Effective pore diffusion coefficient

Diffusivity and tortuosity affect resistance to diffusion caused by collision with other molecules (bulk diffusion) or by collision with the walls of the pore (Knudsen diffusion). Actual diffusivity in common porous catalysts is intermediate between the two types. Measurements and correlations of diffusivities of both types are Known. Diffusion is expressed per unit cross section and unit thickness of the pellet. Diffusion rate through the pellet then depends on the porosity d and a tortuosity faclor 1 that accounts for increased resistance of crooked and varied-diameter pores. Effective diffusion coefficient is D ff = Empirical porosities range from 0.3 to 0.7, tortuosities from 2 to 7. In the absence of other information, Satterfield Heterogeneous Catalysis in Practice, McGraw-HiU, 1991) recommends taking d = 0.5 and T = 4. In this area, clearly, precision is not a feature. 

The first thing to notice about these results is that the influence of the micropores reduces the effective diffusion coefficient below the value of the bulk diffusion coefficient for the macropore system. This is also clear in general from the forms of equations (10.44) and (10.48). As increases from zero, corresponding to the introduction of micropores, the variance of the response pulse Increases, and this corresponds to a reduction in the effective diffusion coefficient. The second important point is that the influence of the micropores on the results is quite small-Indeed it seems unlikely that measurements of this type will be able to realize their promise to provide information about diffusion in dead-end pores. 

As a reactant molecule from the fluid phase surrounding the particle enters the pore stmcture, it can either react on the surface or continue diffusing toward the center of the particle. A quantitative model of the process is developed by writing a differential equation for the conservation of mass of the reactant diffusing into the particle. At steady state, the rate of diffusion of the reactant into a shell of infinitesimal thickness minus the rate of diffusion out of the shell is equal to the rate of consumption of the reactant in the shell by chemical reaction. Solving the equation leads to a result that shows how the rate of the catalytic reaction is influenced by the interplay of the transport, which is characterized by the effective diffusion coefficient of the reactant in the pores, and the reaction, which is characterized by the first-order reaction rate constant. 

Figure 10 shows that Tj is a unique function of the Thiele modulus. When the modulus ( ) is small (- SdSl), the effectiveness factor is unity, which means that there is no effect of mass transport on the rate of the catalytic reaction. When ( ) is greater than about 1, the effectiveness factor is less than unity and the reaction rate is influenced by mass transport in the pores. When the modulus is large (- 10), the effectiveness factor is inversely proportional to the modulus, and the reaction rate (eq. 19) is proportional to k ( ), which, from the definition of ( ), implies that the rate and the observed reaction rate constant are proportional to (1 /R)(f9This result shows that both the rate constant, ie, a measure of the intrinsic activity of the catalyst, and the effective diffusion coefficient, ie, a measure of the resistance to transport of the reactant offered by the pore stmcture, influence the rate. It is not appropriate to say that the reaction is diffusion controlled it depends on both the diffusion and the chemical kinetics. In contrast, as shown by equation 3, a reaction in solution can be diffusion controlled, depending on D but not on k. 

In the case that the effective diffusion coefficient approach is used for the molar flux, it is given by N = —Da dci/dr), where Dei = (Sp/Tp)Dmi according to the random pore model. Standard boundary conditions are applied to solve the particle model Eq. (8.1). 

FIG. 20 Effective diffusion coefficients using Michaels model [241], Eq. (43), versus porosity for various ratios of pore lengths. 

The aqueous diffusivities of charged permeants are equivalent to those of uncharged species in a medium of sufficiently high ionic strength. The product DF(r/R) is the effective diffusion coefficient for the pore. It is implicit in k that adsorption of the cations does not occur, so that the fixed surface charges on the wall of the pore are not neutralized. Adsorption is more likely to occur with multivalent cations than with univalent ones. 

For reasons of simplicity, the Thiele modulus will be defined and calculated for a catalyst plate with pore access at both ends of the plate and not at the bottom or top. Note that for most cases in real-life applications the assumptions have to be modified using polar coordinates for the calculations. The Thiele modulus q> is therefore defined as the product of the length of the catalyst pore, /, and the square root of the quotient of the constant of the speed of the reaction, k. divided by the effective diffusion coefficient DeS ... 

Clearly, the elimination of the unknown concentration Cs between Equations (27), (28), and (29-31) is difficult. However, since the effective diffusion coefficient within the pores of carbon is considerably smaller than the free diffusion coefficient in the stagnant film (109) and since the thickness of the stagnant film is usually much smaller than R, it can be assumed that for large specimens the reaction in the solid will be mainly in Zone II lief ore (Cg — Cr) becomes appreciable. Therefore, at low rates of reaction... 

In Fig. 42, the full-width at half maximum of the (narrower) exchange propagator provides an estimate of the effective diffusion coefficient of water molecules moving between the pore space of the catalyst and the inter-particle space of the bed. In this example, the value is 2 x lO- m s which gives a lower limit to the value for the mass transfer coefficient of 4x 10 ms This value was obtained by defining a mass transfer coefficient as Djd where d is a typical distance traveled to the surface of the catalyst that we estimate as half a typical bead dimension (approximately 500 pm). This value of the mass transfer coefficient is consistent with the reaction occurring under conditions of kinetic as opposed to mass transfer control. 

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... 

