There really isn't "good" LiIon batteries and better one. There are some mislabeled especially the 18650's which might lie about the 2400mAh. (Unless you are talking about the different Lithium mixes being used like Cobalt, etc.. But for the cheapy batteries they are all the same, and you only need to compare the mAh)
You have to decide how much current you want which sets the heat. This is also probably related to how much air you suck past the heater as well, or how "wet" the coil part gets.
So, say you have a 1" coil and it is 3ohms. And you have another coil from different atty that is 1/2" and comes in at 1.5ohms (same NiCr wire). The heat the coil gets up to is based on the current for both. 1A should be some temp, however in terms of heat generated, it is I^2*R so the larger coil has more heater area and more resistance so twice the heat is applied to the air (a 3W heater instead of 1.5W heater)
Now if you apply a fixed voltage, like 3.7V to both, this sort of compensates, but the temp of the coil is different. For a fixed voltage case, the 3ohm coil, it is 1.2A and 4.6Watts. For the 1.5ohm coil, it is 2.5A and 9.1W. Power for a fixed battery voltage is V^2/R so twice the resistance half the heat. However, in this case the smaller coil gets much hotter with 2.5A and might burn more, it might also vaporize a lot more because of extra convection and radiation (which I'll have to look, but might be proportional to temp squared).
The reason the battery rating matters is besides the longer usage on a charge, as a rule of thumb for LiIon batteries can discharge safely "1C" so a 2200mAh shouldn't exceed 2.2amps or you are reducing the battery life. Though, for this application it is very short bursts over a long period of time.
This is partly why I suspect the stock AAAA size batteries don't live up to 200 charges.. because the smaller capacity 3.7V batteries can't handle the 1000mA or so.
Battery University The high-power lithium-ion
batteryuniversity.com/partone-5A .htm
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My current ideal solution is to take two heater coils in one atty and then use one 3.7V battery and the size of it depends on how long between recharges. I also think using a L200C voltage regulator with adjustable current control would be the simplest solution
Also make sure you use the protected LiIon. In fact for the people using the 7.4V stack of two, they sell voltage protection mini PCB for those applications for like $1-$2 and it will keep the batteries charging better. Un-protected ones could be really bad in your pocket.
