PARAMETER ESTIMATION AND MODELING OF SPIRAL-WOUND

(E)Pressure Drop
The flow pattern in feed channel and permeate channel are oriented in orthogonal directions (as described in section before). The pressure drop in these two channels over an infinitesimal element ‘Δx’ & ‘Δy’ can be calculated by the following equations (Senthilmurugan, Ahluwalia et al., 2005):

Pb(i-1,j) – Pb(i,j) = 2kfbnFU¬b(i,j)nF-1μ Δx2∑1i-1[Jv(i,j)/hb] (7)

Pp(i,j+1) – Pp(i,j) = 2kfp μ Δy2 ∑j+1n[Jv(i,j)/hp] (8)

where Pb = pressure in bulk solution side [Pa; ML-1T-2]; Pp = pressure in permeate channel (permeate side) [Pa; ML-1T-2]; Δx = element along x axis [m; L]; Δy = element along y axis [m; L]; μ = dynamic viscosity [Pa•s; ML-1T-1]; Jv = permeate flux [m3/(m2•s);LT-1]; hb = depth of feed channel [m; L]; hp = thickness of permeate channel [m; L]; nF =dimensionless constant parameter[dimensionless]; kfb = friction parameter for feed channel [(1/m2); L-2]; kfp = friction parameter for permeate channel [(1/m2); L-2].
The applicable boundary conditions for the above equations are as follows:
Pf = Pf0, at x = 0 (9)
Pp = Patm, at y = 0 (10)
Where Patm = atmospheric pressure (assumed to be equal to pressure in permeate at outlet) = 1.013×105 Pascal. Above pressure drop equations are derived based on the assumption that the Darcy’s law is applicable for flow through narrow channels and the dimensionless constant nF = 1.


(F) Material Balance and Performance
The local grid area through which the fluid flows and the feed flow rate at the entry grid point of the feed channel may be written as:-
S(i,j) =2*(L*W)/(m*n) (11)
Qb(1,j) = Qf/n*leaves (12)
Where S=membrane (grid) surface area [m2; L2]; Qb=flow rate in bulk solution side [m3/s; L3T-1]; Qf = input feed flow velocity [m3/s; L3T-1]; leaves =3.
The finite difference form of the overall material balance is given by :
Qb(i+1,j) – Qb(i,j) = -Jv(i,j)*S(i,j) (13)
And the bulk solution at inlet is:
Cb(1,j) = Cf0 (14)
Similarly the solute material balance equation and the permeate concentration in finite difference formulation may be written as follows:
Cb(i,j) Qb(i,j) = Cb(i+1,j) Qb(i+1,j) + S(i,j) Cp(i,j) Jv(i,j) (15)
Cp(i,j) = [φ(i,j) (1-R(i,j)Cb(i,j)]/[ φ(i,j) + R(i,j) (1- φ(i,j) )] (16)



Where,
R(i,j) = [(1 – F(i,j))σ]/[1- σ F(i,j)] (17)
F(i,j) = exp[-Jv(i,j)(1- σ)/B] (18)
Where Cb & Cp are solute concentrations at bulk solution side and permeate side respectively [kg/m3; ML3]; R= rejection [dimensionless]; F=flow parameter defined by eq. (18); B=solute permeability of the membrane [m/s; LT-1].
Finally, the total quantity of permeate Qpt and the average concentration of permeate Cpt may be estimated by following equations:
Qpt =∑mi=1 ∑nj=1 (Jv(i,j) S(i,j))*leaves (19)
Cpt =∑mi=1 ∑nj=1 Cp Jv(i,j) S(i,j)*leaves/Qpt (20)
Equation (1) to (18) was solved using C++ program to obtain the values of Qpt and Cpt. The program code is given in the Appendix A. The spiral-wound module Roga 4160-HR was studied. The specifications of this module can be found in Appendix B.