Deriving the Bernardi Heat Generation Equation from First Principles

The Bernardi equation (1985) gives the volumetric heat generation rate of a lithium-ion cell in measurable quantities alone:

$$q = \underbrace{\frac{I}{V_b}(E_{oc} - V_T)}_{q_{irr}} \;-\; \underbrace{\frac{I}{V_b}\,T\frac{\partial E_{oc}}{\partial T}}_{q_{rev}}$$

Symbol	Meaning	Units
$I$	Current (positive on discharge)	A
$V_b$	Cell volume	m³
$E_{oc}$	Open-circuit voltage at current SOC and $T$	V
$V_T$	Terminal voltage under load	V
$T$	Cell temperature	K
$\partial E_{oc}/\partial T$	Entropic coefficient at fixed SOC	V K⁻¹

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Energy balance

Define the system as the entire cell. Two quantities cross the boundary: electrical work $V_T I$ leaves through the load; heat $\dot{Q}_{in}$ exchanges with the environment. All matter stays inside. The First Law in rate form:

$$\frac{dU_{sys}}{dt} = \dot{Q}_{in} - V_T I \tag{1}$$

The internal energy splits as $U_{sys} = U_{chem} + U_{therm}$. Evaluating $dU_{chem}/dt$ requires the reaction rate — moles of lithium transferred per second — which is set by the current.

Faraday's Law. Define:

• $N_A = 6.022\times10^{23}$ mol⁻¹ — Avogadro's number
• $e = 1.602\times10^{-19}$ C — charge per electron → Faraday constant $F = N_A e = 96\,485$ C mol⁻¹
• $n = 1$ — electrons per Li atom ($\text{Li} \to \text{Li}^+ + e^-$, one electron per formula unit for all Li-ion)
• $N_{Li}$ — moles of Li transferred (variable)

Total charge: $Q = nFN_{Li}$. Differentiating ($dQ/dt = I$):

$$\dot{N}_{Li} = \frac{I}{nF} \tag{2}$$

Each mole transferred changes chemical energy by $\Delta H_{rxn}$ (using $\Delta U_{rxn} \approx \Delta H_{rxn}$ since $P\Delta V \approx 0$ for solid/liquid electrodes). Defining $\dot{H}_{rxn} \equiv -(I/nF)\,\Delta H_{rxn} > 0$ (positive because discharge is exothermic, $\Delta H_{rxn} < 0$):

$$\frac{dU_{chem}}{dt} = -\dot{H}_{rxn} \tag{3}$$

At steady state ($dT/dt = 0$), $dU_{therm}/dt = 0$, so $dU_{sys}/dt = -\dot{H}_{rxn}$. Substitute into Eq. 1 and define $\dot{Q}_{gen} = -\dot{Q}_{in}$:

$$\boxed{\dot{Q}_{gen} = \dot{H}_{rxn} - V_T I} \tag{4}$$

Heat generated = enthalpy released by reaction − electrical work extracted. $\dot{H}_{rxn}$ is still unknown. The next section expresses it in measurable electrical terms.

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$\dot{H}_{rxn}$ in measurable form

Nernst. Define the Gibbs free energy $G \equiv H - TS$ — the portion of a system's energy available to do useful work at constant temperature and pressure. For the discharge reaction, $\Delta G_{rxn} = G_{\text{products}} - G_{\text{reactants}} < 0$ (spontaneous). The Clausius inequality shows the maximum non-PV work the reaction can deliver is $W'_{max} = -\Delta G_{rxn}$. Per mole, $n$ electrons flow through $E_{oc}$, so $W'_{max} = nFE_{oc}$. Equating:

$$E_{oc} = \frac{-\Delta G_{rxn}}{nF} \tag{5}$$

OCV is Gibbs energy per coulomb — voltage is thermodynamics in electrical units.

Entropic coefficient. The Gibbs relation $(\partial G/\partial T)_P = -S$ applied to the reaction gives $(\partial\Delta G_{rxn}/\partial T)_P = -\Delta S_{rxn}$. Substituting Eq. 5:

$$\frac{\partial E_{oc}}{\partial T} = \frac{\Delta S_{rxn}}{nF} \tag{6}$$

The OCV slope with temperature at fixed SOC directly measures $\Delta S_{rxn}$ — no calorimetry needed.

Enthalpy. Substitute Eq. 5 and Eq. 6 into $\Delta G_{rxn} = \Delta H_{rxn} - T\Delta S_{rxn}$, then multiply by $-\dot{N}_{Li} = -I/nF$ (Eq. 2); $nF$ cancels:

$$\boxed{\dot{H}_{rxn} = I\!\left[E_{oc} - T\frac{\partial E_{oc}}{\partial T}\right]} \tag{7}$$

All quantities on the right are directly measurable.

