PARAMETER ESTIMATION AND MODELING OF SPIRAL-WOUND

(C) Concentration Polarization
Concentration polarization is one of the most important factors limiting performance of nearly all membrane separation processes (Strathmann, 1981). CP effect leads to increased flow resistance and the solute passage through the membrane, thereby reducing permeate flux and rejection of solutes. Therefore, designing of a reverse osmosis system requires a fair prediction of CP phenomenon. The steady state concentration of solute at the membrane wall (i.e. in the CP layer) can be determined by following nonlinear relationship (Marinas and Urama, 1996).

The equation for concentration polarization in finite difference form can be given by :
φ(i,j)=[Cm(i,j) – Cp(i,j)]/[Cb(i,j) – Cp(i,j)]=exp[Jv(i,j)/k(i,j)] (4)
where Cm = concentration of solute adjacent to membrane wall [mol/m3; ML-3]; Cp = concentration of solute in permeate [mol/m3; ML-3]; Cf = concentration of solute in the feed solution [mol/m3; ML-3]; Jv = volumetric flux of water through membrane [m3/ (m2•s); LT-1]; and k = mass transfer coefficient [m/s; LT-1]. In this study, CP layer is assumed to be fully developed, and a simple non-linear mathematical equation with a lumped parameter approach is used to predict the degree of concentration polarization, as compared to various numerical models based on distributed parameter approaches (Bhattacharya and Hwang, 1997; Kim and Hoek, 2005).

(D) Mass Transfer Coefficient
Several mass transfer coefficient relationships have been developed in the past, as briefly reviewed by Gekas and Hallstrom (1987). The mass transfer through RO membrane is influenced by geometry of spacers (turbulence promoters), geometry of feed flow channel (thickness), fluid properties (dynamic viscosity and velocity) and the solute properties (diffusivity). The presence of the turbulence promoters in spiral wound module needs to be accounted for, because it leads to increased mass transfer as the laminar boundary layer thickness reduces due to turbulence. Considering above factors, an empirical equation that includes the effect of all of the above parameters would best serve the purpose of developing a design and simulation model.
Therefore, a generic empirical equation developed by Winograd et al. (1973) to predict the mass transfer in narrow channels in presence of turbulence promoters is used in this study, based on a sensitivity analysis presented in later part of this study (Winograd, Solan et al., 1973):
k = 0.753[K/(2-K)]^0.5(D/hb) Sc^(-1/6)(Pe.hb/Lmix)^0.5 (5)
Also the variation in the mass transfer coefficient at local grid points may be given as:
k=ka [Ub(i.j)]^kb (6)
where, k = mass transfer coefficient [m/s; LT-1]; K = 0.5 = efficiency of mixing net [dimensionless]; D = diffusion coefficient [m2/s; L2T-1]; hb = thickness of the feed channel [m; L]; hp = thickness of permeate channel [m; L]; Sc = Schmidt number =μ/ρD,[dimensionless]; Pe = Peclet number = 2hbUb/D, [dimensionless]; Lmix= 0.006 m=characteristic length of mixing net [m; L].Ub =feed solution velocity on bulk side [m/s;LT-1].