The power to overcome aerodynamic drag increases with the cube of velocity. This is why freeway speeds drastically impact your EV’s range. If you are math averse, you may want to stop reading here. For the rest of you, it’s time to nerd up.
The Nitty Gritty Details of Aerodynamic Drag
In 2005, David Foster Wallace told a story at the Kenyon College commencement that went like this: “There are these two young fish swimming along, and they happen to meet an older fish swimming the other way, who nods at them and says “Morning, boys. How’s the water?” And the two young fish swim on for a bit, and then eventually, one of them looks over at the other and goes, “What the hell is water?”
I’m going to tweak that a bit, and in my version of the story, two EV enthusiasts ask, “What the hell is air?”
Air is really easy to take for granted, even though it influences everything we do. Humans move through air rather effortlessly for day-to-day movement, and it is forgiven if one thinks of air as inconsequential. But as it turns out, air can be a substantial force; it can hold up an airplane and tear apart buildings. The simple act of moving through the air can take an incredible amount of energy.
Let’s peel this apart.
The Power Required to Overcome Aerodynamic Drag
Propelling a car down the road ﹘any car﹘ requires overcoming the friction of the tires, any spinning shafts and other mechanical innards of the car’s propulsion system, and overcoming aerodynamic drag. Friction is proportional to speed, but the power to overcome aerodynamic drag is proportional to speed cubed (as we shall soon see). Once a car approaches freeway speeds, aerodynamic drag quickly dominates sources of (typical) friction (ignoring the application of brakes). Let’s take a look at what makes up aerodynamic drag and the power to overcome it.
The power, Pd, required to overcome aerodynamic forces as a car travels down the road is given by the following equation:
Pd = Fd V
Where v is the velocity of the car, and Fd is the aerodynamic force which in turn is calculated as:
Fd = ½Cd ρAv2
It therefore follows the that the power to overcome aerodynamic drag simplifies to
Pd = ½Cd ρAv3
But what do all those variables mean?
- Fd is the force required to overcome aerodynamic drag, in Newtons
- Pd is the power required to overcome aerodynamic drag, in Watts
- Cd is the all-important coefficient of drag, which is specific to the car’s geometry
- ρ is the density of air, in kilograms per cubic meter
- A is the area of the front of the car, in square meters
- v is the velocity of the car, in meters per second
The two variables Cd (coefficient of drag) and the frontal area of the car, A, depend on the design of the car in question. Air density, ρ, depends on local conditions (i.e. the weather). Velocity, v, is simply how fast the car happens to be traveling. When you have all that information, you can calculate the power, in Watts, to overcome aerodynamic drag.
I took an aerodynamics course in the 1980s and recall the professor relating a story about how golf balls became dimpled. “Golf balls used to be smooth spheres, but golfers realized that used balls which had been hit a few times flew further and straighter.” Before writing this column, I spent a full 5 minutes on Google trying to determine if this story is true and found multiple sources quoting similar stories, but no real definitive source, though I did find the patent for the dimpled golf ball. While the story may be apocryphal, dimpled golf balls absolutely have better aerodynamics than an otherwise smooth golf ball.
What has this to do with aerodynamics, you ask? Quite a lot, actually, but we’ll return to the golf ball in a bit. Let’s look at what this means for a car.
The coefficient of drag, Cd, is a dimensionless number and is distinct from the cross-sectional area, A. The coefficient of drag and the cross-sectional area depends on the configuration and geometry of the object in question. A full-sized jumbo jet and a detailed scale model can have the same coefficient of drag, yet the cross-sectional area is vastly different.
For a given car, the shape isn’t going to change, but the air density will change due to temperature, elevation, and local weather. The air density given in the table below is at what is known as the standard pressure and temperature. Basically, sea level exists at 32℉ (Fun fact, the density of air is a pretty strong function of altitude, being less dense at higher elevations. It therefore follows that the aerodynamic drag will be about 15% less at Utah’s Slat Flats as compared to sea level. And at Bolivia’s Salar de Uyuni, it’d be a whopping 35% less.).
