Calculation of effective diffusivities

Example 1.11 Calculation of Effective Diffusivity in a Multicomponent Gas Mixture... 

Example 1.13 Calculation of Effective Diffusivity of a Dilute Solute in a Homogeneous Mixture of Solvents... 

D. Maki, M. Stevens, Calculation of Effective Diffusion and Solubility Coefficients in Non-Fickian Materials, in TAPPI2002 PLACE Conferraice, p. 5 (2002)... 

Wolf, Pilchowski, and Karch have studied the adsorption of n-hexanol from toluene by NaA zeolite partially ion exchanged with Ca and Mg. The isotherms rise to a sharp plateau, which is reached at a concentration of about 15 mg hexanol/g toluene (x = 0.013). Pure NaA exhibits no adsorptive capacity for hexanol but, as divalent ions are introduced, this rises, initially slowly, then more sharply to reach, at about 30% degree of ion exchange, a maximum value that remains unchanged from 40% upwards. The paper also deals with the kinetics of the adsorption process, and presents calculations of effective diffusion coefficients and break-through curves under dynamic adsorption conditions. Some experiments with butan-l-ol and ethanol are also reported. 

In this way MTPM distinguishes between transport properties of gases (gas viscosity, binary bulk-diffusion coefficients of all gas pairs) and textural properties of porous materials characterized by the set of transport parameters (, , ly). Transport parameters represent material properties of the porous solid, and, thus do not depend on temperature, pressure and the kind of used gases. The obtained transport characteristics have a wide practical use for simulation and prediction in many industrial processes (e.g. calculation of effective diffusion coefficients for any pairs of gases in automotive catalytic converter [11]). 

The most important parameters in mass transfer modeling are the effective diffusion coefficient Defrand the mass transfer rate dXp/dt, which is linked to the mass transfer coefficient k in the balance equation for external transport. As for the calculation of effective diffusion coefficients, simple equations have been proposed in literature [10]. 

A method of calculating the effective diffusivity /> in terms of each of the binary diffu-... 

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]. 

The solute concentration Wg from each run has been expressed as a step function of the distance z, from which (-Wg/Wo) is calculated. The values of all the experimental parameters, D/uL, Dg/uL, Lu, /LD, Cpyti/k and (h f g/yM. ) ( AT) etc. are calculated from the values of the related physical properties in the literature (19). In Figure 3, g is correlated with z/L for all the thirteen runs, for which the values of effective diffusivity Dg in the melted zone have been predicted from the Kraussold correlation (7) using the experimental values of PrGr number and the values of the parameter Pe calculated show that little improvement has been made by using Pe instead of P in the correlation. 

When the solution is dilute, the three diffusion coefficients in Eq. (40a, b) may be calculated only by taking the intramolecular hydrodynamic interaction into account. In what follows, the diffusion coefficients at infinite dilution are signified by the subscript 0 (i.e, D, 0, D10> and Dr0). As the polymer concentration increases, the intermolecular interaction starts to become important to polymer dynamics. The chain incrossability or topological interaction hinders the translational and rotational motions of chains, and slows down the three diffusion processes. These are usually called the entanglement effect on the rotational and transverse diffusions and the jamming effect on the longitudinal diffusion. In solving Eq. (39), these effects are taken into account by use of effective diffusion coefficients as will be discussed in Sect. 6.3. 

Hints and Help Estimate the transport time by assuming that the chemicals diffuse from a reservoir of constant concentration into the liner that at time t = 0 is assumed to be uncontaminated. Equation 25-41 may give you an idea how to calculate the effective diffusion coefficient through the liner. Select the most critical compound among the four. What is the criterion ... 

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 ... 

The data of Table 17.8 exhibit a fairly narrow range of xp, an average of about 4, but there seems to be no pattern to xrn, which is not surprising since the diffusions actually are intermediate between bulk and Knudsen in these cases. In order to be able to calculate the effective diffusivity, it is necessary to know the pore size distribution, the specific surface, the porosity, and bulk diffasivity in the reaction mixture under reaction conditions. Such a calculation is primarily of theoretical interest. Practically it is more useful to simply measure the diffusivity directly, or even better to measure the really pertinent property of catalyst effectiveness as defined next. 

To calculate the effective diffusivity in the region of molecular flow, the estimated value of D must be multiplied by the geometric factor e/x which is descriptive of the heterogeneous nature of the porous medium through which diffusion occurs. 

The low-temperature enhanced diffusion of B can be modeled by calculating an effective diffusivity that is then applied to the calculation of the B profile by using the PREDICT program (59). The duration of enhanced diffusion is related to the damage annealing time. Empirically, the removal... 

