Roles of Facilitated Transport Through HFSLM in Engineering Applications

For a number of manufacturing processes, separation, concentration and purification are important to handle intermediates, products, by-products and waste streams. In this regards mass and heat transfer play a significant role to attain efficient results. Concern to the separation operations, they can be classified as energy-intensive interphase mass transfer processes and less energyor less material-intensive intraphase mass transfer processes (Henley & Seader, 1981). With environmental and energy constraints in these days, for sustainability it is of much concern the requirements of process intensification and looking for the most effective operation based on green chemistry concepts (Badami, 2008; Escobar & Schafer, 2010; Matthews, 2007). Membrane technologies are a potential sustainable solution in this point of view. In contrast to the energy-intensive interphase mass transfer processes as distillation and extraction, membrane separation is an intraphase-mass-transfer process without the energy-intensive step of creating or introducing a new phase. It involves the selective diffusion of target species through the membrane at different rates. Although membrane operations are a relatively new type of separation process, several of them are fast-growing and successfully not only in biological systems but also a large industrial scale, e.g., food and bioproduct processing (Jirjis & Luque, 2010; Lipnizki, 2010). They can apply for a wide range of applications and provide meaningful advantages over conventional separation processes. In applications of controlling drug delivery, a membrane is generally used to moderate the permeation rate of a drug from its reservoir to the human body. In applications for safety regulations of food packaging, the membrane controls the permeation of undesirable constituents completely. In separation purposes, the membrane allows one component in a feed mixture to permeate itself but prohibits permeation of others. Among several membrane types, supported liquid membranes (SLMs) or immobilized liquid membranes (ILMs) containing carriers or extractants to facilitate selective transport of gases or ions draw high interest of the researchers and users in the industry as they are advanced economical feasible for pre-concentration and separation of the target species. So far, four types of supported liquid membrane modules (spiral wound, hollow fiber, tubular and flat sheet or plate and frame) have been used in the industry (Baker, 2007; Cui et al., 2010). The


Introduction
For a number of manufacturing processes, separation, concentration and purification are important to handle intermediates, products, by-products and waste streams. In this regards mass and heat transfer play a significant role to attain efficient results. Concern to the separation operations, they can be classified as energy-intensive interphase mass transfer processes and less energy-or less material-intensive intraphase mass transfer processes (Henley & Seader, 1981). With environmental and energy constraints in these days, for sustainability it is of much concern the requirements of process intensification and looking for the most effective operation based on green chemistry concepts (Badami, 2008;Escobar & Schäfer, 2010;Matthews, 2007). Membrane technologies are a potential sustainable solution in this point of view. In contrast to the energy-intensive interphase mass transfer processes as distillation and extraction, membrane separation is an intraphase-mass-transfer process without the energy-intensive step of creating or introducing a new phase. It involves the selective diffusion of target species through the membrane at different rates. Although membrane operations are a relatively new type of separation process, several of them are fast-growing and successfully not only in biological systems but also a large industrial scale, e.g., food and bioproduct processing (Jirjis & Luque, 2010;Lipnizki, 2010). They can apply for a wide range of applications and provide meaningful advantages over conventional separation processes. In applications of controlling drug delivery, a membrane is generally used to moderate the permeation rate of a drug from its reservoir to the human body. In applications for safety regulations of food packaging, the membrane controls the permeation of undesirable constituents completely. In separation purposes, the membrane allows one component in a feed mixture to permeate itself but prohibits permeation of others. Among several membrane types, supported liquid membranes (SLMs) or immobilized liquid membranes (ILMs) containing carriers or extractants to facilitate selective transport of gases or ions draw high interest of the researchers and users in the industry as they are advanced economical feasible for pre-concentration and separation of the target species. So far, four types of supported liquid membrane modules (spiral wound, hollow fiber, tubular and flat sheet or plate and frame) have been used in the industry (Baker, 2007;Cui et al., 2010). The www.intechopen.com hollow fiber supported liquid membrane (HFSLM) is renowned as a favorable system to separate valuable compounds or pollutants at a very low concentration and has specific characteristics of simultaneous extraction and stripping of the low-concentration target species in one single stage, non-equilibrium mass transfer, high selectivity and low solvent used. This chapter describes transport mechanisms in HFSLM and shows some applications with reference to our up-to-date publications, for example the effective extraction and recovery of praseodymium from nitrate solutions of mixed rare earths, RE(NO 3 ) 3 . Mass transfer phenomena in the system, the extraction equilibrium constant (K ex ), distribution ratio (D), permeability coefficient (P), aqueousphase mass-transfer coefficient (k i ) and organic-phase mass-transfer coefficient (k m ) were reported (Wannachod et al., 2011). -the enhancement of uranium separation from trisodium phosphate (a by-product from monazite processing) by consecutive extraction with synergistic extractant via HFSLM (Lothongkum et al., 2009). -a mathematical model describing the effect of reaction flux on facilitated transport mechanism of Cu(II) through the membrane phase of the HFSLM system. The model was verified with the experimental separation results of Cu(II) in ppm level by LIX84I dissolved in kerosene. The model results were in good agreement with the experimental data at the average percentage of deviation of 2%. (Pancharoen et al., in press).

