What is the Flow Rate of Water Through a Square Tube?

[blaster]

Find: Flow rate of water through square tube

Given:

Cross-sectional wall and area of square tube, A=w^2
Length of square tube, L
Pressure at start, P1
Pressure at finish, P2
Water

***
If you need specifics I am pushing water (room temp) at 1 bar through
1200mm of square tube that is 1mm x 1mm. The end of the tube is open to
atmosphere.

What is the flow rate in liters per hour that will flow through this
tube?

 

[Clausius2]

Find: Flow rate of water through square tube

Given:

Cross-sectional wall and area of square tube, A=w^2
Length of square tube, L
Pressure at start, P1
Pressure at finish, P2
Water

***
If you need specifics I am pushing water (room temp) at 1 bar through
1200mm of square tube that is 1mm x 1mm. The end of the tube is open to
atmosphere.

What is the flow rate in liters per hour that will flow through this
tube?

My question is: how is it possible to push water at 1 bar being the end of the pipe open to the atmosphere (1 bar)?. In order to generate flow you need a pressure gradient. There is only one case in which you wouldn't need a pressure gradient. And that case is one that belongs only to the literature. That will be an ideal pipe with no friction. If you are assuming an ideal straight pipe with no friction then any mass flow fits with the solution of your problem. I hope that you are not assuming an ideal pipe, since your dimensions indicate that the flow surely would be a low Reynolds flow. The velocity profile inside your tube is going to be some kind of Poiseuille flow corrected for the square section. If you know the pressure at the pipe inlet then you know the mass flow by only applying Poiseuille flow relations. If the pipe inlet is connected to a pump, then the pressure at the inlet is given by the jump of pressures of the pump, whereas if the pipe inlet is connected to a reservoir, the pressure is given by the pressure of the reservoir adecuately corrected.

Here are some dimensional analysis that an engineer should know how to do:

Reynolds Number:
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.

 

[blaster]

Let me clarify P2=0bar.

How do you do the "corrected for a square section" part

 

[Clausius2]

You didn't clarify anything. You don't know what intake pressure you have. Think about it a little bit more and come out with the Reynolds Number by yourself. It will give you what kind of flow best addresses your problem. I won't be helping you farther.

 

[Q_Goest]

Hi Blaster,
Is this homework? I assume it is. Have you reveiwed "hydraulic diameter" yet? You can equate a square tube to a round one using hydraulic diameter. Once you do that, you can apply the Darcey-Weisbach equation directly.

If this isn't homework, I'd be glad to run through this for you.

 

[FredGarvin]

I am still confused about the pressure. If the end of the pipe is open to atmosphere, there is no way that P2=0 bar.

 

[Q_Goest]

Hi Fred. The way I read this is there is a long square tube, open to atmosphere. The pressure at the outlet is therefore at atmospheric pressure as it opens to atmosphere (ie: 0 barg). At a point 1200 mm upstream of the open end the pressure is 1 barg.

Note: barg = bar gauge pressure

 

[blaster]

Hi Blaster,
Is this homework? I assume it is. Have you reveiwed "hydraulic diameter" yet? You can equate a square tube to a round one using hydraulic diameter. Once you do that, you can apply the Darcey-Weisbach equation directly.

If this isn't homework, I'd be glad to run through this for you.

Actually this is not homework. I am a mechanical engineer designing irrigation products. And yes my pressures are gauge pressures.

Thanks for your help. I am excited to see you run through the solution.

 

[FredGarvin]

Hi Fred. The way I read this is there is a long square tube, open to atmosphere. The pressure at the outlet is therefore at atmospheric pressure as it opens to atmosphere (ie: 0 barg). At a point 1200 mm upstream of the open end the pressure is 1 barg.

Note: barg = bar gauge pressure

Hey Q. Good to see you again. I completely missed the 1200 mm upstream part. That makes some sense now.

In regards to the OP, you do indeed need to use the hydraulic diameter in your calcs. The hydraulic diameter is defined as:

D_h = \frac{4*\mbox{cross sect area}}{\mbox{wetted perimeter}}

 

[Q_Goest]

Hi Fred. Nice award, I think you earned it!

Hi Blaster,
Equations for fluid flow, such as Darcy-Weisbach, Hagen-Poiseuille, Colebrook equations, the Moody chart and similar flow equations all assume flow in circular pipes. These equations can be used for other cross sectional shapes such as square cross sections, but these cross sections have to be equated to a circular cross section that produces the same restriction. That cross section is called the "hydraulic diameter". Note that the hydraulic diameter is not the same as an "equivalent diameter" that results in the areas being equal.

