Question

 Water is circulating through a continuous solid pipe in a house. If the water is pumped at a velocity of 0.6 m/s through a diameter of 10 cm in the basement under a pressure of 3.039 x 10⁵ N/m², what are the flow speed and pressure in a 4 cm diameter pipe 6m above? Note that Water has a density of 1000 kg/m3 . 

Answer 1

a) Let's first find the flow speed in a 4 cm diameter pipe 6m above from the Law of Continuity:

$$A_1v_1=A_2v_2,$$ $$v_2=\dfrac{A_1v_1}{A_2}=\dfrac{\pi r_1^2v_1}{\pi r_2^2},$$ $$v_2=\dfrac{\pi(0.1\ m)^2\cdot0.6\ \dfrac{m}{s}}{\pi(0.04\ m)^2}=3.75\ \dfrac{m}{s}.$$

b) We can find the pressure in a 4 cm diameter pipe 6m above from Bernoulli’s equation:

$$P_1+\rho gh_1+\dfrac{1}{2}\rho v_1^2= P_2+\rho gh_2+\dfrac{1}{2}\rho v_2^2,$$ $$P_2=P_1+\rho g(h_1-h_2)+\dfrac{1}{2}\rho(v_1^2-v_2^2),$$

$P_2=3.039\cdot10^5\ Pa+1000\ \dfrac{kg}{m^3}\cdot9.8\ \dfrac{m}{s^2}\cdot(0\ m-6\ m)+\dfrac{1}{2}\cdot1000\ \dfrac{kg}{m^3}\cdot((0.6\ \dfrac{m}{s})^2-(3.75\ \dfrac{m}{s})^2)=2.4\cdot10^5\ Pa.$