# How do I calculate the self discharge rate of a lead acid battery?

Okay, like the title suggests, I need a method of calculating self discharge rates of Lead-Acid batteries. Here's the catch: I varied the electrolyte which the batteries were using, replacing sulphuric acid with hydrochloric acid, another one with nitric, and another one with phosphorous acid. Anybody have any idea how I can get around this?

The emperor has no clothes on.

There is absolutely no certainty that this will work.
Asking people about fine details of cell performance tends to legitimatise and hide the fact that a process of no real certainty is being investigated.
Until the underlying issues have been at least lightly investigatred the question is premature and ill advised.

In the absence of a good grasp of the chemistry there is no certainty that extrapolated results will be in any way accurate.

• A battery may start off well and then fade rapidly.

• Results for one cycle may not be repeatable. The batteries may fail rapidly after a few cycles. Unless you have worked though the chemistry it is not certain that there will be a satisfactory outcome.

A "certain" method, for at least the current cycle, is to use a number of cells of each type and discharge them to endpoint after selected periods.

For example, a total of 7 batteries would allow all periods of 1 to 12 months to be checked. ie 12, 11+1, 10+2, 9+3, 8+4, 7+5, 6 + 3 + 2 + 1

If you did the shorter term measurements first you get the following results after X months

• 1 month - 1 months x 2 = 1, 1
2 months - 2, 1, 1
3 months - 3, 2, 2, 1, 1, 1
4 months - 4, 3, 2, 2, 1, 1, 1
5 months - 5, 4, 3, 2, 2, 1, 1, 1

The new large result per month starts to go wrong as 6+3+2+1 6 months result until after a year.

More batteries will fix this.

A key issue is, are the cells going to have increasing capacity over the first few cycles (as eg NimH do) or will they fall progressively with cycles?

I think the answer goes something like "Die die die !!!" :-)

This sounds like all good fun but

• You'd need to tell us far more about what you had done - acid molarity plate compositions (calcium lead or ...), plate construction and "rather more"

• What do you know? Why are you doing this? What are you trying to achieve? Other?

• Given that lead-sulphuric chemistry is very well known and is very likely the optimum one available using lead (or the industry would be hieng after summat else)

• There is no doubt that you will get some sort of battery in each case, but as the capacity you achieve will be lower at best and probably much lower, then a long self discharge life may not return a better net capacity that a standard lead acid battery for at least 12 months. After 12 months you MAY get more capacity than std lead acid. But certainly not certain.

Determining State of Charge from Voltage:

Determination of battery state of charge from loaded or open circuit voltage is notionally possible, but depends on many factors - with major ones being temperature & specific gravity of electrolyte. Here are some curves for various discharge rates. The unloaded self discharge curve will be slightly above the C/100* curve. You would probably have to lightly load the battery during measurement as Voc will probably be less representative of the real state of charge. (* C/100 = discharge at a current equal to 100th of the nominal Ampere hour capacity.) All of the above "probablys" and "slightly aboves" are well understood for lead acid with lead / sulphuric acid but are a whole new area with different acids (let alone unknown optimum concentrations etc).

The above diagram is a rearrangement of the diagram on page 68 from the excellent Lead Acid battery state of charge versus voltage - Home power #36, Aug-Sep 1993. (They insisted on flipping their graph right-left which made comparison with more normal plotting standards difficult. Note the shape of the curve is not a concave exponential one, as might be expected, but a convex curve - ie voltage change per % of capacity change increases with increasing discharge (and doesn't decrease as in a normal exponential decay).

Standard lead-acid cells have a low self-discharge, about 5% per month, so continuously monitoring makes little sense. To measure this I would take a reading with a DMM every few days, and you may need to take readings over a period of more than a month to get a decent graph.

But you changed the electrolyte, god knows why. Materials and construction of lead accus have been optimized long ago now, and have proven their worth. So I don't think you will gain from changing the electrolyte. Expect a lower capacity, possibly a higher self-discharge and maybe shorter life.

Anyway, self-discharge is exponential, so that the largest change will be right after the cell is charged. Take measurements at a few hours interval, and decide if one measurement every few days is still sufficient.
You should get a graph looking a bit like this:

The equation is

$V = V_{FULL} \left( 1 - a (1 - e^{-bt})\right)$

where $a$ and $b$ can be derived from the readings you've taken. $a$ is the level of self-discharge, in the graph 0.2 or 20%. $b$ is the rate of self-discharge.

I think you would have to take regular readings of the voltage, using a setup that has less current draw than normal expected self discharge rate, so as not to skew the result.

A FET input opamp (often with pA input current, probably way lower than the SDR) followed by an ADC/uC would probably work okay. Or simply use a calibrated/accurate multimeter and take readings at required intervals.

I guess, the self discharge has a function similar to the discharge of a capacitor+resistor circuit (but of course with much larger time constants). Or you can choose one of the existing mathematical models out there.

To determine the parameters of your model, you will have to make experiments. I don't know however, how to get around the time required for some of the tests.