26" vs 700c

The question which wheel is more efficient - 26" or 700C - has already attracted much intellectual energy, probably enough to power a small city for a year. I hope that my little addition to this wasteful practice won't turn you off, especially as you know that up to now the answer to this crucial question was still ambiguous.

The first stumble-stone is in the question itself. There are a lot of different 26" wheels, as much as 700C's, and certainly you will find a pair to show that either 26" or 700c is the correct answer. We can answer the question unambiguously only if we make some further assumptions - and restrictions. In the following we will consider only the difference which comes from the size - all other things being equal. That means both wheels have the same hubs, spokes (apart from length), rims with tyres (apart from different diameter) and tyre pressure. Both wheels also carry the same weight. Secondly, we consider the efficiency in the following sense: a rider is coasting on a bicycle on flat hard surface (asphalt) with no energy input, so that the bicycle will, from the initial velocity, come to a stop due to various resistances, the wheel which stops later is more efficient.

The resistances are: resistance from tyres rolling on ground (also known as rolling resistance), air resistance, and resistance in the hub bearings.

Let's concentrate first on rolling resistance (RR). The cause of the RR is the flexion (deformation) of the tyre. The bigger the flex, the bigger the RR. Both 26" and 700c have the same contact patch A. I conclude this simply because Ap=W and since both tyres have same pressure p and bear the same weight W, the area is also the same. Since the width of the tyre w is the same then from A=wc, c (the length of the contact patch) must be the same for both wheels. But, 26" flexes more. To see this, look at the segment of a circle with a chord of length c. You can see it as flattened part of the tyre in contact with the ground. The area of this segment is equivalent to the amount of the flex. Given the fixed chord length c, bigger circle has smaller segment - or smaller flex. And opposite, smaller circle (26") has bigger area=flex. In other words, rolling resistance of 26" is greater.

The formulas at the Wikipedia page can be used to calculate segment area (equivalent of RR) for any combination of W,w,p,R. I did this for a typical touring situation: W=50 kg, w=35 mm, p=4 kg/cm2 and both wheels (diameters ISO 559 and 622) and I get 10% bigger RR for the 26" (ISO 559). Surprisingly, the result (1.10034) is practically the same as the ratio of wheel diameters (with tyres on): (622+2w)/(559+2w)=1.10016.
BTW, using the same equations we can prove the often stated argument that wider tyres of same diameter and pressure have smaller RR then thinner ones.

Now, let's look at the resistance in the bearings. It is also in favor of 700c wheels, since on a fixed road distance bigger wheels rotate less times. The bearing resistance of a 26" wheel is then bigger for the factor depending on wheel diameters (with tyres on) - the same factor as above: (622+2w)/(559+2w) =1.10016.

The air resistance is proportional to the frontal area. Considering just the wheels, the front area of 700C is greater by the same ratio of the wheel diameters: (622+2w)/(559+2w). But we are considering a rider and a whole bicycle (not just the wheels), so this disadvantage is smaller. If the percent of the wheel frontal area to the whole riding frontal area is f, I estimate f=7%, then the advantage of 26" due to smaller air resistance is by a factor 1.0065.

Our finding is than, that 700C wheels have smaller rolling and bearing resistances, both of about 10%, but have a greater air resistance of about 0.7% because of greater frontal area. It is not difficult now to predict when the 26" wheel will be more efficient. It will be when air resistance force is 0.1/0.07=14,3 times greater then combined rolling resistance + bearing resistance force. The air resistance force increases (nonlinearly) with speed, so we expect that 26" will me more efficient above a certain speed. Using this calculator, starting with the default data, setting slope 0% and weight 100 kg, and playing a bit with the speed, we can see that above 70 km/h the air force is greater than rolling resistance force for a factor 14.3.
So, the conclusion: 26" are more efficient than 700C only at extreme (downhill) speeds.

P.s. Just before 700c (or "bigger is better") enthusiasts start screaming of joy and ordering rounds of beers, let me remind you that situations such as "just size - all other things being equal" don't exist in the real world. Depending on tyre width, tyre pattern, weight, pressure and possibly year of your birthday, 26" might well be preferred.