Q: How many candles does it take to power a bicycle?
A: Twelve. More if you want to go faster.
This is only an estimate, but it's a good one. These are my assumptions:
A person powers a bicycle with between 200-400 Watts. Split the difference, and it's right around 1/2 Horsepower (which is 745.7 Watts.)
A middle of the road candle produces around 75 Watts of heat energy, and I'll ignore the light energy because it is so minimal. I can just round this to 1/10 HP if i want a quick easy estimate.
I need to convert heat energy to useful work. Any time you want to do this, you basically need to use some kind of heat engine. Theoretically speaking, the most efficient theoretical type of engine is the Carnot Engine... no engine is actually as efficient as a Carnot engine, but this should give a good maximum ceiling. It's easy to calculate the 'Carnot Efficiency' if you know the maximum and minimum temperatures your system will use. (Heat energy is a funny thing; it won't do anything if it is sitting there by itself. It must move from one place to another to actually do anything. Oh, and the only thing that makes heat energy move is a difference in temperature... heat energy flows from hot to cold. Never in the other direction. Not unless you push it.. but that's another story...)
You can calculate the efficiency of a Carnot cycle engine with the following equation:
n=1- (Tc/Th )
n is the efficiency
'Tc' is the cold temperature (in Kelvin).
'Th' is the hot temperature (in Kelvin).
Tc = Let's just say it's room temp, which is 295K.
Th = I've had a candle go as hot as 2550˚F, which is 1670K.
n = 1- (295/1670) = 82%
So, if I had an actual Carnot cycle engine, I could expect to have access 82% of the power from a given candle. Carnot cycle engines don't exist, but we can use the next best thing, which is the Stirling Cycle engine. By the time you've got belts on it and have it driving a bicycle, a practical Stirling cycle engine might only be half as efficient as the Carnot engine. I'll just assume that it can transform 40% of a candle's heat into work.
So, we can just put it all together like this:
A candle that produces 75W of heat power can be used to generate 75W * 40% = 30W of mechanical power.
I want 1/2 horsepower to push my bicycle. That's 373W.
Divide 373W by 30W per candle and you get 12.45. That's twelve candles. Yaay! Science is fun! :-) Mostly because it answered my question...