Fluid dynamics for dummies (water flow in pipes)

All the Google searching in the world isn’t helping me, because I don’t have time to study three semesters of fluid dynamics to make sense of everything that I’m finding in this query… so, can some kind Doper help me out with a “good enough,” “back of the envelope” type of way to get a rough calculation of the following?:

I’d like to find the relationship between differential psi, pipe/hose diameter, and fluid volume. Specifically, in a closed system that exhibits a fluid pressure differential of x (i.e., say I have z psi of static incoming pressure and y psi of back pressure [outgoing pressure], and x is the net pressure), a diameter of d, and the fluid in question is plain water at room temperature, and flow is “turbulent” (according to everything I’ve found out). Let’s assume that friction is negligible. Gravity has no influence. It’s water so density and viscosity are whatever they are.

So… is there any way to roughly say that at 40 psi net with a 1/2" pipe I should normally expect some type of water flow rate?

In exchange, I can offer any number of electrical or energy calculations…

Well, neglecting viscosity effects, what you are looking for is Bernoulli’s Equation. However, Bernoulli assumes (and this is typically true for water flowing through a pipe) that the flow is laminar, not turbulent. Turbulent flows get into a good deal more complexity and, in general, have to be dealt with by the application of emperical models rather than analytical. (It is possible to simulate turbulent effects using Navier-Stokes formulations, and this is commonly done with computational fluid dynamic analysis, but the model is highly sensitive to initial and boundary conditions and therefore needs to be “tweaked” to reproduce the correct effect. This is non-trivial and for most engineering applications not relating to the aerospace industry empirical data is used, though CFD applications are becoming more widespread if somewhat questionable in accuracy.)

In the real world you do have some amount of viscosity effects; for water at normal flow rates the internal viscosity can be ignored; for large diameter tubes at nominal flow rates and pressures boundary effects have a small effect that can often be ignored in areas of fully developed flow, provided that the pipe walls are sufficiently smooth. The controlling metric here is what is called the Reynolds Number, which tells you whether the flow in laminar, transitional, or turbulent. Theoretically you can calculate the number (at least for fully developed flows) by knowing the fluid viscosity properties, flow velocity, density, and “surface roughness”; in reality there are a number of conditions which can perturb the flow and affect the resulting Reynolds characteristic, so it is generally determined by application of emperical models and data. From Fundamentals of Fluid Mechanics, 2nd Ed (Munson, Young, Okiishi):

Again, for normal residential water pressure in a pipe (which I assume is your application) and ignoring viscosity, Bernoulli’s Equation should get you a reasonable answer. (Since your inlet and outlet are at the same height , ρgh just cancels out.)

Stranger

Absolutely I can state, not roughly, but with 100% confidence that you should normally expect some type of water flow rate!
How much is another question.
WAG Say about a big PPot full.

Rough guess maybe 20 gallons.

I’m not sure if this is what you are looking for, but it sounds as if a Hazen-Williams calculation might help.

Doesn’t Bernoulli’s equation assume inviscid flow (no boundary layers, not laminar or turbulent)?

Well, formally, yes. In reality, there are, except for certain low temperature superfluids, no inviscid fluids and there are always boundary layers and undeveloped flows. Bernoulli’s Equation gives accurate numbers when considering a fluid with very small viscosity and moving slow enough such that shear effects aren’t significant. The point I intended to make, as Balthisar mentioned turbulent flow, is that Bernoulli doesn’t give good results when considering turbulent or transitional flows, but it is a suitable approximation for many laminar flow conditions.

But you are correct. Bernoulli idealizes the fluid as having no shearing or variation of speed across the section whatsoever. Mea cupla.

Stranger

I don’t have time to go into detail, but what I meant to point out is that Bernoulli works well for some turbulent flows too. Because the boundary layers are much thinner in turbulent flow, it is more like inviscid plug flow than many laminar flows are. While the viscous effects are certainly higher in turbulent flows, you may get good estimates from Bernoulli’s equation for both regimes. As always with equations based on numerous assumptions, use cautiously.

How long is the pipe?

The flow isn’t necessarily turbulent. If you are calculating a Reynolds number and seeing that it is higher than - I forget, 1500 or something - well, higher than some value, implying that the flow is turbulent - what you need to know is that this calculation determines whether turbulence will develop given infinite length. Or given enough length, better to say.

But depending what the entrance to the pipe looks like, the streamlines entering it are typically not swirling around one another and the flow is laminar, even if the Reynolds number is quite high. Watch cigarette smoke entering a vacuum cleaner hose if you don’t believe it (but don’t suck the lit cigarette into the vacuum cleaner bag or you will be posting queries about fire insurance and rebuilding). It can take quite a few pipe diameters in the downstream direction for turbulence to develop.

Also, the entrance to the pipe may matter quite a lot (more so if the pipe is shorter). If it looks like the bell of a trumpet, the pipe will deliver more flow at a given pressure (relative to if it is just a squared off entrance, or worse an entrance through a fitting with an ID restriction).

You can’t neglect friction. In the system you describe friction is a primary factor. What you said was, given voltage of 110 V, and neglecting resistance, how much current will flow through an 8 gauge wire.

What will happen is that when you open the valve, the water flow will increase until the friction forces equal the pressure differential. I left my Cameron’s Handbook at the office, but I can tell you that the frictional forces in a 1/2" pipe are not negligible. Water pressure is usually measured in feet, and 40 psi is equivalent to just over 92 feet.

In a one inch pipe, at a flow rate 20 gpm, the head loss is 25.2 feet/100 feet of pipe, assuming carbon steel schedule 40 pipe. That 100 feet is equivalent length. You have to add in pressure drops across valves and fittings, elbows and whatnot, as equivalent feet of pipe. For example, an 90 degree elbow might be equivalent to 25 feet of straight pipe(that’s a guess, don’t quote me on that, and it’s different for every diameter pipe).

Since in a 1/2" pipe you more than double the velocity, the friction loss will increase by more than 4 times, being roughly equivalent to the square of the velocity.

Thanks everyone for their help. I never realized that this was Ph.D. material. Both of the above links to calculations worked “well enough,” meaning that my actual measured results were within an order of magnitude of the calculated expectation. I think that in the future, experimental data will definitely be the way to go! This is actually for a project at work, and our expert that would know all of this stuff intuitively retired a few weeks ago, and I doubt he’ll be replaced. Which means I have to attempt to wade through it. And boy, was I naïve as I came into this!

Back when I was a student learning electricity, the instructors always tried to exemplify the details by comparing it with hydraulic circuits, so I’d always assumed that there were very, very strong parallels. Hah! Give me parallel matrix calculations any day.