Euler’s Numerical Method for Ions Rejection Reassessment of a Defect-Free Synthesized Nanofiltration Membrane with Ultrathin Titania Film as the Selective Layer
Abstract
:1. Introduction
2. Mathematical Modeling
2.1. Model Assumptions
- All the solutions used are assumed to be ideal.
- The effective charge density of the membrane is identical at all points of the NF membrane under study.
- The nanofiltration membrane consists of a bundle of straight cylindrical pores, all identical, each having a uniform radius and depth (with ).
- The NPs layer thickness is negligible towards the substrate thickness.
- All the solute particles in the solution are transportable.
- The electric potential inside the membrane and the Na2SO4, MgSO4, NaCl, CaCl2, and MgCl2 solutions are defined as centrifugal averaged quantities.
- The Donnan equilibrium is applied not only at the interface of membrane/feed-solution but also at the interface of membrane/permeate solution.
2.2. Model Equations
2.3. Description of The Computation Procedure
- Using Equation (10), the feed concentration Ci,f enables the initial concentration at the fees–solution/membrane interface ci,1 calculation, and even the practical integration of both Equations (3) and (7).
- Using the Euler numerical method, ci,1, ci,2, ci,3, ci,4, …, and ci,N are estimated (integrating Equations (3) and (7)).
- From the estimated ci,N* value, and applying Equation (10), the permeate concentration Ci,p is calculated.
- Finally, the solute particle rejection (R) can be calculated using Equation (11).
2.4. Euler Numerical Method and Ion Transport Inside the Membrane Active Layer
3. Experimental Section
3.1. Materials
3.2. Novel Organic-Inorganic Nanofiltration Membrane NF_PAN_Ti Preparation
3.3. NF_PAN_Ti Membrane Characterization
3.4. Filtration Performance of Organic-Inorganic NF_PAN_Ti Membrane
3.5. Long Test Stability on NF_PAN_Ti Membrane
3.6. Validation of the Predicted Results with Experimental Data
4. Results and Discussion
4.1. NF_PAN_Ti Structures Characterization
4.2. Flux Behavior and Experimental Salts Rejection
4.3. Model Reassessment of Salts Rejection
4.4. Membrane Long-Term Stability
4.5. Validation of the Predicted Results with Experimental Data
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
Abbreviations
concentration of ion i within pore, mol·m−3 | |
bulk feed concentration, mol·m−3 | |
ionic solute bulk solution concentration, mol·m−3 | |
uncharged solute bulk permeate concentration, mol·m−3 | |
/ | (Un)charged solute pore diffusion coefficient, m2·s−1 (=) |
solute bulk diffusion coefficient, m2·s−1 | |
I | ionic strength (mol·m−3) |
uncharged solute flux, pore area basis, mol·m−2·s−1 | |
k | feed-side mass transfer coefficient, m/s |
P | Pressure N/m2 |
effective pore radius, m | |
V | solvent velocity, m/s |
effective charge density, mol/m3 | |
valence of ion i | |
activity coefficient of ion i within pore, dimensionless | |
bulk activity coefficient of ion i, dimensionless | |
applied pressure, N·m−2 | |
effective pressure driving force, N·m−2 | |
membrane thickness, m | |
osmotic pressure difference, N·m−2 | |
Donnan potential, V | |
Bulk/pore dielectric constant, dimensionless | |
η | solvent viscosity within pores, N·s·m−2 |
λ | ratio of ionic or uncharged solute radius to pore radius, dimensionless |
ratio of effective membrane charge density to bulk feed concentration, di-mensionless () | |
steric partition coefficient of ion i, dimensionless | |
𝜓 | potential within the pore (V) |
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Parameters | Value |
---|---|
Faraday’s constant (F) | 96,487 C·mol−1 |
Operating temperature (T) | 303.15 K |
pH in the oxidant unit | 10.2 |
) | 0.60 MPa |
Crossflow velocity | |
Boltzmann constant (k) | 1.38066 × 10−23 J·K−1 |
Parameter | Unit | Value | Refs |
---|---|---|---|
rejection | This study | ||
rejection | |||
Water flux | |||
Membrane geometry | |||
Membrane surface area | |||
Membrane thickness | |||
Equation (27) |
Particle | References | |||
---|---|---|---|---|
2.03 | 0.121 | 17.82 | [49,50,51] | |
0.72 | 0.348 | −21.57 |
Membranes | Water Flux (L·m−2·h−1) | Rejection (%) | References |
---|---|---|---|
58 | 89.3 | This study | |
75 | 80 | [52] | |
60 | >90 | [6] | |
30 | 95 | [53] | |
4.2 | 81.9 | [54] | |
24 | 80 | [55] | |
20.4 | 96 | [56] | |
42 | 90 | [57] | |
22.9 | 90.8 | [58] |
NF_PAN_Ti membrane | |||
---|---|---|---|
24.1 | 24.0191 | 0.33 | |
89.3 | 89.2152 | 0.09 |
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Worou, C.N.; Kang, J.; Shen, J.; Degan, A.; Yan, P.; Wang, W.; Gong, Y.; Chen, Z. Euler’s Numerical Method for Ions Rejection Reassessment of a Defect-Free Synthesized Nanofiltration Membrane with Ultrathin Titania Film as the Selective Layer. Coatings 2021, 11, 184. https://doi.org/10.3390/coatings11020184
Worou CN, Kang J, Shen J, Degan A, Yan P, Wang W, Gong Y, Chen Z. Euler’s Numerical Method for Ions Rejection Reassessment of a Defect-Free Synthesized Nanofiltration Membrane with Ultrathin Titania Film as the Selective Layer. Coatings. 2021; 11(2):184. https://doi.org/10.3390/coatings11020184
Chicago/Turabian StyleWorou, Chabi Noël, Jing Kang, Jimin Shen, Arcadius Degan, Pengwei Yan, Weiqiang Wang, Yingxu Gong, and Zhonglin Chen. 2021. "Euler’s Numerical Method for Ions Rejection Reassessment of a Defect-Free Synthesized Nanofiltration Membrane with Ultrathin Titania Film as the Selective Layer" Coatings 11, no. 2: 184. https://doi.org/10.3390/coatings11020184
APA StyleWorou, C. N., Kang, J., Shen, J., Degan, A., Yan, P., Wang, W., Gong, Y., & Chen, Z. (2021). Euler’s Numerical Method for Ions Rejection Reassessment of a Defect-Free Synthesized Nanofiltration Membrane with Ultrathin Titania Film as the Selective Layer. Coatings, 11(2), 184. https://doi.org/10.3390/coatings11020184