1. Attachments are working again! Check out this thread for more details and to report any other bugs.

MPG vs MPH

Discussion in 'Gen 2 Prius Fuel Economy' started by brosnan, Jul 24, 2004.

  1. brosnan

    brosnan Member

    Joined:
    May 2, 2004
    159
    11
    0
    Location:
    Silicon Valley
    Vehicle:
    2010 Prius
    Model:
    IV
    Hi - does anyone know what the expected relation between MPG and speed is? I'm thinking about the mileage you'd get at a constant speed on level ground on a really long trip. I know it depends on temperature, whether the A/C is on, etc and that this question ignores regeneration and acceleration issues and the details of when the battery is being charged and discharged.

    I was guessing that there are at least three components:

    - When not moving, or moving very slowly, there is an average rate of gasoline consumption per hour independent of speed (to run the A/C, keep the engine hot, etc).

    - When moving moderately slowly there is some extra gasoline needed per hour to overcome friction (tires, wheel bearings, etc). Is this proportional to speed?

    - When moving quickly there is an extra need for gasoline consumption per hour to overcome wind resistance. This is proportional to the cube of speed, right?

    I've put an example graph below thinking it would be great to know the real relationship in this format. The values are just made up by me. Can anyone fine tune this toward the correct values?
     
    #1 brosnan, Jul 24, 2004
  2. 8AA

    8AA Active Member

    Joined:
    Feb 29, 2004
    550
    62
    0
    Location:
    Maryland
    Vehicle:
    2004 Prius
    It's intuitive that bearing friction and wheel resistance would be proportional to speed, but they actually aren't. Their resistance is based on their physical qualities and the load pushing on them. For example, the wheel resistance will be proportional to the weight of the vehicle. You would also need to take the road condition into consideration, a rougher pavement will produce more rolling resistance. And don't forget about tire inflation.

    As for wind drag, it's actually related to the square of the wind speed, not the cube. Keep in mind that it's not the vehicle's velocity, but the apparent wind speed (you'll get better mileage with a tail wind).

    Like you said, this is only predictable for flat roads. Unless you're in Kansas, hills are going to throw all of the calculations about milage.
     
    #2 8AA, Jul 24, 2004
  3. brosnan

    brosnan Member

    Joined:
    May 2, 2004
    159
    11
    0
    Location:
    Silicon Valley
    Vehicle:
    2010 Prius
    Model:
    IV
    VP,
    Thanks for the corrections. Here's a revised graph using a square term for the wind drag. I shrunk the linear term but wasn't sure if I should delete it based on your friction comments. I do see how the friction would vary as you said (weight, road roughness, tires inflation etc). But for a given set of these is there any dependence of speed?

    Maybe we should get someone in Kansas to go for a few 5 hour drives on deserted roads at a few speeds! :wink:
     
    #3 brosnan, Jul 24, 2004
  4. mikepaul

    mikepaul Senior Member

    Joined:
    Dec 2, 2003
    1,763
    6
    0
    Location:
    Columbia, SC
    Vehicle:
    2004 Prius
    I *know* my car pulls 50MPG @ 70MPH on a FLAT road, as proven in Florida. However, I'd hate to estimate the work involved in establishing a graph for each 1 degree increase in incline one could be driving up...
     
    #4 mikepaul, Jul 24, 2004
  5. efusco

    efusco Moderator Emeritus
    Staff Member

    Joined:
    Nov 26, 2003
    19,891
    1,193
    9
    Location:
    Nixa, MO
    Vehicle:
    2004 Prius
    Model:
    N/A
    Brosnan,
    Have you seen/tried out Wayne Brown's THSII Simulator for PDA?

    This does exactly what you're talking about and he's done ALL the work and takes into account factors you probably never considered. The link above will download the program that can then be installed into your PDA.
     
    #5 efusco, Jul 24, 2004
  6. brosnan

    brosnan Member

    Joined:
    May 2, 2004
    159
    11
    0
    Location:
    Silicon Valley
    Vehicle:
    2010 Prius
    Model:
    IV
    No - I hadn't seen that. It sounds interesting. I'll have to track down a friend with a PDA and try it out.
    Thanks
     
    #6 brosnan, Jul 24, 2004
  7. LeVautRien

    LeVautRien Member

    Joined:
    Jul 23, 2004
    379
    2
    0
    Location:
    New Orleans
    Vehicle:
    2004 Prius
    Model:
    N/A
    I would think the MPG dropoff at the higher MPH rates would have to be exponential. I'm no math whiz by any means, and I definatly didn't look at your formula there, but the curve doesn't seem exponential.

