# Battery Single Particle

**Libraries:**

Simscape /
Battery /
Cells

## Description

The Battery Single Particle block models a battery using a single-particle approach. This implementation considers the ohmic and mass transport overpotentials both in the liquid electrolyte phase and in the solid electrode phase. Additionally, it considers the reaction kinetics and the current collector resistance.

The battery comprises three sections: two electrodes, the anode and cathode, and a porous separator between the electrodes. In this block, the anode refers to the negative electrode during discharge and the cathode refers to the positive electrode during discharge. The block models the ohmic overpotentials of the electrodes and electrolyte, as well as the concentration across the cell cross section from the anode current collector to the cathode current collector, in a one-dimensional framework.

This figure illustrates a representative concentration in the electrolyte during the
discharge. The model comprises the anode (*x*=[0 …
*L*^{-}]), the separator
(*x*=[*L*^{-} …
*L*^{-}+*L*^{sep}])
and the cathode
(*x*=[*L*^{-}+*L*^{sep}…
*L*^{-}+*L*^{sep}+*L*^{+}]).

The block calculates the concentration in the electrodes in representative spherical
particle across the radial dimension *r*. This figure shows the concentration
gradient in the representative particles during a continuous discharge of the battery:

### Species Conservation in Solid Phase

**Note**

The superscripts in these equations refer to the respective electrodes. A
*+* superscript refers to the cathode. A *-*
superscript refers to the anode. A *sep* superscript refers to the
separator. A *±* superscript means that the equation applies to both
anode and cathode. For example,
*c ^{+}_{s}* is the
solid-phase concentration of the cathode and

*c*is the solid-phase concentration of the anode.

^{-}_{s}In solid phase, the single-particle approach models the positive and negative electrodes as a single representative spherical particle.

This equation describes the concentration *c* of the cation in the
negative or positive electrode by using Fick's law. The block uses the radial coordinates
only to calculate the concentration in the electrodes. The diffusion in the spherical
particle drives the mass transfer,

$$\frac{\partial {c}_{s}^{\pm}}{\partial t}\left(r,t\right)=\frac{\partial}{\partial r}\left[{D}_{s}^{\pm}\frac{\partial {c}_{s}^{\pm}}{\partial r}\left(r,t\right)\right],$$

where:

*c*is the solid-phase concentration._{s}*D*is the diffusion coefficient in solid phase._{s}*r*is the radius.*t*is the time.

At the center of the particle, the concentration gradient is equal to 0:

$${D}_{s}^{\pm}\frac{\partial {c}_{s}^{\pm}}{\partial r}\left(0,t\right)=0.$$

This equation calculates the ion concentration gradient at the surface of the particle:

$${D}_{s}^{\pm}\frac{\partial {c}_{s}^{\pm}}{\partial r}\left({R}_{s}^{\pm},t\right)=\mp \frac{{J}_{s}^{\pm}}{{a}_{s}^{\pm}\text{}F}.$$

In this equation,

*J*is the molar flux:$${J}^{\pm}(t)=\frac{I(t)}{A\text{}{L}^{\pm}},$$

where:

*I*is the current applied to the cell.*A*is the total area of the current collector.*L*is the length of the respective electrode.

*a*is the active surface area per electrode unit volume:$${a}^{\pm}=\frac{3{\epsilon}^{\pm}}{{R}^{\pm}},$$

where:

*ε*is the active material fraction of the electrode.*R*is the total radius of the active particle.

*F*is the Faraday's constant.

To solve the differential equation, the Battery Single
Particle block discretizes the particle with the radius
*R* into *n* shells. Each shell has a radial distance
equal to $$\delta r=\frac{R}{(n-1)}$$ from the adjacent spheres.

