Bike Doping Analysis

I am finally catching up on the accusation/rumors/suggestions that Fabian Cancellara was using electrically assisted race bikes for his recent victories in the Paris-Roublaix and Ronde van Vlaaderen. An avid cyclist, my regular group made comment, I assumed immediately the batteries are such that it would be a waste of energy to haul them around.

Some articles:
http://www.velonation.com/News/ID/4415/Fabian-Cancellaras-so-called-motorized-doping-debunked.aspx
http://www.velonation.com/News/ID/4412/Saxo-Bank-denies-its-riders-used-motorised-bikes.aspx

You Tube Video explaining the controversy, and how it works:
http://www.youtube.com/watch?v=8Nd13ARuvVE

A product, Gruber Assist available since 2008 (I want one, just because I AM TRONBIKES!) - what awesome advertising, sort of evil I guess.

Specs:

100 watts for 90 minutes, 900g + battery pack, Gruber has a table of battery capacity

Pro riders churn out 500-600 watts in regular hard efforts, likely more peak. Cancellara is a horse, lets assume 800 for the accelerations, a 10% boost would
indeed help - as the late Edmund R. Burke of "Cycling Science" once published, time-trial athletes with 10% better times put out 11% more power. So in theory,
any additional power to a rider is significant - a 100 watt boost = 100 watts faster!

The big question to me is, would the weight of the system be worth hauling along for distance before putting it to use - would the system help?

The Gruber people have a big chart (though a bit confusing, maybe backwards?) for a hybrid bike pulling different amperages, with the resulting range - for some reason, maybe its backwards(? sending email) the bike runs farther when running faster in this chart - I'll send email on this.

Paris-Roubaix  =262 km/hr averaging 32.9 km/hr (http://www.steephill.tv/classics/paris-roubaix/), but its really flat, like 490 ft peak (http://www.runsaturday.com/Maps/3462).

Assuming the system is invisible, we can only assume additional work to haul
this around is due to gravity. With a peak of 490 ft, lets triple this in my test model to 1500 feet, about 500meters.

Gravity pulls (acceleration) at 9.81 m/s^2, Force = Mass * Acceleration, and work is force through a distance expressed as joules

Work = (9.81 m/s^2 * Mass) * 500m

Mass = weight of system, and beyond the motor itself, the weight is batteries, with power storage usually noted in Amp-Hours, lets use the 900g number

Work = 9.81 m/s^2 * 1kg * 500m = 4905 m2 kg/s2 = 4905 Joules = 4905 Watts - sec of work. A watt-sec is not power, but a unit of capacity/work that requires
energy.

With 100% efficiency, 5000 W-sec = 5000 W-sec/3600s = 1.4 W-hr expended to haul this extra 1 Kg system around by gravity alone. A single 18650 Li-Ion battery
can store up to 3000 mAh at 3.7 volts - enough to cover most of the weight of the system - each 18650 itself only weighs about 50g

1 Watt - Hour = Volts * Amps * Time = 3.7 Volts * 3 A-hr = 11.1 Watt - Hour

Gruber calls out capacities up to 10 Ah, which would weigh 10 Ah=3000Mah * X batts, X=10Ah/3Ah = 3.33 batteries, * 0.50g = 170g...very light


Working off the chart, 12 Amps
10Amps-Hr/12Amps = 0.83 Hr = 50 minutes, as on the Gruber chart. Paris-Roubaix was about 6.5 hours long.

100 watts for 50 minutes = 100 watt (50/60) hours = 83 Watt-Hours, only 8 batteries, not much weight penalty

So far I'm not convinced Cancellara was bike doping(or would need too), but I am convinced bike doping could be a real advantage in a flat race like Paris-Roubaix if any of my calculations above are right using the "perfectly efficient" energy conversions I am assuming. I'll sleep on the generic model parts above and maybe generate a spreadsheet to model possible scenarios more closely.