The zero wind tunnel option for serious cyclists

How a cyclist can discover his horsepower and Cd with no tools except his bike and a road, and use the information to improve his performance

by Andre Jute

A cyclist can easily measure his velocity across the road. But if he wants to improve his speed, he must consider the factors which influence the speed. The pros have access to wind tunnels, and all kinds of specialist tools like accelerometers, to help them determine which factors matter most. But the cyclist without such expensive aids can come surprisingly close to the pros in discovering the same answers, given only that he first puts his mind in gear.

A cyclist’s speed on the road is determined by the area of his frontal profile on the bike, by the friction with the road which results from moving the combined mass of rider and bicycle, by the resistance of air to the moving mass, and by the power he can apply to the pedals. The problem is that if, for instance, we adapt a formula from economy cars, such as this one:

HPv = (V/375)*(CrW + 0.0026CdAV^2 + GW + FW)

where

HPv is the horsepower required for some desired speed V

V is the desired speed, mph

Cr is the coefficient of rolling resistance

W is weight of rider and bike, lb

Cd is the coefficient of aerodynamic drag

A is the frontal area of bicycle and rider, sq ft

G is the road gradient expressed as a fraction

F is the acceleration as a fraction of one gravity

-- we see instantly that the weight of the rider and bike appears multiple times, separately influencing the products of friction, gradient and acceleration; this is the origin of the belief that nothing speeds up a cyclist as fast as a diet! Complication and multiplicity is not what we want, so we have to simplify and break up the problem a bit more, march the long way around it so we can use the same resources twice.

However, we can calculate some of the factors separately, determine others on test by observation, and then use those to determine the remainder. Let’s start with the easy ones and work our way up.

Assume a coefficient of rolling resistance: Cr (dimensionless)

The best racing tires running on a reasonably smooth road will have a Cr of around 0.006. We will be able to check this later in a result that does not depend on the assumption. As an additional layer of confidence in our calculation, we shall not calculate directly with this assumed number but, once it is proven appropriate by road tests, or adjusted on hand of them, it will be available to the rider for additional calculations.

Weigh the bike and rider: W (lbs)

Use scales to determine the mass of rider and bike accurately. We shall plug this number into various formulae to determine those factors on which we cannot get such a direct handle.

Calculate the frontal area: A (sq ft)

The rider’s frontal area A can be approximated as

A = 0.85 x W x H

where

A is the area in square feet,

H is the height and

W is the width of his frontal profile on the bike

A photograph with a height reference might be used for calculation, or a helper with a tape measure.

It helps accuracy to split the frontal profile into

a) the wheel(s) and mechanics below the pedal which can be precisely calculated as rectangular parts,

b) the rider’s body from pedal-top to the top of his shoulders which can be factored at 0.85 of the rectangle as above, and

c) the rider’s head which can be factored at 0.70 of the rectangle as above.

Derive a formula for the rider’s power: HP (horsepower)

What we want is a method that can be used to measure separately the power to overcome aerodynamic drag and the power to overcome friction at the tire’s junction with the road. There doesn’t appear to be a way to do it without either making an assumption about Cr (and we’re trying to reduce the number of assumptions used) or special instruments. But there is a way to approach the separation between the two resistances very closely, and that is to consider surplus tractive effort at each stage of acceleration, and the manner in which on coastdown each resistance, in the speed band where it comprises the overwhelming part of total resistance, decelerates the mass of the rider and the bike.

Muscles in the cyclist’s legs provide force to accelerate his mass:

Force = Mass x Acceleration

Acceleration is most easily measured in gravities of 32.2ft/sec^2, read as “32.2 feet per second per second”. So first we must put the desired or achieved velocity V into comparable ft/sec:

V, ft/sec = (V,mph x 5280)/3600

where 5280 are the number of feet in one mile and 3600 the number of seconds in one hour.

We know that one horsepower equals 550 ft/lb/sec, so we can now say that

HP = (Force x Velocity)/550

and from this it follows that

HP = ((W x (acceleration g + deceleration g) x g)/g) x ((V,mph x 5280)/(3600x550))

Now we possess a formulation which takes both aerodynamic and frictional resistances into account. We shall be able to use it after testing with a real rider on a real road.