For the *i*th sphere, this equation calculates the rate of change of
concentration, *δc/δt*:

$${\dot{c}}_{{s}_{i}}=\frac{{D}_{s}}{\delta {r}^{2}}\left\{\left(\frac{i-1}{i}\right){c}_{{s}_{i-1}}-2{c}_{{s}_{i}}+\left(\frac{i+1}{i}\right){c}_{{s}_{i+1}}\right\}.$$

For the innermost shell in the particle, the block implements this boundary condition:

$${\dot{c}}_{1}=\frac{2{D}_{s}}{\delta {r}^{2}}\left\{{c}_{{s}_{2}}-{c}_{{s}_{1}}\right\}.$$

To implement the boundary condition at the surface of the particle, the block adds an additional "ghost" node around the surface. The concentration of this ghost node is not calculated as it does not physically exist. The addition of this ghost node allows the block to calculate the boundary condition between the outermost shell in the particle and the additional non-existent shell around it by using the Neumann boundary condition. This equation describes the discretized result at the surface of the particle:

$${\dot{c}}_{{s}_{end}}=\frac{2{D}_{s}}{\delta {r}^{2}}\left\{{c}_{{s}_{end-1}}-{c}_{{s}_{end}}\right\}-2\frac{n}{n-1}\frac{J}{F\text{}A\text{}\delta r}.$$

### Mass Transport Overpotential in Solid Phase

The open-circuit potential depends on the concentration. To calculate the mass transport overpotential at the electrodes, the Battery Single Particle block subtracts the open-circuit potential of the average relative concentration in the particle from the open-circuit potential of the average relative concentration at the surface,

$${\eta}^{\pm}{}_{\text{diffusion},s}={\text{ocp}}^{\pm}\left({c}_{s,\text{surface},\text{relative}}^{\pm}\right)-{\text{ocp}}^{\pm}\left({\overline{c}}_{s,\text{relative}}^{\pm}\right),$$

where:

*η*_{diffusion}_{,s}is the solid-phase mass transport overpotential.*ocp(c*,_{s}_{surface,relative}*)*is the open-circuit potential for the concentration at the surface of the particle.*ocp(c*_{s}_{relative}*)*is the open-circuit potential for the average concentration of the particle.

The block uses the same equation to calculate the mass transport overpotential in solid phase for both the anode and the cathode.

### Ohmic Overpotential in Solid Phase

To calculate the ohmic overpotential in the solid phase, the Battery Single Particle block linearly approximates the current across the electrodes. The block also approximates the current at the current collector to a value equal to the electric current applied to the cell. The current at the interface between the current separator and the electrode is zero. This equation defines the ohmic overpotential in the solid phase,

$${\eta}^{\pm}{}_{ohmic,s}=\frac{{I}_{batt}}{2A}\ast \frac{{L}^{\pm}}{{\kappa}^{\pm}},$$

where:

*η*is the solid-phase ohmic overpotential._{ohmic,s}*I*is the cell cross section._{batt}/A*L*is the length of the respective electrode and depends on the thickness of the anode or cathode.*κ*is the conductivity. The conductivity depends on the temperature of the cell.*κ*is equal to the**Anode conductivity**parameter when the block calculates the ohmic overpotential of the anode and is equal to the**Cathode conductivity**parameter when the block calculates the ohmic overpotential of the cathode.

The block uses the same equation to calculate the ohmic overpotential in solid phase for both the anode and the cathode.

### Species Conservation in Liquid Phase

This equation describes the concentration in the electrolyte at both electrodes and at the separator. To calculate the concentration across the separator, the model considers the diffusive flow induced by concentration gradient:

$$\frac{\partial {c}_{\epsilon}^{\pm}}{\partial t}\left(x,t\right)=\frac{\partial}{\partial x}\left[{D}_{\epsilon}\frac{\partial {c}_{\epsilon}^{\pm}}{\partial x}\left(x,t\right)\right],$$

where:

*c*is the concentration in the electrolyte._{ε}*D*is the diffusion coefficient in liquid phase._{ε}*x*is the location in the thickness of the battery, from the anode current collector to the cathode current collector.*t*is the time.

