The easiest and most accurate formula to calculate pipe flow is the Churchill equation. It is a noniterative solution, whereas the other typical formula, Colebrook-White, requires iteration. You can find both of them here,
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"Pumping accounts for 20% of the world’s energy used by electric motors and 25% to 50% of the total electrical energy usage in certain industrial facilities." - DOE statistic (Note: Make that 99.99% for pipeline companies)
BigInch, you're right, it most likely isn't laminar flow, but the "Mr.Teacher" style response goes into some flow basics that probably would help anyone asking a question of this nature.
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"Pumping accounts for 20% of the world’s energy used by electric motors and 25% to 50% of the total electrical energy usage in certain industrial facilities." - DOE statistic (Note: Make that 99.99% for pipeline companies)
The method for calculating the flow rate of liquid through a pipe from the pressure drop across it requires an iterative procedure. It is easy, but tedious. You could use the web site referred by BigInch and find the answer by guessing flowrates until you get the known pressure drop. I will give you the steps here, but I suggest that once you get an answer you visit the site recommended by BigInch to validate your answer.
The first formula you need is known as Darcy-Weisbach. I have re-arranged it here from the usual form to give you the flowrate.
Q = 3.51 x 105 x [√](( [Δ]P x D5 ) / ( [ƒ] x L x [ρ] )) {Eq 1}
where
Q is flowrate in liter per second
[Δ]P is the pressure drop over the pipe in bar (= 3 bar)
D is the pipe inside diameter in meter (= 0.1023 m)
[ƒ] is the Moody friction factor (see below)
L is the pipe length in meters (not given)
[ρ] is the fluid density in kg/m3 (= 1000 kg/m3 for water)
All of the terms on the right hand side of this equation are known, except for the Moody friction factor [ƒ]. In order to calculate this you kneed to know the Reynolds number, and strictly you need to know the pipe roughness as well.
The Reynolds number is calculated (in volumetric rather than velocity terms) as
Re = 1.27 x [ρ] x Q / ( [μ] x D ) {Eq 2}
where [μ] is the viscosity in centipoise (= 1 cP for water)
The Moody friction factor can be estimated for clean commercial pipe using the Drew, Koo and McAdams equation. This is not as accurate as the Churchill equation, but has the advantage that it does not require the pipe roughness.
[ƒ] = 0.0123 + 0.754 / ( Re0.38 ) {Eq 3}
Now we have all the formulae and we can outline the solution procedure
Step 1 - guess a Moody friction factor, say 0.015
Step 2 - Calculate Q using Eq 1
Step 3 - Calculate Reynolds number using Eq 2. If Re < 4000 then flow is not turbulent and revert to method for laminar flow given by danw2
Step 4 - Calculate Moody friction factor using Eq 3
Step 5 - If calculated friction factor is different from previous value then apply new value in Step 2 and repeat until Q and [ƒ] stop changing within the required accuracy.
Katmar, Why not Churchill or Chen (to avoid having to do the iteration for f)?
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"Pumping accounts for 20% of the world’s energy used by electric motors and 25% to 50% of the total electrical energy usage in certain industrial facilities." - DOE statistic (Note: Make that 99.99% for pipeline companies)
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"Pumping accounts for 20% of the world’s energy used by electric motors and 25% to 50% of the total electrical energy usage in certain industrial facilities." - DOE statistic (Note: Make that 99.99% for pipeline companies)
Churchill, Chen and indeed the Drew, Koo and McAdams method depend on the Reynolds number. If you know the flowrate, pipe ID and fluid properties (and therefor the Reynolds number) then any of these methods will give you an answer for the friction factor and pressure drop without any iteration.
But navas72's problem is that he knows the pressure drop and not the flowrate so we have no way to calculate the Reynolds number (and therefor the friction factor) except by trial and error. So I am really iterating on the flowrate rather than on the friction factor, but the friction factor changes every time the flowrate changes so we can use either as the measure of convergence.
Verma (1979), and Gulyani and Agarwal (2000) have published explicit methods that do avoid the iteration but they are so complicated that I prefer to iterate. For most of us these calcs are done by a program that does the iteration for us and we take no notice of it. The method I have proposed for navas72 is really only valid for a "one off" calc or for academic interest.
Churchill isn't complicated. Nice and easy, even for spreadsheet work.
Yes I understand the OPs problem. No method or program will work without the appropriate boundary conditions. I figured rather than trying to draw that out for him, we just give him the typical tables and a spreadsheet and let him worry about assuming a P2 somewhere between 0 and 3 barg.
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"Pumping accounts for 20% of the world’s energy used by electric motors and 25% to 50% of the total electrical energy usage in certain industrial facilities." - DOE statistic (Note: Make that 99.99% for pipeline companies)
BigInch, FYI: Churchill is certainly acceptable, but there's an even more accurate method just published by Goudar in Hydrocarbon Processing Aug. 2008. A little more involved than Churchill, but still spreadsheetable.
Sounds interesting. I'll have to evaluate that increase in accuracy for that extra complication though.
Thanks.
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"Pumping accounts for 20% of the world’s energy used by electric motors and 25% to 50% of the total electrical energy usage in certain industrial facilities." - DOE statistic (Note: Make that 99.99% for pipeline companies)
I will skip the calculations and maybe point you a little different direction to research if all else fails. Of course, I don't know your specific application so I don't know if this will help.
Flow can be calculated when flow is through a known size orfice (assume pipe ID for rough number), venting to atmosphere and feet of head at a point before the orfice is known. That said....
I use commercial well drillers that use a "cheat sheet" for calculating flow rates during well/pump performance tests. Not sure if the sheet comes from Orfice manufacturers or "Well Engineers". If you know any, they might be able help or google it (lol) if you would prefer to avoid math calculations.
etech08,
Flow from well head across an orifice is nothing similar to pipe flow capacity. Run it through a 1/2" pipe and you'll see that there's quite a difference.
Look above for a diameter - flow capacity chart link.
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"Pumping accounts for 20% of the world’s energy used by electric motors and 25-50% of the total electrical energy usage in certain industrial facilities."-DOE statistic (Note: Make that 99% for pipeline companies)
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