Mass Continuity and Torricelli's Law

[UMath1]

Based on the law of mass continuity, when a pipe narrows then the speed of the fluid increases. Then why is it that when draining a tank the speed of the fluid only depends on the height of water above and not on the size of the hole? Wouldn't a narrower hole mean that that the speed must be greater to provide the same flow rate?

 

[Orodruin]

The key point is that when the pipe narrows the speed increases relative to where the pipe is wider in that particular pipe. It tells you nothing about the actual flow rate compared to if the entire pipe was narrower.

 

[UMath1]

Yes but the case of the tank drainage, the hole's area is smaller than the cross sectional area of the tank. As per the mass continuity principle, the ratio of these two areas should influence the relative speed of the water exiting the hole. However, in all the example problems I have seen, the areas seem to have no bearing on the velocity of the exiting fluid.

 

[russ_watters]

A change in cross sectional area does result in different velocities in different parts of a system. But that has little to do with the exit velocity, which is based on height (hydraulic head).

 

[nasu]

The speed depends on the area. The formula that gives a speed independent of area (Toricelli's law) is just the first approximation.
It relies on the fact that the ratio between the area of the hole and the area of the big tank is usually very small. The elementary derivation of the formula assumes that the top surface of the water has zero speed.