This average speed reduction is due to a number of factors including an increase in bicycle mass, a more upright riding position and increased rolling resistance from running lower pressured tyres.

*more*average speed I would lose when the Nomad was loaded up with panniers and 24 kg of luggage? I thought I'd do some maths...

#### The Equations

There are two major components of resistance to overcome when riding a bicycle. One is air resistance and the other is road resistance (a combination of friction and gravity). The power (in Watts) required to overcome these resistances can be represented approximately as the sum of equations (1) & (2) below: P

_{air }= ½ρ(v_{c}+v_{wind})^{2}C_{w}A_{f}v_{c}(1)where ρ = Air density = 1.293 kg/m

^{3}, v

_{c}= speed of cyclist (m/s), v

_{wind}= wind speed (m/s), C

_{w}= Drag Coefficient, A

_{f}= frontal area of bike and cyclist (m

^{2})

P

_{road }= gmv_{c}(C_{r}+s) (2)where g = gravity = 9.81 m/s

^{2}, m = mass of cyclist and bike (kg), v

_{c}= speed of cyclist (m/s), Cr = Combined Frictional Coefficient (mostly tyres, but also chain, hubs etc.), s = grade (slope) (E.g. 8% = 0.08)

#### The Assumptions

I've made some assumptions for the constants C_{w}A

_{f}and C

_{r}based on information available here. I weigh about 72 kg in my cycling kit. Mass below is for the bike and rider combined.

Bicycle | Mass (kg) | C_{w} (-) | A_{f} (m^{2}) | C_{r} (-) |

Seven Axiom Ti | 81.5 | 0.9 | 0.45 | 0.003 |

Raven Nomad (Unloaded) | 88 | 1.0 | 0.60 | 0.007 |

Raven Nomad (Loaded) | 112 | 1.0 | 0.70 | 0.007 |

I've also made the assumption that I have a sustainable upper power output of around 230 Watts. I don't own a power meter, but I've reached this figure based on some test numbers I've calculated whereby I matched power output to speeds and conditions I was familiar with for both the Seven and the Nomad (unloaded).

#### Cycling along the flat

In the case of cycling along the flat with no headwind, grade (s) and V_{wind}are both 0. From the two power figures it can be seen that much more power is required to combat air resistance than road resistance. Speeds for a power output of around 230 Watts range from 33 km/h (9.3 m/s) for the Seven, down to 26 km/h (7.3 m/s) for the loaded Nomad.

Bicycle | P_{air} (W) | P_{road} (W) | v_{wind} (m/s) | grade (m/m) | v_{c} (m/s) | kmh_{c} (km/h) | Diff_{seven} (%) |

Seven | 207 | 22 | 0 | 0 | 9.3 | 33.3 | |

Nomad (U) | 183 | 47 | 0 | 0 | 7.8 | 28.0 | 0.84 |

Nomad (L) | 174 | 56 | 0 | 0 | 7.3 | 26.2 | 0.79 |

#### Cycling up a hill

In the case of cycling up a 8% hill, grade (s) is 0.08. This time, much more power is required to combat road resistance (chiefly gravity) than air resistance. Speeds for a power output of around 230 Watts range from 12 km/h for the Seven, down to 8.5 km/h for the loaded Nomad.Bicycle | P_{air} (W) | P_{road} (W) | v_{wind} (m/s) | grade (m/m) | v_{c} (m/s) | kmh_{c} (km/h) | Diff_{seven} (%) |

Seven | 10 | 221 | 0 | 0.08 | 3.3 | 12.0 | |

Nomad (U) | 10 | 221 | 0 | 0.08 | 2.9 | 10.6 | 0.88 |

Nomad (L) | 6 | 226 | 0 | 0.08 | 2.4 | 8.5 | 0.71 |

#### Cycling into a headwind

In the case of cycling into a 20km/h headwind, V_{wind}is 5.6 m/s. An even greater proportion of power is required to combat air resistance as would be expected. Speeds for a power output of around 230 Watts range from 22 km/h for the Seven, down to 16 km/h for the loaded Nomad.

Bicycle | P_{air} (W) | P_{road} (W) | v_{wind} (m/s) | grade (m/m) | v_{c} (m/s) | kmh_{c} (km/h) | Diff_{seven} (%) |

Seven | 218 | 15 | 5.6 | 0 | 6.1 | 22.0 | |

Nomad (U) | 202 | 29 | 5.6 | 0 | 4.8 | 17.4 | 0.79 |

Nomad (L) | 199 | 34 | 5.6 | 0 | 4.4 | 15.9 | 0.72 |

#### Cycling into a headwind up a hill

Probably the worst situation in cycling! Again cycling into a 20km/h headwind, this time on a 3% slope. Now it's pretty much an even split between power required to combat air resistance and power required to combat road resistance.Bicycle | P_{air} (W) | P_{road} (W) | v_{wind} (m/s) | grade (m/m) | v_{c} (m/s) | kmh_{c} (km/h) | Diff_{seven} (%) |

Seven | 114 | 116 | 5.6 | 0.03 | 4.4 | 15.8 | |

Nomad (U) | 114 | 114 | 5.6 | 0.03 | 3.6 | 12.8 | 0.81 |

Nomad (L) | 104 | 125 | 5.6 | 0.03 | 3.1 | 11.1 | 0.70 |

#### Coasting down a hill

No effort is required here! Assuming the hill is long enough, the bike and rider will reach a terminal velocity when P_{air}and P

_{road}cancel each other out. The slope is -8% and assume no wind. For the first time, the loaded Nomad is not the slowest! It's extra mass triumphs, but it's not enough to beat the Seven's better aerodynamics and frictional properties.

Bicycle | P_{air} (W) | P_{road} (W) | v_{wind} (m/s) | grade (m/m) | v_{c} (m/s) | kmh_{c} (km/h) | Diff_{seven} (%) |

Seven | 943 | -943 | 0 | -0.08 | 15.3 | 55.2 | |

Nomad (U) | 803 | -803 | 0 | -0.08 | 12.8 | 45.9 | 0.83 |

Nomad (L) | 1066 | -1066 | 0 | -0.08 | 13.3 | 47.9 | 0.87 |

#### Average Speed for the Loaded Nomad

For the test cases, a rider capable of generating 230 Watts should be able to propel the Unloaded Nomad between 79%-88% of the speed of the Seven and the Loaded Nomad between 70%-87% the speed of the Seven.I can make this (very crude) deduction for myself. If I can manage 28 km/h average for the Seven, I should be able to average about 83% of that on the Nomad Unloaded and 76% of that on the Nomad Loaded. This equates to about 23 km/h for the Unloaded Nomad (which agrees with my recorded averages) and 21 km/h for the Loaded Nomad (as yet untested).

I've scheduled my trip at a leisurely 18-20 km/h average, so I hope my rather assumptive mathematics is reasonably valid!

References:

[1] Cycling Speed Maths and Ruminations

[2] Bicycle Performance

[3] Coefficients of friction. Rolling Resistance, Air resistance, Aerodynamics

[4] Energy consumption during cycling

## 2 comments:

19-20km/h is about average for me fully loaded. it does take a while to adjust to the slower pace, but after a little bit it becomes the norm!

Jees!, I think you need to get out more :-) I only know that gravity is a bummer sometimes.

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