Internal transport effects effectiveness factors

Tlie reaction is first order with respect to hydrogen peroxide and the effectiveness factor is found to be equal to 0.24. This effectiveness factor accounts only for internal transport effects. Due to the dilute feed of hydrogen peroxide, the operation can be considered isothermal. 

Baddour [26] retained the above model equations after checking for the influence of heat and mass transfer effects. The maximum temperature difference between gas and catalyst was computed to be 2.3°C at the top of the reactor, where the rate is a maximum. The difference at the outlet is 0.4°C. This confirms previous calculations by Kjaer [120]. The inclusion of axial dispersion, which will be discussed in a later section, altered the steady-state temperature profile by less than O.S°C. Internal transport effects would only have to be accounted for with particles having a diameter larger than 6 mm, which are used in some high-capacity modern converters to keep the pressure drop low. Dyson and Simon [121] have published expressions for the effectiveness factor as a function of the pressure, temperature and conversion, using Nielsen s experimental data for the true rate of reaction [119]. At 300 atm and 480°C the effectiveness factor would be 0.44 at a conversion of 10 percent and 0.80 at a conversion of 50 percent. 

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 ... 

The effectiveness factor can be used to account for internal transport effects in the sizing and analysis of heterogeneous catalytic reactors. At this point, it is no longer necessary to assume that internal concentration gradients are negligible. 

Inspection of Fig. 15.3 reveals that while for jo 0.1 nAcm , the effectiveness factor is expected to be close to 1, for a faster reaction with Jo 1 p,A cm , it will drop to about 0.2. This is the case of internal diffusion limitation, well known in heterogeneous catalysis, when the reagent concentration at the outer surface of the catalyst grains is equal to its volume concentration, but drops sharply inside the pores of the catalyst. In this context, it should be pointed out that when the pore size is decreased below about 50 nm, the predominant mechanism of mass transport is Knudsen diffusion [Malek and Coppens, 2003], with the diffusion coefficient being less than the Pick diffusion coefficient and dependent on the porosity and pore stmcture. Moreover, the discrete distribution of the catalytic particles in the CL may also affect the measured current owing to overlap of diffusion zones around closely positioned particles [Antoine et ah, 1998]. 

When a solid acts as a catalyst for a reaction, reactant molecules are converted into product molecules at the fluid-solid interface. To use the catalyst efficiently, we must ensure that fresh reactant molecules are supplied and product molecules removed continuously. Otherwise, chemical equilibrium would be established in the fluid adjacent to the surface, and the desired reaction would proceed no further. Ordinarily, supply and removal of the species in question depend on two physical rate processes in series. These processes involve mass transfer between the bulk fluid and the external surface of the catalyst and transport from the external surface to the internal surfaces of the solid. The concept of effectiveness factors developed in Section 12.3 permits one to average the reaction rate over the pore structure to obtain an expression for the rate in terms of the reactant concentrations and temperatures prevailing at the exterior surface of the catalyst. In some instances, the external surface concentrations do not differ appreciably from those prevailing in the bulk fluid. In other cases, a significant concentration difference arises as a consequence of physical limitations on the rate at which reactant molecules can be transported from the bulk fluid to the exterior surface of the catalyst particle. Here, we discuss... 

The support has an internal pore structure (i.e., pore volume and pore size distribution) that facilitates transport of reactants (products) into (out of) the particle. Low pore volume and small pores limit the accessibility of the internal surface because of increased diffusion resistance. Diffusion of products outward also is decreased, and this may cause product degradation or catalyst fouling within the catalyst particle. As discussed in Sec. 7, the effectiveness factor Tj is the ratio of the actual reaction rate to the rate in the absence of any diffusion limitations. When the rate of reaction greatly exceeds the rate of diffusion, the effectiveness factor is low and the internal volume of the catalyst pellet is not utilized for catalysis. In such cases, expensive catalytic metals are best placed as a shell around the pellet. The rate of diffusion may be increased by optimizing the pore structure to provide larger pores (or macropores) that transport the reactants (products) into (out of) the pellet and smaller pores (micropores) that provide the internal surface area needed for effective catalyst dispersion. Micropores typically have volume-averaged diameters of 50 to... 

Paul Weisz suggested in a lucid note published in 1973 that cells, and indeed even entire organisms, have evolved in a way that maintains unity effectiveness factor [24]. That is, the size of the catalytic assembly is increased in nature as the overall rate at which that assembly operates decreases, and the relationship between characteristic dimension and activity can be well approximated by the observable modulus criterion for reaction limitation. It is possible that Weisz s arguments may fail under process conditions, and internal gradients within a compartment or cell may be important. However, at present it appears that the most important transport limitations and activities in cells are those that operate across cellular membranes. Therefore, to understand and to manipulate key transport activities in cells, it is essential that biochemical engineers understand these membrane transport processes and the factors influencing their operation. A brief outline of some of the important systems and their implications in cell function and biotechnology follows. 

