Pressure flow as a function of area

[DocZaius]

Homework Statement

Please note that I am making this problem up and it is not "assigned" to me:

A large tank of water has two small holes at the same height, near the bottom. Hole A's area is twice hole B's area. Is the volumetric flow rate (volume per time) of water out of hole A bigger than B? If so, by how much?

* The height (and width) of the tank is much greater than the radii of the holes.
* You may ignore the pressure difference between the top and bottom of each hole.

Homework Equations

I am not sure! Hagen-Poiseuille seems to be about volumetric flow along a pipe with a pressure difference across it. This problem does not seem like it could be approximated as such.

Hagen-Poiseuille equation:

$\Delta P = \frac{8 \mu L Q}{\pi r^{4}}$

Bernoulli's principle seems more appropriate, but it seems to imply that the velocity is constant between each hole and that twice the area might mean twice the flow? Unsure about this, but it feels too simple.

Bernoulli's principle:

$\frac{v^{2}}{2} + g z +\frac{p}{\rho} = constant$

The Attempt at a Solution

See above. A most likely wrong application of Bernoulli's principle might mean that the velocity of water out each hole is equal and that there is thus twice as much flow out a hole twice the size. The reason this doesn't feel right is that, from experience, a smaller hole seems to lead to a higher velocity. Is the higher velocity enough to make up the reduction in area? I don't know. I don't think so, but I'd like to know.

edit: As a side note, this would be an interesting experiment to do. Fill a large tank of water and create a hole at the bottom. The hard part is making a hole whose area can be accurately measured. Measure the amount of water that comes out of the hole over some time. Then increase the size of the hole and repeat. With enough data points, plot them all with flow rate on the y-axis and area on the x axis. Then fit a y = C1*(x^C2) curve to it to find a scaling factor and power dependence.

 

[rude man]

Use Bernoulli. Experience makes you believe the velocity is greater for the smaller hole? Not mine.

 

[DocZaius]

Use Bernoulli. Experience makes you believe the velocity is greater for the smaller hole? Not mine.

I misspoke when I said experience. My experience involved putting my thumb over the end of a hose. But that particular example would be more like the Hagen-Poiseuille situation.

So just to be clear, you are saying that the velocity of water out of either hole is the same and that the conclusion would be that flow rate would scale linearly with area?

 

[rude man]

I misspoke when I said experience. My experience involved putting my thumb over the end of a hose. But that particular example would be more like the Hagen-Poiseuille situation.

So just to be clear, you are saying that the velocity of water out of either hole is the same and that the conclusion would be that flow rate would scale linearly with area?

Yes, right. Of course, there may be edge effects along the hole perimeters, but to 1st order that is so.

Not sure why when you pinch off a water hose end it gives you a much faster velocity. Whereas if you slowly turn on a faucet you don't seem to get that. Something to do with nozzle dynamics? Non-laminar flow?

As for Hagen-Poiseuille, according to Wikipedia, "The equation fails in the limit of low viscosity". Water falls (pardon pun) into that category I would think.

Maybe we can get our fluid dynamics experts to weigh in on this.

 

[Chestermiller]

Yes, right. Of course, there may be edge effects along the hole perimeters, but to 1st order that is so.

Not sure why when you pinch off a water hose end it gives you a much faster velocity. Whereas if you slowly turn on a faucet you don't seem to get that. Something to do with nozzle dynamics? Non-laminar flow?

As for Hagen-Poiseuille, according to Wikipedia, "The equation fails in the limit of low viscosity". Water falls (pardon pun) into that category I would think.

Maybe we can get our fluid dynamics experts to weigh in on this.

The water pressure at the hose inlet is nearly constant when you open the valve, like a constant voltage source. The added resistance from pinching the hose is negligible compared to the overall resistance of the hose. It's like an electric circuit. So pinching the hose doesn't substantially influence the current (volumetric flow rate). So, because of the smaller cross section, the water velocity is higher. This is the first order picture.

Chet

 

[rude man]

The water pressure at the hose inlet is nearly constant when you open the valve, like a constant voltage source. The added resistance from pinching the hose is negligible compared to the overall resistance of the hose. It's like an electric circuit. So pinching the hose doesn't substantially influence the current (volumetric flow rate). So, because of the smaller cross section, the water velocity is higher. This is the first order picture.

Chet

Hi Chet, thanks for wading in! So this is some kind of Hagen-Poiseuille effect after all? So you're saying that if I attached a faucet at the end of a hose & turned it on slowly I would also get a much greater velocity than if I turned the faucet on full blast? Does a hose afford that much greater resistance to water flow than plumbing? Because if I turn on my bathroom faucet a little bit I don't get that great an increase in water velocity compared to turning it on more - quite the contrary I would have said.
.

 

[Chestermiller]

Hi Chet, thanks for wading in! So this is some kind of Hagen-Poiseuille effect after all? So you're saying that if I attached a faucet at the end of a hose & turned it on slowly I would also get a much greater velocity than if I turned the faucet on full blast?
.

No. The faucet can have an even larger resistance than the entire hose. So when it's mostly shut down, it isn't a small resistance like when you are trying to squeeze off the flow by hand. With the faucet, if you just narrow down the area a little, the velocity will increase. But, once the pressure drop across the faucet starts to be on the same order as the hose, the volumetric flow rate drops. You need to consider both the resistance and the cross sectional area.

Chet

 

[rude man]

No. The faucet can have an even larger resistance than the entire hose. So when it's mostly shut down, it isn't a small resistance like when you are trying to squeeze off the flow by hand. With the faucet, if you just narrow down the area a little, the velocity will increase. But, once the pressure drop across the faucet starts to be on the same order as the hose, the volumetric flow rate drops. You need to consider both the resistance and the cross sectional area.

Chet

OK, will mull it over.
rudy