MaplesoftBattery

Nickel-Metal Hydride $—$ Electrochemical model of a nickel-metal hydride battery

Description

The Nickel-Metal Hydride component is a model of a nickel-metal hydride battery based on a planar electrode approximation. The mass-balance of active materials, the kinetics of electrochemical reactions, internal resistance, and the energy balance of the cell are incorporated. See [1].

There are two main redox reactions at the positive and negative electrodes: the reaction of the nickel active material at the positive electrode and the reaction of the metal hydride material at the negative electrode.  Besides that, Ni-MH cells are also known to have side reactions which cause gases to form inside the cell casing. The most significant side reactions are the oxygen evolution reaction at the positive electrode and the oxygen reduction reaction at the negative electrode. The main and side chemical reactions are described by the following equations:

Positive nickel electrode:

$\mathrm{NiOOH}+{H}_{2}0+{e}^{-}⇄\mathrm{Ni}\left(\mathrm{OH}\right){}_{2}+{\mathrm{OH}}^{-}$

$2{\mathrm{OH}}^{-}\to \frac{1}{2}{\mathrm{O}}_{2}+{H}_{2}\mathrm{O}+2{e}^{-}$

Negative metal hydride electrode:

$\mathrm{MH}+{\mathrm{OH}}^{-}⇄{H}_{2}\mathrm{O}+M+{e}^{-}$

$\frac{1}{2}{\mathrm{O}}_{2}+{H}_{2}\mathrm{O}+2{e}^{-}\to 2{\mathrm{OH}}^{-}$

Governing Equations

Butler-Volmer's equation describes the kinetics of reactions for both positive and negative electrodes:

${j}_{1}={i}_{0,1}\left(\mathrm{exp}\left(0.5\frac{F}{RT}\left({\mathrm{\Phi }}_{p}-{\mathrm{\Phi }}_{\mathrm{eq1}}\right)\right)-\mathrm{exp}\left(-0.5\frac{F}{RT}\left({\mathrm{\Phi }}_{p}-{\mathrm{\Phi }}_{{\mathrm{eq}}_{1}}\right)\right)\right)$

${j}_{2}={i}_{0,2}\left(\mathrm{exp}\left(0.5\frac{F}{RT}\left({\mathrm{\Phi }}_{p}-{\mathrm{\Phi }}_{\mathrm{eq2}}\right)\right)-\mathrm{exp}\left(-0.5\frac{F}{RT}\left({\mathrm{\Phi }}_{p}-{\mathrm{\Phi }}_{{\mathrm{eq}}_{2}}\right)\right)\right)$

${j}_{3}={i}_{0,3}\left(\mathrm{exp}\left(0.5\frac{F}{RT}\left({\mathrm{\Phi }}_{p}-{\mathrm{\Phi }}_{\mathrm{eq3}}\right)\right)-\mathrm{exp}\left(-0.5\frac{F}{RT}\left({\mathrm{\Phi }}_{p}-{\mathrm{\Phi }}_{{\mathrm{eq}}_{3}}\right)\right)\right)$

${j}_{4}=-\frac{{p}_{\mathrm{O2}}}{{p}_{\mathrm{O2},\mathrm{ref}}}{i}_{4,\mathrm{ref}}\mathrm{exp}\left(\frac{{E}_{a,4}}{R}\left(\frac{1}{T}-\frac{1}{{T}_{\mathrm{ref}}}\right)\right)$

where

 ${\mathrm{\Phi }}_{p},{\mathrm{\Phi }}_{n}$ are the electrical potentials in the positive and negative electrodes,
 ${\mathrm{\Phi }}_{{\mathrm{eq}}_{k}}$ is the equilibrium potential of reaction $k$ at the standard conditions,
 ${i}_{0,k}$ is the exchange current density of reaction $k$,
 ${i}_{4,\mathrm{ref}}$ is limiting current density of the oxygen reduction reaction, and
 ${p}_{{\mathrm{O}}_{2}},{p}_{{\mathrm{O}}_{2},\mathrm{ref}}$ are the pressure and reference pressure of oxygen in the cell.

The exchange current densities vary with the nickel hydroxide concentration ( + ) in the nickel active material, and the hydrogen concentration in metal hydride material ( ) and are described by the following equations.

