Issue 035 - Motorsports - Aerodynamic power
How much power does it take to drive 406 mph?
JCB's hydrogen-powered Hydromax set a hydrogen land-speed record at the Bonneville Salt Flats, averaging 406.320 mph over two timed runs. The car uses two hydrogen-burning engines derived from JCB construction equipment.
The problem
Estimate the mechanical power required to keep the Hydromax moving at roughly 406 mph, and then estimate how much hydrogen it would consume during a full-speed one-mile run.
Use your result to compare the record with something familiar: an ordinary highway car, gasoline energy, or household electricity consumption.
At 406 mph, is the main engineering challenge producing enough engine power, carrying enough fuel, or overcoming aerodynamic drag?
Because Fermi problems target an order of magnitude, I normally use no more than two significant digits and write most calculations in scientific notation; the Fermi reference explains both conventions.
Before checking sources
Matt's first pass
I do not have a great recollection of the formula for work done to overcome air resistance. I thought it was something like a friction coefficient x front-facing surface area x velocity squared x time or distance. I thought it was time.
Starting with the vehicle, I assumed it was lighter than a typical passenger car, maybe 700 kg total. I also converted 406 mph to about 200 m/s.
I thought there were two phases of the 1-mile run: getting up to speed, and then maintaining speed for the rest of the distance. For the first phase, I modeled it as about 2.5x the kinetic energy at peak speed, to account for friction and engine inefficiency:
KE ~= 1/2 x m x v^2
~= 1/2 x 700 x (200)^2
~= 1.4 x 10^7 J
fuel-energy phase ~= 2.5 x KE
~= 3.7 x 10^7 JI guessed that phase could be accomplished in about half a mile, leaving half a mile for maintaining speed.
For the next phase I expected it would be dominated by overcoming drag. Using my assumed equation, taking frontal area as about 2 m2, coefficient as about 0.5, and velocity as 200 m/s, it would take about 4 seconds to finish the run:
rough drag energy ~= 0.5 x 2 x (200)^2 x 4
~= 1.6 x 10^5 J
with efficiency ~= 3 x 10^5 JThat was about 1/100th of the energy required to get the car up to speed.
I did not know the energy density of hydrogen, but if it were anywhere close to gasoline, about 4.5 x 10^7 J/L, then all of this could be accomplished with about 1 L of fuel. Based on this, I thought the main challenge would be producing enough engine power, since carrying enough fuel and overcoming aerodynamic drag seemed trivial by comparison.
Calibration Score
Matt's Calibration Score: 40 / 100
Higher is better: earn points for accurate pegs, sound models, correct math, and a result close to the sourced answer. The image shows percent full of it: 100 minus the Calibration Score.
Pegs: 10/30. Vehicle mass, hydrogen-energy, and drag pegs were weak.
Model: 0/30. The drag formula was not applied correctly as force, then power or energy.
Math: 10/10. The arithmetic was clean for the chosen setup.
Result: 20/30. The final conclusion about engine power was passable and the result stayed within an order of magnitude.
Grounding facts
A 1 MW mechanical load is about 1,300 horsepower and roughly 800 average U.S. households worth of continuous power demand. Ordinary highway cruising might require only tens of kilowatts at the wheels; pushing air at 406 mph is a different planet.
The cube-law is the lesson: doubling speed requires about four times the drag force and eight times the drag power, if the shape and air density stay the same.
After checking sources
Check and recalibrate
The missing equation is the drag equation. It gives drag force, not energy:
drag force ~= 1/2 x rho x Cd x A x v^2
drag power ~= drag force x v
~= 1/2 x rho x CdA x v^3Convert the record speed:
406 mph ~= 406 x 0.447 m/s
~= 1.8 x 10^2 m/sUse a Bonneville-ish air density around 1 kg/m3, and use combined drag area CdA rather than trying to separately guess drag coefficient and frontal area. A streamlined land-speed car might plausibly have CdA around 0.3 to 0.5 m2.
v^3 ~= (1.8 x 10^2)^3
~= 5.8 x 10^6
power ~= 1/2 x 1 x (0.3 to 0.5) x 5.8 x 10^6
~= 9 x 10^5 to 1.5 x 10^6 WSo the corrected mechanical power estimate is about 1 megawatt, or roughly 1,300 to 2,000 horsepower. That agrees well with the reported 1,600 hp engine package.
The timed record is a flying mile, so the car is already near top speed before entering the measured mile. At 406 mph:
one mile ~= 1.6 x 10^3 m
time ~= distance / speed
~= 1.6 x 10^3 / 1.8 x 10^2
~= 9 sThe mechanical energy needed to overcome drag during the measured mile is:
mechanical energy ~= power x time
~= (9 x 10^5 to 1.5 x 10^6 W) x 9 s
~= 8 x 10^6 to 1.4 x 10^7 JIf the hydrogen combustion engines are roughly 30% efficient at turning fuel energy into wheel/crankshaft work:
fuel energy ~= mechanical energy / 0.3
~= 3 x 10^7 to 5 x 10^7 JHydrogen has about 120 MJ/kg of lower-heating-value energy, so the measured-mile fuel use is:
hydrogen mass ~= 3 x 10^7 to 5 x 10^7 J / 1.2 x 10^8 J/kg
~= 0.25 to 0.4 kg H2That is only about one liter of gasoline equivalent in energy terms. Carrying enough fuel for the measured mile is not the hard part.
If we include acceleration from rest, the kinetic energy of a few-thousand-kilogram streamliner at 180 m/s is tens of megajoules, comparable to or larger than the flying-mile drag energy. But record timing is based on the measured mile after a run-up, so the cleanest article estimate is the full-speed mile.
The main engineering challenge is aerodynamic drag plus producing reliable power at that speed. Fuel mass for one run is small; the difficult part is pushing enough air out of the way while keeping the vehicle stable, cooled, fueled, and mechanically intact.
Post-check reflection
Matt's reflection
Looks like a bunch of my mental models were weak on this one.
The drag formula comes up often, and I really should commit it to memory. It is an important one to make a peg out of. Also, I need to remember that the air resistance formula gives the force necessary to overcome it, and then that needs to be multiplied by distance to get total energy, or by velocity to get power.
My estimate for the vehicle size was skewed. I should have expected a higher mass than a typical passenger car rather than a lower one.
My mental model for what was going on in the speed trials was not correct either. I had not heard of a "flying mile" before.
Once corrected for what was going on in the problem, the energy to overcome air resistance is quite a bit higher, closer to the same order of magnitude as the energy to reach the max speed, even though that acceleration phase does not apply to the flying-mile trial.
Finally, I was off in terms of energy density of hydrogen. It is about 3x as dense by mass as gasoline, but much less dense by volume when compressed. That is probably an important peg to remember as well.
Ultimately, while my answer was passable in identifying engine power as the major limiter, I did not give air resistance the attention it deserved. My calculations were within an order of magnitude, so technically correct, but this definitely illustrated some weak points that deserve attention.
Recommended memory peg
For high-speed vehicle problems, remember drag force ~= 1/2 rho CdA v^2, drag power ~= 1/2 rho CdA v^3, 1 hp ~= 750 W, 1 mph ~= 0.45 m/s, and hydrogen ~= 120 MJ/kg but low energy per liter.
Reader results
Bars show how submitted estimates sort into the answer choices from the gut-check prompt.