Solute flux
According to the study of Zydney et al. 23, the solute flux (Ns ) across a porous membrane is defined by contributions from both convective and diffusive transport
\(N_{s}=K_{c}JC_{w}-\varepsilon_{m}K_{d}D_{\infty}\frac{dC_{w}}{\text{dz}}\)(23)
where Kc and Kd are convective and diffusive hindrance factors, respectively, Cw is the concentration of solute near the membrane surface, and D is the diffusion coefficient of the solute. The diffusion coefficient of the dextran is calculated using the following correlation 24
\(D_{\infty}=7.667\times 10^{-9}\times M_{w}^{-0.47752}\) (24)
where Mw is the molecular weight in Da.
The observed sieving coefficient of solute (So =Cp /Cf , where Cp and Cf are concentrations in the permeate and feed, respectively) can be calculated by using a stagnant film model with the actual membrane sieving coefficient (Sa =Cp /Cw )25
\(S_{o}=\frac{S_{a}}{\left(1-S_{a}\right)\exp\left(\frac{J}{k_{m}}\right)+S_{a}}\)(25)
where km is bulk mass transfer coefficient, and the value of km in a stirred cell is calculated using the following correlation 26
\(k_{m}=0.23\left(\frac{\omega_{o}R^{2}}{\nu}\right)^{0.567}\left(\frac{\nu}{D_{\infty}}\right)^{0.33}\frac{D_{\infty}}{R}\)(26)
For a membrane with one layer, that is no fouling layer, theSa can be obtained with eq. (27)
\(S_{a}=\frac{S_{\infty}\exp\left(\text{Pe}_{m}\right)}{S_{\infty}-1+\exp\left(\text{Pe}_{m}\right)}\)(27)
Once a fouling layer forms on the membrane surface, that is the membrane has two layers, the expression of Sa is modified as 27
\(S_{a}=\frac{S_{\infty f}\exp\left(\text{Pe}_{\text{mf}}\right)S_{\infty d}\exp\left(\text{Pe}_{\text{md}}\right)}{\left\{S_{\infty f}\left[S_{\infty d}+\exp\left(\text{Pe}_{\text{md}}\right)-1\right]+S_{\infty d}\exp\left(\text{Pe}_{\text{md}}\right)\left[\exp\left(\text{Pe}_{\text{mf}}\right)-1\right]\right\}}\)(28)
where the subscripts ‘f ’ and ‘d ’ denote the fouling and dense layers of the membrane, respectively. Pemand S are the Peclet number and asymptotic sieving coefficient, respectively, and can be calculated as follows
\(\text{Pe}_{m}=\frac{\text{JK}_{c}\tau\delta_{m}}{K_{d}\text{εD}}\)(29)
\(S_{\infty}=\phi K_{c}\) (30)
where τ is tortuosity, δm is the thickness of functional layer. For the fouling layer, the value ofδm is equal to the cake thickness, ϕ = (1–λ )2 is the equilibrium partition coefficient, λ is the ratio of the solute radius (Rs ) to the pore radius (Rp ). For the irregular pore structures, e.g. the cake layer consists of different pore geometries, the λ can be obtained as 28
\(\lambda=R_{s}\left(\frac{{2V}_{p}}{S_{e}}\right)^{-1}\) (31)
where Vp is the pore volume andSe is the pore surface area.
The radius of dextran is evaluated using the following experimental correlation 24
\(R_{S}=3.1\times 10^{-11}\left(M_{w}\right)^{0.47752}\) (32)
Kc and Kd can be calculated using following expressions 28
\(K_{c}=\frac{3-\left(1-\lambda\right)^{2}}{2}\left(1-\frac{\lambda^{2}}{3}\right)\)(33)
\(K_{d}=1-1.004\lambda+0.418\lambda^{3}+0.21\lambda^{4}-0.169\lambda^{6}\)(34)