Multiple source Pipe Network: Given a known outflow, can we deduce the inflows?

Question

I have the following simple Y shape water pipe network:

Given $S_{out}$ ( in $m^3/s$), can we compute $S_1$ and $S_2$?

For the pipe, we know for each pipe their corresponding diameter $d$, length $L$, roughness coefficient $C$, we also know the height for each node relative to ground, and hence I guess we do know about pressure in this case.

All the pipe properties ( such as $d$, $L$ and $C$) are assumed constant regardless of their position in the pipe layout.

An additional remark here (this remark has nothing to do with my current question; but it serves to identify what the kinds of inputs I have with this question.) For a close loop pipe network, we can use the hazen william formula and hardy cross method to compute the head loss. This means that whatever that is required to be known about the pipe is already known, as long as they are used in hazen william formula and hardy cross method.

My question is, given the above information, is there anyway we can uniquely compute $S_1$ and $S_2$ for the Y shape network above? From what I know, we have one equation but two unknowns ( the conservation of source):

$$S_{out}=S_1+S_2$$

And all the pipe properties are not relevant as far as this Y Shape pipe network is concerned.

My conclusion is that above network is unsolvable. But I hear the claim that it is actually solvable as long as we take into account of the aforementioned pipe properties. But I think this is not the case.

What do you think?

Bonus: It would be great it one can provide a formulation for the solution of a general shape water pipe network, with or without loops.

Answer 1

As you say, the above problem stated barely is not solvable. I add that this is in the sense that there are many solutions, based on a thought experiment approach.

It is unstated in the problem what the pressures at the three nodes of the network are. If the pressures are all the same, the flow will be for practical purposes zero. If the pressures are different, there will be flow. One aspect of the pressure differences is the geometry relative to the local gravitational field, which again is unstated. If this is flat, with no other pressures acting, no flow, if it's vertical, flow.

If the sources you see claiming that this is a solvable problem are knowledgeable, I would assume that they mean "given the relative pressures and geometry". It's clear also that if the way in which the pipes are joined is nontrivially asymmetric, in a practical setting, that may modify the effective C-values of the various pipes (in other words, it may not be enough to measure the C-values of the three pipes before the whole is constructed; knowing whether the effects of the join will have to be taken into account in a given situation is a matter of engineering experience).

Thanks for the links to the hazen william formula and the hardy cross method, which were interesting. Needless to say, my answer is ab initio, I do not have experience in pipework. I hope it's helpful, but I guess I'm taking a risk, having just downvoted kleingordon.

Answer 2

Pipe network problems of this sort are isomorphic to circuit problems. You mention in your question that you are considering closed loop pipe networks. So assuming your diagram forms a closed loop, then in addition to the junction equation expressing conservation of flow (analogous to conservation of current in a circuit junction), you've got loop equations to express that head loss is independent of path taken (analogous to Kirchoff's loop law for circuits). You can find the head loss along each path in terms of the unknown currents, using the pipe geometries to compute the hydraulic resistances.