Energy storage devices¶
Only the electrical current and voltage of the device are measurable. Several operating conditions are possibles. One may want to impose:
- The voltage across the device.
- The electrical current that flows through it.
- The load the device is subject to.
- The power .
The class pycap.EnergyStorageDevice
is an abstract representation
for an energy storage device. It can evolve in time at various operating
conditions and return the voltage drop across itself and the electrical
current that flows through it.
The rest of this section describes the energy storage devices that are available in Cap, namely:
- Equivalent circuits
- Supercapacitors
Equivalent circuits¶
Series RC¶
A resistor and a capacitor are connected in series (denoted and in the figure above).
type SeriesRC
series_resistance 5.0e-3 ; [ohm]
capacitance 3.0 ; [fahrad]
Above is the database to build a capacitor in series with a resistance.
stands for the voltage across the capacitor. Its capacitance, , represents its ability to store electric charge. The equivalent series resistance, , add a real component to the impedance of the circuit:
As the frequency goes to infinity, the capacitive impedance approaches zero and becomes significant.
Parallel RC¶
An extra resistance is placed in parallel of the capacitor. It can be instantiated by the following database.
type ParallelRC
parallel_resistance 2.5e+6 ; [ohm]
series_resistance 50.0e-3 ; [ohm]
capacitance 3.0 ; [fahrad]
type
has been changed from SeriesRC
to ParallelRC
.
A leakage resistance is specified.
corresponds to the “leakage” resistance in parallel with the capacitor. Low values of imply high leakage currents which means the capacitor is not able to hold is charge. The circuit complex impedance is given by:
Supercapacitors¶
type
is set to SuperCapacitor
.
dim
is used to select two- or three-dimensional simulations.
device {
type SuperCapacitor
dim 2
geometry {
[...]
}
material_properties {
[...]
}
}
Geometry¶
geometry {
type supercapacitor
anode_collector_thickness 5.0e-4 ; [centimeter]
anode_electrode_thickness 50.0e-4 ; [centimeter]
separator_thickness 25.0e-4 ; [centimeter]
cathode_electrode_thickness 50.0e-4 ; [centimeter]
cathode_collector_thickness 5.0e-4 ; [centimeter]
geometric_area 25.0e-2 ; [square centimeter]
}
The thickness of each layer in the sandwich (anode collector, anode electrode, separator, cathode electrode, cathode current collector) can be adjusted independently from one another. The specified cross-sectional area applies to the whole stack.
Governing equations¶
collector | electrode | separator |
---|---|---|
collector-electrode interface | electrode-separator interface |
---|---|
boundary collector tab |
---|
or or or |
Ignoring the influence of the electrolyte concentration, the current density in the matrix and solution phases can be expressed by Ohm’s law as
and represent current density and potential; subscript indices and denote respectively the solid and the liquid phases. and are the matrix and solution phase conductivities.
The total current density is given by . Conservation of charge dictates that
where is the interfacial area per unit volume and the current transferred from the matrix phase to the electrolyte is the sum of the double-layer the faradaic currents
is the double-layer capacitance. is the exchange current density, and the anodic and cathodic charge transfer coefficients, respectively. , , and stand for Faraday’s constant, the universal gas constant and temperature. is the overpotential relative to the equilibrium potential
Material properties¶
material_properties {
anode {
type porous_electrode
matrix_phase electrode_material
solution_phase electrolyte
}
cathode {
type porous_electrode
matrix_phase electrode_material
solution_phase electrolyte
}
separator {
type permeable_membrane
matrix_phase separator_material
solution_phase electrolyte
}
collector {
type current_collector
metal_foil collector_material
}
separator_material {
void_volume_fraction 0.6 ;
tortuosity_factor 1.29 ;
pores_characteristic_dimension 1.5e-7 ; [centimeter]
pores_geometry_factor 2.0 ;
mass_density 3.2 ; [gram per cubic centimeter]
heat_capacity 1.2528e3 ; [joule per kilogram kelvin]
thermal_conductivity 0.0019e2 ; [watt per meter kelvin]
}
electrode_material {
differential_capacitance 3.134 ; [microfarad per square centimeter]
exchange_current_density 7.463e-10 ; [ampere per square centimeter]
void_volume_fraction 0.67 ;
tortuosity_factor 2.3 ;
pores_characteristic_dimension 1.5e-7 ; [centimeter]
pores_geometry_factor 2.0 ;
mass_density 2.3 ; [gram per cubic centimeter]
electrical_resistivity 1.92 ; [ohm centimeter]
heat_capacity 0.93e3 ; [joule per kilogram kelvin]
thermal_conductivity 0.0011e2 ; [watt per meter kelvin]
}
collector_material {
mass_density 2.7 ; [gram per cubic centimeter]
electrical_resistivity 28.2e-7 ; [ohm centimeter]
heat_capacity 2.7e3 ; [joule per kilogram kelvin]
thermal_conductivity 237.0 ; [watt per meter kelvin]
}
electrolyte {
mass_density 1.2 ; [gram per cubic centimeter]
electrical_resistivity 1.49e3 ; [ohm centimeter]
heat_capacity 0.0 ; [joule per kilogram kelvin]
thermal_conductivity 0.0 ; [watt per meter kelvin]
}
}
The specific surface area per unit volume is estimated using
where is the pore’s geometry factor ( for spheres, for cylinders, and for slabs) and is the pore’s characteristic dimension. [M. W. Verbrugge and B. J. Koch, J. Electrochem. Soc., 150, A374 2003]
The solution electrical conductivity incorporates the effect of porosity and tortuosity
where is the liquid phase (free solution) conductivity, is the void volume fraction, and is the tortuosity factor.
The solid phase conductivity is also corrected for porosity (and tortuosity???)
Batteries¶
NOT IMPLEMENTED