Effects of different potential functions on modeling of RO membrane performance by use of an advanced model
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Effects of different potential functions on modeling of RO membrane performance by use of an advanced model A. Golnari • A. Moradi • A. Soltani
Received: 3 May 2012 / Accepted: 18 August 2012 / Published online: 14 September 2012 Ó Springer Science+Business Media B.V. 2012
Abstract Modeling of reverse osmosis membranes is essential for progress in membrane separation technology. Effects of a new potential function on the modified surface force pore flow (MD-SF-PF) model were investigated by Moradi et al. in 2012. The main purpose of this research was to study the effects of other potential functions on the advanced and extended form of the MD-SF-PF model (i.e. the EXMD-SF-PF model). The results obtained were compared by use of an objective function and by using the root-mean-square deviation. The best potential function was finally selected and a new model was proposed. The complex and nonlinear equations of the EX-MD-SF-PF model were solved by use of advanced numerical methods, for example finite difference and the finite volume. The proposed model, besides enabling excellent prediction of the performance of membranes, for example separation factor, and solvent and pure fluxes, has much better root-meansquare deviation than the EX-MD-SF-PF model. Profiles of potential function against pore radius and pore length were plotted and good trajectories were obtained. Profiles of solute concentration at the beginning, middle, and end of a pore were calculated and compared. The velocity profile of the permeating solvent was calculated and plotted. Most of the experimental data were predicted by use of the model. Keywords RO membrane Modeling EX-MD-SF-PF model Potential function Numerical solution List of symbols b(q) Friction function defined in Eq. (25), dimensionless C Molar density of solution (kmol/m3) A. Golnari A. Moradi (&) A. Soltani Chemical Engineering Department, Shahid Bahonar University of Kerman, P.O. Box 76175-133, Kerman, Iran e-mail: [email protected]; [email protected]
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CA(r,z) DAB DAM E f0 FAB JA;r ðrÞ JA;z ðrÞ l1 and l3 Ni P r R T Ui(r) z k ðqÞ OF RMSD Greek aðqÞ b1 DP e g h1 h2 h3 k f p(r,z) pi q s U(q,f) Xij r2(q)
A. Golnari et al.
Concentration of solute inside a pore (kmol/m3) Solute diffusivity in the free solution (m2/s) Solute diffusivity inside the pore (m2/s) Friction variable defined by Eq. (25) (m) Theoretical separation defined by Eq.(14), dimensionless Frictional force between solute and solvent (kJ/m kmol) Radial component of solute flux through a single pore (kmol/m2 s) Axial component of solute flux through a single pore (kmol/m2 s) Definite integrals defined by Eqs. (19a) and (19b), dimensionless Flux of i through membrane (kmol/m2 s) Hydrostatic pressure (kPa) Cylindrical coordinate normal to the pore wall (m) Gas constant (kJ/kmol K) Temperature (K) Velocity of i inside the pore (m/s) Cylindrical coordinate parallel to the pore wall (m) Ratio of local partition coefficients at the ends of a pore for solute, defined by Eq. (7) Objective function de
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