# Flow rate through a squre tube

AlephZero
Science Advisor
Homework Helper
I can't do anything else and I don't get paid for convince you. If I were in your company it would be different though.
Well, I've never done ANY formal university-level courses in fluid mechanics. But I have spend 20 years working on coupled nonlinear fluid/mechanical/thermal problems (working together with CFD specialists) and I've interviewed a lot of graduate or postgraduate engineers looking for jobs - so I've come to my own conclusions about who is talking the most sense in this thread.

IMO the practical thing to do here is not get into arguments about teminology or who wrote the best textbook, nor is it to start coding up formulas in a spreadsheet or Mathematica.

Why not go and buy some reputable professional software that does the calcs? If your company is so strapped for cash you can't pay a few hundred dollars for software, maybe it's time to start looking for another job.

Or http://www.efunda.com/formulae/fluids/calc_pipe_friction.cfm looks fairly credible to me. What it does is clearly documented, which is usually a good sign.

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minger
Science Advisor
OK, back on topic, I guess I will try to end the fighting. I have an 'exact' solution to the problem, so I WIN!!! I'm surprised no one else has seen this, its out of a book called "Viscous Fluid Flow" by White (2006) in a chapter titled Exact Solutions. He references Berker (1968?) for these exact solutions to combined Couette-Posieulle Flows through non-circular ducts.

I was going to type it out, but decided it would just be easier to scan the pages instead, that way you not only can see the solutions, but there are several other interesting equations on the page.

Enjoy

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In practice, I would treat 1 bar as pressure drop due to friction in the pipe and then calculate velocity from D-W equation. For a start, I use f as 0.02. Once we get the Reynolds number, then we can get the actual flowrate by iteration. I am getting a flowrate of 0.35 lph. I generally don't deal with such lower flow rates (Reynolds number of 124). This is a viscous flow problem and my method may be wrong. However, can we have a feel about the flowrate from first principles also? This way I can have a vanity check of my spreadsheet results for a wider range:-).

Q_Goest,

As a practising engineer, I agree with you fully. Infact, I wrote a lengthy thread about my views on the discussion but an error erased that matter. There is an interesting thread at eng-tips (Crane TP410 fittings) and a subsequent reference to Hooper and Darby (these two engineers modified the equations for k factor by curve fitting the actual pressure drop values). I prepared a pressure drop calculator in excell, using these two methods, that will work including trasition regime (or I think so). PM me your e-mail id if you are interested to have a look into it.

The development of fluid mechanics at academic level may be more advanced and accurate. But if I have responsibility of many other things in a days work apart from doing some pipe size calculations, I will depend upon emperical equations rather. The redundancy we require (for the future) and the control technology available to us will offset the disadvantage of the inaccuracy. There may be inertia in this but things can be managed.

Oops, mistake in entering pipe length. 5.6 lph at NRe 2000.

On a lighter note, I would never call a thing with 1mm x 1mm size as a pipe. At the most I will call it a straw:-)

FredGarvin
Science Advisor
We usually call them tubes, no matter what the wall thickness.

I remember you commenting on that thread idea Quark. I believe you had an issue with your laptop? Practicing engineering vs. academic...the debate rages on.

Q_Goest
Science Advisor
Homework Helper
Gold Member
Seems like one of those threads that never dies.
Aleph said: Why not go and buy some reputable professional software that does the calcs?
Just a few thoughts. I'd agree that professional software is necessary, especially for larger and more complex anaylsis. Our company for example created their own piping software because we do a lot of cryogenic systems and also because we do a lot of off the wall chemical processes. Thermo, heat transfer and two phase flow have to be integrated into the package along with fluid properties. I don't know how many fluid properties there are but I'd guess it's a few hundred.

One problem with these kinds of software is you loose touch with how the analysis is performed. That leads to problems with garbage in = garbage out. Another problem is that folks never learn how to do the basics to begin with. Seems with a lot of these tools, people can turn into machine coders because you don't need to understand how any of the math is done. Another problem is that when modeling other devices such as an eductor for example where there is no readily available software, you are forced to create your own tools, so if you don't know how to do the analysis you're stuck. The last problem I see is that these things are also very time consuming. A problem as simple as this one can be done in a minute on a spread sheet, but when I did it on the full blown pipe software, it took me about 10 minutes because there are so many extra steps. So it's nice to be able to get an answer very quickly and change parameters a few dozen times to find the perfect solution.

Anyway, I'm a strong believer in spread sheets for analysis. I probably have 150 or so, and that's not uncommon. Eng Tips forum has a forum dedicated exclusively to spread sheets used by practicing engineers. You can find it here:
http://www.eng-tips.com/threadminder.cfm?pid=770

Hi Minger,
Thanks for posting your reference. I'm afraid it's not an exact solution to the flow problem, it's only an exact solution to the hydraulic diameter. In this case, it might be a bit more work than some people are interested in to determine Dh to that degree of accuracy though. Have you tried calculating the flow rate for this?