The hydraulic diameter is can be calculated from:

Dh = 4A/U
Where A = cross sectional area
U = wetted perimeter
Ref: http://en.wikipedia.org/wiki/Hydraulic_diameter"

This is the equation Fred provided above. For example, a 1 mm square tube (inner dimensions) has a hydraulic radius of 1 mm.

Once you've found the hydraulic diameter, you can attack this just like any other circular pipe flow problem. For incompressible flow, I'd suggest applying Darcy-Weisbach directly.

Pressure drop is called head loss or frictional head loss and a few other names too. That equation is:

h = f L V^2 / ( 2 D g)

where f = friction factor
L = pipe length
V = fluid velocity
D = hydraulic diameter
g = constant (acceleration due to gravity = 32.174 ft/s2 = 9.806 m/s2)
Ref: http://www.lmnoeng.com/darcy.htm" - I'm using this reference because they have a calculator at this site. You may not want to use it, but it would serve as a check of your own numbers. I'd suggest creating your own program using a spreadsheet.

The only variable above you won't have is friction factor, which is a function of Reynolds number. Wikipedia has a good article on Reynolds number here. It's the same equation provided by Clausius above.
http://en.wikipedia.org/wiki/Reynolds_Number"

Of course, you still need kinematic or absolute viscosity. The values for viscosity can be found in any textbook or on the web, for example here:
http://www.engineeringtoolbox.com/water-dynamic-kinematic-viscosity-d_596.html"

Once you do that, you still need to find friction factor. There are many ways to calculate that, including taking it directly off of the Moody diagram. I'd recommend you create a spreadsheet that does all this for you, so I'd suggest using an equation as described at Engineering Tips Forum here:
http://www.eng-tips.com/faqs.cfm?fid=1236"

Note that for the above friction factor you'll need pipe roughness, which is dependant on your actual hardware. If you don't know what it is for your square tube, I'd suggest using 0.00015 which is commonly used for clean pipe. Note also, D in these equations is the hydraulic diameter.

The last thing to do is determine flow as a function of pressure drop per the Darcy-Weisbach equation. Note that head (h) is the pressure created by a given column of the fluid in question. Velocity is a function of flow rate. You'll need to separate out those portions and treat them separately, or possibly use the calculator given on the internet.

~

If I do these calculations, I come up with a Reynolds number of 2577, which is somewhere in the transition zone. It may be turbulent, and it may be laminar.
- If I assume it is laminar, the flow rate for your case will be .0324 gallons per minute. Convert that how you need to.
- If I assume the flow is turbulent, I get .0178 GPM.

Couple things to note here. I've neglected exit losses, which in this case are fairly small, but to explain how to calculate exit losses would take another post this size. I've also assumed there are no elbows or other restrictions in this line. Finally, I've assumed the line is horizontal. If there is any elevation change in the line, you can correct for that using Bernoulli's equation.

This is a lot to cover in a single post, so I may have been overly brief. Feel free to ask questions, I'm sure myself or others here can help out. Hope that helps.

 

[Clausius2]

Hi Blaster,
Is this homework? I assume it is. Have you reveiwed "hydraulic diameter" yet? You can equate a square tube to a round one using hydraulic diameter. Once you do that, you can apply the Darcey-Weisbach equation directly.

If this isn't homework, I'd be glad to run through this for you.

It seems that we have forgotten the lesson about viscous flows. I hope you are not planning to calculate the mass flow using the -Bernouilli Corrected equation with losses- or whatever is the name. Looking at the left hand part of the Moody chart the only thing that you find for Low Re is the friction coefficient that by the way is automatically and analitically included in the Poseuille relations, so there is no need of Moody Chart. The first thing that I said is that he should get a Reynolds Number, and I still don't see it. If the Reynolds Number is large (and I don't think it's going to be large given the 1mm section), then the Moody Chart is going to be helpful and the Bernouilli-corrected equation is helpful too. If the Re is small then it is misleading to employ the Bernouilli-corrected equation that assummes fully turbulent flow with a laminar friction coefficient.

One more time for the OP, please provide Reynolds Number. That is the FIRST thing you should know when dealing with pipes and pressure losses. If you don't know that, then you know nothing, and we cannot help you.

The thing of the Hydraulic diameter is a good advice though.

 

[Q_Goest]

I hope you are not planning to calculate the mass flow using the -Bernouilli Corrected equation with losses- or whatever is the name.

