Solute mass transfer

Various mathematical concepts and techniques have been used to derive the functions that describe the different types of dispersion and to simplify further development of the rate theory two of these procedures will be discussed in some detail. The two processes are, firstly, the Random Walk Concept [1] which was introduced to the rate theory by Giddings [2] and, secondly, the mathematics of diffusion which is both critical in the study of dispersion due to longitudinal diffusion and that due to solute mass transfer between the two phases. The random walk model allows the relatively simple derivation of the variance contributions from two of the dispersion processes that occur in the column and, so, this model will be the first to be discussed. 

BATCH VOLUMES AND SOLUTE MASS TRANSFER COEFFS... 

In order to increase the overall extraction efficiency during SFE sonication has been applied [352]. Ultrasound creates intense sinusoidal variations in density and pressure, which improve solute mass transfer. Development of an SFE method is a time-consuming process. For new methods, analysts should refer the results to a traditional sample preparation method such as Soxhlet or LLE. 

The driving force of the mass-transfer process now can be related to the concentration gradient of the reacting species, or to the concentration difference between electrode and bulk solution. Mass-transfer rates then can be related in a general way to the concentration driving force. For example, if... 

Efficiency the organic modifier can be used to adjust solvent selectivity as normally practiced in reversed-phase chromatography. Lowers mobile-phase viscosity and improves solute mass-transfer kinetics. 

Concerning the structure of dispersed CLAs, the model originally proposed by Sebba [57] of a spherical oil-core droplet surrounded by a thin aqueous film stabilized by the presence of three surfactant layers is, in our opinion, essentially correct. However, there is still little direct evidence for the microstructure of the surfactant interfaces. From an engineering point of view, however, there is now quantitative data on the stability of CLAs which, together with solute mass transfer kinetics, should enable the successful design and operation of a CLA extraction process. 

Since the dimensionless equations and boundary conditions governing heat transfer and dilute-solution mass transfer are identical, the solutions to these equations in dimensionless form are also identical. Profiles of dimensionless concentration and temperature are therefore the same, while the dimensionless transfer rates, the Sherwood number (Sh = kL/ ) for mass transfer, and the Nusselt number (Nu = hL/K ) for heat transfer, are identical functions of Re, Sc or Pr, and dimensionless time. Most results in this book are given in terms of Sh and Sc the equivalent results for heat transfer may be found by simply replacing Sh by Nu and Sc by Pr. 

A/ Separation length between successive eddy promoters (m) A//7B Solute mass transferred into compartment C (kg)... 

In a great number of papers dealing with the design of ED stacks, and especially in the recent and comprehensive paper by Lee et al. (2002d), the solute mass transfer coefficient (km) is expressed as a nonlinear function of the superficial flow velocity (vs) ... 

EMPIRICAL CORRELATION FOR PREDICTING THE SOLUTE MASS TRANSFER COEFFICIENT (km) IN ED CELLS WITH EDDY PROMOTERS... 

The water transport number (tw) accounts for water transport associated to ion electromigration and thus controls the maximum solute weight concentration theoretically achievable in the concentrate. By dividing the instantaneous solute mass transferred into the C compartment by the actual volume accumulated into tank C for 6 tending to infinite, the following can be obtained ... 

Current-voltage tests to determine the limiting current intensity (/lim), ion transport numbers (ta, tc+), and surface resistances (ra, rc) in anionic and cationic membranes, as well as solute mass transfer coefficient (Am). 

The solute mass transfer coefficient (km) in ED stacks approximately varies with the square root of the liquid superficial velocity (vs) in agreement with the correlations reported in Table III, even if they can differ from those predicted within a 30% deviation band because of the different cell and spacer configuration used. 

Furthermore, these characterizations will be completed in determining mass transfer parameters a and Ps, Cconv and Jdiff, respectively, the reflection coefficient and the solute permeability of the membranes, the part of solute mass transfer dedicated to convection and Jam the part of mass transfer dedicated to hydration-diffusion, for two synthetic chlorides and sulphates sodium salts solutions under different concentrations, 10-3 and 10-1 M. 

By following Cp vs. the reverse of the permeate flux, it is possible to quantify separately both part of the solute mass transfer occurring in NF convection and solvation (hydration)/diffusion as developed recently [9], The results are expected to be valid only in some limited domains of operating conditions (Jdiff and Cconv = Ctes) but may be useful for the comparison of the behaviour of different membranes. 

Weavers LK, Hoffmann MR. Sonolytic decomposition of ozone in aqueous solution mass transfer effects. Environ Sci Technol 1998 32 3941. 

The bonded hydrocarbon packings...are very hydrophobic...Therefore, in reversed phase separations...it is desirable to use aqueous mobile phases containing > 10% of a miscible organic solvent...to improve wetting characteristics. Mobile phases with no or low concentrations of organic solvent produce broad peaks because of the slow equilibrium resulting from the resistance to solute mass transfer across the interface of the two very unlike phases."... 

For spherical particles, the solute mass transfer rate from the bulk to the solute inside the pores is described by a linear kinetic law ... 

Results such as these suggest that dissolution may be treated mathematically via generalized solution mass transfer correlations, with surface reaction having a negligible effect on determining the over-all rate. However, more systems should be tested before sweeping conclusions are drawn. 

The problem of transferring corrosion rate data from one hydrodynamic system to another has also been considered in some depth by Chen et al. [18], by using the corrosion of 90 10 Cu Ni alloy in aerated 1 m NaCl solution at 25 °C in pipe-flow, annular-flow, and rotating-cylinder systems. The authors recognized that two mass-transfer processes should be distinguished transfer through the diffusion boundary layer in e solution (mass-transfer coefficient, h), and transfer through the corrosion product film ( f). The overall mass-transfer coefficient was defined as... 

The fundamental mechanisms for solute mass transfer in liquid-liquid extraction involve molecular diffusion driven by a deviation from equilibrium. When a liquid feed is contacted with a liquid solvent, solute transfers from the interior of the feed phase across a liquid-liquid interface into the interior of the solvent phase. Transfer of solute will continue until the solute s chemical potential is the same in both phases and equilibrium is achieved. 

Limitations and problems additional to those of the speciation codes The models do not consider solid-solution mass transfer and provide only limited information on ion exchange/adsorption mass transfer. All the programs except PHREEQE, PHRQPITZ, and MINTEQA2 keep track of water mass. Except for EQ3/6 and the Geochemist s Workbench, rate laws for mass transfer kinetics cannot be specified. Convergence problems occur more often than for the speciation codes. 

As outlined earlier, hemodialysis and hemofiltration require the removal of solutes smaller than albumin from blood. Solute mass transfer rates across hemodialysis membranes cannot exceed the diffusivity of the solute In water. Solute diffusivity decreases with Increasing molecular diameter (Stokes-Einstein relationship) consequently, solute mass transfer rates for hemodlalyzers intrinsically decrease with increasing molecular size. In addition to limitations Imposed by diffusion In solution, mass transfer is further limited by diffusion resistance in the membrane as well as boundary layer effects resulting from laminar flow both of these effects are also functions of molecular size. The quantitation of mass transfer In hemodlalyzers has been reviewed extensively (22). 

As explained in Sec. 4.4.4, the movement of components through a chromatography column can be modelled by a two-phase rate model, which is able to handle multicomponents with nonlinear equilibria. In Fig. 1 the column with segment n is shown, and in Fig. 2 the structure of the model is depicted. This involves the writing of separate liquid and solid phase component balance equations, for each segment n of the column. The movement of the solute components through the column occurs by both convective flow and axial dispersion within the liquid phase and by solute mass transfer from the liquid phase to the solid. 

---