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

.. index::
   pair: watertap.unit_models.MD;MD

.. currentmodule:: 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	B_0	permeability_coef	[t]	\text{kg/m/Pa/s}
Membrane thickness	\sigma	membrane_thickness	None	\text{m}
Membrane thermal conductivity	k_m	membrane_tc	None	\text{W/K/m}
Mass density of solvent	\rho_{solvent}	dens_solvent	[p]	\text{kg/}\text{m}^3
Mass flux across membrane	J	flux_mass	[t, x]	\text{kg/s}\text{/m}^2
Conduction heat flux across membrane	q_{cond}	flux_conduction_heat	[t, x]	\text{W}\text{/m}^2
Evaporation heat flux from hot channel	q_{evap}	flux_enth_hot	[t, x]	\text{W}\text{/m}^2
Condensation heat flux to cold channel	q_{conden}	flux_enth_cold	[t, x]	\text{W}\text{/m}^2
Membrane area	A_m	area	None	\text{m}^2
Recovery rate	R	recovery_mass	[t]	\text{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	\Delta P	deltaP	[t]	\text{Pa}

if temperature_polarization_type is set to TemperaturePolarizationType.fixed:

Description	Symbol	Variable Name	Index	Units
Hot channel Convective heat transfer coefficient	h_{conv,h}	hot_ch.h_conv	[t]	\text{W/K}\text{/m}^2
Cold channel Convective heat transfer coefficient	h_{conv,c}	hot_ch.h_conv	[t]	\text{W/K}\text{/m}^2

if temperature_polarization_type is set to TemperaturePolarizationType.calculated:

Description	Symbol	Variable Name	Index	Units
Hot channel Prandtl number	Pr_h	hot_ch.N_Pr	[t]	\text{dimensionless}
Cold channel Prandtl number	Pr_c	cold_ch.N_Pr	[t]	\text{dimensionless}
Hot channel Nusselt number	Nu_h	hot_ch.N_Nu	[t]	\text{dimensionless}
Cold channel Nusselt number	Nu_c	cold_ch.N_Nu	[t]	\text{dimensionless}

if concentration_polarization_type is set to ConcentrationPolarizationType.fixed:

Description	Symbol	Variable Name	Index	Units
Concentration polarization modulus in hot channel	CP_{mod,h}	hot_ch.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 hot channel	k_h	hot_ch.K	[t, x, j]	\text{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	h_{ch,h}	hot_ch.channel_height	None	\text{m}
Hot channel Hydraulic diameter	d_{h,h}	cold_ch.dh	None	\text{m}
Hot channel Spacer porosity	\epsilon_{sp,h}	hot_ch.spacer_porosity	None	\text{dimensionless}
Hot channel Reynolds number	Re_{h}	hot_ch.N_Re	[t, x]	\text{dimensionless}
Cold channel height	h_{ch,c}	cold_ch.channel_height	None	\text{m}
Cold channel Hydraulic diameter	d_{h,c}	cold_ch.dh	None	\text{m}
Cold channel Spacer porosity	\epsilon_{sp,c}	cold_ch.spacer_porosity	None	\text{dimensionless}
Cold channel Reynolds number	Re_{c}	cold_ch.N_Re	[t, x]	\text{dimensionless}

if mass_transfer_coefficient is set to MassTransferCoefficient.calculated:

Description	Symbol	Variable Name	Index	Units
Schmidt number	Sc_h	hot_ch.N_Sc	[t, x]	\text{dimensionless}
Sherwood number	Sh_h	hot_ch.N_Sh	[t, x]	\text{dimensionless}
Schmidt number	Sc_c	cold_ch.N_Sc	[t, x]	\text{dimensionless}
Sherwood number	Sh_c	cold_ch.N_Sh	[t, x]	\text{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	\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 hot channel	(\frac{\Delta P}{\Delta x})_{avg,h}	hot_ch.dP_dx	[t]	\text{Pa/m}
Average pressure drop per unit length of cold channel	(\frac{\Delta P}{\Delta x})_{avg,c}	cold_ch.dP_dx	[t]	\text{Pa/m}

if pressure_change_type is set to PressureChangeType.calculated:

Description	Symbol	Variable Name	Index	Units
Hot channel velocity	v_h	hot_ch.velocity	[t, x]	\text{m/s}
Hot channel Friction factor	f_h	hot_ch.friction_factor_darcy	[t, x]	\text{dimensionless}
Pressure drop per unit length of hot channel at inlet/outlet	(\Delta P/\Delta x)_h	hot_ch.dP_dx	[t, x]	\text{Pa/m}
Cold channel velocity	v_c	cold_ch.velocity	[t, x]	\text{m/s}
Pressure drop per unit length of cold channel at inlet/outlet	(\Delta P/\Delta x)_c	cold_ch.dP_dx	[t, x]	\text{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	q_{\text{evap}}(t, x) = J(t, x) \times \widehat{H}_{\text{h}}(t, x, Vap)
Condensation heat flux to cold channel	q_{\text{conden}}(t, x) = J(t, x) \times \widehat{H}_{\text{c}}(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	h_{\text{conv}, h}(t, x) \left( T_{\text{bulk}, h}(t, x) - T_{\text{interface}, h}(t, x) \right) = q_{\text{cond}}(t, x) + q_{\text{evap}}(t, x) - J(t, x) \cdot \widehat{H}_{\text{bulk, h}}(t, x, Liq)
Cold channel convective heat transfer	h_{\text{conv}, c}(t, x) \left( T_{\text{interface}, c}(t, x) - T_{\text{bulk}, c}(t, x) \right) = q_{\text{cond}}(t, x) + q_{\text{conden}}(t, x) - J(t, x) \cdot \widehat{H}_{\text{bulk}, 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	M_p = A \cdot J_{\text{avg}}
Total conduction heat transfer	q_{\text{cond,total}} = - A \cdot q_{\text{cond,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	CP_{mod} = C_{interface}/C_{bulk}
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	A_c = h_{ch}W\epsilon_{sp}
Membrane area	A_m = LW
Pressure drop	\Delta P = (\frac{\Delta P}{\Delta x})_{avg}L
Hot channel velocity	v_h = Q_h/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_h^{2}
Recovery rate	R = \frac{M_{p}}{M_{h,in}}

Class Documentation

• :mod:`watertap.unit_models.MD.membrane_distillation_0D`
• :mod:`watertap.unit_models.MD.membrane_distillation_base`
• :mod:`watertap.unit_models.MD.MD_channel_0D`
• :mod:`watertap.unit_models.MD.MD_channel_base`