22” wheel require less energy
A 26 wheel covers 207,47078 cm in a turn.
A 22 wheel covers 175,5522 cm in a turn.
The relation between these distances is 1,1818182 (which is the same as 26/22 of course).
In a single chainring turn, a 26 bike with a gear relation 1:1 will cover the same distance as a 22 bike with a gear relation 1,1818182:1.
Smaller wheels are said to accelerate faster. Thus the 22 will cover the distance on a shorter time if pedals are pushed with the same force (intensity) than the 26. In case this is true, to cover the same distance in the same time, the 22 requires less energy. Is this correct?
For wheels, the mass (and mass distribution) matters more than the radius. You can find lighter 700c wheels than the 451 wheels on the quest and they take less energy to get spun up to speed.
Once you consider the whole system, the rider, the bike, the wheels, and the random crap, the wheels are such a small part of this that you can ignore pretty much all mass/size effects.
What isn't irrelevant is rolling resistance and here different tires can play a huge difference.
If I were to buy a Quest now, I"d buy the 451. You can pack it in a suitcase and I think the 559 Quest is geared too high. The disadvantage of 451 is the lack of tire choices (and for really rough roads or off-road, bigger wheels help).
The shape of the tire contact patch on the ground is the key. You can have a round shaped one, from a very small wheel, which shows you a lot of distortion of the tire, which equals energy loss. or you can have a long narrow elliptical contact patch from a large wheel, which shows only slight distortion of the tire and only slight energy loss. For any weight and pressure, the contact path areas will be the same.
or maybe I am missing the point.
the only way I see to solve this is to use speed as a constant and measure the amount of force needed to generate that speed with different sized wheels and different gearing to compensate.
I don't even want to think about the different wheel weights and overcoming inertia to spin, so a constant speed to measure force ?
Makes my head spin,,, I think I will just ask to borrow a copy of your spreadsheet when you are done with the calculations.
If you solve the question you ask: How much force (or power) does it take to go a given speed, you'll find that neither the gearing*, nor the wheel size has a very large effect on the answer. At high speeds, it's all a question of aerodynamics. At low speeds, it's a question of rolling resistance.
John is right that wider and taller tires have less rolling resistance. It's not a huge effect and tire composition matters more than tire size (the world record holding bicycle uses 406 tires that have less rolling resistance than anything you'd find on a 700c wheel).
The question of " does the 22 require less energy " will not be solved by adding variables.
In my mind this would have to be done in a lab environment.
on a fixed trainer equivalent
with a fixed amount of resistance on the wheels
at speed to remove overcoming differing inertia or weights.
with power being measured as its applied directly to the crankshaft ( no pedals )
The tires are another variable that would need to be equalized or removed,, maybe just run rims and the resistance part of the trainer would have to be the rubber.
but this is just rambling.
someone else might be able to use math to get the same answers,,, but it makes my head spin
And as you pointed out drivetrain power has to overcome aero drag ( and many other variables ) before it can be applied to bike speed in the real world.
Just to be clear, I'm not "guessing" here, I've done the math. Both linear and rotational kinetic energy of bicycles, riders, and wheels are very well understood phenomena and there really isn't any mystery here. This is advanced high school/university freshman physics.
You're quite welcome to ignore all of this, but it's pretty much at your own peril. You can run the experiment, but long-story-short, if you find out that what I'm saying isn't true, then it's your experimental technique that needs to be questioned. This video, for example, is a great example of exactly that (or, how difficult it is to use a power meter correctly).
I knew somebody could do the math. Very very cool.
So by the math does the 22 require less energy than the 26 ???
Please pick one ( Yes / No )
And, mostly, it doesn't matter
* I'm assuming your comparing the Quest 451 to the Quest 559. What matters is that the 451 wheel weighs less than the 559 wheel. As far as answering the original question, the answer is yes. (Again, you can get CF 700c wheels and tires that weigh less than the 451.)
* Rolling resistance is a bigger effect than the mass difference of the wheels. You can probably get lower rolling resistant tires in 559 than 451 (although I'm not positive; for 406, you can actually get very low rolling resistance tires). So no.
