Disclosure: I make and sell brushed aluminum beer can pots. Originally I brushed the cans for aesthetic reasons. I continue to brush them because of test results like those below.
One huge variable that is difficult to control is relative humidity, unless you have both a humidifier and a de-humidifier, and a hygrometer to measure it. I get a difference of about 30 seconds in boil times in 65% vs. 85% humidity.
At any rate, I did 5 boils each using the cans pictured below over the course of two and a half hours.
Same air, fuel and surface temperature: 63.5* F.
Same quantity of water: 500.0 grams (500.0 ml).
Same quantity of fuel: 12.62 grams (16.00 ml).
Same relative humidity: 68%
Doing it in a two and half hour period with no apparent change in the weather, I presume essentially the same atmospheric pressure as well.
"Boil" measured as 212* F with a digital thermometer.
I did one burn for the series of cans (one shiny, one black, one brushed), then a second burn in the same sequence, etc. Used a series of air-temperature ceramic tiles for the resting surfaces, to avoid any surface temperature variables.
Here are the results I got:
1. Shiny original can:
6:56 (416 sec)
7:02 (422 sec)
7:06 (426 sec)
6:58 (418 sec)
7:05 (425 sec)
Average = 421.4 sec = 7:01.4 min
Maximum deviation = 10 sec = 2.4%
2. Bottom of brushed can blackened to top of windscreen by soot from crappy de-natured alcohol brand (do NOT buy E-nrg alcohol, it burns dirty and stinks too):
7:20 (440 sec)
7:15 (435 sec)
7:24 (444 sec)
7:18 (438 sec)
7:22 (442 sec)
Average = 439.8 sec = 7:19.8 min
Maximum deviation = 9 sec = 2.0%
Clean brushed can:
6:36 (396 sec)
6:29 (389 sec)
6:38 (398 sec)
6:31 (391 sec)
6:40 (400 sec)
Average = 394.8 sec = 6:34.6 min
Maximum deviation = 11 sec = %2.8
I expected the brushed can to absorb heat faster than the shiny can, but I did not expect that the soot-blackened can would be slowest. I had thought that because it was black on the bottom, it should have been the fastest. Perhaps the soot acts as an insulator, or emits IR faster than it absorbs it (is that even possible?), or....?
I did not have any high-temp black paint to try a can with just the bottom painted black. Will try that soon, when I get the same air temperature and humidity. If I was really smart, there is probably some formula that lets you account for those variables and get comparable results, but I don't know what it is.
[edited to add disclosure]