Nanofiltration- Donnan Steric Pore Model with Dielectric Exclusion (0D)
This unit model implement the Donnan Steric Pore Model with Dielectric Exclusion (DSPM-DE).
Note
The DSPM-DE model is still undergoing validation and refinement, as is this documentation.
Variables
Description |
Symbol |
Variable |
Index |
Units |
|---|---|---|---|---|
Water flux, solute flux |
\(jv, js_i\) |
flux_mass_phase_comp |
[p] |
\(\text{kg/m^{2}/s}\) |
Pore diffusivity of ion |
\(Di, p\) |
diffus_pore_phase_comp |
[p] |
\(\text{m^{2}/s}\) |
Convective hindrance factor |
\(ki, c\) |
hindrance_factor_term_comp[[convective, diffusive],c] |
None |
\(\text{dimensionless}\) |
Diffusive hindrance factor |
\(ki, d\) |
hindrance_factor_term_comp[[convective, diffusive],c] |
None |
\(\text{dimensionless}\) |
Pore Diffusivity of ion |
\(Di, p\) |
diffus_pore_phase_comp |
[p] |
\(\text{m^{2}/s}\) |
Pore radius |
\(rp\) |
radius_pore |
[p] |
\(\text{m}\) |
Stokes radius |
\(rs_i\) |
radius_stokes_comp[c] |
[p] |
\(\text{m}\) |
rs/rp |
\(λ_i\) |
lambda_comp[c] |
[p] |
\(\text{dimensionless}\) |
Effective membrane thickness |
\(dx_e\) |
membrane_thickness_effective |
None |
\(\text{m}\) |
Membrane porosity |
\(Ak\) |
membrane_porosity |
None |
\(\text{dimensionless}\) |
Active layer thickness |
\(dx\) |
membrane_thickness_active_layer |
None |
\(\text{m}\) |
Valency |
\(z_i\) |
charge_comp[c] |
[p] |
\(\text{dimensionless}\) |
Streic partitioning factor |
\(φ_i\) |
partitioning_factor_steric_comp |
None |
\(\text{dimensionless}\) |
Born solvation contribution to partitioning |
\(φ_{b_i}\) |
partitioning_factor_born_solvation_comp |
[p] |
\(\text{dimensionless}\) |
Gibbs free energy of solvation |
\(dGsolv\) |
gibbs_solvation_comp |
None |
\(\text{J}\) |
Membrane charge density |
\(cx\) |
membrane_charge_density |
None |
\(\text{mol/m^3}\) |
Dielectric constant of medium (pore) |
\(Σ_p\) |
dielectric_constant_pore |
[p] |
\(\text{dimensionless}\) |
Dielectric constant of medium (feed) assumed equal to that of water |
\(Σ_f\) |
dielectric_constant_feed |
[p] |
\(\text{dimensionless}\) |
Concentration |
\(C_{i,j}\) |
[feed,interface,pore_entrance,pore_exit,permeate].conc_mol)phase_comp |
[p,j] |
\(\text{kg/m^{3}\) |
Electric potential gradient between feed/interface |
\(xi\) |
electric_potential_grad_feed_interface |
None |
\(\text{dimensionless}\) |
Electronic charge |
\(e_o\) |
electronic_charge |
[p] |
\(\text{C}\) |
Absolute permittivity of vacuum |
\(Σ_o\) |
vacuum_electric_permittivity |
None |
\(\text{F/m}\) |
Boltzmann constant |
\(k_b\) |
boltzmann_constant |
None |
\(\text{J/K}\) |
Faraday’s constant |
\(F\) |
faraday_constant |
None |
\(\text{dimensionless}\) |
Ideal gas constant |
\(R\) |
gas_constant |
None |
\(\text{Check}\) |
Relationships
Description |
Equation |
|---|---|
Solvent flux in active layer (pore) domain |
\(X_j = -D_{i,p}\frac{c_{i,j+1}-c_{i,j}}{δx_{j}}-0.5z_{i}(c_{i,j}+c_{i,j+1})D_{i,p}\frac{F}{RT}\frac{ψ_{j+1}-ψ_{j}}{δx_{j}}+0.5K_{i,c}(c_{i,j}+c_{i,j+1})J_{v}\) |
Solute flux at feed/interface domain |
\(J_i = -k_{i}(C_{i,m}-C_{i,f})+J_{w}C_{i,m}-z_{i}C_{i,m}D_{i,∞}\frac{F}{RT}ξ\) |
Solute flux - solvent flux relationship |
\(J_i = J_{v}c_{i,p}\) |
Diffusive hindered transport coefficient \((λ_{i} ≤ 0.95)\) |
\(K_{i,d} = \frac{1+(\frac{9}{8})λ_{i}ln(λ_{i})-1.56034λ_{i}+0.528155λ_{i}^{2}+1.91521λ_{i}^{3}-2.81903λ_{i}^{4}+0.270788λ_{i}^{5}-1.10115λ_{i}^{6}-0.435933λ_{i}^{7}}{(1-λ_{i})^{2}}\) |
Diffusive hindered transport coefficient \((λ_{i} > 0.95)\) |
\(K_{i,d} = 0.984(\frac{1-λ_{i}}{λ_{i}})^{(5/2)}\) |
Convectve hindered transport coefficient |
\(K_{i,c} = \frac{1+3.867λ_{i}-1.907λ_{i}^{2}-0.834λ_{i}^{3}}{1+1.867λ_{i}-0.741λ_{i}^{2}}\) |
Stokes pore radius ratio |
\(λ_{i} = \frac{r_{i,stokes}}{r_{pore}}\) |
Pore diffusion coefficient |
\(D_{i,p} = K_{i,d}D_{i,∞}\) |
Steric partitioning factor |
\(Φ_i = (1-λ_{i})^2\) |
Born solvation partitioning |
\(Φ_b = \frac{-ΔG_{i}}{k_{b}T}\) |
Gibbs free energy of solvation |
\(ΔG = \frac{z_{i}^{2}e_{0}^{2}}{8πε_{0}r_{i}}(\frac{1}{ε_{pore}}-\frac{1}{ε_{f}})\) |
Solvent flux (Hagen-Poiseuille) |
\(J_w = ΔP_{net}\frac{r_{pore}^{2}}{8vρ_{w}\frac{Δx}{A_{k}}} =((P_{f}-P_{p})-Δπ)\frac{r_{pore}^{2}}{8vρ_{w}\frac{Δx}{A_{k}}}\) |
Membrane-solution interface equilibrium |
\(γ_{i,1}c_{i,1} = γ_{i,m}c_{i,m}Φ_{i}Φ_{b}exp(\frac{-z_{i}FΔψ_{D,m}}{RT})\) |
Membrane-solution interface equilibrium |
\(γ_{i,N}c_{i,N} = γ_{i,p}c_{i,p}Φ_{i}Φ_{b}exp(\frac{-z_{i}FΔψ_{D,p}}{RT})\) |
Scaling
This DSPM-DE model includes support for scaling, such as providing default or calculating scaling factors for almost all variables.