Shortening (QR) skewer

BEFORE             and               AFTER
a small bicycle plastic surgery

Skewer is a rod (essentially a long bolt) that fits into the hole of a hollow wheel axle and fixes the axle in the fork dropouts. On one end the rod is threaded and has a corresponding nut, on the other end it usually has QR (quick release) handle, but can also have just a hex head. The skewer is often too long, so that the threaded end extends well past the end of the nut. This is an aesthetic fault more then anything else, although in some extreme cases – as in the picture here– it could be considered as a cold weapon, potentially causing serious injury or damage to the property. The obvious remedy is to cut the excess end of the threaded end, but many will be reluctant to undertake this task, fearing that the cut threads would damage the nut. I was sceptical too, but now, after I pacticed it a few times I can say it is quite safe, if you follow some simple, logical steps:  

• With the skewer on the bike note the length of the threaded end that you want to remove.
• Remove the skewer from the axle and re-thread the nut past the desired cut point. It is advisable to use another,“heavy duty” nut for this purpose, but the original nut is good too.
• Hold the skewer in the bench vise and cut the desired length with fine hacksaw.
• File the cut surface so that no sharp edges remain.
• Remove the nut to “clear” the remaining threaded part. Then install the skewer.

By the way, the procedure is applicable to any bolt that you want to shorten. Weight weenies will be particularly interested in this, and that is why I add a table showing how much mass (not  weight!) in grams you will drop by shortening a bolt of diameter D by length L= 1cm. For different length L multiply the number from the table by L expressed in cm.

Material	Diameter D
	2 mm	3 mm	4 mm	5 mm	6 mm	8 mm
Steel	0.25	0.55	0.98	1.53	2.21	3.92
Titanium	0.14	0.32	0.57	0.88	1.27	2.26
Aluminium	0.08	0.19	0.34	0.53	0.76	1.36

Ultralight bicycle bag

What do garlic, plums and Brussels sprouts have in common?

Have you ever wondered what garlic, plums and Brussels sprouts have in common? They are all edible by humans of course, but other than that they have another significant common point: they are all packed in similar plastic mesh bags. Now, a plastic mesh or a plastic bag immediately rings a warning bell. Certainly, you've heard emotional stories about huge amount of plastic in the oceans, endangering the survival of ocean species like Caretta caretta. The numbers are huge and menacing. Well, you don't need those stories to realize that people are producing too much garbage; any week-end wannanbe minimalist can tell you that. And after seeing thousands of plastic bags hanging from trees after autumn floods in your country, you can tell that recycling plastics is not working either. The only way to reduce the garbage madness is to stop using unnecessary packaging.

If you think – like many - that a single person can't do much about it, just look at the numbers: each of the 8 billions people on Earth using (or reusing) 1 less plastic bag only once in a year (the bag weighing 0.25 g) means more then 5 tons less of garbage every day. My addition to this recommended practice is here: a hint how to make an ultra light and practically costless bicycle handlebar/under seat/frame bag. Let it be my present to you for Christmas.

So, lets get back to garlic, plums and Brussels sprouts. The meshes they are packed in are quite tough and light: three of them weigh less then a resolution of my scale, which is 2 g. Add an unpaired shoelace that you somehow forgot throwing away and a buckle of a cinch chord from a trouser's legging (that any credible ultralighter had cut away and stored for possible future use) and you've got a not-too-bad-looking handlebar bag weighing around 4 g that can store almost 500 g of content. In this example the content is a light wind/rain jacket, a compact camera and a set of tools (all together 360 g) – which makes a world record content/container ratio of 90.

Mesh is attached to the handlebar with a shoelace and at the lower end to the brake/shifter cables.

And here is the mesh in action: holding a wind jacket, compact camera and a set of (mini)tools.

P.S. If you are truly dedicated world-saver, you won't go and buy Brussels sprouts to get the mesh; instead, take a tour in your nearest garbage bin and find your mesh there. It might also be an eye opening experience, the one you've haven't had for quite some time.

Numb and/or cold fingers

Do you have problems with numb and/or cold fingers? I do! My latest effective solution to this problem comes from psychotherapy. You may know the term "progressive muscle relaxation" (PMR). If not, goggle it. It's an exercise in tensing and relaxing individual muscles of your body. PMR in itself is a good relaxing technique to start or end a day of cycling. But it's also very effective when applied to your numb or cold fingers. It's simple:

1. Tense or squeeze your fingers and/or fist. You can do this either by holding to the handlebar or not, keeping the fingers straight or bent in the fist. Whatever turns out better for you.
2. Keep your fingers tensed for 7 to 10 seconds.
3. Slowly relax the tensed muscles (in 5 to 10 seconds).
4. Keep the hand relaxed for another 5 to 10 seconds.
5. Repeat points 1,2,3,4 several times. 5 repeats should do in most cases.

