### C 2. Regular versus Recumbent cycling

We assume the 1-hour pedaling power on a regular racing bicycle of Chris Froome to be 450 watt. But the various stages in the model Tour take approx. 5 hours and thus 80% of his pedaling power seems to be the maximum that can be attained. In the case of Froome: 0.8 × 450 = 360 watt (table 1). The pedaling power of professional cyclists on a recumbent bicycle is unknown. It is assumed in the model-Tour calculations that their 1-hour pedaling power, following thorough training on a recumbent bicycle, still is 20% lower than on a regular racing bicycle. And as a 5-hours pedaling power, it is once again diminished by 20%. Thus, in the case of Froome on the recumbent bicycle, we reckon with 0.8 x 360 = 288 watt (table 1).

Level stages
On a regular racing bicycle, the average speed in the 7 level stages at a pedaling power of 360 watt reaches 41.55 km/hour (11.541 m/s). 5% of pedaling power is lost to the intrinsic resistance of the bicycle:
Pb = 0.05 × 360 = 18.0 watt. The power needed to overcome the rolling resistance is: Pr = m × g × Cr × v
Pr = (70 + 8) × 9.81 × 0.005 × 11.541 = 44.2 watt. The air resistance requires: Pd = 0.5 × ρ × A × Cd × v³
Pd = 0.5 × 1.23 × 0.35 × 0.9 × 11.541³ = 297.8 watt
Ppe = Pb + Pr + Pd = 360 watt
At an average speed of 41.55 km/hour, a stage of 191.3 km in length will take 4.604 hours (191.3/41.55) (table 1).

Table 1 Pedaling power and speed in the level stages

 racing bicycle Ppewatt Pbwatt Prwatt Pdwatt vkm/h r.timehours regular 360 18.0 44.2 297.8 41.55 4.604 recumbent 288 20.2 49.2 218.6 45.16 4.236 difference (%) +8.7

On the recumbent high racer, a pedaling power of 288 watt brings the average speed in these stages to 45.16 km/hour (12.545 m/s). Overcoming the intrinsic resistance requires 7% of pedaling power:
Pb = 0.07 × 288 = 20.2 watt. The following applies to the rolling resistance and air resistance:
Pr = (70 + 10) × 9.81 × 0.005 × 12.545 = 49.2 watt
Pd = 0.5 × 1.23 × 0.2 × 0.9 × 12.545³ = 218.6 watt
Ppe = Pb + Pr + Pd = 288 watt
At an average speed of 45.16 km/hour, a stage of 191.3 km in length takes 4.236 hours (191.3/45.16) (table 1).

Hill stages
Uphill
For pedaling uphill the 1-hour pedaling power is taken into account in the calculations because an uphill stroke takes less than 1 hour of climbing and every uphill stroke is followed by a descent, being a period of relative rest for the cyclist.
On his regular racing bicycle is reckoned for Chris Froome with 450 watt. That brings his climbing speed to 31.34 km/hour (8.705 m/s) on the 4% slopes.
Pb = 0.05 × 450 = 22.5 watt
Pr = (70 + 8) × 9.81 × 0.005 × 8.705 = 33.3 watt
Pd = 0.5 × 1.23 × 0.35 × 0.9 × 8.705³ = 127.8 watt
The pedaling power required to overcome the slope resistance is: Psl = m × g × % × v
Psl = (70 + 8) × 9.81 × 0.04 × 8.705 = 266.4 watt
Ppe = Pb + Pr + Pd + Psl = 450 watt
At an average speed of 31.34 km/hour, the 3 climbs combined in each hill stage take 1.248 hours (39.1/31.34) (table 2).