Membrane structures, materials and modules
The performance of membrane relates closely to its structure, material and module. It is known that porous membranes can be classified according to their structures into microporous and asymmetric membranes. Microporous membranes are designed to reject the species above their ratings. They can get blocked easily compared to asymmetric membranes. In case of membrane materials, polymeric or organic membranes made of various polymers (e.g., cellulose acetate, polyamide, polypropylene, etc) are cheap, easy to manufacture and available of a wide range of pore sizes. However, some limitations like pH, temperature, pressure, etc can impede the applications of polymeric membranes. On the other hand, ceramic or inorganic membranes have advantages of high mechanical strength, high chemical and thermal stability over the polymeric membranes but they are brittle and more expensive. In terms of membrane modules, the development of membrane module with large surface areas of membrane at a relatively low manufacturing cost is very important. Resistance to fouling, which is a particularly critical problem in liquid separation, depends on the membrane module. Of four types of the SLM modules (spiral wound, hollow fiber, tubular and flat sheet or plate and frame), hollow fiber module ( Fig.1) has the greatest surface areato-volume ratio resulting in high mass transfer coefficient and is the most efficient type of Fig. 1 Baker, 2007) In principle, important performance characteristics of membranes are 1) permeability, 2) selectivity and retention efficiency, 3) electrical resistance, 4) exchange capacity, 5) chemical resistance, 6) wetting behavior and swelling degree, 7) temperature limits, 8) mechanical strength, 9) cleanliness, and 10) adsorption properties (Kislik, 2010). We will discuss in section 3 relevant to permeability, resistance and mass transfer across the HFSLM.

Hollow fiber supported liquid membrane (HFSLM)
The characteristics of the hollow fiber module are shown in Table 2. 2. simultaneous extraction and stripping of very low-concentration target species (either precious species or toxic species) in one single stage; 3. mild product treatment due to moderate temperature operation; 4. compact and modular design for easy installation and scaling up for industrial applications; 5. low energy consumption; 6. lower capital cost; 7. lower operating cost (consuming small amounts of extractant and solvent and low maintenance cost due to a few moving parts); 8. higher flux; 9. non-equilibrium mass transfer. As stated, the extremely important disadvantage of HFSLM is the fouling of the hollow fibers causing a reduction in the active area of the membrane and therefore a reduction in flux and process productivity over time. Fouling can be minimized by regular cleaning intervals. The concepts of membrane fouling and cleaning were explained by Li & Chen (Li & Chen, 2010). Active research includes, for example, membrane surface modification (to reduce fouling, increase flux and retention), new module designs (to increase flux, cleanability), etc should be further studied. In short, flux enhancement and fouling control were suggested by different approaches separately or in combination (Cui et al., 2010;Scott & Hughes, 1996): 1. hydrodynamic management on feed side; 2. back flushing or reversed flow and pulsing; 3. membrane surface modification; 4. feed pretreatment; 5. flux control; 6. regular effective membrane cleaning.

