The Fisker Ocean

I've published previously on the seeming futility of solar panels on the roofs of vehicles. But Fisker has announced the "Ocean" in various configurations. It's an SUV style vehicle with the "Fisker Ocean Extreme" boasting solar panels for the full length of the passenger cabin. The claim is that solar charging will produce 1,500 miles worth of charge, or even up to 2,000 miles. Let's investigate!

First, how much energy is needed to travel 1,500 miles in the Fisker? Unlike internal combustion engine powered vehicles, there's no curve with a peak in terms of energy mileage as a function of speed. For the IC vehicle going very slowly uses a lot of the energy from burning fuel to keep the engine turning over, and going very fast has a high drag penalty. The sweet spot differs for various models but might be in the range of 50 m.p.h.

For a battery electric vehicle, there's no such function. The faster you go, the worse your energy economy since it's only a matter of overcoming drag. So, in earlier data collection of my own driving, my overall block speed was on the order of 30 m.p.h. with a blend of city driving, freeway driving, and freeway driving in traffic. I'll use that number, but convert it to 13.41 meters/second.

We'll go to the naive drag equation, ~D=1/2 \rho C_dAv^2~ where D is drag force, ~\rho~ is air density (I'm using sea level, at altitude density would be lower and insolation would be slightly higher), ~C_d~ is the vehicle's drag coefficient, ~A~ is flat plate area, and ~v~ is speed. All are in SI base units. I can't find a drag coefficient spec for the Ocean, I'll go with 0.3. The vehicle's height is 1.631 meters, its width is 1.995 meters. Sea level atmospheric density is about ~1.225 kg/m^s~. Multiplying, we get ~D=0.595 (kg/m) v^2 Nt.~

The other drag factor is rolling resistance. This is, to first order, linearly dependent only on the vehicle's weight (NOT mass!). The curb weight is 2,250 kg force or 22,065 Nt. Add, say, 250 kg of people and luggage for a traveling weight of 2,500 kg force or 24,516 Nt. We'll use 0.014 as the coefficient of rolling resistance, resulting in a rolling resistance of 343 Nt. The result is a total drag of ~D=0.595 (kg/m) v^2+343 Nt~.

Next, power (work/time) is force times speed, so, at 13.41 meters/second, we need ~((0.595*13.41^2)+343)*13.41~ or 6,034 Watts or 8.09 horsepower. This is surprisingly small but, to first order, I'm confident that it's close. Call it 7 kW for our purposes.

Then, we'll assume the electric motor operates at 95% efficiency and that the drivetrain is 85% efficient, so we need 6,352 watts from whatever energy source we're utilizing. Now, 1,500 miles at 30 m.p.h. will take 50 hours or 180,000 seconds. And power times time is energy so the Ocean's solar panel will need to deliver 6,352 watts * 180,000 seconds, 1.14*10^9 joules, or 317 kWh. OK, can the panel on the Ocean's roof deliver 317 kWh in a year?

I'll estimate that the dimensions of the panel are 1.5 meters X 3 meters, or 4.5 m^2. In my Southern California area, the average solar insolation is about 5 kWh/(day*meter^2). This has to be reduced because the panel on the Ocean sits horizontally rather than following the sun. We'll use 50%, so if the Ocean sits outside in the sun all day, we might average 11.25 kWh delivered to the panels. Next, we'll estimate that the panels are 18% efficient, so about 739 kWh ~(11.25*0.18*365)~ are delivered to either the motor or the battery pack over the course of a year. And here, we're assuming that either the car is in motion and the panels are delivering energy to the motor or that there is capacity in the battery pack to accept the energy.

Now, speeds above 30 m.p.h. will hurt more than those below will help due to the dependence of drag on the square of speed (refer to plot at right). And this doesn't account for use of accessories, losses due to climbing hills (not all the gravitational potential energy is regained on the downhill), and stopping and starting (even regenerative braking doesn't recapture all of the kinetic energy). It doesn't include being blocked by buildings and trees, and many other factors. And Minnesota, New York, and other Northern states don't receive the insolation of Southern California. That said, I can't say that the claim is irresponsibly exaggerated so, using the Mythbusters' scale, I'll call it plausible.