Walter Zorn's Kreuzotter Bicycle Speed and Power Calculator

Walter Zorn unfortunately passed away in 2009. During his life he was responsible for much innovation in cycling, and had a very methodical approach. Amongst the many things he created was an excellent speed and power calculator which many people have credited as being quite accurate. Unfortunately, the kreuzotter website is no longer on the web and I could only find the calculator on the wayback machine.

This version of the Kreuzotter calculator is as Walter Zorn's original, but modified in two ways. Firstly, the weight of the Quest velomobile has been altered to 38 kg - this being the advertised weight of Quests in 2010 (and born out by those I've weighed). Secondly, the Mango velomobile has been added, in three forms. My personal Mango Classic (with winter tyres), the current touring oriented Mango+ and the lightweight Mango Sport. The figures for aerodynamics of the Mango vs. Quest were taken from this web page and the weights for the Mangos are that of my own Mango and current production weights for the Mango+ and Mango Sport. Also, I modified the description of one of the two wheel recumbents which I previously noticed had much the same performance as my two wheel recumbent.

For me personally, this means I now have a calculator which seems to make sense for my own bikes. You may find that some of the example bikes are similar to your own.

There's more text at the bottom.

Racing Bicycles Recumbents
Upright ("Dutch") bike
hands on the tops
(top of handlebar)
under seat steering,
commuting equipped
hands on the drops
(bottom of handlebar)
under seat steering,
commuting equipped (close to PDQ)
with racing bars
Triathlon Bicycle ShortWheelBase
above seat steering,
racing equipped

Superman Position
(Racing Bicycle-
1h Record)
above seat steering
Kreuzotter race

Lowracer with
streamlining tailbox
Kreuzotter race

Streamlined Lowracer
White Hawk
(1h World Record)

Quest (closed)
David's Mango (winter tyres, slow)
Mango Sport
Alligt Alleweder A4
Flevobike Versatile/Orca
The comma as well as the point
may be used as decimal point.
Rider's Height in / cm
Rider's Weight lb / kg
Bicycle Weight lb / kg
Air Temperature °F / °C
Height above SeaLevel ft / m
Slope of Road %
uphill positive, downhill negative values
Wind Speed mph / km/h
headwind positive, tailwind negative values
Pedaling Cadence /min
"Change" the sort of tires?
(Recumbents: calculation is based
on 20inch front wheels.)
Front Wheel Tire Rear Wheel Tire
The input field of the variable to be calculated must be empty. The result will appear in that field.
With both fields filled, the variable evaluated previously will be calculated again (facilitates quick comparisons).
Power Watts  
Speed mph / km/h
Further results: Effective Drag Area Cw*A square feet Rolling Resistance Coeff. Cr 
In case you enter (before clicking the "Calculate"-Button):
either the TripDistance miles / km ft / m
or the TripDuration h min sek,
the amount of Calories Burnt by the Rider =kcal   (assumed efficiency: 25 percent) will be calculated.
Besides, the program will evaluate the variable whose fields are empty (Trip Distance or Trip Duration, respectively).

For coast-down simulations, set the Power value to zero and the Slope to the desired negative value.


About the Speed&Power Calculator:

The rider's frontal area is evaluated approximately from the rider's body height and weight, and a parameter which depends on the selected kind of bicycle. These assumptions yield good matches with frontal area measurements, and with measurements done with SRM Power Measuring cranks. Inside the fully streamlined bicycles, of course, the rider's frontal area is assumed to have no influence.
The rolling frictions of the front and rear wheel tires each are taken into account separately.
The calculation also regards the following influences:
Load distribution front/rear wheel. At wider tire tends to generate less rolling friction but more air resistance (not true with the streamliners White Hawk and Quest whose fairings enclose the wheels almost entirely). A thicker tire wall (touring tire) tends to generate higher rolling friction. Tire thread induces air vortices and thus speed-dependent additional resistance. The front wheel has more share in the air drag than the rear wheel. Smaller front wheels of recumbents generate more rolling friction but less air resistance. The air drag share of the bicycles themselves is taken into account too.
At low speeds a wider tire (less rolling friction) may be advantageous while, with higher velocity, a narrower one (less air resistance) increasingly gets the upper hand.
The applied rolling resistances refer to asphalt road pavement. On a smooth velodrome surface, rolling resistances may be essentially lower. Consider this also for the values that this calculator delivers for the Superman Position.
Most of the data and assumptions used for the calculator are based on (and match well with) frontal area measurements, and, first of all, measurements done with SRM Power Measuring cranks.


The most essential of the equations the Speed&Power Calculator is based on:

The following equations take into account all of the relevant resistance components: Rolling friction including the dynamic (speed-dependent) rolling friction, air drag including the influence of wind speed, mechanical losses, and uphill/downhill forces.
P Rider's power
V Velocity of the bicycle
W Wind speed
T Air temperature, in ° Kelvin (influences air density)
HNN Height above sea level (influences air density)
rho Air density
rho0 Air density on sea level at 0° Celsius (32°F)
p0 Air pressure on sea level at 0° Celsius (32°F)
mbike Mass of the bicycle (influences rolling friction, slope-dependent pull-down force, and normal force)
mrider Mass of the rider (influences rolling friction, pull-down force, and the rider's frontal area via body volume)
A Total frontal area (bicycle + rider)
Cd Air drag coefficient
g Gravitational acceleration
Cr Rolling resistance coefficient
Crdyn Coefficient for velocity-dependent dynamic rolling resistance, here approximated by 0.06 * number-of-wheels
Cm Coefficient for power transmission losses and losses due to tire slippage (the latter can be heard while pedaling powerfully at low speeds)
grade Inclination (grade) of road, in percent
Frg Rolling friction (normalized on inclined plane) plus pull-down force on inclined plane

