Buyer Beware!
How to Perform Back-of-the-Envelope Power Saving Calculations
The holiday shopping season is rapidly approaching, and you might be considering a new bike or other cycling equipment as a gift to your self or others. How do you evaluate a manufacturer’s claims to make the most informed purchase decision possible?
Cyclingnews recently published an article about the launch of the new Orbea Orca bicycles that serves as an excellent example case. The article repeats a bold claim from Orbea about the power savings from the focus on weight in lieu of aerodynamics in the new Orbea Orca bike design:
Orbea’s tests suggest that the 500g reduction in frame weight for a lightweight bike over an aero bike will save around three watts at a gradient of 5%, or six watts at 10%.
Is that true? Is that possible?
Luckily, performing quick, back-of-the-envelope power savings calculations to double check this claim is a pretty easy task. First, grab an envelope. Next, write down the equation for potential energy, which is mass multiplied by the gravitational acceleration constant and height, and note that power corresponding to a change in potential energy is found simply by recognizing that the change in height divided by time - or the vertical velocity - leads to the power corresponding to a change in potential energy.
Power is potential energy change per time
In trigonometry, there is a simplification known as the small angle approximation that helps simplify the calculations of angles from road gradients. For small angles tan(φ) ≈ φ and sin(φ) ≈ φ. If you are riding at 10 mph up a 10% grade climb, and would like to determine the vertical component of velocity, you normally have to convert the 10% grade to an angle. But, since grade = tan(φ), and the vertical component of velocity is v × sin(φ), the vertical component of velocity can be simplified to v × grade. So, your vertical velocity riding at 10 mph up a 10% grade is approximately 1 mph. This small angle approximation works reasonably well for road grades up to 25%.
Grade with small angle approximation
As a brief aside for those adventurous enough to plug these values into your calculator to confirm this result, make sure you are using radians instead of degrees.
Power is the product of mass, gravity, grade, and velocity
You now have enough information to finish off the back-of-the-envelope calculation. Orbea states that mass reduction is 500 g, or 0.5 kg. The gravitational acceleration constant is 9.8 m/s2 as standard on Earth. G is the 10% grade. Velocity, v, is 10 mph, or 4.47 m/s.
Complete back-of-the-envelope estimate of power savings from 500 g weight savings
Plugging the values into the formula results in…about 2.2 watts savings, which is quite a bit less than the claimed 6 watts. However, it is some savings, albeit small, and if you feel your greatest limiter is your uphill climbing speed, the Orbea Orca might still be worth considering. Just don’t expect to see quite the benefits that they claim.
— James