Thursday, April 3, 2014

Bicycle bodywork (2 of 4) Power numbers

OK, now that I've got your attention, I'll say the usual "It's easy!"
to make sure that all of you who don't like math are properly scared.
Equation credit: Peter Cox, Energy and the Bicycle – Human powered vehicles in perspective

If you look at the equation above, you'll notice there are five plus signs in it.  That's because there are several things that drain your energy while riding a bike, and they are all being added together.  From biggest to smallest when riding on an average road at 15 mph without any wind, they are:
-air resistance (by far the biggest load)
-rolling resistance (tires and bearings)
-gravity (hills)
-acceleration (stopping and starting)
There are other things that drain energy (riding next to a mountain of magnetic ore, air turbulence from cars, eating too much at the church supper before riding), but they aren't usually entered into the equation.

The reason why the equation looks so complicated is because everything but the kitchen sink has been put into it and common terms have been canceled out, so it doesn't make sense anymore.  If you pull the things back out, you can get a much better idea of where the energy goes.  It's also possible to leave out the gravity or acceleration parts if you don't need them and get a much simpler equation.

Air resistance depends on how fast you go, how big you are, how slippery of a shape you have, and whether you are pedaling in thick air at sea level or thin air in the mountains.  The equation is like this:
Air resistance = Velocity cubed x Frontal Area x Coefficient of drag x Air density
Notice that everything is simply multiplied together, so that if you are twice as slippery (drag is half as big), you can be twice the size and still use the same power.  The part that should stick out like a red flag is Velocity, because it gets cubed.  (The physical drag increases proportional to the square of velocity, but because there is less time for doing a task at a faster speed, the power (watts) that must be used increases by the cube.)  Riding along at 6 mph would be a cube of 216, but riding at 20 mph is a cube of 8000, meaning 37 times as much energy is needed.  You'd have to be 37 times smaller, or 37 times slipperier, to stay at the same power level.

Added April 19, 2014:
Thinking back over the last several years of town energy committee work, I've realized the above energy explanation needs to be better.  Many people I've talked with haven't known the difference between kWh (kilo Watt hours, or the quantity of energy), and kW (kilo Watts, or power - the rate the energy is flowing) on their household electric bill.  Using the common analogy to plumbing,  kWh is a quantity similar to a gallon, and kW is similar to the rate the gallons are flowing through the pipe, (gallons per hour).  Since we can't collect a bucketful of electrons, we define their equivalent to a gallon as the quantity of electricity that has flowed at a specified rate (kW) for one hour.  Changing over to electric bikes, we get rid of the kilo (1000) prefix because a bike uses less electricity than a house, and say that the electricity is flowing to the motor at for example 200 Watts.  Then if the trip lasts an hour, the energy used was 200 Watt hours.

As a practical example for bicycle Air Resistance vs battery range, let's compare bike riding at 10 mph (running speed), and 20 mph.  The slower speed might use 30 Watts (depending on aerodynamics), and riding for one hour would use 30 Watt hours of energy.  But air drag increases as the square of speed, and because the time for traveling the same distance is half as long, the actual power needed increases as the cube.  Two thirds way down this Wikipedia page:  is this equation:

P = gmVg(K1+s) + K2Va(squared)Vg

We can toss out the first part, gmVg(K1+s), because it is about friction and hills, and simplify the second Air Resistance part to Power = K2Va(cubed).   Since K2 is a constant, power needed from the motor is simply proportional to the cube of speed Va.

Doubling the speed to 20 mph would need 8 times the power (2 cubed), or 240 Watts.  (You can also calculate the numbers using K2 = 0.03 for mph and watts units.  Note that K2 would be larger for more air resistance, and smaller for less.)  However the trip would take only half an hour, not a whole hour, so the total energy used would be half of 240, or 120 Watt hours.   Thus to go twice as fast, the motor would need to be 8 times as powerful (240 Watts/30 Watts), and the battery would need to have 4 times as much energy capacity (120 Wh/30 Wh).

