A simple example involving a pipe network having 2 loops.
To demonstrate how Hardy-Cross method is applied to find the flow in individual pipe of the network.
Three methods used to illustrate. May be suitable for beginners.
However, this solution can be easily obtained using 'FindRoot' method also, but is not shown in the example.
Cheers.
Pipe_Network_EX_HARDY_CROSS.sm (287 KiB) downloaded 501 time(s).
Pipe_Network_EX_HARDY_CROSS.pdf (1020 KiB) downloaded 773 time(s).

The earlier example was for 2 loops
Attached example is for 3 loops.
Hope will be useful.

Cheers!

Pipe_Network_EX_HARDY_CROSS_3 LOOPS.sm (550 KiB) downloaded 200 time(s).
Pipe_Network_EX_HARDY_CROSS_3 LOOPS-compressed.pdf (647 KiB) downloaded 267 time(s).

Wrote

The earlier example was for 2 loops
Attached example is for 3 loops.
Hope will be useful.

Thanks for that exceptional work, saved in my Instrumentation compendium.
Take care ... Jean

ndtma
Not sure is you can combine this with your previous post from 5 years ago https://en.smath.com/forum/yaf_postst7459_Pipe-Network-Analysis.aspx
That one was one loop, this one has 2 and 3 loops

Wrote

Thanks for that exceptional work, saved in my Instrumentation compendium.
Take care ... Jean

Thank you Jean for your kind words and encouragement.
Take care.

Hello ndtma

I have used Epanet to solve your exercise, by the Hazen Williams method with the following assumptions:

Minimum tank height 66 meters
the roughness of the pipe 120
node height 0 meters
diameter in millimeters
pipe length in meters





Best Regards

Carlos

Thank you Carlos.

I am extremely sorry that in my last example on 3 LOOPS, I have made a careless mistake in Program 2 to calculate the appropriate corrections in the network. The results were wrong.

Furthermore, for simplicity, I assumed a fixed friction factor instead of the method given in the original EXCEL file in the web.

The revised SM and EXCEL files are attached. They both tally. The original EXCEL file had only 5 trials and I extended to 10 trials.

Also, many thanks to Carlos for his example using EPANET which prompted me to check and correct my program.

Pipe_Network_EX_HARDY_CROSS_3 LOOPS-R2.sm (261 KiB) downloaded 210 time(s).
Pipe_Network_EX_HARDY_CROSS_3 LOOPS-R2-compressed.pdf (469 KiB) downloaded 208 time(s).
hardy cross_3 LOOPS_R2.xlsx (763 KiB) downloaded 244 time(s).

Hardy Cross method revisited.
The friction factor for turbulent flow is calculated using the Colebrook-White equation.
Due to the implicit nature of this equation, calculation of the friction factor requires an iterative solution via numerical methods. Hence, many explicit approximations have been proposed such as Haaland, Moody, Jain etc. to simplify the calculation process.

The attached sm file shows the iterative method of solving the Colebrook-White equation to calculate ONLY the friction factor ‘f’, for use in the Hardy Cross method. The calculated pipe flows were well within the values obtained in my last example.

Cheers.

Colebrook-White-Numerical Soltion.sm (44 KiB) downloaded 110 time(s).
Colebrook-White-Numerical Soltion.pdf (176 KiB) downloaded 113 time(s).

Thanks for sharing the improved project method.
Take care ... Jean

ndtma...

This is very fine work indeed and much better than doing Hardy-Cross by hand like I had to do a few times in college back in the late 1970s. That 4-loop Hardy-Cross model I had to solve was almost enough to make me want to become a lawyer. Almost. :-) However, I now use EPANET and WATERCAD for my modeling needs and life is good. I also use Hazen-Williams far more often than Darcy-Weisbach/Colebrook-White and life is even better.

You mention above explicit approximations to Colebrook-White. This is a topic that has interested me since I learned of the Blasius Equation in college. Blasius does a decent job approximating the smooth pipe curve on the Moody Diagram, but only over a limited range of Reynolds Numbers. Over the years, I have run across at least two dozen published explicit approximations to CW. I even created my own improvement to Blasius, although it pales in comparison to some of the others.

