Models fronts

Fig. 15 Attachment of L = 100 chain onto L = 50 growth front model. Growth front chains are immobilized. The chain exhibits significant mobility (a-e) before it establishes perfect registration with the surface (f). The values of time t are indicated in each frame... 

Fig. 9. Illustration of moving front model for swelling of a glassy polymer gel. Gel assumed to be thin slab. t0 Initially dry glassy state. t, At early and intermediate times after immersion in swelling solution gel contains glassy core and swollen rubbery periphery, with fronts separating the two phases. Core constraints swelling of periphery to occur only normal to front. t2 After fronts meet, swelling constraint vanishes, and swelling permitted in all directions. [Adapted with permission from Ret 24. Copyright CRC Press, Inc. Boca Raton, FL]...

Low-volume and covered-boom sprayers in general increase efficiency and are therefore more enviromnentally safe. Low-volume sprayers come as pull-behind or out-front models and apply the proper amount of active ingredient, using less water and chemical. 

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). 

Pales and Stroeve [31] investigated the effect of the continuous phase mass transfer resistance on solute extraction with double emulsion in a batch reactor. They presented an extension of the perturbation analysis technique to give a solution of the model equations of Ho et al. [29] taking external phase mass transfer resistance into account. Kim et al. [5] also developed an unsteady-state advancing reaction front model considering an additional thin outer liquid membrane layer and neglecting the continuous phase resistance. 

Dutta et al. [32] modified the pseudo-steady-state advancing reaction front model of Stroeve and Varanasi [30] by considering the polydispersity of the emulsion globules and the external phase mass transfer resistance. They also included the outer membrane film resistance in their model [5]. Their results were in good agreement with experimental data for phenol extraction. 

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... 

Lorbach and Hatton [56] analyzed the polydispersity and back mixing effects in terms of the advancing reaction front model by assuming pseudosteady-state diffusion within the macrodrop so that the zero order solution to the perturbation expansion could be used. Mok et al. [57] proposed a... 

A third possibility, is that the reaction is taking place according to a moving reaction front model, i.e., the reaction rate is so fast, that the adsorption and reaction take place immediately when the gas pulse arrives at the catalyst bed. This model is described in detail in the next paragraph. 

Figure 6. Schematical representation of the moving-reaction-front model. A pulsing CO over O-precovered platinum. B Pulsing CO (excess) and Oj alternately over initial O-covered platinum. Gray O-covered White CO covered or clean platinum. The numbers denote the pulse cycle number.

The moving-reaction-front model is able to describe the occurrence of the double pulse responses obtained in measurements. 

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. 

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