Two-Dimensional Mathematical Model of Flows in Thin Film Composite Membranes

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This study presents a mathematical model for flows in thin film composite membranes, focusing on the permeation of solvent flux and solute rejection. Assumptions include incompressible fluid, constant diffusion of chemical species, and isothermal conditions. Equations describe water flux, solute flux, fluid flows in feed/permeate channels, continuity, and Navier-Stokes equations. The model provides insights into the hydrodynamics within the membrane system.

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  1. TWO-DIMENSIONAL MATHEMATICAL MODEL OF FLOWS IN THIN FILM COMPOSITE MEMBRANES Aatma Maharajh1, Prakash Persad2, Denver Cheddie3, Edward Cumberbatch4 1,2,4 Design and Manufacturing Systems, The University of Trinidad and Tobago, Trinidad and Tobago 3 Utilities Engineering Group, The University of Trinidad and Tobago, Trinidad and Tobago IConETech-2020, Faculty of Engineering, The UWI, St. Augustine, Trinidad and Tobago

  2. Modelling RO Membranes Flows across membrane Analytical Models Numerical Models Concentration Polarisation Hydrodynamics within the Feed and Permeate Channels IConETech-2020, Faculty of Engineering, The UWI, St. Augustine, Trinidad and Tobago

  3. Model Assumptions 1. Only two (2) dimensions were considered. Cartesian coordinates system was used. 2. The Solution-Diffusion model was assumed true and used to describe permeation of solvent flux through the membrane. 3. Water permeability was assumed constant. 4. Bulk rejection of solute occurred in the active layer of the membrane. 5. Laminar flow conditions persisted across the feed and permeate channels. IConETech-2020, Faculty of Engineering, The UWI, St. Augustine, Trinidad and Tobago

  4. Model Assumptions 6. The fluid was assumed to be incompressible with constant viscosity and density. 7. No reactions took place between chemical species in the feed channel, membrane and permeate channels. 8. Chemical species were assumed to have constant diffusivity that were not concentration dependent. 9. The process was isothermal. IConETech-2020, Faculty of Engineering, The UWI, St. Augustine, Trinidad and Tobago

  5. Model Equations Water flux across the membrane is given by pseudo Darcy Equation: ??= ?? ? ? Solute flux across the active layer of the membrane is assumed purely diffusive and is given by: ??(?)?2? = 0 IConETech-2020, Faculty of Engineering, The UWI, St. Augustine, Trinidad and Tobago

  6. Model Equations Fluid flows in feed/permeate channels are given by incompressible forms of the continuity and Navier-Stokes Equations. Solute flows in the feed, porous support layer of the membrane and permeate channels given by transport equation assuming both advective and diffusive components. IConETech-2020, Faculty of Engineering, The UWI, St. Augustine, Trinidad and Tobago

  7. Model Validation Validated against Kim & Hoek (2005) Considered: 3 operating pressure (790 kPa, 1136 kPa and 1481kPa) at 3 average crossflow velocities (0.017 m s-1, 0.042 m s-1 and 0.068 m s-1) at 3 concentrations (10 mol m-3, 20 mol m-3 and 50 mol m-3) IConETech-2020, Faculty of Engineering, The UWI, St. Augustine, Trinidad and Tobago

  8. Model Geometry INLET FEED CHANNEL OUTLET Navier-Stokes, Continuity and Transport Equations MEMBRANE Pseudo Darcy Equation and Transport Equations PERMEATE CHANNEL Navier-Stokes, Continuity, Transport Equations IConETech-2020, Faculty of Engineering, The UWI, St. Augustine, Trinidad and Tobago

  9. Grid Independence 13.6 4.5 Average Permeate Flow ( m/s) 4 Concentration (mol/m3) 13.4 Average Permeate 3.5 13.2 3 13 2.5 2 12.8 1.5 12.6 1 12.4 0.5 12.2 0 0 1000000 2000000 3000000 4000000 5000000 6000000 7000000 Number of Elements Average Permeate Flux Average Permeate Concentration IConETech-2020, Faculty of Engineering, The UWI, St. Augustine, Trinidad and Tobago

  10. Results Avg. Crossflow Vel. = 0.017ms-1 25.000 7 Average Permeate Concentration (mol/m3) 6 Average Permeate Flow ( m/s) 20.000 5 15.000 4 3 10.000 2 5.000 1 0.000 0 700 800 900 1000 1100 Inlet Pressure (kPa) 1200 1300 1400 1500 1600 10 mol/m^3 20 mol/m^3 50 mol/m^3 Simulations Perm. Flow 10 mol/m^3 20 mol/m^3 50 mol/m^3 Simulations for Perm. Conc. IConETech-2020, Faculty of Engineering, The UWI, St. Augustine, Trinidad and Tobago

