Coefficients in Liquids

Since diffusion in hquids involves one molecule in close contact with other molecules, Einstein (1905) developed the following relationship  

Equation (3.16) is generally accurate to within 20%, with some notable exceptions, and is one of the standards by which other equations are compared. Note that equation (3.16) also provides a characterization of Dn with the following parameters  

Because absolute temperature does not change much for environmental applications, this is the least variable characterization of equation (3.16). 

Highly viscous solvents, like molasses or glycerin, will have a lower diffusivity than solvents like kerosene or alcohol, all other factors being equal. This is also where temperature becomes important, because the viscosity of a liquid is highly dependent on temperature. 

Wilke and Chang (1955) developed an empirical relationship that was based on the temperature and viscosity characterization of the Stokes-Einstein relationship. It deviates from the equivalent diameter characterization by using another parameter, and incorporates the size of the solvent molecule and a parameter for polarized solvents. It is the most generally used of the available equations (Lyman et al., 1990) and is given as 

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Slater, Rate Coefficients in Liquid-Liquid Extraction Systems in Godfrey and Slater, Liquid-Liquid Extraction Equipment, Wiley, 1994, pp. 45-94. 

Dunlap, J. R. Jr., and J. IT. Rushton, Heat-Transfer Coefficients in Liquid Mixing Using Vertical-Tube Baffles, A-I.Ch.E. Symposium Series No. 5, Vol. 49, 1953, 137. 

In Section HI, we discussed the relation between fugacities and activity coefficients in liquid mixtures, and we emphasized that we have a fundamental choice regarding the way we wish to relate the fugacity of a component to the pressure and composition. This choice follows from the freedom we have in choosing a convention for the normalization of activity coefficients. 

Chemical Reaction Engineering Aspects of Homogeneous Hydrogenations Table 45.4 Estimation of the diffusion coefficient in liquid phase. 

Othmer, D.F. and M.S. Thakar (1953), Correlating diffusion coefficients in liquids, Industrial and... 

Hixson, A. W. and Baum, S. J. Ind. Eng. Chem. 33 (1941) 478, 1433. Agitation mass transfer coefficients in liquid-solid agitated systems. Agitation heat and mass transfer coefficients in liquid-solid systems. 

When one or more of the chemical reactions is sufficiently slow in comparison with the rate of diffusion to and away from the interface of the various species taking part in an extraction reaction, such that diffusion can be considered instantaneous, the solvent extraction kinetics occur in a kinetic regime. In this case, the extraction rate can be entirely described in terms of chemical reactions. This situation may occur either when the system is very efficiently stirred and when one or more of the chemical reactions proceeds slowly, or when the chemical reactions are moderately fast, but the diffusion coefficients of the transported species are very high and the thickness of the two diffusion films is close to zero. In practice the latter situation never occurs, as diffusion coefficients in liquids generally do not exceed 10 cm s, and the depth of the diffusion films apparently is never less than 10 cm. 

The Wilke and Chang (36) correlation equation, whi h is somewhat approximative, is used most frequently to evaluate difft ion coefficients in liquids. This can be written as... 

L denotes overall mass transfer coefficient in liquid film Im logarithmic mean... 

The mass transfer coefficient in liquid membrane, k , can be obtained from the plot of concentration vs. time. 

The molecular diffusion coefficients in liquid phase were estimated from the correlations Wilke and Chang (9) for organic solutions and Hayduk and Minhas (10) for aqueous solutions, respectively. The solvent viscosities needed in the correlations were obtained from the empirical equation based on the experimental data,... 

The double lines in Figure 3.44 represent the Sh number based on the mass transfer coefficient, in the case of a single-particle fall in water, for three different particle densities (Harriot, 1962). This value is considered to be the minimum mass-transfer coefficient in liquid-solid films in agitated vessels. Taking into account the fact that the actual Sh value in an agitated vessel is 1.5 -8 times its minimum value, it is apparent that the mass transfer coefficients are much higher in the case of agitated vessels. 

