Cycling Aerodynamics: Wider tires and Rims examined in Engineering Detail

Introduction

This post will cover the science and marketing relating to wider tires and rims, there is a general push by the bike wheel manufacturers and to an extent tire suppliers to push wider rims and tires. The argument is they are faster, puncture less and produce more comfort. This post is worth reading in conjunction with a topic on modern real world cycling aerodynamics

Aerodynamic Losses

This is perhaps the most contentious subject and it is based on aerodynamics that many would argue are not applicable for a bicycle wheel and tire combination.

The basic formula that many use for determining drag is an application of Bernoulli’s with dynamic pressure and a non dimensional coefficient for drag C_d. This formula is perfectly acceptable when flow is considered to be steady state (parallel with little deviation in incidence angle or 2D velocity).

\begin{equation} 
D = C_{D} \cdot \frac{1}{2}  \sigma  V^{2} \cdot A
\end{equation} 

If this is broken down into the fundamental terms and compared between two rim and tyre combinations (23mm and 30mm), an indicative conclusion can be garnered. In this general equation, the fixed terms are the density and the rider velocity. The parameters that vary between the wheel sets are the A term (wetted area) and the Cd (Drag coefficient. The Cd and A terms are functions of wheel geometry and this is where the math starts to get more involved.

Using an ISO chart for raddii sizes, this gives the following

Considering the two popular combinations, the previous favorite (23mm) and the new favorite (30mm), there is a stark difference in wetted area. This is a simple formula

\begin{equation} 
A = D \cdot W 
\end{equation} 

which for these two equations gives

\begin{equation} 
A_{23mm } = 668 \cdot 23 = 15364 \cdot mm^{2} 

\end{equation} 

\begin{equation} 
A_{30mm} = 682 \cdot 30 = 20460 \cdot mm^{2} 
\end{equation} 

The difference in purely wetted area between a 30mm tire and rim combo is a staggering 33%. This is due to a combination of not only increased width but also height. Henceforth, a 30mm tire and rim combo would require the C_d term in equation (1) to be ~33% lower to “break even” aerodynamically.

The Cd for bike wheels is almost entirely determined by the width and depth of the tire and rim combination. Narrower wheels with deeper rims are more aerodynamic. In this mathematical case, the width is fixed so the only variable that can be changed is the depth. A graph based on wind tunnel data shows a clear correlation between depth and Cd (*in this case proven power is equated to Cd). The absolute values should not be considered accurate as many variables can influence the absolute reading but the relative readings are valid as they were conducted in the same tunnel and other variables remained relatively constant.

At 30km/h the trend line equation is shown below. Very loosely, every mm of rim depth is worth about 0.25W reduction in power

\begin{equation}
Power_{30km/h} = -0.2533 \cdot RimDepth + 199.16
\end{equation}

At 50km/h the trend line equation is shown below. At this speed, every mm of rim depth is worth 0.84W of power reduction

\begin{equation}
Power _{50km/h} = -0.8397\cdot RimDepth + 641.71

\end{equation}

The graphs above show clearly (using verified empirical data) that the increase of width in a wheel and rim combination can only be overcome with an increase in rim depth. Effectively the increase in wetted area can only be mitigated by improving Cd – to improve Cd requires an increase in rim depth. It is possible to add in some aerodynamic “trick” features but their effectiveness is small compared to rim depth with is by far the dominant parameter.

The base data is Dnitriev derived and can be found here 30km/h and 50km/h

Wider Tires have Lower Rolling Resistance

An often cited comment from various internet forums is that wider tires have a lower rolling resistance. This is true providing one key variable is maintained, they must be pumped to the same pressure.

For absolute minimal rolling resistance, an infinitesimally small contact patch (and thus infinite contact stress) would be required and this is the subject of Hertzian contact mechanics. The limitation with total application of Hertzian mechanics to this problem is the fact that road surfaces are rarely perfectly smooth and the objects involved are not sufficiently stiff enough.

