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Cycling: Determining The Optimal Altitude For Maximizing Aerobic Cycling Speed
George D. Swanson
Division Of Science And Health
College Of The Redwoods
Eureka, Ca 95501
Endurance cycling speed can be enhanced by road racing at higher altitudes. That is, mountain air is "thinner" reflecting a lower air density at altitude as compared to sea level. However, the disadvantage at high altitude is a lower aerobic capacity of the rider. Therefore, a trade-off exists between the reduced drag and the reduced aerobic power associated with the lower oxygen availability of altitude.
These concepts suggest there may be an optimal altitude for maximizing aerobic cycling speed. This altitude would balance diminished oxygen availability with enhanced drag characteristics to produce a maximum endurance speed. Our approach for determining this optimum, utilizes the known physics of air density, a mathematical model of cycling and literature data about the altitude effect on reduced aerobic capacity. The result is a mathematical model from which the optimal altitude is determined. A variety of assumptions go into the formulation of this model The data in the form we need on the decrease in aerobic capacity with altitude are not generally available. Therefore, we must rely on data from two subjects from a simulated altitude study (2). We have chosen to use the average response from these two subjects. However, until more data are available, the exact form of this response is uncertain. Our present results must be viewed as tentative.
Air Density Physics
The density of air is proportional to pressure and inversely proportional to temperature. Barometric pressure is given by an exponential decrease with altitude (Boyle's law):
PB = P0 e-x
where PB is the pressure (mmHg), P0 is the corresponding pressure at sea level,is a rate constant (1.161 x 10-4 /m) and x is the altitude (m).Air density is also proportional to air temperature (Charles Law). Thus, air density, (kg/m3), is given by
= PB 133.29/(g R T)
where g is the gravitational field constant, R is the universal gas constant and T is the air temperature(K).
We shall assume temperature and the gravitational field do not vary with altitude. Obviously, as we go up in altitude, the air temperature may drop depending on the time of day. Also, the gravitational field constant changes from 9.807 at seal level to 9.791 at 5000 m. However, in our initial analysis, we shall neglect these effects. Therefore,
= Ke-x
where K incorporates all other constants.
Model Of Aerobic Capacity
A power-duration curve has been used to characterize the aerobic and non-aerobic regions of exercise performance (Poole et al, 1990). This curve is of the form
(P-Pc)t = W'
where P is the applied power (watts), t is the duration time (s) until the exercise stops due to fatigue or exhaustion and W' is the amount of non-aerobic work (J) available above Pc. This work is accomplished at the expense of a finite energy store and is thought to depend upon factors, which comprise the non-aerobic stores (phosphagen stores, non-aerobic glycolysis, and venous blood oxygen stores). The critical power parameter, Pc, represents the upper limit of the aerobic region. Exercise below Pc is essentially sustainable without exhaustion.
Power-duration experiments, with a number of discrete power levels in the non-aerobic region, yield a set of data pairs (duration and power). This set is fit to the power-duration curve via computer to yield the best fit parameter estimates. The critical power parameter is estimated via this process and characterizes the aerobic region. A variety of studies have utilized this approach to characterize the effects of training and other interventions (Poole et al, 1990).
This approach has been utilized to assess the of acute simulated altitude effects via lowered oxygen in a gas mixing system (2). At the present time, this is the only data available for analysis. These data are summarized in Table 1. The average data are remarkably linear (r2 is essentially one) and when expressed as a relative fraction can be fit to the regression line
y = -x
where x is the altitude (m), = 7.5 x 10-5 m-1 and y is the relative fraction.
Although these average data are linear, the individual data are concave upwards and concave downwards. Therefore, this linear decrease with altitude is tentative. Furthermore, these data represent acute simulated altitude experiments. Data from subjects that have acclimated to given altitudes are not available at this time. However, we can speculate that acclimatized critical power data would show less of a decrease with altitude than acute data. That is, the acclimatized decrease in aerobic capacity should be less than the acute exposure decrease in aerobic capacity.
Model Of The Cycle
We shall use a model of the cycle as developed by Weaver (6) for a human powered vehicle designed for maximum speed. This particular cycle is a recumbent design with a faring used to minimize drag. However, the model is also valid for an upright rider with no windshield or faring. The "effective" frontal area of the cycle-rider allows us to paramaterize the particular design. The model is given by
dV/dt = (1/M)[P/V-CrrW-1/2CdAV2-W sin(tan-1(slope)]
where V is velocity (m/s), M is mass (kg), is mechanical efficiency, P is power (watts), Crr is the coefficient of rolling resistance, W is weight (N), Cd is the coefficient of drag, A is the frontal area and slope is the relative change of the roadway elevation.
Analysis
We shall now confine our analysis to the steady state so that dV/dt = 0. Therefore,
P = CrrW V + 1/2 CdV3 +W V sin(tan-1(slope)).
To further simplify the analysis, we shall neglect the rolling friction and assume a flat terrain (slope = 0). Then,
P = 1/2CdV3 or V3 = 2P/(Cd).
Substituting for the density as a function of altitude and critical power as a function of altitude,
V3 = 2Pcû (-x)/(CdKe-xA)
Where Pcû is the sea level critical power. We now want to determine the optimal altitude. Taking the partial derivative of velocity with respect to altitude and setting that term equal to zero, the optimal altitude is given by
xopt = 1/ -1/.