* The 451 Quest has less aerodyamic drag than the 559. Because the wheels have less aero drag. I don't know how large this effect is (not large) and this becomes more important at high speed. So, yes.
* When you add linear energy + rotational energy, you can (almost) double the mass of the wheel as far as accelerating. This is less true of the dual drive wheels as while they are heavier, the mass is located close to the center of the wheel. So, stop worrying about it.
* For real-sized people (you, me, etc), the effect of the wheels is dwarfed by the rest of the weight of the system (us + bike, etc). If you are racing and seconds matter on the hour, then you can worry about these effects. So, wheel size doesn't matter much on speed.
The difference between my 20" Cruzigami Mantis (406) and 26" Sofrider (559) is rolling resistance. When I had the original tires on my Sofrider, my Mantis was faster. When I put Schwalbe Kojacs on the Sofrider (as compared to Marathon racer and Comet), the Sofrider is faster.
How's that for a yes/no answer.
Put it this way, the Quest 451 accelerates a little better and has more nimble handling. The 26" is slightly better on a faster ride, partly because the larger wheel size reduces harshness, and the harshness that you feel slows you down. Why is a very different but equally interesting expression of physics.
There are two implicit halves to your question, start-up transients and long-run behavior. Or another way to put it: energy to get up to a specific speed and power to remain at that speed. (Power is energy over time, an important distinction.)
This was touched on before, if you had two wheels which were the same except one had extra weights on the rim, the heavier rims would store more energy at a given rpm and hence would require more energy to get to that rpm, but once at that rpm wouldn't necessarily require more power to stay spinning at the same speed, though in the real world the more weight on the bearing would increase friction so there would be more friction losses so yes all else being equal it would require some more power.
The energy stored in a spinning wheel is proportional to the moment of inertia (which is how the mass is distributed in the wheel) and the angular velocity squared. A 26" wheel will probably have a higher moment of inertia than a 22" wheel (even if they were the same weight and a 26" wheel probably would be heavier, all else being equal), but at a given bike speed the 22" wheel is spinning faster so without knowing the bike speed it isn't obvious which wheel took more energy to get there. At low speeds the 22" wheel will take less energy but you'd have to know more to compute the cross-over point.
The power needed to remain at a specific speed depends on various losses, with wind resistance and rolling resistance of the tires being major factors. The diameter of the wheel will probably have only small effects on power losses.
At high speeds wind resistance becomes the largest factor, it increases roughly with the square of the velocity, which is why even small amounts of headwind vs tailwind makes a big difference. The diameter of the wheel changes the shape of the bike+rider system which may slighly change the wind resistance, intuitively the smaller diameter wheel would have slightly lower wind resistance but this intuition could easily be wrong.
Rolling resistance is a complicated topic.
The angular momentum of the wheels I w (where w is omega angular speed and I is moment of inertia) will be different for the two sized wheels.
The kinetic energy stored in the wheels, it turn out, doesn't depend on the radius of the wheel at all and only on the relative mass distribution. I = k m r^2 (where k depends on mass distribution and is close to 1 if all the mass is at the rim and 0 if the mass is at the center) and rotational energy is 1/2* I * w^2 = 1/2 I (v/r)^2 = 1/2 k m v^2.
Note that r dropped out and that this term looks very similar to linear energy of the wheel 1/2 m r *2.
As you very correctly alluded to, you "pay" for the extra rotational energy when accelerating up to speed, but once at speed, this is stored energy and takes no more effort.
You can also see why I say that for accelerating, you can almost count the mass of the wheel twice (since k is almost 1), but when climbing at a constant speed, you only count it once.
As far rolling resistance, yup, complicated.
The very low rolling resistance of Kojak tyres, while still having good grip and resistance to tyre cuts is amazing!
After fitting them to my trike, I visited a Aust. Pedal Prix event, of 1,000 school and adult racing velomobiles, and 95% had 20" and 16" 1.3" Kojaks, so they must be fast!
I think tyre selection, Make, Model, and width, is WAY more important than wheel diameter or weight.