Table 2 Climbing speed on 4% slopes

 racing bicycle Ppewatt Pbwatt Prwatt Pdwatt Pslwatt vkm/h r.timehours regular 450 22.5 33.3 127.8 266.4 31.34 1.248 recumbent 360 25.2 31.1 55.0 248.7 28.52 1.371 difference (%) -20.0 -9.0

On the recumbent high racer, Froome’s 1-hour pedaling power when riding uphill (less than 1 hour each time) is 360 watt. This brings his climbing speed on the 4% slopes to 28.52 km/hour (7.921 m/s).
Pb = 0.07 × 360 = 25.2 watt
Pr = (70 + 10) × 9.81 × 0.005 × 7.921 = 31.1 watt
Pd = 0.5 × 1.23 × 0.2 × 0.9 × 7.921³ = 55.0 watt
Psl = (70 + 10) × 9.81 × 0.04 × 7.921 = 248.7 watt
Ppe = Pb + Pr + Pd + Psl = 360 watt
At an average speed of 28.52 km/hour, the 3 climbs combined on the high racer take 1.371 hours (39.1/28.52) (table 2).

Downhill
If holding the legs still while descending, the pedaling power plays no role. The chain is not in motion and so the intrinsic resistance is limited to the friction in the axles, which requires 1% of the driving force (chapter Intrinsic resistance). The driving force stems from the slope resistance (Psl), which does not counteract, but rather generates drive. If the cyclist refrains from braking, his speed of descent will increase until the driving force and resistance (the sum of intrinsic, rolling resistance and air resistance) balance one another.
On his regular racing bicycle in the descent, Chris Froome lies flat on the top tube (in ‘ski-position’); as a result, his front surface is ± 0.3 m². His speed of descent is then 45.46 km/hour (12.627 m/s).
Psl = (70 + 8 ) × 9.81 × 0.04 × 12.627 = 386.5 watt
Pb = 0.01× 386.5 = 3.9 watt
Pr = (70 + 8) × 9.81 × 0.005 × 12.627 = 48.3 watt
Pd = 0.5 × 1.23 × 0.3 × 0.9 × 12.627³ = 334.3 watt
Ppe = Pb + Pr + Pd = 386.5 watt
At an average speed of descent of 45.46 km/hour, descending the 3 slopes combined takes 0.860 hours (39.1/45.46) (table 3).

Table 3 Descending speed on 4% slopes

 racing bicycle Pslwatt Pbwatt Prwatt Pdwatt vkm/h r.timehours regular 386.5 3.9 48.3 334.3 45.46 0.860 recumbent 491.7 4.9 61.5 425.3 56.38 0.694 difference (%) +24.0

On the recumbent high racer, his speed of descent on the 4% slopes, without pedaling or braking, is 56.38 km/hour (15.662 m/s).
Psl = (70 + 10) × 9.81 × 0.04 × 15.662 = 491.7 watt
Pb = 0.01 × 491.7 = 4.9 watt
Pr = (70 + 10) × 9.81 × 0.005 × 15.662 = 61.5 watt
Pd = 0.5 × 1.23 × 0.2 × 0.9 × 15.662³ = 425.3 watt
Psl = Pb + Pr + Pd = 491.7 watt
At an average speed of descent of 56.38 km/hour, descending the 3 slopes combined takes 0.694 hours (39.1/56.38) (table 3).

Riding time per hill stage
On the regular racing bicycle the 3 climbs combined take 1.248 hours (table 2); the descents take 0.860 hours (table 3). On the level stages, his speed is 41.55 km/hour (table 1). At this speed, the flat sections between the hills (95 km per stage) are covered in 2.286 hours (95/41.55). This brings the total riding time of a hill stage to:
1.248 (climbing) + 0.860 (descending) + 2.286 (flat) = 4.394 hours (table 7).
And so the average speed on the regular racing bicycle in each hill stage (173.2 km long) is 39.42 km/hour (173.2/4.394).

On the recumbent high racer the 3 climbs combined take 1.371 hours (table 2); descending takes 0.694 hours (table 3). The speed during the flat sections between the hills is 45.16 km/hour (table 1). With that speed the 95 flat km are covered in 2.104 hour (95/45.16). This brings the total riding time of the average hill stage to:
1.371 (climbing) + 0.694 (descending) + 2.104 (flat) = 4.169 hours (table 7).
With that, the average speed on the recumbent bicycle during each hill stages is 41.54 km/hour (173.2/4.169).

On balance, during the hill stages Chris Froome is an average of 5.4% faster on the recumbent high racer compared to his regular racing bicycle (41.54/39.42). Indeed, he is 9% slower during the climbs on the recumbent bicycle (table 2), but he is 24% faster when descending (table 3), with an additional 8.7% on the flat sections (table 1).