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The Bernardi equation

Write $\dot{Q}_{gen} = q\,V_b$. Substitute Eq. 7 into Eq. 4 and divide by $V_b$:

$$\boxed{q = \frac{I}{V_b}(E_{oc} - V_T) - \frac{I}{V_b}\,T\frac{\partial E_{oc}}{\partial T}} \tag{9}$$

$q_{irr} = \frac{I}{V_b}(E_{oc}-V_T)$ — the overpotential gap $(E_{oc}-V_T)$ is all the voltage lost inside: ohmic resistance, charge-transfer activation, concentration polarisation. Since $E_{oc}-V_T \approx IR_{int}$, this scales as $I^2$. Always $\geq 0$.

$q_{rev} = -\frac{I}{V_b}T\frac{\partial E_{oc}}{\partial T}$ — not a dissipation loss. It is the thermodynamically required heat exchange $T\,\Delta S_{rxn}$ per mole (the Clausius reversible heat). A cell with zero resistance still has it. It scales as $I^1$, so at low C-rates it dominates the quadratic $q_{irr}$. Its sign depends on SOC — the same cell can cool the environment at one SOC and heat it at another. Viswanathan et al. (2010) measured $q_{rev}$ at 20–40% of total heat at C/5 and below.

Why $\partial E_{oc}/\partial T$ changes sign with SOC

Full cell: $\partial E_{oc}/\partial T = dU_+/dT - dU_-/dT$. The graphite anode dominates because it intercalates lithium in ordered staging phases — in Stage $n$, Li occupies every $n$-th graphene layer (Stage 4: dilute, $x < 0.12$; Stage 3; Stage 2: $x \approx 0.25$–$0.5$; Stage 1: dense, $x > 0.5$). Each stage has a distinct configurational entropy.

• 0–47% SOC — graphite is in Stage 4/3 (dilute, disordered). Removing Li collapses disorder sharply → $dU_-/dT \gg 0$ → $\partial E_{oc}/\partial T < 0$ → $q_{rev} > 0$ (extra heat).
• 47–100% SOC — graphite enters Stage 2/1 (ordered, single-phase). Anode entropy contribution shrinks; NMC cathode (solid-solution, no staging) slightly dominates → $\partial E_{oc}/\partial T > 0$ → $q_{rev} < 0$ (cell absorbs heat).

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Numerical example

18650 NMC/graphite, 1C, 50% SOC, 298 K ($I=3$ A, $V_b=16.5\times10^{-6}$ m³, $E_{oc}=3.85$ V, $V_T=3.72$ V, $\partial E_{oc}/\partial T = -1.0\times10^{-4}$ V K⁻¹):

$$q_{irr} = \frac{3}{16.5\times10^{-6}}\times 0.13 = 23\,636\;\text{W m}^{-3}$$ $$q_{rev} = -\frac{3}{16.5\times10^{-6}}\times 298\times(-1.0\times10^{-4}) = +5\,418\;\text{W m}^{-3}$$ $$q = 29\,054\;\text{W m}^{-3}$$

The Joule-only model ($I^2R_{int}/V_b$) gives 23 636 W m⁻³ — a 23% underestimate.

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References

1. D. M. Bernardi, E. Pawlikowski, J. Newman, "A General Energy Balance for Battery Systems," J. Electrochem. Soc., 132(1), 5–12, 1985. DOI: 10.1149/1.2113792.
2. V. V. Viswanathan et al., "Effect of entropy change of lithium intercalation in cathodes and anodes on Li-ion battery thermal management," J. Power Sources, 195(11), 3720–3729, 2010. DOI: 10.1016/j.jpowsour.2009.12.034.
3. K. O'Regan, F. Brosa Planella, W. D. Widanage, E. Kendrick, "Thermal-electrochemical parameters of a high energy lithium-ion cylindrical battery," Electrochimica Acta, 425, 140700, 2022. DOI: 10.1016/j.electacta.2022.140700.
4. A. Jokar et al., "Evaluation of accuracy for Bernardi equation under pulse-discharge protocols," Appl. Therm. Eng., 201, 117794, 2022. DOI: 10.1016/j.applthermaleng.2021.117794.
5. G. L. Plett, Battery Management Systems, Vol. 1, Artech House, 2015.
6. J. Newman, K. E. Thomas-Alyea, Electrochemical Systems, 3rd ed., Wiley, 2004.