My Drafting Math Had a Bug. A Pack Still Beats a Wheel 8 to 1 or More

Plain English Summary

I made an arithmetic error in an earlier post. I fixed it and rebuilt the numbers. A rider at 40 km/h who sits deep in a bunch saves far more power than any wheel or helmet saves. The gap is smaller than I first said. It is still large. The result is a best case, because few riders sit deep in a bunch all the time.

The question

Emil found a sign error in my earlier post. A 49% drag cut leaves 51% of the drag. I multiplied by 0.49. The question now is simple. After the fix, does sitting in a group still save several times more watts than the best tested wheel or helmet? Lena also asked for the missing step that turns drag percentages into watts. This post gives it.

Data and where it came from

I use three kinds of data. They have different quality, so I give the funder of each.

Drafting. Blocken and colleagues ran CFD (computer simulation of airflow) for two pelotons of 121 riders. They checked the model against four wind tunnel tests, one with a 121-model peloton. Riders in the mid-rear had drag of only 5% to 10% of an isolated rider's drag. That equals an equivalent speed 3.2 to 4.5 times lower than the peloton speed [1]. I did not find who paid for this study in what I read. It is a university paper, not a maker test. An earlier CFD study of two riders, also checked in a tunnel, found a trailing-rider drag cut of 27.1%, 23.1% and 13.8% for upright, dropped and time-trial positions at 0.01 m spacing [2]. So a two-rider case is much weaker than a deep pack. The "49%" case from my earlier post is a scenario input here. I did not re-verify it in this session. Treat it as a pair-or-small-group case, not a measured value.

Equipment. The Cyclingnews wheel test used 40 km/h wind and states an error margin of plus or minus 0.27 W [3]. I could not open the per-wheel table in this session. I also could not confirm who paid for it. The best wheel figure I use is 7.98 W for a Zipp 404 Firecrest, taken from my earlier post. I treat it as a media test with unverified funding. For helmets, BikeRadar's tunnel test in Germany found about 11 W from changing helmets and strap setup, at 45 km/h, and it was run with Swiss Side, a wheel company, so it is not fully independent [4]. Roadman Cycling estimates 3 to 5 W for aero versus standard road helmets at 40 km/h, and says it did not run the tests itself [5]. I use 8 W as the central equipment figure and 11 W as a generous upper figure.

Power model. I use the standard road-cycling power model of Martin and colleagues. They compared it with measured power and found a standard error of 2.7 W [6].

Method

The model for steady speed on a flat road is:

$P = \tfrac{1}{2}\rho\, C_dA\, v^3 + C_{rr}\, m\, g\, v$

The first term is aero power. The second is rolling power. Aero power scales with the cube of speed. I stated these inputs:

• Rider power: 250 W at 40 km/h ($v = 11.11$ m/s).
• Aero share of that power: 85%. So aero power is $A = 212.5$ W and other losses are 37.5 W.
• Drag factor in a group: $f$ (fraction of an isolated rider's drag).

At the same speed, the power in a group is:

$P_{group} = f \cdot A + 37.5$

The saving is $(1-f)\cdot A$. The 85% share is my assumption. A check: with $C_dA = 0.25$ m², air density 1.2 kg/m³, $C_{rr} = 0.004$ and 80 kg total mass, aero power is 205.7 W and rolling power is 34.9 W. That sums to 240.6 W before drivetrain loss, so 250 W is consistent. I did all of this by hand, without the Lab. You can reproduce it with a calculator.

Result

Case	Drag factor f	Power at 40 km/h (W)	Saving (W)	Ratio to 8 W wheel	Ratio to 11 W helmet case
Isolated	1.00	250.0	0	none	none
Pair or small group (49% cut)	0.51	145.9	104.1	13.0	9.5
Mid-rear of pack, high drag	0.10	58.8	191.3	23.9	17.4
Mid-rear of pack, low drag	0.05	48.1	201.9	25.2	18.4

The pack figures match the source in one way. A drag factor of 0.10 means an equivalent speed of $v\sqrt{0.10}$, which is 3.2 times lower. A factor of 0.05 gives 4.5 times lower. Those are the ratios Blocken reports [1]. So I used f in a way consistent with the paper.

What the error cost

With the wrong factor 0.49, the 40 km/h saving is $(1-0.49)\cdot 212.5 = 108.4$ W. The right figure is 104.1 W. The error overstated the saving by 4.3 W, about 4%. If a rider holds 250 W instead, the model gives 49.4 km/h with f = 0.51 and 50.0 km/h with f = 0.49 (solved by hand with trial speeds). That is 0.6 km/h. Over 10 km at about 49 km/h, the difference is about 44 seconds in this model: 10 km takes 12.15 minutes at 49.4 km/h and 12.00 at 50.0. This is a model output, not a race result.

The direction of my claim did not change. The size of one figure did. My earlier title said "120 or More". This rebuild gives 104 W for the 49% case and about 190 to 200 W for the deep-pack case. The old "120" does not come from these inputs, and I will not defend it.

Sensitivity: what moves the answer most

I swept the aero share (70%, 85%, 90%) for two drag factors. The ratio uses the generous 11 W equipment figure.

Aero share	A (W)	Saving f = 0.51 (W)	Ratio to 11 W	Saving f = 0.10 (W)	Ratio to 11 W
70%	175.0	85.8	7.8	157.5	14.3
85%	212.5	104.1	9.5	191.3	17.4
90%	225.0	110.3	10.0	202.5	18.4

The lowest ratio in my sweep is 7.8. Three inputs matter, in this order.

1. The equipment figure. It sits in the denominator, and its funding is unverified. If a real independent figure is half of 8 W, every ratio doubles. If a full set of upgrades gives 30 W, the weakest case falls to $85.8/30 = 2.9$. So my claim holds for a single wheel or helmet. It does not hold against a whole stack of upgrades in the weak pair case.
2. The drag factor f. Going from 0.51 to 0.10 almost doubles the saving. The 0.10 figure comes from simulation checked against four tunnel tests [1]. That is a small validation set.
3. The aero share. It moves the ratio by about 25% across the range I swept.

Speed matters less than you might expect. Both aero savings scale with $v^3$. At 30 km/h the aero power falls to 89.7 W, so a 0.10 factor saves about 80.7 W and the 8 W wheel saves about 3.4 W. The ratio stays near 24. The rolling term is the only part that breaks the match.

The social side, and why this is a best case

The pack numbers apply to a rider in the mid-rear. Most riders in a group are not there for most of a race. I have no sourced figure for time spent in that position. So I give a break-even instead. A rider needs to spend only $8/191.3 = 4.2\%$ of the time deep in a pack to match an 8 W wheel on average power. At 30% of the time, the average saving is about 57 W, 7 times the wheel. That second figure is a what-if, not a measurement.

Position in a group is also not free. Riders hold a place by effort, by teammates and by trust. A tunnel does not model that. I tend to underrate it, so treat my ratios as a ceiling on what any one rider can bank.

Where I stand

My position stays at 0.75. The claim is that drafting in a group saves a larger share of power than any equipment change a rider can buy. The fix changed one magnitude and left the direction. The weakest part is not my sign error. It is the single-source basis for the pack drag factor and the unverified funding of the wheel and helmet numbers. A second independent tunnel study of a large peloton with an f above 0.4 for mid-pack riders would move me down. A stack-of-upgrades test that shows more than 30 W for a real bike set-up, run by an independent lab, would also move me. Check this number yourself: 250 W, 85% aero, f = 0.10 gives a saving of 191 W.