Cdg Concentration of D in gas phase, moles/volume Cdp Concentration of D in crystallite phase, moles/zeolite pore volume Deff Effective diffusion coefficient, (length) 2/time F Volumetric feed rate to reactor, volume/time H Henry s law-type constant relating gas phase mole fraction to crystallite phase mole fraction... 

One should take into account the specific features of gas diffusion in porous solids when measuring effective diffusion coefficients in the pores of catalysts. The measurements are usually carried out with a flat membrane of the porous material. The membrane is washed on one side by one gas and on the other side by another gas, the pressure on both sides being kept... 

Diffusion and adsorption studies with Boscan VO-porphyrin extracts and pure VO-TPP in C0M0/AI2O3 catalysts have been reported by Morales and co-workers (Galiasso and Morales, 1983 Morales and Galiasso, 1982 Andreu et al., 1981 Morales et al., 1984). The Boscan extract contained up to 30 wt. % vanadyl DPEP and vanadyl etioporphy-rin. Effective diffusion coefficients at 300°C in a catalyst with an average pore diameter of 150 A (A < 0.1) are on the order of 10-5 cm2/sec. Configurational effects are minor for this system. 

Here L (1 — e) is equivalent to the number of binding sites available in the adsorbent bed. If during fluidization L is increased at higher U, (1 — e) is reduced, which is consistent with the fact, that the amount of matrix in the bed, which is available for protein binding, is independent from the fluidization conditions. Thus increased bed expansion does not affect pore diffusion as expressed by Np in spite of longer liquid residence time. The main influence on Np is found from the effective diffusion coefficient De and from the particle diameter dp. 

An important problem in catalysis is to predict diffusion and reaction rates in porous catalysts when the reaction rate can depend on concentration in a non-linear way.6 The heterogeneous system is modeled as a solid material with pores through which the reactants and products diffuse. We assume for diffusion that all the microscopic details of the porous medium are lumped together into the effective diffusion coefficient De for reactant. 

In cases where well-defined pores ranging in sizes from a few hundredths to several hundred microns exist throughout the matrix, the kinetics of drug release can still be described by Equations (1)—(3) provided that an effective diffusion coefficient is used. When the drug diffusion only takes place through the solvent filled porous network, the effective diffusion coefficient is further related to the matrix structure by ... 

Unsteady state diffusion in monodisperse porous solids using a Wicke-Kallenbach cell have shown that non-equimolal diffusion fluxes can induce total pressure gradients which require a non-isobaric model to interpret the data. The values obtained from this analysis are then suitable for use in predicting effectiveness factors. There is evidence that adsorption of the non-tracer component can have a considerable influence on the diffusional flux of the tracer and hence on the estimation of the effective diffusion coefficient. For the simple porous structures used in these tests, it is shown that a consistent definition of the effective diffusion coefficient can be obtained which applies to both the steady and unsteady state and so can be used as a basis of examining the more complex bimodal pore size distributions found in many catalysts. 

For mono-disperse pore size distributions a combination of steady state diffusion and flow permeability measurements can be used to characterize the structural parameters which enable consistent values for tortuosity to be defined. These results can be used to predict the dynamic response of a Wicke-Kallenbach cell to short pulses of a tracer gas having a comparatively high diffusivity and enable a reasonable estimate of the effective diffusion coefficient to be obtained. 

The use of the effective diffusion coefficients in situations where a pressure gradient arises from non-equimolal fluxes, such as when chemical reactions occur, should then be based on the non-isobaric equations. Although this means that the models to be used are more complex, the parameters will be consistent. Where the pore size distribution is not monodisperse, the additional structural parameters which influence the effective diffusion coefficient will make the problem even more complex and requires further study. 

The characteristics of pore structure in polymers is a key parameter in the study of diffusion in polymers. Pore sizes ranging from 0.1 to 1.0 pm (macroporous) are much larger than the pore sizes of diffusing solute molecules, and thus the diffusant molecules do not face a significant hurdle to diffuse through polymers comprising the solvent-filled pores. Thus, a minor modification of the values determined by the hydrodynamic theory or its empirical equations can be made to take into account the fraction of void volume in polymers (i.e., porosity, e), the crookedness of pores (i.e., tortuosity, x), and the affinity of solutes to polymers (i.e., partition coefficient, K). The effective diffusion coefficient, De, in the solvent-filled polymer pores is expressed by ... 

For pore sizes ranging from 50 to 200 A, which are comparable to the sizes of the diffusing solute molecules and are called microporous, the diffusion of solutes may be substantially restricted by polymer materials. A diffusing molecule may be hindered from entering the pores and be chafed against the pores walls. Equation (6.30) incorporates these factors into the effective diffusion coefficient as ... 

The diffusion path is often altered by the presence of solid boundaries. For example, in the subsurface organic chemicals must diffuse around soil and sediment grains. Within soil and sediment grains, organic chemicals must diffuse inside narrow and possibly undulating pores. To account for these effects the effective diffusion coefficient is modified by a restrictivity factor, Kr [-], and a tortuosity factor, r [-], as follows ... 

Figure 6. Example of two-dimensional 10x10 lattice model used to examine the effects of pore blockage on the effective diffusion coefficient in zeolites. Reprinted with permission from Chem. Eng. Sci., vol. 41, p. 703, W. T. Mo and J. Wei, Effective Diffusivity in Partially Blocked Zeolite Catalyst, copyright 1986 [18], Pergamon Press PLC.

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