Back in 2014, I published on the PTC website two Mathcad Prime 3.0 worksheets that evaluated these explicit approximations for accuracy and relative calculation speed. For accuracy, I used CW as the basis, even though it is itself empirical and includes an empirical friction factor. Computers are so fast now that relative calculation speed is not as relevant as it was when I started my career in 1980. However, it matters for calculator programs. I used an HP-41CX and it's built in time functions to measure and compare calculation speeds.

One MCP worksheet is for hydraulically smooth pipe (the equations simplify) and the other is for hydraulically rough pipe (the usual case). Y'all can find these worksheets here: https://community.ptc.com/t5/PTC-Mathcad/Lusk-Darcy-Friction-Factors-v2-zip/m-p/450068 . Don't worry if you don't have Mathcad Prime 3.0 or later. The .zip file contains Acrobat .pdf files for your reading pleasure. And, please excuse the funky page formatting near the end of each worksheet. These are big files and I had pushed MCP3 to its limits.

Fred

Wrote

This is very fine work indeed and much better than doing Hardy-Cross by hand like I had to do a few times in college back in the late 1970s. That 4-loop Hardy-Cross model I had to solve was almost enough to make me want to become a lawyer. Almost. :-) However, I now use EPANET and WATERCAD for my modeling needs and life is good. I also use Hazen-Williams far more often than Darcy-Weisbach/Colebrook-White and life is even better.

Thank you. Your explanation is welcome.

I don't have any experience in EPANET and WATERCAD.
My aim was to demonstrate basics of Hardy Cross method using SMath and and the importance of
the Vectorize function when dealing with nested matrices.

Stay safe. Take care.

Cheers.

Wrote

You mention above explicit approximations to Colebrook-White. This is a topic that has interested me since I learned of the Blasius Equation in college. Blasius does a decent job approximating the smooth pipe curve on the Moody Diagram, but only over a limited range of Reynolds Numbers. Over the years, I have run across at least two dozen published explicit approximations to CW. I even created my own improvement to Blasius, although it pales in comparison to some of the others.

In my Engineering times, before 1990, the Bible was Moody from standards.
There is nothing to explicit CW, it solves on the running Reynolds,
and for a family of diameters, just like in the Bible diagram.
Smath attached is 1/1 Mathcad 8 Pro [2000].
Cheers ... Jean

Inst_Colebrook MOODY.sm (16 KiB) downloaded 102 time(s).

Wrote

I am extremely sorry that in my last example on 3 LOOPS, I have made a careless mistake in Program 2 to calculate the appropriate corrections in the network. The results were wrong.

Furthermore, for simplicity, I assumed a fixed friction factor instead of the method given in the original EXCEL file in the web.

The revised SM and EXCEL files are attached. They both tally. The original EXCEL file had only 5 trials and I extended to 10 trials.

Also, many thanks to Carlos for his example using EPANET which prompted me to check and correct my program.

Pipe_Network_EX_HARDY_CROSS_3 LOOPS-R2.sm (261 KiB) downloaded 210 time(s).
Pipe_Network_EX_HARDY_CROSS_3 LOOPS-R2-compressed.pdf (469 KiB) downloaded 208 time(s).
hardy cross_3 LOOPS_R2.xlsx (763 KiB) downloaded 244 time(s).

Hello, I am very happy to read your publication, thank you for generating these documents. I want to download your files, however, they do not allow me access, please upload them again. Thank you very much, greetings from Mexico.

Wrote

I want to download your files, however, they do not allow me access, please upload them again. Thank you very much, greetings from Mexico.

Just click the links you quoted. Attached files are hosted within this website, so there's no expiration. Also check if there is some policy that restrict downloads on your system.

Wrote

Hello, I am very happy to read your publication, thank you for generating these documents. I want to download your files, however, they do not allow me access, please upload them again. Thank you very much, greetings from Mexico.

Hello Valentino,

Were you able to download the files in my post as suggested by Davide?
I am sorry for the late reply.

Thank you very much for sharing your knowledge.

Wrote

Wrote

Hello, I am very happy to read your publication, thank you for generating these documents. I want to download your files, however, they do not allow me access, please upload them again. Thank you very much, greetings from Mexico.

Hello Valentino,

Were you able to download the files in my post as suggested by Davide?
I am sorry for the late reply.

Thank you all very much, now if I can download them, I do not know why there are files that do not allow me to download them in certain blogs.

Now I can see the document Thank you very much