Energy storage devices¶

Only the electrical current $I$ and voltage $U$ of the device are measurable. Several operating conditions are possibles. One may want to impose:

The voltage $U$ across the device.

The electrical current $I$ that flows through it.

The load $R=U/I$ the device is subject to.

The power $P=UI$ .

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 $\mathrm{ESR}$ and $\mathrm{C}$ in the figure above).

type                SeriesRC
series_resistance     5.0e-3 ; [ohm]
capacitance           3.0    ; [fahrad]

Above is the database to build a $\mathrm{3\ F}$ capacitor in series with a $50\ \mathrm{m\Omega}$ resistance.

$U = U_C + R I I = C \frac{dU_C}{dt}$

$U_C$ stands for the voltage across the capacitor. Its capacitance, $C$ , represents its ability to store electric charge. The equivalent series resistance, $R$ , add a real component to the impedance of the circuit:

$Z = \frac{1}{jC\omega} + R$

As the frequency goes to infinity, the capacitive impedance approaches zero and $R$ 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 $2.5\ \mathrm{M\Omega}$ leakage resistance is specified.

$U = U_C + R I I = C \frac{dU_C}{dt} + \frac{U_C}{R_L}$

$R_L$ corresponds to the “leakage” resistance in parallel with the capacitor. Low values of $R_L$ imply high leakage currents which means the capacitor is not able to hold is charge. The circuit complex impedance is given by:

$Z = \frac{R_L}{1+jR_LC\omega} + R$

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.

Schematic representation of the supercapacitor conventional sandwich-like configuration. 1: anode electrode, 2: separator, 3: cathode electrode, 4: anode collector, 5: cathode collector.

Governing equations¶

collector

electrode

separator

$i_1 = -\sigma \nabla \Phi_1$

$\nabla \cdot i_1 = 0$

$i_1 = -\sigma \nabla \Phi_1$

$i_2 = -\kappa \nabla \Phi_2$

$-\nabla \cdot i_1 = \nabla \cdot i_2 = a i_n$

$i_2 = -\kappa \nabla \Phi_2$

$\nabla \cdot i_2 = 0$

collector-electrode interface

electrode-separator interface

$0 = -\kappa \left. \frac{\partial \Phi_2}{\partial n} \right|_e$

$-\sigma \left.\frac{\partial \Phi_1}{\partial n}\right|_c = -\sigma \left.\frac{\partial \Phi_1}{\partial n}\right|_e$

$-\kappa \left.\frac{\partial \Phi_2}{\partial n}\right|_e = -\kappa \left.\frac{\partial \Phi_2}{\partial n}\right|_s$

$-\sigma \left. \frac{\partial \Phi_1}{\partial n} \right|_e = 0$

boundary collector tab

$\Phi_1 = U$

or

$-\sigma \frac{\partial \Phi_1}{\partial n} = I/S$

or

$-\sigma \frac{\partial \Phi_1}{\partial n} \Phi_1 = P/S$

or

$-\sigma \frac{\partial \Phi_1}{\partial n} R S = \Phi_1$

Ignoring the influence of the electrolyte concentration, the current density in the matrix and solution phases can be expressed by Ohm’s law as

$i_1 = -\sigma \nabla \Phi_1 i_2 = -\kappa \nabla \Phi_2$

$i$ and $\Phi$ represent current density and potential; subscript indices $1$ and $2$ denote respectively the solid and the liquid phases. $\sigma$ and $\kappa$ are the matrix and solution phase conductivities.

The total current density is given by $i = i_1 + i_2$ . Conservation of charge dictates that

$-\nabla \cdot i_1 = \nabla \cdot i_2 = a i_n$

where $a$ is the interfacial area per unit volume and the current transferred from the matrix phase to the electrolyte $i_n$ is the sum of the double-layer the faradaic currents

$i_n = C \frac{\partial}{\partial t} \left(\Phi_1 - \Phi_2\right) + i_0 \left( e^{\frac{\alpha_a F}{RT}\eta} - e^{-\frac{\alpha_c F}{RT}\eta} \right)$

$C$ is the double-layer capacitance. $i_0$ is the exchange current density, $\alpha_a$ and $\alpha_c$ the anodic and cathodic charge transfer coefficients, respectively. $F$ , $R$ , and $T$ stand for Faraday’s constant, the universal gas constant and temperature. $\eta$ is the overpotential relative to the equilibrium potential $U_{eq}$

$\eta = \Phi_1 - \Phi_2 - U_{eq}$

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 $a$ is estimated using

$a = \frac{(1+\zeta)\varepsilon}{r}$

where $\zeta$ is the pore’s geometry factor ( $\zeta=2$ for spheres, $1$ for cylinders, and $0$ for slabs) and $r$ is the pore’s characteristic dimension. [M. W. Verbrugge and B. J. Koch, J. Electrochem. Soc., 150, A374 2003]

The solution electrical conductivity $\kappa$ incorporates the effect of porosity and tortuosity

$\kappa = \frac{\kappa_\infty \varepsilon}{\Gamma}$

where $\kappa_\infty$ is the liquid phase (free solution) conductivity, $\varepsilon$ is the void volume fraction, and $\kappa$ is the tortuosity factor.

The solid phase conductivity is also corrected for porosity (and tortuosity???)

$\sigma = \sigma_\infty (1-\varepsilon)$

Batteries¶

NOT IMPLEMENTED