Diffusion overall effectiveness factor

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. 

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 ... 

Exemplary results of modeling processes inside the catalytic layer are presented in Fig. 9. The solid lines show the dependency of the overall effectiveness factor on the relative distribution of the catalyst between the comers and the side regions. The two cases represent two levels of the first-order rate constants, with the faster reaction in case (b). As expected, the effectiveness factor of the first reaction drops as more catalyst is deposited in the comers. The effectiveness factor for the second reaction increases in case (a) but decreases in case (b). The latter behavior is caused by depletion of B deep inside the catalytic layer. What might be surprising is the rather modest dependency of the effectiveness factor on the washcoat distribution. The explanation is that internal diffusion is not important for slow reactions, while for fast reactions the available external surface area becomes the key quantity, and this depends only slightly on the washcoat distribution for thin layers. The dependence of the effectiveness factor on the distribution becomes more pronounced for consecutive reactions described by Langmuir-Hinshelwood-Hougen-Watson kinetics [26]. 

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. 

Intraparticle Diffusion and External Mass-Transfer Resistance For typical industrial conditions, external mass transfer is important only if there is substantial intraparticle diffusion resistance. This subject has been discussed by Luss, Diffusion-Reaction Interactions in Catalyst Pellets, in Carberry and Varma (eds.), Chemical Reaction and Reactor Engineering, Dekker, 1987. This, however, may not be the case for laboratory conditions, and care must be exerted in including the proper data interpretation. For instance, for a spherical particle with both external and internal mass-transfer limitations and first-order reaction, an overall effectiveness factor r, can be derived, indicating the series-of-resistances nature of external mass transfer followed by intraparticle diffusion-reaction ... 

The term D/ g(d Cj j,/dz ) is used to represent either diffusion and/or dishow to use n and fl persion in the axial direction. Consequently, we shall use the symbol D, for the dispersion coefficient to represent either or both of these cases. We will come back to this form of the diffusion equation when w e discuss dispersion in Chapter 14. The overall reaction rate within the pellet, -r, is the overall rate of reaction within and on the catalyst per unit mass of catalyst. It is a function of the reactant concentration within the catalyst. This overall rate can be related to the rate of reaction of A that would exist if the entire surface were exposed to the bulk concentration through the overall effectiveness factor fi ... 

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. 

Wang and Yang (1991b) have proposed a more realistic model for triphase catalysis in a batch reactor, where they consider mass transfer of reactants in the bulk aqueous and organic phases, diffusion of reactants within the pores of the solid catalyst particle, and intrinsic reactivities of the ion-ex-change and organic reactions at the active sites within the solid catalyst. An apparent overall effectiveness factor of the catalyst is obtained in this case by applying the pseudo-... 

The catalyst efficiency decreases strongly at small mass Biot numbers as seen in Figure 2.30. This is because of the reduced reactant concentration on the external pellet surface. In contrast, external mass transfer influences can be neglected at > 100. In practice, catalytic particles are in the range of several millimeters and the mass Biot numbers are in the order of 100— 200. Hence, the overall effectiveness factor is almost entirely determined by the intraparticle diffusion. 

Expressions for the overall rate of reaction in differentially operated slurry reactors in case of other reaction type can be found in (2l. In the same paper also diagrams are presented for overall effectiveness factors (including also intraparticle diffusion) and different kinetic models are given. 

When the substrate is first transported in a boundary layer surrounding the particle, before diffusing within the catalyst support where reaction occurs, external resistance needs to be considered (Calabro et al, 2008 Truskey et al., 2004). An example is the case of a packed bed bioreactor, where fluid-dynamics play a significant role in the optimization of system performances. In such a case the kinetic contribution has to be expressed in terms of overall effectiveness factor ri y. To estimate it, the mass balance. Equation [1.29], has to be solved by imposing the continuity of mass flux at the wall. For a flat-sheet support it corresponds to ... 

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. 

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... 

Based on the equations derived in Sections 4.5.3 and 4.5.4, we now define an overall effectiveness factor tjoveraih vvhich includes external and internal diffusion resistances. Here we only consider irreversible and reversible first-order isothermal reactions for more complex cases see Baerns et al. (2006) or Westerterp, van Swaaij, and Beenackers (1998). 

Example 4.5.8 Comparison of the overall effectiveness factor with those for pore diffusion and external mass transfer for a single spherical particle and a fixed bed... 

Figure 4.5.34 shows the influence of the term (kmPmicro/Deff.micro) f the overall effectiveness factor r/p (Figure 4.5.34a) and on the factors for macro- and micropore diffusion for different values of dmicro/dp, D ff. micro/Deftmacro = 0.5, and 3 particle diameter of 1 cm. 

The effective reaction rate tm eff (related to the mass of catalyst/solid) already considers all extra- and intraparticle mass and heat transfer effects (Sections 4.5-4.7). Thus the pseudo-homogeneous model does not distinguish between the conditions in the fluid and in the sohd phase, as more sophisticated heterogeneous models do, as discussed, for example, in Baerns ef al. (2006), Froment and Bischoff (1990), and Westerterp, van Swaaij, and Beenackers (1998). Thus, gradients of temperature and concentration within the particle and in the thermal and diffusive boundary layers are combined by the use of an overall effectiveness factor that enables the system of four equations (mass and heat balances for solid and fluid phase) to be replaced by just two equations, Eqs. (4.10.125) and (4.10.126). 