Rain shell gloves

On my other blog (ultralightcycling) I tried to give some advice about cycling in the rain. And I failed miserably! So, the final solution to cycling in rain is still pending. In the meantime we must satisfy ourselves with partial solutions.

I have here a solution for your hands; the solution that is both effective and fulfills the rigorous minimalist requirements of ultralightcycling. The weight of a pair of such gloves is 2 g. They are also inexpensive, costing about 2 Euro cents. Here's what to do:

1.      Bend an old spoke to form a U.

2.      Take a plastic envelope of a magazine (European A4 size).

3.      Heat the spoke over a heat source.

4.      With the heated spoke make three cuts in the plastic to make a cubist's approximation of a glove.

5.      If you have two hands, repeat the procedure 2,3,4 once again.

6.      Wait for a rainy day, put the plastic gloves on (optionally over regular gloves), tuck them under your sleeves and enjoy the feeling of dry hands while cycling in rain.

You might have some problems with condensation. To overcome them I suggest to make few holes on the palm side of the plastic gloves and to wear only cycling gloves (i.e. not full-finger gloves) under the plastic ones. If the plastic gloves start to tear (this usually happens in the corner between thumb and index finger)
they can be repaired with any heat source.

Bent spoke with umbrella tips glued to the ends.



Plastic cover of a magazine.



1-2-3 and your plastic glove is ready-made.



Your plastic glove in action

... and wears better then the usual plastic glove from petrol station of fruit-stand in supermarkets.

Cleaning cycling water bottles

Plastic cycling water bottles (bidons) are notorius good growing ground for mould and fungus. However, I am not particularly upset about it, partly because I am not too sensible to dirt, but mostly because I don't recommend the use of cycling bidons in the first place. Ordinary plastic 1L bottle is almost twice lighter then 0.7 L bidon and has more volume. It also has simpler mouthpiece that doesn't attract mould.

But if, for whatever reason, you insist on using cycling bidons, I have a simple tip how to clean it.

You don't need any special chemicals, heat treatment or dish washer. Just put a paper towel or a few pieces of (unused) toilet paper in the bottle and stir and rub with a long stiff stick. Instead of a stick you can use the hooked end of a spoke.

Tyre boot

Tyre boot is a temporary solution for a tyre that is cut or badly damaged, when one doesn't have a spare one. It's a piece of material that is put between the tube and the tyre and which prevents the tube to bulge out through the cut in the tyre and explode. The boot material could be of any kind, as long as it is strong enough to hold the pressure and soft enough not to puncture the tube. It's often recommended to use a piece of  the old tyre, although from my experience even a boot made with duct tape did last more then 1500 km (on tarmac).

My hint is that before putting the boot, it very much helps to sew the tyre. If the tyre casing is cut or damaged, I sew up that part with the needle and thread, just as patching a hole in a sock. Then I apply the tyre boot, which in this case can be as simple as a piece of duct tape or a tube patch, applied from the inside of the tyre. A piece of wrapping material (for chocolate bars, for example) might be just as useful.

I have such patched tyres on my commuter currently, both at front and rear, as an experiment of how long this would last. This is a road bike with a pressure of 6 bar. I've been riding 18 months now with patched front tyre and 14 months with the back one, but after I'd confessed it, and Murphy is reading it, I suppose I'll have a blowout tomorrow - will let you know.

Climbing a hill on a bicycle

Deep down everybody fears hill climbing: the beginners, the veterans and the pro's. It is one of those activities which many people - often including cyclists themselves - find unexplainably masochistic. Why do you climb a hill, when you will come back down again? And if you really insist in climbing, why don't you do it in a car?
I won't go deeper into philosophic essays regarding this subject, although, if you care to think enough about it, you may come to some profound quintessential matter like Universe, Big Bang, Meaning of Life, etc. I will only state some physical facts that may help you to climb the hill easier.