${i}_{0,1}={i}_{0,1,\mathrm{ref}}{\left(\frac{{c}_{H}}{{c}_{H,\mathrm{ref}}}\right)}^{0.5}{\left(\frac{{c}_{e}}{{c}_{e,\mathrm{ref}}}\right)}^{0.5}{\left(\frac{{c}_{H,\mathrm{max}}-{c}_{H}}{{c}_{H,\mathrm{max}}-{c}_{H,\mathrm{ref}}}\right)}^{0.5}\mathrm{exp}\left(\frac{{E}_{a,1}}{R}\left(\frac{1}{T}-\frac{1}{{T}_{\mathrm{ref}}}\right)\right)$

${i}_{0,2}={i}_{0,2,\mathrm{ref}}{\left(\frac{{c}_{e}}{{c}_{e,\mathrm{ref}}}\right)}^{0.5}{\left(\frac{{p}_{\mathrm{O2}}}{{p}_{\mathrm{O2},\mathrm{ref}}}\right)}^{0.5}\mathrm{exp}\left(\frac{{E}_{a,2}}{R}\left(\frac{1}{T}-\frac{1}{{T}_{\mathrm{ref}}}\right)\right)$

${i}_{0,3}={i}_{0,3,\mathrm{ref}}{\left(\frac{{c}_{\mathrm{MH}}}{{c}_{\mathrm{MH},\mathrm{ref}}}\right)}^{0.5}{\left(\frac{{c}_{e}}{{c}_{e,\mathrm{ref}}}\right)}^{0.5}{\left(\frac{{c}_{\mathrm{MH},\mathrm{max}}-{c}_{\mathrm{MH}}}{{c}_{\mathrm{MH},\mathrm{max}}-{c}_{\mathrm{MH},\mathrm{ref}}}\right)}^{0.5}\mathrm{exp}\left(\frac{{E}_{a,3}}{R}\left(\frac{1}{T}-\frac{1}{{T}_{\mathrm{ref}}}\right)\right)$

The open-circuit potential curves based on the Nernst equation are utilized:

${\mathrm{\Phi }}_{\mathrm{eq1}}={U}_{1}+\left(T-{T}_{\mathrm{ref}}\right)\frac{\partial {U}_{1}}{\partial T}+\frac{RT}{F}\mathrm{ln}\left(\frac{{c}_{H,\mathrm{max}}-{c}_{H}}{{c}_{e}{c}_{H}}\right)$

${\mathrm{\Phi }}_{\mathrm{eq2}}={U}_{2}+\left(T-{T}_{\mathrm{ref}}\right)\frac{\partial {U}_{2}}{\partial T}+\frac{RT}{F}\mathrm{ln}\left(\frac{{p}_{\mathrm{O2}}^{0.5}}{{c}_{e}^{2}}\right)$

${\mathrm{\Phi }}_{\mathrm{eq3}}={U}_{3}+\left(T-{T}_{\mathrm{ref}}\right)\frac{\partial {U}_{3}}{\partial T}+\frac{RT}{F}\mathrm{ln}\left({c}_{e}\right)+9.7{10}^{-4}+0.23724\mathrm{exp}\left(-28.057\frac{{c}_{\mathrm{MH}}}{{c}_{\mathrm{MH},\mathrm{max}}}\right)-\frac{2.7302{10}^{-4}}{{\left(\frac{{c}_{\mathrm{MH}}}{{c}_{\mathrm{MH},\mathrm{max}}}\right)}^{2}+0.1768}$

The charge and mass balances on the electrodes are given by

$+{i}_{\mathrm{cell}}={A}_{\mathrm{pos}}{a}_{\mathrm{pos}}{\ell }_{\mathrm{pos}}\left({j}_{1}+{j}_{2}\right)$

$-{i}_{\mathrm{cell}}={A}_{\mathrm{neg}}{a}_{\mathrm{neg}}{\ell }_{\mathrm{neg}}\left({j}_{3}+{j}_{4}\right)$

$\frac{{L}_{\mathrm{Ni}\left(\mathrm{OH}\right){}_{2}}}{{\mathrm{\rho }}_{\mathrm{Ni}\left(\mathrm{OH}\right){}_{2}}{\ell }_{\mathrm{pos}}{a}_{\mathrm{pos}}}\frac{\mathrm{d}{c}_{H}}{\mathrm{d}t}=-\frac{{j}_{1}}{F}$

