# Calculate liters of water flowed on a pipe at a given pressure and time

I am having a really hard time getting the answer to this apparently simple question! I have a vertical pipe on my house with cross section area A and lenth L. The pipe is completely filled with water. At the bottom of the pipe the pressure must be $$L \cdot \text{water density} \cdot \text{gravity}$$, right?

So how do I calculate how many $$m^3$$ (cubic meters) has flown from this tube in, for example, 2 seconds? To simplify things, assume the tube is being continuously filled with water as soon it drops its level, so assume the pressure is constant at the nozzle (bottom of the pipe).

I know I will probably have to use water viscosity and friction (from water to the tube) in the formula, but I cant come up with a formula to that.

• The pressure at the bottom is not $\rho g L$. This only applies if the fluid is at static equilibrium. Feb 12, 2021 at 14:41

A force balance on the water (if the flow is steady) is given by $$W=F$$where W is the weight of the water and F is the wall friction force. This reduces to $$\frac{\pi D^2L}{4}\rho g =\tau_w \pi D L$$or$$\tau_w=\frac{\rho g D}{4}$$where $$\tau_w$$ is the shear stress at the wall. The wall shear stress is related to the the fluid velocity by means of the Fanning friction factor f: $$\tau_w=\frac{1}{2}\rho v^2f$$The Fanning friction factor is related to the fluid flow Reynolds number and the pipe surface roughness factor $$\epsilon/D$$ by means of the Fanning friction correlation (available in the literature) $$f=f\left(Re,\frac{\epsilon}{D}\right)$$where $$Re=\frac{\rho v D}{\mu}$$with $$\mu$$ representing the fluid viscosity.