Tortuosity model effectiveness

More detailed aspects of transport in heterogeneous media have been given in the excellent reviews of the subject by Barrer (55) and Petropolus (51) The models described by these authors and others include the effects of size, shape, and anisotropy of the crystalline phase on the tortuosity. Models of highly ordered anisotropic media have been demonstrated to have tortuosities in the range of 30, which reflects the rather dramatic role that orientation can have on the barrier properties of semicrystalline polymers. 

However, the data clearly suggest that the effective diffusion coefficient depends on molecular weight this effect is not predicted by these models of porous structure (i.e., the tortuosity models in Figure 4.18 do not depend on molecular size). The tortuosity for a small water-soluble molecule, the tracer ion TMA, is 2 (Figure 4.22). Large molecules have tortuosity values greater than 2 and the tortuosity increases with molecular size. For larger molecules, this tortuosity —which is estimated from the effective diffusion coefficient— must reflect a decrease in diffusion rate due to actual tortuosity in the extra-... 

Numerical simulations have been made possible by the availability of experimental data on a well-characterized geometry and with accurate concentration measurementst The simple geometry of ceramic monoliths is essential for accurate numerical modelling with no independent adjustable parameters such as tortuosities or effective diffusions within porous media. The only adjustable parameter, the peak diffusion at the moving contact line, will be eliminated when an acceptable, simple flow model of split-ejection streamlines is available. 

In an effective properties model, the porous microstructures of the SOFC electrodes are treated as continua and microstructural properties such as porosity, tortuosity, grain size, and composition are used to calculate the effective transport and reaction parameters for the model. The microstmctural properties are determined by a number of methods, including fabrication data such as composition and mass fractions of the solid species, characteristic features extracted from micrographs such as particle sizes, pore size, and porosity, experimental measurements, and smaller meso- and nanoscale modeling. Effective transport and reaction parameters are calculated from the measured properties of the porous electrodes and used in the governing equations of the ceU-level model. For example, the effective diffusion coefficients of the porous electrodes are typically calculated from the diffusion coefficient of Eq. (26.4), and the porosity ( gas) and tortuosity I of the electrode ... 

Xu and Froment (1989b) developed the most widely accepted kinetic model, deriving the intrinsic parameters and incorporating diffusional limitations through the evaluation of the tortuosity factor, effective diffusivities, and the effectiveness factor. These parameters were used in the simulation of commercial reactors and industrial steam reformers with satisfactory results. 

Diffusion within the largest cavities of a porous medium is assumed to be similar to ordinary or bulk diffusion except that it is hindered by the pore walls (see Eq. 5-236). The tortuosity T that expresses this hindrance has been estimated from geometric arguments. Unfortunately, measured values are often an order of magnitude greater than those estimates. Thus, the effective diffusivity D f (and hence t) is normally determined by comparing a diffusion model to experimental measurements. The normal range of tortuosities for sihca gel, alumina, and other porous solids is 2 < T < 6, but for activated carbon, 5 < T < 65. 

Since theoretical calcination of effectiveness is based on a hardly realistic model of a system of equal-sized cylindrical pores and a shalq assumption for the tortuosity factor, in some industrially important cases the effectiveness has been measured directly. For ammonia synthesis by Dyson and Simon (Ind. Eng. Chem. Fundam., 7, 605 [1968]) and for SO9 oxidation by Kadlec et aJ. Coll. Czech. Chem. Commun., 33, 2388, 2526 [1968]). 

Many theoretical embellishments have been made to the basic model of pore diffusion as presented here. Effectiveness factors have been derived for reaction orders other than first and for Hougen and Watson kinetics. These require a numerical solution of Equation (10.3). Shape and tortuosity factors have been introduced to treat pores that have geometries other than the idealized cylinders considered here. The Knudsen diffusivity or a combination of Knudsen and bulk diffusivities has been used for very small pores. While these studies have theoretical importance and may help explain some observations, they are not yet developed well enough for predictive use. Our knowledge of the internal structure of a porous catalyst is still rather rudimentary and imposes a basic limitation on theoretical predictions. We will give a brief account of Knudsen diffusion. 

For hydrophilic and ionic solutes, diffusion mainly takes place via a pore mechanism in the solvent-filled pores. In a simplistic view, the polymer chains in a highly swollen gel can be viewed as obstacles to solute transport. Applying this obstruction model to the diffusion of small ions in a water-swollen resin, Mackie and Meares [56] considered that the effect of the obstruction is to increase the diffusion path length by a tortuosity factor, 0. The diffusion coefficient in the gel, )3,i2, normalized by the diffusivity in free water, DX1, is related to 0 by... 