Rolling resistance is mathematically quoted as a coefficient Crr. It is a unless value. Some typical Crr’s are shown below.

INSERT TABLE

In practice the maximum pressure and even reasonable working pressure is limited by hoop stresses in the radial fibres of the tire and strength of the hook on clinchers. Hoop stress is a circumferential stress that is present on pressure vessels (typically cylinders eg. Coke cans and Aircraft fuselages) at the walls of the pressure containment object.

Mathematically, the formula for thin walled pressure vessel hoop stress is

\begin{equation} 
\sigma _{hoop} =  \frac{P\cdot D}{2 \cdot t} 
\end{equation} 

Where P = Internal Pressure, D = Diameter, t = Wall Thickness

There is also an axial or longitudinal component, this is generally ignored for this type of analysis as the stress is half of the hoop component.

\begin{equation} 
\sigma _{axial} =  \frac{P\cdot D}{4 \cdot t} 
\end{equation} 

For a more thorough analysis, a mohr’s circle of stress would be required but for the purposes of this blog post, the axial stress will be ignored because it is half of the stress of the hoop component (denominator 4t as opposed to 2t).

Rearranging, the formula, the ramifications are quite clear. An increase in tire width will result in a linear reduction in maximum tire pressure

\begin{equation} 
P=  \frac{\sigma _{hoop}\cdot2 \cdot t} { D} 
\end{equation} 

The relationship between maximum tire pressure and tire width is inverse so the delta will get smaller as the tire width increases, changes. eg. it only takes 1mm of differential to produce a 0.5bar pressure drop between 22mm and 23mm but it requires 3mm of differential to get the same drop at 37mm of tire width. In short, a narrower tire allows a higher pressure with lower rolling resistance and simultaneously a lighter tire which allows faster acceleration.

If a rider runs their tires at the maximum allowable pressure, a narrower tire will always have lower rolling resistance – this is due to the nature of hoop stress.

Vibration or Suspension Losses

Vibration or suspension losses come in two primary forms. There is a loss caused from energy of the rider being absorbed (moving them up and down), deformation and a loss of traction (or a slippage loss). These losses are affected by:

• Rider Weight
• Rider horizontal position with reference to the front and rear wheels
• Tire width
• Handlebar position (stem)
• Spoke pattern, tension
• Wheel stiffness
• Wheelbase of the bike
• headtube angle

The VERTICAL energy loss with regards a bike in this regard is disputed. Theoretically, the energy lost (technically transferred) will be the same between a narrow tire and a wider tire and this is due to the basic equation of energy transfer being a product of the excitation force and the displacement. In reality, the perceived loss is due to the bounce or poor NVH which makes it less “comfortable” and more difficult to pedal.

Mathematically, there is a clear difference between the damping and spring constant of a wide tire against a narrow tire. Tire constructions are generally unchanged between the sizes and the disparity is a result of the tire pressure differentials and volume of air contained within the assembly.

Plenty of simulations can be run to predict the difference between a wider tire and a narrower tire in terms of response. The general thought is that most paved road surfaces are not bumpy enough to provide a tangible suspension difference between a wider tire and a narrower tire. The suspension component could (for instance) be moved to the seat rails and the final outcome would not present a hugely different outcome. The exception to this is if the rider is extremely heavy. In this scenario a wider tire at a lower pressure would likely result in a much improved traction and comfort. If this applied at a complete bike level, very few riders would actually have any advantage from going from 23mm to 28mm on the FRONT of their bike, there is essentially insufficient weight on the front axle for a tangible difference.

The caveats are quite extensive as these losses are heavily influenced by bike geometry. A pure analysis of rider vertical movement is not realistic but it is often used by wheel companies to sell the advantage of “wider wheels”. For example, Moving the rider handlebars (and thus shifting their weight) is likely to have more of an effect than a change in tire size.