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I have a fluid channel through which a fluid with known properties (viscosity, density etc.) flows at a known velocity with a known pressure and temperature and mass flow rate. The external pressure (on the outside of the channel) is also known. The channel has a rectangular cross section with known dimensions.

I want to open a hole on one side of this channel such that I will get a predetermined mass flow rate leaking out (that is much smaller than the flow rate through the channel). How can I determine the size of the hole I need to open? (A sample calculation would be very nice although not absolutely necessary.)

Description of the problem

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  • $\begingroup$ What material is flowing in the pipe, what is the pressure drop through the hole, and what flow rate are you looking for? Also, is it permissible to install a quick opening valve just downstream of the hole? $\endgroup$
    – David White
    Commented May 4, 2022 at 19:19
  • $\begingroup$ @DavidWhite You could just use random values, so I'm going to give you some. Let's say that the fluid is water (since it is a well-studied fluid), pressure difference between the inside of the pipe and the outside ambience is 5 bars. The mass flow rate through the pipe is 0.02 kg/s, and I'm looking for a leak of 0.006 kg/s. It is not permissible to install a quick opening valve. $\endgroup$
    – arda güler
    Commented May 4, 2022 at 21:53
  • $\begingroup$ Just to be clear regarding your random assumptions, there is potentially a huge difference between the flow rate estimation of a compressible fluid and an incompressible fluid experiencing a pressure difference of 5 bar. Is the fluid incompressible? $\endgroup$
    – David White
    Commented May 5, 2022 at 0:25
  • $\begingroup$ @DavidWhite The fluid is incompressible. $\endgroup$
    – arda güler
    Commented May 5, 2022 at 7:32
  • $\begingroup$ You can start here: instrumentationandcontrol.net/… $\endgroup$
    – David White
    Commented May 5, 2022 at 15:27

1 Answer 1

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Here is how to obtain a simple estimate, and a design recommendation for your project.

First, redraw your diagram to show the pipe splitting into two parallel pipes of different cross-sectional area. Make the cross section of the smaller pipe equal to the area of the opening you want to make in the larger pipe.

For slow flow rates, the sum of all flows at the splitting junction will equal zero (continuity equation) which means sum of the flows in the two split pipes will equal the original flow rate entering the split junction.

Estimate the flows in the two exit pipes by apportioning the inlet flow between them in direct proportion to the ratio of their cross-sectional areas.

The flow through the smaller pipe in your original drawing exits the main pipe at a sharp 90 degree angle, which is going to restrict the flow exiting there slightly, so we know that this method will yield a slight overestimate of the flow rate coming out of the hole.

Now, as a piping system designer, your time is worth a certain amount per hour, and it is going to take you a certain amount of time to do a careful calculation or even a computer modeling run of your design problem- which will cost more than buying an adjustable valve and teeing it into the main pipe, and then turning the valve handle until you get exactly the flow rate out of it that you desire.

I have used this approach (using extremely basic physics to quickly get a "back of the envelope" estimate of one significant digit accuracy, and then running an experiment to get two significant digits) over and over in my former career, and was blessed by having a professional physicist sitting nearby who was amenable to accepting caffeine bribes for the back-of-the-envelope part of the process.

Very cost-effective, even taking into account the price of coffee.

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  • $\begingroup$ Thank you so much for the answer. Unfortunately I can not attach valves to this particular problem, and I can not undrill the holes I make, that's why I'm resorting to maths. Maybe I'll have to solve it with CFD eventually. $\endgroup$
    – arda güler
    Commented May 4, 2022 at 22:04
  • $\begingroup$ But Arda, you could build a small model section of the pipiong system in which you could drill holes, test valves, etc., could you not? how big is the main pipe? $\endgroup$
    – niels nielsen
    Commented May 6, 2022 at 0:54
  • $\begingroup$ The pipe is actually more like a "tube" or "fluid channel" (I think I've been using the wrong word, English is my second language, sorry), and there are many of these tubes that are fed by a manifold, which is fed by a pipe, which absolutely must have the given flow rate. An individual tube is not larger than 5mm x 5mm (rectangular). I've edited the question to not use the word "pipe". $\endgroup$
    – arda güler
    Commented May 6, 2022 at 9:50

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