Liquid membranes mathematical models

An important number of references have been published dealing with many applications of supported liquid membranes. Mathematical modeling of the process has been developed and it can be generalized and applied to the determination of the response of different systems containing more than one solute. After evaluation of the parameters, process optimization can be applied using common optimization procedures, as described in the text. 

Ultrasound can thus be used to enhance kinetics, flow, and mass and heat transfer. The overall results are that organic synthetic reactions show increased rate (sometimes even from hours to minutes, up to 25 times faster), and/or increased yield (tens of percentages, sometimes even starting from 0% yield in nonsonicated conditions). In multiphase systems, gas-liquid and solid-liquid mass transfer has been observed to increase by 5- and 20-fold, respectively [35]. Membrane fluxes have been enhanced by up to a factor of 8 [56]. Despite these results, use of acoustics, and ultrasound in particular, in chemical industry is mainly limited to the fields of cleaning and decontamination [55]. One of the main barriers to industrial application of sonochemical processes is control and scale-up of ultrasound concepts into operable processes. Therefore, a better understanding is required of the relation between a cavitation coUapse and chemical reactivity, as weU as a better understanding and reproducibility of the influence of various design and operational parameters on the cavitation process. Also, rehable mathematical models and scale-up procedures need to be developed [35, 54, 55]. 

In this paper an overview of the developments in liquid membrane extraction of cephalosporin antibiotics has been presented. The principle of reactive extraction via the so-called liquid-liquid ion exchange extraction mechanism can be exploited to develop liquid membrane processes for extraction of cephalosporin antibiotics. The mathematical models that have been used to simulate experimental data have been discussed. Emulsion liquid membrane and supported liquid membrane could provide high extraction flux for cephalosporins, but stability problems need to be fully resolved for process application. Non-dispersive extraction in hollow fib er membrane is likely to offer an attractive alternative in this respect. The applicability of the liquid membrane process has been discussed from process engineering and design considerations. 

None of these indirect techniques relate the breakage of an individual particle to its mechanical properties. To achieve this, direct techniques are required. These are described later, and their capabilities and limitations discussed. Direct techniques also allow more sophisticated mathematical modelling to be undertaken, which is particularly valuable when the particles are not homogeneous, for example, cells with walls and membranes surrounding cytoplasm, or a liquid-filled microcapsule. 

Following on from this work two types of mathematical model were developed that do not rely on measuring the contact area. These models are the "liquid-drop" model (Yoneda, 1973) and the elastic membrane model (Cheng, 1987a Feng and Yang, 1973 Lardner and Pujara, 1980). 

A mathematical model to be solved numerically has been developed and used to predict the separation effects caused by nonstationary conditions for a liquid membrane transport. Numerical calculations were made to compute pertraction characteristics such as input and output membrane selectivity (ratio of respective fluxes), concentration profiles for cations bound by a carrier in a liquid membrane phase, and the overall separation factors. These quantities are discussed as dependent... 

Huang CR, Wang KC, and Zhou DW. Mathematical modelling of carrier-facilitated transport in emulsion liquid membranes. In Bartsch RA, Way JD, eds. Chemical Separations with Liquid Membranes, Washington, DC American Chemical Society, ACS s3miposium series 642, 1996 115-122. 

The mathematical model described here has illustrated that electrochemical effects can significantly influence protein flux in an affinity-mediated transport system. The system considered consists of a supported liquid membrane containing a pH-sensitive monoclonal antibody as carrier and human growth hormone as permeant. On a microscopic scale, Donnan inclusion of the hormone can increase the flux of hormone into the membrane. This allows more complex to be formed and simultaneously generates a steep hormone concentration gradient which drives a greater flux of free hormone than would occur in the absence of inclusion. 

An overview chapter by Hamel and Hunter presents the state of the art of research on bioseparations. Extraction processes using biphasic aqueous systems, liquid membranes, reversed-micellar systems, and membrane processes are all being actively studied. Significant advances in these topics, including predictive mathematical models, are presented in the first section. The second section includes several papers on affinity and other interaction techniques that are finding uses in protein purification. In the last section, we offer several reports that delineate advances in isolation and purification processes such as electrophoresis and chromatography. 

The bond-graph network of liquid membrane process can be successfully exploited for modeling the separation and transport ability of complex reaction-diffusion phenomena. However, such models involving appropriate mathematical formulations are especially useful in predicting the system s response to the changes in operating conditions and specific characteristics of the liquid membrane components. In general, such models are not... 

Ho et al.2 30 and Stroeve and Varanasi22 have developed rophisticaied mathematical models of liquid -membrane iranspon. These models ate of useful theonsjea) interest and provide some imponant insights into the most imponenl perameters affecting the rate and efficiency of liquid-membrane extraction process. 

Also, Reinhoudt and co-workers reported the data on guanidinium (thiocyanate) transport with various crowns through bulk [43] and supported [44] liquid membranes. They developed corresponding mathematical models and found that not only the host-guest complexation constant but also the lipophilicity of the host per se control ionophoric properties. Omitting mathematics, the natural reason is the possibility of carrier leakage from the membrane phase (see related work on partition coefficients and their increments for crown ethers, K-octanol/water [127]). This interplay leads, for example, to better ionophoric properties of more lipophilic... 

The mathematical modeling of liquid membrane separations Is essential to accurate prediction and scale-up of these systems. Also, accurate and complete models Identify the Important physical properties and operating conditions. Models can be used to Identify and guide the pertinent experimental program which should be followed. 

However, in spite of the known advantages and applications of liquid membrane separation processes in hollow-fiber contactors, there are scarce examples of industrial application. The industrial application of a new technology requires a reliable mathematical model and parameters that serve for design, cost estimation, and optimization purposes allowing to accurate process scale-up. " The mathematical modeling of liquid membrane separation processes in HFC is divided into two steps (1) the description of the diffusive mass transport rate and (2) the development of the solute mass balances to the flowing phases. 

The design and scale-up of liquid-membrane separation processes need separation and concentration mathematical models as reported in Section 29.2.1. When complex solutions such as wastewaters are treated, several simplifications according to the specific characteristics of the system are usually assumed in order to reduce the number of parameters and mathematical complexity of the EPT model. From a kinetic point of view, the transport through the membrane... 

Keshavarz, R, Ayatollah , S., Fathikalajahi, J. 2008. Mathematical modeling of gas-liquid membrane contactors using random distribution of fibers. J. Membr. Sci. 325 98-108. 

Bringas, E., San Roman, M.F., Irabien, J.A., and Ortiz, I. 2009. An overview of the mathematical modeling of liquid membrane separation processes in hollow fibre contactors. J. Chem. Technol. Biotechnol. 84 1583-1614. 

Leiber, J.P., Noble, R.D., Way, J.D., and Bateman, B.R. 1985. Mathematical modeling of facilitated liquid membrane transport systems containing ionicaUy charged species. Sep. Sci. Technol. 20 231. 

S. Suren, T. Wongsawa, U. Pancharoena, T. Prapasawat, A.W. Lotbongkum, Uphill transport and mathematical model of Pb(n) from dilute synthetic lead containing solutions across hollow fiber supported liquid membrane, Chem. Eng. 7. 191 (2012) 503-511. 

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