S. P. Agashichev, Modeling the influence of temperature on gel-enhanced concentration polarization in reverse osmosis, Desalination, vol.236, issue.1-3, pp.252-258, 2009.
DOI : 10.1016/j.desal.2007.10.074

A. L. Ahmad, F. , C. M. , S. , and B. , Mathematical modeling of multiple solutes system for reverse osmosis process in palm oil mill effluent (POME) treatment, Chemical Engineering Journal, vol.132, issue.1-3, pp.183-193, 2007.
DOI : 10.1016/j.cej.2006.12.022

N. M. Albastaki and A. Abbas, Predicting the performance of ro membranes, Desalination, vol.88, pp.181-187, 2010.

A. Alexiadis, D. E. Wiley, A. Vishnoi, R. H. Lee, D. F. Fletcher et al., CFD modelling of reverse osmosis membrane flow and validation with experimental results, Desalination, vol.217, issue.1-3, pp.242-250, 2007.
DOI : 10.1016/j.desal.2007.02.014

M. A. Ali, A. Ajbar, E. Ali, and K. Alhumaizi, Modeling the transient behavior of an experimental reverse osmosis tubular membrane, Desalination, vol.245, pp.194-204, 2009.

V. Alvarez, S. Alvarez, F. A. Riera, and R. Alvarez, Permeate flux prediction in apple juice concentration by reverse osmosis, Journal of Membrane Science, vol.127, issue.1, pp.25-34, 1997.
DOI : 10.1016/S0376-7388(96)00285-2

A. S. Berman, Laminar Flow in Channels with Porous Walls, Journal of Applied Physics, vol.3, issue.9, pp.1232-1235, 1953.
DOI : 10.1017/S0305004100019575

B. Bernales, Modélisation de l'hydrodynamique et des transferts dans les procédés de filtration membranaire, 2013.

B. Bernales and P. Haldenwang, Laminar flow analysis in a pipe with locally pressure-dependent leakage through the wall, European Journal of Mechanics - B/Fluids, vol.43, pp.100-109, 2014.
DOI : 10.1016/j.euromechflu.2013.07.006
URL : https://hal.archives-ouvertes.fr/hal-01053307

D. Bhattacharyya, S. L. Back, R. I. Kermode, and M. C. Roco, Prediction of concentration polarization and flux behavior in reverse osmosis by numerical analysis, Journal of Membrane Science, vol.48, issue.2-3, pp.231-262, 1990.
DOI : 10.1016/0376-7388(90)85007-8

A. Chatterjee, A. Ahluwalia, S. Senthilmurugan, and S. K. Gupta, Modeling of a radial flow hollow fiber module and estimation of model parameters using numerical techniques, Journal of Membrane Science, vol.236, issue.1-2, pp.1-16, 2004.
DOI : 10.1016/j.memsci.2004.01.006

J. Choi and J. Kim, Modeling of full-scale reverse osmosis desalination system: Influence of operational parameters, Journal of Industrial and Engineering Chemistry, vol.21, pp.261-268, 2015.
DOI : 10.1016/j.jiec.2014.02.033

M. L. Costa and J. M. Dickson, Modelling of modules and systems in reverse osmosis. Part I: Theoretical system design model development, Desalination, vol.80, issue.2-3, pp.251-274, 1991.
DOI : 10.1016/0011-9164(91)85162-N

J. Fernández-sempere, F. Ruiz-beviá, P. García-algado, and R. Salcedo-díaz, Experimental study of concentration polarization in a crossflow reverse osmosis system using Digital Holographic Interferometry, Desalination, vol.257, issue.1-3, pp.36-45, 2010.
DOI : 10.1016/j.desal.2010.03.010

D. F. Fletcher and D. E. Wiley, A computational fluids dynamics study of buoyancy effects in reverse osmosis, Journal of Membrane Science, vol.245, issue.1-2, pp.175-181, 2004.
DOI : 10.1016/j.memsci.2004.07.023

T. Fujioka, S. J. Khan, J. A. Mcdonald, A. Roux, Y. Poussade et al., Modelling the rejection of N-nitrosamines by a spiral-wound reverse osmosis system: Mathematical model development and validation, Journal of Membrane Science, vol.454, pp.212-219, 2014.
DOI : 10.1016/j.memsci.2013.12.008

D. V. Gauwbergen and J. Baeyens, Modelling reverse osmosis by irreversible thermodynamics, Separation and Purification Technology, vol.13, issue.2, pp.117-128, 1998.
DOI : 10.1016/S1383-5866(97)00065-8