Although there are a lot of data in the literature regarding diffusion coefficients in liquids or then calculation from molecular properties (Appendix I, Section 1.2), it is not the case for diffusion coefficients in solids, where the phenomena appearing are more complex. In solids, the molecule may be forced to follow a longer and tortuous path due to the blocking of the cross-sectional area, and thus the diffusion is somehow impaired. Several models have been developed to take into consideration this effect in the estimation of diffusion coefficients, leading, however, to a variety of results. 

A number of useful empirical and semiempirical relations have been proposed for predicting diffusion coefficients in liquids a rather complete summary of these has been given by Treybal (T6, pp. 102 et seg.). Most of these are valid only for nonionic systems at very low concentrations. 

Equation (2) may be used for the rate constant k of a chemical reaction or applied to the diffusion coefficient in liquid or solid phases or to the fluidity of liquids (reciprocal of dynamic viscosity) or to the specific electrical conductivity of semiconductors. 

In chemical heterogeneous catalysis, it is common to use highly porous catalysts that come in particles of millimeter to centimeter size to increase the effective catalyst surface. In practical electrocatalysis, in particular applying electrocatalysis in fuel cells, it is also usual to use highly porous— although accounting for the low diffusion coefficients in liquid electrolytes compared to gases, 10 5 cm2/sec vs 1 cm2/sec, much smaller—catalyst particles. 

A.ctivity Coefficients. Activity coefficients in liquid mixtures are direcdy related to the molar excess Gibbs energy of mixing, AGE, which is defined as the difference in the molar Gibbs energy of mixing between the real and ideal mixtures. It is typically an assumed function. Various functional forms of AGE give rise to many of the different activity coefficient models found in the literature (1—3,18). Typically, the liquid-phase activity coefficient is a function of temperature and composition explicit pressure dependence is rarely included. 

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... 

Pressure is more directly connected to the concept of explosion nevertheless, it is less directly connected to the reactor status, since, for liquid-phase reactors, pressure nonlinearly depends on temperature (trough the vapor pressure relationship) and concentration (through the activity coefficients in liquid phase). Moreover, since pressure measurements are usually less accurate than temperature measurements, they are to be considered in particular for gassy reactions, i.e., when the runaway produces small temperature effects but large amounts of incondensable products in gaseous phase. 

Mass transfer coefficients in liquid cyclohexane can be predicted using available correlations. In spite of the many methods available for measuring k a in stirred vessels, the overall accuracy of the methods leaves a lot to be desired. 

Because of the low diffusion coefficients in liquids, the particle size for packed columns in LC needs to be very small (see section 7.2.1). For the same reason, all external volumes and diameters need to be minimized. This may easily be understood if we express the standard deviation in volume units (ctv) in the parameters that represent the dimensions of the column ... 

Fundamental to the description of the friction coefficient in liquids is Stokes law, which shows that / is proportional to the radius a of spherical particles undergoing transport and proportional to the viscosity 17 of the medium [24,25]... 

In relation 8.6 - rA is the reaction rate in kmol/(kg catalyst s), pc density of solid catalyst, R particle radius, I ) i diffusion coefficient in liquid and CAs reactant concentration at the catalyst surface. If CWp 1 there are no diffusion limitations, but if CWP 1 the catalyst effectiveness is severely affected. 

Dunlap Jr., I.R., J.H. Rushton, Chem. Engng. Progr. Symp. Series 49 (1953) 5,137-151 Heat-transfer coefficients in liquid mixing using vertical-tube baffles... 

A modern set of methods which can be used to estimate partition coefficients is the group contribution method. These methods were developed to allow chemical engineers to estimate activity coefficients in liquid and polymeric systems. Of the numerous methods developed, UNIFAC, the oldest and most thoroughly tested method, is probably the most universally applicable to a wide variety of substances and sytems despite its known weaknesses (Baner, 1999). The use of UNIFAC and example calculations will be described later in this chapter. 

The most common basis for estimating diffusion coefficients in liquids is the Stokes-Einstein equation ... 

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