At the positive and negative electrodes, the block considers both the diffusive flow and the cation flux from the solid electrode into the electrolyte:

$${\in}_{\epsilon}^{\pm}\frac{\partial {c}_{\epsilon}^{\pm}}{\partial t}\left(x,t\right)=\frac{\partial}{\partial x}\left[{D}_{\epsilon}^{eff}\frac{\partial {c}_{\epsilon}^{\pm}}{\partial x}\left(x,t\right)\pm \frac{\left(1-{t}_{+}^{0}\right)J}{F}\right],$$

where:

*∈*is the volume fraction of the electrolyte.*D*is the diffusion coefficient in liquid phase that considers the porosity of the material. The diffusivity of the liquid electrolyte depends on the properties of the surrounded solid electrode material. The electrode comprises multiple components, such as the active material and the filler, which form a characteristic porous material.^{eff}_{ε}*J*is the molar flux.*t*is the transference number of the cation.^{+}*F*is the Faraday's constant.

As the electrolyte is a continuous fluid, the cation concentration at the border between the negative and positive electrode and the separator must be equal. For the concentration in the electrolyte at the anode-separator and cathode-separator interfaces, the block must define the boundary conditions between the three sections of the battery. The block represents both electrodes and the separator as cuboids.

This block considers the electrolyte as a continuous medium across the electrodes and the separator. As the concentrations on both sides of the interface must be equal, a continuity boundary condition exists for the interface between the electrodes and the separator.

This equation describes the continuity boundary condition for the concentration at the interface between the anode and the separator,

$${c}_{\epsilon}^{-}\left({L}^{-},t\right)={c}_{\epsilon}^{sep}\left({L}^{-},t\right),$$

where:

$${c}_{\epsilon}^{-}\left({L}^{-},t\right)$$ is the concentration in the anode at the border between the anode and the separator.

$${c}_{\epsilon}^{sep}\left({L}^{-},t\right)$$ is the concentration in the separator at the border between the anode and the separator.

This equation describes the continuity boundary condition for the concentration at the interface between separator and cathode,

$${c}_{\epsilon}^{sep}\left({L}^{-}+{L}^{sep},t\right)={c}_{\epsilon}^{+}\left({L}^{-}+{L}^{sep},t\right),$$

where:

$${c}_{\epsilon}^{sep}\left({L}^{-}+{L}^{sep},t\right)$$ is the concentration in the separator at the border between the separator and the cathode.

$${c}_{\epsilon}^{+}\left({L}^{-}+{L}^{sep},t\right)$$ is the concentration in the cathode at the border between the separator and the cathode.

The block also applies a flux boundary condition to the interfaces between the electrodes and the separator. There, the flux at both sides of the interface must be equal,

$$\begin{array}{l}{D}_{eff}^{-}\frac{\partial {c}_{\epsilon}^{-}}{\partial x}\left({L}^{-},t\right)={D}_{eff}^{sep}\frac{\partial {c}_{\epsilon}^{sep}}{\partial x}\left({L}^{-},t\right)\\ {D}_{eff}^{+}\frac{\partial {c}_{\epsilon}^{+}}{\partial x}\left({L}^{-}+{L}^{sep},t\right)={D}_{eff}^{sep}\frac{\partial {c}_{\epsilon}^{sep}}{\partial x}\left({L}^{-}+{L}^{sep},t\right)\end{array}$$

where:

$${D}_{eff}^{-}$$ is the diffusion coefficient of the anode.

$${D}_{eff}^{+}$$ is the diffusion coefficient of the cathode.

$${D}_{eff}^{sep}$$ is the diffusion coefficient of the separator.

$$\frac{\partial {c}_{\epsilon}^{-}}{\partial x}\left({L}^{-},t\right)$$ is the concentration gradient of the anode at the border between the anode and the separator.

$$\frac{\partial {c}_{\epsilon}^{sep}}{\partial x}\left({L}^{-},t\right)$$ is the concentration gradient of the separator at the border between the anode and the separator.

$$\frac{\partial {c}_{\epsilon}^{+}}{\partial x}\left({L}^{-}+{L}^{sep},t\right)$$ is the concentration gradient of the cathode at the border between the cathode and the separator.

$$\frac{\partial {c}_{\epsilon}^{sep}}{\partial x}\left({L}^{-}+{L}^{sep},t\right)$$ is the concentration gradient of the separator at the border between the cathode and the separator.