V. Geraldes, V. Semiao, and M. De-pinho, The effect on mass transfer of momentum and concentration boundary layers at the entrance region of a slit with a nanofiltration membrane wall, Chemical Engineering Science, vol.57, issue.5, pp.735-748, 2002.
DOI : 10.1016/S0009-2509(01)00441-9

P. Haldenwang, Laminar flow in a two-dimensional plane channel with local pressure-dependent crossflow, Journal of Fluid Mechanics, vol.15, pp.463-473, 2007.
DOI : 10.1063/1.864735
URL : https://hal.archives-ouvertes.fr/hal-00907372

P. Haldenwang and P. Guichardon, Pressure runaway in a 2D plane channel with permeable walls submitted to pressure-dependent suction, European Journal of Mechanics - B/Fluids, vol.30, issue.2, pp.177-183, 2011.
DOI : 10.1016/j.euromechflu.2010.09.007
URL : https://hal.archives-ouvertes.fr/hal-00905831

P. Haldenwang, P. Guichardon, G. Chiavassa, and N. Ibaseta, Exact solution to mass transfer in Berman flow: Application to concentration polarization combined with osmosis in crossflow membrane filtration, International Journal of Heat and Mass Transfer, vol.53, issue.19-20, pp.3898-3904, 2010.
DOI : 10.1016/j.ijheatmasstransfer.2010.05.008
URL : https://hal.archives-ouvertes.fr/hal-00907275

L. Hung, S. J. Lue, and J. You, Mass-transfer modeling of reverse-osmosis performance on 0.5???2% salty water, Desalination, vol.265, issue.1-3, pp.67-73, 2011.
DOI : 10.1016/j.desal.2010.07.033

K. Jamal, M. A. Khan, and M. Kamil, Mathematical modeling of reverse osmosis systems, Desalination, vol.160, issue.1, pp.29-42, 2004.
DOI : 10.1016/S0011-9164(04)90015-X

A. S. Kahdim, S. Ismail, and A. A. Jassim, Modeling of reverse osmosis systems, Desalination, vol.158, issue.1-3, pp.323-329, 2003.
DOI : 10.1016/S0011-9164(03)00471-5

A. S. Kim, Permeate flux inflection due to concentration polarization in crossflow membrane filtration: A novel analytic approach, The European Physical Journal E, vol.13, issue.4, pp.331-341, 2007.
DOI : 10.1007/978-94-009-1766-8

S. Kim and E. M. Hoek, Modeling concentration polarization in reverse osmosis processes, Desalination, vol.186, issue.1-3, pp.111-128, 2005.
DOI : 10.1016/j.desal.2005.05.017

C. Liu, A. Morse, K. Rainwater, and L. Song, Modeling of Concentration Polarization in a Reverse Osmosis Channel with Parabolic Crossflow, Water Environment Research, vol.86, issue.1, pp.56-62, 2014.
DOI : 10.2175/106143013X13778144233972

G. Lopes, B. B. Chavez, N. Ibaseta, P. Guichardon, and P. Haldenwang, Prediction of Permeate Flux and Rejection Rate in RO and NF Membrane Processes: Numerical Modelling of Hydrodynamics and Mass Transfer Coupling, Procedia Engineering, vol.44, pp.1934-1936, 2012.
DOI : 10.1016/j.proeng.2012.09.001
URL : https://hal.archives-ouvertes.fr/hal-01299944

G. Lopes, N. Ibaseta, P. Guichardon, and P. Haldenwang, Predicting Permeate Fluxes and Rejection Rates in Reverse Osmosis and Tight-Nanofiltration Processes, Chemical Engineering & Technology, vol.39, issue.1, pp.585-594, 2015.
DOI : 10.1016/j.progpolymsci.2013.07.001
URL : https://hal.archives-ouvertes.fr/hal-01135689

E. Lyster, J. Au, R. Rallo, F. Giralt, and Y. Cohen, Coupled 3-D hydrodynamics and mass transfer analysis of mineral scaling-induced flux decline in a laboratory plate-and-frame reverse osmosis membrane module, Journal of Membrane Science, vol.339, issue.1-2, pp.39-48, 2009.
DOI : 10.1016/j.memsci.2009.04.024

E. Lyster and Y. Cohen, Numerical study of concentration polarization in a rectangular reverse osmosis membrane channel: Permeate flux variation and hydrodynamic end effects, Journal of Membrane Science, vol.303, issue.1-2, pp.140-153, 2007.
DOI : 10.1016/j.memsci.2007.07.003

