TWODIMENSIONAL MATHEMATICAL MODEL OF FLOWS IN THIN FILM

TWO-DIMENSIONAL MATHEMATICAL MODEL OF FLOWS IN THIN FILM COMPOSITE MEMBRANES Aatma Maharajh 1, Prakash Persad 2, Denver Cheddie 3, Edward Cumberbatch 4 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 ICon. ETech-2020, Faculty of Engineering, The UWI, St. Augustine, Trinidad and Tobago

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

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. ICon. ETech-2020, Faculty of Engineering, The UWI, St. Augustine, Trinidad and Tobago

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. ICon. ETech-2020, Faculty of Engineering, The UWI, St. Augustine, Trinidad and Tobago

Model Equations • ICon. ETech-2020, Faculty of Engineering, The UWI, St. Augustine, Trinidad and Tobago

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. ICon. ETech-2020, Faculty of Engineering, The UWI, St. Augustine, Trinidad and Tobago

Model Validation • Validated against Kim & Hoek (2005) • Considered: – 3 operating pressure (790 k. Pa, 1136 k. Pa and 1481 k. Pa) – 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) ICon. ETech-2020, Faculty of Engineering, The UWI, St. Augustine, Trinidad and Tobago

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

13, 6 4, 5 13, 4 4 13, 2 3, 5 13 3 12, 8 2, 5 12, 6 2 12, 4 1, 5 12, 2 1 12 0, 5 11, 8 11, 6 0 1000000 2000000 3000000 4000000 5000000 6000000 0 7000000 Average Permeate Concentration (mol/m 3) Average Permeate Flow (µm/s) Grid Independence Number of Elements Average Permeate Flux Average Permeate Concentration ICon. ETech-2020, Faculty of Engineering, The UWI, St. Augustine, Trinidad and Tobago

Results – Avg. Crossflow Vel. = 0. 017 ms-1 7 6 20, 000 5 15, 000 4 3 10, 000 2 5, 000 1 0, 000 700 800 900 1000 1100 1200 1300 1400 1500 Average Permeate Concentration (mol/m 3) Average Permeate Flow (µm/s) 25, 000 0 1600 Inlet Pressure (k. Pa) 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. ICon. ETech-2020, Faculty of Engineering, The UWI, St. Augustine, Trinidad and Tobago

Results – Avg. Crossflow Vel. = 0. 042 ms-1 Average Permeate Flow (µm/s) 4, 5 4 20, 000 3, 5 3 15, 000 2, 5 2 10, 000 1, 5 1 5, 000 0, 5 0, 000 700 800 900 1000 1100 1200 1300 1400 1500 Average Permeate Conentration (mol/m 3) 5 25, 000 0 1600 Inlet Pressure (k. Pa) 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 Simulation of Perm. Conc. ICon. ETech-2020, Faculty of Engineering, The UWI, St. Augustine, Trinidad and Tobago

30, 000 4, 5 4 25, 000 3, 5 20, 000 3 2, 5 15, 000 2 10, 000 1, 5 1 5, 000 0, 5 0, 000 700 800 900 1000 1100 1200 1300 1400 1500 0 1600 Average Permeate Concentration (mol/m 3) Average Permeate Flow (µm/s) Results – Avg. Crossflow Vel. = 0. 068 ms-1 Inlet Pressure (k. Pa) 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. ICon. ETech-2020, Faculty of Engineering, The UWI, St. Augustine, Trinidad and Tobago

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%. ICon. ETech-2020, Faculty of Engineering, The UWI, St. Augustine, Trinidad and Tobago

Feed Spacers • ICon. ETech-2020, Faculty of Engineering, The UWI, St. Augustine, Trinidad and Tobago

FEED SPACER Table 1 Simulation Results for Submerged Spacer Types Spacer Geometric Sim Jw Sim. Cp Type Ratio (μm/s) (mol/m 3) Avg. Channel Velocity (m/s) Feed Max. Wall Channel Shear Stress Pressure (N m-2) Loss (Pa/m) No Spacer n/a 11. 97 3. 738 0. 0420 158. 5 0. 1977 Submerged 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 ICon. ETech-2020, Faculty of Engineering, The UWI, St. Augustine, Trinidad and Tobago

FEED SPACERS Table 2 Simulation Results for Cavity and Zigzag Spacer Types Spacer Geometric Sim Jw Type Ratio (μm/s) Cavity Zigzag 7 8 9 13. 33 13. 67 13. 83 14. 30 14. 25 14. 19 Avg. Feed Max. Wall Sim. Cp Channel Shear Stress (mol/m 3) Velocity Pressure (N m-2) (m/s) Loss (Pa/m) 3. 688 0. 0426 488. 7 0. 2197 3. 481 0. 0422 446. 6 0. 2432 3. 367 0. 0419 418. 0 0. 2435 3. 015 0. 0427 477. 8 0. 5523 3. 005 0. 0422 440. 6 0. 5549 2. 983 0. 0419 415. 5 0. 5585 ICon. ETech-2020, Faculty of Engineering, The UWI, St. Augustine, Trinidad and Tobago

Conclusions • Developed a 2 D 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. ICon. ETech-2020, Faculty of Engineering, The UWI, St. Augustine, Trinidad and Tobago

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. ICon. ETech-2020, Faculty of Engineering, The UWI, St. Augustine, Trinidad and Tobago

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