# Flow rate in split bottleneck

Water flows from top to bottom.

First there are two pipes, one with $$10$$ LPM max flow rate, and another with $$20$$ LPM max flow rate.
They both go to another pipe, which is $$20$$ LPM max flow rate.

The total flow rate will be $$20$$ LPM, because of the lower pipe bottleneck.
What would be the flow rate in the two upper pipes?

Picture of this problem: I thought it may be $$10$$ LPM each or $$13.333333$$ in the fatter, $$6.6666667$$ in the slimmer. Both make sense to me.

I would also like to know which field does this question belong to? I mean more specific than just fluid mechanics. What theoretical material should I read? Any specific topics / equations?

• You claim the flow through the lower, single pipe is 20 LPM, yet you don't know that. Without detailed knowledge of pipe diameters and pipe lengths (and assuming smooth pipes) this cannot be solved. – Gert Nov 22 '15 at 20:05
• let's say that upper fat pipe is twice the area of the upper slimmer pipe, and same area as the lower pipe. is that solvable now? – user1952686 Nov 22 '15 at 20:08
• Using Darcy-Weisbach, I get: $\frac{Q_1}{Q_2}=\bigg(\frac{A_1}{A_2}\bigg )^{5/2}$ for the volumetric flow rates in the upper pipes ($A$ are the cross-sections of the pipes) but without lengths and friction coefficients it's as far as this goes. Total flow rate would of course be $Q=Q_1+Q_2$. – Gert Nov 22 '15 at 21:13
• The continuity equation requires that the flow rates from the fat and thin pipe exiting the upper tank equal the flow rate of the pipe entering the lower tank. In addition, the pressure drop in each of the top pipes must be equal. Finally, as Gert pointed out, you need quite a bit more information before you can set up and solve this problem. – David White Oct 9 '16 at 15:53

## 1 Answer

It completely depends on how the maximum flow rates are enforced.

For one extreme, I could imagine a sensor that losslessly watches the flow and, if the max flow is exceeded, closes a valve to limit the flow. In that case, there'd be no difference between the two pipes up to the smaller pipe's 10LPM limit, so the flow would be equally split.

On the other hand, I can imagine a turbulence-based device where the device's back-pressure soars as you approach the configured limit. In that case, the 10LPM pipe would offer greater restriction than the 20LPM pipe, so the flow would be asymmetric.

On the other other hand, perhaps the flow is shut down if the rate is exceeded, as it is in new propane tanks' flow limiting devices. In that case, the flow rate might be split anywhere from equal (if the 10LPM limit isn't exceeded) to all in the 20LPM tube.

Summary: you need to add more information to your question.

• hi. no active enforcement on flow rate. no measurement. gravity makes water flow downwards. surface area of the pipes are: 2x fox upper fat, x for upper slim, 2x for down pipe. lets say water flow rate in the down pipe is 2y LPM. what is the flow in the upper pipes? how these 2y LPM are distributed between them? – user1952686 Nov 22 '15 at 20:16
• Still no good. A bare pipe has no intrinsic flow limit; it depends on the force behind the flow overcoming the resistance of the pipe. If you put two pipes that you say have a 20LPM limit in series, you'll get less than 20LPM total because the resistance has increased. – Daniel Griscom Nov 22 '15 at 20:24