You can see that a little bit of work on aerodynamics goes a long way...

I should mention that no one has a clue what the Coefficient of drag (Cd) of a design is, until it is built and tested in a wind tunnel.  I've included a couple of generic charts below to help you make a guess for your project, and compare it with other projects.

Rolling resistance is the tire deforming as it hits the ground (bigger), and the friction in the bearings and chain (smaller- as long as you oil your chain).  The Coefficient of rolling resistance (Crr) is the force needed to move the bike forward divided by the force downward (weight) and is effectively independent of speed, but again velocity must be included to calculate the power needed because of the effect of a shorter amount of time:
Power used for rolling = Velocity x Crr x weight + Velocity x bearing drag
Crr is often given values like this:
     0 is no drag at all
     0.001  for a wooden indoor bike track with slick tires
     (i.e. it takes 0.001 pound of pushing to move 1 pound of weight forward)
     0.004  for a typical 100 psi road tire
     0.0044 for 27"x 1.25" road racing clincher tires at 95 psi on a smooth road
     0.007  average bike- good
     0.008  bad asphalt
     0.010  average bike- not so good
     0.013  27" x 2.25" 45 psi BMX knobby tires
Clunker average tires can have twice the rolling resistance that racing road tires have.  Weight is included because the heavier you press down the more the tire deforms, and the larger rubber flexing absorbs more energy.  High tire pressure reduces deforming, but when the tires are so hard that they cannot flex, the bike and rider are pushed upwards over every little bump.  This is lost energy because it isn't converted to forward motion by coasting back down the other side of the bump (you just fly through the air for an instant), so a very tiny amount of tire flex will actually give the lowest energy use.  I should mention here that narrow tires aren't faster because they flex less, it's because they have a smaller frontal area with less air resistance, (actually fat tires have less energy loss due to flexing, and if you ride only at slow speeds then fat tires are a better choice).  Aerodynamic rims are used to further save energy.
Air turbulence (i.e. lost energy) around a tire traveling towards the left.
Photo credit: Schwalbe Ironman Triathlon tire simulation
On racing motorcycles that are in an open bodywork class (front and rear axles
and driver must be visible from the side) there is often a very tightly fitting
aerodynamic front fender to try to smooth out this turbulence.  This type of fender
is not practical on the street, where there must be a larger gap so that mud buildup
doesn't rip the fender off.  
Bearing drag is so small that it can be ignored for many general calculations.  As long as your chain isn't a rusted solid piece of metal, then doubling a very small number is still a very small number, with not much chance of using it to improve the design of a cargo bike.  (Just don't put a dozen bearings, chains, and gears in your project.)

Gravity can be thought of as you lifting yourself and your bike up or down a hill.  The faster you lift per unit of time, the more power that is needed, so Velocity is included again:
Power needed for hills = Velocity x (Mass x Gravity constant) x % Grade
As I briefly mentioned in the Regeneration post, you do not get all the energy back going down the other side of the hill that you put in when climbing.  Some is lost to braking, and some to air resistance, so the useful energy in the downhill momentum is less.  The best thing you can do here is to make the bike (and yourself) lighter.

Acceleration is the same equation as gravity, just in a horizontal direction, so you remove the % grade number, and the fixed gravity constant is changed to your desired acceleration:
Power needed for acceleration = Velocity x Mass x Rate of acceleration
Sometimes the rotational mass of the wheels is included in more complete equations, but if you are not accelerating it doesn't matter for other considerations.
Like gravity there isn't much to work with here other than keeping things light.

Drag forces on a bike relative to speed.  (Wind is air resistance,
Roll is the tires, Drivetrain is bearing drag)
This is rolling down the road numbers, gravity and acceleration have been left out.
Credit: Drag Forces in Formulas, Article for Radfahren magazine, Rainer Pivit

As you can see by looking at the parts of the power equation and the above graph, most of the energy lost is due to air resistance, and aside from keeping your tires inflated, oiling the chain and bearings, and traveling slower, the best power efficiency improvements are found in reducing air resistance..  (Tightening up loose clothing is an easy fix.)

