Newtons are a unit of mileage

The conventional units for expressing the mileage of an electric vehicle, like Wh/km (“watt-hours per kilometer”), are dimensionally equivalent to force, N (“newtons”). It seems this has a useful physical interpretation, and we can do some rough predictive calculations by estimating the drag and rolling resistance forces that align to real-world observed efficiencies.

We recently got a battery electric vehicle. We took a road trip which both left me with new dials and sensors to play with, and involved many hours of boring highway driving… the perfect time for some dimensional analysis!⁰

Following the units led to the surprising-to-me conclusion that energy consumption is a force. I think it has the sensible interpretation of being some sense of average force on the vehicle as it travels. This automatically captures how some energy in the battery is converted and recovered¹, rather than thrown away:

• driving up a hill and then back down is just temporarily outsourcing energy storage from the battery to gravitational potential energy
• accelerating and then regeneratively-braking is outsourcing to kinetic energy

The remaining force are non-recoverable losses like drag, rolling resistance, the entertainment system, and climate control.

EquiValent units? Guilty as charged

Electric vehicle (EV) efficiency or mileage seems to be conventionally expressed in either:

• Wh/km or kWh/100km, similar to L/100km² for internal-combustion-engine (ICE) vehicles
• km/kWh or miles/kWh, similar to miles per gallon for ICE vehicles

The first ones have their components cancel down to just N, a unit of force, in two steps³:

1. Wh is equivalent to Newton-meters: $$\begin{align*} 1 \unit{Wh} &= 3\,600 \unit{Ws} \\ &= 3\,600 \unit{J} \\ &= 3\,600 \unit{Nm} \end{align*}$$

2. Substituting that back gives us the result, Wh/km is equivalent to N, and the specific conversion factor: $$\begin{align*} 1 \frac{\unit{Wh}}{\unit{km}} &= \frac{3\,600 \unit{Nm}}{1\,000 \unit{m}} \\ &= 3.6 \unit{N} \end{align*}$$

Similarly, 1 kWh/100km = 36 N. Thus, for instance, 200 Wh/km = 20 kWh/100km = 720 N.

On the other hand, km/kWh ≡ N^-1, the reciprocal. This is easily converted, but harder to work with and doesn’t seem like it has direct physical interpretation that’s as “nice”, so I’ll focus on the other two.

Back of the EnVelope

We’ve got connection between units on paper, but does it mean anything? Can we predict EV efficiency by adding up some forces to get a N number?

It certainly seems like it.

At highway speeds on flat ground, I believe the dominant forces on the car are:

• rolling resistance of the tyres against the road
• drag/air resistance of moving through the atmosphere

Other factors like climate control and the entertainment system seem like they’d be much smaller.

For that road trip, we averaged approximately 190 Wh/km = 684 N. There was a mix of slow roads and highways, so using that number isn’t perfect… but most of the distance was spent on highways, and we’re just doing rough estimates here, so I reckon it’s good enough!

Let’s substitute the car’s numbers into the equations for those forces³:

• The formula for rolling resistance combines the normal force $N$ (in this case, just the weight of the car: mass $m$ and gravity $g$) with a coefficient $C_{rr}$ that captures the behaviour of tyres and the surface and their interaction, which I estimated as 0.0077 (the specific tyres on the vehicle are class B on the EU fuel efficiency rating, which is for rolling resistance coefficients between 0.0066 and 0.0077): $$\begin{align*} F_{rr} &= \textcolor{red}{C_{rr}} \overbrace{N}^{\textcolor{blue}{m}\textcolor{green}{g}} \\ &= \textcolor{red}{\overbrace{C_{rr}}^{0.0077}} \ \textcolor{blue}{\overbrace{m}^{2100 \unit{kg}}}\ \textcolor{green}{\overbrace{g}^{9.8 \unit{m/s^{2}}}} \\ &\approx 158 \unit{N} \end{align*}$$

• The formula for drag combines the density of air $\rho$, with the velocity $v$ and the aerodynamics of the car. Hyundai suggests the drag coefficient $c_d$ is 0.228 for our Ioniq 5, and I estimated the frontal area $A$ from the width (1890 mm) and the height of the car (1605 mm), pretending it’s a rectangle:

  $$\begin{align*} F_d &= \frac{1}{2} \textcolor{red}{\overbrace{ρ}^{1.225 \unit{kg/m^2}}}\ \textcolor{blue}{\overbrace{v^2}^{110 \unit{km/h}}}\ \textcolor{green}{\overbrace{c_d}^{0.288}}\ \textcolor{purple}{\overbrace{A}^{3.0 \unit{m^2}}} \\ &\approx 500 \unit{N} \end{align*}$$

Adding those up gives a total force of 658 N, only 4% away from the 684 N average from my road trip! Potentially a coincidence, but I hope not. Maybe there is something to this energy consumption = force equivalence.

Speeding: assault on battery

Our numbers are close, so we’ve “proved” the rule. Let’s go all in, and make predictions and generalisations! Yay! Ignoring nuance!

In particular, the $v^2$ term in the drag equation is interesting: the drag force is related to speed quadratically: increasing speed 2× increases drag by 4×… and increases (that part of) energy consumption by 4×.

Continuing to ignore all the other real factors that influence it, we can predict the energy consumption of the vehicle at any velocity $v$ (expressed in km/h)³:

$$\begin{align*} E(v) &= 158 + 0.0413 v^2 \unit{N} \\ &\equiv 44.0 + 0.0115v^2 \unit{Wh/km} \\ &\equiv 4.40 + 0.00115v^2 \unit{kWh/100km} \end{align*}$$

That makes for a classic quadratic curve, where increasing velocity results in a non-linear increase in estimated energy consumption. The upward curve of the quadratic drag component is visible at these reasonable speeds, it isn’t just at huge numbers that it starts being relevant.

That’s suggesting that accelerating +30 km/h has noticeably different impacts on energy consumption, depending what the base speed is:

• going from 30 km/h to 60 km/h increases energy usage by 111 N = 31 Wh/km = 3.1 kWh/100km;
• going from 110 km/h to 140 km/h increases energy usage by 310 N = 86 Wh/km = 8.6 kWh/100km, far more.

When you’re already going fast, going even faster results in higher energy consumption and less range.

This is true for all vehicles, ICE or electric.

⚡ Lightning summary

Dimensional analysis implies EV efficiency can be measured in Wh/km, kWh/100km or… N. That is, it’s a force.

Back of the envelope calculations seem to back this up: combing rolling resistance and drag gave a prediction of 658 N, aligning fairly closely to observed real world usage of 684 N.

Generalising that calculation shows usage is quadratically related to speed, and that the quadratic (drag) component is noticeable at normal driving speeds.