    From what I remember, (and this certainly isn't absolutely right, it's just something I remember reading) drag increases exponentially against speed. The drag increase from 75-76 MPH is much much more than from 10-11.

    Again, I could be totally wrong...but that graph doesn't seem to strike me as too accurate when you start getting into 80, 90, and 100 (Not like it matters all that much).
     
    #7 LeVautRien, Jul 24, 2004
  8. brosnan

    brosnan Member

    Joined:
    May 2, 2004
    159
    11
    0
    Location:
    Silicon Valley
    Vehicle:
    2010 Prius
    Model:
    IV
    You may be right that at high speed the mpg value would fall off faster than in the graphs above. In my younger days I'd probably already have some personal data on high speed mileage, but I don't drive like that anymore!

    I looked around on the web and found quite a few sites explaining that the force needed to overcome wind resistance is proportional to the square of the relative velocity and the the power (in horsepower or watts) needed is proportional to the cube of the relative velocity. So I think the first graph/formula is of the right form - but the constants could be way off. When I try to adjust the constants so that there's a lot of falloff from 65mph (say 50mpg) to 100mph (say 20mpg), I get silly(?) results at 20-30 mph saying we could get 80 or 90 mpg. Hmmm...


    http://www.uwm.edu/~horowitz/PropulsionResistance.html
    http://www.atkinsopht.com/row/windfact.htm
    http://www.physics.gmu.edu/~amin/phys251/T...tileMotion.html
    http://www.exploratorium.edu/cycling/aerod...odynamics1.html


    For wind resistance component only (still air):
    Force ~ v^2
    Power ~ v^3
    Gasoline per hour ~ v^3 (Energy per time is Power) (if efficiency constant)
    MPG = MPH/GPH = v/GPH
    so MPG ~ v^2
     
    #8 brosnan, Jul 25, 2004
  9. LeVautRien

    LeVautRien Member

    Joined:
    Jul 23, 2004
    379
    2
    0
    Location:
    New Orleans
    Vehicle:
    2004 Prius
    Model:
    N/A
    Hmmm...I simply remember this all from when I studied for a pilot's license. The max speed of an airplane is limited because the drag increases faster than the engine power can...or something along those lines.
     
    #9 LeVautRien, Jul 25, 2004
  10. DaveinOlyWA

    DaveinOlyWA 3rd Time was Solariffic!!

    Joined:
    Apr 13, 2004
    15,140
    611
    0
    Location:
    South Puget Sound, WA
    Vehicle:
    2013 Nissan LEAF
    Model:
    Persona
    actually the dropoff in mpg would have to take into account the power output of the ICE at various RPM's.

    that is another huge reason why gas mileage drops fast after a certain speed.
     
    #10 DaveinOlyWA, Jul 25, 2004
  11. brosnan

    brosnan Member

    Joined:
    May 2, 2004
    159
    11
    0
    Location:
    Silicon Valley
    Vehicle:
    2010 Prius
    Model:
    IV
    Turns out there's a good discussion of this on another site: http://www.priusonline.com/viewtopic.php?t...=asc&highlight=

    The graph below now shows
    -some data extracted from the Wayne Brown PDA simulator (light blue -19 to 81 mph) according to the excel sheet posted by smoothchat ( http://www.priusonline.com/viewtopic.php?p=7187#7187 )

    -My formula with revised constants which best match with Wayne's values (dark blue 0 to 105mph). This formula is mpg=s/(0.184+0.0065*s+2.26e-6*s^3) s is in mph

    -PriusPhysics's values from http://www.priusonline.com/viewtopic.php?p=9903#9903 (dark red from 25 to 80mph)

    -The apparent formula used by PriusPhysics (magenta 0 to 105mph). The only parts that weren't clear from his posts were W2 (seems to be 0.0105*15362*speed_in_m_per_s) and W1(seems to be 0.26*2.5/1.5*speed_in_m_per_s^3)

    -Measurements from buggman (black dots) from http://www.priusonline.com/viewtopic.php?p=9880#9880

    -Measurement from dan (green dot) http://autos.groups.yahoo.com/group/Prius-...2G/message/4945
     
    #11 brosnan, Aug 1, 2004
(You must log in or sign up to post here.)