Various techniques are available for determining the effective diffusivity of solute in gel (Itamunoala, 1988). One of the most reliable techniques is the thin-disk method which uses a diffusion cell with two compartments divided by a thin gel. Each compartment contains a well-stirred solution with different solute concentrations. Effective diffusivity can be calculated from the mass flux verses time measurement (Hannoun and Stephanopoulos, 1986). A few typical values of effective diffusivities are listed in Table 3.2. 

The sorption coefficient (K) in Equation (2.84) is the term linking the concentration of a component in the fluid phase with its concentration in the membrane polymer phase. Because sorption is an equilibrium term, conventional thermodynamics can be used to calculate solubilities of gases in polymers to within a factor of two or three. However, diffusion coefficients (D) are kinetic terms that reflect the effect of the surrounding environment on the molecular motion of permeating components. Calculation of diffusion coefficients in liquids and gases is possible, but calculation of diffusion coefficients in polymers is much more difficult. In the long term, the best hope for accurate predictions of diffusion in polymers is the molecular dynamics calculations described in an earlier section. However, this technique is still under development and is currently limited to calculations of the diffusion of small gas molecules in amorphous polymers the... 

Since reactive absorption systems often contain electrolyte species, the calculation of relevant diffusion coefficients is crucial. The effective diffusion coefficients for electrolyte components can be obtained from the Nernst-Hartley equation (see... 

Due to the lack of a reliable description, the diffusion of an ionic species in a molecular species is usually represented by the effective ionic diffusivity in the liquid phase [52]. The calculation of the diffusion coefficient for an ionic component in another ionic species is reduced to the arithmetical mean of both effective ionic diffusivities [52]. 

The well known Thiele modulus of the reaction. This is defined as the ratio of the intrinsic chemical rate, calculated at bulk fluid phase conditions, to the maximum rate of effective diffusion at the external pellet surface. For spherical catalyst pellets, the Thiele modulus is given by... 

Here it is assumed that it is possible to use the concept of an effective diffusion coefficient without making too large an error. Hence the effect of micro properties will not be studied here and it is assumed the value of De is known. The discussion is restricted to the impact of the macro properties and reaction properties on the effectiveness factor. Furthermore only simple reactions are discussed. Generalized formulae are provided that enable calculation of effectiveness factor for varying properties of the catalyst or the reacting system. 

Cybulski and Moulijn [27] proposed an experimental method for simultaneous determination of kinetic parameters and mass transfer coefficients in washcoated square channels. The model parameters are estimated by nonlinear regression, where the objective function is calculated by numerical solution of balance equations. However, the method is applicable only if the structure of the mathematical model has been identified (e.g., based on literature data) and the model parameters to be estimated are not too numerous. Otherwise the estimates might have a limited physical meaning. The method was tested for the catalytic oxidation of CO. The estimate of effective diffusivity falls into the range that is typical for the washcoat material (y-alumina) and reacting species. The Sherwood number estimated was in between those theoretically predicted for square and circular ducts, and this clearly indicates the influence of rounding the comers on the external mass transfer. 

For a further check on the practical applicability of the theory, we may for any point, e.g., the fitted point, calculate the effective diffusivity Dell of this catalyst from the observed value of rj, the particle size ro, and the actual reaction velocity k, (obtained from measurement on fines). This diffusivity was then compared to the diffusivity measured directly by a gas flow method as described in Section IV.5. 

The Thiele modulus for the mesoporous structure of the eatalyst, ( ) i, was calculated using the following parameters particle size, Rp = 0.0137 cm mean pore radius, rpore.ave = 20 10 cm catalyst porosity, e = 0.52 catalyst density, pg = 1210 g 1. N2 adsorption-desorption isotherms were used for measurement. The calculated value of effective diffusivity coefficient in the mesoporous structure of the catalyst is Dg = 9.71 10 2 cm min . This value is not affeeted by coke deposition. 

It is worth noting that the almost constant value of the composition of nitrogen would have been predicted with any of the formulas used in Example 6.1.1 to calculate the effective diffusivity. Thus, we have our first demonstration of the inability of the effective diffusivity approach to model multicomponent diffusion processes. ... 

Wilke [103] proposed a simpler model for calculating the effective diffusion coefficients for diffusion of a species s into a multicomponent mixture of stagnant gases. For dilute gases the Maxwell-Stefan diffusion equation is reduced to a multicomponent diffusion flux model on the binary Pick s law form in which the binary diffusivity is substituted by an effective multicomponent diffusivity. The Wilke model derivation is examined in the sequel. 

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