Given a tank with a certain height of water above the discharge pipe of know length, open to the atmosphere at the other end. I know the flow rate can be calculated if the pipe was horizontal by using Poiseuille's equation (No entry losses, no turbine or pump work, pipe cross-section is constant) by considering the pressure loss to be equal to the difference between the hydrostatic pressure at the inlet of the pipe and the atmospheric pressure.
If the pipe was not horizontal.. ie the pipe outlet is at a certain distance below the inlet to the pipe, by knowing the length of the pipe and not considering any pipe bend losses can we calculate the flow rate?….I considered using Bernoulli's equation by considering the pressure drop to be equal to the difference between the hydrostatic pressure at the inlet of the pipe and the atmospheric pressure and as the height difference between the inlet and outlet is know. I couldn't figure out how to calculate the pressure loss term in the equation. Is this data sufficient to calculate the flow rate…if not what other data would be required?
Best Answer
Assuming an open vessel and the outlet to atmospheric pressure also:
Then the total pressure available to overcome pipe pressure losses is:
$$\Delta P=\rho g h$$
(with $\rho$ the density of the fluid)
As regards the pressure loss in the pipe, assume no losses in the bends (if any) and assume laminar flow, so you can use the Hagen Poiseulle equation:
$$\Delta P=\frac{8\mu Q}{\pi R^4}L$$
Then calculate the flow rate. Check the Reynolds number $Re$ to see if flow is indeed laminar.
Should flow prove not to be laminar ($Re>2600$), you will need the Darcy Weisbach equation for turbulent flow to calculate the pressure loss in the pipe:
$$\Delta P=f_D\frac{8Q^2}{\pi^2 g D^5}L$$