Forward solute diffusion

The mass transport (both water and solute) in the FO processes is illustrated in Figure 14.1. The water in the FS of higher water chemical potential will permeate through the membrane into the DS of lower water chemical potential. Meanwhile, the draw solute in the DS will reversely diffuse into the FS due to the concentration gradient across the membrane. Solute diffusion in this fashion is referred to as reverse solute diffusion. Coupled with reverse solute diffusion, solute in the FS will forwardly diffuse into the DS if its concentration in FS is greater than that in DS. Solute diffusion in this fashion is referred to as forward solute diffusion. This section will introduce the fundamentals of the mass transport and solute rejection in FO processes. 

To introduce the forward solute diffusion, boron, a contaminant of interest in the desalination industry, is selected as the representative feed solute following Jin s study (Jin et al., 2011). A schematic of boron transport into DS in the FO processes is shown in Figure 14.3. 

She, Q., Jin, X., Li, Q. Tang, C.Y. (2012a) Relating reverse and forward solute diffusion to membrane fouling in osmotically driven membrane processes. Water Research, 46 (7), 2478-2486. 

In order to solve the mathematical model for the emulsion hquid membrane, the model parameters, i. e., external mass transfer coefficient (Km), effective diffu-sivity (D ff), and rate constant of the forward reaction (kj) can be estimated by well known procedures reported in the Hterature [72 - 74]. The external phase mass transfer coefficient can be calculated by the correlation of Calderback and Moo-Young [72] with reasonable accuracy. The value of the solute diffusivity (Da) required in the correlation can be calculated by the well-known Wilke-Chang correlation [73]. The value of the diffusivity of the complex involved in the procedure can also be estimated by Wilke-Chang correlation [73] and the internal phase mass transfer co-efficient (surfactant resistance) by the method developed by Gu et al. [75]. 

As the sample plug starts being pushed forward, axial diffusion is the main component of the dispersion process, due to the high concentration gradients at the sample/carrier stream interface. The hypothetical peak shape associated with the flowing sample is shown in Fig. 5.9b, which corresponds to the first theoretical Taylor solution [28,29] for the diffusive-convective equation (Eq. 3.4). Situations associated with Fig. 5.9a,b never occur in practice in flow injection analysis. 

S. C. Choo, Theory of a forward-biased diffused-junction p-i-n rectifier—Part 1 Exact numerical solution, IEEE Trans. Electron Devices ED-19 (1972) 954-966. 

The driving force for the transfer process was the enhanced solubility of Br2 in DCE, ca 40 times greater than that in aqueous solution. To probe the transfer processes, Br2 was recollected in the reverse step at the tip UME, by diffusion-limited reduction to Br . The transfer process was found to be controlled exclusively by diffusion in the aqueous phase, but by employing short switching times, tswitch down to 10 ms, it was possible to put a lower limit on the effective interfacial transfer rate constant of 0.5 cm s . Figure 25 shows typical forward and reverse transients from this set of experiments, presented as current (normalized with respect to the steady-state diffusion-limited current, i(oo), for the oxidation of Br ) versus the inverse square-root of time. 

As reversible ion transfer reactions are diffusion controlled, the mass transport to the interface is given by Fick s second law, which may be directly integrated with the Nernst equation as a boundary condition (see, for instance. Ref. 230 232). A solution for the interfacial concentrations may be obtained, and the maximum forward peak may then be expressed as a function of the interfacial area A, of the potential scan rate v, of the bulk concentration of the ion under study Cj and of its diffusion coefficient D". This leads to the Randles Sevcik equation [233] ... 

In the case of diffusion in the absence of advection, the stability of a forward-intime solution in one dimension is given by the von Neuman criterion,... 

A further important step forward was the work of Nemst [73, 74] and Planck [81, 82] on transport in electrolyte solutions. Here the concept of the diffusion potential was defined diffusion potential arises when the mobihties of the electrically-charged components of the electrolyte are different and is important both for description of conditions within membranes as well as for quantitative determination of the liquid-junction potential. 

This section describes the experimental methods and focuses on the estimation of diffusivity after the experiment. The analytical methods are not described here. Estimation of diffusivity from homogeneous reaction kinetics (e.g., Ganguly and Tazzoli, 1994) is discussed in Chapter 2 and will not be covered here. Determination of diffusion coefficients is one kind of inverse problems in diffusion. This kind of inverse problem is relatively straightforward on the basis of solutions to forward diffusion problems. The second kind of inverse problem, inferring thermal history in thermochronology and geospeedometry, is discussed in Chapter 5. 

Dungan et al. [186] have measured the interfacial mass transfer coefficients for the transfer of proteins (a-chymotrypsin and cytochrome C) between a bulk aqueous phase and a reverse micellar phase using a stirred diffusion cell and showed that charge interactions play a dominant role in the interfacial forward transport kinetics. The flux of protein across the bulk interface separating an aqueous buffered solution and a reverse micellar phase was measured for the purpose. Kinetic parameters for the transfer of proteins to or from a reverse micellar solution were determined at a given salt concentration, pH, and stirring... 

Sherwood and Pigford (S9) have discussed the problem of the absorption of a solute A by a solvent S upon solution, A may be converted into B according to the reaction A = B (k/ and krf being the forward and reverse reaction-rate constants, and K = k//k/). The concentration of A is maintained at cAo at the surface of the liquid S, and it is assumed that S is semiinfinite in extent. It is further assumed that B is nonvolatile that is, it cannot escape from solvent S. Equation (51) is then used to explain the diffusion of A and B, with DAg and DBs taken as concentration independent, and the term containing the molar average velocity w is neglected. Hence the mathematical statement of the problem is (for very dilute solutions of A and B)... 

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