Diffusivity ratio

Furtliennore, since tlie bifurcation must occur from a stable homogeneous steady state we must have D ID < 1 i.e. tlie diffusion coefficient of tlie inhibitor is greater tlian tliat of tlie activator. The critical diffusion ratio at tlie bifurcation is... 

The mobility ratio equal to the diffusion ratio in this equation would naturally follow from application of the Nemst-Einstein equation, Eq. (88), to transport gels. Since the Nemst-Einstein equation is valid for low-concentration solutes in unbounded solution, one would expect that this equation may hold for dilute gels however, it is necessary to establish the validity of this equation using a more fundamental approach [215,219]. (See a later discussion.) Morris used a linear expression to fit the experimental data for mobility [251]... 

FIG. 16-13 Effect of Re Sc group, distribution ratio, and diffusivity ratio on height of a transfer unit. Dotted lines for gas and solid lines for liquid-phase systems. 

Lennard-Jones gas dimensionless thermal diffusion ratio, 25 307 Lennard-Jones molecules, 25 302 Lennard-Jones potential, 7 620 23 94 Lennard-Jones potential energy function, 25 302... 

MRs, with 10-MRs the para ortho raho is typically >2, the smaller the pore size, the higher the para-xylene selectivity. More recent molecular dynamics simula-hons verify that the diffusivity ratio for para ortho is much higher for ZSM-5 (7.4) than for Beta (2.3), consistent with the higher para-xylene selectivity obtained over ZSM-5 [76]. 

In the second expression the thermal diffusion ratio kr has been introduced it is defined for binary mixtures to be... 

Why can the diffusivity ratio of two chemicals in water be approximated by a power law of their molecular mass ratio What is the exponent of the power law ... 

Because the diffusivity ratio, Dia/Dja, is not exactly identical for air and water and since Eq. 20-3 also contains Kia,w, Eq. 20-19 does not hold for the composite (overall) exchange velocity v,Ww. It can be applied to classes of substances which are either solely water-phase- or air-phase-controlled. 

Therefore, instead of Sc/a we use the diffusivity ratios to compare v,a of different substances. According to the empirical observations of Mackay and Yeun (1983), the appropriate exponent is 2/3. That is, it lies between the film model and the surface replacement model ... 

Equation 13 reduces to the Rayeigh equation (3) when the ratio of the gas-phase diffusivities, , is unity. Since gas-phase diffusivity is inversely proportional to the square root of the reduced mass, in the case of fission product-sodium systems where sodium has the smallest molecular weight, the above diffusivity ratio is less than unity. Therefore, the Rayleigh equation, which was derived on the basis of equilibrium vaporization, in fact represents an upper limit for the fractional fission-... 

In equation IV. 6, Dik represents the ordinary diffusion coefficient for binary interactions, and a, is the thermal diffusion ratio. The reactants are often present in small amounts (<1%) relative to the carrier gas thus, the multicomponent diffusion expression (equation IV. 6) may be replaced by a simple Fickian diffusion expression that includes thermodiffusion... 

Another well-known example is the coupling between mass flow and heat flow. As a result, an induced effect known as thermal diffusion (Soret effect) may occur because of the temperature gradient. This indicates that a mass flow of component A may occur without the concentration gradient of component A. Dufour effect is an induced heat flow caused by the concentration gradient. These effects represent examples of couplings between two vectorial flows. The cross-phenomenological coefficients relate the Dufour and Soret effects. In order to describe the coupling effects, the thermal diffusion ratio is introduced besides the transport coefficients of thermal conductivity and dififusivity. 

Table 7.1 shows some experimental values of thermal diffusion ratios for liquids and gases at low density and pressure. If K-n is positive, component 1 diffuses to a cooler region otherwise, it diffuses to a hotter regioa The thermal diffusion factor for component 1 is mainly independent of concentration for gases, and is given by... 

Table 7.1. Experimental values of thermal diffusion ratios for liquids and gases at low density and pressure3...

Table 7.2 shows the viscosity, mutual diffusion coefficient, and thermodynamic factor for aqueous solutions of ethylene glycol and polyethylene glycol (PEG) at 25°C the diffusivity decreases considerably with increasing molecular weight, while the viscosity increases. Table 7.2 shows the thermal diffusion ratios for liquids and gases at low density and pressure the thermal diffusion ratios are relatively larger in liquids. 

Table 7.2b. Thermal diffusion ratio, KJt thermal diffusion coefficients Dj, and heats of transport Q for aqueous ethylene glycol and polyethylene glycol (PEG) solutions at 25°Ca...

We can describe the degree of coupling q and the thermal diffusion ratio of component 1 KTl in terms of the transport coefficients and thermodynamic factor (F)... 

Figure 7.2. Change of the thermal diffusion ratio /CT1 with the alkane concentrations xx at 30°C and ambient pressure (a) straight chain alkanes, (—) /7-hexane, (—), /7-heptane, (—) n-octane (b) branched-chain alkanes, (—) 3-methylpentane, (—) 2,2-dimethylpentane, (—) 2,2,4-trimethylpentane. Reprinted with the permission from Elsevier, Y. Demirel and S.l. Sandler, Int. J. Heat Mass Transfer, 43 (2002) 75.

Concentration effects on the heats of transport and the thermal diffusion ratio of chloroform with various alkanes at 30°C and 1 atm are seen in Table 7.6. Table 7.7 shows the experimental heats of transport at various concentrations and at temperatures 298 and 308 K for binary mixtures of toluene (1), chlorobenzene (2), and bromobenzene (3) at 1 atm. The absolute values of heats of transport decrease gradually as the concentrations of the alkane increase. Table 7.7 also contains values of cross coefficients obtained from easily measurable quantities and the thermodynamic factor. 

Table 7.8 shows the thermal diffusion ratios and thermal diffusion coefficients obtained from Onsager s reciprocal rules for toluene, chlorobenzene, and bromobenzene at 1 atm and at 298 and 308 K. Thermal diffusion or heats of transport may be extremely sensitive to the molecular interactions in solutions (Rowley et al., 1988). 

Table 7.6. Heats of transports and thermal diffusion ratio Kja of chloroform in binary mixtures with selected alkanes at 30°C and 1 atmb...

Table 7.11. Thermal diffusion ratios from Onsagefs reciprocal rules for toluene (1), chlorobenzene (2), and bromobenzene (3) at 1 atm and 35oa...

Table 7.11 shows the thermal diffusion ratios obtained from Onsager s reciprocal rules for toluene (1), chlorobenzene (2), and bromobenzene (3) at 1 atm and 35°C. The heats of transport for the ternary mixtures are shown in Tables 7.12 and 7.13. For the ternary mixture of toluene (l)-chlorobenzene (2)-bromobenzene (3), the heats of transport are tabulated at 298 and 308 K. The temperature- and composition-dependent heats of transport values are fitted by the following equations by Platt et al. (1982) with a deviation below 5% ... 

The separation of polymers due to thermal diffusion may be quite large. For example, the thermal diffusion ratio for dilute solutions of polystyrene in tetrahydrofuran is around 0.6 K1. This indicates that the change of polystyrene concentration per degree is 60%. The type of solvent and polymer pair may have a considerable effect on both the thermal diffusion ratio and the thermal diffusion coefficient. 

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