Add method to ED1d for calculating membrane resistance as a function of ion concentration

docs/technical_reference/unit_models/electrodialysis_1D.rst

@@ -120,7 +120,9 @@ are parameters that should be provided in order to fully solve the model.
    "Cell pair number", ":math:`n`", "cell_pair_num", "None", "dimensionless", 1
    "Current utilization coefficient", ":math:`\xi`", "current_utilization", "None", "dimensionless", 1
    "Channel height", ":math:`d`", "channel_height", "none", ":math:`m` ", 1
-   "Membrane areal resistance", ":math:`r`", "membrane_areal_resistance", "['cem', 'aem']", ":math:`\Omega m^2`", 2
+   "Membrane areal resistance independent on ion concentration", ":math:`r_{const}`", "membrane_areal_resistance_const", "['cem', 'aem']", ":math:`\Omega m^2`", 2
+   "Membrane areal resistance coefficient to ion concentration", ":math:`r_{coef}`", "membrane_areal_resistance_coef", "['cem', 'aem']", ":math:`\Omega mol m^{-1}`", 2
+   "Spacer conductivity coefficient", ":math:`r`", "spacer_conductivity_coefficient", "None", "dimensionless", 1
    "Cell width", ":math:`b`", "cell_width", "None", ":math:`\text{m}`", 1
    "Cell length", ":math:`l`", "cell_length", "None", ":math:`\text{m}`", 1
    "Thickness of ion exchange membranes", ":math:`\delta`", "membrane_thickness", "['cem', 'aem']", ":math:`m`", 2
@@ -186,7 +188,8 @@ Additionally, several other equations are built to describe the electrochemical
 
    "Electrical input condition", ":math:`i(x) = \frac{I}{bl}`, for 'Constant_Current';  :math:`u(x) =U` for 'Constant_Voltage'"
    "Ohm's law", ":math:`u(x) =  i(x) r_{tot}(x)`"
-   "Resistance calculation", ":math:`r_{tot}(x)=n\left(r^{cem}+r^{aem}+\frac{d}{\kappa^C(x)}+\frac{d}{\kappa^D(x)}\right)+r_{el}`"
+   "mebrane resistance calculation", ":math:`r^{iem}(x)=r^{iem}_{const}+\frac{r^{iem}_{coef}}{c_b^D}`"
+   "Total resistance calculation", ":math:`r_{tot}(x)=n\left(r^{cem}(x)+r^{aem}(x)+\frac{d}{\sigma \kappa^C(x)}+\frac{d}{\sigma \kappa^D(x)}\right)+r_{el}`"
    "Electrical power consumption", ":math:`P(x)=b\int _0 ^l u(x)i(x) dx`"
    "Water-production-specific power consumption", ":math:`P_Q=\frac{P(x=l)}{3.6\times 10^6 nQ_{out}^D}`"
    "Current efficiency for desalination", ":math:`bi(x)\eta(x)=-\sum_{j \in[cation]}{\left[\left(\frac{\partial N_j ^D(x)}{\partial x}\right) z_j F\right]}`"
@@ -238,7 +241,8 @@ Some other modifications to previously defined equations are made to accommodate
    :header: "Original equation description", "Equation replacement", "Condition"
 
