# Battery Freezing Math

Quote of the Day

The best argument against democracy is a five-minute conversation with the average voter.

— Winston Churchill

## Introduction

Figure 1: 7.2 A-hr Sealed Lead Acid Battery
(Source). This battery is a workhorse products
for many markets, including telecom and home
security.

I live in a cold climate -- so cold that under certain circumstances we can freeze our lead-acid batteries (Figure 1). A customer who lives in my region called recently and was wondering if I thought any of his batteries would have frozen over the winter. A number of his Internet service subscribers have vacation homes that are unoccupied over the winter. All of these vacation home owners turn off their AC power for the winter. Since all of our Optical Network Terminals (ONT) are connected to Uninterruptible Power Sources (UPS), they will begin operating off of their battery when the AC power goes away. If the home owner does not disconnect the battery, the ONT will run discharge the battery. This is important because a discharged battery will freeze -- a charged battery will not freeze. A battery that has been frozen is very likely a dead battery and you will need a replacement battery.

Note that car and rv batteries rarely freeze because a car and rv battery is rarely fully discharged. These batteries will freeze if they are allowed to get cold enough when discharged. Here is a typical situation:

• You have a car or rv with remote start and standard electronics. This suite of hardware puts a 100 mA parasitic load on the battery: 70 mA for the remote starter and 30 mA for the car computer.
• The car or rv is parked in a cold garage for five days.
• The battery discharges and the battery freezes.

This EXACT situation just happened to my son. He now must ensure he drives the car every few days to keep the battery charged.

Let's examine why my ONT customers need to worry about discharged batteries over the winter.

## Background

Lead-acid batteries contain a solution of sulfuric acid (H2SO4) and water -- the solution is referred to as the battery's electrolyte. Adding a solute (in this case, H2SO4) to a solvent (in this case, H2O) will lower the freezing point of a solution. A fully charged battery has more H2SO4 than a discharged battery. The additional H2SO4 depresses the freezing point of the batteries electrolyte to around -70 °C. This is a temperature we do not see in Minnesota. However, a discharged battery's freezing point rises to ~-10 °C. Unfortunately, the temperature in Minnesota frequently drops below -10 °C.

## Analysis

This blog post is going to focus on presenting empirical data. However, I do want to spend a bit of time discussing the various ways of expressing the concentration of battery acid.

### Battery Acid Concentration

There are three common ways of expressing the concentration of battery acid.

• Specific Gravity (symbolized by SG)

Specific gravity compares the density of the battery electrolyte to that of water. Specific gravity is readily measured using a hydrometer, which almost every auto mechanic has used – even I have a hydrometer. I see inexpensive hydrometers for sale every time I go into an automotive parts store.

• Mass Fraction (symbolized by w)

The mass fraction expresses the acid concentration in terms of the mass of acid divided by the total mass of the acid -water mixture. This measure of concentration is a convenient measure because only a scale is required to mix up a properly measured solution. Unfortunately, there is no inexpensive instrument for measuring the mass fraction directly after it has been mixed. Once mixed, we use SG.

• Molality (symbolized by m)

Molality is the number of moles of solute per kg of solvent. The advantage of using molality as a measure of battery acid concentration is that you can create a properly mixed solution using only a scale. The problem is that there is no readily available instrument for measuring molality after it has been mixed. Again, we usually use SG.

The mass fraction and molality are related by the equation $latex m=\frac{w}{{MM\cdot \left( {1-w} \right)}}&s=-1$, where MM is the molar mass of the solute (98 grams/mol for H2SO4). Specific gravity can be used to relate mass fraction to molarity (symbolized by M) by the equation $latex M=\frac{w \cdot SG\left( w \right)}{MM}&s=-1$, where SG is assumed to be equal to the solution's density (close enough for most applications). I do not see molarity used by battery people, but chemists use it all the time. I cover these formulas in more depth in this post.

### Cell Voltages: Open Circuit, Charging, and Discharging

Figure 2 (Source) shows the terminal voltages experienced by "12 V", 6-cell, lead-acid battery when at different levels of charge and discharge currents.

Figure 2: Voltages During Charging and Discharging for a 12 V Battery.

Because of the variation in battery terminal voltage with charge or discharge current, I will plot (Figure 3) the open-circuit terminal voltage. This will make the graph simpler.

### Freezing Point and Open-Circuit Cell Voltage Versus Acid Concentration

Figure 3 (Source) shows both the electrolyte's freezing point and the open-circuit cell voltage as a function of mass fraction, specific gravity, and molality. We usually define a fully charged battery has having an electrolyte with a molality of 6.0 moles/kg. Similarly, a discharged battery is usually defined as a battery with an electrolyte having a molality of 2.0 moles/kg.

 Figure 3(a): Battery Freezing Point Versus Specific Gravity. Figure 3(b): Cell Voltage Versus Specific Gravity.

## Conclusion

I have dealt with this issue for a number of years. I thought it was worth documenting why the UPS batteries can freeze. The solution is simple -- disconnect the charged battery from the UPS. A lot of other things will freeze (e.g. the water/anti-freeze mixture often used to winterize vacation home plumbing) before that charged battery will freeze.

To cross-check my information, I also consulted additional sources, which I document here.

## Appendix A: Primary Battery Source Material

In addition to the data presented in Figure 2, I also used the following table from Vinal's "Storage Batteries". (Google Books Reference)

Figure 4: Battery Data from Vinal, 1951 (Source).