I assume you're referring to the use of Bernoulli's equation to resolve pressure changes along a pipe where pressure drop due to friction is incorporated. I'll have to reference http://www.flowoffluids.com/tp410.htm" on this:

All practical formulas for the flow of fluids are derived from Bernoulli's theorem, with modifications to account for losses due to friction.

~

Looking at the left hand part of the Moody chart the only thing that you find for Low Re is the friction coefficient that by the way is automatically and analitically included in the Poseuille relations, so there is no need of Moody Chart.

Per the Crane paper:

If this quantity [equation for friction factor under laminar flow conditions] is subsituted into Equation 1-4 [Darcy-Weisbach Equation], the pressure drop in pounds per square inch is:
{simplified equation shown}
which is the Poiseuille's law for laminar flow.

So you are correct that the Poiseuille law is applicable where laminar flow is concerned and one wishes to determine frictional pressure losses. But as you can see, this law is simply a very specific case of the Darcy-Weisbach equation. There is no difference. That is why I would not suggest using the P' equation. It is too specific. The D-W equation is much more general.

~

If the Re is small then it is misleading to employ the Bernouilli-corrected equation that assummes fully turbulent flow with a laminar friction coefficient.

The "Bernoulli corrected equation" doesn't assume fully turbulent flow. Nor does the Darcy-Weisbach equation. Friction factor is a function of whether or not the flow is laminar or turbulent, and friction factor is the only correction that needs to be made to account for the two types of flow. The type of flow (ie: laminar or turbulent) has nothing to do with the Bernoulli corrected equation you're referring to.

~

One more time for the OP, please provide Reynolds Number.

How is the OP going to provide Reynolds number without velocity? And of course, you can't determine velocity without flow rate.

One thing I skipped over in explaining all this is that one needs to iterate to determine frictional pipe loss, which is why a spread sheet or other computer program is so valuable.

One input to the calculation is a guess at the flow rate. With this 'guess' one can determine velocity. With velocity you can then determine Reynolds number so that a friction factor can be determined. Only then can you calculate dP. So it's not a simple task. It requires iteration, but of course a computer can do all these iterations in a fraction of a second.

 

[FredGarvin]

I have not seen the energy equation being restricted to fully turbulent flow. The only restrictions I have seen are fully developed, steady state, incompressible.

 

[Clausius2]

The holly molly, it is impossible to deal with Q_Goest. It seems we are coming from Universes totally different, where the physics laws are completely different. I wish I could know the other Universe, if it exists. I will try to answer you rather than to answer the OP later, now I have work to do. I honestly get dissapointed and discouraged to talk with someone that uses only the reference of the "Cranepaper". I think that in these world there are better references to cite. But anyways, I don't feel like discussing with a wall again. So I may be not replying to your considerations. I will try to make an effort though, but not for wanting you to change your mind, rather for not letting the rest of the people (included students) to believe that you're right.

 

[Q_Goest]

Hi Clausius. People often resort to insults when disagreements arise. I'm not a wall. I don't come from another universe. In fact, aren't you still a student? I graduated almost 20 years ago and since then have done flow analysis, and more complex thermodynamic, heat transfer and two phase flow analysis on piping systems ever since. When working for General Dynamics Space Systems Division, I wrote a paper that was used as a standard that covered all of the piping flow analysis we've discussed in this thread. If I didn't understand this stuff, almost every launch pad in the US would have serious problems. I also hold a number of patents which required this knowledge. And I continue to analyze complex piping systems today, working in the air separation and chemical processing industry.

I'd apreciate it if you'd use thoughtful arguments instead of resorting to insults.

~

Regarding the Crane paper, if you do any piping analysis you'll find the Crane paper is the formost authority on the subject. If you don't believe that I can also reference others.

Regarding the Poiseuille law for example:

Poiseuille's law (or the Hagen-Poiseuille law also named after Gotthilf Heinrich Ludwig Hagen (1797-1884) for his experiments in 1839) is the physical law concerning the voluminal laminar stationary flow Φ of incompressible uniform viscous liquid (so called Newtonian fluid) through a cylindrical tube with the constant circular cross-section, experimentally derived in 1838, formulated and published in 1840 and 1846 by Jean Louis Marie Poiseuille (1797-1869),

… The law can be derived from the Darcy-Weisbach equation, developed in the field of hydraulics and which is otherwise valid for all types of flow, and also expressed in the form: …

Ref: "[URL

If you don't understand the use of Bernoulli's equation and how frictional pressure drop is also accounted for, but you don't like the Crane paper reference you can try this reference also:
http://www.lmnoeng.com/DarcyWeisbach.htm"

The analysis of pressure drop through a straight, horizontal pipe with no fittings, valves, or other restrictions is the most simple of all analysis. Until you understand this basic analysis, you won't be able to move on to how other restrictions affect flow. Then there are thermodynamic and heat transfer considerations which also must rely on understanding these basics. Two phase fluid flow then relies on understanding all of this.

