### B 5. Air resistance

Figure 1 From pedaling power to cycling speed pedaling power (Ppe), intrinsic resistanc (Rb), rolling resistance (Rr),
air resistance (Rd), slope resistance (Rsl) and speed (v)

The pedaling power required to overcome the air resistance (Rd) is calculated as follows: Pd = 0.5 × ρ × A × Cd × v³ (1).
The letter d stands for drag (air resistance). The Greek letter ρ (roo) stands for the relative density of air; this is ±1.23 kg/m³ in the Netherlands and 1 kg/m³ at a height of 1800 m (1). A (area) is the front surface in m². The air resistance coefficient (Cd) is a dimensionless number. The speed (v) is expressed in meters per second (m/s).

Front surface (A)
A non-competitive cyclist with an average height and build, who is riding a regular racing bicycle with his hands on the handlebars, has a front surface (A) of ± 0.5 m² (figure 2). This is approx. 0.4 m² if the hands are placed in the curve of the handlebars. On a recumbent racing bicycle, the rider leans far back, with his legs, feet, underarms and hands positioned in front of the body (figure 2). This reduces his front surface to an estimated 0.2 m².

Figure 2 Front surface hands on handlebars A = ± 0,5 m² hands in curves A = ± 0,4 m² recumbent bicycle A = ± 0,2 m²

Air resistance coefficient (Cd)
The air resistance coefficient is a measure for the streamlining. It is measured in a wind tunnel (2). Because that is expensive, estimated values are used for Cd when calculating air resistance. Our cyclist has an estimated Cd of 0.9 for all three bicycle positions (figure 2).

Regular racing bicycle
A 75-kg male is riding on a 10-kg regular racing bicycle with his hands in the curves of the handlebars on a level road. The weather conditions are calm and his speed is 10 km/hour (2.778 m/s). The pedaling power required to overcome the rolling resistance is (previous chapter):
Pr = (75 + 10) × 9.81 × 0.006 × 2.778 = 13.9 watt
Overcoming the air resistance requires: Pd = 0.5 × ρ × A × Cd × v³
Pd = 0.5 × 1.23 × 0.4 × 0.9 × 2.778³ = 4.8 watt
Pr + Pd = 18.7 watt
The intrinsic resistance of the racing bicycle costs 5% of the total pedaling power. Which means that the total pedaling power is:
Ppe = 18.7/0.95 = 19.7 watt
Pb = 0.05 × 19.7 = 1.0 watt (table 1).

Table 1 Pedaling power at 10 km/hour on a level road

 racing bicycle vkm/h Pbwatt Prwatt Pdwatt Ppewatt regular 10.0 1.0 13.9 4.8 19.7 recumbent 10.0 1.2 13.9 2.4 17.5 difference (%) -50.0 -11.2

Ppe = Pb + Pr + Pd

Recumbent high racer
With the same cyclist as the rider the recumbent high racer and the regular racing bicycle have the same rolling resistance. So, when riding 10 km/hour (2.778 m/s) it requires again a 13.9 watt pedaling power to overcome that resistance:
Pr = (75 + 10) × 9.81 × 0.006 × 2.778 = 13.9 watt
Overcoming air resistance requires a pedaling power of:
Pd = 0.5 × 1.23 × 0.2 × 0.9 × 2.778³ = 2.4 watt
Pr + Pd = 16.3 watt
In the high racer, 7% of the pedaling power is lost to the intrinsic resistance. The pedaling power is therefore:
Ppe = 16.3/0.93 = 17.5 watt
Pb = 0.07 × 17.5 = 1.2 watt
To ride an average speed of 10 km/h on the recumbent high racer needs 11.2% less pedaling power compared to the regular racing bicycle (17.5/19.7). Only due to the 50% smaller front surface on the recumbent bicycle (2.4/4.8) (table 1).

Speed and air resistance
Cycling a constant speed of 30 km/hour (8.333 m/s) on the regular racing bicycle requires a pedaling power:
Pd = 0.5 × 1.23 × 0.4 × 0.9 × 8.333³ = 128.1 watt
Riding the same speed on the recumbent high racer requires:
Pd = 0.5 × 1.23 × 0.2 × 0.9 × 8.333³ = 64.0 watt (table 2)
So it can be said of both bicycles that a speed that is 3 times higher requires approx. 27 times (3³) more pedaling power to overcome the air resistance (4.8³ and 2.4³ respectively). This is shown as v³ in the formula for calculating Pd. On balance, the recumbent high racer requires 36.4% less pedaling power on a level road compared to the regular racing bicycle (113.7/178.6) when cycling 30 km/hour (table 2).

Table 2 Pedaling power at 30 km/hour on a level road

 racing bicycle vkm/h Pbwatt Prwatt Pdwatt Ppewatt regular 30 8.9 41.7 128.1 178.7 recumbent 30 8.0 41.7 64.0 113.7 difference (%) -50.0 -36.4

Ppe = Pb + Pr + Pd

Conclusions
1. On a regular racing bicycle, with your hands in the curves of the handlebars, your front surface is approx. twice as large as the front surface on a recumbent high racer; consequently, the air resistance is also twice as high.
2. Overcoming the air resistance at a speed of 30 km/hour requires approx. 27 times (3³) more pedaling power than is the case at 10 km/hour; this applies to both the regular as well as the recumbent racing bicycle.
3. A recumbent high racer requires a third less pedaling power compared to a regular racing bicycle to realize a constant speed of 30 km/hour on a level road.

Sources
1. Wiel van den Broek: Technische artikelen over de fiets: Vermogen en krachten. juni 2013
2. Wikipedia: Weerstandscoëfficiënt