> ## Documentation Index
> Fetch the complete documentation index at: https://docs.ionworks.com/llms.txt
> Use this file to discover all available pages before exploring further.

# Thermal Calculations

> Model Arrhenius temperature dependence, heat capacity, and heat generation for lithium-ion battery thermal simulations.

Temperature affects nearly every aspect of battery behavior: reaction kinetics, transport properties, degradation rates, and safety. This guide covers both thermal property calculations and temperature-dependent parameter modeling.

## Why Thermal Modeling Matters

Temperature influences batteries through multiple mechanisms:

* **Kinetics**: Reaction rates increase exponentially with temperature (Arrhenius)
* **Transport**: Diffusivity and ionic conductivity are strongly temperature-dependent
* **Degradation**: Higher temperatures accelerate aging mechanisms
* **Safety**: Thermal runaway is the primary safety concern for lithium-ion batteries

Understanding and modeling these effects is essential for accurate simulations and safe designs.

## Arrhenius Temperature Dependence

The Arrhenius equation describes how rate-limited parameters vary with temperature:

$$
k(T) = A \exp\left(-\frac{E_a}{RT}\right)
$$

where:

* $A$ is the pre-exponential factor
* $E_a$ is the activation energy \[J/mol]
* $R = 8.314$ J/(mol·K) is the gas constant
* $T$ is absolute temperature \[K]

### Physical Interpretation

The activation energy $E_a$ represents the energy barrier for the process:

* **Diffusion**: Energy barrier for ions hopping between sites
* **Reaction kinetics**: Energy barrier for electrochemical reactions
* **Conductivity**: Energy for ion transport through the material

Higher activation energies mean stronger temperature sensitivity.

### Reference Temperature Formulation

A more practical form uses a reference temperature:

$$
k(T) = k_{\text{ref}} \exp\left[-\frac{E_a}{R}\left(\frac{1}{T} - \frac{1}{T_{\text{ref}}}\right)\right]
$$

where $k_{\text{ref}} = k(T_{\text{ref}})$ is the value at the reference temperature (typically 298.15 K).

<Tip>
  Using a reference temperature makes parameters more intuitive—$k_{\text{ref}}$ is the value at room temperature rather than an abstract pre-exponential factor.
</Tip>

### Fitting Arrhenius Parameters

Taking the logarithm linearizes the relationship:

$$
\ln k = \ln A - \frac{E_a}{R} \cdot \frac{1}{T}
$$

Plotting $\ln k$ vs $1/T$ gives a straight line with slope $-E_a/R$ — the basis of the `ArrheniusLogLinear` calculation, which fits $(k_{\text{ref}}, E_a)$ from a table of $(T, k)$ measurements.

<Note>
  To run an Arrhenius fit or specific-heat calculation, see [Pipelines → Calculations → Thermal](/pipelines/calculations/thermal).
</Note>

### Typical Activation Energies

| Parameter                        | Typical $E_a$ Range |
| -------------------------------- | ------------------- |
| Solid-state diffusion (graphite) | 20-40 kJ/mol        |
| Solid-state diffusion (NMC)      | 30-60 kJ/mol        |
| Electrolyte conductivity         | 10-20 kJ/mol        |
| Exchange current density         | 20-50 kJ/mol        |

### When Arrhenius Doesn't Apply

The Arrhenius model assumes a single mechanism across all temperatures. It may fail when:

* Phase transitions change the mechanism
* Multiple processes compete at different temperatures
* Non-thermal effects (concentration, stress) also matter

For non-Arrhenius behavior, use [piecewise interpolation](/guide/calculations/piecewise) with temperature as the independent variable.

## Thermal Properties

### Heat Generation

Batteries generate heat through several mechanisms:

$$
\dot{Q} = \dot{Q}_{\text{reversible}} + \dot{Q}_{\text{irreversible}}
$$

| Component                   | Formula                                         | Description                      |
| --------------------------- | ----------------------------------------------- | -------------------------------- |
| Irreversible (Joule)        | $I^2 R$                                         | Ohmic heating from current flow  |
| Irreversible (polarization) | $I \cdot \eta$                                  | Overpotential losses             |
| Reversible (entropic)       | $I \cdot T \cdot \frac{\partial U}{\partial T}$ | Entropy change during lithiation |

At high rates, irreversible heating dominates. At low rates, reversible heating can be significant and may cause local cooling during discharge.

### Heat Capacity

The specific heat capacity $c_p$ determines temperature rise for a given heat input:

$$
\Delta T = \frac{Q}{m \cdot c_p}
$$

A `SpecificHeatCapacity` calculation converts between cell heat capacity \[J/K] and specific heat capacity \[J/(kg·K)] given the cell mass.

### Typical Thermal Property Values

| Component          | Heat Capacity (J/(kg·K)) | Thermal Conductivity (W/(m·K)) |
| ------------------ | ------------------------ | ------------------------------ |
| Graphite electrode | 700-900                  | 1-5 (in-plane)                 |
| NMC electrode      | 700-1000                 | 1-5 (in-plane)                 |
| Separator          | 1000-1400                | 0.3-0.5                        |
| Electrolyte        | 1500-2000                | —                              |

## Lumped vs. Distributed Thermal Models

<Tabs>
  <Tab title="Lumped Thermal">
    Treats the cell as a single temperature:

    $$
    m c_p \frac{dT}{dt} = \dot{Q} - h A (T - T_{\text{ambient}})
    $$

    **Use when**: Cell is small, gradients negligible, or fast simulation needed. A `LumpedHeatCapacityAndDensity` calculation combines the specific heat and density into the lumped heat-capacity term used by this model.
  </Tab>

  <Tab title="Distributed Thermal">
    Solves the heat equation with spatial variation:

    $$
    \rho c_p \frac{\partial T}{\partial t} = \nabla \cdot (k \nabla T) + \dot{q}
    $$

    **Use when**: Large cells, fast rates, or thermal gradients matter.
  </Tab>
</Tabs>

## Thermal Safety

<Warning>
  Thermal runaway occurs when heat generation exceeds dissipation, causing self-accelerating temperature rise. This is the primary safety concern for lithium-ion batteries.
</Warning>

### Onset Temperatures

| Event                 | Typical Temperature |
| --------------------- | ------------------- |
| SEI decomposition     | 90-120°C            |
| Separator shutdown    | 130-150°C           |
| Thermal runaway onset | 150-200°C           |

Accurate thermal modeling helps design cells and systems that stay well below these thresholds.

## Practical Guidelines

<Tip>
  For initial modeling, lumped thermal with Arrhenius temperature dependence is usually sufficient. Add distributed thermal modeling only when investigating thermal management or large-format cells.
</Tip>

### Measurement Methods

| Property           | Method                                                 |
| ------------------ | ------------------------------------------------------ |
| Cell heat capacity | Accelerating rate calorimetry (ARC)                    |
| Specific heat      | Differential scanning calorimetry (DSC)                |
| Activation energy  | Measurements at multiple temperatures + log-linear fit |