$\frac{{L}_{\mathrm{MH}}}{{\mathrm{\rho }}_{\mathrm{MH}}{\ell }_{\mathrm{neg}}{a}_{\mathrm{neg}}}\frac{\mathrm{d}{c}_{\mathrm{MH}}}{\mathrm{d}t}=-\frac{{j}_{4}}{F}$

$\frac{{V}_{\mathrm{gas}}}{RT}\frac{\mathrm{d}{p}_{{\mathrm{O}}_{2}}}{\mathrm{d}t}=\frac{1}{F}\left({A}_{\mathrm{pos}}{a}_{\mathrm{pos}}{\ell }_{\mathrm{pos}}{j}_{2}+{A}_{\mathrm{neg}}{a}_{\mathrm{neg}}{\ell }_{\mathrm{neg}}{j}_{4}\right)$

Thermal Effects

Select the thermal model of the battery from the heat model drop-down list.  The available models are: isothermal, external port, and convection.

 Isothermal The isothermal model sets the cell temperature to a constant parameter, ${T}_{\mathrm{iso}}$.
 External Port The external port model adds a thermal port to the battery model. The temperature of the heat port is the cell temperature. The parameters ${m}_{\mathrm{cell}}$ and ${c}_{p}$ become available and are used in the heat equation ${m}_{\mathrm{cell}}{c}_{p}\frac{\mathrm{d}{T}_{\mathrm{cell}}}{\mathrm{d}t}={P}_{\mathrm{cell}}-{Q}_{\mathrm{cell}}$ ${Q}_{\mathrm{flow}}={n}_{\mathrm{cell}}{Q}_{\mathrm{cell}}$ ${P}_{\mathrm{cell}}={i}_{\mathrm{cell}}^{2}{R}_{\mathrm{cell}}+{\mathrm{Vol}}_{\mathrm{cell}}\left({\sum }_{k=1}^{4}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}{a}_{k}{j}_{k}\left({U}_{k}-{T}_{\mathrm{ref}}\left(\frac{{\mathrm{dU}}_{k}}{\mathrm{dT}}\right)\right)\right)$ ${a}_{1}={a}_{3}={a}_{p}$ ${a}_{2}={a}_{4}={a}_{n}$ where ${P}_{\mathrm{cell}}$ is the heat generated in each cell, including chemical reactions and ohmic resistive losses, ${Q}_{\mathrm{cell}}$ is the heat flow out of each cell, and ${Q}_{\mathrm{flow}}$ is the heat flow out of the external port.
 Convection The convection model assumes the heat dissipation from each cell is due to uniform convection from the surface to an ambient temperature. The parameters ${m}_{\mathrm{cell}}$, ${c}_{p}$, ${A}_{\mathrm{cell}}$, $h$, and ${T}_{\mathrm{amb}}$ become available, as does an output signal port that gives the cell temperature in Kelvin. The heat equation is the same as the heat equation for the external port, with ${Q}_{\mathrm{cell}}$ given by ${Q}_{\mathrm{cell}}=h{A}_{\mathrm{cell}}\left({T}_{\mathrm{cell}}-{T}_{\mathrm{amb}}\right)$
 State of Charge A signal output, soc, gives the state-of-charge of the battery, with 0 being fully discharged and 1 being fully charged. The parameter ${\mathrm{SOC}}_{\mathrm{min}}$ sets the minimum allowable state-of-charge; if the battery is discharged past this level, the simulation is terminated and an error message is raised. This prevents the battery model from reaching non-physical conditions. A similar effect occurs if the battery is fully charged so that the state of charge reaches one. The parameter ${\mathrm{SOC}}_{0}$ assigns the initial state-of charge of the battery.
 Capacity The capacity of the battery can either be a fixed value, $\mathrm{CA}$, or be controlled via an input signal, ${C}_{\mathrm{in}}$, if the use capacity input box is checked.
 Resistance The resistance of each cell can either be a fixed value, ${R}_{\mathrm{cell}}$, or be controlled via an input signal, ${R}_{\mathrm{in}}$, if the use cell resistance input box is checked.

