Bernoulli's Equation & Equation of Continuity

[Alexander83]

Hi all,
I'm considering an fluids example that's giving me an apparent contradiction when I consider it from the perspective of Bernoulli's Equation vs. the Equation of Continuity.

What I'm thinking of is the common observation that putting one's thumb over a garden hose results in an increase in the velocity of the water leaving the hose. I've always understood this in terms of the Equation of Continuity under the assumption that the volume flow rate (R) is a constant.

R = v*A

Where v is the speed of the flow in the hose and A is its cross-sectional area. From this approach, the speed of water leaving the hose would increase as A is reduced by putting a thumb over the outlet of the hose.

Where I'm getting confused is in trying to apply Bernoulli's equation to explain what's going on. I'm imagining a very conceptually simple case where the water in the hose is originating from a large storage tank that is open to the atmosphere at the top. In this case, Bernouilli's equation could be used to work out the speed of the water at the hose outlet and it essentially boils down to Torricelli's theorem here and one obtains:

v = (2gh)^0.5

where h is the depth of the water in the tank feeding the hose. This equation indicates that the flow speed through the end of the hose should be a constant and independent of the diameter of the hose.

The two results clearly contradict one another, so there's clearly something I'm missing here... I suspect I'm missing something in trying to apply Bernoulli's theorem to this situation, but I can't see what it is that I'm missing.

Any help would be gratefully appreciated!

Cheers,

Alex

 

[BvU]

v = (2gh)^0.5

when leaving the hose would mean the water spouts up to a height equal to the level h in your water tower ! In reality the transition from tank to hose and the hose itself form resistances, so you don't reach that.

When you reduce the nozzle size you get a bigger delta P for the Bernoulli equation, so a higher speed.

 

[russ_watters]

The primary issue here is that Bernoulli's Principle applies along a streamline and the two points you are comparing aren't even part of the same system. So you cannot apply it in the way you are trying to apply it.

The secondary issue, as described in the previous post, is that for water coming out of an open garden hose the frictional losses are near 100%.

 

[boneh3ad]

I guess I will point out that, in addition to what has been mentioned here, Torricelli's law and the garden hose example come with a different set of assumptions. For example, the garden hose example assumes a constant mass flow rate regardless of how much of the hose you cover with your finger. This turns out to be a pretty good assumption that gets worse once you cover up about 2/3 of the exit (see this thread to see where I went through the calculations on this problem; the calculation starts in post #25, though you can certainly browse the discussion in the whole thread if you'd like). Torricelli's law, on the other hand, assumes that all of the hydrostatic pressure in a tank is converted to dynamic pressure at the exit and allows the mass flow rate to float freely to accommodate the exit velocity (the accuracy of this assumption depends on your system).

As mentioned before, the addition of a hose to the system, for example, introduces significant losses and so not all of the static pressure in the system is converted into dynamic pressure. In that case, you have to take into account the losses in the system given a constant supply pressure, and according to the post I linked up there, it turns out that the constant mass-flow assumption is a very good one for most cases. In the case of Torricelli's law, the fact that it assumes a simple hole in the side of a bucket (or other storage vessel) means that the losses in the flow are very close to zero and the assumptions made in that situation are valid, namely that all of the static pressure is converted to dynamic pressure and the flow rate can vary.

 

[Alexander83]

Thank you all for your responses... I think I need to do some additional reading to better understand the assumptions made in the derivation of each of these equations. Is there a particularly good introductory fluid dynamics text that you could recommend? I'm looking for something that's not too high level. I've been doing most of my reading on this topic from first-year level Physics textbooks, so something accessible with that as a starting point would be great. I'm an engineering student and would also be interested in a text that's got some good applied examples.

Thanks again for all your help! Much appreciated.

Alex