If external diffusion also has to be considered, Eq. (6.9.5) has to be extended. The overall effectiveness factor Jjoveraii is then given by Eq. (4.5.103), which reads for coke burn-off as ... 

In the kinetic control regime (where the overall effectiveness factor t = 1), the rate is directly proportional to the concentration of active sites, L, which is incorporated into the rate constant. In the regime of internal (pore) diffusion control, the rate becomes proportional to and when external diffusion controls the rate there is no influence of L, i.e., there is a zero-order dependence on L. This can be seen by examining equations 4.47 and 4.68. This observation led to the proposal by Koros and Nowak to test for mass transfer limitations by varying L [62]. This concept was subsequently developed further by Madon and Boudart to provide a test that could verify the absence of any heat and mass transfer effects as well as the absence of other complications such as poisoning, channeling and bypassing [63]. 

An overall catalyst effectiveness factor no can be derived to take into account the effective surface wetting and the external reactant supply. This factor is defined as the ratio of the actual conversion rate and that obtained without diffusion resistance. This overall effectiveness factor may be approximated by the weighted average value for the differently wetted fractions of the pellet surface (Capra et al. [8]). If the wetting situation can be simplified into one wetted and one non-wetted surface fraction, with pellet volume fractions equivalent to surface fractions, then ... 

It is clear from the foregoing discussion that the general problem of diffusion-reaction for the overall effectiveness factor is quite involved. Fortunately, however, the physical and chemical processes at work under realistic conditions favor isothermal pellets and negligible external mass transfer resistances. A more detailed examination of this is in order. Combining Eqs. 4.32 and 4.33 results in ... 

The selectivities for any complex reaction networks of first-order reactions can be compactly analyzed using the approach of Wei (1962) discussed earlier. All that is required for the analysis is the expressions for the effectiveness factor vector % which in this case is the overall effectiveness factor representing both deactivation and diffusion effects. In the domain of transformed concentration Ct, the elements of i), rik are given, for instance, for a slab-like pellet by ... 

The ratio of the overall rate of reaction to that which would be achieved in the absence of a mass transfer resistance is referred to as the effectiveness factor rj. SCOTT and Dullion(29) describe an apparatus incorporating a diffusion cell in which the effective diffusivity De of a gas in a porous medium may be measured. This approach allows for the combined effects of molecular and Knudsen diffusion, and takes into account the effect of the complex structure of the porous solid, and the influence of tortuosity which affects the path length to be traversed by the molecules. 

The concentration of gas over the active catalyst surface at location / in a pore is ai [). The pore diffusion model of Section 10.4.1 linked concentrations within the pore to the concentration at the pore mouth, a. The film resistance between the external surface of the catalyst (i.e., at the mouths of the pore) and the concentration in the bulk gas phase is frequently small. Thus, a, and the effectiveness factor depends only on diffusion within the particle. However, situations exist where the film resistance also makes a contribution to rj so that Steps 2 and 8 must be considered. This contribution can be determined using the principle of equal rates i.e., the overall reaction rate equals the rate of mass transfer across the stagnant film at the external surface of the particle. Assume A is consumed by a first-order reaction. The results of the previous section give the overall reaction rate as a function of the concentration at the external surface, a. ... 

The effect of catalyst particle size was investigated by two different catalyst particle size fractions 63-93 pm and 150-250 pm, respectively. The effect of the particle size is very clear as demonstrated by Figure 47.2. The overall hydrogenation rate was for smaller particles 0.17 mol/min/gNi while it was 0.06 mol/min/gNi, for the larger particles, showing the presence of diffusion limitation. This kind of studies can be used to determine the effectiveness factors. The conversion levels after 70 min time-on-stream were 21% and 3%, respectively, for these two cases. 

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. 

The catalyst activity depends not only on the chemical composition but also on the diffusion properties of the catalyst material and on the size and shape of the catalyst pellets because transport limitations through the gas boundary layer around the pellets and through the porous material reduce the overall reaction rate. The influence of gas film restrictions, which depends on the pellet size and gas velocity, is usually low in sulphuric acid converters. The effective diffusivity in the catalyst depends on the porosity, the pore size distribution, and the tortuosity of the pore system. It may be improved in the design of the carrier by e.g. increasing the porosity or the pore size, but usually such improvements will also lead to a reduction of mechanical strength. The effect of transport restrictions is normally expressed as an effectiveness factor q defined as the ratio between observed reaction rate for a catalyst pellet and the intrinsic reaction rate, i.e. the hypothetical reaction rate if bulk or surface conditions (temperature, pressure, concentrations) prevailed throughout the pellet [11], For particles with the same intrinsic reaction rate and the same pore system, the surface effectiveness factor only depends on an equivalent particle diameter given by... 

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... 

If, however, both reactions were influenced by intraparticle diffusion effects, the rate of reaction of a particular component would be given by the product of the intrinsic reaction rate, fecg, and the effectiveness factor, Tj. Substituting eqn. (6) for the effectiveness factor gives (for a first-order isothermal reaction) the overall rate as 0tanh< >. As is often the case, the molecular weights of the diffusing reactants are similar and can be... 

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