Mass transfer across HFSLM
Mass transfer plays significant role in membrane separation. The productivity of the membrane separation processes is identified by the permeate flux, which represents rate of target species transported across the membrane. In general practice, high selectivity of membranes for specific solutes attracts commercial interest as the membranes can move the specific solutes from a region of low concentration to a region of high concentration. For example, membranes containing tertiary amines are much more selective for copper than for nickel and other metal ions. They can move copper ions from a solution whose concentration is about 10 ppm into a solution whose concentration is 800 times higher. The mechanisms of these highly selective membranes are certainly different from common membranes which function by solubility mechanism or diffusion. The selectivity of these membranes is, therefore, dominated by differences in solubility. These membranes sometimes not only function by diffusion and solubility but also by chemical reaction. In this case, the transport combines diffusion and reaction, namely facilitated diffusion or facilitated transport or carrier-mediated transport (Cussler, 1997). For an in-depth understanding of the facilitated transport through liquid membrane, we recommend to read (Kislik, 2010). The facilitated transport mechanisms can be described by solute species partitioning (dissolving), ion complexation, and diffusion. The detailed steps are as follows: www.intechopen.com Step 1. Metal ions or target species in feed solution or aqueous phase are transported to a contact surface between feed solution and liquid membrane, subsequently react with the organic extractant at this interface to form complex species.
Step 2. The complex species subsequently diffuse to the opposite side of liquid membrane by the concentration gradient. It is assumed that no transport of target species passes this interface.
Step 3. The complex species react with the stripping solution at the contact surface between liquid membrane and the stripping solution and release metal ions to the stripping phase.
Step 4. Metal ions are transferred into the stripping solution while the extractant moves back to liquid membrane and diffuses to the opposite side of liquid membrane by the concentration gradient to react again with metal ions in feed solution. The facilitated transport mechanisms through the hollow fiber module are shown in Fig. 2. The facilitated transport through an organic membrane is used widely for the separation applications. The selectivity is controlled by both the extraction/ stripping (back-extraction) equilibrium at the interfaces and the kinetics of the transported complex species under a non-equilibrium mass-transfer process (Yang, 1999). The distribution ratio should be derived as a function of the extraction equilibrium constant as Mass transfer through HFSLM for the separation of the target species in terms of permeability coefficient (P) depends on the overall mass transfer resistance. To determine the overall mass transfer coefficient for the diffusion of the target species through HFSLM, the relationship between the overall mass transfer coefficient and the permeability coefficient is deployed. The permeability coefficient is reciprocal to the mass transfer coefficients as follows (Urtiaga et al., 1992;Kumar et al., 2000;Rathore et al., 2001) ii il m mo s 11 r1r 1 Pk rP r k   where k i and k s are the feed-phase and stripping-phase mass transfer coefficients, r lm is the log-mean radius of the hollow fiber in tube and shell sides, r i and r 0 are the inside and outside radius of the hollow fiber, P m is membrane permeability coefficient relating to the distribution ratio (D) in Eq. (4) and can be defined in terms of the mass transfer coefficient in liquid membrane (k m ) as Three mass transfer resistances in Eq. (5) are in accordance with three steps of the transport mechanisms. The first term represents the resistance when the feed solution flows through the hollow fiber lumen. The second resistance relates to the diffusion of the complex species through liquid membrane that is immobilized in the porous wall of the hollow fibers. The third resistance is due to the stripping solution and liquid membrane interface outside the hollow fibers. The mass transfer resistance at the stripping interface can be disregarded as the mass transfer coefficient in the stripping phase (k s ) is much higher than that in the feed phase (k i ) according to the following assumptions (Uedee et al., 2008): 1. The film layer at feed interface is much thicker than that at the stripping interface. This is because of a combination of a large amount of target species in feed and co ions in buffer solution at the feed interface while at the stripping interface, a few target species and stripping ions exist. In Eqs. (7) and (8), thick feed interfacial film (l if ) makes the mass transfer coefficient in feed phase (k i ) much lower than that in the stripping phase (k s ).