This equation specifies the concentration at the boundaries between the electrodes and the current collectors. The flux is proportional to the flux at the current collector, which is equal to zero since the model does not store any ions there. Hence the resulting flux at the interface is zero,

$$\frac{\partial {c}_{\epsilon}^{-}}{\partial x}\left(0,t\right)=\frac{\partial {c}_{\epsilon}^{+}}{\partial x}\left({L}^{-}+{L}^{sep}+{L}^{+},t\right)=0,$$

where:

$$\frac{\partial {c}_{\epsilon}^{-}}{\partial x}\left(0,t\right)$$ is the concentration gradient in the anode at the border between the anode and the leftmost current collector.

$$\frac{\partial {c}_{\epsilon}^{+}}{\partial x}\left({L}^{-}+{L}^{sep}+{L}^{+},t\right)$$ is the concentration gradient in the cathode at the border between the cathode and the rightmost current collector.

Similar to the solid phase, to solve the differential equation, the block divides the
electrolyte into *n* sections of equal size. This equation expresses the
concentration in the *i*th section with a distance *δx*
between sections, and it is valid for all [1,M_{s}-1] sections:

$$\frac{\partial {c}_{\epsilon}}{\partial t}\left(x,t\right)={D}_{\epsilon ,eff}^{\pm}\frac{{c}_{i+1}+{c}_{i-1}-2{c}_{i}}{\partial {x}^{2}}+\frac{\left(1-{t}_{+}^{0}\right)J}{F}.$$

The block discretizes the separator using this equation:

$$\frac{\partial {c}_{\epsilon}}{\partial t}\left(x,t\right)={D}_{\epsilon ,eff}^{sep}\frac{c{}_{i+1}+{c}_{i-1}-2{c}_{i}}{\delta {x}^{2}}.$$

To calculate the concentrations at the interfaces between the electrodes and the separator, the block applies all the boundary conditions. For example, for the interface between the anode and the separator, the block applies this equation:

$${c}_{\epsilon ,i=1}^{sep}=\frac{{\epsilon}_{\epsilon}^{-}}{{\epsilon}_{\epsilon}^{sep}}\frac{{c}_{\epsilon ,i=0}-{c}_{\epsilon ,i=-1}^{sep}}{\partial x}\partial x-{c}_{\epsilon ,i=0}.$$

### Mass Transport Overpotential in Liquid Phase

Using the concentrations at the interfaces between the current Collector and the anode and at the interfaces between the cathode and the current collector, the block calculates the mass transport overpotential in the electrolyte using this equation,

$${\eta}_{diffusion,\epsilon}=\frac{2RT}{F}\left(1-{t}_{\epsilon}^{0}\right)\mathrm{ln}\frac{{c}_{\epsilon}\left({L}^{-}+{L}^{sep}+{L}^{+},t\right)}{{c}_{\epsilon}\left(0,t\right)},$$

where:

*R*is the universal gas constant.*T*is the temperature.*F*is the Faraday's constant.

### Ohmic Overpotential in Liquid Phase

To calculate the ohmic overpotential in the liquid phase, the block linearly
approximates the ionic current in each section of the battery. For the electrodes, the ionic
current at the interface to the current collector is zero. At the interface to the
separator, the ionic current is equal to the electric current of the battery
*I _{batt}*. Across the separator, the block
approximates the ionic current as constant and equal to the electric current applied to the
battery. Considering these approximations, the block calculates the ohmic overpotential
using this equation,

$${\eta}_{ohmic,\epsilon}=-\frac{{I}_{batt}}{2A}\ast \left(\frac{{L}^{+}}{{\kappa}_{eff}{}^{+}}+2\frac{{L}^{sep}}{{\kappa}_{eff}{}^{sep}}+\frac{{L}^{-}}{{\kappa}_{eff}{}^{-}}\right),$$

where *κ _{eff}* is the effective
conductivity that the block calculates by using the Bruggeman coefficient. For more
information about effective parameters, see the Effective Electrolyte Properties section.

