Thought Experiment - Full Recharge Going Down Imaginary Steep Hill

[James Anders]

Friend and I were having one of those what-if thought experiment conversations about Tesla recharging.

We wondered how high/steep/long an imaginary incline would have to be in order to fully recharge a Tesla.

I guess a part of this would be to determine how steep a hill has to be in order to effectively regeneratively charge continuously.

Anyone have any ideas how best to come up with an approximate answer?

Obviously this question is just for fun and not intended to be anything other than a thought experiment.

 

[swaltner]

Let's assume

85 kWh battery pack
70% efficiency in the regen process
Car weighs 4,960 pounds (2,250 kg)

85 kWh / 70% = 121.4 kWh energy required = 437,000,000 Joules

Potential Energy of an object is mgh (mass * gravity* height), solve for h, you get h equals PE / ( mass * gravity).

437000000J / (5000 kg * 9.8 m/sec2) = 437,000,000/49,000 = 8,918m = 29,258 feet high

So, you'd have to start at the summit of Mt Everest and coast all the way down to sea level to fully charge a Model S battery.

 

[Max*]

Shouldn't the 5000 kg be 2,250kg in the above equation?

 

[SabrToothSqrl]

I think you need to factor the real world regen, it puts 30kW into the pack at full regen, correct? you have to be doing approx 60 mph?

If you assume full regen, you need a hill steep enough to keep that 30kW pegged for a duration at 60 mph, which should give you length...

And, what if the dash is showing power generated, not power going into the pack? Like.. charging at 10kW, doesn't mean 10kW is being added to the battery... more like 80%?

 

[swaltner]

Shouldn't the 5000 kg be 2,250kg in the above equation?

Doh! Only excuse was I was typing that on the iPad, so shuffling back and forth between Tapatalk and google.com to run calculations. Also, looks like the average mass for the Model S is closer to 2,100 kg. While the P90D is close to this mass, the other models are lower. I was just trying to get close without doing a lot of research. Running the calculation again yields:

437,000,000 Joules / (2250 kg * 9.8 m/sec2) = 19,818 m = 65,000 feet of altitude

If you went by the rule of thumb proposed at Mountain driving - some numbers | Tesla Motors of regaining 5 miles of rated range per thousand foot descent, that would yield a descent down from a 48,000' tall mountain (240 miles / 5 miles/1000'descent).

Both those methods of calculating the hill hight are relatively close to each other (same order of magnitude), especially since I'm just guessing at what the efficiency is for regen converting potential energy into battery energy. Since my proposed calculation comes up requiring more altitude, the conversion losses due to regen are likely less than 30%.

No matter what, it doesn't seem like there's a mountain you could come down on Earth that would fully recharge the Model S battery from regen. You'd need to drain all the water from the ocean and then descend from the top of Mt Everest (29,000') to the bottom of the Mariana Trench (36,000' depth) to get the altitude drop required to fully charge the Model S 80 kWh battery. While these two points aren't close enough to make such a descent feasible, it's a fun way to visualize the amount of energy in the battery.

 

[James Anders]

Does anyone know the % grade incline that the S or X can maintain a coasting regen?

 

[Nikxice]

A physics or math person could probably shred these calculations, but here's my real world experience.

Last summer descended the Mt. Washington Auto Road. Elevation change from 6,145 ft to 1,527 ft., a drop of 4,618 ft. The average gradient is 11.6% over 7.6 miles. On the descent I gained back 24 miles of range, used no brake pedal (makes those switchback turns interesting!). So if my rated range is 265 miles, I would be looking at driving down a 51,000 foot hill, with an 84 mile long road, to fully recharge my Model S.