Feed-mass transfer coefficient
Stripping-mass transfer coefficient 2. The difference in the concentration of target species in feed phase (C f ) and the concentration of feed at feed-membrane interface (C f *) is higher than the difference in www.intechopen.com the concentration of stripping phase at membrane-stripping interface (C s *) and the concentration of target species in stripping phase (C s ). At equal flux by Eq. (9), k i is, therefore, much lower than k s .
3. From Eq. (5), we can ignore the third mass transfer resistance. This is attributed to the direct contact of stripping ions with the liquid membrane resulting in rapid dissolution and high mass transfer coefficient of the stripping phase. P m in Eq. (5) can be substituted in terms of the distribution ratio (D) and the mass transfer coefficient in liquid membrane (k m ) in Eq. (6) In addition, from the permeability coefficient (P) by Danesi (Danesi, 1984): We can calculate the permeability coefficient from the slope of the plot between Table 3 shows some applications of HFSLM and their mass transfer related.

HFSLM in engineering applications
Compared to conventional separations, membrane separations are attractive for the processing of food, bioproducts, etc where the processed products are sensitive to temperature since most membrane separations involve no chemical, biological, or thermal changes (or moderate temperature changes) of the target component during processing. For environmental-related applications, membrane separation has developed into an important technology for separating volatile organic compounds (VOCs), e.g., acetaldehyde, BTXs, ethylene oxide, trichloroethylene, etc and other gaseous air pollutants from gas streams (Schnelle and Brown, 2002;Simmon et al., 1994). The following works show the applications of using HFSLM and the role of facilitated transport in separation of praseodymium from nitrate solution of mixed rare earths RE(NO 3 ) 3 (Wannachod et al., 2011), separation of uranium from trisodium phosphate from monazite ores processing (Lothongkum et al., 2009) supplied by the Rare Earth Research and Development Center, Office of Atoms for Peace, Bangkok, Thailand, and separation of Cu(II) by LIX84I.

Effective extraction and recovery of praseodymium from mixed rare earths
Praseodymium (Pr), one of the elements recovered from mixed rare earths (REs), is very useful, e.g., as a composition in mischmetall alloy and a core material for carbon arcs in film www.intechopen.com  (Ortiz et al., 1996) Cr ( Table 3. Applications of HFSLM and mass transfer related studio light and searchlights. Praseodymium produces brilliant colors in glasses and ceramics. The composition of yellow didymium glass for welding goggles derived from infrared-heat absorbed praseodymium. Currently, the selective separation and concentration of mixed rare earths are in great demand owing to their unique physical and chemical properties for advanced materials of high-technology devices. Several separation techniques are in limitations, for example, fractionation and ion exchange of REs are time consuming. Solvent extraction requires a large number of stages in series of the mixer settlers to obtain high-purity REs. Due to many advantages of HFSLM and our past successful separations of cerium(IV), trivalent and tetravalent lanthanide ions, etc by HFSLM Patthaveekongka et al., 2006;Ramakul et al., 2004Ramakul et al., , 2005Ramakul et al., , 2007, we again approached the HFSLM system for extraction and recovery of praseodymium from mixed rare earth solution.
The system operation is shown in Fig. 3. Of three extractants, Cyanex 272 in kerosene found to be more suitable for high praseodymium recovery than Aliquat 336 and Cyanex 301 as shown in Fig. 4. Higher extraction of 92% and recovery of 78% were attained by 6-cycle continuous operation about 300 min as shown in Fig. 6. In this work, the extraction equilibrium constant (K ex ) obtaining from Fig. 7 was 1.98 x 10 −1 (Lmol -1 ) 4 . The distribution ratio (D) at Cyanex 272 concentration of 1.0-10 (%v/v) were calculated and found to be increased with the extractant concentration and agreed with Pancharoen et al., 2010. We obtained the permeability coefficients for praseodymium at Cyanex 272 concentration of 1.0-10 (%v/v) from Fig.8. The mass transfer coefficients in feed phase (k i ) and in liquid membrane (k m ) of 0.0103 and 0.788 cm s -1 , respectively were www.intechopen.com obtained from Fig.9. Because k m is much higher than k i , it indicates that the diffusion of praseodymium ions through the film layer between the feed phase and liquid membrane is the rate-controlling step.

