The inner diameter of the human aorta is about 2.50 cm, while that of a typical capillary is about 6 = 6.00 × 10−6 m. 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 the capillary. calculate;
(a) the volume flow rate of blood in the aorta
(b) the volume flow rate in a single capillary
(c) the total number of open capillaries into which the blood from the aorta is distributed at any one time?

a) Flow rate of aorta = 9.818 x/s

b) Flow rate of capillariy = 2.827 x  /s

c) Number of capillaries = 3.473 x

Explanation:

This question focuses on the flow rate of a fluid. So let us look what flow rate is.

Flow rate is the volume of fluid that flows through the tube per unit time.

i.e. Flow rate =

where,

V = Volume of the fluid

t = time

We know that Volume = Area x distance

therefore,

Flow rate =

where,

A = cross sectional area of the tube

x = distance

Since in this problem area is a constant, we can take A out of differentiation.i.e.

Flow rate =

Now, we got which is equal to velocity of the fluid.

Thus,

Flow rate = = Av

where, v = velocity

a) diameter of aorta = 2.50cm = 2.5x

Area, A = π = π

= 4.909 x

v = 20.0 cm/s = 0.2m/s

We know that Flow rate = Av = 4.909 x x 0.2 = 9.818 x/s

b) diameter of capillary = 6.00 × 10−6 m

Area, A = π = π = 2.827x

v = 1.00 mm/s = m/s

We know that Flow rate = Av = 2.827x x   = 2.827 x  /s

c)Since the fluid is incompressible, the flow rate before and after should be same. Let there be n capillaries.

Then total flow rate in the capillaries = n x 2.827 x  /s

Flow rate in aorta = 9.818 x/s

These two should be equal. i.e.

n x 2.827 x    = 9.818 x

n =

n = 3.473 x

0.11 m/s and 0.28 m/s should be correct