 

[Clausius2]

Hi Clausius. People often resort to insults when disagreements arise. I'm not a wall. I don't come from another universe. In fact, aren't you still a student? I graduated almost 20 years ago and since then have done flow analysis, and more complex thermodynamic, heat transfer and two phase flow analysis on piping systems ever since. When working for General Dynamics Space Systems Division, I wrote a paper that was used as a standard that covered all of the piping flow analysis we've discussed in this thread. If I didn't understand this stuff, almost every launch pad in the US would have serious problems. I also hold a number of patents which required this knowledge. And I continue to analyze complex piping systems today, working in the air separation and chemical processing industry.

Here we go, on the contrary I am not going to throw here my CV, that is probably much better than yours one 20 years ago. I will use scientific arguments instead of medals:

All practical formulas for the flow of fluids are derived from Bernoulli's theorem, with modifications to account for losses due to friction.

Even though the Crane paper says that, that makes no sense sir!. What a naive generalization!. What a reduction of the spectra of the fluid dynamics science!.

The D-W equation is much more general.

What is a Darcy Weisbach equation?. You mean the experimental correlations written on the Moody Chart??. Or you mean the Darcy Weisbach
coefficient??. Do you mean as DW equations the correlations of Colebrook, Prandtl or Von Karman?. There is no exact solution for the DW coefficient for the whole range of Re, so your comments about the DW equation are naive and shows that your rockets may not be that good designed.

But as you can see, this law is simply a very specific case of the Darcy-Weisbach equation. There is no difference. That is why I would not suggest using the P' equation. It is too specific. The D-W equation is much more general.

Those correlations of Prandtl, Von Karman and Colebrook DON'T match with the laminar solution of 64/Re at low Re, so I don't see the generality of something that cannot be handled generally.

The "Bernoulli corrected equation" doesn't assume fully turbulent flow. Nor does the Darcy-Weisbach equation. Friction factor is a function of whether or not the flow is laminar or turbulent, and friction factor is the only correction that needs to be made to account for the two types of flow. The type of flow (ie: laminar or turbulent) has nothing to do with the Bernoulli corrected equation you're referring to.

That is not true sir. And in particular reveals that you don't know where those correlations of the Moody Chart are coming from. In particular those correlations such that the Colebrook one come from the Turbulent Flow in a Channel analysis. Do you know what is the defect law of Von Karman?. Here there is an excellent ppt from a guy of the U of Taiwan (not that prestigious as the Crane paper) but it may be helpful for us, especially from slide 48 on:

www.esoe.ntu.edu.tw/courses/50522100/Chapter_08(compact%20version).ppt[/URL]

There you are going to find familiar comments.

[quote]
How is the OP going to provide Reynolds number without velocity? And of course, you can't determine velocity without flow rate.
[/quote]

This is the culmination. I gave you a way of calculating the Reynolds number in my first thread. The velocity is an OUTCOME of an external force, so that the Reynolds number is intrinsically defined even if you don't know the solution. That's the task of an engineer and you should know it. To guess the unknown variable estimating it by using the flow equations (NAVIER STOKES EQUATIONS).

I'm going to finish with you saying something. I appreciate your experience and your background, and I am pretty sure that you have developed an excellent job. But don't cover your ears never when hearing something that does not fit with your stuff, despites how old are you.

Now I will answer Fred:
[quote=Fred] I have not seen the energy equation being restricted to fully turbulent flow. The only restrictions I have seen are fully developed, steady state, incompressible.
[/quote]

My questions are:
Given this equation (the Bernouilli corrected or whatever Crane calls it):

$$P_1+ \rho g z_1+ \rho \frac{U_1^2}{2}=\lambda L \rho \frac{U_1^2}{2D}+P_2+ \rho g z_2+ \rho \frac{U_2^2}{2}$$

or the similar version you use, that is coming from a differential version:

First:
How are you integrating for obtaining U_1 and U_2 constants when the profiles are not constants? (recall Poseuille profile in viscous flow)

Second and much more important:
How do you know that the energy dissipated on the RHS is proportional to the Kinetic Energy?.

