Cyclist Power

Basic Thermodynamics ~ J. Pohl ©

www.THERMOspokenhere.com (162-C550)

The events of a bicycle race, simplistically explained are: "the cyclist expends physical (metabolic) energy to move along a road and through the surrounding air. In places the road is flat and further along hills must be climbed. Descents are the easiest and most dangerous parts of a race." This example studies each of the types but from a special perspective.

Cycling at Constant Speed: For the majority of a race the rider and cycle will move essentially at constant speed. Of course short bursts of effort occur during which speed is changed but for most of the time: as during a time trial, as during a climb or during a downhill, the cycle and rider move together at near constant speed. We will analyze these "constant speed" aspects of racing.

Three stages of "constant speed" racing.

Time Trial: At the start, the racer rolls down a ramp peddling at a maximum to attain road speed. The course is likely to have dangerous turns requiring reduced speed but once past them, the racer re-assumes maximum, nearly constant, speed.

Climb: Racers are often required to "climb" a hill, as they say. For a long hill, having constant grade, cyclists (usually riding upright) attain a steady, constant but slow speed of ascent.

Downhill: Gravity assists the racer in coming downhill. We approximate this as constant speed.

Masses: The cyclist mass is 70 kilograms; the cycle mass is 6 kilograms.

System: The first step of an analysis is selection of a system. While "cycle and rider" is the system of racing events, this discussion (being a tutorial) begins with analysis of the "CYCLE" as system. RIDER as system is next and finally CYCLE plus RIDER" will be analyzed.

System: CYCLE Mass of the cycle is the initial consideration. Expensive racing bicycles have small masses which remain constant for all events. The Rate Form Energy Equation (constant mass) is written below.(A Review of Sign Conventions?)

(1) Energy Equation: CYCLE.

Next we expand the summation of works (first term right-of-equality). First there is "thermodynamic or pdV" work. This is zero (the cycle has zero volume changes). The cycle is a composite of simple compressible materials. Next, we know, a work effect occurs to the cycle at the tops of its pedals. The rider powers the cycle by applying force to each pedal as it displaces downward.

(2) Energy Equation: CYCLE.

All events involve friction. While friction is very difficult to address in general, some aspects can be included. Friction is a work effect in which mechanical energy becomes thermal and is then dissipated to the environment. With motion of the cycle two types of friction are anticipated.

i) Internal Friction: There will be a friction internally that we call "cycle friction." As a cycle moves contacting parts within it rub: the frame flexes, teeth of the sprocket and chain gnash, tires impact irregularities of the road, etc.

ii) External Friction: Also there is friction of movement of the cycle through the surrounding air. The cycle moves through air, which opposes the passing of the cycle. We call this effect drag. More later but, this drag depends upon speed and weather conditions of the air.

(2) Energy Equation (cycle as system).

In writing Equation (2), the friction-work terms have a special notation. For an event, the usual thermodynamic work modes might entail increase or decrease of energy of the system. Property changes of the event tell which. Thus, ideally, "energy as work to a system" might happen with no system energy change because "energy as work from the system" passed to surroundings as work (not heat). Friction changes "large-scale" motion to "molecular-scale" motion. have can be associated with , The usual case is that some terms act together. Shown as Equation (3) and discussed below.

(3) (3a)

Left-of-equality is the rate-change of internal energy of the cycle. The first term, right-of-equality is negligibly small. The second term is friction of the cycle in consequence of flexing of its metal, gnashing of gears, impacts of tires with road roughness, etc. These realities of the ride cause temperature increase, increase of internal energy. The last term, heat, occurs with the surroundings. These four terms constitute a valid equation.

When Equation (3) is subtracted from Equation (2), the resulting equation contains the principle terms of the energy equation with the cycle taken as system - Equation (4).

(3b) (3b)

System: RIDER Since the rider is a human being, thermodynamics applies only in a limited way. To obtain the energy equation of the cyclist we begin with the long, general form as we did for "cycle as system." The long-form is cumbersome but it helps us avoid mistakes.

The rate of change of kinetic energy of the cyclist (for constant speed events) equals zero. The cyclist experiences inconsequential compression work. The cyclist will experience heat. For now we assume that energy transfer to be negligibly small.

