Pressure Buildup Model in a Battery Pack During Thermal Runaway

The tool treats the sealed battery pack as a single control volume of gas and tracks one state variable: the absolute pressure inside it. This page derives the equation that state variable obeys, so you can see exactly what is being calculated and defend it in a design review.

The Control Volume

Everything inside the enclosure that is not solid is one well mixed volume of gas at a single pressure and a single temperature. Gas enters it from venting cells and leaves it through the relief valve.

The three influences on pack pressure, and the term each contributes.

Starting Point

The gas obeys the ideal gas law:

P V = n R T

The equation describes an equilibrium state, and we apply it to a transient event. That is justified because pressure equalises across a pack at the speed of sound, which is far faster than the pressure changes we care about, so at every instant the gas is effectively in equilibrium with itself. This and the other assumptions are set out in Assumptions and Limits.

Deriving the Rate of Change

Differentiate both sides with respect to time. The volume $V$ is constant, so it comes straight out of the left side:

V \frac{dP}{dt} = R \frac{d(nT)}{dt}

Apply the product rule on the right, since both the amount of gas and its temperature change during the event:

V \frac{dP}{dt} = R \left( n \frac{dT}{dt} + T \frac{dn}{dt} \right)

Divide through by V:

\frac{dP}{dt} = \frac{R}{V} \left( n \frac{dT}{dt} + T \frac{dn}{dt} \right)

The first term still contains $n$ , which we would rather not track separately. Substituting $nR = PV/T$ from the ideal gas law removes it:

\frac{dP}{dt} = \frac{P}{T} \frac{dT}{dt} + \frac{RT}{V} \frac{dn}{dt}

Splitting the Molar Term

$\frac{dn}{dt}$ is the net rate of change of gas in the pack, which is what is generated minus what escapes:

\frac{dn}{dt} = \frac{dn_{gen}}{dt} - \frac{dn_{out}}{dt}

$\frac{dn_{gen}}{dt}$ is the slope of the venting curve you supply, scaled by the number of venting cells.

$\frac{dn_{out}}{dt}$ needs converting, because valve manufacturers publish flow as a volumetric rate, not a molar one. For the gas passing through the valve at the current pack pressure and temperature, the ideal gas law relates the two:

\frac{dn_{out}}{dt} = \frac{P Q}{R T}

where $Q$ is the volumetric flow rate through the valve. Substituting it back in, the $RT/V$ factor cancels neatly and gives the governing equation.

The Governing Equation

\frac{dP}{dt} = \frac{P}{T} \frac{dT}{dt} + \frac{RT}{V} \frac{dn_{gen}}{dt} - \frac{PQ}{V}

Symbol	Meaning	SI unit
$P$	Absolute pressure inside the pack	Pa
$T$	Gas temperature inside the pack	K
$V$	Pack void volume	m³
$n$	Amount of gas inside the pack	mol
$R$	Universal gas constant	J·mol⁻¹·K⁻¹
$\dfrac{dn_{gen}}{dt}$	Gas generation rate, all venting cells combined	mol·s⁻¹
$Q$	Volumetric flow through the valve, all valves combined	m³·s⁻¹

Reading the Three Terms

$\frac{P}{T}\frac{dT}{dt}$ , the thermal term. Heating a fixed quantity of gas in a fixed volume raises its pressure. This term is zero when Temperature Mode is Assumed Constant, and it is the reason the other mode exists. See Temperature Modes.

$\frac{RT}{V}\frac{dn_{gen}}{dt}$ , the source term. This is what drives the event. Note the two things it is proportional to: the gas generation rate, and the inverse of the void volume. A small pack pressurises faster on the same gas.

$\frac{PQ}{V}$ , the relief term. The only negative term, and the only one you have design authority over. $Q$ is not a constant: it is read off the valve’s pressure to flow curve at the current differential pressure, and it depends on whether the valve is currently open or closed. That is what makes this a coupled problem rather than an integration you could do by hand, and it is covered in PRV Behaviour.

What the Peak Actually Is

Reading the equation tells you what the reported peak pressure means physically. Pressure rises while the source term dominates the relief term, and turns over at the moment they balance. The peak is therefore not set by the total gas released. It is set by the rate of gas release measured against the valve’s flow capacity at that instant.

Two practical consequences follow, and both show up in the tutorials:

Doubling the total gas produced, over twice the time, may barely move the peak.
A valve whose flow rises steeply just above its setpoint can hold the peak close to the setpoint almost regardless of how much gas arrives.

Units

The solver works entirely in SI. The interface lets you enter values in whatever unit your data is in, including litres, bar, psi, °C, °F, L/min, and CFM, and converts on your behalf. If you are checking hand calculations against the tool, convert to the SI column above first.

What This Page Does Not Cover

How the equation is integrated numerically, how the valve curves are interpolated, and how the returned points are selected are part of the implementation and are not documented here. Nothing you need in order to interpret, defend, or reproduce a result depends on those details: the model above is the complete physical statement of the problem.