Figure 1. From pedaling power to cycling speed
pedaling power (Ppe), intrinsic resistance (Rb), rolling resistance (Rr),
air resistance (Rd), slope resistance (Rsl) and speed (v)
When climbing steep slopes, the pedaling power is largely needed to overcome the slope resistance. That portion of the pedaling power is calculated as follows: Psl = m × g × % × v (1).
The % represents the slope percentage. Overcoming the rolling resistance and air resistancde requires little pedaling power on a steep slope because the speed is limited.
A 75-kg non-competitive cyclist intends to climb a 3% slope on a 10-kg regular racing bicycle. His 1-hour pedaling power is 225 watt, realizing a climbing speed of 20.63 km/hour (5.730 m/s).
The pedaling power required to overcome the slope resistance is: Psl = m × g × % × v
Psl = (75 + 10) × 9.81 × 0.03 × 5.730 = 143.3 watt. The intrinsic resistance costs 5% of the pedaling power:
Pb = 0.05 × 225 = 11.3 watt. The pedaling power to overcome the rolling resistance is: Pr = m × g × Cr × v
Pr = (75 + 10) × 9.81 × 0.006 × 5.730 = 28.7 watt. The air resistance requires: Pd = 0.5 × ρ × A × Cd × v³
Pd = 0.5 × 1.23 × 0.4 × 0.9 × 5.730³ = 41.7 watt (table 1).
Added up: Ppe = Psl + Pb + Pr + Pd = 225 watt.
Table 1 Pedaling power and climbing speed on a 3% slope
|225 ||11.3 ||28.7 ||41.7 ||143.3 ||20.63|
|180 ||12.6 ||25.5 ||14.6 ||127.3 ||18.32|
Ppe = Pb + Pr + Pd
We assume that our cyclist on the recumbent high racer can achieve a 1-hour pedaling power of 180 watt. That is 20% less than what he achieved on the regular racing bicycle. The reason for that difference seems to be that the pedaling power on a recumbent bicycle is not increased by pulling on the handlebars. And it is not possible to stand on the pedals either. Consequently, cycling on a recumbent high racer can be compared to riding a regular racing bicycle ‘with no hands’. Now that the power meters are on the verge of sweeping the world of cyclists, we can expect clarity on the real magnitude of this difference in the pedaling power of the same individual when cycling on a regular racing bicycle and recumbent.
On a 3% slope, with a pedaling power of 180 watt, our cyclist can achieve a speed of 18.32 km/hour (5.089 m/s) on a recumbent high racer. The pedaling power needed to overcome the slope resistance is: Psl = m × g × % × v
Psl = (75 + 10) × 9.81 × 0.03 × 5.089 = 127.3 watt. The intrinsic resistance costs 7% of the pedaling power:
Pb = 0.07 × 180 = 12.6 watt. Overcoming the rolling resistance takes: Pr = m × g × Cr × v
Pr = (75 + 10) × 9.81 × 0.006 × 5.089 = 25.5 watt. The air resistance requires: Pd = 0.5 × ρ × A × Cd × v³
Pd = 0.5 × 1.23 × 0.2 × 0.9 × 5.089³ = 14.6 watt (table 1).
Added up: Ppe = Psl + Pb + Pr + Pd = 180 watt.
Slope and climbing speed
On the regular racing bicycle with 225 watt pedaling power, the speed of our non-competitive cyclist on a level road (0% slope) is 32.84 km/hour (table 2). On the recumbent high racer with a pedaling power of 180 watt, his speed is 36.62 km/hour, which is 11.5% faster (36.62/32.84) despite 20% less pedaling power. But on a slight 1% slope, the difference in speed is already reduced to 2.9% (table 2). And when climbing a 3% slope, he is 11.2% slower on the recumbent bicycle (18.32/20.63). That difference increases to 20.4% on a 8% slope. The explanation for that increasing difference in speed is as follows: the steeper the slope, the less climbing speed. That diminishes the advantage of lower air resistance on the recumbent high racer with increasing slope percentage. However, the disadvantage of 20% less pedaling power for the recumbent cyclist remains constant and starts to prevail with increasing slope. As a result, the difference in speed between the recumbent and the regular racing bicycle (both equal in weight) changes from +11.5% on a level road to -20.4% when climbing an 8% slope.
With a 10.46 km/hour cycling speed on the regular racing bicycle, a 10 km long 8% climb takes 0.956 hours (10/10.46) or 57 min. and 22 sec. The same 10 km climb costs, at a speed of 8.33 km/hour on the recumbent high racer, 1.200 hours (10/8.33) or 1 hour, 12 min. Which is almost 15 minutes more.
Table 2 Slope and climbing speed (km/hour)
|225 ||32.84 ||28.39 ||20.63 ||15.20 ||10.46|
|180 ||36.62 ||29.21 ||18.32 ||12.54 ||8.33|
Figure 2. When climbing slopes over 1% the recumbent high racer
is slower than the regular racing bicycle
During a time trial on the Alpe d’Huez (mean slope 7.9%), my climb on the regular racing bicycle took 1.333 hours (1 hour 20 min). On a recumbent low racer (figure 3), the same time trial cost 1.583 hours (1 hour 35 min). That is 15 minutes or 18.8% more (1.583/1.333). These are the only paired observations of this kind so far. Prior to the time trial, it was necessary to equip the recumbent bicycle with an extra-large sprocket on the rear axle, because it was otherwise impossible for me to make it up the mountain. This renders a recumbent bicycle (with the handlebar construction as it is today) less suitable for riding in high mountains. Therefore, it is not surprising that I never came across any (other) recumbent cyclist in the mountains during my cycling holidays, covering a total distance of about 11,000 km.
Figure 3 Paired observations on the Alpe d’Huez
on the recumbent racing bicycle: 1 hour 35 min
on the regular racing bicycle: 1 hour 20 min
1. The pedaling power is estimated to be 20% less on a recumbent high racer compared to a regular racing bicycle, because one can neither pull up on the handlebars; moreover, one can not stand on the pedals when cycling recumbent.
2. The disadvantage of about 20% less pedaling power for the recumbent cyclist is valid both on level roads as well as during climbs.
3. The steeper the slope, the less climbing speed; thus the advantage of lower air resistance on the recumbent high racer diminishes when climbing slopes of increasing steepness.
4. The disadvantage of less pedaling power when cycling recumbent prevails over the advantage of lower air resistance when climbing steeper slopes.
5. A recumbent bicycle (with the handlebar construction as it is today) is less suitable for climbing steep slopes.
1. Wiel van den Broek: Technische artikelen over de fiets; Vermogen en Krachten, juni 2013. http://www.velofilie.nl/vermogen.htm
© Leo Rogier Verberne