Mountain stages
Uphill
On the regular racing bicycle, Chris Froome reaches a speed of 21.74 km/hour (6.040 m/s) when climbing the 8% slopes with his 1-hour pedaling power of 450 watt.
Pb = 0.05 × 450 = 22.5 watt
Pr = (70 + 8) × 9.81 × 0.005 × 6.040 = 23.1 watt
In the mountains, we assume a value of 1 kg/m³ for the relative density of the air (ρ = 1), corresponding to a height of 1800 m (1).
Pd = 0.5 × 1.0 × 0.35 × 0.9 × 6.040³ = 34.7 watt
Psl = (70 + 8) × 9.81 × 0.08 × 6.040 = 369.7 watt
Ppe = Pb + Pr + Pd + Psl = 450 watt
At an average speed of 21.74 km/hour, the 35 km of climbing take 1.610 hours (35/21.74) on the racing bicycle (table 4).

Table 4 Climbing speed on 8% slopes

 racing bicycle Ppewatt Pbwatt Prwatt Pdwatt Pslwatt vkm/h r.timehours regular 450 22.5 23.1 34.7 369.7 21.74 1.610 recumbent 360 25.2 19.1 10.4 305.4 17.51 1.999 difference (%) -20.0 -19.5

On the recumbent high racer, the 1-hour pedaling power during climbs is 360 watt and his climbing speed comes to 17.51 km/hour (4.864 m/s).
Pb = 0.07 × 360 = 25.2 watt
Pr = (70 + 10) × 9.81 × 0.005 × 4.864 = 19.1 watt
Pd = 0.5 × 1.0 × 0.2 × 0.9 × 4.864³ = 10.4 watt
Psl = (70 + 10) × 9.81 × 0.08 × 4.864 = 305.4 watt
Ppe = Pb + Pr + Pd + Psl = 360 watt
At an average speed of 17.51 km/hour, the 35 kilometers of climbing take 1.999 hours (35/17.51) on the high racer (table 4).

Downhill
In the mountain stages of the model Tour the average distance of descending is a mere 25.7 km per stage due to the fact that 4 out of 6 mountain stages finish on a mountain top (foregoing chapter).
On his regular racing bicycle in the 8% descents(in ski-position A = 0.3 m²), Froome’s speed (without braking) reaches 73.83 km/hour (20.508 m/s). The driving force is:
Psl = (70 + 8) × 9.81 × 0.08 × 20.508 = 1255.4 watt
Pb = 0.01 × 1255.4 = 12.5 watt
Pr = (70 + 8) × 9.81 × 0.005 × 20.508 = 78.5 watt
Pd = 0.5 × 1.0 × 0.3 × 0.9 × 20.508³ = 1164.4 watt
Psl = Pb + Pr + Pd = 1255.4 watt
At an average speed of 73.83 km/hour, descending 25.7 km of 8% slopes takes 0.348 hours (25.7/73.83) (table 5).

Table 5 Descending speed on 8% slopes

 racing bicycle Pslwatt Pbwatt Prwatt Pdwatt vkm/h r.timehours regular 1255.4 12.6 78.5 1164.4 73.83 0.348 recumbent 1596.9 16.0 99.8 1481.1 91.57 0.281 difference (%) +24.0

On the recumbent high racer, his speed during the 8% descent reaches 91.57 km/hour (25.436 m/s). The driving force is:
Psl = (70 + 10) × 9.81 × 0.08 × 25.436 = 1597.0 watt
Pb = 0.01 × 1597.0 = 16.0 watt
Pr = (70 + 10) × 9.81 × 0.005 × 25.436 = 99.8 watt
Pd = 0.5 × 1.0 × 0.2 × 0.9 × 25.436³ = 1481.1 watt
Psl = Pb + Pr + Pd = 1596.9 watt
At an average speed of descent of 91.57 km/hour, descending 25.7 km of 8% slopes takes 0.281 hours (25.7/91.57) (table 5).

