Reverse Osmosis (0D)
- This reverse osmosis (RO) unit model
is 0-dimensional
supports a single liquid phase only
supports steady-state only
is based on the solution-diffusion model and film theory
assumes isothermal conditions
Degrees of Freedom
Aside from the inlet feed state variables (i.e. temperature, pressure, component flowrates), the RO model has at least 4 degrees of freedom that should be fixed for the unit to be fully specified.
- Typically, the following variables are fixed, in addition to state variables at the inlet:
membrane water permeability, A
membrane salt permeability, B
permeate pressure
membrane area
On the other hand, configuring the RO unit to calculate concentration polarization effects, mass transfer coefficient, and pressure drop would result in 3 additional degrees of freedom. In this case, in addition to the previously fixed variables, we typically fix the following variables to fully specify the unit:
feed-spacer porosity
feed-channel height
membrane length or membrane width or inlet Reynolds number
Model Structure
This RO model consists of 1 MembraneChannel0DBlock for the feed-side, a StateBlock indexed by time and space for the permeate-side (permeate_side[t, x]), and a StateBlock for the final permeate at the outlet (mixed_permeate).
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_in and properties_interface_out).
The permeate-side includes 3 StateBlocks (properties_in, properties_out, and mixed_permeate). The inlet and outlet StateBlocks are used to only determine the permeate solute concentration for solvent and solute flux at the feed-side inlet and outlet, while the mixed StateBlock is used for mass balance based on the average flux.
Sets
Description |
Symbol |
Indices |
|---|---|---|
Time |
\(t\) |
[0] |
Inlet/outlet |
\(x\) |
[‘in’, ‘out’] |
Phases |
\(p\) |
[‘Liq’] |
Components |
\(j\) |
[‘H2O’, ‘NaCl’]* |
*Solute depends on the imported property model; example shown here is for the NaCl property model.
Variables
Description |
Symbol |
Variable Name |
Index |
Units |
|---|---|---|---|---|
Solvent permeability coefficient |
\(A\) |
A_comp |
[t, j] |
\(\text{m/Pa/s}\) |
Solute permeability coefficient |
\(B\) |
B_comp |
[t, j] |
\(\text{m/s}\) |
Mass density of solvent |
\(\rho_{solvent}\) |
dens_solvent |
[p] |
\(\text{kg/}\text{m}^3\) |
Mass flux across membrane |
\(J\) |
flux_mass_phase_comp |
[t, x, p, j] |
\(\text{kg/s}\text{/m}^2\) |
Membrane area |
\(A_m\) |
area |
None |
\(\text{m}^2\) |
Component recovery rate |
\(R_j\) |
recovery_mass_phase_comp |
[t, p, j] |
\(\text{dimensionless}\) |
Volumetric recovery rate |
\(R_{vol}\) |
recovery_vol_phase |
[t, p] |
\(\text{dimensionless}\) |
Observed solute rejection |
\(r_j\) |
rejection_phase_comp |
[t, p, j] |
\(\text{dimensionless}\) |
Over-pressure ratio |
\(P_{f,out}/Δ\pi_{out}\) |
over_pressure_ratio |
[t] |
\(\text{dimensionless}\) |
Mass transfer to permeate |
\(M_p\) |
mass_transfer_phase_comp |
[t, p, j] |
\(\text{kg/s}\) |
The following variables are only built when specific configuration key-value pairs are selected.