  11. Results Avg. Crossflow Vel. = 0.042ms-1 5 25.000 4.5 Average Permeate Conentration (mol/m3) 4 Average Permeate Flow ( m/s) 20.000 3.5 3 15.000 2.5 2 10.000 1.5 1 5.000 0.5 0.000 0 700 800 900 1000 1100 1200 1300 1400 1500 1600 Inlet Pressure (kPa) 10 mol/m^3 20 mol/m^3 50 mol/m^3 Simulations for Perm. Flow 10 mol/m^3 20mol/m^3 50 mol/m^3 Simulation of Perm. Conc. IConETech-2020, Faculty of Engineering, The UWI, St. Augustine, Trinidad and Tobago

  12. Results Avg. Crossflow Vel. = 0.068ms-1 30.000 4.5 Average Permeate Concentration Average Permeate Flow ( m/s) 4 25.000 3.5 20.000 3 (mol/m3) 2.5 15.000 2 10.000 1.5 1 5.000 0.5 0.000 0 700 800 900 1000 1100 1200 1300 1400 1500 1600 Inlet Pressure (kPa) 10 mol/m^3 20 mol/m^3 50 mol/m^3 Simulations for Perm. Flow 10 mol/m^3 20 mol/m^3 50 mol/m^3 Simulations for Perm. Conc. IConETech-2020, Faculty of Engineering, The UWI, St. Augustine, Trinidad and Tobago

  13. Results The average predicted error for permeate flow across all simulations was -0.6% 2.6%. The average predicted error for permeate concentration was 0.7% 7.6%. IConETech-2020, Faculty of Engineering, The UWI, St. Augustine, Trinidad and Tobago

  14. Feed Spacers Used to keep the membrane(s) apart from the flow channel. Promote shearing at the membrane surface reducing effect of concentration polarization. Geometric Ratio by Koutsou, Yiantsios, Karabelas (2007): ?? ? ?.?.= IConETech-2020, Faculty of Engineering, The UWI, St. Augustine, Trinidad and Tobago

  15. FEED SPACER Table 1 Simulation Results for Submerged Spacer Types Avg. Channel Velocity (m/s) Feed Channel Pressure Loss (Pa/m) Max. Wall Shear Stress (N m-2) Spacer Type Geometric Ratio Sim Jw ( m/s) Sim. Cp (mol/m3) No Spacer n/a 11.97 3.738 0.0420 158.5 0.1977 7 8 9 13.15 13.03 12.92 3.064 3.128 3.187 0.0453 0.0448 0.0446 1039.0 930.3 857.1 1.1781 1.1777 1.1773 Subme- rged IConETech-2020, Faculty of Engineering, The UWI, St. Augustine, Trinidad and Tobago

  16. FEED SPACERS Table 2 Simulation Results for Cavity and Zigzag Spacer Types Avg. Channel Velocity (m/s) 0.0426 0.0422 0.0419 0.0427 0.0422 0.0419 Feed Channel Pressure Loss (Pa/m) 488.7 446.6 418.0 477.8 440.6 415.5 Max. Wall Shear Stress (N m-2) Spacer Type Geometric Ratio Sim Jw ( m/s) Sim. Cp (mol/m3) 7 8 9 7 8 9 13.33 13.67 13.83 14.30 14.25 14.19 3.688 3.481 3.367 3.015 3.005 2.983 0.2197 0.2432 0.2435 0.5523 0.5549 0.5585 Cavity Zigzag IConETech-2020, Faculty of Engineering, The UWI, St. Augustine, Trinidad and Tobago

  17. Conclusions Developed a 2D mathematical model for thin film RO Membranes. Model accounts for various layers of thin film RO Membranes. Model accounts for the effects of feed spacers. IConETech-2020, Faculty of Engineering, The UWI, St. Augustine, Trinidad and Tobago

  18. REFERENCES S. Kim, E.M. Hoek (2005). Modeling concentration polarization in reverse osmosis processes. Desalination, 186(1-3), 111-128. C.P. Koutsou, S.G. Yiantsios, A.J. Karabelas (2007). Direct numerical simulation of flow in spacer-filled channels: Effect of spacer geometrical characteristics. Journal of Membrane Science, 291(1-2), 53-69. IConETech-2020, Faculty of Engineering, The UWI, St. Augustine, Trinidad and Tobago

  19. THANK YOU! IConETech-2020, Faculty of Engineering, The UWI, St. Augustine, Trinidad and Tobago

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