Tuesday, 25 May 2010

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.

Saturday, 1 May 2010

Digital maps

Do you like to hang on at the end of the day, with a map spread in front of you, tracing on it the route you've just cycled and contemplating the part that lies before you? It's one of the best feelings isn't it? Well, that's just about all there is good about a map! From a point of view of a light-weight unsentimental cycling warrior - a map is a waste of space. The first urge after this realisation is to cut the map, leaving only the part where you wish to travel. But, there is a much more thorough approach: digitizing a map.

The idea behind it is rather simple. You can view a map as a colection of points of interest and connections between them. This info can be stored in "digital" form, using alphanumeric characters, in much smaller space then the map itself. The point of it is to store all the information on a small plastified card which is accessible "on the fly", say from a cycling jersey's pocket, without any unfolding, searching, folding and storing. Larger itineraries can be stored on several cards with only one used at any time. Such digital info may not be visualy as clear as a map, but all the info will be there. In fact your card will have much more info then a map, since there will usually be enough space on the card to store info on water points, food, accomodation, camping spots, altitute, passes, check points, usefull phrases, important telephone numbers, repair tips, ...

There are many ways to make a digitization. I will explain how I do it currently by an example of digitizing Himalaya itineraries in Pakistan, India, China and Nepal.

  1. Let me start with some definitions. A crossroad is a point where more then two roads meet. A point of interest is either a crossroad or any other feature (e.g. a town, a camping spot) that is on the road and is not a crossroad. A connection is a part of the road between two points of interests, A and B, with the condition that it has no other crossroads on it. Any network of roads can be described as a collection of connections.
  2. The first step is to draw a diagram of the selected routes, showing all possible connections and major points of interests. In the diagram the points of interests are presented as numbers, 0,1,2,...
  3. We represent a point of interest as a name and its number immediately following it. The points of interests are, for example: Islamabad0, Kashi1, Ali2, turn off to north4, Saga3, etc. The points of interests are fully defined by their numbers, the name is not necessary and may be used for clarity.
  4. We represent a connection by its ending points of interest, separated by a dash: Islamabad0-Kashi1, Ali2-Saga3, Ali2-turn off to north4. The connection is fully defined by ending points' numbers, the name may be used for clarity: 0-1, 2-Saga3, Ali2-4, Saga3-t.o.north4.
  5. We first list all the possible connections, using their numbers and names when they are introduced for the first time. As an additional information we add distances of the connections in parenhteses. All the distances will be in kilometers. Example: Islamabad0-Kashi1(1182); 1-Ali2(1323); 2-Saga3(752); 3-t.o.north4(62); 2-4(978). 3-HWY318t.o.5(167); 5-Kathmandu7(231); 5-Tingri.t.o.6(73); 6-EBC8(67). 6-Shegar.t.o.9(48); 8-9(96); 9-Lhatze10(133). 4-10(257);
In the second part we add detailed information about the individual connections. We can do that for all of the connections or just the selected ones. The detailed info for one connection looks like this:
1-2: #315: 200/TL:Yarkand. 65/Yecheng,TR. #219: 25,1500/armyB. 31,1840/V. 15/F. 41,3150. 14/F. 5/armyB. 30,3000/Kudi(426). 18,3280/armyB,F. 27,4300/RRS. 11,s217,4825. 24,3675/Mazar,TL. 13/F. 35,3860/RRS. 3/F. 17,s309,4795. 15/rrs. 16/RRS. 24,3560/Xaidulla(629). 41,3700/t.o.KKH. 20,s425,4180/Kosbel. 12,3800/Kangxiwar,RRS. 50,4000/Dahongliutan. 25,4500/RRS. 22,s534,5080,Aksai Chin. 9,4900/tent,f. 34,4735/Tianshuihai. 93,s670,5125. 2,5005/Tielong. 16,s688,5185. 12,5000,Lungmo tso. 15,s715,5090/NP. 5,5140. 10,5090/Sumxi. 10,s740,5400/!. 88,4275/Domar(1093). 17,s845,4520. 8,4300/LP. 40,4170/house,f. 2,4150/Nyak tso. 23/H,boats,fish. 15,4150/Rutok Xian,TR. 27,4240/tent. 2,4250/houses. 3,4250/t.o.Rabang. 2,4250/petrogl. 53,s1020,4715/Lame. 32,4350/cp. 6,4200/Ali2.

The meaning of it is as follows:

1-2:                   info on connection 1-2 (Kashi-Ali) follows.
#315:                  road number 315.
200/TL:Yarkand.        after 200 km turn left for Yarkand.
65/Yecheng,TR.         65 km to Yecheng, after that turn right.
#219:                  road number 219.
25,1500/armyB.         25 km to army base, it's at the altitude 1500 m.
31,1840/V.             31 km to a village, its altitude is 1840 m.
15/F.                  15 km to guaranteed food.
41,3150.               after 41 km there is a pass, at 3150 m a.s.l. Passes are represented in bold.
14/f.                  after 14 km there is probable food
5/armyB.               5 km to army base.
30,3000/Kudi(426),CP.  30 km to Kudi. Kudi is at 3000 m altitude and 426 km from the starting point of this connection (e.g. from Kashi).
                       There is also a Checkpoint.
18,3280/armyB,F.       18 km to army base at altitude 3280 m and guaranteed food.
27,4300/RRS.           27 km to Road Repair Station, at 4300 m.
11,s217,4825.          11 km to a pass at 4825 a.s.l. There is a stone kilometer marker "217" at the top.
24,3675/Mazar,TL.      24 km to Mazar, at 3675 m. Mazar is underlined, which means you can get everything there (water, food, accomodation).
                       After Mazar turn left.
The extent of the detail for individual connections can be different. For difficult routes like the above one, you need detail on food points and altitude. For an easy connection only a connection length may be needed, all other info may not be necessary or might be found while cycling the road itself. The info on all of the usual connections in Hymalaya region can be stored on a card with dimensions 13x8 cm - for a person with good eyesight, that is. The front page of this card would look like this:

The opposite side would have info on connections in India and Nepal (between Lahore, Amritsar, Srinagar, Leh, Delhi and Kathmandu). There would be enough space left on that side for personal info and few grace-saving chinese phrases.

"Digital maps" are an evolving subject: here is the latest one, covering the tour in Vietnam and China
Iik's latest Navigation App. 
Note some minor, ingenious upgrades to his cue-sheet (i.e. digital map) concept, 
like color-coded items and top-left notch for quick card orientation
(which could possibly be used as bottle opener).