UTAlumnus — Your examples are fallacious, I'm afraid. Since you seem to want the whole enchilada, I'll give you the whole enchilada! Here's the reasoning.

Metric example: During a trip, let L (liters) denote the amount of fuel used up to an arbitrary point x (kilometers). Then in L/100 km:

Trip FC = (trip liters) / (100's of trip kilometers)
= (integral dL) / (trip kilometers/100)
= (100 * integral dL) / (trip kilometers)
= (100 * integral {dL/dx}*dx) / (trip kilometers)
= (integral {100*dL/dx}*dx) / (trip kilometers)
= distance-weighted average of the instantaneous FC in L/100 km

To clarify: The second-to-last line is the mathematical definition of the "x-weighted average of {100*dL/dx}," and {100*dL/dx} is just the instantaneous FC in L/100 km. The car's metric FC gauge shows you the instantaneous L/100 km as you drive. You need to average this with respect to distance in order to arrive at the trip-average FC. This is feasible.

US example: During a trip, let G (gallons) denote the amount of fuel used up to an arbitrary point x (miles). Then in mpg:

Trip FE = (trip miles) / (trip gallons)
= (integral dx) / (trip gallons)
= (integral {dx/dG}*dG) / (trip gallons)
= volume-weighted average of the instantaneous FE in mpg

To clarify: The second-to-last line is the mathematical definition of the "G-weighted average of {dx/dG}," and {dx/dG} is just the instantaneous FE in mpg. The car's US FE gauge shows you the instantaneous mpg as you drive. You need to average this with respect to volume in order to arrive at the trip-average FE. This isn't feasible.

Time does not enter into these trip-average calculations at all.

Stan

 

You're right that I'm using imaginary numbers for amount of fuel used. This was to make calculations simpler for posting. But time does still come into the equation.

Since you are basing the calculation on distance and where you are along the distance determines which dL/dx function you are integrating time has to be a factor. Acceleration is the second order differential of distance with respect to time. The equation needs to be broken up such that your equation integratges instantaneous FC over the accelleration distance + instantaneous FC over the cruising distance.

Acceleration would be approximately constant with a manual transmission. Integrate w/ respect to time = Velocity. Integrate velocity w/ respect to time = Distance. This is only approximate due to the CVT and the effect of speed on air resistance.

 

UTAlumnus — Sorry, but my response is still "No!" Try to construct a proof of your claim, and you'll see that you are indeed wrong. Time enters only indirectly, since both speed (and hence distance covered during a time interval) and fuel used (during the same time interval) are functions of time. But this time dependence exactly cancels out in computing the trip FC. The trip FC depends only on the instantaneous FCs and the distances over which they applied.

Stan

 

Just as a random note on this, there is one instance where time and fuel rate are completely disconnected from distance so instantaneous FC is only relevant based on fuel usage/time--when you're idling. Any time you're doing 0 mpg, there's no way to say how much fuel you're using except as a rate vs time computation, because 0 m/gal = 0 mpg no matter how many gallons you burn. Not sure where this plugs into the discussion, as I've just been skimming the math, but did want to remind anybody scanning the thread that you can't only take the instantaneous FE/FC gauge and determine the trip result from the curve of distance vs fuel, because any time the ICE runs while stopped you have no way to measure what 0 mi/? gallons really means. And little or no view into real fuel usage. So clearly the trip result is based on total fuel used vs total distance traveled. And isn't that enough?

Otherwise, don't mind me. Didn't mean to interrupt your argument fun.

 

Actually, maybe I am stupid and wrong, but this would fit in easily. Wouldn't it be factored into the FE equation kind of in the reverse that the Synergy system is. When you are running on pure electric the miles driven get thrown into the mileage used to calculate your MPG. So say you had number like 32m\1g while using the ICE and then went another 3m on electric only then the equation would be 35m\1g. Basically when you are idling you are just increasing M. But I think the only way you could truly measure this would be to have an accurate scaled volume measurement device hooked to the fuel tank and not the "guessgauge".

 

chestr — A good try, but no dice! Idling is an interesting case, though, as you point out.

• For FE, as I showed, you need to average the instantaneous FE (in mpg) with respect to fuel usage in gallons. If you're not moving, your instantaneous FE is 0 mpg, and averaged with respect to gallons of fuel consumed, it does indeed give the correct answer of 0 mpg overall.
• For FC, the instantaneous FC (in L/100 km) needs to be averaged with respect to distance in kilometers. If you're not moving, the instantaneous FC is infinity. When averaged with respect to distance covered (zero) the integral becomes mathematically an "indeterminate form" whose value should be interpreted as infinity by using the original definition of trip FC: (liters used) / (kilometers travelled) = (liters used) / (zero) = infinity.

Again, time does not directly enter into these equations. Please note that in the FE case, the integral of instantaneous FE with respect to gallons of fuel used gives precisely the slope of the chord joining the endpoints of the graph of miles vs gallons, from the mathematical definition of "average value of a function." The slope of this chord is (total miles) / (total gallons) = trip FE in mpg of course! A corresponding statement applies to FC in L/100 km. These statements are just tautologies, however, since they amount to simply restating the definitions of trip FE and trip FC. The whole point of this discussion is to relate these overall measures to the respective instantaneous measures.

Stan

 

coolshock1 — You raise the converse case to chestr: moving in EV mode, no fuel being used. Then the instantaneous FE = infinite mpg, and the instantaneous FC = 0 L/100 km. The "zero" and the "infinity" change places in the descriptions I gave in the two bullets in post #26. Corresponding conclusions apply.

Stan

 

That's what I tried to state with the bolded statement except that I mistyped. M should have been G. When Idling you add more Gallons or volume into the equation.

 

Well that was my point, it's easy to determine your instantaneous mpg at rest while idling is 0mpg, but for some variable amount of fuel burned over a distance of 0 mi, you have no way of using that 0 mpg instantaneous FE to determine an overall result. Your distance doesn't change at all and your instantaneous FE doesn't change at all, but your trip FE is steadily decreasing as long as your ICE keeps burning fuel. There's no way to tie that fuel burn to any "total miles" just looking at the instantaneous results, the only way to plug that fuel into the bigger general equation of delta-d/delta-V is to have a non-instantaneous view of how much fuel was burned during that time of stationary fuel burn, and/or have a fuel-burn/time view to produce that information. If you and I both have the exact same drive with the exact same distance covered and the exact same fuel usage curve while we're in _motion_, the one who sits idling for more _time_ is still going to be the one who loses by burning more fuel and getting lower overall FE. Basically your "with respect to fuel usage in gallons" line is the problem for me, because you can't instantaneously know that unless you're monitoring fuel usage vs time. There's no "I'm going 0mpg now so that means since I'm going 0 miles I'm using x gallons", the way you can do while in motion.

Counterargument?

 

My brain is swelling....some of us might be too smart for our own good....actually I am jealous in a sense....