Advancing-front model

Models for type 1 facihtation can be divided into two main categories the state of the art advancing front models and the reversible reaction models. 

Chan and Lee [24] assumed that a reaction equilibrium existed in both the internal and external continuous phases. They also incorporated the overall mass transfer resistance in their model as weU as accounting for leakage of the internal phase into the external phase as did Borwankar et al. [25], Liu and Liu [26], and Boyadzhiev et al. [27]. The reversible model was later extended by Baird et al. [22] to predict the extraction rate for multicomponent systems. A comparison study of the advanced front model and the reversible reaction model for multicomponent systems undertaken by Wang and Bunge [28] found the latter to be signihcantly better for mixtures of organic acids. 

Used the advancing front model and three reversible reaction models Bunge and Noble [16] model, Lin and Lxtng [23] model, modified Bunge and Noble model. 

Unsteady-state mathematical model based on the advancing front model of Ho et al. [3] considers a reaction front to exist within the emulsion globule and assumes instantaneous and irreversible reaction between the solute and the internal reagent at the membrane-internal droplet interface. 

The second mode of CSTR operation is that used by Thien (17) and by Li and Shrier (10). Here, both the external phase and the LM emulsion are in a continuous flow mode. The reactor effluents are sent to gravity settlers where the exterior phase is separated from the emulsion phase. The emulsion phase is then demulsified to recover the product followed by remulsification and recycle back to the reactor. Hatton and Wardius (48) have developed the advancing front model for the analysis of such staged LM operations. Thien (17) employed this scheme to remove the amino acid L-phenylalanine from simulated fermentation broth (dilute aqueous solution). 

Chakraborty ef al. [38] also used the advancing front model to analyze simultaneous extraction of copper(II) and nickel(II) using D2EIIPA as extractant and hydrochloric acid as stripping membrane phase... 

Advancing front model and three reversible reaction models were applied to describe 2-chlorophenol permeation from aqueous solutions [47]. The numerical implementation seemed more stable in the Bunge and Noble [8] model than in reversible reaction models that allowed changes of effective diffusivity with solute concentration in membrane phase, although results were quite similar for the three models. Kargari et al. [48] studied the selective separation of gold(III) ions from acidic aqueous solutions, using MIBK as carrier and LK-80 as emulsifier. They found only Au + ions is transported across the liquid membrane and nearly all (Pd2+, Cu2+, and Fe ) of other ions remained in the external... 

Sauter mean diameter was used for characterization of globules and internal droplets described by advanced front model and modehng of facihtated transport... 

Wardius [51] extended the advancing front model to be employed to multistage mixer-settler systems for liquid membrane operations. They presented a zero order solution to the perturbation equations based on the model developed by Ho et al. [29]. The emulsion globule residence time distribution in each mixer was assumed to be exponential and the fractional utilization of internal reagent was given by... 

Kopp et al. ( ) used this approach to examine the analogous planar problem with constant bulk solute concentration. Ho et al. (90), Kim et al. (9J[) and Stroeve and coworkers (92, 93) formulated advancing-front models which include both spherical geometry and depletion of solute in the continuous phase. All three models assume homogeneous distribution of noncirculating internal droplets within the globule, although Kira et al. assume a thin outer liquid... 

Hatton and coworkers have also analyzed processes involving ELMs. Using their advancing-front model as a basis, they have studied staged operations (10 1), continuous stirred tank reactors (105), and mixer cascades (106). One interesting aspect of their analysis is the effect of emulsion recycle. They analyzed the effect on extraction rate of recycling used emulsion and combining this with new emulsion. 

Following the approach of Hatton, Reed and coworkers (107) have analyzed continuous stirred tank extractors when reaction reversibility contributes. They have developed a simple way to extend the simpler pseudo steady state advancing front model to predict extractor performance even when reaction reversibility may be significant. 

The next chapter Is a modeling study of a continuous flow extraction system utilizing ELHs by Reed et al. (107). The authors consider the extraction of a solute which Is trapped In the Inner droplet phase by a chemical reaction. The paper compares predictions of the reversible reaction model of Bunge and Noble (96) to the advancing front model of Ho et al. (90) for a continuous flow ELM extractor. The calculatlonal results show that assuming Irreversible reaction can lead to underdesign of the process under conditions of high solute recovery where the outlet solute concentration Is low. Under these conditions, an exact analytical solution to the reversible reaction model can be obtained. 

