# Battery Single Particle

**Libraries:**

Simscape /
Battery /
Cells

## Description

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

The battery comprises 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
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
particles 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

When the block is in solid phase, the single-particle approach models the positive and negative electrodes as a single representative spherical particle.

**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}This equation uses Fick's law to describe the concentration, *c*, of
the cation in the negative or positive electrode. 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, *F* is Faraday's constant, and
*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.

Additionally, *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.

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 $$\delta r=\frac{R}{n}$$ 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 node around the surface. The block does not calculate the concentration of this node because it does not physically exist. The block uses this node to calculate the boundary condition between the outermost shell in the particle and the non-existent shell around it by using the Neumann boundary condition. This equation describes the discretized result at the surface of the particle:

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

### Mass Transport Overpotential in Solid Phase

When the block is 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 for both the anode and the cathode.

### Ohmic Overpotential in Solid Phase

To calculate the ohmic overpotential when the block is in the solid phase, the Battery Single Particle block linearly approximates the current across the electrodes and 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

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

Because the electrolyte is a continuous fluid, the cation concentration at the border between the negative and positive electrodes 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. Because 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. 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 because the block 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, the block solves the differential equation by dividing 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 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 the 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 the 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

When the block is in the liquid phase, it uses the concentrations at the interfaces between the current collector and the anode and at the interfaces between the cathode and the current collector to calculate the mass transport overpotential in the electrolyte using the 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 Faraday's constant.

### Ohmic Overpotential in Liquid Phase

When the block is in the liquid phase, it calculates the ohmic overpotential by linearly
approximating 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. The block calculates the ohmic overpotential using the 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. 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 by using the 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 by using 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 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 the 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

Set the values of these parameters based on the microstructure of the porous electrodes you want to model:

**Diffusion coefficient of electrolyte**— Set this parameter to the value of the diffusion coefficient of the electrolyte that influences the mass transport in the electrolyte.**Electrolyte conductivity**— Set this parameter to the value of the 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. This value 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. This value is equal to the value of the**Anode Bruggeman exponent**,**Separator Bruggeman exponent**, and**Cathode Bruggeman exponent**parameters, accordingly.

### Thermal

The block considers the temperature constant across the cell. These block parameters depend on the temperature of the cell:

**Diffusion coefficient of anode active material**and**Diffusion coefficient of cathode active material**— These parameters are the diffusion coefficients of electrodes that influence the mass transport in the electrodes.**Diffusion coefficient of electrolyte**— This parameter is the diffusion coefficient of electrolyte that influences the mass transport in the electrolyte.**Electrolyte conductivity**— This parameter is the conductivity of the electrolyte.**Anode conductivity**and**Cathode conductivity**— These parameters are the conductivity of the electrodes.**Charge transfer rate constant for Anode**and**Charge transfer rate constant for Cathode**— These parameters are the 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._{block}*E*is the activation energy and is equal to the value of the activation energy parameters in the_{a}**Thermal**settings.*T*is the value of the_{ref}**Arrhenius reference temperature**parameter.*T*is the battery temperature.

### Heat Generation of Battery

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

You can use the Probe block to access these variables in the Battery Single Particle block. The units are the default values.

`anodeModel.averageStoichiometry`

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

— Mass transport overpotential, in volts.`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 volts.`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 volts.`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 amperes.`batteryTemperature`

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

, the`batteryTemperature`

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

, 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 volts.`cathodeModel.averageStoichiometry`

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

— Mass transport overpotential, in volts.`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 volts.`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 volts.`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 volts.`electrolyteModel.ohmicOverpotential`

— Ohmic overpotential of the electrolyte, in volts.`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 watts. The block calculates the heat generation rate by adding the resistive losses and the reversible heating contributions.`power_dissipated`

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

— Charge transfer overpotential of the battery, in volts.`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 output by the cell.`thermalModel.heatDissipationRate`

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

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

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

## Examples

## Ports

### Conserving

## Parameters

## References

[1] Prada, E., D. D. Domenico, Y.
Creff, J. Bernard, V. Sauvant-Moynot, and F. Huet. “Simplified Electrochemical and Thermal
Model of LiFePO_{4}-Graphite Li-Ion Batteries for Fast Charge
Applications.” *Journal of The Electrochemical Society* 159, no. 9
(August 2012): A1508–A1519. https://doi.org/10.1149/2.064209jes.

[2] Kemper, P. and D. Kum. “Extended
Single Particle Model of Li-Ion Batteries Towards High Current Applications”. In
*2013 IEEE Vehicle Power and Propulsion Conference (VPPC)*, 1–6, 2013.
https://doi.org/10.1109/VPPC.2013.6671682.

[3] Weaver, T., A. Allam, and S.
Onori. “A Novel Lithium-Ion Battery Pack Modeling Framework - Series-Connected Case Study.” In
*2020 American Control Conference (ACC)*, 365–372. Denver, CO, USA:
IEEE, 2020. https://doi.org/10.23919/ACC45564.2020.9147546.

## Extended Capabilities

## Version History

**Introduced in R2024a**