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If the mods are set at 10 watts and you vape the same amount of time per hour, the batteries will last about the same time.
so if you are pushing more volts to achieve the same wattage it is still the same?
Hmm -- That clearly has not been my experience, but I use stock atty's and cartos. It was significant enough that I stopped using LR on my vari devices.
I don't have a clue how the gauge of wire and type of atty used would affect the above....but I haven't fully studied subohm 101 and there could be some explanation?
My logic says if exact time and other conditions were the same, the higher ohm would have longer battery life.
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Watts(Power)=Voltage(Energy)^2 / Ohms(Resistance)
A variable wattage controller really just gives you a broader control over voltage, and it also adjusts the voltage accordingly to what coil you have on your device. I say broader because most mods increase in .5 watts and .1 volts. You can be a little more specific in your power range with voltage control in that circumstance.
Wattage isn't a determination of your battery life. Current is the determination to how long your battery will last. Using your specifications:
Amps=Square root(W/R)
Square root(10/0.7)=3.78 Amps
Square root(10/2.1)=2.18 Amps
I have to also state, that with a lower resistance I don't take as long of a pull as a higher resistance. The cooler vape makes me take a larger pull, so that could also be a factor in determining your battery life.
A variable wattage controller really just gives you a broader control over voltage, and it also adjusts the voltage accordingly to what coil you have on your device. I say broader because most mods increase in .5 watts and .1 volts. You can be a little more specific in your power range with voltage control in that circumstance.
Wattage isn't a determination of your battery life. Current is the determination to how long your battery will last. Using your specifications:
Amps=Square root(W/R)
Square root(10/0.7)=3.78 Amps
Square root(10/2.1)=2.18 Amps
I have to also state, that with a lower resistance I don't take as long of a pull as a higher resistance. The cooler vape makes me take a larger pull, so that could also be a factor in determining your battery life.
Lower resistance will result in a higher amperage draw and shorter battery life.
Yes, that's the ticket 
This uses 3.8A to attain 10W
This uses 2A to attain 10W, a batteries capacity is measured in mAh = milli Amperes/hour, so the less Amperage it uses the longer the battery will last.
Volts are determined by your batteries charge level and are not variable. Watts output is variable and will be set by changing the amp draw from the battery by your VW device. Yes, it's pretty close to the same.
Yes, of course. But we're talking set wattage (power), not set voltage here.
Here's a much better explanation of that than I could ever give anyone myself.
Yes -- VW. I learned this with my Darwin, which was the first VW device and my first vari device, via personal vaping experience. It was noticeable.
I would say perceived differences between VV and VW have probably been due to the equipment, not Ohms Law.
As I understand it Evolv chips measure the ohms and determine the voltage required to deliver the set amount of watts(power) to the atomizer. Provape VV delivers the voltage set to the atomizer. Perhaps too simplistic---but it is basically so. Many will agree that both do their job pretty darn accurately and consistently.
For all practical purposes, if the equipment was delivering the same voltage to the same atty, the vape would be identical.....whether it was regulated by Watts (power) or Volts.
Now -- adding in triple snakeskin xyq twisted flat fibre coil in a diametric pattern of the stars -- changes previous concepts on the practical use of very low ohm devices.....and the whole amp limit and voltage output becomes much more significant..........and any discussion on that is certainly above my pay grade.
Maybe I get to those studies next year?
PS -- and using the current Chinese VW/VW chips as the norm to compare VV and VW is due to the equipment, not the science. I believe those devices may use more battery power for VW than VV settings as I've seen some complaints to that effect.
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Tearose, have you read Rader's explanation of this stuff? The difference between VW, VV and battery output is significant.
VV and VW are conceptually the same as far as them both being regulated. Regulated devices draw the requested wattage (voltage) from the battery's voltage. Rader's blog post is definitive.
So the answer to the OP is that with a theoretically consistently efficient regulator, they would draw the same power from the battery at the same rate. Any difference in rate would be the result of differing regulator efficiency at different voltage.
The rate of usage may vary due to different coil performance profiles between the coils.
So the answer to the OP is that with a theoretically consistently efficient regulator, they would draw the same power from the battery at the same rate. Any difference in rate would be the result of differing regulator efficiency at different voltage.
The rate of usage may vary due to different coil performance profiles between the coils.
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Seems my work here is done. I will echo the performance difference between builds. I found that my battery charge life increased going from .7Ω to .5Ω. Granted this was on a mech mod and actual battery drain current went for ~6A to ~8A, but I was taking much shorter pulls to get the same effect. So even when battery drain current increased, the coil efficiency and overall effect increased by a positive net factor that resulted in better battery life.
I will most certainly have to study it more carefully. I continue to see VV and VW are conceptually the same, and the perceived differences would mostly be based on the equipment/electronics/chip.
I've certainly experienced various battery drain times depending on the vari device --- and in my layman's language ---- Clearly, some controllers are more efficient than others when it comes to battery drain.
That's as far as my studies will take me tonight.
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Watts is Watts, but the efficiency of buck/boost (or boost only such as the DNA20) has different efficiency for different volts out. Not to mention that you have IR (current through resistance) loss as you pass more current through the flyback inductor on the switching circuit and any (if any) wiring between the switcher and the 510-801/ego atomizer threading...
I will echo the performance difference between builds. I found that my battery charge life increased going from .7Ω to .5Ω. Granted this was on a mech mod and actual battery drain current went for ~6A to ~8A, but I was taking much shorter pulls to get the same effect. So even when battery drain current increased, the coil efficiency and overall effect increased by a positive net factor that resulted in better battery life.
Surely, vape style makes a difference in how long a battery lasts. If that factor is entered into the OP's question, imho there could be no answer.......as it is highly unlikely that the vape styles could be identical in the scenario's in the real world.
Citing one's personal observations is irrelevant to the discussion that was occurring.
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While my personal observation may not be directly relevant to the OP's question (that had already been answered by others), it does have relevance to discussion points that were introduced by other members. If the OP is concerned with battery life, electricity of one one piece of the pie. Improving coil efficiency or making slight changes in one's vape style, while still yielding the same net effect, can have greater effects on battery life than improving electrical efficiency.
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