    "Ohm's law", ":math:`u(x) =  i(x) r_{tot}(x) + \phi_m(x) + \phi_d^{ohm}(x) + \phi_d^{nonohm}(x)` \ :sup:`1`", "`has_nonohmic_potential_membrane == True` and/or \ `has_Nernst_diffusion_layer==True`"
-   "Resistance calculation", ":math:`r_{tot}(x)=n\left(r^{cem}+r^{aem}+\frac{d- \Delta_{cem}^L(x) - \Delta_{aem}^R(x)}{\kappa^C(x)}+\frac{d- \Delta_{cem}^R(x) - \Delta_{aem}^L(x)}{\kappa^D(x)}\right)+r_{el}`", "`has_Nernst_diffusion_layer==True`"
+   "mebrane resistance calculation", ":math:`r^{iem}(x)=r^{iem}_{const}+\frac{r^{iem}_{coef}}{c_b^D}`"
+   "total resistance calculation", ":math:`r_{tot}(x)=n\left(r^{cem}(x)+r^{aem}(x)+\frac{d- \Delta_{cem}^L(x) - \Delta_{aem}^R(x)}{\sigma \kappa^C(x)}+\frac{d- \Delta_{cem}^R(x) - \Delta_{aem}^L(x)}{\sigma \kappa^D(x)}\right)+r_{el}`", "`has_Nernst_diffusion_layer==True`"
    "mass transfer flux, concentrate, solute", ":math:`J_j^{C} = \left(t_j^{cem}-t_j^{aem} \right)\frac{\xi i(x)}{ z_j F}-\left(\frac{D_j^{cem}}{\delta ^{cem}}\left(c_{s,j}^{L,cem}(x)-c_{s,j}^{R,cem}(x) \right) +\frac{D_j^{aem}}{\delta ^{aem}} \left(c_{s,j}^{R,aem}(x)-c_{s,j}^{L,aem}(x) \right)\right)`", "`has_nonohmic_potential_membrane == True` and/or \ `has_Nernst_diffusion_layer==True`"
    "mass transfer flux, diluate, solute", ":math:`J_j^{D} = -\left(t_j^{cem}-t_j^{aem} \right)\frac{\xi i(x)}{ z_j F}+\left(\frac{D_j^{cem}}{\delta ^{cem}}\left(c_{s,j}^{L,cem}(x)-c_{s,j}^{R,cem}(x) \right) +\frac{D_j^{aem}}{\delta ^{aem}} \left(c_{s,j}^{R,aem}(x)-c_{s,j}^{L,aem}(x) \right)\right)`", "`has_nonohmic_potential_membrane == True` and/or \ `has_Nernst_diffusion_layer==True`"
    "mass transfer flux, concentrate, H\ :sub:`2`\ O", ":math:`J_j^{C} = \left(t_w^{cem}+t_w^{aem} \right)\frac{i(x)}{F}+\left(L^{cem} \left(p_{s, osm}^{cem, L}(x)-p_{s, osm}^{cem, R}(x) \right)+L^{aem} \left(p_{s, osm}^{aem, R}(x)-p_{s, osm}^{aem, L}(x) \right)\right)\frac{\rho_w}{M_w}`", "`has_Nernst_diffusion_layer==True`"
@@ -301,9 +305,12 @@ Nomenclature
    ":math:`p_{osm}`", "Osmotic pressure", ":math:`Pa`"
    ":math:`r_{tot}`", "Total areal resistance", ":math:`\Omega m^2`"
    ":math:`r`", "Membrane areal resistance", ":math:`\Omega m^2`"
+   ":math:`r_const`", "Membrane areal resistance independent on ion concentration", ":math:`\Omega m^2`"
+   ":math:`r_coef`", "The dependent cofficient of membrane areal resistance to :math:`1/c_b`", ":math:`\Omega mol m^{-1}`"
    ":math:`r_{el}`", "Electrode areal resistance", ":math:`\Omega m^2`"
    ":math:`d`", "Channel height", ":math:`m`"
    ":math:`\kappa`", "Solution conductivity", ":math:`S m^{-1}\ or\  \Omega^{-1} m^{-1}`"
+   ":math:`\sigma`", "Spacer conductivity coefficient", ":math:`S m^{-1}\ or\  \Omega^{-1} m^{-1}`"
    ":math:`\eta`", "Current efficiency for desalination", "dimensionless"
    ":math:`P`", "Power consumption", ":math:`W`"
    ":math:`P_Q`", "Specific power consumption", ":math:`kW\ h\  m^{-3}`"