Comparison of Four Different EVs
For any given car, and in any given place, all of the variables reduce down to a constant times the velocity cubed. This result tells us what power is required, in Watts, to overcome aerodynamic drag. The table below gives the results for some popular EVs.
|Car||½ρ (kg/m3)||Cd||A (m2)||Pd = ½CdρAv3|
|BMW i3||0.6465||0.30||2.41||Pd = 0.4474v3|
|Mercedes EQS||0.6465||0.20||2.51||Pd = 0.3245v3|
|Rivian R1T||0.6465||0.32||3.04||Pd = 0.6289v3|
|Ford F-150 Lightning||0.6465||0.44||2.93||Pd = 0.8335v3|
|Porsche Taycan||0.6465||0.22||2.33||Pd = 0.3314v3|
Table showing the relationship of power and velocity for different cars.
This discussion ignores any wind. If there is a 20 MPH headwind and you’re traveling at 60 MPH, then the power required is equivalent to driving at 80 MPH. In a similar manner, if there is a 20 MPH tailwind and you’re traveling at 60 MPH, then the power required is equivalent to driving at 40 MPH.
If there is one thing I would the reader to take away from the above data, it’s that to travel at the same velocity, it takes the F-150 2.5 times (0.8335/0.3314) more power than the Taycan. That kind of power drain just to maintain freeway speeds has a huge impact on range and therefore the design of the entire vehicle, requiring larger batteries to go the same distance, with more weight and a higher cost.
For those reading the table closely, you may realize that all of the cars are German. There is a good reason for that, and that’s because Germans give the data for Cd and A right on their website. ‘Muricans are proud of their drag coefficient, and it is prominently displayed on their website, but they protect the area, A, as if it were nuclear launch codes; I simply couldn’t find a reliable source for A on any American car. However, I really wanted to compare cars to trucks, so I turned to a favorite YouTube channel to tease out the needed data for the F150 and R1T.
I was unable to find the drag coefficient for the F-150 Lightning and the cross-sectional area for either truck on the web (the ICE F-150 Cd is online, but the Lightning has a flat underside impacting the drag). So I turned to AirShaper, which I highly recommend if you are the least bit interested in this topic. This is an app you can access online (though it does cost money); however, you can learn just as much from watching this Youtube channel. The analysis revealed not only the data I needed, but also an unnoticed panel gap in the Rivian R1T that was scanned to make his 3D model.
It is fascinating that something as simple as a panel gap not only impacts the drag coefficient, but that the computational fluid dynamics behind AirShaper is sophisticated enough to tease it out.
Nuances of the Drag Coefficient
Let’s close this discussion by returning to the example of the golf ball and taking a 40,000-foot view of the drag coefficient, Cd.
The coefficient of drag, Cd, is strongly influenced by many factors. The anecdote about the golf ball did not occur by accident. Let’s take that one step further. If we shine a flashlight at a raindrop, a golf ball, a ping pong ball, and a circular disk, the shadow these objects will cast on a wall are identical. While these may have the same cross-sectional area (that’d be a big raindrop, but stay with me), they all have vastly different aerodynamic drag because they all have a vastly different coefficient of drag, Cd, which is 0.04 for the raindrop, 0.24 for the golf ball, 0.5 for the ping-pong ball, and 1.1 for the flat disc. (The raindrop is the most aerodynamic shape as it is shaped by nature.)
What makes all these shapes so different is how the air flows around them and if a turbulent wake is formed. Intuition can’t substitute for knowledge and experience when it comes to aerodynamics. The small change of adding dimples to the golf ball reduced the drag by half! The sleek “tail” of the raindrop reduces drag an astonishing amount.
The point of all this talk of drag coefficients, equations, and computer simulations and what I hope readers will remember months later is really two things. First, small changes to the shape of something can have large and non-obvious impacts on the overall drag. Last but not least, air is a force to be reckoned with, and as speed increases, it can take a lot of power to overcome the drag of “just air.” This is why the manufacturers of the most efficient EVs spend months of effort to reduce Cd by another 0.1, as it will pay huge dividends in making the energy stored in the battery take the car further down the road.
Next time you find yourself a bit short on range to make it to your next destination, remember that not only does speed kill range, but it also does so to the power of three.
Next week will be a much lighter subject – one pedal driving.