Thermal Runaway in a Battery

In a damaged battery cell, heat speeds up the chemistry and the chemistry makes more heat. In the simplest model the temperature doesn't just grow, it reaches infinity at a finite time. The solution only exists up to that moment, and the moment depends on where you start.

Click or drag anywhere to start a new solution there.

How it works

Intervals of validity

In one sentence

If $f$ and $\partial f/\partial y$ are continuous near $(t_0, y_0)$ , then $y' = f(t, y)$ , $y(t_0) = y_0$ has a unique solution on some interval around $t_0$ ; that interval can end in finite time, as for $y = 1/(1/y_0 - kt)$ at $t = t_0 + 1/(ky_0)$ .

General formula

{\htmlData{sym=3}{\frac{dT}{dt}}} = {\htmlData{sym=1}{k}}\,{\htmlData{sym=0}{T}}^2 \quad\Rightarrow\quad {\htmlData{sym=0}{T}}(t) = \frac{1}{1/{\htmlData{sym=2}{T_0}} - {\htmlData{sym=1}{k}}\,t}, \qquad \text{valid only for } t < \frac{1}{{\htmlData{sym=1}{k}}{\htmlData{sym=2}{T_0}}}

The growth rate is the square of the value, so growth feeds on itself faster than exponentially, fast enough to reach infinity in finite time.

What the symbols mean

$T$
Excess temperature
How far the cell is above its safe temperature.
$k$
Self-heating rate
How strongly heat speeds up the reactions that make more heat.
$T_0$
Starting temperature
The excess temperature when the runaway begins.
$\frac{dT}{dt}$
Heating rate
Degrees per second the cell is warming.

Model the excess temperature $T$ with self-heating that grows like $T^2$ : $T' = kT^2$ .
Separate and integrate: $-1/T = kt - 1/T_0$ , so $T = 1/(1/T_0 - kt)$ .
The formula has a vertical asymptote at $t^* = 1/(kT_0)$ : the solution exists only on $t < t^*$ .
Nothing in the equation warns of this. The interval of validity depends on the starting value: hotter starts blow up sooner.

Learn the math

Intervals of ValidityDiff Eq · Lesson 9, First Order Differential Equations