When you cycle, and in particular when you cycle up hill, the nature is working against you. It is trying to stop you from cycling by imposing various forces that act against your movement. There are three major forces acting against you: the force of gravity (G), the force of rolling resistance (R) and the force of air resistance (A). To move against these forces you have to do some work (W). In mathematical terms the work is a (scalar) product of the force(s) and the path (L) of your movement. When you climb a hill of road length L and of grade N (in %), the work against the gravity is:

W=M*g*L*sin(arctan(N/100))

where g is the acceleration of gravity (9.81 m/s^2) and M is a mass of yourself, bicycle and everything on you and on it. You are doing also the work against the rolling and air resistance, but when climbing steeper hills these two represent only about 5% to 10% of the total work, so, for now, we will disregard them for the sake of simplicity.
The energy to do the work is provided by your body, by your muscles. To climb a hill you must produce at least as much energy as the work W. In reality you are producing much more energy, because some of it (some say even 70% of it) is lost in the dissipative processes in your body, and, of course, some is required to keep you alive, even when not moving at all. We will denote this additional energy as internal body energy (Eb). The total energy required is thus (Eb+W).
You will input that energy during some period of time (t) - the time that you climb the hill. Energy divided by time in physical terms is called power (P):

P=(Eb+W)/t

When measuring and reporting bicycling power we usually only measure the part that corresponds to the energy W; the part Eb, which is lost due to small muscle efficiency, is usually not measured or reported. So, we are left with:

P=W/t                           

The ability to produce a certain amount of power is in accordance with everyday conception of physical power: trained or stronger people can produce more power. The pro cyclist can produce around 400 Watt to 500 Watt of power when climbing hard. Ordinary cyclists produce 100 Watt up to 250 Watt of power over a period of few hours. According to the above equation, you need more power if you climb the hill faster (in less time), and if you climb the same hill slower, you need less power. Think of it in exaggerated terms: if you had the whole week to climb the hill, you would still need to input the same amount of work W, but you would need so little power that you would probably not even notice there is a hill. The first point to remember is thus: you can climb practically every hill by climbing it slower.

The speed of cycling (v) can be expressed as:

v = L/t = 2*π*R*c*T

where (R) is the radius of the wheel, (c) is cadence of pedaling and (T=F/B) is the transmission ratio (where F and B are number of teeth on front and back rings). By substituting the above equations, we have:

P=M*g*sin(arctan(N/100))*2*π*R*c*T

What can we say about this equation? It connects the physical quantities (P,M,N,R,c and T) in a definite way. If we change one of those quantities, then at least one of the rest of them will change too. For example, with a maximum power (P) that we can produce, we can climb a steeper hill (=higher N), if we have lower mass, lower cadence, lower transmission or smaller wheel. Conversely, if we cannot change M, c, T or R, then the only way to climb a steeper hill is to produce more power. Incidentally, from the equation you can even see another possible solution: lowering the effective gradient (N) by zigzagging the climb. Of course, any self-respecting cyclist would never do such a thing - it would be as if stepping down on a climb.

Now, you probably have the feeling that this is perfectly logical, so you may ask why did you have to go through all this math to come to the conclusions that you already know? Well, first of all, it's the proof that physics and common sense sometimes do agree. Second, you can use the equation to calculate, for example, what gearing you require on a certain hill, or what is your power output, or how much will you gain by buying lighter bike, or by loosing 2 kg of fat, or climbing with a MTB instead of road bike. We can very well say that it is the "mother of all cycling equations". Third - and this may be the most important - you may realize that you can climb a certain hill that you considered too hard, and not even that, you may even plan how to climb it without much effort.

To see that, lets first write the above equation in sipmlified form, like this:

P=M*g*v*sin(arctan(N/100))

From this simpler form, you can see that you can climb a steeper hill either by loosing weight (difficult), producing more power (impossible) or lowering climbing speed (quite possible, as we already mentioned).
Lowering the speed probably instinctively makes you think of lower gearing. But even when you run out of lower gears, you still have another ace in your sleeve: lower the cadence. If you lower the cadence down to around 35 rpm you will be surprised how easy it is to climb even the hills that you previously considered impossible.

However, there is of course a limit: you can climb just as slow as the lowest speed before toppling over. For me it's about 4.5 km/h. By inserting this minimal speed into the equation you can calculate whether you can climb the hill at all (without zigzagging).

There is another interesting thing that we can conclude from the above equations: that the transmission ratio doesn't really matter. If you could pedal at, say, 10 rpm, you may be able to climb in the big front ring, without the need for any more power than you are usually able to produce. Does it mean that you need only one ring in front? No derailleur, no shifter? It is well worth a try!

Flipping a stem

If you are reading this, then you most likely know what is a stem and what "flipping a stem" means and why it is done. For those who came to this page by accident, here are few clarifications:

• A stem is a part of a bicycle that connects the steering tube (which is the top part of the front fork) with the handlebar. There are two types of stem. The "quill stem" is found on older bicycles and in cheaper today's bikes. Most of today's quality bicycles have the stem as the one in the picture in this post. It connects to the steering tube with two bolts. Usually this kind of stem is not perpendicular to the axis of the steering tube, but is at a certain angle to it - angle α in the picture. When this is the case, you can raise (or lower) the handlebar by flipping the stem around. 

