membrane_distillation_0D.rst

Membrane Distillation (0D)

This Membrane Distillation (MD) unit model

• is developed for direct contact configuration (other configurations are under development)
• is developed for couterflow mode (parallel flow is under development)
• is 0-dimensional
• supports steady-state only
• Assumes heat loss in equipment is negligible
• Assumes permeate exits the membrane pores with zero salinity
• Assumes no concentration polarization for the cold channel
• Assumes complete vapor condensation on the cold channel

pair: watertap.unit_models.MD;MD

watertap.unit_models.MD

Degrees of Freedom

In addition to the hot channel and cold channel inlet state variables (i.e, temperature, pressure, and component flowrates), the MD 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 for the MD model:

• Membrane permeability coefficient
• Membrane thickness
• Membrane thermal conductivity
• Recovery or membrane area

Configuring the MD unit to calculate temperature polarization, concentration polarization, mass transfer coefficient, and pressure drop would result in five 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:

• Hot channel spacer porosity
• Hot channel height
• Cold channel spacer porosity
• Cold channel height
• Membrane length or membrane width

Model Structure

This MD model consists of a separate MDchannel0Dblock for the hot channel and the cold channel of the module.

• Each MDchannel0Dblock includes 6 stateblocks: 2 stateBlocks for the bulk properites at the inlet and outlet (properties_in and properties_out), which are used for mass, energy, and momentum balances; 2 StateBlocks for the conditions at the membrane interface, and 2 stateblocks for the vapor phase at the membrane interface. Property packages must be declared for each MD channel block for the liquid (bulk and interface) and vapor Phases.

Sets

Description	Symbol	Indices
Time	t	[0]
Inlet/outlet	x	['in', 'out']
Phases	p	['Liq', 'Vap']
Components	j	['H2O', solute]*

*Solute depends on the imported property model.

Variables

Description	Symbol	Variable Name	Index	Units
Membrane permeability coefficient	B0	permeability_coef	[t]	kg/m/Pa/s
Membrane thickness	σ	membrane_thickness	None	m
Membrane thermal conductivity	km	membrane_tc	None	W/K/m
Mass density of solvent	ρsolvent	dens_solvent	[p]	kg/m3
Mass flux across membrane	J	flux_mass	[t, x]	kg/s/m2
Conduction heat flux across membrane	qcond	flux_conduction_heat	[t, x]	W/m2
Evaporation heat flux from hot channel	qevap	flux_enth_hot	[t, x]	W/m2
Condensation heat flux to cold channel	qconden	flux_enth_cold	[t, x]	W/m2
Membrane area	Am	area	None	m2
Recovery rate	R	recovery_mass	[t]	dimensionless

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 temperature_polarization_type is set to TemperaturePolarizationType.fixed:

Description	Symbol	Variable Name	Index	Units
Hot channel Convective heat transfer coefficient	hconv, h	hot_ch.h_conv	[t]	W/K/m2
Cold channel Convective heat transfer coefficient	hconv, c	hot_ch.h_conv	[t]	W/K/m2

if temperature_polarization_type is set to TemperaturePolarizationType.calculated:

Description	Symbol	Variable Name	Index	Units
Hot channel Prandtl number	Prh	hot_ch.N_Pr	[t]	dimensionless
Cold channel Prandtl number	Prc	cold_ch.N_Pr	[t]	dimensionless
Hot channel Nusselt number	Nuh	hot_ch.N_Nu	[t]	dimensionless
Cold channel Nusselt number	Nuc	cold_ch.N_Nu	[t]	dimensionless

if concentration_polarization_type is set to ConcentrationPolarizationType.fixed:

Description	Symbol	Variable Name	Index	Units
Concentration polarization modulus in hot channel	CPmod, h	hot_ch.cp_modulus	[t, j]	dimensionless

if concentration_polarization_type is set to ConcentrationPolarizationType.calculated:

Description	Symbol	Variable Name	Index	Units
Mass transfer coefficient in hot channel	kh	hot_ch.K	[t, x, j]	m/s

if temperature_polarization_type is set to TemperaturePolarizationType.calculated: or mass_transfer_coefficient is set to MassTransferCoefficient.calculated or pressure_change_type is set to PressureChangeType.calculated:

Description	Symbol	Variable Name	Index	Units
Hot channel height	hch, h	hot_ch.channel_height	None	m
Hot channel Hydraulic diameter	dh, h	cold_ch.dh	None	m
Hot channel Spacer porosity	ϵsp, h	hot_ch.spacer_porosity	None	dimensionless
Hot channel Reynolds number	Reh	hot_ch.N_Re	[t, x]	dimensionless
Cold channel height	hch, c	cold_ch.channel_height	None	m
Cold channel Hydraulic diameter	dh, c	cold_ch.dh	None	m
Cold channel Spacer porosity	ϵsp, c	cold_ch.spacer_porosity	None	dimensionless
Cold channel Reynolds number	Rec	cold_ch.N_Re	[t, x]	dimensionless

if mass_transfer_coefficient is set to MassTransferCoefficient.calculated:

Description	Symbol	Variable Name	Index	Units
Schmidt number	Sch	hot_ch.N_Sc	[t, x]	dimensionless
Sherwood number	Shh	hot_ch.N_Sh	[t, x]	dimensionless
Schmidt number	Scc	cold_ch.N_Sc	[t, x]	dimensionless
Sherwood number	Shc	cold_ch.N_Sh	[t, x]	dimensionless

if temperature_polarization_type is set to TemperaturePolarizationType.calculated: or 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 hot channel	$(\frac{ΔP}{Δx})_{avg,h}$	hot_ch.dP_dx	[t]	Pa/m
Average pressure drop per unit length of cold channel	$(\frac{ΔP}{Δx})_{avg,c}$	cold_ch.dP_dx	[t]	Pa/m

if pressure_change_type is set to PressureChangeType.calculated:

Description	Symbol	Variable Name	Index	Units
Hot channel velocity	vh	hot_ch.velocity	[t, x]	m/s
Hot channel Friction factor	fh	hot_ch.friction_factor_darcy	[t, x]	dimensionless
Pressure drop per unit length of hot channel at inlet/outlet	(ΔP/Δx)h	hot_ch.dP_dx	[t, x]	Pa/m
Cold channel velocity	vc	cold_ch.velocity	[t, x]	m/s
Pressure drop per unit length of cold channel at inlet/outlet	(ΔP/Δx)c	cold_ch.dP_dx	[t, x]	Pa/m

Equations

Description	Equation
Vapor flux across membrane	$J(t, x) = \frac{B_0(t)}{\sigma} \times \left( P_{\text{sat, hot}}(t, x) - P_{\text{sat, cold}}(t, x) \right)$
Average flux across membrane	$J_{avg, j} = \frac{1}{2}\sum_{x} J_{x, j}$
hot channel membrane-interface solute concentration	$C_{\text{interface, j, h}}(t, x) = C_{\text{bulk, j, h}}(t, x) \times \exp\left( \frac{J(t, x)}{\rho_{\text{solvent}} \times k_h(t, x, j)} \right)$
Evaporation heat flux from hot channel	qevap(t, x) = J(t, x) × Ĥh(t, x, Vap)
Condensation heat flux to cold channel	qconden(t, x) = J(t, x) × Ĥc(t, x, Vap)
Average evaporation flux from hot channel	$\overline{q}_{\text{evap}}(t) = \frac{1}{2} \sum_{x} q_{\text{evap}}(t, x)$
Average condensation flux to cold channel	$\overline{q}_{\text{conden}}(t) = \frac{1}{2} \sum_{x} q_{\text{conden}}(t, x)$
Hot channel convective heat transfer	hconv, h(t, x)(Tbulk, h(t,x)−Tinterface, h(t,x)) = qcond(t, x) + qevap(t, x) − J(t, x) ⋅ Ĥbulk, h(t, x, Liq)
Cold channel convective heat transfer	hconv, c(t, x)(Tinterface, c(t,x)−Tbulk, c(t,x)) = qcond(t, x) + qconden(t, x) − J(t, x) ⋅ Ĥbulk, c(t, x, Liq)
Conduction heat flux across membrane	$q_{\text{cond}}(t, x) = \frac{k_{\text{m}}}{\sigma} \left( T_{\text{interface}, h}(t, x) - T_{\text{interface}, c}(t, x) \right)$
Average conduction heat across membrane	$q_{\text{cond, avg}}(t) = \frac{1}{N} \sum_{x} q_{\text{cond}}(t, x)$
Total permeate production	Mp = A ⋅ Javg
Total conduction heat transfer	qcond,total =  − A ⋅ qcond,avg
Hot channel total evapration heat	$q_{\text{evap,total}} = - A \cdot \overline{\widehat{H}_h}$
Cold channel total condensation heat	$q_{\text{conden,total}} = A \cdot \overline{\widehat{H}_c}$
Convective heat transfer coefficient	$h_{\text{conv},(t, x)} = \frac{\kappa_{(t, x)} \cdot \text{Nu}_{(t, x)}}{d_h}$
Nusselt number	Nu[t, x] =  = 0.162 * (Re[t, x] *  * 0.656) * (Pr[t, x] *  * 0.333)
Prandtl number	$Pr(t, x) = \frac{\mu(t, x) \cdot C_p(t, x)}{\kappa}$
Effectiveness	$\epsilon(t) = \frac{T_{\text{cold, first}}(t) - T_{\text{c, last}}(t)}{T_{\text{h, first}}(t) - T_{\text{c, last}}(t)}$
Thermal efficiency	$\eta(t) = \frac{q_{\text{evap,total}}(t)}{q_{\text{evap,total}}(t) + q_{\text{cond,total}}(t)}$
Concentration polarization modulus	CPmod = Cinterface/Cbulk
Mass transfer coefficient	$k_h = \frac{D Sh}{d_h}$
Sherwood number	Sh[t, x] =  = 0.2 * (Re[t, x] *  * 0.57) * (Pr[t, x] *  * 0.4)
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	$ΔP = (\frac{ΔP}{Δx})_{avg}L$
Hot channel velocity	vh = Qh/Ac
Friction factor	$f = 0.42+\frac{189.3}{Re}$
Pressure drop per unit length	$\frac{ΔP}{Δx} = \frac{1}{2d_h}f\rho v_h^{2}$
Recovery rate	$R = \frac{M_{p}}{M_{h,in}}$

Class Documentation

• watertap.unit_models.MD.membrane_distillation_0D
• watertap.unit_models.MD.membrane_distillation_base
• watertap.unit_models.MD.MD_channel_0D
• watertap.unit_models.MD.MD_channel_base