Effectiveness factors diffusion-limited regime

The study of transport schemes by means of Green function renormalization reveals the analogies existing between sorption properties and the scaling of the effectiveness factor vs the Thiele modulus in a diffusion-limited regime [9-10]. 

Hence, it is not possible to redefine the characteristic length such that the critical value of the intrapellet Damkohler number is the same for all catalyst geometries when the kinetics can be described by a zeroth-order rate law. However, if the characteristic length scale is chosen to be V cataiyst/ extemai, then the effectiveness factor is approximately A for any catalyst shape and rate law in the diffusion-limited regime (A oo). This claim is based on the fact that reactants don t penetrate very deeply into the catalytic pores at large intrapellet Damkohler numbers and the mass transfer/chemical reaction problem is well described by a boundary layer solution in a very thin region near the external surface. Curvature is not important when reactants exist only in a thin shell near T] = I, and consequently, a locally flat description of the problem is appropriate for any geometry. These comments apply equally well to other types of kinetic rate laws. 

Two expressions are given below to calculate the effectiveness factor E. The first one is exact for nth-order irreversible chemical reaction in catalytic pellets, where a is a geometric factor that accounts for shape via the surface-to-volume ratio. The second expression is an approximation at large values of the intrapellet Damkohler number A in the diffusion-limited regime. 

When the intrapellet Damkohler number for reactant A is large enough and the catalyst operates in the diffusion-limited regime, the effectiveness factor is inversely proportional to the Damkohler number (i.e., Aa, intrapeiiet)- Under these conditions, together with a large mass transfer Peclet number which minimizes effects due to interpellet axial dispersion, the following scaling law is valid ... 

Simulations in the Diffusion-Limited Regime p = 10. None of the profiles listed in Table 23-9 is more effective than uniform deposition at any aspect ratio when the Damkohler number p = 10. This is understandable in the diffusion-controlled regime where the rate of chemical reaction on the catalytic surface is not the primary factor that governs the conversion of reactants to products. However, some profiles perform poorly relative to uniform deposition when diffusion of reactants toward the active surface is slow. Differences among the profiles are more pronounced at higher aspect ratios. A qualitative summary at each aspect ratio is provided in Table 23-10. Identification numbers are underlined when... 

First to be considered are the limiting forms of this dependence. In the kinetic regime, that is without any diffusion limitations (2 1, and hence usually An0 1 and Anx 1), the effectiveness factor approaches unity and the mean reaction rate according to Equation 8.1 is proportional to the specific surface area ... 

After drying and reduction, the Pd-Ag/C catalysts are composed of bimetallic Eilloy nanoparticles ( 3 nm). CO chemisorption coupled to TEM and XRD analysis showed that that, for catalysts 1.5% wt. in each metal, the bulk composition of the alloy is close to 50% in each metal, whereas the surface is 90% in Ag and 10% in Pd [9]. Mass transfer limitations can be detected by testing the same catalyst with various pellet sizes [18]. Indeed, if the reactants diffusion is slow due to small pore sizes, the longer the distance between the pellet surface and the metal particle, the larger the influence of the difiusion rate on the apparent reaction rate. Pd-Ag catalysts with various pellet sizes were thus tested in hydrodechlorination of 1,2-dichloroethane. Results were compared to those obtained with a Pd-Ag/activated charcoal catalyst. Fig. 4 shows the evolution of the effectiveness factor of the catalysts, i.e. the ratio between the apparent reaction rate and the intrinsic reaction rate, as a function of the pellet size. The intrinsic reaction rate was considered equal to the reaction rate obtained with the smallest pellet size. When rf = 1, no diffusional limitations occur, and the catalyst works in chemical regime. When j < 1, the observed reaction rate is lower than the intrinsic reaction rate due to a slow diffusion of the reactants and products and the catalyst works in diffusional regime [18]. 

Fig. 4.32 represents the dependence of the effectiveness factor rj. on the modulus gxt at different values of s/K. One observes the kinetic regime with rj 1 and the diffusion regime, which approaches a limiting value of... 

The same asymptote is shown in Figure 3.4 for the region mainly limited by intraphase diffusion. The regime of strong interphase mass transfer resistance is also depicted in the same representation for two values of the mass Biot number. The asymptotic behavior of the effectiveness factor in this limit 1 and low is obtained from (power-law... 

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]. 

The kinetic model used is shown in Figure 6.15 (Chapter 6). It consists of five lumps and was previously reported in the literature (Sfinchez et al., 2005). The lumps are gases, naphtha (IBP— 204°C), middle distillates (204°C-343°C), VGO (343 C-538 C), and unconverted vacuum residue (538°C+). As mentioned earlier, experiments were performed under a disguised kinetic regime due to the presence of internal diffusion limitations this made necessary the introduction of the effectiveness factor (n) since internal gradients are indeed present. 

Heat transfer is an extremely important factor in CVD reactor operation, particularly for LPCVD reactors. These reactors are operated in a regime in which the deposition is primarily controlled by surface reaction processes. Because of the exponential dependence of reaction rates on temperature, even a few degrees of variation in surface temperature can produce unacceptable variations in deposition rates. On the other hand, with atmospheric CVD processes, which are often limited by mass transfer, small susceptor temperature variations have little effect on the growth rate because of the slow variation of the diffusion with temperature. Heat transfer is also a factor in controlling the gas-phase temperature to avoid homogeneous nucleation through premature reactions. At the high temperatures (700-1400 K) of most... 

Numerous studies on inorganic membranes have shown that the separation factor is limited and not far from that predicted by the Knudsen diffusion. This primarily reflects the current status of material developments of inorganic membranes. The majority of commercial and developmental inorganic membranes contain macropoies or mesopores. These pore sizes fall within the dominant regime of Knudsen diffusion which is of limited use for many gas separation applications from the standpoint of process economics. To break this barrier, finer pore sizes or transport mechanisms more effective than Knudsen diffusion for gas separation is essential. 

The analysis in Eq. (24) does not take into account the transport limitations. Therefore these predictions of the diffuse-layer effects are only valid for the interval of the potentials in which the rate of the process is determined by the electron-transfer step. If this kinetic regime corresponds to high negative electrode charges, the Kfi factor in Eq. (25) varies slowly and the deviations from the Tafel behavior are rather weak. On the contrary, the anion electroreduction wave that starts at positive electrode charges may demonstrate a complicated curve a usual behavior within this potential range, with approach to the limiting current at less positive potentials. 

Liquid diffusivity is smaller than gas diffusivity by a factor of two orders of magnitude. Considering the typical size (a few mm) of a channel (internal side of a tubular membrane) or of the cavity on the other membrane side (between the membrane and the housing wall) the typical density and viscosity of solvents used in multi-phase reactions, high liquid phase flow rates have to be used in order to be in a turbulent regime (a high Reynolds number) which minimizes the effect of the external mass transfer limitations. 

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