Simple flow through a pipe question. 2

[ziptron]

Hey Everyone,

I have what I think should be a relatively simple problem, unfortunately I cannot find a good formula to help me do the calculations. Each formula I find does not resonates well in my head because I can't seem to grasp the basic principals..any input would be greatly appreciated!

Lets say I have a 1/2 in. copper tube of some length X connected to a large water main which has essentially constant pressure (50 PSI) and infinite water flow capacity.

If this copper pipe was capped at its end so that it had no flow through it, I know I could stick a gauge at every inch of this copper tube length and measure 50 PSI, that part is clear in my head. However, if I was to remove the cap and let water flow, what formula could I use to calculate the pressure within the copper tube at various lengths say 0.25X, 0.5X, 0.75X and 0.9999X (right before exit). I'd love to know the formula for this so that it can help me visual what is actually happening inside that pipe and at it's exit.

The above theory is just so that I can learn how to calculate how much water (GPM) will actually come out of the copper tube assuming its just open without the cap given that the source of the water is 50 PSI constantly.

Help with any of the above would be of great help.

Thanks in advance!

 

[SNORGY]

In my mind you don't need anything other than Bernoulli, adjusted for the fact that you will have 1.5 velocity head losses in addition to the friction head loss. The pressure at the entrance is 50 psig, the pressure at the exit is 0 psig, and your flow will be fully developed turbulent flow.

Unless I'm missing something...

 

[desertfox]

hi
Look at this site:-

http://www.princeton.edu/~asmits/Bicycle_web/Bernoulli.html

 

[katmar]

The equation you want is called the Darcy-Weisbach formula. You can read about it in Wikipedia or many other web references or fluids text books. If you can get your hands on a Crane TP410 handbook that would be a good start. The basic form of the equation as shown in Wikipedia (in pressure terms) can be modified as mentioned by SNORGY above to include the resistance coefficients (K-values) for the entrance and exit losses. The exit loss is the kinetic energy which is in the exiting stream, expressed in pressure terms.

The D-W formula, including the resistance coefficients is

ΔP = ( (ƒ_D x L/D) + ΣK) x (ρ.V²/2)

Unfortunately there are 2 unknowns - ƒ and V. But ƒ is only a function of the pipe and fluid properies and the velocity so it is not really independent. You guess an ƒ, solve for V and then test that V to see if it gives the correct ƒ. If not, correct ƒ and re-solve for V. Repeat until ƒ doesn't change any more.

The tricky part is getting the units right. There are plenty of online calculators and downloadable spreadsheets and calculators that will do this for you.

Katmar Software - AioFlo Pipe Hydraulics

http://katmarsoftware.com

"An undefined problem has an infinite number of solutions"

 

[zdas04]

Star for Katmar, not so much for Snorgy. Bernoulli cannot be used if "any work is done on or by the fluid" and friction is irreversible work done on the fluid.

David Simpson, PE
MuleShoe Engineering

Law is the common force organized to act as an obstacle of injustice Frédéric Bastiat

 

[SNORGY]

True enough, zdas04. My bad.

I tend to lump "Bernoulli" and "Darcy" together...which, as you state, is technically wrong. The formula provided by Katmar is indeed what I had in mind, but I messed up.

Too long on the tractor yesterday.

 

[zdas04]

My wife and kids always say I'm terribly pedantic. I just run into so many wrong decisions being made based on trying to calculate real flows down long pipes with Bernoulli.

David Simpson, PE
MuleShoe Engineering

Law is the common force organized to act as an obstacle of injustice Frédéric Bastiat

 

[BigInch]

Listen to your wife.
Use the "modified Bernoulli equation" that incorporates f_L and it works even better than D'arcy alone. Facts are that you can't ignore either one at the expense of the other and understand what the heck is going on in anything except a horizontal pipe with flow, or a pipe at any orientation but with no flow at all.

I hate Windowz 8!!!!

 

[ziptron]

Thanks everyone, I definitely think the Darcy-Weisbach formula helps with the first part of my problem, however can I also use it for the second part where I wish to calculate the flow out of the open ended pipe?

I see I can use the Darcy-Weisbach formula to calculate pressure loss per length of pipe, but that assumes I know velocity of the liquid going through the pipe right? How do I get this velocity?

I figure also that once I know velocity, I can calculate the flow rate..

 

[ziptron]

Right.. so my total pressure drop is Gauge to Atmospheric..

Seems like I can also just use Hazen-Williams equation right and where it requires pressure drop, I just substitute gauge - atmospheric. Am I correct with that?

http://en.wikipedia.org/wiki/Hazen–Williams_equation

 

[katmar]

I didn't state it, but you have to re-arrange the D-W formula to be able to solve for V as a function of pressure drop (you stated that you knew the total pressure drop). But it has to solved iteratively because the friction factor also changes with V. With each iteration you would have to find a new friction factor from the Moody Chart (or with a friction factor equation like Churchill's).

Hazen Williams is not as accurate as D-W. H-W is a simplified solution where the roughness coefficient is not a function of Reynolds number (i.e. unaffected by velocity). This is not true in reality, but H-W is useful in water reticulation systems where the Reynolds numbers fall in a relatively narrow band.

You could use Hazen Williams as a first estimate to start the trial and error solution with D-W, or if your need for accuracy is not too great you could accept the HW solution. In your case, a major shortcoming of H-W is that it does not have a good way to deal with the "minor losses" due to the pipe fittings - the inlet and outlet in this instance. Because you have a short piece of pipe the minor losses are likely to be significant and whether or not you include them depends only on how accurate you want to be. As mentioned by SNORGY earlier, your ΣK will be 1.5 (0.5 for the inlet and 1.0 for the outlet).

Katmar Software - AioFlo Pipe Hydraulics

http://katmarsoftware.com

"An undefined problem has an infinite number of solutions"

 

[BigInch]

If you simply start with a first guess for the friction factor of 0.02 it will usually converge within 3 or 4 iterations.

I hate Windowz 8!!!!

 

[zdas04]

That is not a bad approach BigInch. I find it to work much better than what I learned in school (i.e., assume a Reynolds Number of 1 million and look up [or calculate] a friction factor with actual ε/D), but my guess for Fanning Friction factor is 0.02, for Moody I use 0.08.

David Simpson, PE
MuleShoe Engineering

Law is the common force organized to act as an obstacle of injustice Frédéric Bastiat