Nanofiltration- Donnan Steric Pore Model with Dielectric Exclusion (0D)

This unit model implements the Donnan Steric Pore Model with Dielectric Exclusion (DSPM-DE) for nanofiltration.

Note

Documentation for the DSPM-DE model is undergoing refinement.

Model Structure

The model consists of 1 ControlVolume0DBlock for the feed-side of the membrane and includes 11 StateBlocks overall.

  • The feed-side includes 2 StateBlocks (properties_in and properties_out) which are used for mass, energy, and momentum balances, and 2 additional StateBlocks for the conditions at the membrane interface (properties_interface).

  • 2 StateBlocks are attributed to the membrane pore entrance (pore_entrance) at the inlet and outlet of the module.

  • 2 StateBlocks are attributed to the membrane pore exit (pore_exit) at the inlet and outlet of the module.

  • The permeate-side of the membrane includes 3 StateBlocks; 2 of them are attributed to the permeate side at the inlet and outlet (permeate_side), and the 3rd StateBlock is attributed to the final, mixed permeate (mixed_permeate) exiting the module. The inlet and outlet StateBlocks of the permeate are used to only determine the permeate solute concentration for solvent and solute flux at the feed-side inlet and outlet, while the mixed_permeate StateBlock is used for mass balance based on the average flux.

Variables

Description

Symbol

Variable

Index

Units

Water flux, solute flux

\(j_v, j_s,_i\)

flux_mass_phase_comp

[p]

\(\text{kg/m}^2\text{/s}\)

Pore diffusivity of ion

\(D_i,_p\)

diffus_pore_phase_comp

[p]

\(\text{m}^2\text{/s}\)

Convective hindrance factor

\(k_i,_c\)

hindrance_factor_term_comp[[convective, diffusive],c]

None

\(\text{dimensionless}\)

Diffusive hindrance factor

\(k_i,_d\)

hindrance_factor_term_comp[[convective, diffusive],c]

None

\(\text{dimensionless}\)

Pore Diffusivity of ion

\(D_i,_p\)

diffus_pore_phase_comp

[p]

\(\text{m}^2\text{/s}\)

Pore radius

\(r_p\)

radius_pore

[p]

\(\text{m}\)

Stokes radius

\(r_s,_i\)

radius_stokes_comp[c]

[p]

\(\text{m}\)

rs/rp

\(λ_i\)

lambda_comp[c]

[p]

\(\text{dimensionless}\)

Effective membrane thickness

\(Δx_e\)

membrane_thickness_effective

None

\(\text{m}\)

Valency

\(z_i\)

charge_comp[c]

[p]

\(\text{dimensionless}\)

Steric 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

\(dG_{solv}\)

gibbs_solvation_comp

None

\(\text{J}\)

Membrane charge density

\(c_x\)

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

\(J_s,_j = -D_{i,p}\frac{c_{i,j+1}-c_{i,j}}{δx_{j}}-0.5z_{i}(c_{i,j}+c_{i,j+1})D_{i,p}\frac{F}{RT}\frac{ψ_{j+1}-ψ_{j}}{δx_{j}}+0.5K_{i,c}(c_{i,j}+c_{i,j+1})J_{v}\)

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 \((λ_{i} > 0.95)\)

\(K_{i,d} = 0.984(\frac{1-λ_{i}}{λ_{i}})^{(5/2)}\)

Convective 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

\(Φ_b = exp(\frac{-ΔG_{i}}{k_{b}T})\)

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)

\(J_w = ΔP_{net}\frac{r_{pore}^{2}}{8vρ_{w}Δx_e} =((P_{f}-P_{p})-Δπ)\frac{r_{pore}^{2}}{8vρ_{w}Δx_e}\)

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

The DSPM-DE model includes support for scaling, such as providing default or calculating scaling factors for almost all variables.

Class Documentation

References

Geraldes and Alves, 2008 https://doi.org/10.1016/j.memsci.2008.04.054

Roy et al., 2015 http://dx.doi.org/10.1016/j.memsci.2015.06.030

Labban et al., 2017 http://dx.doi.org/10.1016/j.memsci.2016.08.062

Wang and Lin, 2021 https://doi.org/10.1016/j.memsci.2020.118809