For first-order reactions we can use an overall effectiveness factor to help us analyze diffusion, flow, and reaction in packed beds. We now consider a situation where external and internal resistance to mass transfer to and within the pellet are of the same order of magnitude (Figure 12-9). At steady state, the transport of the reactant(s) from the bulk fluid to the external surface of the catalyst is equal to the net rate of reaction of the reactant within and on the pellet. 

An example would be the dehydration of ethanol to ethylene and its dehydrogenation to acetaldehyde. If both reactions are first order, selectivity is unaffected by internal mass transport the ratio of the rates of reactions, 1 and 2 is k jkj at any position within the pellet. Equation (11-89) cannot be applied separately to the two reactions because of the common reactant A. The development of the effectiveness-factor function would require writing a differential equation analogous to Eq. (11-45) for the total consumption of A by both reactions. Hence k in Eq. (11-89) would be k- + k2 and Fp would be (Tp) -1- (rp)2- Such a development would shed no light on selectivity. 

For process modeling proposes the effective chemical reaction rate has to be expressed as a function of the liquid bulk composition x, the local temperature T, and the catalyst properties such as its number of active sites per catalyst volume c, its porosity e, and its tortuosity t. As discussed in Section 5.4.2, the chemical reaction in the catalyst particles can be influenced by internal and external mass transport processes. To separate the influence of these transport resistances from the intrinsic reaction kinetics, a catalyst effectiveness factor p is introduced by... 

More complex cases, such as cases with S- and P-inhibition kinetics, have been solved numerically by Moo-Young and Kobayashi (1972). Recently criteria have been developed specifically for Monod-, S-inhibited-, Teissier-, and maintenance-type kinetics to quantify and predict diffusional control within whole cells and cell floes (Webster, 1981). Further details concerning the use of the effectiveness factor concept for the quantification of biological processes will be given in Sect. 5.8, presenting simple formal kinetics in the case of internal transport limitation. 

In the literature are many articles on porous diffusion, especially in connection with carrier-bound enzymes or cells (for example. Pitcher, 1978). These are directly connected to the principles expressed in Sect. 4.5 concerning the influence of internal and external mass transport. The results are presented in the same graphical form as Fig. 4.36 in which the effectiveness factor of the reaction rj. is presented as a function of the Thiele modulus. For formulating an appropriate moduls one needs knowledge of the difficult to measure value. The following equation has shown itself useful in that the volume-based reaction rate is obtainable directly from the experimental measurements (Pitcher, 1978) (cf. Equ. 4.74)... 

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 concept of effectiveness developed separately for external or internal transport resistances can be extended to an overall effectiveness factor for treating the general diffusion-reaction problem where both external and internal concentration and temperature gradients exist The overall effectiveness factor, D, is defined for relating the actual global rate to the intrinsic rate, that is, -Ra)p to (-Ra)6- To stun up the definitions for y, 7], and D,... 

The general problem of diffusion-reaction for the overall effectiveness factor D is rather complicated. However, the physical and chemical rate processes prevailing under practical conditions promote isothermal particles and negligible external mass transfer limitations. In other words, the key transport limitations are external heat transfer and internal mass transfer. External temperature gradients can be significant even when external mass transfer resistances are negligibly small. 

As discussed in Sections 4.5 and 4.6, mass and heat transport may influence the effective rate of heterogeneously catalyzed and gas-solid reactions. External profiles of concentration and temperature may be established in the boundary layer around the outer surface of the particles, and internal gradients may develop in the particles. Deviations from the ideal zero-gradient situation are usually considered by effectiveness factors. 

In industrial reactors, particle diameters in the range 1-10 mm are used (Appl, 1999). Thus, pore diffusion of the reactants and of ammonia may influence the effective rate as discussed by Akehata et al. (1961), Bokhoven and van Raayen (1954), and by Jennings and Ward (1989) [see also Appl (1999) and Nielsen, (1971)]. The ratio of the effective rate to the intrinsic (maximum) rate in the absence of internal mass transport restrictions is characterized by the pore effectiveness factor >jpore (Section 4.5.4). Figure 6.1.6 shows values of >jpore determined in a laboratory reactor at different temperatures and particle sizes. For technically relevant temperatures of 400-500 °C and particles up to 10 mm, rjpore is in a range of 1 down to 0.2. 

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