the reactant concentration and thus it will depend upon the place inside the catalyst pellet. 

Sylvester and Pitayagulsarn53,54 considered combined effects of axial dispersion, external diffusion (gas-liquid, liquid-solid), intraparticle diffusion, and the intrinsic kinetics (surface reaction) on the conversion for a first-order irreversible reaction in an isothermal, trickle-bed reactor. They used the procedure developed by Suzuki and Smith,51,52 where the zero, first, and second moments of the reactant concentration in the effluent from a reactor, in response to a pulse introduced, are taken. The equation for the zero moment can be related to the conversion X, in the form... 

The Effectiveness Factor. The effectiveness factor used in Eq. (1) depends on the reactant concentration in the catalyst pores as defined by Eq. (4), which is affected by diffusion in a porous medium. 

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