S. Ma, L. Song, S. L. Ong, and W. J. Ng, A 2-D streamline upwind Petrov/Galerkin finite element model for concentration polarization in spiral wound reverse osmosis modules, Journal of Membrane Science, vol.244, issue.1-2, pp.129-139, 2004.
DOI : 10.1016/j.memsci.2004.06.048

L. Malaeb and G. M. Ayoub, Reverse osmosis technology for water treatment: State of the art review, Desalination, vol.267, issue.1, pp.1-8, 2011.
DOI : 10.1016/j.desal.2010.09.001

A. Malek, M. N. Hawlader, and J. C. Ho, A lumped transport parameter approach in predicting B10 RO permeator performance, Desalination, vol.99, issue.1, pp.19-38, 1994.
DOI : 10.1016/0011-9164(94)00117-0

P. P. Mane, H. H. Park, J. C. Brown, and J. Kim, Modeling boron rejection in pilot- and full-scale reverse osmosis desalination processes, Journal of Membrane Science, vol.338, issue.1-2, pp.119-127, 2009.
DOI : 10.1016/j.memsci.2009.04.014

H. Mehdizadeh, K. Molaiee-nejad, and Y. C. Chong, Modeling of mass transport of aqueous solutions of multi-solute organics through reverse osmosis membranes in case of solute???membrane affinityPart 1. Model development and simulation, Journal of Membrane Science, vol.267, issue.1-2, pp.27-40, 2005.
DOI : 10.1016/j.memsci.2005.03.059

R. Mellis, W. N. Gill, and G. Belfort, FLUID DYNAMICS IN A TUBULAR MEMBRANE: THEORY AND EXPERIMENT, Chemical Engineering Communications, vol.21, issue.37, pp.103-125, 1993.
DOI : 10.1002/aic.690360406

M. Moresi, B. Ceccantoni, and S. L. Presti, Modelling of ammonium fumarate recovery from model solutions by nanofiltration and reverse osmosis, Journal of Membrane Science, vol.209, issue.2, pp.405-420, 2002.
DOI : 10.1016/S0376-7388(02)00330-7

C. Prabhavathy and S. De, Treatment of fatliquoring effluent from a tannery using membrane separation process: Experimental and modeling, Journal of Hazardous Materials, vol.176, issue.1-3, pp.434-443, 2011.
DOI : 10.1016/j.jhazmat.2009.11.048

T. Y. Qiu and P. A. Davies, Concentration polarization model of spiral-wound membrane modules with application to batch-mode RO desalination of brackish water, Desalination, vol.368, pp.36-47, 2015.
DOI : 10.1016/j.desal.2014.12.048

S. A. Regirer, On the approximate theory of the flow of a viscous incompressible liquid in a tube with permeable walls, Sov. Phys. Tech. Phys, vol.5, pp.602-605, 1960.

E. Roth, B. Fabre, A. Accary, and G. Thomas, Modelling of stimulus response experiments in the feed channel of spiral-wound reverse osmosis membranes, Desalination, vol.127, issue.1, pp.69-77, 2000.
DOI : 10.1016/S0011-9164(99)00193-9

C. Sagne, C. Fargues, B. Broyart, M. Lameloise, and M. Decloux, Modeling permeation of volatile organic molecules through reverse osmosis spiral-wound membranes, Journal of Membrane Science, vol.330, issue.1-2, pp.40-50, 2009.
DOI : 10.1016/j.memsci.2008.12.038
URL : https://hal.archives-ouvertes.fr/hal-01266517

R. Salcedo-diaz, P. Garcia-algado, M. Garcia-rodriguez, J. Fernandez-sempere, and F. Ruiz-bevia, Visualization and modeling of the polarization layer in crossflow reverse osmosis in a slit-type channel, Journal of Membrane Science, vol.456, pp.21-30, 2014.
DOI : 10.1016/j.memsci.2014.01.019

M. Sekino, Precise analytical model of hollow fiber reverse osmosis modules, Journal of Membrane Science, vol.85, issue.3, pp.241-252, 1993.
DOI : 10.1016/0376-7388(93)85278-5

M. Sekino, Study of an analytical model for hollow fiber reverse osmosis module systems, Desalination, vol.100, issue.1-3, pp.85-97, 1995.
DOI : 10.1016/0011-9164(96)00010-0

