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

pair: watertap.unit_models.reverse_osmosis_0D;reverse_osmosis_0D

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 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]	m/Pa/s
Solute permeability coefficient	B	B_comp	[t, j]	m/s
Mass density of solvent	ρsolvent	dens_solvent	[p]	kg/m3
Mass flux across membrane	J	flux_mass_phase_comp	[t, x, p, j]	kg/s/m2
Membrane area	Am	area	None	m2
Component recovery rate	Rj	recovery_mass_phase_comp	[t, p, j]	dimensionless
Volumetric recovery rate	Rvol	recovery_vol_phase	[t, p]	dimensionless
Observed solute rejection	rj	rejection_phase_comp	[t, p, j]	dimensionless
Over-pressure ratio	Pf, out/Δπout	over_pressure_ratio	[t]	dimensionless
Mass transfer to permeate	Mp	mass_transfer_phase_comp	[t, p, j]	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]	Pa

if concentration_polarization_type is set to ConcentrationPolarizationType.fixed:

Description	Symbol	Variable Name	Index	Units
Concentration polarization modulus	CPmod	cp_modulus	[t, j]	dimensionless

if concentration_polarization_type is set to ConcentrationPolarizationType.calculated:

Description	Symbol	Variable Name	Index	Units
Mass transfer coefficient in feed channel	kf	Kf	[t, x, j]	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	hch	channel_height	None	m
Hydraulic diameter	dh	dh	None	m
Spacer porosity	ϵsp	spacer_porosity	None	dimensionless
Reynolds number	Re	N_Re	[t, x]	dimensionless

if mass_transfer_coefficient is set to MassTransferCoefficient.calculated:

Description	Symbol	Variable Name	Index	Units
Schmidt number	Sc	N_Sc	[t, x]	dimensionless
Sherwood number	Sh	N_Sh	[t, x]	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	m
Membrane width	W	width	None	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]	Pa/m

if pressure_change_type is set to PressureChangeType.calculated:

Description	Symbol	Variable Name	Index	Units
Feed-channel velocity	vf	velocity	[t, x]	m/s
Friction factor	f	friction_factor_darcy	[t, x]	dimensionless
Pressure drop per unit length of feed channel at inlet/outlet	ΔP/Δx	dP_dx	[t, x]	Pa/m

Equations

Description	Equation
Solvent flux across membrane	Jsolvent = ρsolventA(Pf − Pp − (πf − πp))
Solute flux across membrane	Jsolute = B(Cf − Cp)
Average flux across membrane	$J_{avg, j} = \frac{1}{2}\sum_{x} J_{x, j}$
Permeate mass flow by component j	Mp, j = AmJavg, 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	CPmod = Cinterface/Cbulk
Mass transfer coefficient	$k_f = \frac{D Sh}{d_h}$
Sherwood number	Sh = 0.46(ReSc)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	Ac = hchWϵsp
Membrane area	Am = LW
Pressure drop	$ΔP = (\frac{ΔP}{Δx})_{avg}L$
Feed-channel velocity	vf = Qf/Ac
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

watertap.unit_models.reverse_osmosis_0D