Transport mechanism of electrolytes across charge-mosaic membrane

1 THEORY
1.1 Concentration polarization
Usually, the apparent retention R_a is used to express the separating capability of a membrane, that is:
    (1)
    Here, C_b and C_p represent the feed concentration and permeate concentration, respectively, mol/L.
    For the separation process by using charge mosaic membrane, the concentration at the membrane surface C_m is greater than C_b owing to the influence of concentration polarization. The real retention R_s can be defined as follows:
  （2）
    According to the concentration polarization theory , the following equation can be obtained:
（3）
    Where Jv is membrane flux, m³/(m²·s¹)；k is mass transfer coefficient, m/s.

1.2 Phenomenological equation              
For the charge-mosaic membrane with high quality, the anion and cation exchange materials are highly permselective. Since none of the practical applications envisaged for the mosaic involve the passage of net current across the membrane, we need only to concern with the two-process phenomenological equations for salts and volume flow. According to the analysis by Weinstein^[5], volume flux and salt flux, Jv and Js, are expressed by the following equations, respectively, related to the driving forces hydrostatic pressure , osmotic pressure , and chemical potential difference :
（4）
（5）
    Where Lp is the filtration coefficient,  s is the reflection coefficient, w is the solute permeability, and Cs is the average concentration.
    Considering that the retention of salt is lower for the charge-mosaic membrane, the average concentration of salt can be adopted as the arithmetic average concentration instead of logarithmic average concentration. When hydrostatic pressure DP>>Dp, we suppose . Thus the following equation can be obtained by substituting Eq. (4) and (5) into Eq. (2) :
（6）
Where ; T is temperature, K; R=8.314 J/(mol·K). K_S is the ions transported by the membrane, which is based on the type of salts.( For NaCl，K_S=2；Na₂SO₄ and MgCl₂，K_S=3)

Fig. 1 Scheme of the experimental apparatus
1.feed reservoir 2.pump 3.manometer 4.membrane module 5.tank for permeation

2 EXPERIMENTAL               
The charge-mosaic membrane was prepared by using interfacial polymerization technique. Polyethersulfone (PES) hollow fiber membrane was used as the support membrane. Firstly, the aqueous solution containing 2, 5-diaminobenzene sulfonic acid and polyethylenimine (PEI) was introduced into the fiber from the lumen and kept for about 30 minutes. Different additives and/or acid acceptors were also added into the aqueous amine solution. Then, the dodecane solution containing trimesoyl choride acid (TMC) and 4-(chloromethyl) benzoyl chloride was gently fed into the fiber for the interfacial polymerization reaction. After the polymer layer was formed, the membrane was immersed with an aqueous trimethylamine solution and kept for 24 hours to convert the chloromethylated groups into cationic quaternary ammonium groups. The membrane finally was immersed in distilled water until use in the test.
    The membrane separating experiments were carried with NaSO₄ and MgCl₂. The initial concentration of salts was 0.01 mol/L. The concentrations of the permeated salts were measured with microprocessor conductivity meter. Fig.1 is the experimental scheme for measuring the volume flow and the apparent retention of salts.

Figure 2 The filtration coefficient

3.2 Model Parameter Fitting
The filtration coefficient can be obtained by fitting the experimental data of water flux in Figure 2, that is:
Lp= 2.976×10^-11 m³/(m²·s¹·Pa¹ )
    The correlation coefficient R is 0.9998.
   According to Eq. (6), the reflection coefficient and the solute permeability can be calculated based on nonlinear curve fitting to the experimental data of Na₂SO₄ and MgCl₂ by using Origin software ^[8]:
for Na₂SO₄，s=0.46 ，w=3.93×10^-10 mol/(N¹·s¹);
for MgCl₂，s=0.37，w=5.177×10^-10 mol/(N¹·s¹).
    Figure 3 shows the comparison of theoretical values with experimental data, in which the solid lines are the theoretical values calculated by equation (6) and the dispersing dots are the experimental data. It shows that the theoretical values fit well with the experimental data.

Figure 3 Comparison of the theoretical values with experimental data

4 CONCLUSIONS
The transport mechanism of electrolytes across a charge-mosaic membrane is discussed. The membrane coefficients can be determined by fitting the experimental data. The mathematical model is in good agreement with the experimental data.

REFERENCES
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[5] Weinstein J N, B. Bunow J, Caplan S R. Desalination,1972, 11: 341-377.
[6] Mulder M . (translated by Li L). Basic Principles of Membrane Technology (Mojishu Jiben Yuanli), 2nd edition, Beijing: Tsinghua University Publisher, 1996:270-275.
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[8] Hao H W, Shi G K. Origin 6.0 Professional (Origin 6.0 Shili Jiaocheng), Beijing: China Electric Power Publisher, 2000: 88-104.