Variables

 Name Units Description Modelica ID ${T}_{\mathrm{cell}}$ $K$ Internal temperature of battery Tcell $i$ $A$ Current into battery i $v$ $V$ Voltage across battery v

Connections

 Name Type Description Modelica ID $p$ Electrical Positive pin p $n$ Electrical Negative pin n $\mathrm{SOC}$ Real output State of charge [0..1] SOC ${C}_{\mathrm{in}}$ Real input Sets capacity of cell, in ampere hours; available when use capacity input is true Cin ${R}_{\mathrm{in}}$ Real input Sets resistance of cell, in Ohms; available when use resistance input is true Rin ${T}_{\mathrm{out}}$ Real output Temperature of cell, in Kelvin; available with convection heat model Tout $\mathrm{heatPort}$ Thermal Thermal connection; available with external port heat model heatPort

Basic Parameters

 Name Default Units Description Modelica ID ${N}_{\mathrm{cell}}$ $1$ Number of cells, connected in series ncell $\mathrm{CA}$ $1$ $\mathrm{A·h}$ Capacity of cell, in ampere-hours C ${\mathrm{SOC}}_{0}$ $1$ Initial state-of-charge [0..1] SOC0 ${\mathrm{SOC}}_{\mathrm{min}}$ $0.01$ Minimum allowable state-of-charge SOCmin ${R}_{\mathrm{cell}}$ $0.005$ $\mathrm{\Omega }$ Internal resistance of one cell; available if use cell resistance input is not enabled Rcell

Basic Thermal Parameters

 Name Default Units Description Modelica ID ${T}_{\mathrm{iso}}$ $298.15$ $K$ Constant cell temperature; used with isothermal heat model Tiso ${c}_{p}$ $750$ $\frac{J}{\mathrm{kg}K}$ Specific heat capacity of cell cp ${m}_{\mathrm{cell}}$ $0.014$ $\mathrm{kg}$ Mass of one cell mcell $h$ $100$ $\frac{W}{{m}^{2}K}$ Surface coefficient of heat transfer; used with convection heat model h ${A}_{\mathrm{cell}}$ $0.0014$ ${m}^{2}$ Surface area of one cell; used with convection heat model Acell ${T}_{\mathrm{amb}}$ $298.15$ $K$ Ambient temperature; used with convection heat model Tamb