The answer to both of them is that it does not make sense to use it in a Low Re flow. On the other hand, we do know that energy dissipation is proportional to the kinetic energy of the large scales in the bulk of a turbulent flow. As we approach to the wall, the small scales are dominant and the roughness of the wall starts to be of the order of the viscous length. There the energy dissipation is proportional to the viscous stresses, but such contribution is of second order in a fully turbulent flow.

 

[Q_Goest]

Hi Clausius.

What is a Darcy Weisbach equation?. You mean the experimental correlations written on the Moody Chart??. Or you mean the Darcy Weisbach
coefficient??. Do you mean as DW equations the correlations of Colebrook, Prandtl or Von Karman?.

I mean the Darcy-Weisbach equation. Not the Darcy friction factor f, the equation. Please see my post above and the reference to the LMNO web site. You can also find it on Wikipedia, here:
http://en.wikipedia.org/wiki/Darcy-Weisbach_equation
and here:
http://biosystems.okstate.edu/darcy/DarcyWeisbach/Darcy-WeisbachHistory.htm

Note also this equation is listed on your own reference from the gentleman in Tiawan on sheet 23 and again on 24, except he fails to mention the equation by name. He does however, mention the "Darcy-Weisbach Friction Factor" though this is generally simply called a "friction factor" or at best, Darcy friction factor. See also this Wikipedia article which correctly defines the "Darcy friction factor".
http://en.wikipedia.org/wiki/Darcy_friction_factor

There is no exact solution for the DW coefficient for the whole range of Re, so your comments about the DW equation are naive and shows that your rockets may not be that good designed.

Please refrain from insults, Clausius. If you don't, I'll have to mention this to one of the moderators.

Regarding the DW equation, it is valid for laminar and turbulent flow. It is not a friction factor equation. It is an equation for calculating pressure (ie: head) loss. Friction factor is a variable in it.

I suspect however, you've confused the Darcy friction factor with the DW equation. The friction factor may be obtained from the Moody diagram. Unfortunately, charts such as this must be interpreted mathematically for use in computer programs. I'd strongly recommend the use of explicit equations as given by the Engineering Tips forum FAQ I posted above. From that post:

To make our life easier, some great engineers developed explicit expressions for the friction factor. As I went on reading the subject, I came to know that there were many explicit expressions that equal no. of hairs on my head (no, I am not bald FYI). Out of those, the following are the famous equations.

Reference: http://www.eng-tips.com/faqs.cfm?fid=1236

Those correlations of Prandtl, Von Karman and Colebrook DON'T match with the laminar solution of 64/Re at low Re, so I don't see the generality of something that cannot be handled generally.

From this remark, it appears obvious you've mistaken the DW equation for the "Darcy" friction factor. Von Karman for example has an equation that is only applicable to highly turbulent flow where Reynolds number aproaches infinity. This isn't applicable to any of my comments thus far.

And in particular reveals that you don't know where those correlations of the Moody Chart are coming from. In particular those correlations such that the Colebrook one come from the Turbulent Flow in a Channel analysis.

Here's another good reference regarding the Darcy-Weisbach equation and friction factors.
http://biosystems.okstate.edu/darcy/DarcyWeisbach/Darcy-WeisbachHistory.htm
From that reference:

In about 1770 Antoine Chézy (1718-1798), an early graduate of l'Ecole des Ponts et Chaussées, published an equation for flow in open channels that can be reduced to the same form. Unfortunately, Chézy's work was lost until 1800 when his former student, Prony published an account describing it. Surprisingly, Prony developed his own equation, but it is believed that Weisbach was aware of Chézy's work from Prony's publication. Darcy, (Prony's student) in 1857 published new relations for the Prony coefficients based on a large number of experiments.

So yes, it seems some work was done using open channels early on, but that's not where the relationships for friction factor come from, which is from pipe as this reference continues on:

Darcy thus introduced the concept of the pipe roughness scaled by the diameter; what we now state as the relative roughness when applying the Moody diagram. Therefore, it is traditional to call f, the "Darcy f factor", even though Darcy never proposed it in that form. Fanning apparently was the first effectively put together the two concepts in (1877). He published a large compilation of f values as a function of pipe material, diameter and velocity. His data came from French, American, English and German publications, with Darcy being the single biggest source. However, it should be noted that Fanning used hydraulic radius, instead of D in the friction equation, thus "Fanning f" values are only 1/4th of "Darcy f" values.