watertap/flowsheets/electrodialysis/electrodialysis_1stack.py

@@ -205,8 +205,10 @@ def set_operating_conditions(m):
     m.fs.EDstack.cell_pair_num.fix(100)
     m.fs.EDstack.current_utilization.fix(1)
     m.fs.EDstack.channel_height.fix(2.7e-4)
-    m.fs.EDstack.membrane_areal_resistance["cem"].fix(1.89e-4)
-    m.fs.EDstack.membrane_areal_resistance["aem"].fix(1.77e-4)
+    m.fs.EDstack.membrane_areal_resistance_const["cem"].fix(1.89e-4)
+    m.fs.EDstack.membrane_areal_resistance_const["aem"].fix(1.77e-4)
+    m.fs.EDstack.membrane_areal_resistance_coef["cem"].fix(0)
+    m.fs.EDstack.membrane_areal_resistance_coef["aem"].fix(0)
     m.fs.EDstack.cell_width.fix(0.1)
     m.fs.EDstack.cell_length.fix(0.79)
     m.fs.EDstack.membrane_thickness["aem"].fix(1.3e-4)
@@ -220,6 +222,7 @@ def set_operating_conditions(m):
     m.fs.EDstack.ion_trans_number_membrane["cem", "Cl_-"].fix(0)
     m.fs.EDstack.ion_trans_number_membrane["aem", "Cl_-"].fix(1)
     m.fs.EDstack.spacer_porosity.fix(1)
+    m.fs.EDstack.spacer_conductivity_coefficient.fix(1)
 
     # check zero degrees of freedom
     check_dof(m)
@@ -266,7 +269,7 @@ def optimize_system(m, solver=None, checkpoint=None, fail_flag=True):
     # Choose and unfix variables to be optimized
     m.fs.EDstack.voltage_applied[0].unfix()
     m.fs.EDstack.cell_pair_num.unfix()
-    m.fs.EDstack.cell_pair_num.set_value(10)
+    m.fs.EDstack.cell_pair_num.set_value(30)
     # Give narrower bounds to optimizing variables if available
     m.fs.EDstack.voltage_applied[0].setlb(0.5)
     m.fs.EDstack.voltage_applied[0].setub(20)
@@ -280,8 +283,9 @@ def optimize_system(m, solver=None, checkpoint=None, fail_flag=True):
     print("---report model statistics---\n ", report_statistics(m.fs))
     if solver is None:
         solver = get_solver()
-    results = solver.solve(m, tee=True)
-    check_solve(results, checkpoint=checkpoint, logger=_log, fail_flag=fail_flag)
+    solve(m, solver=solver, tee=True)
+    m.fs.EDstack.cell_pair_num.fix(round(value(m.fs.EDstack.cell_pair_num)))
+    solve(m, solver=solver, tee=True)
 
 
 def display_model_metrics(m):
@@ -319,6 +323,7 @@ def display_model_metrics(m):
         data=[
             value(m.fs.EDstack.recovery_mass_H2O[0]),
             value(m.fs.mem_area),
+            value(m.fs.EDstack.cell_pair_num),
             value(m.fs.EDstack.voltage_applied[0]),
             value(m.fs.costing.specific_energy_consumption),
             value(m.fs.costing.LCOW),
@@ -327,6 +332,7 @@ def display_model_metrics(m):
         index=[
             "Water recovery by mass",
             "Total membrane area (aem or cem), m2",
+            "Cell pair number",
             "Operation Voltage, V",
             "Specific energy consumption, kWh/m3",
             "Levelized cost of water, $/m3",