Air density via barometric formula:
Air density via barometric formula
Rolling friction plus pull-down force:
Equation for rolling friction force and pull-down force

Equation for the required human power

In order to solve this Power equation for Velocity V, we write it in the implicit form
Equation for the required human power
so we can use the cardanic formulae to obtain the solutions:
If a2 + b3 ≥ 0:
Velocity equation

If a2 + b3 < 0 (casus irreducibilis; in case of sufficient downhill slope or tailwind speed):
Velocity equation (casus irreducibilis)

Expression "a" of the velocity equation
Expression "b" of the velocity equation


Relation between Body Weight, Body Height and BodyMassIndex (BMI)

The comma as well as the point may be used as decimal point.
The field of the variable to be calculated must be empty. The result will appear there.
Body Weight = lb / kg  Body Height = in / cm BMI =
Note:   BMI = weight[kg] / (height[m] * height[m])
BMI<19 signifies underweight,  BMI between 25 and 30 slight overweight...

With the values from this BMI-Calculator you can check how much a change of body weight or height (or both) would influence the bicycle speed or the required propulsion power. For example, you might want to compare riders with different heights at identical BMI values, meaning "adequately" changed body weights. It may become noticeable that also on a recumbent bicycle the rider's tallness has influence on the air drag. A change of the rider's weight (meaning changed body surface area and thus frontal area) has more influence on aerodynamics than a different tallness alone.


Last modified: 15.4.2010

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More text...

I find it interesting to note the interactions of the effects of weight and aerodynamics on the speed of cycling. The 38 kg Quest initially appears to beat the 27.5 kg Mango Sport due to its better aerodynamics. However, select just a 0.5% hill to climb and the Mango Sport gains an advantage over the Quest due to the difference in weight becoming a greater influence than the difference in aerodynamics.

Outside of the rather artificial environment of a velodrome, there are gradients everywhere. A 0.5% gradient is really not very much at all. It's almost too gradual to see. Even "flat" areas of the world like the Netherlands have plenty of gradients at least as steep as this. Between work and home I have a number of (very) small hills which I go up and down, and my home is about 15 m higher in altitude than my work.

Add in the effect of the lower weight every time you have to build up speed again after traffic lights or when filtering through other traffic, and a Mango is relatively spritely in real life conditions.

If you're comparing with a different bike, be realistic about weight. Some bikes are advertised with very attractive low weights, but by the time you've added mudguards, racks, lights, a toolkit, spare tubes, a pump, a little luggage etc. the weight is probably higher than you think it is. Remember that the velomobiles come with mudguards, racks, lights, and even the equivalent of a raincoat.

The figures I've added to the calculator can of course only approximate real life, as can also be said for the original figures in the calculator. Don't expect to be able to cycle in any real world situation at exactly the speed the calculator gives.

The code which generates this web page can be easily viewed. Simply save the web page, and also fetch the javascript and css files. If you believe I've made any mistakes in adding the Mango figures, let me know.

Saturday 17th update. I found figures for the Alleweder (from the same source as above), and also comparisons with the aerodynamics of the Versatile/Orca and Leitra (from Adventures of Greg) so I've added these to the calculator too. I'm a little less sure of their accuracy than of the Mango figures, but they seem about right. i.e. Alleweder which I've ridden a little is quicker than my PDQ but slower than a Mango. What I find most interesting with these extra figures is the comparison between Versatile and Leitra. The Leitra generally is shown to be a little quicker than the Versatile, and it's almost entirely a function of its lower weight. Add an incline and it does quite well, amongst the velomobiles it comes second only to the Mango Sport with a 5% hill.

Sunday 17th July 2011 update. Yesterday, a blogger accused me (not via email directly to me, but on his own blog without even telling me) of changing the results of this calculator to favour the Mango over his Quest. He based this comparison on the re-appearance of the calculator on the Kreuzotter website, saying that this "original" calculator gives different results to my modified version. He is right that the results are different, but there is a much more innocent explanation than subterfuge. The calculator on the Kreuzotter website has a different version of code and gets different results to that which I retrieved from the wayback machine and modified. Other people also have copies of same version as I do, including (no longer available). Their calculators give the same results for the same inputs as mine does (be careful, not all default values are the same). I am not sure as yet whether the version of the calculator on the Kreuzotter website is an updated version of what I have or an older version which has been restored. However, unfortunately it does not have the extra velomobiles defined. If you want to make a comparison between different bikes then probably either calculator will work with a reasonable degree of accuracy. Any inaccuracy in one calculator vs. another will be approximately equal for one bike vs. another in the same calculator. Making comparisons between one calculator's results for one bike and another calculator's results for another bike will not give an accurate impression. Therefore, to compare a Quest with a Mango or other velomobile, please use my version of the calculator as it is the only one which has data for these other machines. Finally, note that this is just an online calculator. It can only ever give an estimate of actual performance.