If you flip the stem from the picture around, the point T1 (intersection of the central axis of the stem with the handlebar) will move to point T2.

The question is, how much will you raise the handlebar by flipping the stem. Will it be too high, too low, or is it just not worth the trouble flipping the stem at all? 

Well, I've got the answer for you. I will not go into detail of developing all the equations, I'll just state the final result:

If the steering tube is at the angle β, the stem has a flip-flop angle α and length L, then, after flipping the stem, the handlebar will move vertically by ΔY and horizontally by ΔX, calculated by:

ΔY=L[cos(β-α)-cos(β+α)],     ΔX=L[sin(β-α)-sin(β+α)]

Positive values mean that the handlebar will move up and forward, negative values move the handlebar down and backward.

In the following table are the values of ΔY and ΔX in mm, for some common values of β and α and for the stem length L=100 mm. If you have a stem of different length, multiply the values in the table by L/100, where L is the length of the stem in mm.

	α = 6 º		α = 8 º		α = 10 º		α = 17 º
	ΔY	ΔX	ΔY	ΔX	ΔY	ΔX	ΔY	ΔX
β = 72 º	19,9	-6,5	26,5	-8,6	33,0	-10,7	55,6	-18,1
β = 73 º	20,0	-6,1	26,6	-8,1	33,2	-10,2	55,9	-17,1
β = 74 º	20,1	-5,8	26,8	-7,7	33,4	-9,6	56,2	-16,1
β = 75 º	20,2	-5,4	26,9	-7,2	33,6	-9,0	56,5	-15,1

Packing a bicycle for a flight

There are several schools of thought on how to pack a bicycle for a flight. Some like to leave it unpacked so that the baggage handlers will see it's a bike and will treat it gently. Others are convinced that only a hard shell case with steel bracing will do the job. Most of the cyclists take the middle way and put the bicycle in a cardboard box. So let us first start with:

Recipe for packing a bicycle in a bike box

Ingredients:
- 1 road bicycle, size 622 (or 700c) wheels, frame of the size L (or 58), drop bar.
- 1 cardboard box of dimensions 138x78x20 cm.
- several pieces of cardboard (a middle sized cardboard box will do).
- 1 spacer for the fork (or cardboard).
- 1 roll of packaging tape.
- 2 meters of duckt tape.
- 2 or 4 plastic bottles.
- Allen keys 4, 5 and 6 mm.
- 15 mm pedal spanner.
- scissors.
- 1 large beer.

Procedure:

1. Put the beer in the fridge.
2. Tape the bottom of the box with packaging tape, from inside and outside.
3. Reinforce the carrying openings of the box with cardboard from the inside.
4. Shift the chain to the largest rear cog and largest front ring.
5. Put one plastic water bottle in a bottle cage.
6. Unscrew the pedals.
7. Fix one crank arm by taping it to the seat tube with duckt tape. The other crank arm should point down below the front rings to protect them.
8. Remove the seat and fasten back the seat clamp.
9. Deflate the tires slightly. The rear one a bit more then the front one.
10. Lower the rear wheel a bit below the dropout and move the wheel toward the seat tube. Tighten the rear Quick Release. This will give you some more room to fit the bike in the box. You may not need to do this if the box is long enough.
11. Cut the bottom of plastic bottles and protect the rear axle and the rear derailleur with it. This is done in order to prevent the axle puncturing the box and as a protection of rear derailleur. If you have only 2 bottles, cut them in quarters. 
12. Unscrew the stem-steerer bolts and turn the handlebar for 90 degrees in line with the bike.
13. Fasten the stem-steerer bolts back.
14. Turn the bike upside down.
15. Remove the front wheel and remove the front quick release axle.
16. Put a spacer in fork dropouts. You can use a rolled piece of cardboard.
17. Protect the front wheel axle with plastic bottles.
18. Put the bicycle (without the front wheel) in a box.
19. Unscrew the stem-handlebar bolts and remove the handlebar.
20. Screw the stem-handlebar bolts back in.
21. Put the front wheel in the box so that it fits somewhere between the fork and the seat tube.
22. Put the handlebar in the box.
23. Put additional pieces of cardboard between the wheel axles and box sides, and between the front wheel and bicycle frame.
24. From a couple of pieces of cardboard a bit wider then the box width make rolls and put them in the box as spacers. This is not a crucial step, but will give you some protection against squashing the box. 
25. Put the saddle, the pedals and front QR axle into the box and fix them to the bike or the rack.
26. If you have anything else to put in a box, do it now.
27. Shake the box and look for any loose parts. Everything should fit tightly without moving.
28. Close the box and seal it with packaging tape.
29. Have a beer.