Enhancement of uranium separation from trisodium phosphate
Two grades of trisodium phosphate, food and technical grades, are extensively used for various purposes. Food grade is used as an additive in cheese processing. Technical grade is used for many applications, e.g., in boiler-water treatment, testing of steel parts after pickling, industrial detergents such as degreasers for steels, and heavy-duty domestic cleaners. As trisodium phosphate is a by-product from the separation of desired rare earths in monazite processing, it is contaminated by some amount of uranium which is often found with the monazite. Uranium is a carcinogen on the other hand it is useful as a radioactive element in the front and back ends of the nuclear fuel cycle, therefore the separation method to recover uranium from trisodium phosphate is necessary. For 45-ppm-uraniumcontaminated trisodium phosphate solution, HFSLM is likely a favorable method as it can simultaneously extract the ions of very low concentration and can recover them in one single operation. Undoubtedly, the facilitated transport across the HFSLM accelerates the extraction and recovery of uranium. Eq. 13 shows that uranium species form complex species with Aliquat 336 (tri-octyl methyl ammonium chloride: CH 3 R 3 N + Cl -) in modified leaching and extraction of uranium from monazite (El-Nadi et al., 2005).  Fig. 12 shows percentage of uranium extraction by different extractants. We can see that D2EHPA (di (2-ethylhexyl) phosphoric acid) obtained high percentage of extraction, however its extractability abruptly decreased with time. Thus, Aliquat 336, of which its extractability followed D2EHPA and decreased slightly with time, was considered the most appropriate extractant for uranium. It can be attributed that uranium ions in trisodium phosphate solution are in [UO 2 (CO 3 ) 3 ] 4-and Aliquat 336, a basic extractant, is good for cations while D2EHPA, an acidic extractant, is good for anions form of UO 2 2+ . The percentage of uranium extraction at different concentrations of Aliquat 336 is shown in Fig. 13. and 53%, respectively by 7-cycle separation in 350 min. The percentage of uranium stripping was much lower than the percentage of extraction presuming that uranium ions accumulated in liquid membrane phase of the hollow fiber module. This is a limitation of the HFSLM applications. For higher stripping, a regular membrane service is needed. In conclusion, the remaining amount of uranium ions in trisodium phosphate solution was 0.22 ppm, which stayed within the standard value 3-ppm uranium of the technical-grade trisodium phosphate. Further study on a better stripping solution for uranium ions is recommended.

