This paper presents an overview to the state-of-the-art area of mathematical modeling for reverse osmosis (RO) process. As a liquid-solid process, RO constitutes a valuable and vital physical solution for seawater desalination, wastewater depollution, and water treatment comparatively to the largely controversial and polluting chemical processes such as chlorination and coagulation/flocculation. Great works are required in modeling of RO technique in order to obtain a complete and exhaustive model. Complicated and varying raw water qualities and quantities parameters through time and space are rendering RO modeling hard. Be facing the increasing pollution levels and trying to satisfy the drinking water guidelines, RO membrane modification is often required. This situation made modeling this highly dynamic process more difficult to accomplish.
Reverse Osmosis Process Membranes Modeling – A Historical Overview, Journal of Civil, Construction and Environmental Engineering.
Vol. 2, No. 4,
2017, pp. 112-122.
S. A. Creber, T. R. R. Pintelon, D. A. W. Graf von der Schulenburg, J. S. Vrouwenvelder, M. C. M. van Loosdrecht, M. L. Johns, Magnetic resonance imaging and 3D simulation studies of biofilm accumulation and cleaning on reverse osmosis membranes, Food Bioprod. Process. 88 (2010) 401-408.
M. Verhuelsdonk, T. Attenborough, O. Lex, T. Altmann, Design and optimization of seawater reverse osmosis desalination plants using special simulation software, Desalination 250 (2010) 729-733.
G. Srivathsan, Modeling of fluid flow in spiral wound reverse osmosis membranes, PhD Thesis, University of Minnesota, 2013.
S. M. S. Ghiu, Mass transfer of ionic species in direct and reverse osmosis processes, PhD Thesis, University of South Florida, 2003.
H. Mehdizadeh, Modeling of transport phenomena in reverse osmosis membranes, PhD Thesis, McMaster University, Ontario, 1990.
L. E. Applegate, Membrane separation processes, Chem. Eng., June 11, (1984) 64-89.
S. Sourirajan, Reverse osmosis, Academic Press, New York, 1970.
S. Sourirajan, Reverse osmosis and synthetic membranes: Theory-technology-engineering, National Research Council Canada, Ottawa, Canada, 1977.
B. R. Min, Experimental and theoretical study of NaCl and Ca Cl2 transport through cellulose acetate membranes using a reverse osmosis batch cell, PhD Thesis, State University of New York at Buffalo, 1983.
Z. Hu, A. Antony, G. Leslie, P. Le-Clech, Real-time monitoring of scale formation in reverse osmosis using electrical impedance spectroscopy, J. Membr. Sci. 453 (2014) 320-327.
C. Ba, Design of advanced reverse osmosis and nanofiltration membranes for water purification, PhD Thesis, University of Illinois at Urbana-Champaign, 2010.
W. Rohlfs, G. P. Thiel, J. H. Lienhard V, Modeling reverse osmosis element design using superposition and an analogy to convective heat transfer, J. Membr. Sci. 512 (2016) 38-49.
A. E. Anqi, N. Alkhamis, A. Oztekin, Numerical simulation of brackish water desalination by a reverse osmosis membrane, Desalination 369 (2015) 156-164.
Y. Wang, W. He, H. Zhu, Computational fluid dynamics (CFD) based modelling of osmotic energy generation using pressure retarded osmosis (PRO), Desalination 389 (2016) 98-107.
J. I. Marriott, Detailed modelling and optimal design of membrane separation systems, PhD Thesis, University College London, 2001.
A. K. Gautam, T. J. Menkhaus, Performance evaluation and fouling analysis for reverse osmosis and nanofiltration membranes during processing of lignocellulosic biomass hydrolysate, J. Membr. Sci. 451 (2014) 252-265.
S. Jamaly, N. N. Darwish, I. Ahmed, S. W. Hasan, A short review on reverse osmosis pretreatment technologies, Desalination 354 (2014) 30-38.
K. Jamal, M. A. Khan, M. Kamil, Mathematical modeling of reverse osmosis systems, Desalination 160 (2004) 29-42.
S. Sourirajan, Thirty years of membrane research – A few highlights, Proc. Int. Memb. Conf. 25th Anniv. Memb. Res. Canada, M. Malalyandi, O. Kutowy, F. Talbot (Eds.), Ottawa, September 24-26, 3-32 (1986).