if has_pressure_change is set to True:
Description |
Symbol |
Variable Name |
Index |
Units |
|---|---|---|---|---|
Pressure drop |
\(ΔP\) |
deltaP |
[t] |
\(\text{Pa}\) |
if concentration_polarization_type is set to ConcentrationPolarizationType.fixed:
Description |
Symbol |
Variable Name |
Index |
Units |
|---|---|---|---|---|
Concentration polarization modulus |
\(CP_{mod}\) |
feed_side.cp_modulus |
[t, j] |
\(\text{dimensionless}\) |
if concentration_polarization_type is set to ConcentrationPolarizationType.calculated:
Description |
Symbol |
Variable Name |
Index |
Units |
|---|---|---|---|---|
Mass transfer coefficient in feed channel |
\(k_f\) |
feed_side.K |
[t, x, j] |
\(\text{m/s}\) |
if mass_transfer_coefficient is set to MassTransferCoefficient.calculated
or pressure_change_type is set to PressureChangeType.calculated:
Description |
Symbol |
Variable Name |
Index |
Units |
|---|---|---|---|---|
Feed-channel height |
\(h_{ch}\) |
feed_side.channel_height |
None |
\(\text{m}\) |
Hydraulic diameter |
\(d_h\) |
feed_side.dh |
None |
\(\text{m}\) |
Spacer porosity |
\(\epsilon_{sp}\) |
feed_side.spacer_porosity |
None |
\(\text{dimensionless}\) |
Reynolds number |
\(Re\) |
feed_side.N_Re |
[t, x] |
\(\text{dimensionless}\) |
if mass_transfer_coefficient is set to MassTransferCoefficient.calculated:
Description |
Symbol |
Variable Name |
Index |
Units |
|---|---|---|---|---|
Schmidt number |
\(Sc\) |
feed_side.N_Sc_comp |
[t, x, j] |
\(\text{dimensionless}\) |
Sherwood number |
\(Sh\) |
feed_side.N_Sh_comp |
[t, x, j] |
\(\text{dimensionless}\) |
if mass_transfer_coefficient is set to MassTransferCoefficient.calculated
or pressure_change_type is NOT set to PressureChangeType.fixed_per_stage:
Description |
Symbol |
Variable Name |
Index |
Units |
|---|---|---|---|---|
Membrane length |
\(L\) |
length |
None |
\(\text{m}\) |
Membrane width |
\(W\) |
width |
None |
\(\text{m}\) |
if pressure_change_type is set to PressureChangeType.fixed_per_unit_length:
Description |
Symbol |
Variable Name |
Index |
Units |
|---|---|---|---|---|
Average pressure drop per unit length of feed channel |
\((\frac{ΔP}{Δx})_{avg}\) |
feed_side.dP_dx |
[t] |
\(\text{Pa/m}\) |
if pressure_change_type is set to PressureChangeType.calculated:
Description |
Symbol |
Variable Name |
Index |
Units |
|---|---|---|---|---|
Feed-channel velocity |
\(v_f\) |
feed_side.velocity |
[t, x] |
\(\text{m/s}\) |
Friction factor |
\(f\) |
feed_side.friction_factor_darcy |
[t, x] |
\(\text{dimensionless}\) |
Pressure drop per unit length of feed channel at inlet/outlet |
\(ΔP/Δx\) |
feed_side.dP_dx |
[t, x] |
\(\text{Pa/m}\) |
Equations
Description |
Equation |
|---|---|
Solvent flux across membrane |
\(J_{solvent} = \rho_{solvent} A(P_{f} - P_p - (\pi_{f}-\pi_{p}))\) |
Solute flux across membrane |
\(J_{solute} = B(C_{f} - C_{p})\) |
Average flux across membrane |
\(J_{avg, j} = \frac{1}{2}\sum_{x} J_{x, j}\) |
Permeate mass flow by component j |
\(M_{p, j} = A_m J_{avg,j}\) |
Permeate-side solute mass fraction |
\(X_{x, j} = \frac{J_{x, j}}{\sum_{x} J_{x, j}}\) |
Feed-side membrane-interface solute concentration |
\(C_{interface} = CP_{mod}C_{bulk}=C_{bulk}\exp(\frac{J_{solvent}}{k_f})-\frac{J_{solute}}{J_{solvent}}(\exp(\frac{J_{solvent}}{k_f})-1)\) |
Concentration polarization modulus |
\(CP_{mod} = C_{interface}/C_{bulk}\) |
Mass transfer coefficient |
\(k_f = \frac{D Sh}{d_h}\) |
Sherwood number |
\(Sh = 0.46 (Re Sc)^{0.36}\) |
Schmidt number |
\(Sc = \frac{\mu}{\rho D}\) |
Reynolds number |
\(Re = \frac{\rho v_f d_h}{\mu}\) |
Hydraulic diameter |
\(d_h = \frac{4\epsilon_{sp}}{2/h_{ch} + (1-\epsilon_{sp})8/h_{ch}}\) |
Cross-sectional area |
\(A_c = h_{ch}W\epsilon_{sp}\) |
Membrane area |
\(A_m = LW\) |
Pressure drop |
\(ΔP = (\frac{ΔP}{Δx})_{avg}L\) |
Feed-channel velocity |
\(v_f = Q_f/A_c\) |
Friction factor |
\(f = 0.42+\frac{189.3}{Re}\) |
Pressure drop per unit length |
\(\frac{ΔP}{Δx} = \frac{1}{2d_h}f\rho v_f^{2}\) |
Component recovery rate |
\(R_j = \frac{M_{p,j}}{M_{f,in,j}}\) |
Volumetric recovery rate |
\(R_{vol} = \frac{Q_{p}}{Q_{f,in}}\) |
Observed solute rejection |
\(r_j = 1 - \frac{C_{p,mix}}{C_{f,in}}\) |