**3. 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:

Roles of Facilitated Transport Through HFSLM in Engineering Applications 183

The distribution ratio should be derived as a function of the extraction equilibrium constant as

ex n

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

> i i i lm m o s

1 1 r 1 r1 P k r P rk

where ki and ks are the feed-phase and stripping-phase mass transfer coefficients, rlm is the log-mean radius of the hollow fiber in tube and shell sides, ri and r0 are the inside and outside radius of the hollow fiber, Pm 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

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 (ks) is much higher than that in the feed

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 (lif) makes the mass transfer coefficient in feed phase (ki) much lower than that in the stripping phase (ks).

> if <sup>D</sup> <sup>k</sup>

is <sup>D</sup> <sup>k</sup>

2. The difference in the concentration of target species in feed phase (Cf) and the concentration of feed at feed-membrane interface (Cf\*) is higher than the difference in

phase (ki) according to the following assumptions (Uedee et al., 2008):

Feed-mass transfer coefficient i

Stripping-mass transfer coefficient s

coefficients as follows (Urtiaga et al., 1992; Kumar et al., 2000; Rathore et al., 2001)

K [RH] <sup>D</sup>

n n [MR ] <sup>D</sup>

n

[M ] (3)

[H ] (4)

(5)

Pm = Dkm (6)

<sup>l</sup> (7)

<sup>l</sup> (8)

The distribution ratio (D) is

liquid membrane (km) as


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).

Fig. 2. Facilitated transport mechanisms through the HFSLM

The chemical reaction at the interface between feed phase and liquid membrane phase takes place when the extractant (RH) reacts with the target species (Mn+ ) in the feed Eq. (1).

$$\text{M}^{\text{n}+} + \overline{\text{nRH}} \rightleftharpoons \overline{\text{MR}\_{\text{n}}} + \text{nH}^{+} \tag{1}$$

MRn is the complex species in liquid membrane phase. The extraction equilibrium constant (Kex) of the target species is

$$\mathbf{K}\_{\rm ex} = \frac{\overline{\left[\overline{\mathbf{MR}}\_{n}\right]} \cdot \left[\overline{\left[\mathbf{H}^{+}\right]}\right]^{n}}{\left[\mathbf{M}^{n+}\right] \cdot \left[\overline{\left[\mathbf{RH}\right]}\right]^{n}}\tag{2}$$

The distribution ratio (D) is

182 Mass Transfer in Chemical Engineering Processes

**Step 1.** Metal ions or target species in feed solution or aqueous phase are transported to a

**Step 3.** The complex species react with the stripping solution at the contact surface between

**Step 4.** Metal ions are transferred into the stripping solution while the extractant moves

The chemical reaction at the interface between feed phase and liquid membrane phase takes place when the extractant (RH) reacts with the target species (Mn+ ) in the feed Eq. (1).

> <sup>n</sup> ex <sup>n</sup> <sup>n</sup> [MR ] [H ] <sup>K</sup> [M ] [RH]

 

<sup>M</sup> nRH <sup>n</sup> <sup>n</sup> nHMR (1)

(2)

n

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

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

passes this interface.

non-equilibrium mass-transfer process (Yang, 1999).

Fig. 2. Facilitated transport mechanisms through the HFSLM

MRn is the complex species in liquid membrane phase. The extraction equilibrium constant (Kex) of the target species is

phase.

contact surface between feed solution and liquid membrane, subsequently react

by the concentration gradient. It is assumed that no transport of target species

liquid membrane and the stripping solution and release metal ions to the stripping

back to liquid membrane and diffuses to the opposite side of liquid membrane by

$$\mathbf{D} = \frac{\overline{\{\mathbf{MR}\_n\}}}{\mathbf{[M^{n+}]}} \tag{3}$$

The distribution ratio should be derived as a function of the extraction equilibrium constant as

$$\text{ID} = \frac{\text{K}\_{\text{ex}} \overline{\text{[RH]}}^n}{[\text{H}^+]^n} \tag{4}$$

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)

$$\frac{1}{P} = \frac{1}{\mathbf{k}\_i} + \frac{\mathbf{r\_i}}{\mathbf{r\_{lm}}} \frac{1}{P\_m} + \frac{\mathbf{r\_i}}{\mathbf{r\_o}} \frac{1}{\mathbf{k\_s}} \tag{5}$$

where ki and ks are the feed-phase and stripping-phase mass transfer coefficients, rlm is the log-mean radius of the hollow fiber in tube and shell sides, ri and r0 are the inside and outside radius of the hollow fiber, Pm 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 (km) as

$$\mathbf{P\_m} = \mathbf{Dk\_m} \tag{6}$$

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 (ks) is much higher than that in the feed phase (ki) 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 (lif) makes the mass transfer coefficient in feed phase (ki) much lower than that in the stripping phase (ks).

