. 2023 May 29;9(6):e16797. doi: 10.1016/j.heliyon.2023.e16797

Table 2.

- Equations of the DSPM-DE used in this study.

Solute flux in the solution-membrane interface (polarization layer)
$J_{s} = C_{p, s} J_{v} = - (\frac{J_{v} \exp (J_{v} / k_{s})}{\exp (J_{v} / k_{s}) - 1}) (C_{w, s} - C_{f, s}) - \frac{z_{s} C_{w, s} D_{s}}{R_{g} T} F ξ + C_{w, s} J_{v}$	(T1)
The mass-transfer coefficient in the polarization layer [38]
$k_{s} = 91.5 \cdot D_{s}^{0.67} \cdot v^{0.8}$	(T2)
Partitioning equation
$\frac{C_{m, s}}{C_{w, s}} = {(1 - \frac{r_{s}}{r_{p}})}^{2} \exp (- \frac{z_{s} F}{R_{g} T} \cdot {Δ ψ}_{D}) \exp (- \frac{z_{s}^{2} e^{2}}{8 π k_{B} T ε_{o} r_{s}} \cdot (\frac{1}{ε_{p}} - \frac{1}{ε_{b}}))$	(T3)
Extended Nernst-Planck equation for simulating solute transport through the membrane
$J_{s} = C_{p, s} J_{v} = - D_{s, p} \frac{d C_{m, s}}{d x} - \frac{z_{s} C_{m, s} D_{s, p}}{R_{g} T} F \frac{d ψ_{m}}{d x} + K_{s, c} C_{m, s} J_{v}$	(T4)
The potential gradient inside the pores
$\frac{d ψ_{m}}{d x} = \frac{\sum_{s = 1}^{n} \frac{z_{s} J_{v}}{D_{s, p}} (K_{s, c} C_{m, s} - C_{p, s})}{\frac{F}{R_{g} T} \sum_{s = 1}^{n} z_{s}^{2} C_{m, s}}$	(T5)
Electroneutrality in the solution-membrane interface
$\sum_{s = 1}^{n} z_{s} C_{w, s} = 0$	(T6)
Electroneutrality inside the membrane pores
$\sum_{s = 1}^{n} z_{s} C_{m, s} = - X_{d}$	(T7)
Solute hindrance factor for convection
$K_{s, c} = (2 - {(1 - \frac{r_{s}}{r_{p}})}^{2}) (1.0 + 0.054 (\frac{r_{s}}{r_{p}}) - 0.988 {(\frac{r_{s}}{r_{p}})}^{2} + 0.441 {(\frac{r_{s}}{r_{p}})}^{3})$	(T8)
Solute pore diffusion coefficient
$D_{s, p} = D_{s} (1.0 - 2.30 (\frac{r_{s}}{r_{p}}) + 1.154 {(\frac{r_{s}}{r_{p}})}^{2} + 0.224 {(\frac{r_{s}}{r_{p}})}^{3})$	(T9)
Rejection of uncharged solutes
$R_{u s s} = 1 - \frac{K_{s, c} {(1 - \frac{r_{s}}{r_{p}})}^{2}}{1 - (1 - K_{s, c} {(1 - \frac{r_{s}}{r_{p}})}^{2}) \exp (- \frac{K_{s, c}}{D_{s, p}} \frac{Δ x}{A_{k}} J_{v})}$	(T10)