Redoing the highway math

Posted on May 29, 2008


I haven’t had the time or the patience to repeat the roll-down test again. However, I found an old picture file I took out of the Web a few years ago that has all the measures of my car, and, most importantly, it has a perfect frontal area projection! 🙂

With a little help from “The Gimp” and some basic maths I was then able to get a very precise frontal area for my car: 2,30 m2.

Then I used the spreadsheet again and this time the values appear a little bit better:

Cd = 0,364

Crr = 0,013

I wouldn’t put my hand in fire for the trueness of these values, but they seem very likely.

So, taking Cd = 0,364 and Crr = 0,013

The force to maintain the car at 120 km/h on a horizontal road would be:

  • Velocity = 120 km/h = 33,3 m/s
  • F(air) = – Cd * Frontal_Area * 0,5 * Air_Density * V^2
    • = -0,364 * 2,41 * 0,5 * 1,22 * 1108,89 = -593,38 N
  • F(road) = – Crr * Mass * Gravity
    • = -0,013 * (1080+120+30) * 9,81 = -156,86 N
  • F(air+road)@120 = – 750,24 N

So the necessary power would be:

  • P(120km/h) =F * V
    • = 750,24 * 33,3 = 25 kW (~34 HP)

And the equivalent necessary torque:

  • T(120km/h) =F * Wheel_Radius
    • = 750,24 * 0,273 = 204,82 Nm

So each motor would have to continuously handle 12,5 kW and 102,41 Nm. Pretty easy for today’s technology.

Now let’s think about this for a minute.

The “GM Volt“, it is said, will have:

  • a 16 kWh battery pack (next-generation Lithium-Ion), with a useful charge window of 50% (8 kWh);
  • a 120 kW motor (3-phase induction);
  • a 53 kW generator.

This means a fully charged “Volt” battery (8 kWh) would propel my car at 120 km/h for 19,2 minutes (60*8/25) or 38,4 km, which is not that bad. And the on-board generator would charge the battery in just 9 minutes (60*8/53), so I would enjoy almost 10 minutes of silence followed by 9 minutes of engine roar, and so on. Hmm… it does have the potential to become irritating, if the engine cabinet is not sound-proof enough.

Of course, if I also had the Volt’s electric motor and permanently slammed on the gas (“redlining” the system) the full 120 kW would totally discharge the battery in just 4 minutes (at an unknown speed) and the generator would not be capable of recharging the battery on time, effectively stopping the car!! 🙂 (That won’t happen in the “Volt”, I hear it has a “bypass” mode, so the generator can feed the motor directly if the battery is kaput. But it will still limit the car to 53 kW of power, if that happens.)

I really like these numbers. There is a nice symmetry to them: the generator is rated at half the power of the motor. I wonder what kind of recipe GM followed to get at this result… city driving cycle statistics? Must be. It also makes my life simpler in building my hybrid: I can build 3 equal motors, and use one as the generator (coupled to the combustion engine). 😉