Berner 15-T derailleur pulley upgrade: an analytic model

A lot of fuss was made in the Tour about the Berner SRAM derailleur upgrades used by Alberto Contador and Andy Schleck. By going from 11 teeth to 15 teeth on the lower rear pulley, the chain bends less upon entry and exit to the pulley while the pulley additionally turns more slowly. Each of these effects reduces drivetrain losses, and that means more of the power to the pedals goes to the road instead of to heating up the bike and air.

Berner mod to a Red rear derailleur on a Team Saxo Bank SL/3 (BikeRadar)
SRAM has evaluated this and claims the results are inconclusive. So much for experimental data: what's a model show?

I spent several posts looking at a drivetrain model. I ended up with the following:
P_loss = (K / L) P (1 / N_f + 1 / N_r) + C T₀ K (1 + N_f/N_r + N_f [ 1 / N_dt + 1 / N_db ] ) + K_d C N_f [ 1 / N_dt + 1 / N_db ] ) / 2

where I define:

P_loss = power lost to drivetrain,
N_f = chainring teeth,
N_r = cog teeth,
N_db = bottom pulley teeth,
N_dt = top pulley teeth, and
C = cadence.

I used the constants:

K/L = 0.265,
T₀ K = 0.322 J.

A key parameter is K_d. I previously derived:

K_d = 94 mJ/rev for Shimano Dura-Ace
K_d = 2.4 mJ/rev for CeramicSpeed pulleys

or using data measured by Mark Kelly:

K_d = 37 mJ/rev (standard bearings)
K_d = 6.4 mJ/rev (high quality steel bearings)

Another Red derailleur modification available from Dark Albert with 15-tooth pulley
I assume everyone uses the best ceramic bearings before even considering going to a Berner modification. So I'll use K_d = 2.4 mJ/rev.

The savings are calculated from the loss equation:

ΔP = C T₀ K N_f ( 1 / 11 ‒ 1 / 15 ),

yielding 0.46 watts for the 39-tooth ring and 0.62 watts for the 54 tooth ring. Bigger ring at the same cadence means the chain is moving through the pulleys faster.

Are these savings worth it? I assumed 300 W steady power, so the savings is around 0.2% of total power with this model. Of course, all of the savings are lost if the Berner pulley doesn't spin as well as your 11T pulley, or if you don't keep your drivetrain meticulously clean you'll squander any savings from this hefty investment.

But it does appear the benefit is real, if small.

Of course if the pulleys are clogged with dirt, things may be different. In the limit of an infinitely-mucked-up chain (T₀ very large), the 15-toother saves 10% of lost power in a 53/23. So maybe it's the option of choice for cyclocross, mountain biking, or other races in the mud.