Diffusivity combined

Karonen JO, Vanninen RL, Liu Y, 0stergaard L, Kuikka JT, Nuutinen J, Vanninen EJ, Partanen PL, Vainio PA, Korbonen K, Perkio J, Roivainen R, Sivenius J, Aronen HJ. Combined diffusion and perfusion MRI with correlation to single-photon emission CT in acute ischemic stroke. Ischemic penumbra predicts infarct growth. Stroke 1999 30 1583-1590. 

Darby DG, Barber PA, Gerraty RP, Desmond PM, Yang Q, Parsons M, Li T, Tress BM, Davis SM. Pathophysiological topography of acute ischemia by combined diffusion-weighted and perfusion MRI. Stroke 1999 30 2043-2052. 

Obviously, there will be a range of pressures or molecular concentrations over which the transition from ordinary molecular diffusion to Knudsen diffusion takes place. Within this region both processes contribute to the mass transport, and it is appropriate to utilize a combined diffusivity (Q)c). For species A the correct form for the combined diffusivity is the following. 

The combined diffusivity is, of course, defined as the ratio of the molar flux to the concentration gradient, irrespective of the mechanism of transport. The above equation was derived by separate groups working independently (8-10). It is important to recognize that the molar fluxes (Ni) are defined with respect to a fixed catalyst pellet rather than to a plane of no net transport. Only when there is equimolar counterdiffusion, do the two types of flux definitions become equivalent. For a more detailed discussion of this point, the interested readers should consult Bird, Stewart, and Lightfoot (11). When there is equimolal counterdiffusion NB = —NA and... 

In the case of nonequimolal cpunterdiffusion, equation 12.2.6 suffers from the serious disadvantage that the combined diffusivity is a function of the gas composition in the pore. This functional dependence carries over to the effective diffusivity in porous catalysts (see below), and makes it difficult to integrate the combined diffusion and transport equations. As Smith (12) points out, the variation of 2C with composition (YA) is not usually strong, and it has been an almost universal practice to use a composition independent form of Q)c (12.2.8) in assessing the importance of intrapellet diffusion. In fact, the concept of a single effective diffusivity loses its engineering utility if the dependence on composition must be retained. 

ILLUSTRATION 12.1 ESTIMATION OF COMBINED DIFFUSIVITY FOR CUMENE IN A CRACKING CATALYST... 

For the purposes of this illustrative example, we wish to calculate the combined and effective diffusivities of cumene in a mixture of benzene and cumene at 1 atm total pressure and 510 °C within the pores of a typical TCC (Thermofor Catalytic Cracking) catalyst bead. For our present purposes, the approximation to the combined diffusivity given by equation 12.2.8 will be sufficient because we will see that the Knudsen diffusion term is the dominant factor in determining the combined diffusivity. 

The measured value of k Sg is 0.716 cm3/(sec-g catalyst) and the ratio of this value to k ltTueSg should be equal to our assumed value for the effectiveness factor, if our assumption was correct. The actual ratio is 0.175, which is at variance with the assumed value. Hence we pick a new value of rj and repeat the procedure until agreement is obtained. This iterative approach produces an effectiveness factor of 0.238, which corresponds to a differs from the experimental value (0.17) and that calculated by the cylindrical pore model (0.61). In the above calculations, an experimental value of eff was not available and this circumstance is largely responsible for the discrepancy. If the combined diffusivity determined in Illustration 12.1 is converted to an effective diffusivity using equation 12.2.9, the value used above corresponds to a tortuosity factor of 2.6. If we had employed Q)c from Illustration 12.1 and a tortuosity factor of unity to calculate eff, we would have determined that rj = 0.65, which is consistent with the value obtained from the straight cylindrical pore model in Illustration 12.2. 

Thus a zero-order reaction appears to be 1/2 order and a second-order reaction appears to be 3/2 order when dealing with a fast reaction taking place in porous catalyst pellets. First-order reactions do not appear to undergo a shift in reaction order in going from high to low effectiveness factors. These statements presume that the combined diffusivity lies in the Knudsen range, so that this parameter is pressure independent. 

