Fluid mechanics - Poiseuille Equation

[simonre7]

The question is as follows:

Determine the maximum velocity at which the poiseuille Equation may be used for
a) water flowing in pipes diameter 0.001m, 0.015m & 0.20m, stating f, pressure drop/length in each case.
b) Re-work question a) using air instead of water.

I am a bit confused as to how i reach the maximum velocity in each lengths of pipe any help would be greatly appriciated guys.

thank you in advance.

 

[Astronuc]

Think in terms of transition from laminar to turbulent flow - Poiseuille's law (or the Hagen-Poiseuille law) is valid for laminar flow. The maximum velocity is that velocity (or speed) where the flow begins transition from laminar to turbulent.

 

[piercas]

I can provide some insights on how to determine the maximum velocity at which the Poiseuille Equation can be used for different pipe diameters and fluids. The Poiseuille Equation is used to calculate the flow rate of a fluid through a pipe based on its viscosity, diameter, and pressure drop. It can be written as Q = (πr^4ΔP)/(8ηL), where Q is the flow rate, r is the pipe radius, ΔP is the pressure drop, η is the fluid viscosity, and L is the pipe length.

a) For water flowing in pipes with diameters of 0.001m, 0.015m, and 0.20m, we can use the Poiseuille Equation as long as the velocity does not exceed the critical velocity. The critical velocity is the maximum velocity at which laminar flow can be maintained. It can be calculated using the Reynolds number (Re) which is given by Re = ρvD/η, where ρ is the fluid density, v is the fluid velocity, and D is the pipe diameter.

For a pipe with a diameter of 0.001m, the critical velocity for water can be calculated as:
Re = (1000 kg/m^3)(v)(0.001m)/(0.001 Pa.s)
Solving for v, we get v = 1 m/s. This means that as long as the water velocity is below 1 m/s, the Poiseuille Equation can be used for this pipe diameter. The pressure drop per unit length (f) can be calculated using the Darcy-Weisbach equation, which is given by f = (64/Re). For Re = 1000, f = 0.064 Pa/m.

Similarly, for a pipe diameter of 0.015m, the critical velocity for water is 10 m/s, and the pressure drop per unit length is 0.0043 Pa/m.
For a pipe diameter of 0.20m, the critical velocity for water is 133.3 m/s, and the pressure drop per unit length is 0.00032 Pa/m.

b) If we consider air instead of water, the critical velocity and pressure drop per unit length will be different. Air has a lower viscosity compared to water, which means it can flow at higher velocities before reaching turbulent flow. For a pipe diameter