## Introduction

Very nearly every product we make ships with one or more lead-acid batteries. Since we have built hundreds of thousands of units, that is a lot of batteries. While most people encounter batteries everyday, few really understand the problems that they present.

The least understood problem with a lead-acid battery may be that they are susceptible to thermal runaway. The Wikipedia has a useful definition of thermal runaway.

Thermal runaway refers to a situation where an increase in temperature changes the conditions in a way that causes a further increase in temperature, leading (in the normal case of an exothermic reaction) to a destructive result. It is a kind of positive feedback.

Thermal runaway can be quite destructive. Figure 1 shows a photograph of a car battery in a trunk that experienced runaway. Not a pretty sight.

Figure 1: Car Battery After Thermal Runaway (Source: Voltphreaks).

I have struggled with coming up with a good explanation of thermal runaway that is generally understandable. Today, I saw a paper that did a nice job of explaining thermal runaway using a simple mathematical model. I review that paper in this blog and add explanatory notes that will make the material more accessible to a non-specialist audience.

## Background

### Thermal Runaway Basics

Battery thermal runaway is a positive feedback process.

1. The charging chemical equations are exothermic (i.e. generate heat). As we charge the battery heat is generated.
2. Heat accelerates the exothermic chemical reaction within the battery.
3. The accelerated exothermic reactions generate more heat.
4. go back to step 1.

This process is illustrated in Figure 2.

Figure 2: Illustration of Thermal Runaway Process (Source: Wikipedia)

### Fundamental Equations

Thermal runaway only occurs when a battery is being overcharged. If you want to know how to drive a lead-acid battery into thermal runaway, take a look at the document “Induced Destructive Overcharge Test IEC Standard 952-1:1988.” Its procedure works like a champ.

The thermal overload model presented herein uses two equations:

• Input Power Equation
This equation calculates the electrical power dissipation within the battery as a function of temperature and voltage.
• Output Power Equation
This equation calculates the ability of the battery mechanical design to dissipate the electrical heat generated within as a function of battery temperature and ambient temperature.

The simultaneous solution of the input and output power equations show that there exists an operational region where more power can be generated within the battery than the battery can dissipate to the environment while maintaining a stable temperature. This is the region of thermal runaway.

#### Derivation of the Input Power Equation

Understanding overcharging requires knowing the functional relationship between charge current, float voltage, and temperature. Figure 3 is an excellent illustration of this relationship for an aged battery, which are more susceptible to thermal runaway than new batteries.

Figure 3: Battery Charging Current Versus Float Voltage and Battery Temperature (Source: Battery Technology Handbook, Kiehne, ISBN 9780824742492).

Figure 3 is really a graphic restatement of the Tafel equation, which is an equation fundamental to battery operation. Because Figure 3 is a semilog plot, we can convert Figure 3 into an equation of the form shown in Equation 1.

 Eq. 1 $\log \left( {{I}_{Charge}} \right)=k+\alpha \cdot {{V}_{Float}}+\beta \cdot {{T}_{Battery}}$

Given that we know the charging current, we can compute the power generated within the battery by noting that $P_{IN} = V_{Float} \cdot I_{Charging}$. Equation 2 computes the power generated in the battery.

 Eq. 2 ${{P}_{IN}}={{V}_{Float}}\cdot {{I}_{Charge}}={{V}_{Float}}\cdot {{10}^{k+\alpha \cdot {{V}_{Float}}+\beta \cdot {{T}_{Battery}}}}$

where k, α, β are constants determined through curve fitting.

I will not go through the details of the curve fitting operation, except to say that I captured the graphic data using a tool called Dagra and I processed the data using Mathcad. Figure 4 contains a screen shot of the Mathcad work.

Figure 4: Curve Fit Work in Mathcad.

A 3D surface graph (Figure 5) does a nice job of showing what the function looks like.

Figure 5: 3D Graphic of Battery Charging Power.

This completes Part 1. Part 2 will cover how the power generated within the battery is dissipated. Using equations derived for the power generated and power dissipated within the battery, a condition will be derived for when thermal runaway occurs.