This paper examines theoretically the continuous flow extraction by emulsion globules in which the transferring solute reacts with an internal reagent. The reversible reaction model is used to predict performance. These results are compared with advancing front calculations which assume an Irreversible reaction. A simple criterion which indicates the Importance of reaction reversibility on performance is described. Calculations show that assuming an irreversible reaction can lead to serious underdesign when low solute concentrations are required. For low solute concentrations an exact analytical solution to the reversible reaction problem is possible. For moderate solute concentrations, we have developed an easy parameter adjustment of the advancing front model which reasonably approximates expected extraction rates. 

To estimate the extent of globule utilization for a given exposure time requires a mathematical description of a diffusion-controlled extraction coupled with chemical reaction. Two approaches, the reversible reaction model ( ), and the advancing front model (27), will be described and compared. 

The sole difference between the advancing front and reversible reaction approach is the assumed size of the equilibrium constant for reaction 1. Advancing front models assume that reaction 1 is irreversible K is infinitely large. Finite values for K are assumed in the reversible reaction theory. 

Advancing Front Model. The advancing front model follows a similar approach except that solute diffusion only occurs through the fully reacted outer shell. The no-reaction effective diffusion coefficient, Deff,NR> applies in this case. The solute concentration is zero at the dimensionless location of the reaction front, x which moves from the globule surface (x= 1) toward the globule center. 

Since the normalized advancing front model Is adequate except for small oj values, a legitimate question Is whether oj of 1.0 or less Is likely to occur. Table I shows reaction equilibrium constants for several acidic and basic compounds and concentrations corresponding to oj = 1. Vadues of oj when solute concentrations are 1 ppm are also tabulated. Clearly, removal of trace organic acids or bases with small reaction constants can lead to small a values. 

By contrast, the advancing front model curve indicates that the extraction (o - aj) of (200-100) is not possible in a single stage the reversible reaction model shows a Te of 100 should work. 

Three methods which do not require solution of the nonlinear partial differential equation are presented for estimating extractor performance. The choice of method depends on the value of the dimensionless outlet solute concentration, oj. If (oj + 1)/oJ is close to 1, the reaction is effectively irreversible and the pseudosteady-state solution of the advancing front model satisfactorily predicts performance after normalization to include solute solubility in the globule. If (oj + 1)/oJ is not close to 1, the advancing front results will still apply, provided that the amount of solute extracted by reaction is small and membrane solubility controls. When oj is small enough so that (oj + 1) is close to 1, then the reversible reaction model can be reduced to a linear equation with an analytical solution. Otherwise, for oj values when neither (oj + 1) nor (oj + 1)/o is nearly 1, a reasonable first approximation is made by adjusting the actual concentration of internal reagent to an effective concentration which equals the amount consumed to reach equilibrium. 

Actinides, in nitric acid waste, 182 Advancing-front model, ELMs, 18,68 Alkali metal cations transport across bulk liquid membranes, 89-92 transport across liquid surfactant membranes, 93-95 transport across polymer-supported liquid, 95-96... 

Diffusion-Type Mass Transfer Models for Type 1 FacUitation. The state-of-the-art model for Type 1 facilitation is the advancing front model (2,7,8), In this model, the solute is assumed to react instantaneously and irreversibly with the internal reagent at a reaction surface which advances into the globule as the reagent is consumed. A perturbation solution to the resulting nonlinear equations is obtained. In general, the zero-order or pseudo-steady-state solution alone often gives an adequate representation of the diffusion process. 

Fales and Stroeve (9) extended the advancing front model to include the external phase mass transfer resistance outside the emulsion globules, which becomes significant (>10% error) if the Biot number is less than about 20. In addition to the external phase... 

Acyclic oligoamides, 167-168,170/ Advancing front model, description, 116 Alkali metal cation facilitated transport through supported liquid membranes with fatty acids electrogenic processes, 79-81 electroneutral transport, 76-80 experimental description, 77 kinetics, 81-85... 

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