arguably one wants to measure volume because that is what the fuel tanks are
We could just as easily use mass for the tanks capacity, as Avgas density is pretty stable.
arguably one wants to measure volume because that is what the fuel tanks are
But it is the mass of the fuel (the number of molecules, not their volume) which produces the power.
Avgas expands 0.1%/K so you can get a few % error between filling up at +35C and climbing to -35C (7% in volume actually).
So if you took on say 86.2 USG (TB20 max usable) at +35C and climbed to -35C, your tank contents has shrunk by 7% so very soon after top of climb you have 80.6 USG. You still have the same mass but if you are measuring the volume flow (as all GA flowmeters are, including the pressure based gauges) then you have 5.6 USG less in there than you think. So when the totaliser is saying you have 20 USG you actually have only 14.4.
But since the distance the plane will fly (assuming given airborne conditions) will be according to the mass of the fuel, the error in the flowmeter doesn’t matter. Well, it does if you are flying around the peak EGT area, where changing the fuel flow a bit doesn’t change the power, and you are setting a constant volume flow. Then, 10.5 GPH at -35C will be consuming your tank contents 7% faster than 10.5 GPH at +35C.
I hope I got that right – it’s one of those mind bending games… 
For a liquid, one could correct volume flow to mass flow trivially, with temperature compensation. But nobody in GA does that, AFAIK. And for sure nobody is using coriolis flowmeters.
Incidentally, the 11 GPH fully-open figure for the fuel distributor must have a very interesting impact on collecting the GAMI data. You definitely want to do it above 11 GPH if you normally cruise above 11 GPH. But nobody flying non-turbo above ~ 8000ft will be running 11 GPH. At FL160 it will be about 9 GPH. So clearly a perfect injector match cannot be achieved if you do the test at a low level, as most do.
Or one could do the sensible thing and measure both flow and capacity in mass.
Then you would need fuel tank temp sensors. And would need to either always fill right up, or do something with the fuel you buy, which is never (in GA) measured in mass.
If you measure mass flow and fuel mass vs volume what would the fuel tank temp sensor do?
Just because GA always measures by volume doesn’t mean it’s the only solution.
Airlines and GA jets dont have a problem to operate by mass, almost ever without uploading max fuel.
Airlines and GA jets dont have a problem to operate by mass, almost ever without uploading max fuel.
True, and how do they do it?
Measure block fuel, subtract trip fuel?
If the question referred to “how is quantity on board gauged” – on the types i am familiar it is with a number of capacitors. Density is either assumed from the resistance or obtained from a densimeter (ie you can have 737s with or without densimeter). Fuel gauging tolerance varies based on whether the densimeter is installed or not. The fuel gauge temp serves for fuel temp dispatch limits not fuel-on-board calculations. This is from memory from back when I was dispatching for BA. Technology might have changed since.
True, and how do they do it?
By trusting your fuel gauges. They are quite accurate (+/- 50lb out of 5400lb tank capacity in case of “my” C560). And since we measure and indicate both fuel quantity and fuel flow as mass we don’t have to care about temperature/density/volume. There is a fuel temperature display, but as Shorrick already wrote, this is for dispatch limits (-38 degrees Celsius or similar, not a number that I need to memorise because we never see these figures here).
