Pump Pressure for Fluid Mechanics: H, v, P0

[r.wazlib]

How large should be the pressure P, pumped by a pump to supply water with velocity v to a part of a building at the height H, where it is consumed. The liquid may be considered as incompressible and inviscid and the flow, as steady and irrotational. (Hint: take into account that the water inside the pump has zero velocity and that it is consumed at the atmospheric pressure P0).

 

[Bob S]

This sounds like homework. Even so, in order to solve the real problem, you need to know the dynamic pressure drop in the pipes themselves due to turbulence and Reynold's number. Here is an online calculator (needs Java installed):
http://www.pipeflowcalculations.com/pressuredrop/index.htm

 

[astrokreng]

The pressure P needed to supply water at a velocity v to a height H in a building can be calculated using the Bernoulli's equation for incompressible and inviscid flow. This equation states that the total energy of a fluid remains constant along a streamline. In this case, the total energy is given by the sum of the kinetic energy (1/2 * v^2) and the potential energy (gh), where g is the acceleration due to gravity.

Therefore, the pressure P can be calculated using the equation:

P + 1/2 * v^2 + gh = P0 + gh

Solving for P, we get:

P = P0 + 1/2 * v^2 + gh

Since the water inside the pump has zero velocity, the pressure P0 at the atmospheric pressure is equal to the pressure at the pump inlet. Hence, the pressure P should be pumped by the pump to supply water at a velocity v to a height H can be calculated using the equation:

P = 1/2 * v^2 + gh

This equation shows that the pressure P is directly proportional to the velocity v and the height H. Therefore, to increase the pressure P, the velocity v or the height H needs to be increased. However, it is important to note that there may be limitations to increasing the velocity or the height, depending on the design and capacity of the pump.

In conclusion, the pressure P needed to supply water with a velocity v to a height H in a building should be calculated using the Bernoulli's equation, taking into account the velocity of the water at the pump inlet and the atmospheric pressure. The pump should be designed to provide a pressure that is sufficient to overcome the effects of gravity and friction losses in the pipes to ensure a steady and efficient flow of water to the desired location.