Extended Parameters

 Name Default Units Description Modelica ID ${E}_{\mathrm{a1}}$ $10000$ $\frac{J}{\mathrm{mol}}$ Activation energy of reaction 1 Ea1 ${E}_{\mathrm{a2}}$ $120000$ $\frac{J}{\mathrm{mol}}$ Activation energy of reaction 2 Ea2 ${E}_{\mathrm{a3}}$ $10000$ $\frac{J}{\mathrm{mol}}$ Activation energy of reaction 3 Ea3 ${E}_{\mathrm{a4}}$ $10000$ $\frac{J}{\mathrm{mol}}$ Activation energy of reaction 4 Ea4 ${L}_{\mathrm{MH}}$ $1.13$ $\frac{\mathrm{kg}}{{m}^{2}}$ Loading of nickel active material LMH ${L}_{\mathrm{NiOH2}}$ $0.68$ $\frac{\mathrm{kg}}{{m}^{2}}$ Loading of metal hyrdroxide material LNiOH2 ${U}_{\mathrm{1c}}$ $0.527$ $V$ Apparent open-circuit potential of the redox reaction of nickel active material at standard conditions during the whole range charge process U1c ${U}_{\mathrm{1d}}$ $0.427$ $V$ Apparent open-circuit potential of the redox reaction of nickel active material at standard conditions during the whole range discharge process U1d ${U}_{2}$ $0.4011$ $V$ Equilibrium potential of reaction 2 at standard condition U2 ${U}_{3}$ $-0.8279$ $V$ Equilibrium potential of reaction 3 at standard condition U3 ${U}_{4}$ $\mathrm{U2}$ $V$ Equilibrium potential of reaction 4 at standard condition U4 ${a}_{\mathrm{neg}}$ $3·{10}^{5}$ $\frac{{m}^{2}}{{m}^{3}}$ Specific surface area of negative electrode aneg ${a}_{\mathrm{pos}}$ $4·{10}^{5}$ $\frac{{m}^{2}}{{m}^{3}}$ Specific surface area of positive electrode apos ${c}_{H,\mathrm{max}}$ $37000$ $\frac{\mathrm{mol}}{{m}^{3}}$ Maximum concentration of nickel hydroxide in nickel active material cHmax ${c}_{H,\mathrm{ref}}$ $0.5\mathrm{cHmax}$ $\frac{\mathrm{mol}}{{m}^{3}}$ Reference concentration of nickel hydroxide in nickel active material cHref ${c}_{\mathrm{MH},\mathrm{max}}$ $102500$ $\frac{\mathrm{mol}}{{m}^{3}}$ Maximum concentration of hydrogen in metal hydride material cMHmax ${c}_{\mathrm{MH},\mathrm{ref}}$ $0.5\mathrm{cMHmax}$ $\frac{\mathrm{mol}}{{m}^{3}}$ Reference concentration of hydrogen in metal hydride material cMHref ${c}_{e}$ $7000$ $\frac{\mathrm{mol}}{{m}^{3}}$ Concentration of KOH electrolyte ce ${c}_{e,\mathrm{ref}}$ $1000$ $\frac{\mathrm{mol}}{{m}^{3}}$ Reference concentration of KOH electrolyte ceref $\frac{{\mathrm{dU}}_{1}}{\mathrm{dT}}$ $-1.35·{10}^{-3}$ $\frac{V}{K}$ Temperature coefficient of reaction 1, from Wang (2000) Thermal-Electrochemical Modeling of Battery Systems dU1dT $\frac{{\mathrm{dU}}_{2}}{\mathrm{dT}}$ $-1.68·{10}^{-3}$ $\frac{V}{K}$ Temperature coefficient of reaction 2, from Wang (2000) Thermal-Electrochemical Modeling of Battery Systems dU2dT $\frac{{\mathrm{dU}}_{3}}{\mathrm{dT}}$ $-1.55·{10}^{-3}$ $\frac{V}{K}$ Temperature coefficient of reaction 3, from Wang (2000) Thermal-Electrochemical Modeling of Battery Systems dU3dT $\frac{{\mathrm{dU}}_{4}}{\mathrm{dT}}$ $\frac{{\mathrm{dU}}_{2}}{\mathrm{dT}}$ $\frac{V}{K}$ Temperature coefficient of reaction 4, from Wang (2000) Thermal-Electrochemical Modeling of Battery Systems dU4dT ${i}_{\mathrm{01},\mathrm{ref}}$ $5.3$ $\frac{A}{{m}^{2}}$ Exchange current density of reaction 1 at reference reactant concentrations i01ref ${i}_{\mathrm{02},\mathrm{ref}}$ $2·{10}^{-7}$ $\frac{A}{{m}^{2}}$ Exchange current density of reaction 2 at reference reactant concentrations i02ref ${i}_{\mathrm{03},\mathrm{ref}}$ $7.3$ $\frac{A}{{m}^{2}}$ Exchange current density of reaction 3 at reference reactant concentrations i03ref ${i}_{\mathrm{04},\mathrm{ref}}$ $1$ $\frac{A}{{m}^{2}}$ Exchange current density of reaction 4 at reference reactant concentrations i04ref ${l}_{\mathrm{neg}}$ $5.5·{10}^{-4}$ $m$ Thickness of negative electrode lneg ${l}_{\mathrm{pos}}$ $7.0·{10}^{-4}$ $m$ Thickness of positive electrode lpos ${p}_{{\mathrm{O}}_{2},\mathrm{ref}}$ $1.01325$ $\mathrm{bar}$ Reference oxygen pressure in cell pO2ref ${\mathrm{\rho }}_{\mathrm{MH}}$ $7490$ $\frac{\mathrm{kg}}{{m}^{3}}$ Density of metal hydride rhoMH ${\mathrm{\rho }}_{\mathrm{NiOH2}}$ $3400$ $\frac{\mathrm{kg}}{{m}^{3}}$ Density of nickel active material rhoNiOH2 ${V}_{\mathrm{gas}}$ $1·{10}^{-7}$ ${m}^{3}$ Gas volume in cell Vgas ${\mathrm{Vol}}_{\mathrm{cell}}$ $2.355·{10}^{-6}$ ${m}^{3}$ Volume of cell VolCell

References

 [1] Wu, B., Mohammed, M., Brigham, D., Elder, R., and White, R.E., A non-isothermal model of a nickel-metal hydride cell, Journal of Power Sources, 101 (2001) pp. 149-157.
 [2] Dao, T.S. and McPhee, J., Dynamic modeling of electrochemical systems using linear graph theory, Journal of Power Sources, No. 196, pp.10442-10454, 2011.