S. Senthilmurugan, A. Ahluwalia, and S. K. Gupta, Modeling of a spiral-wound module and estimation of model parameters using numerical techniques, Desalination, vol.173, issue.3, pp.269-286, 2005.
DOI : 10.1016/j.desal.2004.08.034

M. Soltanieh and W. N. Gill, REVIEW OF REVERSE OSMOSIS MEMBRANES AND TRANSPORT MODELS, Chemical Engineering Communications, vol.22, issue.9, pp.279-363, 1981.
DOI : 10.1080/00986448108912583

L. Song, Concentration polarization in a narrow reverse osmosis membrane channel, AIChE Journal, vol.173, pp.143-149, 2010.
DOI : 10.1016/j.desal.2004.06.035

L. Song and M. Elimelech, Theory of concentration polarization in crossflow filtration, Journal of the Chemical Society, Faraday Transactions, vol.91, issue.19, pp.3389-3398, 1995.
DOI : 10.1039/ft9959103389

L. Song and C. Liu, A total salt balance model for concentration polarization in crossflow reverse osmosis channels with shear flow, Journal of Membrane Science, vol.401, issue.402, pp.401-402, 2012.
DOI : 10.1016/j.memsci.2012.02.023

L. Song and K. G. Tay, Performance prediction of a long crossflow reverse osmosis membrane channel, Journal of Membrane Science, vol.281, issue.1-2, pp.163-169, 2006.
DOI : 10.1016/j.memsci.2006.03.026

L. Song and S. Yu, Concentration polarization in cross-flow reverse osmosis . Fluid Mechanics and Transport Phenomena 45, pp.921-928, 1999.

A. Subramani, S. Kim, and E. M. Hoek, Pressure, flow, and concentration profiles in open and spacer-filled membrane channels, Journal of Membrane Science, vol.277, issue.1-2, pp.7-17, 2006.
DOI : 10.1016/j.memsci.2005.10.021

S. Sundaramoorthy, G. Srinivasan, and D. V. Murthy, An analytical model for spiral wound reverse osmosis membrane modules: Part I ??? Model development and parameter estimation, Desalination, vol.280, issue.1-3, pp.403-411, 2011.
DOI : 10.1016/j.desal.2011.03.047

N. Tilton, D. Martinand, E. Serre, and R. M. Lueptow, Incorporating Darcy's law for pure solvent flow through porous tubes: Asymptotic solution and numerical simulations, AIChE Journal, vol.28, issue.7, pp.230-244, 2012.
DOI : 10.1002/(SICI)1097-0363(19980915)28:3<501::AID-FLD730>3.0.CO;2-S

R. I. Urama and B. J. Marinas, Mechanistic interpretation of solute permeation through a fully aromatic polyamide reverse osmosis membrane, Journal of Membrane Science, vol.123, issue.2, pp.267-280, 1997.
DOI : 10.1016/S0376-7388(96)00230-X

B. Van-der-bruggen, M. Manttari, and N. Nystrom, Drawbacks of applying nanofiltration and how to avoid them: A review, Separation and Purification Technology, vol.63, issue.2, pp.251-263, 2008.
DOI : 10.1016/j.seppur.2008.05.010

N. G. Voros, Z. B. Maroulis, and D. Marinos-kouris, Salt and water permeability in reverse osmosis membranes, Desalination, vol.104, issue.3, pp.141-154, 1996.
DOI : 10.1016/0011-9164(96)00037-9

J. Weissbrodt, M. Manthey, B. Digens, G. Laufenberg, and B. Kunz, Separation of aqueous organic multi-component solutions by reverse osmosis ??? development of a mass transfer model, Desalination, vol.133, issue.1, pp.65-74, 2001.
DOI : 10.1016/S0011-9164(01)00083-2

J. G. Wijmans and R. W. Baker, The solution-diffusion model: a review, Journal of Membrane Science, vol.107, issue.1-2, pp.1-21, 1995.
DOI : 10.1016/0376-7388(95)00102-I

D. E. Wiley and D. F. Fletcher, Techniques for computational fluid dynamics modelling of flow in membrane channels, Journal of Membrane Science, vol.211, issue.1, pp.127-137, 2003.
DOI : 10.1016/S0376-7388(02)00412-X

W. Zhou, L. Song, and T. K. Guan, A numerical study on concentration polarization and system performance of spiral wound RO membrane modules, Journal of Membrane Science, vol.271, issue.1-2, pp.38-46, 2006.
DOI : 10.1016/j.memsci.2005.07.007