A. Shrivastava, S. Rosenberg, M. Peery, Energy efficiency breakdown of reverse osmosis and its implications on future innovation roadmap for desalination, Desalination 368 (2015) 181-192.
Saltwater intrusion, https://en.wikipedia.org/wiki/Saltwater_intrusion (accessed on 22/06/2017).
S. T. Yuster, S. Sourirajan, K. Bernstein, Sea water demineralization by the ‘surface skimming’ process, University of California (UCLA), Dept. of Engineering, Rept. 58-26, March, 1958.
S. Loeb, The Loeb-Sourirajan Membrane: How it came about, in: Synthetic membranes, A. F. Turback (Ed.), vol. 1, ACS Symposium Series 153, Washington, 1981.
E. J. Jr., Breton, Water and ion flow through imperfect osmotic membranes, Office of saline water, US Dept. of the Interior, Res. & Dev. Prog. Rept. 16, April, 1957.
C. E. Reid, E. J. Breton, Water and ion flow across cellulosic membranes, J. Appl. Polym. Sci. 1 (1959) 133-143.
S. Loeb, S. Sourirajan, Sea water demineralization by means of an osmotic membrane, Advan. Chem. Ser. 38 (1962) 117-132.
A. Altaee, G. Zaragoza, H. R. van Tonningen, Comparison between Forward Osmosis-Reverse Osmosis and Reverse Osmosis processes for seawater desalination, Desalination 336 (2014) 50-57.
V. Haluch, E. F. Zanoelo, C. J. L. Hermes, Experimental evaluation and semi-empirical modeling of a small-capacity reverse osmosis desalination unit, Chem. Eng. Res. Design 122 (2017) 243-253.
J. S. Kim, J. Chen, H. E. Garcia, Modeling, control, and dynamic performance analysis of a reverse osmosis desalination plant integrated within hybrid energy systems, Energy 112 (2016) 52-66.
J. Lin, Molecular dynamics simulation of osmosis, reverse osmosis and electro-osmosis in electrolyte solutions, PhD Thesis, University of Illinois at Chicago, 2000.
J.-S. Choi, J.-T. Kim, Modeling of full-scale reverse osmosis desalination system: Influence of operational parameters, J. Ind. Eng. Chem. 21 (2015) 261-268.
A. R. Bartman, Control and monitoring of reverse osmosis water desalination, PhD Thesis, University of California, Los Angeles, 2011.
J. Garcia-Aleman, Mathematical modeling of the pressure-driven performance of McMaster pore-filled membranes, McMaster University, Ontario, 2002.
M. Soltanieh, W. N. Gill, Review of reverse osmosis membranes and transport models, Chem. Eng. Commun. 12 (1981) 279-363.
J. Heo, L. K. Boateng, J. R. V. Flora, H. Lee, N. Her, Y.-G. Park, Y. Yoon, Comparison of flux behavior and synthetic organic compound removal by forward osmosis and reverse osmosis membranes, J. Membr. Sci. 443 (2013) 69-82.
A. Saengrung, Modeling of reverse osmosis plants using system identification and neural networks, Master Thesis, Florida Atlantic University, 2002.
K. Kezia, J. Lee, W. Ogieglo, A. Hill, N. E. Benes, S. E. Kentish, The transport of hydronium and hydroxide ions through reverse osmosis membranes, J. Membr. Sci. 459 (2014) 197-206.
T.-U. Kim, Transport of organic micropollutants through nanofiltration (NF) and reverse osmosis (RO) membranes: Mechanisms, modeling, and applications, University of Colorado at Boulder, 2006.
L. K. Boateng, Multiscale modeling of water transport and the influence of water quality parameters on membrane processes, PhD Thesis, University of South Carolina, 2016.
A. F. Corral, Alternative technologies for inland desalination, PhD Thesis, The University of Arizona, 2014.
E. Mancha, Alternatives to piloting in brackish waters: Accuracy and precision of commercial reverse osmosis membrane design model projections, PhD Thesis, The University of Texas at El Paso, 2012.
