Question

The blood was flowing out from the heart at the speed of 0.45 m/s through aorta. The diameter of aorta is 1.6 cm. The aorta was splitted into a number of tiny blood vessels. The blood flows through the vessels to various organs with the speed of 5 x 10-5 m/s. If the diameters of the blood vessels are 99.95% smaller than diameter of aorta and the blood is incompressible. Calculate the number of blood vessels in human body by assuming that all the blood vessels have similar diameter.

Answer 1

Solution:

The flow rate through Aorta, Q₁ = v₁A₁

Q₁ = v₁𝜋($\frac{d₁}{2}$)²

Q₁ = $\dfrac{V_1\pi d_1^2}{4}$

The flow rate through one tiny blood vessel, Q₂ = v₂A₂

Q₂ = v₂𝜋($\frac{d_2}{2}$)²

Q₂ = $\dfrac{V_2\pi d_2^2}{4}$

Given d₂ = d₁ - 99.95%d₁

d₂ = 0.0005d₁

$Q_2=\dfrac{V_2 \pi(0.0005d_1)^2}{4}$

$Q_2=\dfrac{(2.5*10^{-7})V_2\pi d_1^2}{4}$

A number of tiny blood vessels:

$\dfrac{Q_1}{Q_2}=\dfrac{\dfrac{V_1\pi d_1^2}{4}}{\dfrac{(2.5*10^{-7})V_2\pi d_1^2}{4}}=\dfrac{V_1}{(2.5*10^{-7})V_2}=\\\;\\\dfrac{0.45(\frac{m}{s})}{(2.5*10^{-7})(5*10^{-5}(\frac{m}{s}))}=3.6*10^{10}$