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

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 pure water	\(\rho_w\)	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 membrane-interface 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 proteuslib.unit_models.reverse_osmosis_0D.ConcentrationPolarizationType(value)

An enumeration.

class proteuslib.unit_models.reverse_osmosis_0D.MassTransferCoefficient(value)

An enumeration.

class proteuslib.unit_models.reverse_osmosis_0D.PressureChangeType(value)

An enumeration.

class proteuslib.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. RO units do not support dynamic behavior.

  has_holdup

  Indicates whether holdup terms should be constructed or not. default - False. RO units do not have defined volume, thus this must be False.

  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.}

  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 keyword has_pressure_change must also be set to True. 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

  concentration_polarization_type

  Options to account for concentration polarization.

  default - ConcentrationPolarizationType.none

  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.none

  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

  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.}

• 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 proteuslib.unit_models.reverse_osmosis_0D.ReverseOsmosisData(component)

Standard RO Unit Model Class: - zero dimensional model - steady state only - single liquid phase only

build()

Build the RO model.

initialize(initialize_guess=None, state_args=None, outlvl=0, solver=None, optarg=None, fail_on_warning=False, ignore_dof=False)

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