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. 

The effectiveness of the internal O2 transport by diffusion or convection depends on the physical resistance to movement and on the O2 demand. The physical resistance is a function of the cross-sectional area for transport, the tortuosity of the pore space, and the path length. The O2 demand is a function of rates of respiration in root tissues and rates of loss of O2 to the soil where it is consumed in chemical and microbial reactions. The O2 budget of the root therefore depends on the simultaneous operation of several linked processes and these have been analysed by mathematical modelling (reviewed by Armstrong... 

Reid, Sherwood and Prausnitz [11] provide a wide variety of models for calculation of molecular diffusion. Dr is the Knudsen diffusion coefficient. It has been given in several articles as 9700r(T/MW). Once we have both diffusion coefficients we can obtain an expression for the macro-pore diffusion coefficient 1/D = 1/Dk -i-1/Dm- We next obtain the pore diffusivity by inclusion of the tortuosity Dp = D/t, and finally the local molar flux J in the macro-pores is described by the famiUar relationship J = —e D dcjdz. Thus flux in the macro-pores of the adsorbent product is related to the term CpD/r. This last quantity may be thought of as the effective macro-pore diffusivity. The resistance to mass transfer that develops due to macropore diffusion has a length dependence of R]. 

If data are available on the catalyst pore- structure, a geometrical model can be applied to calculate the effective diffusivity and the tortuosity factor. Wakao and Smith [36] applied a successful model to calculate the effective diffusivity using the concept of the random pore model. According to this, they established that ... 

In Equation (9.6), x is the direction of flux, nt [mol m-3 s 1 ] is the total molar density, X [1] is the mole fraction, Nd [mol m-2 s 1] is the mole flux due to molecular diffusion, D k [m2 s 1] is the effective Knudsen diffusion coefficient, D [m2 s 1] is the effective bimolecular diffusion coefficient (D = Aye/r), e is the porosity of the electrode, r is the tortuosity of the electrode, and J is the total number of gas species. Here, a subscript denotes the index value to a specific specie. The first term on the right of Equation (9.6) accounts for Knudsen diffusion, and the following term accounts for multicomponent bulk molecular diffusion. Further, to account for the porous media, along with induced convection, the Dusty Gas Model is required (Mason and Malinauskas, 1983 Warren, 1969). This model modifies Equation (9.6) as ... 

The free parameters of this model are the ratio of porosity to tortuosity, e/r, the binary molecular diffusivities, D-j, the Knudsen coefficient, Ko (which determines specific effective Knudsen diffusivities, D j) and the permeability constant, Bo. 

The literature data on the tortuosity factor r show a large spread, with values ranging from 1.5 to 11. Model predictions lead to values of 1/e s (8), of 2 (parallel-path pore model)(9), of 3 (parallel-cross-linked pore model)(IQ), or 4 as recently calculated by Beeckman and Froment (11) for a random pore model. Therefore, it was decided to determine r experimentally through the measurement of the effective diffusivity by means of a dynamic gas chromatographic technique using a column of 163.5 cm length,... 

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. 

Early efforts to model catalyst deactivation either utilized simplified models of the catalyst s porous structure, such as a bundle of nonintersecting parallel pores, or pseudo-homogeneous descriptions in terms of effective diffusivities and tortuosity... 

Apparently the parameters of stochastic models are quite different from those of classic (deterministic) models where the permeability, the porosity, the pore radius, the tortuosity coefficient, the specific surface, and the coefficient of the effective diffusion of species represent the most used parameters for porous media characterization. Here, we will present the correspondence between the stochastic and deterministic parameters of a specified process, which has been modelled with a stochastic and deterministic model in some specific situations. 

Finally, it is important to notice the effect of the support in the pervaporation flux, analyzed by Bruijn et al. [117] who proposed a model and evaluated the contribution of the support layer to the overall resistance for mass transfer in the selected literature data. They found that in many cases, the support is limiting the flux the permeation mechanism through the support corresponds to a Knudsen diffusion mechanism, which makes improvements in the porosity, tortuosity, pore diameter, and thickness necessary for an increase in the pervaporation flux. 

The approach was extended to include diffusional limitations. The fo3rm of the continuity equations for the reactant A corresponded to that for a continuum model, but the effect of the structure of the catalyst was introduced through P. Blockage increases the tortuosity of the diffusion path, thus affecting the effective diffusivities. This was also accounted for. Both parallel and consecutive coking were investigated. In the latter case non monotonic internal coke profiles were obtained, as for the single... 

---