## Appendix B: Additional Battery Source Material

I consulted numerous sources to corroborate the data presented here. Figure 5 shows data from Sandia, which has gathered together a surprising amount of lead-acid battery data.

Figure 5: Specific Gravity, Terminal Voltage, and State of Charge Data (Sandia Labs). A 12 V battery has 6 cells. To derive the cell voltage, I divided the terminal voltage by 6.

To simplify comparison with Figure 3, I will reformat the Sandia data so that the state of charge and cell voltage are functions of specific gravity (Figure 6).

Figure 6: Sandia Data Reformatted As a Function of Specific Gravity.

The data in Figure 5 is similar to that shown in Figure 2. Since this is empirical data, I expect some differences between sources. If you are curious about how I generated Figure 5, see the Mathcad and PDF file here.

Figure 6 shows another set of data. This data is also consistent with the other data sets I found.

Figure 6: Cell Voltage Versus Specific Gravity and Depth of Discharge (Source).

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### 17 Responses to Battery Freezing Math

1. Drakonomikon says:

Both of your formulas are incorrect. In particular, the mass fraction scale in fig. 2 is wrong.

• mathscinotes says:

You are correct – thank you for helping me improve my blog site. I have corrected the formulas and updated the graphs. As part of re-verifying everything, I put together a blog post with additional checks on the formula correctness. The most interesting part was learning how to make a secondary axis with a nonlinear scale. I will be doing a post on that subject.

Again, thanks.

mathscinotes

P.S.

Another reader said that Figure 3 was too difficult to read with both freezing point and cell voltage on the same chart. I agreed and I have now split Figure 3 into two parts.

2. Frank Edwards says:

This gave me the basic idea, however more to my point is that I live in Ontario, Canada
and the batteries I question are those found in E bikes, scooters etc. My bike has 5 12V small batteries under the seat. If I store my bike in my shed should I ideally try to heat the shed. If I can't would a space heater be good enough? For me it is too difficult to take the batteries out as it means almost stripping the bike right down to the frame. Could you give me some ideas? Thank you.
Frank Edwards

• mathscinotes says:

I use battery heating pads on my products. Would that work in your case? Here is a photo of one on Amazon. There are an endless variety -- some have temperature-sensing turn-on switches.

mark

3. Tamer Rezk says:

I have a E-bike with 6 batteries and I live in Ukraine, during next 2 months temp. may reach -20 C . I store my bike in a simi closed garage and don't know how to protect my batteries from getting freezed.
Any advice highly required

• mathscinotes says:

Assuming they are lead-acid batteries, keep your batteries charged! A charged battery will only freeze at temperatures far lower than -20°C. Some folks do not want to keep their batteries charged, and I recommend they use a battery heater. There are an enormous number of battery heaters available (example).

mark

4. Tamer Rezk says:

So fully charged batteries will be protected till next summer?
how often you recommend to charge the batteries if bike will not be used at all?
thanks for help

• mathscinotes says:

The only thing I do with my batteries over winter is to charge them up and disconnect the cables. If you do not disconnect the battery cables, many vehicles will have a slight draw on the battery that will discharge the battery over the winter and will allow the battery to freeze. My Polaris ATV has this issue.

It is much colder where I live than -20°C — often more like -40°C. I have never frozen a charged battery. I have frozen a few discharged batteries.

mark

5. Tamer Rezk says:

in my bike there is a switch to disconnect the electricity from bike without disconnecting batteries. so if I fully charge it and switch this main switch off is it enough to charge the bike one time till next winter? someone told me to charge it once every 2 weeks.

• mathscinotes says:

If you disconnect it, a charged, lead-acid, AGM battery should keep at least 90% of its charge over a 3 month period (example: Panasonic LC-R127R2PG). When you are ready to drive, just charge it, connect it up, and go.

mark

6. Adam says:

So, if I leave a pair of 6V Deep Cycle batteries on a battery maintainer over the winter, they won't freeze? Also, they would be fully charged when we go to use them next time? It only gets to about -25c here.

7. Hi Mark,
I live in Ontario and drive not so often. Every winter, I face with discharged and dead batteries to struggle with. I could not see the inside to make sure if the sealed battery is frozen. That is why I reached your page.
I cannot readily follow your equations, but I understand the chemistry.
You mentioned that we could not determine the molarity once acid is mixed with water. I guess, we could do that in a simple way, but it requires a laboratory set up. The procedure is called acid/base TITRATION. Simply take a known volume of the solution and dilute it 10 or 100 folds. Using a graduated burette, add 0.1 or 0.01 N (normal) NaOH to known volume of acid solution with unknown molarity. Final end point of titration is determined by adding indicators and seeing the designated color change or following the change in pH thru pH-meter. A pH of 7.0 could be regarded as the end point. Make sure to place the beaker containing the acid over a magnetic stirrer and drop a magnetic stirring bar into the solution. At the end, you need to record the volume of NaOH used to neutralize the acid. Plug the numbers in equation C1V1=C2V2 where C stands for concentration and V represents the volume; one side for acid and another side for base. You would learn this in freshman chemistry course.
Ali.

8. Ali Mahmoudpour says:

Also, since we are dealing with 2 protons in sulfuric acid vs 1 hydroxyl group in sodium hydroxide, for every unit of the acid, we should use two units of base for titration. So, we should consider dividing the base volume in half in calculation.

• mathscinotes says:

Thank you for your reply. I have not thought about titration since I was in chemistry at university. Great suggestion!