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Energy Equation: CYCLE plus RIDER When these energy equations are combined the "cycle as system" work rate, energy leaving the cyclist passing through the pedals to the cycle equals the work rate "cyclist as system" leaving him. The terms add to zero. Stated in another manner, the contacts of feet with pedals once the systems are combined is not a boundary of the combined system. Our combined equation is:

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In developments like this, it is logical to check the equation against cases so obvious we know the answer. If the equation does not "agree," (that is, yield the obvious answer) then something is wrong - we must backtrack and check out logic. That technique will be demonstrated below.

With the cycle as system, drag work is negative. The cycle must have the energy to spend if it is to pass through the surrounding air. Of course more energy is required the faster the cycle moves.

Friction of the cycle and the cyclist as they move through air constitutes work. As they move the air ahead out must be shoved out of the way. Engineering characterizes this physical effect as the action of a drag force that displaces with the rider and cycle. Physically, this amounts to cyclist passing energy from himself to the ambient air (some of it passes from him to the cycle thence to the air). The effect on the air is higher local velocities. Once the system passes the energy of the increased velocity air dissipates to become a very very small local temperature increase of the air.

A classical approach has been developed to quantify the frictional work consequent to wind drag. The work rate is written as the scalar product of the drag force times the velocity of the rider and cycle:

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Wind drag is made quantitative in terms of the system velocity written in a magnitude-direction vector form.

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The work rate required equals the scalar product of the drag force times the system velocity. Racing team sponsors have studied drag extensively. The resisting force is always directed opposite to the velocity of the system. The magnitude of the drag force is directly proportional to the area of a frontal projection of the cycle and rider, A_frontal and to the air density. Also it is proportional to the square of the relative speed. (The square of speed is divided by two as a convenience, to have it pattern kinetic energy). Proportionality, such as this, apply in a general way and are written as shown (below left). Usually expensive experimentation is required to change the proportionality into an equation, to "get the constant," C_drag, which applies to quite specific equipment and conditions.

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Returning to the equation (above) containing "F · V," we substitute for the vectors, F and V.

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Drag force opposes motion in proportion to the square of the speed. The associated work is negative, meaning energy must leave the system.

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Bicycle racing is highly competitive; plenty of money is spent preparing riders and testing equipment. Approximate numbers for the drag coefficient for all aspects of performance are determined for each rider/cycle-combination. The frontal area of this investigation is 6 m², the density of air as 1.18 kg/m³ and the drag coefficient, C_drag equals 0.88. (For the sake of simplicity, We will use only one number for drag coefficient). This information entered into the equation yields a result suitable for any phase of a race.(all_phases)

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Time Trial Power Suppose the first day our rider completes a ride of 6.1 kilometers (flat course with few corners) in 6 minutes and 52 seconds or an average speed of 14 meters per second. Calculate the steady watts of physical energy expended during the race. Since the time trial events occur on horizontal roads, potential energy changes equal zero.

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Maximum Speed of Climb: We assume the rider capable of the same sustained power in climbing as he did during the time trial. Rate of potential energy change must be included. It is the mass times gravity timed the vertical component of the road speed. A side calculation is needed:

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A factor of the rate of potential energy change of the rider and Potential energy of the rider and cycle are related to road speed. The uphill grade is 7%. We Our energy equation we realize rate of climb is part of road speed. We can write dPE/dt in terms of the cycle speed and the grade of the slope.

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These numbers are (assumed constant or average) for all phases of the race. These, substituted into the energy equation, yield:

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We will apply this energy equation for the constant speed events of time trial, downhill coast, and climb.

Time Trial Cyclist Power: Suppose the first day our rider completes a ride of 6.1 kilometers (flat course with few corners) in 6 minutes and 52 seconds or an average speed of 14 meters per second. Calculate the steady watts of physical energy expended during the race. Since the time trial events occur on horizontal roads, potential energy changes equal zero.

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Downhill Speed: The grade at 10% represents a steep, dangerous downhill. Our cyclist (no madman) will rest and coast.

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Climbing speed: Suppose our rider were to apply the same wattage to a 7% climb as he did to the time trial. Estimate the road speed. Climbing is uphill and against the wind. All three terms are non-zero and the constant speed will be the least of the three cases. Run a calculation of this.

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About assumptions: Though we dislike assumptions, the task of engineering analysis is to arrive at some answer. If, while analyzing things that might happen, one arrives in a mental corner, assumption can be a substitute for despair. Reasonable assumptions are preferred, of course. Necessary assumptions distort the consideration to some other situation; what is the new situation. When progress seems impossible. If an absurd assumption comes to mind and is applied to the model of physical reality - mathematical nonsense should result.