Y. Xiang, Understanding membrane fouling mechanisms through computational simulations, PhD Thesis, The George Washington University, 2016.
F. Daniels, R. A. Alberty, Physical chemistry, 3rd Ed., John Wiley and Sons, Inc., New York, 1972.
R. W. Stoughton, M. H. Lietzke, Calaculation of some thermodynamic properties of sea salt solutions at elevated temperatures from data on NaCl solutions, J. Chem. Eng. Data, 10 (1965) 254-260.
R. C. Weast, CRC handbook of chemistry and physics, 56th Ed., CRC Press, 1975.
A. J. Staverman, Recueil Trav. Chim. Pays-Bas, 70 (1951) 344-352.
H. G. Burghoff, K. L. Lee, W. Pusch, Characterization of transport across cellulose acetate membranes in the presence of strong solute – membrane interactions, J. Appl. Polym. Sci. 25 (1980) 323-347.
A. J. Staverman, Structure and function of membranes, J. Membr. Sci. 16 (1983) 7-20.
R. Salcedo-Díaz, P. García-Algado, M. García-Rodríguez, J. Fernández-Sempere, F. Ruiz-Beviá, Visualization and modeling of the polarization layer in crossflow reverse osmosis in a slit-type channel, J. Membr. Sci. 456 (2014) 21-30.
A. Jogdand, A. Chaudhuri, Modeling of concentration polarization and permeate flux variation in a roto-dynamic reverse osmosis filtration system, Desalination 375 (2015) 54-70.
K. M. Sassi, I. M. Mujtaba, Simulation and optimization of full scale reverse osmosis desalination plant, 20th European Symposium on Computer Aided Process Engineering – ESCAPE20, S. Pierucci, G. B. Ferraris (Eds.), Elsevier B. V., 2010.
M. A. Al-Obaidi, I. M. Mujtaba, Steady state and dynamic modeling of spiral wound wastewater reverse osmosis process, Comput. Chem. Eng. 90 (2016) 278-299.
R. B. Bird, W. E. Stewart, E. N. Lightfoot, Transport phenomena, John Wiley and Sons, Inc., New York, 1960.
J. L. Siler, Reverse osmosis membranes concentration polarization and surface fouling: Predictive models and experimental verifications, PhD Thesis, The University of Kentucky, 1987.
G. Srivathsan, E. M. Sparrow, J. M. Gorman, Reverse osmosis issues relating to pressure drop, mass transfer, turbulence, and unsteadiness, Desalination 341 (2014) 83-86.
W. Lawler, J. Alvarez-Gaitan, G. Leslie, P. Le-Clech, Comparative life cycle assessment of end-of-life options for reverse osmosis membranes, Desalination 357 (2015) 45-54.
R. E. Kesting, Synthetic polymeric membranes: A structural perspective, 2nd Ed., John Wiley and Sons, New York, 1985.
T. Fujioka, N. Oshima, R. Suzuki, W. E. Price, L. D. Nghiem, Probing the internal structure of reverse osmosis membranes by positron annihilation spectroscopy: Gaining more insight into the transport of water and small solutes, J. Membr. Sci. 486 (2015) 106-118.
T. Fujioka, S. J. Khan, J. A. McDonald, A. Roux, Y. Poussade, J. E. Drewes, L. D. Nghiem, Modelling the rejection of N-nitrosamines by a spiral-wound reverse osmosis system: Mathematical model development and validation, J. Membr. Sci. 454 (2014) 212-219.
J. Duan, E. Litwiller, I. Pinnau, Preparation and water desalination properties of POSS-polyamide nanocomposite reverse osmosis membranes, J. Membr. Sci. 473 (2015) 157-164.
N. Y. Yip, Sustainable production of water and energy with osmotically-driven membrane processes and ion-exchange membrane processes, PhD Thesis, Yale University, 2014.
B. M. Carter, Design, development, and evaluation of thin-film composite bicontinuous cubic lyotropic liquid crystal polymer membranes for water filtration applications, PhD Thesis, University of Colorado, 2014.
