Solute Diffusion and Mass-Transfer Coefficients

Solute Diffusion and Mass-Transfer Coefficients For a binary system consisting of components A and B, the overall rate of mass transfer of component A with respect to a fixed coordinate is the sum of the rates due to diffusion and bulk flow ... 

Johansson L and Lofroth JE. Diffusion and interactions in gels and solution. I. Method. J. Colloid Interface Sci. 1991 142 116-120. Inoue SK, Guenther RB, and Hoag SW. Algorithm to determine diffusion and mass transfer coefficients. Proceedings of the Conference on Advances in Controlled Delivery, Baltimore (USA), August 19-20, 1996, pp. 145-146. 

Parameters a and b are related to the diffusion coefficient of solutes in the mobile phase, bed porosity, and mass transfer coefficients. They can be determined from the knowledge of two chromatograms obtained at different velocities. If H is unknown, b can be estimated as 3 to 5 times of the mean particle size, where a is highly dependent on the packing and solutes. Then, the parameters can be derived from a single analytical chromatogram. 

Cybulski and Moulijn [27] proposed an experimental method for simultaneous determination of kinetic parameters and mass transfer coefficients in washcoated square channels. The model parameters are estimated by nonlinear regression, where the objective function is calculated by numerical solution of balance equations. However, the method is applicable only if the structure of the mathematical model has been identified (e.g., based on literature data) and the model parameters to be estimated are not too numerous. Otherwise the estimates might have a limited physical meaning. The method was tested for the catalytic oxidation of CO. The estimate of effective diffusivity falls into the range that is typical for the washcoat material (y-alumina) and reacting species. The Sherwood number estimated was in between those theoretically predicted for square and circular ducts, and this clearly indicates the influence of rounding the comers on the external mass transfer. 

For quasi-stationary conditions in the film, a constant diffusion or mass-transfer coefficient, and if the compositon of the bulk solution is constant, the fractional conversion, F t), becomes... 

The first component ofEq. (3), fCsAcs. represents the solute diffusive flux, driven by Donnan equihbrium couphng with facilitation by lEM and LMF potentials. Comparison ofEq. (3) with the equations in the model for the BOHLM systems (see Chapter 5 and [46]) shows that the diffusive mass-transfer coefficient corresponds to the diffusive overaU mass-transfer coefficients, fCp/E on the feed side and Ke/r on the strip side of the BOHLM system with hydrophihc or ion-exchange membranes ... 

The diffusion theory states that matter is deposited in a continuous way on the surface of a crystal at a rate proportional to the difference in concentration between the bulk and the surface of the crystal. The mathematical analysis is then the same as for other diffusion and mass transfer processes and makes use of the film concept. Sometimes, the film theory is considered to be an oversimplification for crystallization and is replaced by a random surface removal theory (20-23). For both theories the rate of crystal growth (dm/dt) is given by equation XVII, where m, is the mass of solid deposited in time t k, the mass transfer coefficient by diffusion. A, the surface area of the crystal, c, the concentration in the supersaturated solution and Cj, the concentration at the crystal-solution interface (3). For the stagnant film and random surface removal model, equations XVIII and XIX can be used, respectively (3,4) D is the diffusion coefficient, x, the film thickness and f, the fractionai rate of surface renewal. 

Solubilities and diffusivities of gas are practically always required for design of gas-liquid process and obtaining solubility and diffusivity data for the gas-liquid system under consideration may be a chalenging problem so wide is the range of solutes and solvent the chemical engineer or researcher may encounter. Moreover the choice of a suitable gas-liquid contactor is also a question of matching these data, those concerning the reaction kinetics and the physical kinetics characteristics of the proposed reactor, i.e., specific gas-liquid interfacial area, heat and mass transfer coefficients and gas or liquid holdup. Some considerations on solubility and diffusivity will be proposed in part 1 of this review and on gas-liquid mass transfer in part 2. 

R is rate of reaction per unit area, a is interfacial area per unit volume, S is solubiHty of solute in continuous phase, D is diffusivity of solute, k is rate constant, kj is mass-transfer coefficient, is concentration of reactive species, and Z is stoichiometric coefficient. When Dk is considerably greater (10 times) than Ra = aS Dk. 

Dl = diffusivity of transferring solute in liquid, m /sec If the diffusivity, Dl, needed for use in the above equations is not known, it can be estimated from data or methods given in the Perry s Chemical Engineers, Handbook (Section 14 in 4th Edition or Section 3 in 5th Edition). Note that the calculation of the mass transfer coefficients for a given regime involves only physical properties and is independent of agitation conditions. 

A phenomenon that is particularly important in the design of reverse osmosis units is that of concentration polarization. This occurs on the feed-side (concentrated side) of the reverse osmosis membrane. Because the solute cannot permeate through the membrane, the concentration of the solute in the liquid adjacent to the surface of the membrane is greater than that in the bulk of the fluid. This difference causes mass transfer of solute by diffusion from the membrane surface back to the bulk liquid. The rate of diffusion back into the bulk fluid depends on the mass transfer coefficient for the boundary layer on feed-side. Concentration polarization is the ratio of the solute concentration at the membrane surface to the solute concentration in the bulk stream. Concentration polarization causes the flux of solvent to decrease since the osmotic pressure increases as the boundary layer concentration increases and the overall driving force (AP - An) decreases. 

A reagent in solution can enhance a mass transfer coefficient in comparison with that of purely physical absorption. The data of Tables 8.1 and 8.2 have been cited. One of the simpler cases that can be analyzed mathematically is that of a pseudo-first order reaction that goes to completion in a liquid film, problem P8.02.01. It appears that the enhancement depends on the specific rate of reaction, the diffusivity, the concentration of the reagent and physical mass transfer coefficient (MTC). These quantities occur in a group called the Hatta number,... 

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