Parallel to the development in hydraulics, viscosity and laminar flow were defined by Jean Poisseuille (1799-1869) and Gotthilf Hagen (1797-1884), while Osborne Reynolds (1842-1912) described the transition from laminar to turbulent flow in 1883. During the early 20th century, Ludwig Prandtl (1875-1953) and his students Th. von Kármán (1881-1963) Paul Blasius (1883-?) and Johnann Nikuradse (1894-1979) attempted to provide an analytical prediction of the friction factor using both theoretical considerations and data from smooth and uniform sand lined pipes. Their work was complimented by Colebrook and White's analysis of pipes with non-uniform roughness in 1939.

So it seems most of the data is from "sand lined pipes".

Do you know what is the defect law of Von Karman?

Are you referring to the von Karman equation? It's given on slide 60 of the reference you provided!

I gave you a way of calculating the Reynolds number in my first thread.

Regarding the aproximation you used for Re, I've honestly not seen that one before. The Crane paper has a list of 14 different variations on Re, but none of them match the aproximation you gave, nor do any of them provide a method of determining Re without knowing either velocity or flow rate in some way. Please provide a reference to the aproximation you gave.

Using some aproximation when it is simple enough to iterate however, seems like a poor method of doing any analysis.

But don't cover your ears never when hearing something that does not fit with your stuff, despites how old are you.

I'm not covering my ears, Clausius. I'll give you a chance to show me what you've learned, but you need to prove to me you really understand all this.

Regarding the equation you left for Fred, have you checked also the reference you provided? Check out slides 27 and 77. They show Bernoulli's equation modified to handle head losses as well as pump head increases. There is no indication anywhere in his presentation that indicates this equation is invalid for laminar flow. Fred is correct. The energy equation is not restricted to fully turbulent flow, and this also agrees with how the Crane paper handles things.

Note also slides 67 through 77. Much of this data and the concepts of resistance coefficient come from the Crane paper which is updated yearly for practicing engineers. Yes, the presentation you provided is a very good one, and I only spotted the one minor oversight I mentioned above.

 

[Clausius2]

Allright, thanks for not feeding the flames, sometimes I take it more seriously than what I should. Anyways I hope the reader is learning something out of all of this. At least we agree that the stuff of my link has been useful, and still it's going to be the workhorse of my answer, since you have pointed out partial truths here.

Hi Clausius.

I mean the Darcy-Weisbach equation. Not the Darcy friction factor f, the equation. Please see my post above and the reference to the LMNO web site. You can also find it on Wikipedia, here:
http://en.wikipedia.org/wiki/Darcy-Weisbach_equation
and here:
http://biosystems.okstate.edu/darcy/DarcyWeisbach/Darcy-WeisbachHistory.htm

Note also this equation is listed on your own reference from the gentleman in Tiawan on sheet 23 and again on 24, except he fails to mention the equation by name. He does however, mention the "Darcy-Weisbach Friction Factor" though this is generally simply called a "friction factor" or at best, Darcy friction factor. See also this Wikipedia article which correctly defines the "Darcy friction factor".
http://en.wikipedia.org/wiki/Darcy_friction_factor

My mistake. I have seen many times that equation but I didn't recall that such a trivial equation may have a name.

Regarding the DW equation, it is valid for laminar and turbulent flow. It is not a friction factor equation. It is an equation for calculating pressure (ie: head) loss. Friction factor is a variable in it.

Sure it is. The friction coefficient there reaches a certain particular states of similarity when the flow is fully turbulent, not depending on the Re, and when the flow is viscous, not depending on the rugosity.

http://biosystems.okstate.edu/darcy/DarcyWeisbach/Darcy-WeisbachHistory.htm
From that reference:

So yes, it seems some work was done using open channels early on, but that's not where the relationships for friction factor come from, which is from pipe as this reference continues on:

So it seems most of the data is from "sand lined pipes".

I meant Turbulent flow in Channels, no in Open Channels. A turbulent flow in a Channel is an essential problem in Turbulence (see Pope's book on Turbulence Flows). From the analysis of the Channel flow it emerges the Von Karman defect law (in slide 50) which is the theoretical basis for the experimentally corrected equation of the slide you posted.

Regarding the aproximation you used for Re, I've honestly not seen that one before. The Crane paper has a list of 14 different variations on Re, but none of them match the aproximation you gave, nor do any of them provide a method of determining Re without knowing either velocity or flow rate in some way. Please provide a reference to the aproximation you gave.