Therefore, the optimal altitude is given by the reciprocal of the fractional slope for the decrease in critical power minus the reciprocal of the rate constant for the decrease in density. For our parameter values (x 10-4 m-1, = 7.5 x 10-5 m-1)
xopt = 4700 m (15,400 ft).
However, as we go up in altitude and the drag is reduced, the rolling resistance term becomes more dominant. Therefore, we can not neglect that term in our analysis. From the steady state equation,
P = CrrW V + 1/2 ACdV3
we can take the partial derivative of velocity with respect to altitude and set that term equal to zero. This yields
V3 = Pcû/(1/2CdA)
at the optimum altitude and velocity. Substituting this result into the equation above we have
V = (Pcû/WCrr)[1-x-].
These two equations (cubic and linear) can be solved simultaneously to determine the optimal altitude.
We shall now proceed assuming the rider and cycle characteristics as described in Weaver (6):
A = 0.26 m2
Cd = 0.125
W = 938.5 N
Pcû = 373 watts
Crr = 0.0035.
Then
V3 = 14100/
V = 108 [0.3534 - 0.75x10-4x].
Figure 1 indicates the simultaneous plot of both equations as a function of altitude. Note the optimal altitude for the case of rolling resistance is given by
xopt = 1740 m (5700 ft).
Implications
This optimal altitude (1740 m) yields an aerobic velocity of 24.2 m/s (54.1 mph). However, sea level (0 m) yields an aerobic velocity of 24.0 m/s (53.7 mph). Therefore, the actual velocity gain from sea level to altitude is only of the order of 0.4 to 0.5 mph. Although this may be important, the more important gain maybe in the non-aerobic region. By definition, this is the region above the limit of aerobic capacity where work is performed using exhaustible stores independent of oxygen supply. For example, Weaver(6) as a rider managed 1120 watts for 30 s (sea level) before exhaustion . From these data, the amount of non-aerobic energy store available is estimated to be
W' = (1120-373) 30 = 22.4 KJ.
Now let us assume the rider applies a 30 s thrust of this non-aerobic energy on top of his maximum aerobic velocity. His actual energy loss due to rolling resistance and air resistance drag is dependent upon on the actual velocity trajectory. However, we can produce a rough estimate by assuming an "average" velocity through out the region of non-aerobic thrust. Suppose this average velocity is 60 mph (26.8 m/s). Then the loss at altitude given by
[CrrVW + 1/2ACdV3] 30 = 12 KJ.
Alternatively, at sea level the loss is given similarly as 14 KJ.
The energy needed to accelerate the cycle and the rider in the non-aerobic phase is given by the kinetic energy
KE = 1/2 M(Vf2 - Vo2).
If we now assume the initial velocity is 54 mph for both cases, the peak velocity is 68.2 mph at altitude and 66.6 at sea level. Thus, the velocity gain during the non-aerobic trust is about 1.5 mph.
Conclusion
On September 13-17 (1993), the Colorado Speed Challenge was held at Alamosa Colorado where the altitude is 7600 ft. Sam Whittingham, a rider from British Columbia, achieved the top speed of 63 mph in the 200 m speed challenge. But, is 7600 ft the optimal altitude for breaking the 70 mph barrier (1)? Proceeding with the analysis as given above, the maximum aerobic velocity at 7600 ft is 22.0 m/s (49.2 mph). Based on this, our estimate of the peak non-aerobic velocity at 7600 ft is 28.2 m/s (63.2 mph). Therefore, it appears that Whittingham would have done at least 5 mph more at 5700 ft.
Acknowledgments
The following students helped with the problem formation and reviewed parts of the manuscript: Eric Berenson, Douglas Boyer, Lemel Lara, Steve Sanchez and Robert Whitmore. The International Human Powered Speed Championships were held in Eureka, California, August 1-7, 1994. The author gratefully acknowledges the opportunity to interview several riders and cycle designers, especially Mad Dog Mike Schuler, San Whittingham and Gardner Martin.
References
1. Boyer, D. R. and G.D. Swanson. The 70 mph quest(ion). In: T. Fahey (Ed.). Encyclopedia of Sports Medicine and Exercise Physiology. New York: Garland Press (in press).
2. Moritani, T., A. Nagata, H.A. DeVries and M. Muro. Critical power as a measure of physical work capacity and anaerobic threshold. Ergonomics. 24:339-350, 1981.
3. Poole D. C., S. A. Ward and B. J. Whipp. The effects of training on the metabolic and respiratory profile of high-intensity cycle ergometer exercise. Eur. J. Appl. Physiol. 59:421-429, 1990.
4. Posth, M. A. The world's fastest bike. Popular Science. 243:79-80, 1993
5. Swanson, G. D. Human powered flight. In: T. Fahey (Ed.). Encyclopedia of Sports Medicine and Exercise Physiology. New York: Garland Press (in press).
6. Weaver, M. The cutting edge streamlined cycle. Cycling Science. 3:17-23, 1991.
Table 1 - Summary of Critical Power Parameter estimates from the data of Moritani et al (2) Ñ Study of Acute Simulated Altitude
SubjectAltitude (Inspired Oxygen %)
0 m (20.9%) / 4285 m (12%) / 6750 m (9%)C.M. / 230 / 130 / 102
T.M. / 197 / 160 / 110
AVERAGE / 213.5 / 145 / 106
RELATIVE FRACTION / 1 / 0.679 / 0.496
Figure 1. Simultaneous solution of cubic and linear equations.