Example:
- ICE car gets 30 miles per gallon at 65 mph at 30% propulsion efficiency.
- 33.7 kWh per gallon of gasoline
What is power of wind drag assuming rolling resistance is negligible?
33.7 kWh potential energy/1 gallon
* 30% propulsion efficiency
* 1 gallon / 30 miles
* 65 miles/1 hour
= 22 kW of wind drag power
If the gas car has an 18-gallon tank, it can sustain 30 mpg * 18 gal = 540 miles of range in this operating condition before running out of fuel.
If the speed is increased to 80 mph, the drag becomes 22*(80/65)^2 = 33 kW.
33 kW of wind drag power
/ 30% propulsion efficiency
* 1 hour/80 miles
* 1 gallon/33.7 kWh
= 0.0408 gallons per mile
--> 24.4 mpg
24.4 mpg * 18 gal = 439 miles range = 19% range loss vs 65 mph.
Now let's take the same car and substitute a Model 3 powertrain with 85 kWh of useable battery and 90% propulsion efficiency.
22 kW of wind drag
/ 90% propulsion efficiency
* 1 hour/65 miles
= 0.376 kWh of battery drain/mile
--> 2.66 miles/kWh
Range at 65 mph is 2.66 miles/kWh * 85 kWh = 226 miles
Again comparing at 80 mph:
33 kW of wind drag power
/ 90% propulsion efficiency
* 1 hour/80 miles
= 0.461 kWh of battery drain/mile
--> 2.17 miles/kWh
2.17 miles/kWh * 85 kWh = 185 miles range = 19% range loss vs 65 mph. Same as the percentage loss for the ICE version, and this will always be true in general as long as propulsion efficiency doesn't vary with speed. This isn't quite true but it's close enough for this approximation.
General formula:
Wind drag power / propulsion efficiency / speed = stored energy used / mile
range = stored energy total / (stored energy/mile)
range = stored energy tot / (wind/efficiency/speed)
range = stored energy tot * efficiency * speed / wind
r2/r1 = (stored energy2 * efficiency2 * speed2 / wind2) / (stored energy1 * efficiency1 * speed1 / wind1)
Stored energy and propulsion efficiency are constant
r2/r1 = (
stored energy2 * efficiency2 * speed2 / wind2) / (
stored energy1 * efficiency1 * speed1 / wind1)
r2/r1 = speed2/speed1 * wind1/wind2
w2/w1 = (speed2/speed1)^2
r2/r1 = speed2/speed1 * (speed1/speed2)^2
r2/r1 = speed1/speed2
Define x = energy/mile
x = Wind drag power / propulsion efficiency / speed
x2/x1 = (Wind2 / efficiency2 / speed2) / (Wind1 / efficiency1 / speed 1)
Propulsion efficiency is constant
x2/x1 = (Wind2 /
efficiency2 / speed2) / (Wind1 /
efficiency1 / speed 1)
x2/x1 = (wind2/wind1) * (speed1/speed2)
w2/w1 = (speed2/speed1)^2
x2/x1 = (speed2/speed1)^2 * (speed1/speed2)
x2/x1 = Speed2 / Speed 1