J. E. Cadotte, R. J. Peterson, Thin-film composite reverse-osmosis membranes: Origin, development, and recent advances, in: Synthetic membranes, A. F. Turbak (Ed.), vol. 1, ACS Symposium Series, Washington, 1980.
A. M. Kamal, T. A. El-Sayed, A. M. A. El-Butch, S. H. Farghaly, Analytical and finite element modeling of pressure vessels for seawater reverse osmosis desalination plants, Desalination 397 (2016) 126-139.
W. He, Y. Wang, A. Sharif, M. H. Shaheed, Thermodynamic analysis of a stand-alone reverse osmosis desalination system powered by pressure retarded osmosis, Desalination 352 (2014) 27-37.
J. E. Cadotte, R. J. Petersen, R. E. Larson, E. E. Erickson, A new thin-film composite sea water reverse osmosis membrane, Desalination 32 (1980) 25-31.
R. E. Larson, J. E. Cadotte, R. J. Petersen, The FT-30 seawater reverse osmosis membrane: Element test results, Desalination 38 (1981) 473-483.
R. E. Larson, R. J. Petersen, P. K. Eriksson, Test results on FT-30 eight-inch-diameter seawater and brackish water reverse osmosis elements, Desalination 46 (1983) 81-90.
J. E. Cadotte, Evolution of composite reverse osmosis membranes, in: Material Science of Synthetic Membranes, D. R. Lloyd (Ed.), ACS Symposium Series 269, Washington, 1985.
M. A. Al-Obaidi, C. Kara-Zaïtri, I. M. Mujtaba, Removal of phenol from wastewater using spiral-wound reverse osmosis process: Model development based on experiment and simulation, J. Water Process Eng. 18 (2017) 20-28.
M. A. Al-Obaidi, C. Kara-Zaitri, I. M. Mujtaba, Scope and limitations of the irreversible thermodynamics and the solution diffusion models for the separation of binary and multi-component systems in reverse osmosis process, Comput. Chem. Eng. 100 (2017) 48-79.
M. A. Al-Obaidi, J-P. Li, C. Kara-Zaïtri, I. M. Mujtaba, Optimisation of reverse osmosis based wastewater treatment system for the removal of chlorophenol using genetic algorithms, Chem. Eng. J. 316 (2017) 91-100.
G. Belfort, Membrane modules: Comparison of different configurations using fluid mechanics, J. Membr. Sci. 35 (1988) 245-270.
J. S. Vrouwenvelder, C. Picioreanu, J. C. Kruithof, M. C. M. van Loosdrecht, Biofouling in spiral wound membrane systems: Three-dimensional CFD model based evaluation of experimental data, J. Membr. Sci. 346 (2010) 71-85.
A. I. Radu, L. Bergwerff, M. C. M. van Loosdrecht, C. Picioreanu, A two-dimensional mechanistic model for scaling in spiral wound membrane systems, Chem. Eng. J. 241 (2014) 77-91.
A. I. Radu, J. S. Vrouwenvelder, M. C. M. van Loosdrecht, C. Picioreanu, Modeling the effect of biofilm formation on reverse osmosis performance: Flux, feed channel pressure drop and solute passage, J. Membr. Sci. 365 (2010) 1-15.
C. Picioreanu, J. S. Vrouwenvelder, M. C. M. van Loosdrecht, Three-dimensional modeling of biofouling and fluid dynamics in feed spacer channels of membrane devices, J. Membr. Sci. 345 (2009) 340-354.
J. M. Dickson, Fundamental aspects of reverse osmosis, in: Reverse osmosis technology: Application for high-purity-water production, B. S. Parekh (Ed.), Marcel Dekker, Inc., New York, 1988.
O. Kedem, A. Katchalsky, Thermodynamic analysis of the permeabilty of biological membranes to non- electrolytes, Biochem. Biophys. Acta 27 (1958) 229-246.
K. S. Spiegler, O. Kedem, Thermodynamics of hyperfiltration (reverse osmosis): Criteria for efficient membranes, Desalination 1 (1966) 311-326.
