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.
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.
| 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 | \lambda _i | lambda_comp[c] | [p] | \text{dimensionless} |
| Effective membrane thickness | \Delta x_e | membrane_thickness_effective | None | \text{m} |
| Valency | z_i | charge_comp[c] | [p] | \text{dimensionless} |
| Steric partitioning factor | \varphi _i | partitioning_factor_steric_comp | None | \text{dimensionless} |
| Born solvation contribution to partitioning | \varphi _{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) | \Sigma _p | dielectric_constant_pore | [p] | \text{dimensionless} |
| Dielectric constant of medium (feed) assumed equal to that of water | \Sigma _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 | \Sigma _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} |
| Description | Equation |
|---|---|
| Solvent flux in active layer (pore) domain | J_s,_j = -D_{i,p}\frac{c_{i,j+1}-c_{i,j}}{\delta x_{j}}-0.5z_{i}(c_{i,j}+c_{i,j+1})D_{i,p}\frac{F}{RT}\frac{\psi _{j+1}-\psi _{j}}{\delta x_{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,\infty }\frac{F}{RT}\xi |
| Solute flux - solvent flux relationship | J_i = J_{v}c_{i,p} |
| Diffusive hindered transport coefficient (\lambda _{i} \leq 0.95) | K_{i,d} = \frac{1+(\frac{9}{8})\lambda _{i}ln(\lambda _{i})-1.56034\lambda _{i}+0.528155\lambda _{i}^{2}+1.91521\lambda _{i}^{3}-2.81903\lambda _{i}^{4}+0.270788\lambda _{i}^{5}-1.10115\lambda _{i}^{6}-0.435933\lambda _{i}^{7}}{(1-\lambda _{i})^{2}} |
| Diffusive hindered transport coefficient (\lambda _{i} > 0.95) | K_{i,d} = 0.984(\frac{1-\lambda _{i}}{\lambda _{i}})^{(5/2)} |
| Convective hindered transport coefficient | K_{i,c} = \frac{1+3.867\lambda _{i}-1.907\lambda _{i}^{2}-0.834\lambda _{i}^{3}}{1+1.867\lambda _{i}-0.741\lambda _{i}^{2}} |
| Stokes pore radius ratio | \lambda _{i} = \frac{r_{i,stokes}}{r_{pore}} |
| Pore diffusion coefficient | D_{i,p} = K_{i,d}D_{i,\infty } |
| Steric partitioning factor | \Phi _i = (1-\lambda _{i})^2 |
| Born solvation partitioning | \Phi _b = exp(\frac{-\Delta G_{i}}{k_{b}T}) |
| Gibbs free energy of solvation | \Delta G = \frac{z_{i}^{2}e_{0}^{2}}{8\pi \varepsilon _{0}r_{i}}(\frac{1}{\varepsilon _{pore}}-\frac{1}{\varepsilon _{f}}) |
| Solvent flux (Hagen-Poiseuille) | J_w = \Delta P_{net}\frac{r_{pore}^{2}}{8v\rho _{w}\Delta x_e} =((P_{f}-P_{p})-\Delta \pi )\frac{r_{pore}^{2}}{8v\rho _{w}\Delta x_e} |
| Membrane-solution interface equilibrium | \gamma _{i,1}c_{i,1} = \gamma _{i,m}c_{i,m}\Phi _{i}\Phi _{b}exp(\frac{-z_{i}F\Delta \psi _{D,m}}{RT}) |
| Membrane-solution interface equilibrium | \gamma _{i,N}c_{i,N} = \gamma _{i,p}c_{i,p}\Phi _{i}\Phi _{b}exp(\frac{-z_{i}F\Delta \psi _{D,p}}{RT}) |
The DSPM-DE model includes support for scaling, such as providing default or calculating scaling factors for almost all variables.
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