Mass transfer tortuosity factor

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. 

Internal diffusion in porous catalysts, if dominant, also reduces the observed activity of the biocatalyst. The decisive coefficient for mass transfer is the effective diffusion coefficient De((, which is defined in Eq. (5.56), where D0is the diffusion coefficient in solution, e the porosity of the carrier, and t the tortuosity factor. 

This example illustrates the distillation of a binary mixture in an open-batch distillery with flowing sweep gas and pervaporation by having a porous plate floating on top of the liquid hold up, as shown in Fig. 4.20. The porous plate was made from inert sintered metal with various pore sizes between 100 and 1 mfi, and had a thickness of 1 mm. The porosity was 40 % and the tortuosity factor was about 2. This results in an effective liquid phase mass transfer coefficient of about hiq = 2 X 10-7 m s-i, which results in Kiiq = 1.9 X 10 22. Therefore, one would expect the distillation process to be nonselective - that is, Si = xi - xi = 0. 

The extraction of toluene and 1,2 dichlorobenzene from shallow packed beds of porous particles was studied both experimentally and theoretically at various operating conditions. Mathematical extraction models, based on the shrinking core concept, were developed for three different particle geometries. These models contain three adjustable parameters an effective diffusivity, a volumetric fluid-to-particle mass transfer coefficient, and an equilibrium solubility or partition coefficient. K as well as Kq were first determined from initial extraction rates. Then, by fitting experimental extraction data, values of the effective diffusivity were obtained. Model predictions compare well with experimental data and the respective value of the tortuosity factor around 2.5 is in excellent agreement with related literature data. 

Some investigators lump eonstrietion and tortuosity into one factor, called the tortuosity factor, and set it equal to r/cr. C. N. Satterfield, Transfer in Heterogeneous Catalysis (Cambridge, Mass. MIT Press, 1970), pp. 33-47, has an excellent diseus-sion on this point. 

The factor in front of Dm is an approximation of the internal tortuosity factor. For a porosity of 0.9 to 0.5, Dpore is 5 times smaller than Dm or even lower, so the contribution to the effective mass transfer coefficient Eq. 6.191 is... 

Figure 5.4 Schematic illustration of the eluite diffusion into a stationary phase particle, ka and fcrf, rate constants for adsorption and desorption ke, mass transfer coefficient at the particle boimdary d, tortuosity factor e,-, internal porosity. Reproduced with permission from Cs. Horvdth and H.J. Lin,. Chromatogr., 149 (1978) 43 (Fig. 1).

Tortuosity factors less than unity can occur when the surface diffusion is significant. This is because is increased, while D as defined in Eq. (11-4) does not include this contribution. See Sec. 11-3 for a corrected D to include surface diffusion. C. N. Satterfield ( Mass Transfer in Heterogeneous Catalysis, Massachusetts Instiute of Technology Press, Cambridge, Mass., 1970) has summarized data from the literature and recommended the use of 5 = 4 when surface diffusion is insignificant. See also P. Carman, Trans. Inst. Chem. Engrs., 15, 150 (1937). 

When the simplified mass transfer Peclet number is very small (i.e., <1), T 0.67 instead of unity because the numerator of T (i.e., lSA.eff. axial disp.) is based on unsteady-state pore diffusion without convection, whereas the denominator of T (i.e., a, ordinary) is measured in an unrestricted bulk fluid phase. In other words, the diffusivity in the numerator of T is reduced by porosity and tortuosity factors. 

Ca, is the fluid reactant concentration in the pore, Rp the pore radius. D,p in this model may be a harmonic mean of the bulk and Knudsen diflusion coefficient with real geometries it would be a true effective difTusivity including the tortuosity factor and an internal void fraction. D p is an effective diffiisivity for the mass transfer inside the solid and is a correction factor accounting for the restricted availability of reactant surface in the region where the partially reacted zones interfere. For Jt(y) < LJ2 (shown in Fig. 4.5-2) or j>2 < J f e factor ( = 1 for L/ > R y) > L/2 or >i < y < yj the factor = 1 — (40/x) where tgB = (2/L) Ji (y) - (L/2) for y < yi the factor C 0, where R(y) is the radial position of the reaction front. It is clear from Eq. 4.S-1 that no radial concentration gradient of A is considered within the pore. 

Van Brakel, J. and Heertjes, P.M., Analysis of diffusion in macroporous media in terms of a porosity, a tortuosity and a constrictivity factor, Int. J. Heat Mass Transfer, 17 1093-1103, 1974. 

The balances of mass of the chemical species i and the terms for the adsorption kinetics (mass transfer, pore diffusion) are listed in Table 9.5-1 for the three systems with Cj as the concentration in the fluid phase and Xj as the mass loading of the adsorbent. J3 denotes the mass transfer coefficient of a pellet and sj, is its internal porosity. The tortuosity factor will be explained later. The derivation of equations describing instationary diffusion in spheres has already been presented in Sect. 4.3.3. With respect to diffusion in macropores it is important to consider that diffusion can take place in the fluid as well as in the adsorbate phase. In Table 9.5-1 special initial and boimdaty conditions valid for a completely unloaded bed (adsorption) or totally loaded bed (desorption) are given. In this section only the model valid for a thin layer in a fixed bed with the thickness dz and the volmne / dz will be derived, see Fig. 9.5-2. 

The mass transfer of a component tmder discussion is very fast (small particle sizes, small tortuosity factors, low viscous flttids)... 

The supposition of negligible internal mass transfer resistances was validated by solving Model A using the obtained kinetics parameters, and evaluating the effectiveness factor (Eq.3). The D jf values were estimated assuming tortuosity factors reported in the literature (Hayes and Kolaczkowski, 1997), i.e. the effective diffusivity was not fitted. The solution of the heterogeneous Model A indicates that the reactor operates with effectiveness factors between 1 and 0.98 for the temperature range 310 - 420 °C. 

Ep, Fj, Radial and axial dispersion coefficients kf Interphase (or external) mass transfer coefficient. q Average adsorbed-phase concentration (on a mass basis) t, r, z Independent variables of time, radial distance, and axial distance, respectively Mj Superficial velocity e Fixed-bed porosity Ep Particle porosity Particle density X Tortuosity factor... 

The inlet conditions for the numerical simulations are based on the experimental conditions. The simulations are performed with the three different models for internal diffusion as given in Section 2.3 to analyze the effect of internal mass transfer limitations on the system. The thickness (100 pm), mean pore diameter, tortuosity (t = 3), and porosity ( = 60%) of the washcoat are the parameters that are used in the effectiveness factor approach and the reaction-diffusion equations. The values for these parameters are derived from the characterization of the catalyst. The mean pore diameter, which is assumed to be 10 nm, hes in the mesapore range given in the ht-erature (Hayes et al., 2000 Zapf et al., 2003). CO is chosen as the rate-limiting species for the rj-approach simulations, rj-approach simulations are also performed with considering O2 as the rate-hmiting species. 

The terms a, P, a, and n are empirical coefficients [-], t [-] is the tortuosity factor of the flow path, L [E] is the dissolution length, dso [E] is the mean grain size, and 0n [-] is the volumetric NAPE content. The mass transfer coefficient K [T ] that appears in Equation 15.31 is the lumped mass transfer coefficient and it contains the NAPE/water interface area as introduced in Equation 15.28. 

With the above model, Wu and Gschwend (1986) also assumed that the entire surface area is available for mass flux and the path length of diffusive transfer is half the particle diameter. The authors introduced a correction factor f(ps, t) for >eff for natural silts, which is a function of intraaggregate porosity or tortuosity (tor), that is,... 

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