$$\text{Feed-mass transfer coefficient} \qquad \qquad \mathbf{k}\_i$$

$$\mathbf{k}\_i = \frac{\mathbf{D}}{\mathbf{1}\_{i\mathbf{f}}} \tag{7}$$

Stripping-mass transfer coefficient s

$$\mathbf{k}\_s = \frac{\mathbf{D}}{\mathbf{l}\_{is}}\tag{8}$$

2. The difference in the concentration of target species in feed phase (Cf) and the concentration of feed at feed-membrane interface (Cf\*) is higher than the difference in

Roles of Facilitated Transport Through HFSLM in Engineering Applications 185

Hexanol/ Decane

Cyanex 923 Toluene - - 0.072

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 (Pancharoen et al., 2005; Patthaveekongka et al., 2006; Ramakul et al., 2004, 2005, 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

In this work, the extraction equilibrium constant (Kex) obtaining from Fig. 7 was 1.98 x 10−<sup>1</sup> (Lmol-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 (ki) and in liquid membrane (km) of 0.0103 and 0.788 cm s-1, respectively were

**(105 m/s)**

**D 103 (-)** 

Aliquat 336 Kerosene - - - 0.0022 The model results

D2EHPA Kerosene 0.1-0.26 - - - The model results

Cyanex 272 Kerosene 27-77.5 4.6-15.5 0.103 7.88 The mass transfer in

**ki (103 m/s)** 


0.107

Toluene 5.5-11.5 0.63-1.5 0.392 0.102 The mass transfer in

Toluene 34-53.1 4.5-8.7 22.1 0.013 The mass transfer in

**km**

diffusion (very low km)

> 34.5 17.9

**(105 m/s) Results** 

agree well with the experiment

reasonably agree with the experiment

The model results agree well with the experiment

The mass transfer in the film layer between the feed phase and liquid membrane is the rate controlling step

the membrane is the rate controlling step

the membrane is the rate controlling step

the membrane is the rate controlling step

**Authors Species Extractants Solvents <sup>P</sup>**

Cu(II) N-decyl- (L)-hydroxy proline

Aliquat 336 Bromo-PADAP, Cyanex 471, Cyanex 923

Aliquat 336 Bromo-PADAP, Cyanex 471, Cyanex 923

Table 3. Applications of HFSLM and mass transfer related

Cr(VI) in synthetic water

Cd(II) in synthetic water

D-Phe and L-Phe in synthetic water

As(III), As(V) in synthetic water

Pr(III) from RE(NO3)3 solution

As from produced water

Hg from produced water

operation about 300 min as shown in Fig. 6.

(Ortiz et al., 1996)

(Marcese & Camderros, 2004)

(Prapasawat et al., 2008)

(Wannachod et al., 2011)

(Lothongkum et al., 2011)

(Huang et al., 2008)

the concentration of stripping phase at membrane-stripping interface (Cs\*) and the concentration of target species in stripping phase (Cs). At equal flux by Eq. (9), ki is, therefore, much lower than ks.

$$\mathbf{J} = \mathbf{k}\_{\mathbf{i}} (\mathbf{C}\_{\mathbf{f}} - \mathbf{C}\_{\mathbf{f}}^{\*}) = \mathbf{k}\_{\mathbf{s}} (\mathbf{C}\_{\mathbf{s}}^{\*} - \mathbf{C}\_{\mathbf{s}}) \tag{9}$$

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.

Pm in Eq. (5) can be substituted in terms of the distribution ratio (D) and the mass transfer coefficient in liquid membrane (km) in Eq. (6) as

$$\frac{1}{P} = \frac{1}{\mathbf{k}\_i} + \frac{\mathbf{r\_i}}{\mathbf{r\_{lm}}} \frac{1}{\mathbf{D}\mathbf{k\_m}} \tag{10}$$

In addition, from the permeability coefficient (P) by Danesi (Danesi, 1984):

$$-\mathbf{V\_{f}}\ln\left(\frac{\mathbf{C\_{f}}}{\mathbf{C\_{f,0}}}\right) = \mathbf{AP}\frac{\beta}{\beta+1}\mathbf{t} \tag{11}$$

$$\text{where}\\
\qquad\qquad\qquad\qquad\beta=\frac{\mathbf{Q}\_{\bar{t}}}{\text{PLNe}\pi\mathbf{r}\_{\bar{i}}}\tag{12}$$

We can calculate the permeability coefficient from the slope of the plot between <sup>f</sup> f f,0 <sup>C</sup> V ln C 

against t.

Table 3 shows some applications of HFSLM and their mass transfer related.
