Bernoulli Equation and Leakage in a Pipe

[Zahid Iftikhar]

Hi boneh3ad
Your discussion on Bernoulli Equation was very impressive and helped me a lot to understand this rather complicated equation. I have a question which puzzles me a lot when I want to solve it using Bernoulli equation.
Here is the statement.
" If there is some fluid flowing thru a pipe and it develops a leak, the fluid starts oozing out of it. If this pipe is in some ditch and the fluid keeps collecting in the ditch and finally covers the whole pipe upto some height, then according to Bernoulli equation, there should not be any further leakage. The velocity being higher inside the pipe will cause low pressure inside than outside and the fluid will leak back to the pipe."
But this does not happen. I am sure I am mistaken somewhere. Please guide on this.
Regards
Zahid

 

[boneh3ad]

So my first question here would be why do you think this doesn't happen?

 

[russ_watters]

H If this pipe is in some ditch and the fluid keeps collecting in the ditch and finally covers the whole pipe upto some height, then according to Bernoulli equation, there should not be any further leakage. The velocity being higher inside the pipe will cause low pressure inside than outside and the fluid will leak back to the pipe."

You are assuming the answer to your question (pressure outside > pressure inside) for no apparent reason, but it isn't true: the pressure inside the pipe is not lower than the pressure outside. If it was, you'd never get water to your house! The pipe is under pressure because you need pressure to both move the water and to overcome the friction inside the pipe from the moving water. So it will always need to be higher than the pressure outside the pipe.

 

[anlon]

In any case where the pressure is lower inside the pipe than outside the pipe, fluid will flow back into the pipe. The reason this does not happen often in real life (i.e. in the case of a burst oil pipeline or water line) is because fluid inside pipes is often pressurized, so that even though its pressure is lower when it is flowing than when it is still, the pressure inside the pipe is greater than the pressure outside the pipe. Also, fluid leaking from a pipe often has somewhere else it can go to avoid building up pressure around the pipe; it can flow into the ground around it or can find some path away from the pipe. In a situation where fluid did build up around a pipe, it would reach a height where the pressure at the level of the leak was equal to the pressure inside the pipe at the leak. The system would then be in equilibrium, and no more fluid would leak out of the pipe.
For a more rigorous mathematical treatment of the problem, consider this. Bernoulli's equation is $$p + \frac{1}{2}\rho V^2 + \rho g h = constant$$
Now, consider the two primary regions in this problem: the fluid inside the pipe, and the fluid outside the pipe. Inside the pipe, the fluid has a velocity $V$. Let us say, for the sake of simplicity, that $\rho = constant$ and at the leak $h = 0$. Then inside the pipe, we have $$p + \frac{1}{2}\rho V^2 = const.$$
Outside the pipe, $V = 0$. Since we are dealing with a continuous fluid, we may equate Bernoulli's equation from both regions: $$p + \frac{1}{2}\rho V^2 = p + \rho g h$$
At equilibrium, no fluid can pass into or out of the pipe, so the two static pressures $p$ must be equal. Subtracting $p$ from both sides of the equation, $$\frac{1}{2} \rho V^2 = \rho g h$$
Dividing by $\rho$, $$\frac{1}{2} V^2 = g h$$
So the height to which the fluid rises depends only on the velocity of the fluid inside the pipe.

Now, if the fluid is accelerated to a higher velocity, static pressure will initially drop inside the pipe, and the condition $p_1 = p_2$ will no longer be true (where $p_1$ is the static pressure inside the pipe and $p_2$ is the pressure outside the pipe). In this case, fluid will enter the pipe until Bernoulli's equation is again satisfied (specifically until $p_1 = p_2$ again). If the fluid is decelerated to a lower velocity, then static pressure will increase inside the pipe ($p_1 > p_2$) and fluid will again leak out of the pipe until equilibrium is reached.

 

[boneh3ad]

You are assuming the answer to your question (pressure outside > pressure inside) for no apparent reason, but it isn't true: the pressure inside the pipe is not lower than the pressure outside. If it was, you'd never get water to your house! The pipe is under pressure because you need pressure to both move the water and to overcome the friction inside the pipe from the moving water. So it will always need to be higher than the pressure outside the pipe.

