We have a cylindrical tank of known dimensions. Attached at the bottom is a pipe perpendicular to the tank of known dimensions. We fill the tank with a liquid and let it drain out through the pipe. What formula will predict the volume of the tank at any time t? I believe the Hagen–Poiseuille equation (involves the viscosity of the liquid) and Pascal's law (involves the density of the liquid) are involved, but i cannot derive an equation for the volume at any time. All working and any explanation where ever necessary would be appreciated.(adsbygoogle = window.adsbygoogle || []).push({});

Heres my attempt at it:

ΔP = 8μLQ / πr^4

ΔP is the pressure drop

L is the length of pipe

μ is the dynamic viscosity

Q is the volumetric flow rate

r is the radius

ΔP = ρgΔh

g is acceleration due to gravity

ρ is the fluid density

Δh is the height of fluid

ΔP = 8μLQ / πr^4

ΔP = ρgΔh

Volume of tank = πr^2h

8μLQ / πr^4 = ρgΔh

Q = ρgΔh ✕ πr^4 / 8μL

This is essentially equal to Q = k Δh where k is = ρg ✕ πr^4 / 8μL

Im fairly sure is is redundant however as of course the rate of flow is proportional to the change in height of water as when the Δh is multiplied by the area of the circular face of the tank it will give the volume of the cylinder of water which has left through the pipe during the Δh. There has to be an equation for the relationship im looking for as the results of the test will be consistent.

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# How to predict the volume of a tank of water at any time? (fluid motion)

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