### Charge Transfer Overpotential

To model the charge transfer overpotential, this block uses the Butler-Volmer equation.
The Butler-Volmer equation describes the relationship between the current density,
*j*, and the overpotential, *η*, which is the difference
between the actual electrode potential and the thermodynamic equilibrium potential. Under
the assumption of equal charge transfer coefficient, *α*, the Butler-Volmer
equation is

$${J}^{\pm}\left(t\right)={j}_{0,k}\left(t\right)\left[\mathrm{exp}\left(\frac{\alpha {n}_{\epsilon}F}{RT}{\eta}^{\pm}\left(t\right)\right)-\mathrm{exp}\left(-\frac{\left(1-\alpha \right){n}_{\epsilon}F}{RT}{\eta}^{\pm}\left(t\right)\right)\right],$$

where:

*α*is the charge transfer coefficient for the oxidation and reduction.*j*is the exchange current density._{0}

Solving the equation for the electrode overpotential results in these equations:

$$\begin{array}{l}{\eta}_{\text{kinetic,s}}=\frac{RT}{\alpha F}\mathrm{ln}\left({\xi}^{\pm}+\sqrt{{\left({\xi}^{\pm}\right)}^{2}+1}\right)\\ {\xi}^{\pm}=\frac{{j}^{\pm}}{2{a}^{\pm}{i}_{0}^{\pm}}\end{array}$$

*i ^{±}_{0}* is the exchange
current density in the anode and in the cathode and is equal to

$${i}_{0}^{\pm}={k}^{\pm}{\left[{\overline{c}}_{\epsilon}^{\pm}\left({c}_{s,\mathrm{max}}^{\pm}-{c}_{s,surf}^{\pm}\right){c}_{s,surf}^{\pm}\right]}^{\alpha},$$

where:

*k*is the charge transfer rate constant and is equal to the value of the**Charge transfer rate constant for Anode**parameter for the anode and to the value of the**Charge transfer rate constant for Cathode**parameter for the cathode.$${\overline{c}}_{\epsilon}^{\pm}$$ is the average electrolyte concentration.

*c*is the maximum electrode concentration._{s,max}*c*is the electrode surface concentration._{s,surf}

To calculate the kinetic overpotential of the complete cell, the block subtracts the kinetic overpotential at the anode from the kinetic overpotential at the cathode:

$${\eta}_{\text{kinetic,s}}={\eta}^{+}{}_{\text{kinetic,s}}-{\eta}^{-}{}_{\text{kinetic,s}}.$$

### Current Collector Resistance

This block models the current collector resistance as a single resistance. You can set
the current collector resistance by specifying the **Current collector
resistance** parameter.

### Cell Voltage

To model the cell voltage, this block considers the potentials at the surfaces of each electrode, the overpotentials, and the voltage loss due to the current collector resistance and uses this equation,

$$V\left(t\right)={\text{ocp}}^{+}({c}_{\text{surface,relative}}^{+})-{\text{ocp}}^{-}({c}_{\text{surface,relative}}^{-})+{\eta}_{\text{diffusion},\epsilon}+{\eta}_{ohmic,\epsilon}+{\eta}_{\text{kinetic},s}+{\eta}_{ohmic}^{-}+{\eta}_{ohmic}^{+}+{I}_{\text{batt}}{R}_{\text{CurrentCollector}},$$

where:

*ocp*is the open-circuit potential for the concentration at the surface of the cathode particle.^{+}(c^{+}_{surface,relative})*ocp*is the open-circuit potential for the concentration at the surface of the anode particle.^{-}(c^{-}_{surface,relative})*η*_{diffusion,ε}is the mass transport overpotential in the electrolyte.*η*_{ohmic,ε}is the ohmic overpotential in the electrolyte.*η*_{kinetic,s}is the charge transfer overpotential in the electrodes.*η*^{-}_{ohmic}is the ohmic overpotential in the anode.*η*^{+}_{ohmic}is the ohmic overpotential in the cathode.*I*_{batt}is the battery current.*R*_{CurrentCollector}is the resistance of the current collector.