Fortunately if you don't wish to sum up equations, there are plenty of Fitness Freaks who ride bikes and only want to know how many watts they are putting out and calories they've burned, without doing the math either.  In the sidebar on the right you'll see five web based calculators in the Tech list to help you pick out one you like:
-Oldest bike calculator
-Kreuzotter bike calculator
-Lamancusa bike calculator
-Geocities bike calculator
-Analytic Cycling bike calculator

Now that we have some basics, we start getting into the fun stuff of how they interact.  Here are a few charts and graphs for roughing out some boundaries for working on this:

How long a rider can last at different power levels.  Do not use these numbers
for design unless you want to arrive exhausted.  Smaller power levels,
such as 50 to 200 watts are probably OK for an average rider on an average trip,
and 300 watts for a short time up a hill.  (746 watts is one horsepower.)
Source: "NASA, 1964", part of motorcycle aerodynamics essay from Kraig Schultz, , original document unknown

Drag and power numbers for different bicycle riding positions.
In the first column of numbers, the upper number is air resistance,
the lower is rolling resistance, both are in pounds of force at a speed of 20 mph.
The second column is the Cd number for that configuration, note that a commuter bike
(Dutch Oma bike) has a Cd of 1.1, and the speed record bike's shape is closer to 0.1.
One person's power, (or one battery's worth), can go 10 times further in the streamlined body.
The sixth column is how much energy that riding position uses relative to a touring bike.
Source: The Aerodynamics of Human Powered Land Vehicles, 
Albert Gross, Chester Kyle, Douglas Malewicki, Human Powered Vehicle Association
The cleanest copy I can find of this chart for downloading is from Kraig Schultz at:

A common shorthand is CdA, which is the Coefficient of drag x the frontal Area.  With it you can compare different body configurations without having to calculate out the rest of the power equation for many different velocities, it's the fourth column in both the chart above and this chart, (note that the wattage columns are for 22 mph only):
Energy use of various bike types.
The fifth column is watts needed due to air drag at 22 mph.
The last column is watts needed due to rolling resistance at 22 mph.
Credit: IHPVA Human Power Issue Number 54

Commuter bike (Dutch Oma bike), Photo credit: Berlin Cycle Chic
The top line, "Upright commuting bike", lists 345 watts to push against air resistance at 22 mph, and 53 watts for rolling resistance.  Since we know an average commuter is good for 150-200 watts, we know that 22 mph isn't going to happen for more than a few seconds.  This is a problem when doing long errands to the next town.  Adding an electric motor would make it possible to travel faster, but at the cost of using 400 watts of energy.  Trying to design something with longer battery range will require paying attention to air resistance.

Looking at the fourth line down "Road bike + Zzipper fairing", the air resistance becomes 157 watts and the rolling resistance is 38 watts, a 200 w improvement from a front mounted bubble, lower riding position, and more efficient tires:
Eric Brill on 1984 Kestrel with Zzipper zz-os fairing, Photo credit:
This bike is using about half the energy of an upright commuter bike.
It doesn't offer much rain protection or place to mount solar panels though,
and I don't want to ride 30 miles bent forward.

Frontal area can be greatly reduced by riding position.
Commuter, touring, road racing, flat track, and touring and
racing recumbent frontal areas.  Photo credit:

Less frontal area and a fairing- a long wheelbase recumbent with Zzipper fairing.
Lowering the riding position with a pedal forward design reduces the frontal area,
and the fairing reduces drag.  My estimate for this bike's frontal area loss
based on the Kestrel above is a little over 100 watts for air resistance,
and about 45 watts of rolling resistance.  This should have well over twice the battery range
of an upright commuter bike.  A problem with this example is the fairing is fork mounted,
and will be steered by crosswinds.  Photo credit:

In the next post I'll look at Cd, air flow, and play with some basic shapes used in airflow design.


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