Hi Quark, I'd be interested in seeing how you handle the transition regimen. When Re is between 2000 and about 4000, or in the 'critical zone', I'm not sure there's a definative value that can be used for friction factor. That would be an interesting topic of discussion. Can you post any references you have for that?

Fred said: Practicing engineering vs. academic...the debate rages on.
I think you've touched on one of the problems there. One problem I had when getting out of college and having to do pipe flow, was the lack of preparation in many practical aspects of engineering. College is generally focused on this academic ideal where we learn about such things as viscous shear and how to set up differential equations that almost invariably can't be solved. This as opposed to teaching how to determine pressure drop through elbows, miter bends, valves and the like. I bet the vast majority of students don't even know what Cv for a valve is, let alone how to calculate pressure drop given this flow coefficient.

I wonder if we shouldn't create a thread that went through pipe flow in a bit more detail and explained how engineers in industry go about this. It would be a lot of work, but maybe at the end, it would provide the background in pipe flow from a practical perspective. We might also include a spreadsheet one could download that did all this analysis for them and served as an example for how such things can be done. I'm not saying I'm volunteering to do that, nor volunteering you (Fred) but just thought I'd throw the idea out there and see if there's any interest from the people here.

FredGarvin
Science Advisor
I will be the first to admit that I never heard of Cv until I got to my first job!

Count me in to help with the thread idea. Maybe we can come up with a managebale way to break it down into sections and have multiple inputs from people and then bring it together at the end.

Clausius2
Science Advisor
Gold Member
Hi Minger,
Thanks for posting your reference. I'm afraid it's not an exact solution to the flow problem, it's only an exact solution to the hydraulic diameter. In this case, it might be a bit more work than some people are interested in to determine Dh to that degree of accuracy though. Have you tried calculating the flow rate for this?
Now it's my turn to talk and to tell you that once again you show a narrow knowledge about the Basics. I will leave to use the calculators to the practising engineers. In the meanwhile I'll be talking about physics. Minger is right, the solution he posted is an exact solution for viscous flow, in the sense that the velocity field obtained there represents the exact distribution of velocity you would measure with PIV on each section of the pipe (or tube). It says:

Since eq 3-32 for fully developed duct flow is equivalent to a classic Dirichlet problem, it is not surprising that an enormous numbers of exact solutions are known.
Do you know what is a Dirichlet problem? I will refresh your memory. For sufficiently small Re, the viscous forces are balanced with pressure forces:

$$\nabla P=\mu \nabla ^2 \mathbf{u}$$

The Poseuille flow assumption reduces the problem to the streamwise coordinate:

$$\frac{\partial P}{\partial x}=\mu\nabla^2 u$$ (1)
where u is the streamwise velocity, P the pressure, $$\mu$$ the dynamic viscosity and $$\nabla^2$$ the Laplacian operator. This equation is complemented with a non slip boundary condition:

$$u=0$$ on the wall. (2)

