Modern Engineering Thermodynamics

17.7 Limits to Biological Growth 713
where E is the elastic modulus of the trunk and γ is its weight density. It can also be shown that the tallest selfsupporting
homogeneous tapering conical column with base diameter d is about twice as tall as the critical
height given by Eq. (17.20). For live wood, the ratio of ðE/γÞ 1/3 is approximately 120. m 1/3 for all trees. Thus,
the critical height of trees varies approximately with their base diameter to the 2/3 power according to
h critical = 68:0 d 2/3 (17.21)
where h critical and d are in meters and the coefficient 68.0 has units of m 1/3 .
EXAMPLE 17.8
Determine the critical buckling height of a small tree whose base diameter is 5.00 × 10 –3 m.
Solution
From Eq. (17.21), we have h critical = 68:0ðd 2/3 Þ = 68:0ð5:00 × 10 − 3 Þ 2/3 = 1:99 m:
Exercises
22. Determine the critical buckling height of a tree in your yard that has a base diameter of 8.00 in (0.203 m).
Answer: h critical = 23.5 m (77.1 ft).
23. A wooden flag pole with a 6.00 in (0.152 m) base diameter is made from fresh wood. Determine its critical buckling
height. Answer: h critical = 19.4 m (63.6 ft).
24. Determine the critical buckling height of a giant redwood tree with a base diameter of 6.00 m (20.0 ft).
Answer: h critical = 225 m (738 ft). Note: Such trees seldom exceed a growth height of about 90 m (300 ft).
Tree limbs are sized to withstand the bending forces due to their own weight. If a branch is considered to be a
cantilever beam attached at an angle α to the trunk, then there exists a critical length l crit that allows the tip of
the branch to extend horizontally. Longer branches droop below the horizontal and shorter branches point
upward at an angle approximately the same as their attachment angle α. It can be shown that the equation for
l crit is identical in form to Eq. (17.20), except with a different multiplying constant (in this case the diameter
d is the limb diameter at the point of attachment). Thus, the shape and size of trees and other plants is proportional
to the 2/3 power of the base diameter of the limbs and trunk.
It can be shown that muscular power for animal locomotion is also proportional to the square of the characteristic
body dimension. Therefore, the work (i.e., power × time) done by a muscle is proportional to L 2 × ðL/VÞ,
or L 3 , where V is the locomotion velocity. The kinetic energy of motion at constant velocity is also proportional
to L 3 because the animal’s mass is proportional to its volume. Since both the work done by the muscle and the
system kinetic energy it produces are proportional to L 3 (if we ignore any aerodynamic drag and acceleration
effects), we see that there can be no significant size effect in the horizontal locomotion of animals. That is,
all animals should be able to run at about the same maximum velocity on a horizontal surface.
Consider now an animal running uphill at constant velocity. The rate of energy expenditure in increasing
its potential energy (again, ignoring aerodynamic and other effects) is proportional to L 3 × (dZ/dt). Since its
muscular power is always proportional to L 2 , an energy rate balance on the animal tells us that its ascent velocity
dZ/dt mustthereforebeproportionaltoL –1 . That is, the speed of an animal running uphill should be
inversely proportional to its size. A hill that a rabbit can easily run up may reduce a dog to a trot and a hunter
to a walk.
A similar argument can be made for large flying animals. It can be shown with an energy rate balance that the
rate of energy expenditure required for hovering or forward flight is proportional to L 3.5 . Since the flight muscles
can supply power only proportional to L 2 (again ignoring aerodynamic drag and inertia), the ratio of required
power to available power is proportional to L 1.5 . Thus, an upper limit to the size of flying animals is quickly
reached. In the case of birds, their aerodynamic design sets this upper limit at about 16 kg (35 lbm).
WHY YOU CANNOT CATCH A RABBIT
Though observations of animals from rabbits to horses show that they can all run about the same maximum speed on a
horizontal surface, smaller animals can accelerate and decelerate (i.e., maneuver) much faster than larger animals
(people).