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Automotive Engineering DISCUSSION
I want a first-principles estimate of motorway range for an electric car: mass 1800 kg, drag coefficient 0.28, frontal area 2.3 m², usable battery energy 60 kWh. Published range figures do not tell me how much is lost at higher speeds.
How do I calculate the power needed at a steady 100 km/h and 130 km/h, turn that into Wh/km and range, and which assumptions have the largest effect on the result?
At steady speed on a level road the force is aerodynamic drag plus rolling resistance. Drag = ½ × ρ × Cd × A × v² = 0.5 × 1.2 × 0.28 × 2.3 × v² = 0.386 × v². At 100 km/h (27.8 m/s) that is 298 N. Rolling resistance = Crr × m × g = 0.010 × 1800 × 9.81 = 177 N, taking a coefficient of 0.010 for low-resistance tyres.
The total is 475 N, and power at the wheels = 475 × 27.8 = 13.2 kW.
From wheels to battery, divide by the drivetrain efficiency (inverter, motor, reduction gear), around 0.88 as a round figure, and add the auxiliary load: 13.2 / 0.88 = 15.0 kW, plus about 0.5 kW for electronics and pumps, 15.5 kW. Over one hour you cover 100 km, so consumption is 155 Wh/km and the range is 60,000 / 155 = about 390 km.
At 130 km/h (36.1 m/s): drag = 0.386 × 1304 = 504 N, total 681 N, wheel power 24.6 kW, battery power about 28.4 kW, consumption 219 Wh/km, range about 275 km. A 30 percent increase in speed costs about 30 percent of the range.
The assumptions that move the answer most, in rough order: speed, because drag power goes with the cube of speed; air density and wind, since a 20 km/h headwind at 100 km/h raises the drag force by 44 percent, (120 / 100)²; and cabin heating, which can be several kilowatts with a resistive heater in cold weather and is a fixed power, so it hurts most at low average speeds.
Rolling resistance also rises in the cold and on wet roads, and the usable battery energy falls at low temperature. A calculation like the one above is a best case for a mild, calm day.
Mass matters less than people expect at steady speed, since it only enters through rolling resistance: 100 kg more adds about 10 N, roughly 2 percent of the total at 100 km/h. It matters in stop-start driving, where kinetic energy ½mv² is put in at each acceleration. Regenerative braking returns a good part of that, but not all, because the energy passes through the motor, inverter and battery losses twice.
Climbing is the other mass term: raising 1800 kg by 500 m takes m × g × h = 8.8 MJ, about 2.5 kWh, of which a part is recovered on the way down.
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