Measure the rider’s acceleration: g

To a good level road on a dry and windless day, bring a tape measure, a stopwatch (there’s a good enough clock on your bike computer or your heart rate monitor) and a movie camera or a tape recorder or even a notebook, and start cycling. What we’re interested in is acceleration in gravities. This is a better test of power than some notional extended one-shot effort. So choose a stretch of road long enough to get up to maximum speed, put up some equidistant markers, and take a reading (speed and time for the segment) at each marker while accelerating, so that we get a reading at a variety of speeds. For each (rolling) pair of speeds:

acceleration, g = ((higher mph – lower mph) x 5280)/(t x3600 x 32.2)

where

t is the time taken over the segment in seconds.

Add and average all the results for average acceleration.

Measure the rider’s deceleration: g

Coast down from speed and measure the time intervals at the markers to calculate

deceleration, g = ((lower mph – higher mph) x 5280)/(t x3600 x 32.2)

Add and average all the results for average acceleration.

Calculate the cyclist’s power output: HP

Now we can enter the numbers into the formula we derived earlier.

HP = ((W x (acceleration g + deceleration g) x g)/g) x ((V,mph x 5280)/(3600x550))

Note that the coastdown g number, the deceleration, is taken as an absolute number.

The resulting horsepower includes separate elements for aerodynamic drag and frictional resistance that are only mildly cross-contaminated. Further analysis of the data can refine the division, so we will take from here only the gross horsepower the rider produced.

The same dataset gives tractive resistance: TR (lbf)

Tractive resistance, lbf = (deceleration g x W)/g

Note that tractive resistance if measured over the entire speed range of the cyclist will include both aerodynamic and frictional resistances. But, clearly. as you slow down, the aerodynamic influence declines as the cube of the fall in speed. I would say that at speeds below 6mph or 10kph it is safe to assume that the aero influence is negligible. TR is thus taken at low speeds.

At slow speeds tractive resistance will then work as a check on the assumption we made about the coefficient of rolling resistance Cr, which can also be read as “6 pounds per thousand of bike and rider mass”.

At high speeds, say 20mph and up, aerodynamic resistance will be preponderant, so you could use high speed tractive resistance as a check on the Cd number we calculate below. But it might be better to save the energy, as by now the Cd number has so many cross-referenced reality checks built in, one more would be superfluous.

We have ignored the inertia of the rotating wheels but they won’t make much difference.

Convert to Watt

Cyclists talk about their output in Watt. To convert from HP to Watt, divide HP by 0.001341.

Multiply tractive resistance in lbf by 0.001818 to get horsepower or by 1.35581796 to get Watt.

At last, we can calculate the aerodynamic coefficient: Cd (dimensionless)

If the rider has taken enough acceleration and deceleration tests, the averaged, individual HP we have now derived for him will be a more reliable number than can be measured without instruments on a single or a few top speed efforts. We can work confidently with such a number.

First, from HP, the total power, remove HPtr, the power consumed in overcoming road friction (measured via low speed deceleration), to leave only HPa, the power spent on overcoming aerodynamic resistance.

HPa = HP - HPtr

The horsepower to overcome aerodynamic drag is proportional to the aerodynamic coefficient, the frontal area to be accelerated and the cube of the speed desired (or achieved!):

Drag horsepower HPa = (CdAV^3)/146,600

Therefore

Cd = (HPa x146,600)/(AV^3)

where

Cd is the dimensionless coefficient of aerodynamic efficiency

HPa is horsepower expended to overcome aerodynamic drag and

V is the highest speed achieved in these tests, mph.

One Watt is 0.001341HP. If you want to work in Watts the formula becomes:

Cd = (W x 197)/(AV^3)

where

W is Watts

and I’ll leave it to you first to remove the tractive resistance in Watts…

If the Cd of a cyclist on the drops of a road bike is very far from 0.50, measure his frontal area A again!

Cd is in any event a near-irrelevant number: the Cd of the square Volkswagen Microbus of the flower power generation is 0.42, and that of the iconic zero-comfort minimalist sportscar the Caterham (neé Lotus Seven) is 0.62. What really matters is the overall drag of the package, CdA.

So what can we conclude?

The cyclist doesn’t need time in a twenty million dollar wind tunnel to determine his coefficient of aerodynamic efficiency, he doesn’t need a whole university medical sports department assisting him to discover how many watts he puts out, and he doesn’t need to employ umpteen engineers to discover that guestimates about the coefficient of rolling resistance are pretty close, or how his pedal-power is divided between overcoming air resistance and road resistance.

***

The formulae above are adapted from

Andre Jute: Designing and Building Special Cars, 1985, B T Batsford, London; Robert Bentley, Boston; several other editions. See more books by Andre Jute.