Therefore, (1) and (2) represents a Dirichlet problem, and the solutions you are seeing in Minger's reference are the exact solutions of $$u(x,y) of that problem (a reference which by the way is a respected one as far as fluid dynamic scientists are concerned). There is no approximation of any hidraulic diameter or similar. I think you've touched on one of the problems there. One problem I had when getting out of college and having to do pipe flow, was the lack of preparation in many practical aspects of engineering. College is generally focused on this academic ideal where we learn about such things as viscous shear and how to set up differential equations that almost invariably can't be solved. This as opposed to teaching how to determine pressure drop through elbows, miter bends, valves and the like. I bet the vast majority of students don't even know what Cv for a valve is, let alone how to calculate pressure drop given this flow coefficient. I hope you are not meaning that I don't know what it is. What I do know, in words of important professors of academia in U.S. of Aerospace Engineering, is that the education that US universities are giving to undergrads is a poor one, in the sense that as time goes by they get out knowing less things. Don't forget that my undergrad education is from Spain, more over I am not a B.S. Engineer, I am an Ingeniero Superior or Ingenieur which corresponds to a M.S. here and now a M.S. in Europe. I don't know if Fred and you are trying to diminish what we do in Science (I was an student when I was an undergrad, now they are paying me for doing Science). I will let you know that here we are a bunch of engineers, we are given projects with fundings of amounts of order 500K, and we have to manage to take the project to a final solution not only having to make the funders happy, but also doing new Science for the world. And we do what is needed to arrive to a solution. People they work in Labs, another work in Computation, and some others work in Theory. And be sure that the current status of the Science does not allow you to pursue ideal dreams far from real applications, because they become unpublishable. So as you may realise, the thing is not so naive, even though we make less money by far than a practising engineer. I have been writing here since 3 or 4 years (i don't remember exactly). I consider I have been wrong sometimes and right anothers. I really think I have given a valuable contribution to various forums of PF, and so the mentors have recognized it. If you and those who meant it, keep on teasing the academia or those who don't dedicate their lives to such honour of being a practising engineer ( so far), then I would have to leave this forum and every writer here will obtain help when calculating pressure drops, but some other questions pertaining to fluid dynamics science will remain unanswered collaborating to a professionalization of this forum and not leaving any place to fundamental questions of students and non students. Last edited: Q_Goest Science Advisor Homework Helper Gold Member Minger is right, the solution he posted is an exact solution for viscous flow, in the sense that the velocity field obtained there represents the exact distribution of velocity … Yes, you're correct. I read through it too quickly - thought it was a calculation for determining Dh. My mistake. Next step is to show how the equation can be resolved for some given example. How about the 1 mm square tube? Can you calculate the flow rate using this equation? I'd honestly like to see how that can be done, and no I'm not being facetious about that. I've provided in some detail how I've calculated the flow rate for the example provided, and even did the calculation. I've also done it using more sophisticated software, and found the same answer. Note that the software I used also uses the same methodology as I've provided here, it doesn't use the equation provided by Minger, and it doesn't use Navier Stokes equations. So I'd like to see how you propose this is supposed to be done. You haven't explained it yet! What I see is a fundamental limitation for obtaining real solutions to practical problems using the more theoretical approaches you've suggested. In fact, without doing CFD analysis, I honestly don't know how one could realistically do a flow analysis on any simple piping system using the approch you're suggesting. And if someone has to resort to a CFD analysis to do a piping run, the entire engineering community is in serious trouble. There is a lot you don't understand about flow through piping systems, that much is obvious. The standard methodology for analyzing pipe flow throughout the industry is something you haven't learned yet because it's not generally taught in college. Furthermore, you're obviously holding a grudge against everyone that uses this methodology. Why I'm not quite sure. I'd like to say Clausius, that I'm sure you understand the theoretical aspect of fluid flow quite well, and at least in some areas I'm sure you understand it better than I do. But you have to understand and respect others in the engineering community that don't share your passion for the theoretical. There are standard methodologies used in engineering and you shouldn't be bad mouthing folks for using those methods, especially when you don't even understand them. Your remarks indicate that you don't respect others that have worked in this field, and you don't respect them simply because they aren't doing what you expect. That attitude, more than any lack of knowledge, is what has created the friction here, and it's why you find yourself on the defensive. I'm sure you don't like being told you don't understand a subject you hold so dear, and you can be sure no one else enjoys being told that either. If you show people respect, you'll get that in return. Engineers that don't think exactly like you are not indicating some "narrow knowledge about the Basics" and I'm not a "grown practicing engineer you have an unstopabble inertia" and Stokes would probably admire the way engineering has resolved piping flow analysis. Clausius2 Science Advisor Gold Member sure you are not going to have the last word here while I'll be able to log on. What I see is a fundamental limitation for obtaining real solutions to practical problems using the more theoretical approaches you've suggested. In fact, without doing CFD analysis, I honestly don't know how one could realistically do a flow analysis on any simple piping system using the approch you're suggesting. And if someone has to resort to a CFD analysis to do a piping run, the entire engineering community is in serious trouble. Sure that using what Minger gave is a little bit out of scope for this problem. But integrating given a pressure loss, integrating the u(x,y) formulas there on a differential element of pipe section dS=dxdy gives you the EXACT mass flow. Your approach is an APPROXIMATE one. Possibly a good one though. It is worthy to use what Minger gave instead the engineering method?. Possibly it is not worthy for an engineering calculation. (keep on reading, please). There is a lot you don't understand about flow through piping systems, that much is obvious. The standard methodology for analyzing pipe flow throughout the industry is something you haven't learned yet because it's not generally taught in college. Furthermore, you're obviously holding a grudge against everyone that uses this methodology. Why I'm not quite sure. Sure there is a lot of particular stuff concerning engineering methods that I don't know (underline this). That's damn right!. Maybe here there is a misunderstanding. The misunderstanding is that what you think is Basics I think is Application and what I think is Basics you think I don't know what. With Basics I mean the Basic laws of Fluid Mechanics, in its differential forms and integral forms and the Theory supported by experiments that holds them. Upon them lyes all the physics of this Science. With all my respect, you know a lot about engineering methods (applications) but you don't know anything about those basics laws. Therefore you are practically blind and driven by your software and the three or four equations that you know (DW and Bernouilli and a couple more). Maybe this is not the case and the OP wants a quick response using engineering methods, but what I will not allow while I'll be writting here is that many students reading these thread get encouraged by you to forget about those basic physics concepts and jump directly to your engineering methods. That's not the way one ends up knowing a little bit of this, and that's not the way to show the power of this field to the rest of the science disciplines. To your information I will tell you that I may end up in industry, because I like applied things. But I am pursuing a Ph.D. degree because my understanding of such applications would be much better afterwards because my knowledge will have a solid basis of theory and computation. And I want to encourage all the students and non students here to understand the basic physical laws before facing a problem rather than picking up a manual. In addition to that, looking things at Cranes paper doesn't take a long time, and understanding your methods neither. On the other hand, it would take you years and years to understand the physical phenomena of what is happening in more complex problems than a simple pipe installation. I think even that's the spirit of the whole Physics Forums, to give insight into nature explaining it with a solid understanding of the physics. That's out of your range. I don't grudge to anyone, but if by any chance an student jumps from the book of Batchelor to your methods because he thinks that it will save him time and he will reach quicker the answer, I will be grudging you for sure. On the contrary, you will be always constrained to your software and engineered methods if you keep on that way, and as you have widely shown in this forum, you are unable to answer basic questions of physics (which is the pillar of the engineering). All the pipe based threads for engineers will be for you though. Bon apetite. Last edited: Q_Goest, There are two excellent papers that give you the theory behind development of explicit equations of friction factor. One is by Churchill (Friction-factor equation spans all fluid-flow regimes) appeared in November, 77 issue of Chemical Engineering Journal. The other one is by Chandra Verma (Solve pipe flow problems directly) appeared in August, 79 issue of Hydrocarbon Processing. When I was comparing various friction factors, Churchill's was deviating significantly from Colebrook's at low Reynolds numbers. One member(katmar) at eng-tips modified Churchill's equation by using the method given by Verma and this single equation is in excellent agreement with both Poiseuille's as well as Colebrook's friction factor. Further, plotting of this equation gives you a continuous curve over a wide range of Reynolds numbers. That is why, I presumed this one single equation can give us correct friction factor in Laminar, Transient and Turbulent regimes as well. I checked my pressure drop calculator with other commercially available software and freeware. The values are matching at higher Reynolds numbers. Please note that, so far, I didn't see any commercial software available based on 2-K and 3-K methods. So, it is difficult to check the results at lower Reynolds numbers. However, the papers of Hooper and Darby show excellent curve fit of the experimental data. Q_Goest Science Advisor Homework Helper Gold Member Hi Quark. OMG, is this the same quark I've been quoting from EngTips? LMAO If so, thanks for the great FAQ post. Regardless, yes I'd like to see a good correlation for the transient zone. Was this one of the equations posted on the FAQ? I've been using an explicit equation handed down to me from a guy I worked with about 10 years ago, and I've cheked the accuracy of it but of course it's only valid in the fully turbulent zone. I'll send you a PM with my email - I'd love to see the paper your refering to. Thanks Reynolds Number: [tex] Re=\frac{\rho Ua}{\mu}\sim \frac{\rho \Delta P a^3}{L\mu^2}$$

