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).

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

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 $-E_a/R$.

import ionworkspipeline as iwp
import pandas as pd

# Experimental data at different temperatures
data = pd.DataFrame({
    "Temperature [K]": [273, 298, 323, 348],
    "Diffusivity [m2.s-1]": [1e-14, 3e-14, 8e-14, 2e-13]
})

# Fit Arrhenius parameters
calc = iwp.calculations.ArrheniusLogLinear(
    data,
    reference_temperature=298.15
)
result = calc.run(None)

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 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:

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

calc = iwp.calculations.SpecificHeatCapacity()
result = calc.run({
    "Cell heat capacity [J.K-1]": 50,
    "Cell mass [kg]": 0.05,
})
# Returns: Cell specific heat capacity [J.kg-1.K-1] = 1000

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

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.

calc = iwp.calculations.LumpedHeatCapacityAndDensity()
result = calc.run({
    "Cell specific heat capacity [J.kg-1.K-1]": 1000,
    "Cell density [kg.m-3]": 2500,
})

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.

Thermal Safety

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

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

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.

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