Time required: 2 hours.

If you have smaller bike or wheels (MTB especially) and flat handlebar you can use a smaller box. If the box is too small, you'll have to take the rear wheel off too: in this case the procedure might be considerably different.

Other options

Of course, the cardboard box is not the only way to pack the bike for a flight. Other options are:

1. Leave the bike unpacked, just turn the handlebar, lower the seat, deflate the tires and remove the pedals. There is an ongoing debate whether by doing so your chances of getting your bike damaged are increased or decreased.
2. Dismantle the bike and put it in a hard shell travel case. This would certainly be the preferred method if you have somewhere to leave the case upon your arrival.
3. After the procedure as in 1, put a minimal protection made of pieces of cardboard or pieces ob plastic bottles on the sensitive parts: shifters and derailleur. Something like this.
4. After the procedure as in 1, wrap the bike in plastic. You can use a roll of food wrapping plastic. A roll of 30cm x 50m should do. The result looks like this.

Which procedure to use might depend on the policy of the airline company and whether you are starting the tour or flying back. I prefer to start the tour by packing the bike in a box as explained and my preferred option for a return trip is to wrap the bike in food-wrapping plastic. This allows me to cycle to the airport with minimal packaging material and without hassles of finding a box or transportation, preparing the bike and wrapping is quite easy and fast (half an hour) and attracts lot of interested attention at the airport. It also complies with rules of some companies that the bike has to be packaged (to avoid damage to other baggage).

On return trips I usually improvise. More so as I like to cycle all the way to the airport.
This is hardly a school example of the proper packaging of a bike for the flight, but, weighing less then 10 kg it was free of charge on the plane and arrived unscratched to destination after 3 connecting flights.
Ingredients/tools required: a roll of scotch tape, a piece of cardboard and a razor blade.
Pedals and seat were in my hand-luggage, tools & tent spikes (not allowed as hand-luggage) were taped to the rack .

Improvisation can save you some money on train (or bus) trips too. As an example, I can tell you my experience on trains Dali-Kunming (seater) and Kunming-Chengdu (hard sleeper) in China. I bought the biggest bag I could find on the market, disassembled the bike and put it (stuffed it really) in the bag and carried it on the train. The bag was too small even only for the frame and rear wheel, I carried second wheel in the other hand, but nobody in the station or in the train said anything. In the seater I put the bag in the place between compartments and in the sleeper under the bed. Everything went smoothly, except that my arm grew few cm longer after carrying the bag around the stations. See the picture of the bike/bag below.

This is hardly a school example of proper packaging of a bike,
but it was free of charge on the train.

Stiff sole myth

Another in the long line of cycling myths goes approximately like this: "When buying shoes for cycling make sure you get the ones with the stiffest sole you can find." The argument seems to be that the energy is lost by the flexion of the sole, so the one that flexes the least is the best. I must admit that this is a very convincing myth. I myself had fallen for it until very recently.

But firstly, a clarification of the terms. If by "cycling shoes" we mean special shoes that are fixed to a pedal (SPD or similar clipless system) than this is not a myth but a reality. In this case the force on the pedal is transmitted through small area and if the sole were not strong (=stiff) it would quickly be ruined. However, I think that the stiff-sole-myth had been around before the clipless system hype, so it doesn't descend directly from it (btw, for the reasons explained above the clipless system is not an option for me).

A recent experience convinced me that stiff sole used with ordinary pedals (no clips or straps) might be unfavorable. One Saturday I rode 150 km with two friends in a relatively fast pace. During and after the ride I felt great, the bike fit was perfect, I had no pains whatever. I felt as if I was at the peak of fitness. Next Sunday I rode 80 km "recovery" ride, same bike, same clothes, different shoes. To my surprise on that ride nothing seemed right: I had back pains and the saddle felt uncomfortable. On Saturday I had trainers with soft rubber sole, on Sunday I had shoes for in-door football (which, btw, were up to now my touring shoes) with stiff sole which became sleek and slippery during three years of use. So I am quite convinced that my uneasiness on Sunday was caused by the constant corrections of the position of my feet because the soles of the shoes were slipping on the pedals. If the soles were a bit more flexible they would grip the pedal preventing the slip.

My conclusion is therefore that the non-slipping contact between the pedal and the sole is much more important then the stiffness of the sole; and this is better attained if the sole is somewhat softer.

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.