where $$a$$ is the pipe hydraulic diameter and $$L$$ is the pipe length.
I have been trying to use this but the dimensions do not cancel out at the end.

using SI units:

for $$\rho$$ I have $$\frac{kg}{m^3}$$

for $$\Delta P$$ I have $$\frac{N}{m^2}=\frac{kg m}{m^2}=\frac{kg}{m}$$

for $$a^3$$ I have $$m^3$$

for $$L$$ I have $$m$$

for $$\mu^2$$ I have $$Poise^2 = Pa^2 s^2 = \frac{N^2 s^2}{m^4}= \frac{kg^2 s^2}{m^2}$$

I always end up with s^2 as the units of this equation. Please help me see where I have gone wrong.

everything cancels out but the s^2

*** Nevermind. Clausius2 pointed out that Newton is not a kgm it is a kgm/s^2

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I actually needed to do the same calculation for my research and came across this thread. Very helpful information from the person who posted the "exact solution" from the text. Now

P2 - P1 = Q * R

I did the calculation from the textbook equation 3-48 for the square tube case and solved for the R. Here's the resutling equation

P2 - P1 = Q * 1.7784 (l*u/a^4)

where l is the length of your tube, a is the width of your channel divided by 2, and u is the dynamic viscosity of your fluid. Plug in your flow rate (volume/time) and you have your pressure drop.

Enjoy. This equation only applies for laminar flow, so do your Reynold's # calculation first.

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