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We are taught that the speed of the fluid through a hole in a water filled tank obeys Torricelli law: $v=\sqrt{2gh}$, thus the speed is independent of the size of the hole. Why is it then that making the hole of the end of a hose smaller (for instance, by using my finger) the speed increases? (note, I was told that this is because the situation is different, and what I have is constant pressure from the street, but this is not the case in my house where we have a tank on the roof)

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Torricelli's law is just a restatement of the conservation of energy of a non-viscous, non-turbulent and incompressible liquid flow. Thus, the maximum speed that can be obtained by making water flow through a hose (purely by the force of gravity, ie. a tank on the roof) is the formula that is given.

The water flows slower when it moves through a hose with a larger diameter because of the turbulence of the flow, reducing it's speed from the ideal limit.

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If you severely reduce the cross-section of the hose by putting your finger into it, you increase the pressure drop across the restriction and thus the flow rate decreases (you can block flow altogether too, of course, if your finger tightly fits or covers the hose's open end, or by 'kinking' the hose).

This causes the volumetric flow rate $\dot{Q}$ ($\mathrm{m^3/s}$) to be reduced but it also needs to flow through a smaller cross-section. Roughly we can calculate $\dot{Q}$ as follows:

$$\dot{Q}=vA,$$

or:

$$v=\frac{\dot{Q}}{A}.$$

with $v$ ($\mathrm{m/s}$) the flow speed and $A$ ($\mathrm{m^2}$) the cross-section.

So to maintain the same volumetric flow rate, at smaller $A$, $v$ needs to increase. Thus, seemingly paradoxically perhaps, you can reach high $v$ at low $\dot{Q}$.

When you're blocking the end of the tube gradually more and more, flow speed $v$ will first increase, until you've blocked the opening completely, so then $\dot{Q}=0$ and thus also $v=0$.

For 'reasonable' values of $A$ Torricelli's Law is respected.

The above also holds true for water drawn from the mains (constant pressure).

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Torricelli's formula can be derived for water jet running out from a tank, where water does not move much and obeys laws of hydrostatics.

Inside a hose, all the water moves roughly with the same velocity and the Toricelli's formula does not apply, because moving water experiences considerable friction as it moves along the hose and behaves in a more complicated way that make laws of hydrostatics inapplicable. The pressure in a water element decreases as it moves along the hose.

When you restrict the opening of the hose, more friction happens at the end, which decreases water throughput and therefore also speed of water inside the hose. The friction then is not as strong and the conditions are closer to hydrostatic case. The speed of water at the end gets closer to the maximum possible value, given by Torricelli's formula.

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