Thursday, July 23, 2009

effect of body mass on wind resistance

I already considered the effect of mass on climbing speed, but body mass (as opposed to equipment mass) also affects wind resistance, which is still important when climbing.

This has been estimated empirically by Heil, European Journal of Applied Physiology, V 93 N 5-6; March, 2005. However, I never really liked Heil's analysis, because it looks at body mass without also considering body height. Obviously if you consider gaining mass without height versus gaining mass in proportion to height it is reasonable to expect different results. I'm reminded of a physics lecture I attended as an undergrad: "First, we model the human body as a sphere." The crowd chuckled. It was a sphere of salt water, BTW, useful for electrostatic calculations, but of course the human body is not a sphere. So shape matters.

So first the usual wind resistance model, in the absence of external wind:

Pw = fw

fw = ½ ρ CD A

where ρ is the mass density of air, CD is the wind drag coefficient, and A is the cross-sectional area normal to the direction of motion. I'll assume CD A is due to two independent parts: one from the body and one from the bike:

CD A = (CD A)bike + (CD A)body.

Obviously the body affects the drag on bike components, but I'll assume this effect for a given bike is independent of the mass or height of the rider. But obviously a taller rider tends to ride a larger bike, although the wind resistance may not scale in strict proportion, since the wheels and fork blades, as well as the components fail to scale with rider size. So I'll assume a linear model:

Abike = A⁰bike + Weff × height

where Weff has units length and describes how the area of the bike increases with rider height. In contrast, Heil estimates the effective cross-sectional area of a road bike is around 0.11 to 0.13 m², with a time trial bike around 0.066 m².

More important is the body. It's been estimated the body accounts for around 70% of the total wind resistance. Heil claims an empirical formula for cross-sectional area which can be written as 0.306 m² (mass / 100 kg)^(0.762). This formula is useful for establishing a relative magnitude, but the analytic form, as I noted, is less than satisfying.

For this analysis, assume CD is fixed (Heil claims this may not be a good assumption). Then assume area is proportional to width × height, but not depth. It's well known an elongated tube has less wind resistance, up to a 3:1 ratio of depth to width, than a cylindrical tube, for wind striking normal to the principal axis of the tube. In certain cases "depth" may even contribute to a propulsive force via the "sail effect". So depth can either increase resistance (due to skin friction) or decrease it (due to reduced turbulance) depending on specifics of the shape relative to the wind direction and the direction of motion.

So the model I'll use is:

Abody = width × height.

I assume we know height. Assuming mass is proportional to width × height × depth, and height and depth increase in proportion, then height and width are each proportional to sqrt(mass / height). So:

Abody = KA × sqrt(mass / height) × height
    = KA × sqrt(mass × height)

where KA is a constant.

So the total CD A is then:

CD A = CD,bike × ( A⁰bike + Weff × height ) + CD,body × KA × sqrt(mass × height).

Since height is closely correlated with mass, but less strongly correlated with body mass index (BMI) = mass / height², it's useful to cast this equiation in terms of BMI:

CD A = CD,bike × ( A⁰bike + Weff × height ) + CD,body × KA × sqrt(BMI × height³).

Note that if BMI is considered relatively constant, then height is proportional to sqrt(mass), and the body's cross-sectional area is proportional to mass^(3/4), which is quite close to Heil's exponent of 0.762.

This model allows for a refinement of the effect of body mass (whether due to BMI or height) on climbing speed. I'll leave that for later.

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