V. Sasidhar, Role of electrokinetic phenomena and of transport facilitation in membrane processes, PhD Thesis, State University of New York at Buffalo 1983.
W. Banks, A. Sharples, Studies on desalination by reverse osmosis: III. Mechanism of solute rejection, J. Appl. Chem. 16 (1966) 153-158.
H. K. Lonsdale, U. Merten, R. L. Riley, Transport properties of cellulose acetate osmotic membranes, J. Appl. Polym. Sci. 9 (1965) 1341-1362.
K. Kezia, J. Lee, A. J. Hill, S. E. Kentish, Convective transport of boron through a brackish water reverse osmosis membrane, J. Membr. Sci. 445 (2013) 160-169.
T. K. Sherwood, P. L. T. Brian, R. E. Fisher, Desalination by reverse osmosis, 1&EC Fund 6 (1967) 2-12.
E. Glueckauf, On the mechanism of osmotic desalting with porous membranes, Proc. First Int. Symp. on water desalination, Washington, US Dept. Interior, Office of Saline Water, 1 (1965) 143-156.
E. Glueckauf, The distribution of electrolytes between cellulose acetate membrane and aqueous solutions, Desalination 18 (1976) 155-172.
L. Onsager, Reciprocal relations in irreversible processes, Phys. Rev. 37 (1931) 405-425.
P. Xie, L. C. Murdoch, D. A. Ladner, Hydrodynamics of sinusoidal spacers for improved reverse osmosis performance, J. Membr. Sci. 453 (2014) 92-99.
P. Xie, Simulation of reverse osmosis and osmotically driven membrane processes, PhD Thesis, Clemson University, 2016.
L. Song, S. Yu, Concentration polarization in cross-flow reverse osmosis, AIChE J. 45 (1999) 921-928.
W. W. Focke, P. G. J. M. Nuijens, Velocity profile caused by a high porosity spacer between parallel plates (membranes), Desalination 49 (1984) 243-253.
S. Karode, A. Kumar, Flow visualization through spacer filled channels by computational fluid dynamics I. Pressure drop and shear rate calculations for flat sheet geometry, J. Membr. Sci. 193 (2001) 69-84.
M. Shakaib, S. M. F. Hasani, M. Mahmood, CFD modeling for flow and mass transfer in spacer-obstructed membrane feed channels, J. Membr. Sci. 326 (2009) 270-284.
A. Subramani, S. Kim, E. M. V. Hoek, Pressure, flow, and concentration profiles in open and spacer-filled membrane channels, J. Membr. Sci. 277 (2006) 7-17.
K. Madireddi, A transient model for predicting concentration polarization in commercial spiral wound membranes, PhD Thesis, University of California, Los Angeles, 1996.
M. F. Gruber, U. Aslak, C. Hélix-Nielsen, Open-source CFD model for optimization of forward osmosis and reverse osmosis membrane modules, Sep. Purif. Technol. 158 (2016) 183-192.
M. Park, J. Lee, C. Boo, S. Hong, S. A. Snyder, J. H. Kim, Modeling of colloidal fouling in forward osmosis membrane: Effects of reverse draw solution permeation, Desalination 314 (2013) 115-123.
S. Kim, E. M. V. Hoek, Modeling concentration polarization in reverse osmosis processes, Desalination 186 (2005) 111-128.
H. Yuan, I. M. Abu-Reesh, Z. He, Mathematical modeling assisted investigation of forward osmosis as pretreatment for microbial desalination cells to achieve continuous water desalination and wastewater treatment, J. Membr. Sci. 502 (2016) 116-123.
T.-U. Kim, J. E. Drewes, R. S. Summers, G. L. Amy, Solute transport model for trace organic neutral and charged compounds through nanofiltration and reverse osmosis membranes, Water Res. 41 (2007) 3977-3988.
S. S Manickam, J. R. McCutcheon, Model thin film composite membranes for forward osmosis: Demonstrating the inaccuracy of existing structural parameter models, J. Membr. Sci. 483 (2015) 70-74.
D. Ghernaout, A. El-Wakil, Requiring reverse osmosis membranes modifications – An overview, Am. J. Chem. Eng. 5 (2017) 81-88.