You can parameterize the open-circuit potential as table data with the
relative concentration as the breakpoints by specifying the **Anode open-circuit
potential**, **Cathode open-circuit potential**, and
**Normalized stoichiometry breakpoints** parameters.

To calculate the relative concentration, the block considers the maximum concentration
and the maximum and minimum stoichiometry of each electrode. The **Anode maximum ion
concentration** and the **Cathode maximum ion concentration**
parameters represent the theoretically possible maximum concentration of each electrode. To
obtain the achievable maximum and minimum concentrations, the block multiplies the values of
these two parameters with the value of the **Anode maximum stoichiometry**,
**Anode minimum stoichiometry**, **Cathode maximum
stoichiometry**, and **Cathode minimum stoichiometry**
parameters, respectively. Then, the block calculates the relative concentration by using
this equation,

$${c}_{\text{s,relative}}=\frac{\frac{{c}_{s}}{{c}_{s,\mathrm{max}}}-{{\rm N}}_{\mathrm{min}}}{{{\rm N}}_{\mathrm{max}}-{{\rm N}}_{\mathrm{min}}},$$

where:

*c*,_{s}_{max}is the maximum concentration.*N*_{max}is the maximum stoichiometry.*N*_{min}is the minimum stoichiometry.

### Effective Electrolyte Properties

These block parameters depend on the microstructure of the porous electrodes:

**Diffusion coefficient of electrolyte**— Diffusion coefficient of electrolyte that influences the mass transport in the electrolyte.**Electrolyte conductivity**— Conductivity of the electrolyte.

To model this dependency, this block uses the Bruggeman correlation,

$${\text{Parameter}}_{\text{effective}}={\text{Parameter}}_{\text{block}}\ast {\phi}_{\epsilon}{}^{\alpha},$$

where:

*φ*is the volume fraction of the electrolyte and is equal to the value of the_{ε}**Volume fraction of electrolyte in anode**,**Volume fraction of electrolyte in separator**, and**Volume fraction of electrolyte in cathode**parameters, accordingly.*α*is the Bruggeman exponent and is equal to the value of the**Anode Bruggeman exponent**,**Separator Bruggeman exponent**, and**Cathode Bruggeman exponent**parameters, accordingly.

### Thermal

Certain block parameters depend on the temperature of the cell. The block considers the temperature constant across the cell. These are the temperature-dependent parameters:

**Diffusion coefficient of anode active material**and**Diffusion coefficient of cathode active material**— Diffusion coefficients of electrodes that influence the mass transport in the electrodes.**Diffusion coefficient of electrolyte**— Diffusion coefficient of electrolyte that influences the mass transport in the electrolyte.**Electrolyte conductivity**— Conductivity of the electrolyte.**Anode conductivity**and**Cathode conductivity**— Conductivity of the electrodes.**Charge transfer rate constant for Anode**and**Charge transfer rate constant for Cathode**— Charge transfer rate constants of the electrodes.

To calculate the temperature-adjusted values of these parameters, the block uses the Arrhenius equation,

$${\text{Parameter}}_{\text{T-adjusted}}={\text{Parameter}}_{\text{block}}\ast {e}^{\frac{{E}_{a}}{R}\left(\frac{1}{{T}_{ref}}-\frac{1}{T}\right)},$$

where:

*Parameter*is the value of the temperature-dependent parameters as you specify them inside the block._{block}*E*is the activation energy and is equal to the value of the activation energy parameters in the_{a}**Thermal**settings of the**Property Inspector**.*T*is the value of the_{ref}**Arrhenius reference temperature**parameter.*T*is the battery temperature.

### Heat Generation

This block models the battery as a lumped thermal mass. The single particle model calculates the irreversible heat generation that the overpotentials cause in the battery by using this equation:

$$Q={I}_{batt}\left({\eta}_{\text{diffusion},\epsilon}+{\eta}_{ohmic,\epsilon}+{\eta}_{ohmic,s}^{-}+{\eta}_{ohmic,s}^{+}+{I}_{\text{batt}}{R}_{\text{CurrentCollector}}+{\eta}_{\text{kinetic},s}+{\eta}^{+}{}_{\text{diffusion}}+{\eta}^{-}{}_{\text{diffusion}}\right).$$

### Public Variables (Visible with Probe Block)

The Battery Single Particle block comprises these public variables that you can probe using the Probe block. The units are the default values.