This equation gives the differential yield of V for a porous catalyst at a point in a reactor. For equal combined diffusivities and the case where hT approaches zero (no diffusional limitations on the reaction rate), this equation reduces to equation 9.3.8, since the ratio of the hyperbolic tangent terms becomes y/k2 A/ki v As hT increases from about 0.3 to about 2.0, the selectivity of the catalyst falls off continuously. The selectivity remains essentially constant when both hyperbolic tangent terms approach unity. This situation corresponds td low effectiveness factors and, in tliis case, equation 12.3.149 becomes... 

At this point it is instructive to consider the possible presence of intraparticle and external mass and heat transfer limitations using the methods developed in Chapter 12. In order to evaluate the catalyst effectiveness factor we first need to know the combined diffusivity for use... 

To employ this relation one needs to estimate a tortuosity factor in order to convert the combined diffusivity to an effective diffusivity by equation 12.2.9. If we assume a value of 3, then... 

From Figure 12.2 it is evident that the catalyst effectiveness factor for isothermal operation will be approximately 0.47. At higher temperatures the effectiveness factor will be smaller because the rate constant will increase more rapidly with temperature than will the combined diffusivity. However, the reactions in question are quite... 

Figure 9.7 shows concentration profiles schematically for A and B according to the two-film model. Initially, we ignore the presence of the gas film and consider material balances for A and B across a thin strip of width dx in the liquid film at a distance x from the gas-liquid interface. (Since the gas-film mass transfer is in series with combined diffusion and reaction in the liquid film, its effect can be added as a resistance in series.)... 

As depicted in Figure 2.8, mass transport of substrate from the bulk water phase takes place through a fluid boundary layer (liquid film) and into a biofilm followed by a combined diffusion and utilization of the substrate in the biofilm. 

In the particular case dealt with now (fully labile complexation), due to the linearity of a combined diffusion equation for DmCm + DmlL ml, the flux in equation (65) can still be seen as the sum of the independent diffusional fluxes of metal and complex, each contribution depending on the difference between the surface and bulk concentration value of each species. But equation (66) warns against using just a rescaling factor for the total metal or for the free metal alone. In general, if the diffusion is coupled with some nonlinear process, the resulting flux is not proportional to bulk-to-surface differences, and this complicates the use of mass transfer coefficients (see ref. [II] or Chapter 3 in this volume). 

The continuity equation for combined diffusion and mass flow is obtained by combining Equations (2.2) and (2.4) ... 

Birks [6] has commented on several approaches to combined diffusion and long-range transfer which do not involve the use of a diffusion equation. These are generally ill-defined and require ad hoc assumptions. For instance, the Perrin [133] active sphere analysis is quite satisfactory once the assumptions are established from experiment [134]. 

Similar coefficients have been introduced by Taylor (1953) for combined diffusion and convection, and by Amundson Aris (1957) for the packed bed. Their value is enhanced by the fact that ultimately the disturbance tends to have the shape of a Gaussian distribution, as is shown in 5. It is easy to see, however, that the change in the third moment m3 is also proportional to x, and hence the absolute skewness m2lm2 is 0(jc 1) for large x, so that the disturbance tends to become more symmetrical. 

If tj - 1 the reaction is not, or not significantly, influenced by pore diffusion. If tj pore diffusion is the sole dominating rate-limiting step. For the determination of Tj, the combined diffusion and reaction equation has to be solved. With a sequential model of the two rate phenomena, diffusion and reaction, and with the assumption of spherical geometry and validity of the Michaelis-Menten equation for the en2yme kinetics, r = kcat[E] [S]/(JCM + [S]), Eq. (5.58) results. 

In order to achieve near-zero-order release from the matrix, a unique geometry, a specific nonuniform initial concentration profile, and/or a combined diffusion/erosion/swelling mechanism provide theoretical basis for such an approach. 

The overall clearance of hemofiltration is more difficult to calculate than the diffusive clearance or the HF clearance, as it combines diffusive and convective transfers. An approximate equation for this clearance, obtained from an exact numerical solution has been given by Jaffrin et al. [12] as... 

Homstein and Crowe (37) prepared a water extract of lean beef, concentrated the extract by freeze-drying and dialyzed the solution at 0°C against an equal volume of water. The dialysis procedure was repeated several times and the combined diffusates were lyophilized to yield a white, fluffy powder that rapidly browned on exposure to air to give a "meaty" odor. 

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