Riding time per mountain stage
On the regular racing bicycle, climbing the 35 km of 8% slopes takes 1.610 hours (table 4); descending 25.7 km takes 0.348 hours (table 5). The speed during the flat sections between the mountains is 41.55 km/hour (table 1). At this speed, the flat sections between the hills (124 km per stage) are covered in 2.984 hours (124/41.55). This brings the riding time of a mountain stage on the regular racing bicycle to:
1.610 (climbing) + 0.348 (descending) + 2.984 (flat) = 4.942 hours (table 7).
So the average speed in each 184.7 km long mountain stage of the model Tour is 37.37 km/hour (184.7/4.942).

On the recumbent high racer, the 35 km of climbing in each model-mountain stage take 1.999 hours (table 4); descending 25.7 km takes 0.281 hours (table 5). The speed on the recumbent high racer during the flat sections is 45.16 km/hour (table 1). And so the 124 km in between the mountains are covered in 2.746 hours (124/45.16). Thus the riding time of a mountain stage on the recumbent high racer is:
1.999 (climbing) + 0.281 (descending) + 2.746 (flat) = 5.026 hours (table 7).
It brings the average speed on the recumbent bicycle during the mountain stages to 36.75 km/hour (184.7/5.026).

With that, Chris Froome is an average of 1.7% slower during the mountain stages on the recumbent compared to his regular racing bicycle (36.75/37.37). Due to the 4 finishes on the mountain top, he is missing 4 descents in which he would have been 24% faster on the recumbent high racer.

Flat km’s and 4% and 8% slopes respectively
The model-tour includes 2558 flat kilometers: 1339 in the level stages and 475 + 744 in the flat sections between the hills and mountains respectively (table 1 foregoing chapter). The 4% climbs in the hill stages have a combined length of 195.5 km, as do the 4% descents. The 8% climbs in the mountain stages make up a total of 210 km, but the descents come to a mere 154 km, because 4 descents are missing in mountain stages finishing on the top of a mountain. With that, the total length of the model-round is 3313 km (table 6). On the recumbent high racer, Chris Froome is 8.7% faster during the flat kilometers compared to his racing bicycle, and he is 24% faster during both, the 4% and 8% descents. But when climbing the 4% and 8% slopes on the high racer, he is 9% and 19.5% slower, respectively (table 6).

Table 6 Flat kilometres, 4% and 8% climbing and descending

 distancekm regularkm/h recumbentkm/h difference% level stages 2558 41.55 45.16 +8.7 4% uphill 195.5 31.34 28.52 -9.0 4% downhill 195.5 45.46 56.38 +24.0 8% uphill 210 21.74 17.51 -19.5 8% downhill 154 73.83 91.57 +24.0 Total 3313

And the winner of the model-tour is …
In the 7 flat stages of the model-Tour the average speed on the regular racing bicycle is 41.55 km/hour. Each stage taking 4.604 hours (table 1). So all of the flat stages combined take 32.228 hours (table 7). At an average speed of 45.16 km/hour on the recumbent bicycle, each of the stages take 4.236 hours (table 1), and combined 29.652 hours (table 7).

In the 5 hill stages the average speed on the regular racing bicycle is 39.42 km/hour and each stage takes 4.394 hours (see riding time per hill stage); combined, the 5 stages take 21.970 hours (table 7). At an average speed of 41.54 km/hour on the recumbent high racer, the hill stages take 4.169 hours each, and a total of 20.845 hours combined (table 7).

In the 6 mountain stages the average speed on the regular racing bicycle is 37.37 km/hour and each stage therefore takes 4.942 hours (see riding time per mountain stage); all of the mountain stages combined take 29.652 hours (table 7). The average speed on the recumbent bicycle is 36.75 km/hour and each mountain stage takes 5.026 hours; the 6 stages combined therefore take 30.156 hours (table 7).

