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
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 |
\(ΔP\) |
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 |
\((\frac{ΔP}{Δx})_{avg,h}\) |
hot_ch.dP_dx |
[t] |
\(\text{Pa/m}\) |
Average pressure drop per unit length of cold channel |
\((\frac{ΔP}{Δx})_{avg,c}\) |
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 |
\((ΔP/Δx)_h\) |
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 |
\((ΔP/Δx)_c\) |
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 |
\(ΔP = (\frac{ΔP}{Δx})_{avg}L\) |
Hot channel velocity |
\(v_h = Q_h/A_c\) |
Friction factor |
\(f = 0.42+\frac{189.3}{Re}\) |
Pressure drop per unit length |
\(\frac{ΔP}{Δx} = \frac{1}{2d_h}f\rho v_h^{2}\) |
Recovery rate |
\(R = \frac{M_{p}}{M_{h,in}}\) |