Liquid phase diffusivities

Liquid phase diffusivities are strongly dependent on the concentration of the diffusing component which is in strong contrast to gas phase diffusivities which are substantially independent of concentration. Values of liquid phase diffusivities which are normally quoted apply to very dilute concentrations of the diffusing component, the only condition under which analytical solutions can be produced for the diffusion equations. For this reason, only dilute solutions are considered here, and in these circumstances no serious error is involved in using Fick s first and second laws expressed in molar units. 

Values of the diffusivities of various materials in water are given in Table 10.7. Where experimental values are not available, it is necessary to use one of the predictive methods which are available. 

A useful equation for the calculation of liquid phase diffusivities of dilute solutions of non-electrolytes has been given by Wilke and CHANG(I6). This is not dimensionally consistent and therefore the value of the coefficient depends on the units employed. Using SI units  

The values are based mainly on International Critical Tables 5 (1928). 

V4 is the molecular volume of the solute (m /kmol). Values for simple molecules are given in Table 10.4. For more complex molecules, V. is calculated by summation of the atomic volume and other contributions given in Table 10.4. 

It may be noted that for water a value of 0.0756 m /kmol should be used. 

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Many more correlations are available for diffusion coefficients in the liquid phase than for the gas phase. Most, however, are restiicied to binary diffusion at infinite dilution D°s of lo self-diffusivity D -. This reflects the much greater complexity of liquids on a molecular level. For example, gas-phase diffusion exhibits neghgible composition effects and deviations from thermodynamic ideahty. Conversely, liquid-phase diffusion almost always involves volumetiic and thermodynamic effects due to composition variations. For concentrations greater than a few mole percent of A and B, corrections are needed to obtain the true diffusivity. Furthermore, there are many conditions that do not fit any of the correlations presented here. Thus, careful consideration is needed to produce a reasonable estimate. Again, if diffusivity data are available at the conditions of interest, then they are strongly preferred over the predictions of any correlations. 

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. 

The calculation of liquid phase diffusivities is discussed further in Volume 6. 

Liquid phase diffusivity Units in SI system m2/s Dimensions in M. N. L, i T... 

Despite the plethora of data in the scientific literature on thermophysical quantities of substances and mixtures, many important data gaps exist. Predictive capabilities have been developed for problems such as vapor-liquid equihbrium properties, gas-phase and—less accmately—liquid-phase diffusivities, aud solubilities of uouelectrolytes. Yet there are many areas where improved predictive models would be of great value. Au accrrrate and rehable predictive model can obviate the need for costly, extensive experimental measurements of properties that are critical in chemical manufactming processes. 

Estimate the liquid phase diffusion coefficient for the following systems at 25 °C ... 

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

Finally, the liquid-phase diffusivities and mass-transfer coefficients are related, as a consequence of equation 9.2-7, by... 

Mass transfer (the C term), which involves collisions and interactions between molecules, applies differently to both packed and capillary columns. Packed columns are mostly filled with stationary phase so liquid phase diffusion dominates. The mass transfer is minimized by using a small mass of low-viscosity liquid phase. Capillary columns are mostly filled with mobile phase, so mass transfer is important in both the gas and liquid phases. A small mass of low-viscosity liquid phase combined with a low-molecular weight carrier gas will minimize this term. 

M is the mass of solute transferred in time t, and k is the diffusion coefficient. (This is approximately equal to the liquid phase diffusivity DL, discussed in Volume 1, Chapter 10, and is usually assumed constant.)... 

The diffusion coefficient allows for both gas and liquid phase diffusion. It is given by (Stephen et al, 1998a,b) ... 

Since the Sorbex process is a liquid-phase fixed-bed process, the selection of particle size is an important consideration for pressure drop and process hydraulics. The exact particle size is optimized for each particular Molex process to balance the liquid phase diffusion rates and adsorbent bed frictional pressure drop. The Sorbex process consists of a finite number of interconnected adsorbent beds. These beds are allocated between the following four Sorbex zones zone 1 is identified as the adsorption zone, zone 2 is identified as the purification zone, zone 3 is identified as the desorption and zone 4 is identified as the buffer zone. The total number of beds and their allocation between the different Sorbex zones is dependent on the desired performance of the particular Molex process. Molex process performance is defined by two parameters extract normal paraffin purity and degree of normal paraffin recovery from the corresponding feedstock. Details about the zone and the bed allocations for each Molex process are covered in subsequent discussions about each process. 

Treatment of systems in which gas-phase diffusion, mass accommodation, liquid phase diffusion, and reaction both in the bulk and at the interface must be taken into account is discussed in Section E.l. 

Analysis of Systems with Gas- and Liquid Phase Diffusion, Mass Accommodation, and Reactions in the Liquid Phase or at the Interface... 

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. 

Here, issues in relation to the trickle flow regime—isothermal operation and plug flow for the gas phase—will be dealt with. Also, it is assumed that the flowing liquid completely covers the outer surface particles (/w = 1 or aLS = au) so that the reaction can take place solely by the mass transfer of the reactant through the liquid-particle interface. Generally, the assumption of isothermal conditions and complete liquid coverage in trickle-bed processes is fully justified with the exception of very low liquid rates. Capillary forces normally draw the liquid into the pores of the particles. Therefore, the use of liquid-phase diffusivities is adequate in the evaluation of intraparticle mass transfer effects (effectiveness factors) (Smith, 1981). 

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. 

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. 

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) ... 

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. 

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