Download PWE 11-12: How Many Capillaries?

 Example 11-12 How Many Capillaries?
The inner diameter of the human aorta is about 2.50 cm, while that of a typical capillary is about 6.00 mm = 6.00 *
1026 m (see Figure 11-24). In a person at rest, the average flow speed of blood is about 20.0 cm>s in the aorta and about
1.00 mm>s in a capillary. Calculate (a) the volume flow rate of blood in the aorta, (b) the volume flow rate in a single capillary, and (c) the total number of open capillaries into which blood from the aorta is distributed at any one time.
Set Up
Figure 11-24 shows the situation. We are given
the dimensions of the aorta and each capillary as well as the flow speed in each of these
pipes. Our goal is to determine the volume flow
rate (in volume per unit time, or m3 >s) in the
aorta and in a capillary as well as the number
of capillaries into which the aorta empties.
The volume flow rate in a pipe is related to its
cross-sectional area and the speed of the fluid
moving in the pipe. Like water, blood acts like
an incompressible fluid. (It will only compress
appreciably under pressures much higher than
those found in the body.) So we can use the
ideas of the equation of continuity, including
the idea that the volume flow rate through the
aorta must be equal to the flow rate through all
of the open capillaries combined.
Solve
(a) The volume flow rate in the aorta is
equal to the product of its cross-sectional
area and the flow speed of aortal blood
1vaorta = 20.0 cm>s = 0.200 m>s2.
(b) Do the same calculations for a
capillary, in which the flow speed is
vcapillary = 1.00 mm>s = 1.00 * 10-3 m>s.
Equation of continuity for steady flow
of an incompressible fluid:
(11-19)
A1v1 = A2v2
1 = aorta
2 = all open capillaries combined
v
volume flow rate = Av
Radius of aorta:
raorta = 1>2 * 1diameter of aorta2 = 1>2 * 2.50 cm
= 1.25 cm = 1.25 * 1022 m
Cross-sectional area of aorta:
Aaorta = pr 2aorta = p 11.25 * 10-2 m2 2
= 4.91 * 1024 m2
Volume flow rate in aorta:
Aaorta vaorta = 14.91 * 10-4 m2 2 10.200 m>s2
= 9.82 * 10-5 m3 >s
Radius of a capillary:
rcapillary = 1>2 * 1diameter of capillary2 = 1>2 * 6.00 * 10-6 m
= 3.00 * 1026 m
Cross-sectional area of a capillary:
Acapillary = pr 2capillary = p 13.00 * 10-6 m2 2
= 2.83 * 10211 m2
(c) Our results from (a) and (b) show that
compared to the volume flow rate through a
single capillary, the volume flow rate through
the aorta is 3.47 * 109 times greater. The idea
of continuity tells us that the combined volume
flow rate through all the open capillaries must
be equal to the volume flow rate through the
aorta. We therefore learn the total number of
open capillaries.
A
Volume flow rate in a capillary:
Acapillary vcapillary = 12.83 * 10-11 m2 2 11.00 * 10-3 m>s2
= 2.83 * 10-14 m3 >s
9.82 * 10-5 m3 >s
Volume flow rate in aorta
=
Volume flow rate in a capillary
2.83 * 10-14 m3 >s
= 3.47 * 109
(Volume flow rate in aorta) = (total volume flow rate in all open
capillaries combined)
. . . so there must be 3.47 * 109 open capillaries.
Reflect
Our results show that the human circulatory
system is truly extensive!
As a check on our results, note that the
combined cross-sectional areas of all capillaries
is 9.82 * 1022 m2, which is 200 times greater
Total cross-sectional area of all open capillaries combined:
than the cross-sectional area of the aorta.
By the equation of continuity, the flow
speed in the capillaries should therefore
be slower than in the aorta by a factor of
1> 12.00 * 102 2: That is, vcapillary = vaorta >
12.00 * 102 2 = 10.200 m>s2 > 12.00 * 102 2 =
1.00 * 10-3 m>s. This gives us back one of the
numbers we started with, so our calculation is
consistent.
Area of all open capillaries combined
9.82 * 10-2 m2
=
Area of aorta
4.91 * 10-4 m2
= 2.00 * 102
Aall open capillaries = (3.47 * 109)Acapillary
= (3.47 * 109)(2.83 * 10211 m2)
= 9.82 * 1022 m2