How can acceleration affect fuel efficiency in a 1991 Geo Metro?

[Amadameus]

Greetings everyone!

Just a few weeks ago, I got it in my head that I'd like to determine some formulas for the efficiency of an engine - particularly mine, a 1991 Geo Metro.

The overall goal is to compare various acceleration rates to show how slower acceleration can produce greater gas mileage(and also to determine the lower bound to this trend - certainly accelerating for an hour to reach 25mph is not going to be very efficient!).

Very quickly I learned that there are many, many factors involved with which I was not immediately familiar. Not the first of this being the many-layered approach to acceleration. Just wrapping my brain around concepts like http://en.wikipedia.org/wiki/Jerk_%28physics%29" , torque, air resistance, and internal friction in enough to make my head spin!

Suffice to say, I've uploaded https://spreadsheets.google.com/ccc?key=0AoLwpZGcpaBqdFJDUmNVRzhzT284VUxhRmd3YXg0dXc&hl=en&authkey=CPSJv-0J" that's quite a mess right now. I haven't accounted for many, many factors - and some of the initial calculations (although they appear correct) seem to imply that you can get better gas mileage by accelerating faster!

Clearly there's quite a bit wrong with this. However, I'll be working on it. I've also opened up permission for anyone to edit the file, if they feel like being generous and helping out.

Updates to the file and such will be posted here. Cheers!

 

[xxChrisxx]

I think you'll find this isn't as simple as you'd think as fuel flow isn't linear. So for example: if you accelerate at 50% throttle in 1st gear you'll accelerate faster than 100% throttle in 3rd (assuming the same starting speed) but 3rd will use far more fuel as the mixture becomes as rich as it's going to go at full throttle acceleration.

There is however one fairly obvious error I can spot in the spreadsheet. You power output increases with increased time, it should drop. Also the convertion from kW to HP is of by a factor of 1000 (ie you've done W -> HP).

Your aero drag numbers also look a bit funny to me, they should raise much much more sharply than they should. I'm not going to work it out, but at 80mph your drag froce in N should be probably 10x what it is.

Gettting on to nit picky stuff, it appears you've assumes as constant acceleration and therefore constant force output. This is fine if the assumption is intentional, but doesn't reflect how a car really accelerates due to the torque curve and gearing.


Looks like a fairly good start though. With all model building i'd start with the very very basics first, then make it more complicated. So I'd ignore all drag for now, and assume constant acceleration, get a spreadsheet with that working first.

 

[Amadameus]

I think you'll find this isn't as simple as you'd think as fuel flow isn't linear. So for example: if you accelerate at 50% throttle in 1st gear you'll accelerate faster than 100% throttle in 3rd (assuming the same starting speed) but 3rd will use far more fuel as the mixture becomes as rich as it's going to go at full throttle acceleration.

There is however one fairly obvious error I can spot in the spreadsheet. You power output increases with increased time, it should drop. Also the convertion from kW to HP is of by a factor of 1000 (ie you've done W -> HP).

Your aero drag numbers also look a bit funny to me, they should raise much much more sharply than they should. I'm not going to work it out, but at 80mph your drag froce in N should be probably 10x what it is.

Gettting on to nit picky stuff, it appears you've assumes as constant acceleration and therefore constant force output. This is fine if the assumption is intentional, but doesn't reflect how a car really accelerates due to the torque curve and gearing.


Looks like a fairly good start though. With all model building i'd start with the very very basics first, then make it more complicated. So I'd ignore all drag for now, and assume constant acceleration, get a spreadsheet with that working first.

The part that's killing me right now is what you noticed: power output increases with time. It should be the other way around, but as I check my numbers I can't see my mistake. Argh!

The spreadsheet I put online is incredibly simple - it models linear acceleration up to a single speed, doesn't account for drag (although it has some data so it can in the future) and doesn't account for engine idle or any of a myriad factors. It assumes engine power output is reflective of gas used, for one.

Point is, I'm still in the process of adding more. Thanks for bringing the drag numbers to my attention - I had tried some shorthand (instead of running the true formula, I found a "quick fix" on a drag racing site) that clearly isn't accurate.

I'm planning to add things like transmission speeds, gear changes, engine output at various RPMs and inefficiency as acceleration increases. (More gas flow doesn't equate to a linear increase in power - in fact it drops off dramatically)

Thanks very much for the help!

So far, here's my checklist:
[X] Power consumed to accelerate to given speed
[ ] Engine efficiency without acceleration at given speeds
[ ] Transmission ratios and possible engine RPMs at speeds
[ ] Engine power as a function of RPMs
[ ] Drag forces (engine idle, wind/rolling resistance, clutch and transmission loss)

 

[Bob S]

Two things:

1) The engine efficiency is best modeled using a BSFC (brake specific fuel consumption) map. See thumbnail for a conventional 2.7L engine. Vertical axis is torque, and horizontal axis is RPM. The contours are for constant fuel consumption (grams of gasoline) per kilowatt-hour of output energy measured at flywheel. Constant power contours are hyperbolas (see lower left corner). The highest efficiency in this map (83 grams/~250 grams = 33%) is at ~ 75% of max torque and ~35% of redline.

2) The total kinetic energy of the car at velocity v is KE = ½mv²

The power output is P = d(KE)/dt = m·v·dv/dt = m·v·a

So at constant power, the maximum acceleration is a = P/(m·v)

So acceleration a for constant power scales as 1 over velocity.

Whenever the acceleration a exceeds ~0.4 g (two-wheel drive) or 0.8 g (4 wheel drive), the horizontal acceleration force is close to the maximum without losing static friction contact with road (burning rubber).

Bob S

 

[Danger]

It certainly seems like a formidable project, Amadameus. If there's no particular reason why you have to undertake it via computer, I'd recommend an empirical study instead. Chart your actual real-world experiences during differing trials. There will, of course, be complicating factors such as varying traffic and weather conditions, but that merely shows the difference between reality and simulation.

 

[Amadameus]

Thanks so much! That's helped solve the most vexing problem I've seen in the entire project: making the transition from physics-based watts to reality-based gallons of gasoline.

Excellent! I'll be doing more work immediately. Expect updates!

(EDIT) Empirical testing will be done as well, have no fear. I intend to compare them to "ideal" math estimates, but for now I'm avoiding the legwork. This is a math project first and a physics project second. (No offense guys!)