References: G.K. Batchelor 'Introduction to Fluid Dynamics'. IF the flow is viscous (the starting pre-assumption of your virtual iterative process), then the velocities expected are going to be of order U\sim \Delta P a^2/\mu L which comes naturally from the Poiseuille profile:

u(x,z)=-\frac{1}{2\mu}\frac{\partial P}{\partial x}z(a-z)

Substitute that characteristic velocity on the Reynolds number and you will obtain what I showed. If the flow is viscous for sure that my velocities are going to be of that order (sorry man that's physics of fluids) and everything would be coherent. If turns out that the Re so calculated is very large then my assumption is wrong and we should consider other regime (still waiting for the input here of the OP).

I'll give you a chance to show me what you've learned, but you need to prove to me you really understand all this.

Man, don't be like that. I don't know where you been working, but I am working in maybe the best department of Mechanical and Aerospace Engineering as far as Fluid Dynamics is concerned on the whole earth, and you will find that many grants here are provided by powerful institutions of U.S. So don't diminish what I have could learned here about simple pipe flows.

Regarding the equation you left for Fred, have you checked also the reference you provided? Check out slides 27 and 77. They show Bernoulli's equation modified to handle head losses as well as pump head increases. There is no indication anywhere in his presentation that indicates this equation is invalid for laminar flow. Fred is correct. The energy equation is not restricted to fully turbulent flow, and this also agrees with how the Crane paper handles things.

Again you are citing partial truths. Take a look at that slide and tell me what are the \alpha. They come from the Integral Momentum Equation and the Integration of a Velocity profile averaged. Even for viscous flows, it represents something less accurated than the exact Poiseuille theory and the equation of energy for viscous flows. Take a look at slide 67. My question is, again, (you didn't answer), how is it possible that in general (for the whole range of Re) the ENERGY loss is going to be directly proportional to the Kinetic ENERGY! (where is the viscosity playing at low Re?). The key, Q-Goest, is that the equation I posted is the result of an integration over a fluid volume, each one of the variables are averaged. And now, how am I going to integrate an hypothetical ENERGY loss which coefficient depends itself on the fluid velocity for not Fully turbulent flow? (If the flow is fully turbulent the DW coefficient reaches a similarity regime in which it does not depend on Re but on rugosity). To my point of view, you are right about the DW equation, but your jump from there to your energy equation is mistaken.

I hope you read carefully this last paragraph, it is my main conclusion.

 

[Clausius2]

And I am going to add an addendum:

Even though you may have much more experience than me on all these stuffs, I think I am satisfied enough if a simple grad student 24 years old with only the experience of the books has been able to challenge your knowledge of several years in well known companies here and to make you develop the best of you in this and past threads.

 

[Q_Goest]

Hi Clausius,

Q said: Regarding the DW equation, it is valid for laminar and turbulent flow. It is not a friction factor equation. It is an equation for calculating pressure (ie: head) loss. Friction factor is a variable in it.

C said: Sure it is. The friction coefficient there reaches a certain particular states of similarity when the flow is fully turbulent, not depending on the Re, and when the flow is viscous, not depending on the rugosity.

sure it is? The Darcy-Weisbach equation is for calculating pressure drop. The friction factor in it changes depending on flow regime and Re. Flow regime is dependant on surface roughness (what you're calling rugosity*). Are we in agreement on this? And once the flow is compleltely turbulent, friction factor no longer is dependant on Re, right? But friction factor is still dependant on surface roughness, right? Let's verify we at least agree on these basics.

Regarding your equation:
U\sim \Delta P a^2/\mu L
and
Re=\frac{\rho Ua}{\mu}\sim \frac{\rho \Delta P a^3}{L\mu^2}
I put both of these into my spreadsheet I use for pipe flow so I could compare these calculations to standard usage. I was very careful to ensure units were properly adjusted.
The result was off by more than an order of magnitude at low Re and 3 orders of magnitude at high Re. You might verify for me the viscosity is absolute. Regardless, it looks like an equation that might be used for something like heat transfer. I checked a text on heat transfer but still haven't found a similar equation to what you've posted. You might also want to check the assumptions used and post them here as well.

I'll have to get to the rest of your post another time.

*Note on terminology: Rugosity generally regards a concrete surface. I've never heard it used in reference to pipe flow, and I'd suggest you use "surface roughness" instead so we can all speak the same language. See http://en.wikipedia.org/wiki/Rugosity" for definition.

