Modern Engineering Thermodynamics

17.8 Locomotion Transport Number 715
where P o is the basal metabolic rate (BMR), P D is the power absorbed by aerodynamic drag, and P m is the power
absorbed by the muscles. From fluid mechanics, we know that the power required to overcome viscous drag is
given by
P D = 1 2 ρAC DV 3
where ρ is the fluid density, A is the frontal projected area of the animal, and C D is its drag coefficient. We can
therefore determine the most efficient transport velocity by minimizing T as follows:
∂T
∂V = 0 = − P o + P m
wV 2 + ρAC D V
+ 1
∂P m
w wV ∂V
(17.23)
where we have assumed the weight to be constant during the locomotion. Equation (17.23) can be rewritten as
ρAC D V 3 + V ∂P
m
− ðP o + P m Þ = 0 (17.24)
∂V
which could be solved for the most efficient locomotion velocity if we knew how P m depends on V. Ifwe
assume that P m increases linearly with V, then we can write
where K is a constant. Then, Eq. (17.24) becomes
then,
P m = KV
ρAC D V 3 − P o = 0
V most efficient =
1/3
P o
(17.25)
ρAC D
Figure 17.12 shows T vs. V for a 70 kg human. The minimum value of T occurs at about V=1.75 m/s, which
corresponds to a fast walk. Locomotion velocities faster or slower than this value require more energy consumption
per distance traveled and are hence less efficient locomotion speeds.
Mechanical locomotion devices have the potential of altering the T vs. V curvebymovingitsminimumtoa
higher velocity. Of course, the weight of the locomotion device must be added to the animal’s weight such that
w in Eq. (17.22) is
w = w animal + w device
therefore, the weight of the locomotion device alone tends to decrease the value of V at minimum T. Bicyclists
are willing to carry along the extra weight of their machines, because at their most efficient velocity, their minimum
value of T is about 0.064, which is about 25% of their minimum value of T in normal leg locomotion
without the bicycle. In fact, the bicyclist has the lowest value of T ever measured for any animal or machineanimal
combination.
1.0
Transport number T
0.8
0.6
0.4
0.2
Minimum at V = 1.75 m/s
0.0 0 1 2 3 4 5
Velocity V (m/s)
FIGURE 17.12
The locomotion transport number vs. velocity calculated from Eq. 17.22 for a 70-kg human.