cars: fun with math and physics

People have a hard time distinguishing between energy and power, it really manifests itself when thinking about electric vehicles (EVs).

Batteries store energy, and people grasp that they don’t store a lot of it. To travel about 75 miles the Nissan Leaf electric vehicle has a battery that weighs 660 pounds and costs $5500 to replace. It stores 24 kW·h of energy.

A kiloWatt hour?

But what is a kW·h?  It’s a “kiloWatt hour”, which means the ability to provide a kiloWatt of power for an hour; the ‘·’ is multiplying power and time. It’s a measure of energy, and it’s the stuff that your electric company sells you. Be more energy-efficient and you use fewer kiloWatt hours.

pay the man

As Googling “1 kW in hp” will tell you, one kiloWatt is a mere 1.3 horsepower. So 24 kW·h is only 31 “horsepower hours,” and then the confusion sets in. “Ugh, electric cars are so weak! Wimpy granola-wearing sandal-eating hippies!” Or the new refrain “Electric cars are so weak, unless you’re a rich person pretending to be green who can afford Tesla’s huge batteries.”

No. Cars don’t need lots of power for hours, they need occasional bursts of power. A 24 kiloWatt hour battery holds the energy to provide 1 kiloWatt of power for an entire day, or 48 kiloWatts for 30 minutes, or…

Performance is a burst of power

Again, energy is not power. Let’s build a hot-rod that throws down 2,000 horsepower for 10 seconds to accelerate. That’s 1,500 kiloWatts for 10 seconds, which is only 4 kW·h. In other words the weakest battery in a plug-in car (currently the Prius plug-in hybrid) has enough energy storage to throw down a brief surge of power to out-accelerate any ultra-supercar from a stoplight! In practice only a big battery can discharge that much power without disintegrating, but there’s no physics preventing it. If and when someone improves battery chemistry so smaller batteries can discharge rapidly, even a mid-range EV will out-accelerate all gasoline sports cars. Most performance car fans haven’t woken up to this yet, but all the engineers working on AMG/M-series/RS-line/-V 600+ horsepower engines are going to be collecting unemployment and taking night classes in battery chemistry in less than a decade.

Recharging is about power and energy, and it isn’t

Let’s recharge our Nissan Leaf with its 24 kW·h battery pack. Firstly, it’s dirt cheap. Electricity averages 12 cents per kW·h (you’re buying energy), so even with some charging losses (the battery gets warm) you’re looking at $3 to travel 75 miles. If you charge overnight you should look into a cheaper rate from your utility. If you have solar panels, you’re laughing.

How do you put 24 kW·h of energy into a battery? As the unit suggests, you could deliver 24 kiloWatts for one hour, which is quite a lot of domestic power; for comparison turning on every light on my house only takes 4 kW. But again energy is power for a period of time; you can vary the power or time. A regular 110 Volt outlet can deliver 15 amps. That’s 1.6 kW, meaning it will take 15 hours, plus an hour or two for charging losses, to completely recharge a fully depleted Leaf. That’s a little tight to fully recharge overnight, So many EV buyers use a 240 Volt outlet. Most houses have one or two 240 Volt circuits to power ovens and dryers, an electrician can install a 240 Volt 30 amp socket for about $300. Now instead of 1.6 kW you can send 7.2 kW to your car, which means you’re done recharging your Leaf in about 4 hours. That means

Every morning you get into a car that you’ve cheaply, and overall-less-bad-for-the-planet, fully charged for your regular driving duties.

(Expressed that way, how come every two-car family with a garage doesn’t have an EV?! Suburban familes of the world, get a clue!)

The confusion between power and energy continues with recharging. Some car makers and fans trumpet how their plug-in car can fully recharge in only three hours from a wall plug, but that’s only because it has a small battery! “You can refill the Diamante in 30 seconds… because it’s got a tiny gas tank” is not a feature!

Similarly, people make fun that recharging a top-of-the-line Tesla from a wall plug can take days. That is stupid. Whether you plug a Leaf or a Tesla into an AC outlet, it’ll take about the same amount of time to recharge either car enough to travel 30 miles. So long as you wake up to a car with enough charge for your commute, who cares?

Fast recharging is power in reverse

SAE J1772 EV recharging plug

Besides billions and billions of regular AC sockets, there are lots of these 240 Volt sockets around the country, and there’s a standard plug for them. This government map shows 23,000 of them at 9,000 “public charging stations” around the USA. (My neighbor has and loves a Chevy Volt and recharges it at work, she doesn’t even have access to a garage at home.) But if you’re on a long road trip, that four hour wait to travel the next 75 miles is onerous.

No problem, just increase the power, and the time to provide the energy goes down. An EV is designed to transfer huge current at high DC voltage from its battery to the motor, so you add a circuit to the car that does the reverse, accepting huge current at high DC voltage to recharge the battery. It doesn’t add much cost to the car, but the station to provide it involves some more elaborate electrical engineering and so a “Fast DC charging station” costs $10,000 rather than $1,000. Nissan promoted the CHAdeMO standard that supplies 50 kW, the other car manufacturers dickered around in standards hell before coming up with the “Frankenplug” that adds extra pins for fast DC charging. If you go to the same map you can see these fast DC charging stations slowly popping up.  Meanwhile Tesla went their own way with the Supercharger that can deliver a staggering 130 kiloWatts. Again, because of battery chemistry limitations, only a big battery can accept that kind of power input, just as only a big battery can output Ferrari-beating power.

Gasoline is amazing!?

Let’s go back to that Leaf with its 24 kW·h battery pack.

Meanwhile a single gallon of gasoline holds 33.4 kW·h of energy and only weighs 6 pounds. It holds 100× more energy by weight than that Leaf battery pack! The comparison looks terrible, but it’s misleading. When an engine burns that fossil fuel, at best only around a third turns into forward motion. The rest turns into useless heat. The efficiency of the Leaf is reflected in its EPA sticker, which has the confusing “99MPG equivalent” number on it. The electric car goes a long long way on the same amount of energy.

Nissan Leaf EPA sticker

99MPG equivalent means what?

And when you discharge an expensive heavy battery pack, you still have an expensive heavy battery pack. You can recharge it over and over before eventually recycling it.

The amount of gasoline a car burns through is staggering. I made a spreadsheet that does the math. A car that gets 35 miles per gallon will consume 3,400 gallons of gasoline over 120,000 miles. That gasoline weighs 10 tons, and it turns into 33 tons of CO2 that contributes to global warming (and ocean acidification, plus smog, plus funds Middle Eastern terrorists). A 1.5 ton car burns through 6 times its own weight in gasoline! So any troll on the internet who bleats about all the pollution from making a hybrid or battery-powered car is a complete f***ing asshole crying crocodile tears while understanding nothing. The idea that making a 600 pound non-toxic recyclable battery pack is as bad as producing, spilling, refining, and shipping all those tons of gasoline is completely ludicrous. Obviously there’s more to the comparison than that, because the energy to recharge the battery pack has to come from somewhere, but remember the battery pack and motor are so much more efficient. These days the power most likely comes from a natural-gas power plant, and the proportion of renewables is going up while coal is going down.

Let us review

  • “Power” is doing hard labor, “Energy” is doing hard labor for some time.
  • Electric cars are incredibly efficient.
  • Anyone driving a conventional car is burning tons of gasoline and it’s bad for the planet.

Class dismissed!

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