 

[FredGarvin]

My questions are:
Given this equation (the Bernouilli corrected or whatever Crane calls it):

P_1+ \rho g z_1+ \rho \frac{U_1^2}{2}=\lambda L \rho \frac{U_1^2}{2D}+P_2+ \rho g z_2+ \rho \frac{U_2^2}{2}

or the similar version you use, that is coming from a differential version:

First:
How are you integrating for obtaining U_1 and U_2 constants when the profiles are not constants? (recall Poseuille profile in viscous flow)

If you want to get into 2D flows, then fine we can integrate over the diameter and play around with the \alpha's to deal with a more precise velocity profile. The basic 1D assumption is using an average velocity across the pipe. In this simple situation (and I am not going to waste my time doing this) I would venture that the %error induced by using the average velocity will be well below 10%. Considering the nature of the problem, this is really not necessary. We're not talking blood or plasma flows here. It's water in a square pipe for Pete's sake.

Second and much more important:
How do you know that the energy dissipated on the RHS is proportional to the Kinetic Energy?.

The assumption is made that the system is conservative. So, we rule out pressure and gravitational potential energies, what is left? The energy accounted for on the RHS has to come from somewhere and since there is no heat transfer, I don't see any other place it can come from.

Anyone that wants to do an error analysis to see what difference there is by using the established engineering methods is more than welcome to do so. In the mean time, I would simply do what has already been suggested and move on.

 

[Clausius2]

allright. I withdraw. Seems ridiculous to me that this discussion is centered around some specific comments or words like roughness and around customer-chosen paragraphs ignoring the rest, and not paying any attention to the physical arguments I have given even for a simple estimation of Re of 1 line. I think that as grown practicing engineer you have an unstopabble inertia, and I'm sure that even if Stokes would be here saying something you would still agree with your pipe manuals. 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.

One more time, nobody can argue that I haven't used physical arguments to support my explanation. That's for sure.

 

[Clausius2]

The assumption is made that the system is conservative. So, we rule out pressure and gravitational potential energies, what is left?

Last thing, I could't stand quiet with this. By that fred you are pre assumming already high Re. Pressure, gravitational, kinetic and no more energies right?. Sounds familiar to me when talking about high Re numbers. Where is the Viscous Dissipation \phi or also called Rayleigh Dissipation Function, that is the energy dissipated by the viscous stresses?. Hint: go and look at the integral energy equation for the kinetic energy. I don't see it anywhere in your analysis. 2nd hint: when non dimensionalized, \phi has a coefficient in front that is...1/Re surprise!. Where is the viscosity in your analysis man?.

There is not much left to talk here. A useful reference for understanding friction for all of us is the book of Pope (a wise guy in Turbulence): Turbulent Flows. That's a serious book.

 

[FredGarvin]

You know, I have 3 pipe flow references on my desk and not one of them talks about Rayleigh dissipation factors. Hmmm. Curious.

Well, after this tirade I want to see your solution to this question. Specifically I want to know just how the viscous dissipation has anything to do with what we are discussing. After that I want to see your method and why you know more than established engineering practice. It should be a walk in the park. I have heard a lot of blustering and wise ass comments from someone who has not stepped one foot out of academia. Now it's put up or shut up. Q has already called you out saying that your method has not even come within an order of magnitude to the most likely answer.

We all await you to come down from the mountain top and prove everyone wrong.

 

[Clausius2]

!

 

[AlephZero]

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.

 

[minger]

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

 

[quark]

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.

 

[quark]

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]

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]

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 into 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]

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

Count me into 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]

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 into 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:

<br /> \nabla P=\mu \nabla ^2 \mathbf{u}<br />

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

<br /> \frac{\partial P}{\partial x}=\mu\nabla^2 u <br /> (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. <br /> <br /> <blockquote data-attributes="" data-quote="" data-source="" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> I think you&#039;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&#039;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&#039;t even know what Cv for a valve is, let alone how to calculate pressure drop given this flow coefficient. </div> </div> </blockquote><br /> I hope you are not meaning that I don&#039;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&#039;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. <br /> <br /> I don&#039;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 realize, the thing is not so naive, even though we make less money by far than a practising engineer.<br /> <br /> I have been writing here since 3 or 4 years (i don&#039;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&#039;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.

 

[Q_Goest]

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]

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.

 

[quark]

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]

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 referring to. Thanks

 

[blaster]

Reynolds Number:
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

 

[KR79]

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.