Phase Diffusion Coefficients

The theory of molecules diffusing in liquids is not very well developed. A rigorous formulation of multicomponent diffusion, such as the Stefan-Maxwell equation for the gas phase, is not successful in describing diffusion in a Uquid phase, because a general theory for calculating binary diffusion coefficients is lacking. However, semiempirical correlations that describe the diffusion of a dissolved component (solute) in a solvent can be used. The concentration of the dissolved component is of course assumed to be low compared with that of the solvent. The diffusion in liquids is very much dependent on whether the molecules are neutral species or ions. 

---

For prediction of gas phase diffusion coefficients in multicomponent hydi ocarbon/nonKydi ocai bon gas systems, the method of Wilke shown in Eq. (2-154) is used. 

It is important to recognize that the effects of temperature on the liquid-phase diffusion coefficients and viscosities can be veiy large and therefore must be carefully accounted for when using /cl or data. For liquids the mass-transfer coefficient /cl is correlated in terms of design variables by relations of the form... 

In addition, it was concluded that the liquid-phase diffusion coefficient is the major factor influencing the value of the mass-transfer coefficient per unit area. Inasmuch as agitators operate poorly in gas-liquid dispersions, it is impractical to induce turbulence by mechanical means that exceeds gravitational forces. They conclude, therefore, that heat- and mass-transfer coefficients per unit area in gas dispersions are almost completely unaffected by the mechanical power dissipated in the system. Consequently, the total mass-transfer rate in agitated gas-liquid contacting is changed almost entirely in accordance with the interfacial area—a function of the power input. 

As compared to HPLC, cSFC shows higher efficiency, universal and selective detection, minimal derivatisation for separation and the ability to separate thermally labile organic compounds. Often, cSFC analyses are also considerably faster. This arises because higher mobile phase diffusion coefficients translate directly into higher optimum velocities. However, sensitivity, detection dynamic range and sample capacity... 

Estimate the gaseous phase diffusion coefficient for the following systems, at 1 atmosphere and the temperatures given ... 

C, calculated from measured liquid-phase diffusion coefficients, measured range -10 to 20°C,... 

The gas-phase diffusion coefficients are estimated from the molecular weights M of the speciesl0 ... 

This equation shows that the saturation greatly affects the effective gas-phase diffusion coefficients. Hence, flooding effects are characterized by the saturation. 

Equation (105) is the basis for the determination of gas-phase diffusion coefficients and ultra low vapor pressures using the methods proposed by Davis and Ray (1977), Ravindran et al. (1979), and Ray et al. (1979). Additional information can be gained by writing the Chapman-Enskog first approximation for the gas-phase diffusivity (Chapman and Cowling, 1970),... 

We emphasize that we are interested in the fluid-phase diffusion coefficient of the reactant A, which we call D, and also the solid-phase diffusion of this species D,4s (bold and subscript s). The diffusion of reactant A in either situation can limit the reaction process. 

Diffusion of 0 ions has also been taken into account as rate determining step (rds) and the value of their solid phase diffusion coefficient has been determined as 2.4 x 10 cm [122]. 

Transport of the gas to the surface and the initial interaction. The first step in heterogeneous reactions involving the uptake and reaction of gases into the liquid phase is diffusion of the gas to the interface. At the interface, the gas molecule either bounces off or is taken up at the surface. These steps involve, then, gaseous diffusion, which is determined by the gas-phase diffusion coefficient (Dg) and the gas-surface collision frequency given by kinetic molecular theory. 

Adapted from Schwartz (1984a), and Shi and Seinfeld (1991). ka = particle radius, Dg = gas-phase diffusion coefficient, D, = liquid-phase diffusion coefficient, H = Henry s law constant, a = mass accommodation coefficient, u.w = mean thermal speed, and k = first-order aqueous-phase rate constant. 

In the case of two fluids, two films are developed, one for each fluid, and the corresponding mass-transfer coefficients are determined (Figure 3.2). In a fluid-solid system, there is only one film whereas the resistance within the solid phase is expressed by the solid-phase diffusion coefficient, however, in many cases an effective mass-transfer coefficient is used in the case of solids as well. Consider the irreversible catalytic reaction of the form... 

C0 = the total counterion concentration in the liquid-phase D = the liquid-phase diffusion coefficient S = the film thickness r0 = the particle radius aA 3 = the separation factor. 

By a tiial-and-eiTor procedure, using the value of U(t) and the model eqs. (4.52)-(4.58), the solid-phase diffusion coefficient is found to be 1.3 X 10 12 m2/s. This value is very close to the one given in the study of Meshko el al. The trial-and-error procedure can be done easily. By changing the value of Ds, the model predicts the values of U(t) for each t. The best value of Ds is the one that results in the lowest mean deviation between the experimental and the model values of U(t). In Figure 4.19, the performance of the model is shown. The average error is 3.1%. 

Again following a trial-and-eiror procedure, the solid-phase diffusion coefficient is found to be 1.82 X 10-9 cm2/s. This value is very close to the one given in the study of Choy and McKay. In Figure 4.21, the performance of the model is shown. The average error is 3%. 

The liquid-phase diffusion coefficient can be estimated from the Nemst-Haskell eq. (1-24) (see Appendix I) ... 

Hashimoto et al. (1977) studied the removal of DBS from an aqueous solution in a carbon fixed-bed adsorber at 30 °C. The dimensions of the bed were D = 20 mm and Z = 25.1 cm. Carbon particles of 0.0322-cm radius were used, with 0.82 g/cm3 particle density, and 0.39 g/cm3 bulk density. The concentration of the influent stream was 99.2 rng/L and the superficial velocity was 0.0239 cm/s. The fixed bed was operated under upflow condition. Furthermore, the isotherm of the DBS-carbon system at 30 °C was found to be of Freundlich type with Fr = 0.113 and = 178 (mg/g)(L/mg)0113. Finally, the average solid-phase diffusion coefficient was found to be 2.1 X 10 10 cm2/s. The approximate value of 10 9 m2/s could be used for DBS liquid-phase diffusion coefficient. 

Murillo et al. (2004) studied the adsorption of phenanthrene (polycyclic aromatic hydrocarbon -PAH) from helium as carrier gas on a coke fixed-bed adsorber, at 150 °C. The isotherm of the phenanthrene-coke system at 150 °C was found to be of Freundlich type with Fr = 0.161 and KF = 1.9 (mol/kg)(m3/mol)0161. The isotherm has been derived for phenanthrene concentrations between 1.71 X 10 4 and 1.35 X 10-2 mol/m3. Finally, the average solid-phase diffusion coefficient, calculated from several experimental runs, was found to be 6.77 X 10-8 cm2/s. 

Die next parameter we need is the diffusion coefficient Df of hydrogen peroxide in water. Here, we can assume the approximate value of 10 9 m2/s. However, this coefficient will be needed further in this example for the determination of the effective solid-phase diffusion coefficient, in a calculation that is extremely sensitive to the value of the liquid-phase diffusion coefficient. For this reason, coefficient should be evaluated with as much accuracy as possible. The diffusion coefficient of solutes in dilute aqueous solutions can be evaluated using the Hayduk and Laudie equation (see eq. (1.26) in Appendix I) ... 

Using this expression, the effective solid-phase diffusion coefficient is found to be equal to 3.46 X 10-10 m2/s. This coefficient is related to the liquid-phase coefficient as follows (eq. (3.602)) ... 

The liquid-phase diffusion coefficient of sulfur dioxide can be found from Table 1.10, Appendix I, and is equal to 1.7 x 10 9 m2/s. 

---