It feels like methane is growing popularity to cook food while camping. I've only used the "classics" like methanol, propane and butane.
Why would I prefer methane gas as my camp stove fuel?
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I've never heard of a methane stove for camping but I can give some justification as to why you would not want to use it.
In summary, methanol, propane, and butane make good camping fuels because they can easily be liquified or compressed in a light container. Methane, has advantages for household use because of its much lower cost, but only if delivery lines for natural gas are available.
Methane is a biogas, so you get it from biological processes that otherwise would escape into the atmosphere.
When you burn methane, you are converting the methane into water and carbon dioxide. Carbon dioxide is a greenhouse gas, but methane is a more destructive greenhouse gas, so it makes sense to capture and use.
Propane, butane and others also convert to carbon dioxide and water, but these gases are mined. So where as methane is already in the biosphere, using petrol based gases is adding carbon to the biosphere from naturally sequestered sources.
Only real thing you need to know is that you need a different regulator for your gas burning equipment. Methane is a lighter density gas.
I should add that methanol is a perfectly good and sustainable fuel, it is the liquid alcohol form of methane. Methanol although coming from other processes rather than gas capture at least still comes from the biosphere in the form of wood as the usual source.