Table 7 Riding time in various stages

 regularhours recumbenthours level stages 7 × 4.604 = 32.228 7 × 4.236 = 29.652 hill stages 5 × 4.394 = 21.970 5 × 4.169 = 20.845 mountain stages 6 × 4.942 = 29.652 6 × 5.026 = 30.156 total riding time 83.850 80.653 difference 3.197 hours 3.8%

On balance, the difference between the regular and the recumbent racing bicycle in total riding time over the entire model Tour is 3.197 hours (table 7), meaning 3 hours 10 minutes and 37 seconds (3h 10' 37"). So, on the recumbent bicycle Chris Froome is 3,8% faster than on his regular racing bicycle. His average speed over the entire model Tour on the regular racing bicycle is 39.51 km/hour (3313/83.850); on the recumbent high racer 41.08 km/hour (3313/80.653).

Difference in speed
It will be evident in view of the above that one cannot possible state that, in general : a regular racing bicycle is faster than a recumbent bicycle (or the other way round). The difference in the total average speed between the two bicycles in a major tour such as the Tour de France is highly dependent upon the route of the Tour itself: the higher the number of flat kilometres, the greater the advantage of the recumbent bicycle. The higher and the steeper the hills or mountains, the better the result of the regular racing bicycle; all the more if many of the mountain stages end on top of a mountain. The Tour de France of 2013 was certainly not unfavourable for climbers. And the same goes for the regular racing bicycle in the comparison of the two bicycles in the model Tour.

Actual Tour and model Tour
The actual Tour de France 2013 was 3403 km in length. Froome’s final time on his racing bicycle was 83.944 hours (83h 56' 40"), his average speed being 40.54 km/hour (2). That actual Tour had 3 time trials (90 km combined). Froome needed a total of 1.693 hours to ride them. If we subtract that from his final time, then he took 82.251 hours to cover the remaining actual 3313 km, bringing his average speed to 40.28 km/hour (3313/82.251). With that, he was 2% faster (40.28/39.51) during the 3313 km of the actual Tour de France in 2013 compared to the model Tour on the same bicycle.
How can this be explained?
In the actual Tour of 2013, but without the time trials, Chris Froome covered 3313 km as part of the pack. That is a serious advantage over the model Tour where he rode all 3313 km solo. Particularly on a regular racing bicycle, with a larger front surface and higher air resistance, riding in a pack is a major save of pedaling power. Depending on speed and (head)wind, this saving can be as high as 20 to 40% compared to the person riding alone. This advantage of riding in a pack exceeds the effect of all other differences between riding in the real Tour de France and the model-Tour.

How significant is a difference of 3.8% to the final result?
In the actual Tour of 2013, Chris Froome’s winning time was 83.944 hours (83h, 56' 40"). Svein Tuft was the 169th and last with a final time of 88.409 hours (2). A difference of 4.465 hours (4h 27' 55"). If Tuft had been the only one to ride a recumbent high racer in 2013 and if he had been 3.8% faster, then he would have arrived in Paris 3.360 hours earlier (0.038 × 88.409 = 3.360), meaning 1.105 hour (1h 6' 18") behind Froome. With that, he would have come in 32nd place (2). If all of the participants in the actual Tour of 2013, except for Chris Froome, had rode high racers, and if each of them had been 3.8% faster, then Froome, with his winning final time of 83.944 hours in 2013, would have finished as number 112. Thus, 3,8% is a huge difference for professional cyclists. So the UCI was, and still is, right not to allow mixed competitions with regular and recumbent bicycles: The contest would be too unequal.

Conclusions
1. In the level stages of the model Tour, Chris Froome is 8,7% faster when riding recumbent than riding regularly; although the recumbent bicycle is 2 kg heavier and its intrinsic resistance is 2% more; and on top of that, Froome’s pedaling power is estimated 20% smaller when riding recumbent.
2. When climbing 4% and 8% slopes, Froome is 9% and 19,5% slower on the recumbent compared to the regular racing bicycle.
3. When descending 4% and 8% slopes, without pedaling or braking, he is 24% faster on both 4% and 8% slopes when riding recumbent.
4. Across the entire route of the model Tour, he is 3,8% faster on a recumbent high racer and his final time is 3h 10' 37" shorter than riding his regular racing bicycle.
5. If all of the participants in the real Tour de France 2013 had rode a recumbent bicycle, with the exception of Froome, and if all of them had realized a 3,8% shorter final time, then the winning final time on the regular racing bicycle in 2013 would have ranked Chris Froome in 112th place (of all 169 participants).

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