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Suppose I have a pipe with holes on its top (kind of like a flute, but with different sized holes). One end of the pipe is open and the other is closed. The pipe is submerged in water horizontally, and I start pumping water into the open end at a known flow rate. How can I determine the pairwise ratios of the holes' areas that will ensure the flow rates out of all holes are equal?

Here are my initial thoughts on how to approach the problem. Suppose the pipe has $n$ holes on top, and the cross-sectional area of the $i^{\text{th}}$ hole from the open end is $A_i$. Using continuity and the condition that all holes have equal flow rates, we know that

$$A_i=\frac{Q}{nv_i}$$

where $Q$ is the flow rate from the pump, and $v_i$ is the velocity of water coming out of the $i^{\text{th}}$ hole. This gives the ratio

$$\frac{A_i}{A_j}=\frac{v_j}{v_i}$$

so to find my desired pairwise ratios, I'll need to know $v_i$ for all $i$. Using Bernoulli's equation, I can show that

$$v_i=\sqrt{v^2+\frac{2}{\rho}(p-p_i)}$$

where $v$ is the (known) velocity of water entering the open end of the pipe, $p$ is the pressure at that end, $\rho$ is the density of water, and $p_i$ is the pressure near the $i^{\text{th}}$ hole. If I'm right so far (a big if, as I'm not a physicist and have little intuition for fluid dynamics), this leaves me needing to determine $p$ and $p_i$ for all $i$ to solve my problem. If we suppose the depth of the pipe below the water's surface is $d$, can we express $p$ and $p_i$ solely in terms of $v$, $d$, and $\rho$? Given some idealized assumptions, it seems like this should be the case, or perhaps I'm missing something (or perhaps my whole approach is flawed).

My motivation here is to design a filter for a fish tank that circulates water evenly from the whole length of the tank.

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    $\begingroup$ @TimWescott: Thanks for your comment. I'm a mathematician, so I didn't view this problem as practically as you did, which is very helpful. I'm still interested in a theoretical solution for various sizes of pipes and holes, so I'll try to migrate the question as you've suggested. As a new contributor here, I can't migrate the question on my own, but I've just flagged it for a moderator's attention to do so for me. Thank you again. $\endgroup$
    – Jared
    Jul 14, 2023 at 17:37
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    $\begingroup$ If your end goal is to design something that has to work in reality, you cannot neglect friction and turbulent losses, especially with fluids, because they tend to change the whole flow field of the fluid. Due to that, finding an analytical solution can get very hard. As an example, if your holes are big enough and not far apart enough, the whole flow becomes kind of messy and the common equations start becoming more and more imprecise. You can read about Darcy-weisbach equation and head loss coefficients to get an idea of how such losses are usually estimated with a nice enough flow. $\endgroup$ Jul 15, 2023 at 22:18
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    $\begingroup$ Another difficulty with finding a precise theoretical solution starting with, e.g., the smallest pipe possible is that -- at least in a fish tank -- it may work as predicted when it is clean and if it is built exactly to specification, but even slight deviations from the design -- whether built-in or acquired -- will cause large deviations from the design intent. So if you do find the necessary math to solve this, be sure to anticipate some possible problems in manufacturing or use, and find the sensitivity of the flow at the holes to these variations. $\endgroup$
    – TimWescott
    Jul 17, 2023 at 1:09

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Similar question from 2020 How to divide the water flow into outlets with equal flow each? and from 2021 How to cut a pipe such that water comes out evenly across the length?

Do you have any specs like Size of tank, size of pipe, and flow rate of pump? Here are several ideas that might work, depending on your situation.

  1. Drip irrigation: Flow is small, pipe is large so same pressure everywhere in pipe. Same small holes will have same small flow.
  2. Pressure compensated irrigation drippers : They are supposed to give same flow at various pressures.
  3. Guess/Check/Adapt Empirical Engineering Exercise: 3a Guess: Start with equally spaced holes small enough that combined area of holes (Ndiadia) is about (D*D/2) == half area of pipe. 3b Check, measure, or eyeball the equality of the flow. 3c Adapt: drill additional small holes in areas of the pipe where flow is less. repeat from 3b until flow is equal enough.
  4. Tapered pipe: Machine or 3dprint a step tapered pipe, where each reduction of cross section is equal to the cross section of the hole just before the reduction.

Alternate thoughts: Your listed goal: "a filter for a fish tank that circulates water evenly..length" Your proposal: (source the incoming water evenly across length of tank) The location of the outgoing water also matters; where is that? Alt proposal: source water evenly across area of left end, outgoing water sink evenly at right end; All the same water goes from left to right, so equal everywhere in tank.

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