`anodeModel.averageStoichiometry`

— Average stoichiometry in the anode.`anodeModel.massTransportOverpotential`

— Mass transport overpotential, in volt.`anodeModel.normalizedAverageStoichiometry`

— Average stoichiometry normalized to the minimum and maximum values.`anodeModel.normalizedSurfaceStoichiometry`

— Surface stoichiometry normalized to the minimum and maximum values.`anodeModel.ohmicOverpotential`

— Ohmic overpotential of the anode, in volt.`anodeModel.shellConcentration`

— Concentration of the modeled shells, in mol/m^3. The number of shells is equal to the value of the**Anode Shell Count**parameter.`anodeModel.shellStoichiometry`

— Stoichiometry of the modeled shells. The number of shells is equal to the**Anode Shell Count**parameter.`anodeModel.surfaceConcentration`

— Concentration at the surface of the particle, in mol/m^3.`anodeModel.surfacePotential`

— Potential at the surface of the particle, in volt.`anodeModel.temperatureAdjustedConductivity`

— Conductivity adjusted to the battery temperature, in S/m.`anodeModel.temperatureAdjustedDiffusionCoefficient`

— Diffusion coefficient adjusted to the battery temperature, in m^2/s.`averageElectrolyteConcentration`

— Average concentration in the particle, in mol/m^3.`batteryCurrent`

— Total current measured through the battery terminals, in ampere.`batteryTemperature`

— Battery average temperature that the block uses for the table look-up of resistances and open-circuit voltage. If you set the**Thermal model**parameter to`Constant temperature`

, then the`batteryTemperature`

variable is equal to the specified temperature value. If you set the**Thermal model**parameter to`Lumped thermal mass`

, then the`batteryTemperature`

variable is a differential state that varies during the simulation.`batteryVoltage`

— Battery terminal voltage, or the voltage difference between the positive and the negative terminals, in volt.`cathodeModel.averageStoichiometry`

— Average stoichiometry in the cathode.`cathodeModel.massTransportOverpotential`

— Mass transport overpotential, in volt.`cathodeModel.normalizedAverageStoichiometry`

— Average stoichiometry normalized to the minimum and maximum values.`cathodeModel.normalizedSurfaceStoichiometry`

— Surface stoichiometry normalized to the minimum and maximum values.`cathodeModel.ohmicOverpotential`

— Ohmic overpotential of the cathode, in volt.`cathodeModel.shellConcentration`

— Concentration of the modeled shells, in mol/m^3. The number of shells is equal to the value of the**Anode Shell Count**parameter.`cathodeModel.shellStoichiometry`

— Stoichiometry of the modeled shells. The number of shells is equal to the**Anode Shell Count**parameter.`cathodeModel.surfaceConcentration`

— Concentration at the surface of the particle, in mol/m^3.`cathodeModel.surfacePotential`

— Potential at the surface of the particle, in volt.`cathodeModel.temperatureAdjustedConductivity`

— Conductivity adjusted to the battery temperature, in S/m.`cathodeModel.temperatureAdjustedDiffusionCoefficient`

— Diffusion coefficient adjusted to the battery temperature, in m^2/s.`electrolyteModel.averageConcentration`

— Average concentration in the electrolyte across the whole cell, in mol/m^3.`electrolyteModel.averageConcentrationAnode`

— Average concentration in the electrolyte inside the anode, in mol/m^3.`electrolyteModel.averageConcentrationCathode`

— Average concentration in the electrolyte inside the cathode, in mol/m^3.`electrolyteModel.averageConcentrationSeparator`

— Average concentration in the electrolyte inside the separator, in mol/m^3.`electrolyteModel.concentrationAnode`

