Question

In an industrial cooling process, water is circulated through a system. If water is pumped with a speed of 4.00 m/s under a pressure of 6.70 × 10⁵ Pa from the first floor through a 3.70-cm diameter pipe, what will be the pressure on the next floor 3.90 m above in a pipe with a diameter of 2.00 cm?

Answer 1

bernoulli principle can be used to determine the pressure.

bernoulli principle states that:

P1+0.5*D*v1^2+D*g*h1=P2+0.5*D*v2^2+D*g*h2

where P1=initial pressure=6.7*10^5 N/m^2

P2=final pressure (to be determined)

D=density of water=1000 kg/m^3

v1=initial speed=4 m/s

v2=final speed

h1=initial height

h2=final height

given that h2-h1=3.9 m

initial area of the pipe=pi*(diameter/2)^2=pi*(3.7*0.01/2)^2=1.0752*10^(-3) m^2

final area of the pipe=pi*(diameter/2)^2=pi*(2*0.01/2)^2=3.1416*10^(-4) m^2

using continuity equation:

initial area*initial speed=final area*final speed

==>1.0752*10^(-3)*4=3.1416*10^(-4)*v2

==>v2=13.6898 m/s

hence putting all the values in bernoulli's equation:

6.7*10^5+0.5*1000*4^2+1000*9.8*h1=P2+0.5*1000*13.6898^2+1000*9.8*h2

==>P2=5.843*10^5+100*9.8*(h1-h2)

==>P2=5.843*10^5-1000*9.8*3.9

==>P2=5.46*10^5 N/m^2

hence pressure on the next floor will be 5.46*10^5 N/m^2 or 5.46*10^5 Pa.