watertap/flowsheets/electrodialysis/electrodialysis_1stack_conc_recirc.py

@@ -277,8 +277,10 @@ def _condition_base(m):
     m.fs.EDstack.water_trans_number_membrane["aem"].fix(4.3)
     m.fs.EDstack.water_permeability_membrane["cem"].fix(2.16e-14)
     m.fs.EDstack.water_permeability_membrane["aem"].fix(1.75e-14)
-    m.fs.EDstack.membrane_areal_resistance["cem"].fix(1.89e-4)
-    m.fs.EDstack.membrane_areal_resistance["aem"].fix(1.77e-4)
+    m.fs.EDstack.membrane_areal_resistance_const["cem"].fix(1.89e-4)
+    m.fs.EDstack.membrane_areal_resistance_const["aem"].fix(1.77e-4)
+    m.fs.EDstack.membrane_areal_resistance_coef["cem"].fix(0)
+    m.fs.EDstack.membrane_areal_resistance_coef["aem"].fix(0)
     m.fs.EDstack.solute_diffusivity_membrane["cem", "Na_+"].fix(3.28e-11)
     m.fs.EDstack.solute_diffusivity_membrane["aem", "Na_+"].fix(3.28e-11)
     m.fs.EDstack.solute_diffusivity_membrane["cem", "Cl_-"].fix(3.28e-11)
@@ -299,6 +301,7 @@ def _condition_base(m):
     # Spacer properties
     m.fs.EDstack.spacer_porosity.fix(0.83)
     m.fs.EDstack.spacer_specific_area.fix(10400)
+    m.fs.EDstack.spacer_conductivity_coefficient.fix(1)
 
     # Electrochemical properties
     m.fs.EDstack.electrodes_resistance.fix(0)

watertap/flowsheets/electrodialysis/electrodialysis_1stack_conc_recirc_ui.py

@@ -240,23 +240,23 @@ def export_variables(flowsheet=None, exports=None, build_options=None, **kwargs)
     )
 
     exports.add(
-        obj=fs.EDstack.membrane_areal_resistance["cem"],
+        obj=fs.EDstack.membrane_areal_resistance_const["cem"],
         name="Areal resistnace of CEM",
         ui_units=pyunits.ohm * pyunits.meter**2,
         display_units="ohm m^2",
         rounding=2,
-        description="Areal resistnace of the cation exchange membrane",
+        description="Constant areal resistance of the cation exchange membrane measured in concentrated electrolyte.",
         is_input=True,
         input_category="Membrane properties",
         is_output=False,
     )
     exports.add(
-        obj=fs.EDstack.membrane_areal_resistance["aem"],
+        obj=fs.EDstack.membrane_areal_resistance_const["aem"],
         name="Areal resistnace of AEM",
         ui_units=pyunits.ohm * pyunits.meter**2,
         display_units="ohm m^2",
         rounding=2,
-        description="Areal resistnace of the anion exchange membrane",
+        description="Constant areal resistance of the anion exchange membrane measured in concentrated electrolyte.",
         is_input=True,
         input_category="Membrane properties",
         is_output=False,

watertap/flowsheets/electrodialysis/tests/test_electrodialysis_1stack.py

@@ -173,7 +173,6 @@ def test_optimization(self, electrodialysis_1D1stack):
         edfs.optimize_system(m)
         isinstance(m.fs.objective, Objective)
         assert m.fs.objective.expr == m.fs.costing.LCOW
-        assert degrees_of_freedom(m) == 1
 
         assert value(m.fs.feed.properties[0].flow_vol_phase["Liq"]) == pytest.approx(
             8.7e-5, abs=1e-6
@@ -188,14 +187,16 @@ def test_optimization(self, electrodialysis_1D1stack):
         assert value(m.fs.disposal_salinity) == pytest.approx(18.1124, rel=1e-3)
 
         assert value(m.fs.EDstack.recovery_mass_H2O[0]) == pytest.approx(
-            0.4846, rel=1e-3
+            0.48455, rel=1e-3
+        )
+        assert value(m.fs.mem_area) == pytest.approx(1.1060, rel=1e-3)
+        assert value(m.fs.EDstack.voltage_applied[0]) == pytest.approx(
+            7.03676, rel=1e-3
         )
-        assert value(m.fs.mem_area) == pytest.approx(1.0980, rel=1e-3)
-        assert value(m.fs.EDstack.voltage_applied[0]) == pytest.approx(7.0325, rel=1e-3)
         assert value(m.fs.costing.specific_energy_consumption) == pytest.approx(
-            2.3062, rel=1e-3
+            2.2922, rel=1e-3
         )
-        assert value(m.fs.costing.LCOW) == pytest.approx(0.42546, rel=1e-3)
+        assert value(m.fs.costing.LCOW) == pytest.approx(0.42547, rel=1e-3)
 
     @pytest.mark.unit
     def test_main_fun(self, electrodialysis_1D1stack):