Reaction flux model for extraction of Cu(II) with LIX84I
In regard to apply the hollow fiber contactor for industrial scale, the reliable mathematical models are required. The model can provide a guideline of mass transfer describing the transport mechanisms of the target species through liquid membrane, and predict the extraction efficiency. Normally, different types of the extractants, their concentration and transport mechanisms (diffusion and facilitated transport or carrier-mediated transport) play important roles on the extraction efficiency. The facilitated transport mechanism relates to the reaction flux of chemical reaction between the target species and the selected single extractant or synergistic extractant to form complex species (Bringas et al., 2009;Kittisupakorn et al., 2007;Ortiz et al., 1996). In principle, the metal-ion transport through the membrane phase occurs when the metal ions react with the selected extractant at the interface between feed phase or aqueous phase and liquid membrane phase, consequently the generated complex species diffuse through the membrane phase. In this work, we developed a mathematical model describing the effect of reaction flux on facilitated transport mechanism of copper ions through the HFSLM system because copper is used extensively in many manufacturing processes, for example, electroplating, electronic industry, hydrometallurgy, etc. Therefore, copper ions, which are toxic and nonbiodegradable, may contaminate wastewaters and cause environmental problems and health effects if no appropriate treatment is taken (Lin & Juang, 2001;Ren et al., 2007). The model was verified with the experimental extraction of copper ions in ppm level using LIX84I dissolved in kerosene by continuous counter-current flow through a single-hollow www.intechopen.com fiber module. It is known that LIX-series compounds are the most selective extractants of high selectivity and widely used for copper ions (Breembroek et al., 1998;Campderros et al., 1998;Lin & Juang, 2001;Parhi & Sarangi, 2008;Sengupta, et al., 2007). The schematic flow diagram of the separation via HFSLM is shown in Fig. 16. The transport mechanism of copper ion in micro porous hollow fiber is presented schematically in Fig. 17. The chemical reaction at the interface between feed phase and liquid membrane phase takes place when the extractant (RH) reacts with copper ions in feed (Eq. (15)).
(o rg ) CuR is the complex species of copper ion in liquid membrane phase.
where A is copper ion, B is LIX84I, C is complex species of copper ion and LIX84I, D is hydrogen ion, and a, b, c, d are stoichiometric coefficients of A, B, C and D, respectively. The reaction rate (r A ) is k f is the forward reaction rate constant and n is the order of reaction.
www.intechopen.com The transport of copper ions through a cylindrical hollow fiber is considered in the axial direction or bulk flow direction and radial direction. In order to develop the model, the following assumptions are made: 1. The inside and outside diameters of a hollow fiber are very small. Thus, the membrane thickness is very thin; therefore the radial concentration profile of copper ions is constant. 2. Only the complex species occurring from the reaction, not copper ions, diffuse through liquid membrane phase. 3. The extraction reaction is irreversible that means only the forward reaction of Eq. (15) is considered. 4. Due to very thin membrane thickness, it is presumed that the reaction occurs only in the axial direction of the hollow fibers. Mass flux of copper ions exists in the axial direction. The conservation of mass for copper ion transport in the hollow fiber is considered as shown in Fig. 18. At a small segment Δx, the conservation of mass can be described below: A ra n d A C are the average values of the reaction rate and the concentration of copper ions, respectively Dividing Eq. (18) by xA c and taking a limit x 0, obtains At the initial condition (t = 0), the conservation of mass in Eq. (19) is considered with regard to 3 cases of the reaction orders as follows: At time t (t  0), the conservation of mass in Eq. (19) in the differential form is Linearize Eq. (23) by taking Laplace transforms and considering 3 cases of reaction orders, we obtain: The reaction rate constant of the second order is taken into consideration for a better curve fitting between the model and the experimental results, as shown in Table 4 by higher Rsquared and less deviation. www.intechopen.com The optimum separation time and separation cycles of the extraction can be estimated. The model was verified with the experimental extraction results and other literature. Fig. 19 is a plot of the integral concentrations of Cu(II) against time to determine the reaction order (n) and the forward reaction rate constant (k f ). The rate of diffusion and/or rates of chemical changes may control the kinetics of transport through liquid membrane depending on transport mechanisms (diffusion or facilitated). The reaction rate constants of first-order (n = 1) and second-order (n = 2) are 0.393 min -1 and 0.708 L/mgmin, respectively.
The optimum separation time for the prediction of separation cycles can be estimated by the model based on the optimum conditions from the plot of percentage of extraction as a function of initial concentration of the target species in feed and also feed flow rate. In this work, at the legislation of Cu(II) concentration in waste stream of 2 mg/L, the calculated separation time is 10 min for about 15-continuous cycles. The percentage of extraction calculated from this reaction flux model is much higher than the results from other works which applied different extractants and transport mechanisms. Types of extractants and their concentrations are significant to the separation of metal ions. For example, a hard base extractant can extract both dissociated and undissociated forms in a basic or weak acidic condition but dissociated forms are high favorable. While a neutral extractant normally reacts with undissociated forms, but in an acidic condition it can react with dissociated forms. It is noteworthy to be aware that not only types of the extractants (single or synergistic), in this case LIX84I for Cu(II), but also the transport mechanism, e.g., facilitated transport mechanism attributes to the extraction efficiency. The model results are in good agreement with the experimental data at the average percentage of deviation of 2%.