I think you missed the part where he said the leak causes the water to pool until its height above the pipe makes its hydrostatic pressure greater than that in the pipe.

 

[Zahid Iftikhar]

You are assuming the answer to your question (pressure outside > pressure inside) for no apparent reason, but it isn't true: the pressure inside the pipe is not lower than the pressure outside. If it was, you'd never get water to your house! The pipe is under pressure because you need pressure to both move the water and to overcome the friction inside the pipe from the moving water. So it will always need to be higher than the pressure outside the pipe.

But the same is happening in a bunsen burner. When the air rushes thru the cylinder covering the jet, the air from outside gets into the cylinder and mixes with the gas. Same also happens in a carburetor.

 

[Zahid Iftikhar]

So my first question here would be why do you think this doesn't happen?

I mean I observe in real life it does not happen. But according to the law, it should happen.

 

[Zahid Iftikhar]

You are assuming the answer to your question (pressure outside > pressure inside) for no apparent reason, but it isn't true: the pressure inside the pipe is not lower than the pressure outside. If it was, you'd never get water to your house! The pipe is under pressure because you need pressure to both move the water and to overcome the friction inside the pipe from the moving water. So it will always need to be higher than the pressure outside the pipe.

But every fluid moves under some pressure difference. Then what is the Bernoulli equation about? Under what circumstances is it applicable? Same thing happens in a Bunsen burner where gas is pushed by the jet, but it creates low pressure inside and as a result air enters thru the vent into the cylinder and mixes with the gas.

 

[boneh3ad]

I mean I observe in real life it does not happen. But according to the law, it should happen.

And I am wondering where you have observed it not happening. I would propose that any situation where this is not happening therefore doesn't represent the situation you've outlined since what you've ruined is correct. However, note that really the pipe would leak and fill up its surroundings until the hydrostatic pressure outside matched the pressure inside, and at that point it would just stop leaking. It wouldn't typical overshoot. If it did overshoot, water would leak back into the pipe.

 

[Zahid Iftikhar]

And I am wondering where you have observed it not happening. I would propose that any situation where this is not happening therefore doesn't represent the situation you've outlined since what you've ruined is correct. However, note that really the pipe would leak and fill up its surroundings until the hydrostatic pressure outside matched the pressure inside, and at that point it would just stop leaking. It wouldn't typical overshoot. If it did overshoot, water would leak back into the pipe.

Well, what you say is in my observation. I don't mean that situation. I am referring to the time when leaked fluid just covers the pipe and same fluid is present outside and inside the pipe. It is the situation before the equilibrium point.

 

[russ_watters]

But every fluid moves under some pressure difference.

Right.

Then what is the Bernoulli equation about? Under what circumstances is it applicable?

Bernoulli's principle is applicable along a streamline. It only applies to points along a flow that are connected (an element of the flow will travel from one place to the other you are comparing).

To say it another way, pressure in faster moving fluid is lower than the same fluid, in the same system, previously moving slower. You can't ignore the rest of the dynamics of the system when analyzing it!

Another key assumption typically in Bernoulli's equation/principle is lack of friction, which creates losses and requires higher pressures in real-world systems. You can add friction to Bernoulli's equation as a separate term, but that is a little deeper than a lot of people get.

 

[russ_watters]

I think you missed the part where he said the leak causes the water to pool until its height above the pipe makes its hydrostatic pressure greater than that in the pipe.

No, I didn't miss that, I contradicted it: it is an assumption and it is almost certainly false in most real-world situations. The OP is talking about pseudo-real-world situations, and that scenario is usually not possible. A domestic water pipe would need something on the order of 150 feet of water pooled above it to equalize the pressure!

 

[russ_watters]

But the same is happening in a bunsen burner. When the air rushes thru the cylinder covering the jet, the air from outside gets into the cylinder and mixes with the gas. Same also happens in a carburetor.

The bunsen burner pipe is open, so the gas is no longer pressurized: it is at atmospheric pressure and moving, causing entrainment.

There is actually a style of water pump called an eductor that works on this principle, so it is possible to make it happen, you just have to get the pressure in the pipe down to enable it. A normal domestic water pipe though is under too much pressure to enable this to happen.

Either way, the key is knowing the pressure inside the pipe and outside the pipe. You can't just assume that because the water is moving the pressure will be "low".