What models are used for battery simulation?

Broadly, equivalent-circuit models that represent the cell with resistors and capacitors, and electrochemical models (such as Newman's pseudo-2D model) that solve lithium diffusion and reactions in the electrodes. Equivalent circuits are fast and suit system-level analysis; electrochemical models suit degradation and fast-charging studies.

How is battery heat generation calculated?

The main terms are irreversible heat from internal resistance (I²R) and reversible heat from the cell reaction (I·T·dU/dT). For an illustrative 3 Ah cell, resistive heat is 0.40 W at 1C, 1.62 W at 2C and 3.65 W at 3C, growing with current squared. Reversible heat can be positive or negative depending on state of charge — about 10–20% here.

How does discharge rate affect temperature rise?

Heat rises with current squared while discharge time falls inversely with current, so heat per discharge grows in proportion to current. For the example cell (10 W/(m²·K) surface cooling), temperature rise per discharge is 9 K at 1C, 31 K at 2C and 56 K at 3C; without cooling 2C alone gives 61 K, so thermal management is essential.

Lithium-Ion Battery Simulation

Category: Coupled Analysis | Revised 2026-10-01

Theory: models linking electricity and heat

Overview

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Our power-tool battery pack gets hot and shuts down under continuous use. Can simulation predict cell temperature?

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Yes. First model the cell's electrical behaviour to find how much heat is generated for a given current, pass that to a thermal analysis for temperature, and feed the temperature-dependent internal resistance back to the electrical model — electrochemical–thermal coupling. The simplest option is an equivalent-circuit model representing the cell as a voltage source with resistors and capacitors; for reactions and diffusion inside the electrodes, use an electrochemical model. For an overheating shutdown, an equivalent circuit gives a good first estimate.

Equivalent-circuit model (1RC)

$$ V = U_{oc}(\mathrm{SOC}) - I R_0 - V_1, \qquad \frac{dV_1}{dt} = \frac{I}{C_1} - \frac{V_1}{R_1 C_1} $$

$U_{oc}$ is the open-circuit voltage (set by state of charge), $R_0$ the instantaneous internal resistance, and $R_1, C_1$ represent delayed effects such as diffusion. Voltage drops by $IR_0$ the moment current flows, then further with time constant $R_1C_1$. Values vary with temperature and SOC, so they are tabulated from tests.

Heat generation

$$ \dot Q = \underbrace{I^2 R_0 + \frac{V_1^2}{R_1}}_{\text{irreversible}} + \underbrace{I\,T\,\frac{\partial U_{oc}}{\partial T}}_{\text{reversible}} $$

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

Irreversible heat comes from resistance and is always positive. Reversible heat comes from the reaction's entropy change and can be positive or negative depending on SOC and current direction. Temperature follows the balance of heat generation and surface cooling.

Coffee Break Trivia

Newman's electrochemical model

The foundation of detailed cell models is the porous-electrode theory developed by Newman and colleagues in the US from the 1970s to the 1990s; the pseudo-two-dimensional (P2D) model applied to Li-ion cells in the 1990s remains the standard. It couples one dimension through the electrode thickness with one radially inside particles to solve lithium diffusion and reactions. Li-ion batteries were commercialised in 1991, and in 2019 Akira Yoshino and others received the Nobel Prize in Chemistry for their development.

Worked example: an illustrative 18650 cell

Capacity 3 Ah, R₀ 30 mΩ, R₁ 15 mΩ, C₁ 2,000 F (30 s time constant), mass 45 g, specific heat 1,000 J/(kg·K), surface area 0.0042 m², surface heat transfer 10 W/(m²·K), 25 °C ambient; constant-current discharge to 95% of capacity:

Rate	Current	Initial drop	Resistive heat	Heat per discharge	Rise (cooled)	Rise (adiabatic)
1C	3 A	0.09 V	0.40 W	1.4 kJ	9 K	31 K
2C	6 A	0.18 V	1.62 W	2.7 kJ	31 K	61 K
3C	9 A	0.27 V	3.65 W	4.1 kJ	56 K	91 K

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Tripling the current makes heat nine times and temperature rise six times higher.

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Right. Heat rises with current squared while discharge time falls inversely, so heat per discharge triples. This cell's thermal time constant is about 18 minutes (1,071 s); a one-hour 1C discharge sheds much of the heat, but a 20-minute 3C discharge can't keep up, so the rise is six times higher. Cells inside a pack, surrounded by others, cool even worse and approach the adiabatic value. Including reversible heat (10–20% here) and resistance falling with temperature improves accuracy.

How to approach the analysis

Procedure

Extract R₀, R₁, C₁ and open-circuit voltage versus temperature and SOC from pulse tests.
Measure dU/dT of open-circuit voltage and include reversible heat.
Validate temperature on one cell, then extend to the pack (cell layout, coolant flow, contact resistances).
Evaluate peak temperature and cell-to-cell spread over realistic duty cycles.

Cautions

Temperature-dependent resistance: at low temperature internal resistance multiplies, increasing heat and voltage drop.
Cell-to-cell spread: temperature differences become ageing-rate differences, widening capacity scatter.
Before thermal runaway: at high temperature side-reaction heat adds and rises with temperature; assess safety separately.

Coffee Break Trivia

“Only the middle cells age fast”

After a few years in service, the middle cells of a pack have lost far more capacity than the end cells — a familiar story. Middle cells, surrounded by others, cool poorly and run a few kelvin hotter every cycle. Ageing is often said to roughly double for every 10 K rise, so small temperature differences become large over years. Pack life is set by the weakest cell, so thermal design must minimise not just peak temperature but cell-to-cell spread.

Common mistakes

Mistakes and fixes

Mistake	Effect	Fix
Constant internal resistance	Wrong low-temperature heat and voltage	Use T–SOC tables
Ignoring reversible heat	Large errors at low current	Measure and include dU/dT
Using single-cell cooling for a pack	Central cells underestimated	Analyse the whole pack
Watching only peak temperature	Ageing spread missed	Evaluate cell-to-cell spread
Constant discharge only	Mismatch with real use	Analyse realistic duty cycles

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I'd like to try battery calculations myself.

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Try the battery model simulator for equivalent circuits, the state-of-charge calculator for state of charge, the battery ageing (Arrhenius) tool for ageing and the battery sizing tool for capacity sizing. Related topics are covered in battery thermal management, thermal runaway, battery degradation and fuel-cell modelling.

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