Conclusions
Facilitated transport of the solutes or target species benefits the separation process by liquid membrane with a non-equilibrium mass transfer and uphill effect. It is more drastic chemical changes of the target species with the presence of a suitable extractant or carrier (sometimes by synergistic extractant) in liquid membrane to form new complex species (dissociated and undissociated forms) to diffuse through the liquid membrane phase. As a result, the efficiency and selectivity of the transport across liquid membrane markedly enhance. Factors that affect the facilitated transport and diffusion through the membrane are, for example, extractant types and properties (e.g., proton donors, electron donors), solvent characteristics, stripping types and properties, life time of membrane due to fouling, operating temperature. Many outstanding advantages of the HFSLM make it the most efficient type of membrane separation for several applications. It is worth to note that the HFSLM can simultaneously extract the target species of very low concentration and recover them in one single operation. For favorable ions (e.g., precious metals), high percentage of recovery is desirable. Despite many advantages, at present the HFSLM is not often used in a large-scale industry because the major drawbacks of hollow fibers are not only fouling but also mechanical stability of the support. However, in regard to apply the HFSLM in industrial scale, the reliable mathematical model is required as the model can foretell the effect of mass transfer as the functions of operating parameters, membrane properties and feed properties on the separation efficiency. However, due to the limitations of applications or unclear phenomena around the membrane surface, no model so far is fully satisfactory and universally applicable. Even though, the model can help to understand and predict the operation as well as the separation performance. In case the separation of metal ions by the HFSLM, as there are several parameters involved, e.g., types of metal ions, extractants and stripping solutions, and the transport mechanisms, therefore the model probably has implications for other metals but it may need some modifications corresponding to such parameters.

Acknowledgments
The authors are highly grateful to the Royal Golden Jubilee Ph.D. Program (Grant No. PHD50K0329) under the Thailand Research Fund, the Rare Earth Research and Development Center of the Office of Atoms for Peace (Thailand), Thai Oil Public Co., Ltd., the Separation Laboratory, Department of Chemical Engineering, Chulalongkorn University, Bangkok, Thailand. Kind contributions by our research group are deeply acknowledged.

Nomenclature
A Membrane area (cm 2 ) A C Cross-sectional area of hollow fiber (cm 2 ) BLM Bulk liquid membrane BTXs Benzene, toluene, xylenes C A Concentration of copper ions <C A > Average value of the concentration of copper ions C f Concentration of target species in feed phase (moles per unit volume) C f* Concentration of target species at feed-membrane interface (moles per unit volume) C f,0 Initial concentration of target species in feed phase (moles per unit volume) C f,in , C f,out Concentration of target species at feed inlet and feed outlet (moles per unit volume) C s Concentration of target species in the stripping solution (moles per unit volume) C s* Concentration of target species at membrane-stripping interface (moles per unit volume) C(0,t) Concentration of target species at liquid membrane thickness = 0 and any time (moles per unit volume) C(x 0 ,t) Concentration of target species at liquid membrane thickness of x 0 and any time (