— Concentration of the modeled layers of the electrolyte in the anode, in mol/m^3. The number of elements is equal to the**Electrolyte layer count of anode**parameter.`electrolyteModel.concentrationCathode`

— Concentration of the modeled layers of the electrolyte in the cathode, in mol/m^3. The number of elements is equal to the**Electrolyte layer count of cathode**parameter.`electrolyteModel.concentrationSeparator`

— Concentration of the modeled layers of the electrolyte in the separator, in mol/m^3. The number of elements is equal to the**Electrolyte layer count of electrolyte**parameter.`electrolyteModel.currentDensityAnode`

— Current density in the anode, in A/m^3.`electrolyteModel.currentDensityCathode`

— Current density in the cathode, in A/m^3.`electrolyteModel.diffusionCoefficientAnode`

— Temperature-adjusted effective diffusion coefficient of the electrolyte in the anode, in m^s/s.`electrolyteModel.diffusionCoefficientCathode`

— Temperature-adjusted effective diffusion coefficient of the electrolyte in the cathode, in m^s/s.`electrolyteModel.diffusionCoefficientSeparator`

— Temperature-adjusted effective diffusion coefficient of the electrolyte in the separator, in m^s/s.`electrolyteModel.conductivityAnode`

— Temperature-adjusted effective conductivity of the electrolyte in the anode, in S/m.`electrolyteModel.conductivityCathode`

— Temperature-adjusted effective conductivity of the electrolyte in the cathode, in S/m.`electrolyteModel.effectiveConductivitySeparator`

— Temperature-adjusted effective conductivity of the electrolyte in the separator, in S/m.`electrolyteModel.massTransportOverpotential`

— Mass transport overpotential of the electrolyte, in volt.`electrolyteModel.ohmicOverpotential`

— Ohmic overpotential of the electrolyte, in volt.`electrolyteModel.temperatureAdjustedConductivity`

— Temperature-adjusted conductivity of the electrolyte, in S/m.`electrolyteModel.temperatureAdjustedDiffusionCoefficient`

— Temperature adjusted diffusion coefficient of the electrolyte, in m^s/s.`heatGenerationRate`

— Total battery heat generation rate, in watt. The block calculates the heat generation rate by adding up all resistive losses, reversible heating contribution, and the exothermic reaction heat if you enabled an exothermic fault.`power_dissipated`

— Resistive heat generation rate or dissipated power, in watt.`reactionKineticsModel.chargeTransferOverpotential`

— Charge transfer overpotential of the battery, in volt.`reactionKineticsModel.exchangeCurrentDensityAnode`

— Exchange current density in the anode, in C/(m^2*s).`reactionKineticsModel.exchangeCurrentDensityCathode`

— Exchange current density in the cathode, in C/(m^2*s).`reactionKineticsModel.temperatureAdjustedChargeTransferRateAnode`

— Temperature-adjusted charge transfer rate constant for the anode, in m^(5/2)/(mol^(1/2) * s).`reactionKineticsModel.temperatureAdjustedChargeTransferRateCathode`

— Temperature-adjusted charge transfer rate constant for the cathode, in m^(5/2)/(mol^(1/2) * s).`stateOfCharge`

— Battery state of charge obtained from coulomb counting.`thermalModel.batteryTemperature`

— Temperature of the battery, in K.`thermalModel.cellTemperature`

— Temperature of the cell, as an output.`thermalModel.heatDissipationRate`

— Heat dissipation rate of the battery, in watt.`thermalModel.heatGeneration`

— Heat that the battery generates, in watt.`thermalModel.thermalMass`

— Thermal mass of the battery, in J/K.

## Ports

### Conserving

## Parameters

## References

[1] Prada, E., et al.
*"Simplified Electrochemical and Thermal Model of
LiFePO _{4-}-Graphite Li-Ion Batteries for Fast Charge
Applications"*. Journal of The Electrochemical Society, vol. 159, no. 9, 2012, pp.
A1508–19. DOI.org (Crossref), https://doi.org/10.1149/2.064209jes.

## Extended Capabilities

## Version History

**Introduced in R2024a**