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Intuitively, when the diameter of pipe is decreased, there will be more friction loss, more water pressure, and a higher flow rate. Is there a direct relationship/equation derived to see the affect of the pipe lengths?

Deriving from Hagen-Poiseuille's equation, we get:

$$\ Q = \frac{\Delta P\pi r^4}{8\mu L} $$

where :

$Q =$ flow rate

$\Delta \ P =$ change in fluid pressure

$r =$ radius of pipe,

$\mu =$ dynamic viscosity of fluid,

$L =$ length of pipe

To keep similar flow rates, are we able to use Poiseuille's derived formula to find the new lengths of pipe with a change in $r$ (pipe radius)?

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  • $\begingroup$ In a pipe the flow rate must be constant throughout the whole pipe in order to satisfy mass continuity / mass conservation. If you then input the known variation of the radius, the pressure change follows direcly (so $\Delta P, \mu$ and $L$ are your free variables) $\endgroup$ Mar 11 at 13:03
  • $\begingroup$ What do you want to hold constant in the Poiseuille equation and what do you want to vary? $\endgroup$ Mar 11 at 16:17

2 Answers 2

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If you want to keep flow rate and pressure drop the same, because your engineering task requires it, you just have to rearrange the Hagen-Poiseuille law so that everything that is known is on the right hand side, and everything that is unknown is on the left hand side: $$\frac{L}{r^4} = \frac{\pi\Delta P}{8\mu Q}=:C$$ Usually, you would know $C$ because you know $\Delta P$ and $Q$, and thus, you could just calculate for example the length $L$ of the pipe if you provide a reasonable pipe radius $r$.

But if you just want to know how $L$ changes if $r$ changes, in general, you can provide an 'initial guess' $(L_1,r_1)$ and eliminate the constant $C$ by equating it to a 'final set of design parameters' $(L_2,r_2)$: $$\frac{L_2}{r_2^4}=\frac{L_1}{r_1^4}$$ Hence, if you rearrange that once again, you obtain $$L_2=L_1\cdot\frac{r_2^4}{r_1^4}$$ In more simple words, the length of the pipe behaves like the fourth power of pipe radius, if pressure drop and current shall stay the same.

By the way, this is all reasonable and a typical engineering question. Say, you want to build a ventilation system for a building, and you need an airflow $Q$ for convenient air quality and you have found an affordable fan that provides a certain pressure drop $\Delta P$ at its maximum fan speed. Now suppose you have chosen a pipe with diameter $r$, but as soon as you calculate the length of the pipe that is compatible with your fan and ventilation requirements (like with the first equation), you notice that this pipe is not long enough (say it is 20% too short) to connect the interior of the ventilated room through the building with the outside air. Now you might well ask the question: if I increase the radius of the pipe, how much length will this gain me.

For small changes of the respective quantities, as a rule of thumb (Taylor's theorem, first order), an increase of the pipe radius of 1% will result in a permissible length increase of 4%. This is due to the power of 4 on the radius, because $$(101\%)^4=(1.01)^4\approx 1.041\approx 104\%$$ In the example above, where the length is 20% too short, you would have to increase the pipe radius by approximately 5%. That is a pretty quick way of estimating what you need.

But of course, this is still all subject to the assumption of Hagen-Poiseuille flow.

Another caveat (if you ever have to design a ventilation system): real fans don't just offer a defined pressure drop at a certain fan speed, but the pressure drop also depends on the flow conditions around the fan, namely, if it is free blowing, or if it is placed inside a duct (which is how normally a fan is used).

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Usually, pipe length in the equation for Q is something which is independent from the diameter. Meaning you can change the diameter and the length of the pipe stays the same as it is bound to the stable solid of the pipe material. Therefore, if you change the diameter, the flow rate changes following your equation - but the length stays what it is.

It's really hard to imagine an experiment where the flow Q and the pressure are forced to be constant, then the diameter is changed and the length is made variable. Which material should that pipe be of? How do you plan to force the flow and the pressure to be constant? Impossible from my point of view.

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