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It is simple to calculate a continuously-on device's theoretical battery life based on the battery datasheet capacity (assuming always nominal voltage, etc.)

An LED @ 20mA..

2800mAh / 20mA = 240 hours.

The issue I am having is verifying calculation of intermittent use of such a device.

Lets say it consumes 20mA for ten seconds three times per day. Would a calculation such as this make sense?

20 * 10 * 3 = 600mAs? / 60 / 60 = 0.166mAh/day

To be novel, resembles $$0.166 \centerdot 10^{-3} \ C \centerdot s \centerdot hour \centerdot day^{-1}$$

Now to extract days from that I would possibly do this:

2800mAh / 0.166mAhd? = 16867 days / 365 = 46 years battery life

Does this seem at all coherent? The only proof I can muster:

16867 days * (30 * 10 * 2 / 60 / 60) = ~2800mAh

Had never seen anyone use this before, maybe I have pioneered some new units.

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Battery capacity with time varies with usage and with battery chemistry.
For very long lifetimes (years) information about battery chemistry is essential to lifetime considerations.

You mention a 2800 mAh capacity - what sort of battery type is that? If it is an AA (14500) of any sort the capacity claim will be higher than actual regardless of load or manufacturer). If it is a LiIon 18650 cell it is also probably higher than actual. For very long lifetime quality and manufacturer experience matter and brand is often but not always an indicator of quality. Batteries with claimed capacities at the top end of the range for their size and chemistry usually have lower capacities than quality competitors with lower claimed capacities.

Primary (non rechargeable) and secondary (rechargeable) batteries have somewhat different factors affecting lifetime. A rechargeable battery has a self discharge rate but can be "refilled". A non-rechargeable battery loses available capacity with time but this is usually seen as a shelf life or degradation rater than loss of charge as such. The two continue to increasingly overlap with the introduction of better performing low self discharge batteries such as the recent versions of the Sanyo Eneloop which retain most of their charge for several years.
The following is biased toward non-rechargeable batteries as these are usually used for very long lifetimes, but to some extent applies to rechargeables.

It is only possible to obtain approximate capacity results for batteries which are subject to occasional small loads.
Where the result indicates an extremely long lifetime, such as the 46 years that you calculate, the self discharge of shelf life of the battery will be the main factor affecting lifetime.
The longest battery lifetimes that you are liable to be able to obtain with commercial products are around 10 years although longer or much longer may be possible in special cases.

Your method of dividing battery capacity by mAh/day to get days of lifetime is valid providing the mAh capacity that you use is correct. However, for very light loads and long lifetimes the mAh capacity will be greater than the manufacturer's rated value.
For extremely long lifetimes the user-utilised mAh will fall because an increasingly large part of the true mAh capacity will begin to be consumed by non load related factors such as self discharge.
In addition to true self discharge, some battery chemistries have "calendar lives" where the battery slowly degrades in capacity due to irreversible chemical and mechanical changes in the battery structure. A good example are standard Lithium Ion batteries which typically have "calendar lives" in the 2 to 5 year range regardless of amount of use.

Manufacturers data sheets are available in some cases that show the effect of loading on both lifetime and capacity. The very wide range of possible loads and usage patterns means that these are only a guide. Very low and very high temperatures, especially those outside the manufacturers stated range, may have very significant effects on available capacity.

Here is a datasheet for Evereday CR2032 coil cell with nominal 240 mAh capacity.
It does not answer your questij but gives some indication of how loading affects capacity.

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