reverse_osmosis_0D.rst

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

.. index::
   pair: watertap.unit_models.reverse_osmosis_0D;reverse_osmosis_0D

.. currentmodule:: watertap.unit_models.reverse_osmosis_0D

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}/\Delta \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	\Delta P	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{\Delta P}{\Delta x})_{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	\Delta P/\Delta x	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	\Delta P = (\frac{\Delta P}{\Delta x})_{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{\Delta P}{\Delta x} = \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

• :mod:`watertap.unit_models.reverse_osmosis_0D`