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 2 ControlVolume0DBlocks: one for the feed-side and one for the permeate-side.
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 properties_mixed). 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_io_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 |
\(CP_{mod}\) |
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 |
\(k_f\) |
Kf_io |
[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 |
\(h_{ch}\) |
channel_height |
None |
\(\text{m}\) |
Hydraulic diameter |
\(d_h\) |
dh |
None |
\(\text{m}\) |
Spacer porosity |
\(\epsilon_{sp}\) |
spacer_porosity |
None |
\(\text{dimensionless}\) |
Reynolds number |
\(Re\) |
N_Re_io |
[t, x] |
\(\text{dimensionless}\) |
if mass_transfer_coefficient
is set to MassTransferCoefficient.calculated
:
Description |
Symbol |
Variable Name |
Index |
Units |
---|---|---|---|---|
Schmidt number |
\(Sc\) |
N_Sc_io |
[t, x] |
\(\text{dimensionless}\) |
Sherwood number |
\(Sh\) |
N_Sh_io |
[t, x] |
\(\text{dimensionless}\) |
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 |
\((\frac{ΔP}{Δx})_{avg}\) |
dP_dx |
[t] |
\(\text{Pa/m}\) |
if pressure_change_type
is set to PressureChangeType.calculated
:
Description |
Symbol |
Variable Name |
Index |
Units |
---|---|---|---|---|
Feed-channel velocity |
\(v_f\) |
velocity_io |
[t, x] |
\(\text{m/s}\) |
Friction factor |
\(f\) |
friction_factor_darcy_io |
[t, x] |
\(\text{dimensionless}\) |
Pressure drop per unit length of feed channel at inlet/outlet |
\(ΔP/Δx\) |
dP_dx_io |
[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 |
\(r_j = 1 - \frac{C_{p,mix}}{C_{f,in}}\) |
Class Documentation
- class watertap.unit_models.reverse_osmosis_0D.ReverseOsmosis0D(*args, **kwds)
- Parameters
rule (function) – A rule function or None. Default rule calls build().
concrete (bool) – If True, make this a toplevel model. Default - False.
ctype (class) – Pyomo ctype of the block. Default - pyomo.environ.Block
default (dict) –
Default ProcessBlockData config
- Keys
- dynamic
Indicates whether this model will be dynamic or not. default = False. Membrane units do not yet support dynamic behavior.
- has_holdup
Indicates whether holdup terms should be constructed or not. default - False. Membrane units do not have defined volume, thus this must be False.
- property_package
Property parameter object used to define property calculations, default - useDefault. Valid values: { useDefault - use default package from parent model or flowsheet, PhysicalParameterObject - a PhysicalParameterBlock object.}
- property_package_args
A ConfigBlock with arguments to be passed to a property block(s) and used when constructing these. default - None. Valid values: { see property package for documentation.}
- material_balance_type
Indicates what type of mass balance should be constructed, default - MaterialBalanceType.useDefault. Valid values: { MaterialBalanceType.useDefault - refer to property package for default balance type **MaterialBalanceType.none - exclude material balances, MaterialBalanceType.componentPhase - use phase component balances, MaterialBalanceType.componentTotal - use total component balances, MaterialBalanceType.elementTotal - use total element balances, MaterialBalanceType.total - use total material balance.}
- energy_balance_type
Indicates what type of energy balance should be constructed. default - EnergyBalanceType.useDefault. Valid values: { EnergyBalanceType.useDefault - refer to property package for default balance type **EnergyBalanceType.none - exclude energy balances, EnergyBalanceType.enthalpyTotal - single enthalpy balance for material, EnergyBalanceType.enthalpyPhase - enthalpy balances for each phase, EnergyBalanceType.energyTotal - single energy balance for material, EnergyBalanceType.energyPhase - energy balances for each phase.}
- momentum_balance_type
Indicates what type of momentum balance should be constructed, default - MomentumBalanceType.pressureTotal. Valid values: { MomentumBalanceType.none - exclude momentum balances, MomentumBalanceType.pressureTotal - single pressure balance for material, MomentumBalanceType.pressurePhase - pressure balances for each phase, MomentumBalanceType.momentumTotal - single momentum balance for material, MomentumBalanceType.momentumPhase - momentum balances for each phase.}
- concentration_polarization_type
Options to account for concentration polarization.
default -
ConcentrationPolarizationType.calculated
Configuration Options
Description
ConcentrationPolarizationType.none
Simplifying assumption to ignore concentration polarization
ConcentrationPolarizationType.fixed
Specify an estimated value for the concentration polarization modulus
ConcentrationPolarizationType.calculated
Allow model to perform calculation of membrane-interface concentration
- mass_transfer_coefficient
Options to account for mass transfer coefficient.
default -
MassTransferCoefficient.calculated
Configuration Options
Description
MassTransferCoefficient.none
Mass transfer coefficient not used in calculations
MassTransferCoefficient.fixed
Specify an estimated value for the mass transfer coefficient in the feed channel
MassTransferCoefficient.calculated
Allow model to perform calculation of mass transfer coefficient
- has_pressure_change
Indicates whether terms for pressure change should be constructed, default - False. Valid values: { True - include pressure change terms, False - exclude pressure change terms.}
- pressure_change_type
Indicates what type of pressure change calculation will be made. To use any of the
pressure_change_type
options to account for pressure drop, the configuration keywordhas_pressure_change
must also be set toTrue
. Also, if a value is specified for pressure change, it should be negative to represent pressure drop.default -
PressureChangeType.fixed_per_stage
Configuration Options
Description
PressureChangeType.fixed_per_stage
Specify an estimated value for pressure drop across the membrane feed channel
PressureChangeType.fixed_per_unit_length
Specify an estimated value for pressure drop per unit length across the membrane feed channel
PressureChangeType.calculated
Allow model to perform calculation of pressure drop across the membrane feed channel
initialize (dict) – ProcessBlockData config for individual elements. Keys are BlockData indexes and values are dictionaries described under the “default” argument above.
idx_map (function) – Function to take the index of a BlockData element and return the index in the initialize dict from which to read arguments. This can be provided to overide the default behavior of matching the BlockData index exactly to the index in initialize.
- Returns
(ReverseOsmosis0D) New instance
- class watertap.unit_models.reverse_osmosis_0D.ReverseOsmosisData(component)[source]
Standard RO Unit Model Class: - zero dimensional model - steady state only - single liquid phase only
- build()[source]
Build the RO model.
- initialize(initialize_guess=None, state_args=None, outlvl=0, solver=None, optarg=None, fail_on_warning=False, ignore_dof=False)[source]
General wrapper for RO initialization routines
- Keyword Arguments
initialize_guess – a dict of guesses for solvent_recovery, solute_recovery, and cp_modulus. These guesses offset the initial values for the retentate, permeate, and membrane interface state blocks from the inlet feed (default = {‘deltaP’: -1e4, ‘solvent_recovery’: 0.5, ‘solute_recovery’: 0.01, ‘cp_modulus’: 1.1})
state_args – a dict of arguments to be passed to the property package(s) to provide an initial state for the inlet feed side state block (see documentation of the specific property package) (default = None).
outlvl – sets output level of initialization routine
optarg – solver options dictionary object (default=None)
solver – solver object or string indicating which solver to use during initialization, if None provided the default solver will be used (default = None)
fail_on_warning – boolean argument to fail or only produce warning upon unsuccessful solve (default=False)
ignore_dof – boolean argument to ignore when DOF != 0 (default=False)
- Returns
None