So I've been trying to find out how to determine what proportion of fluid will flow down a given path when a main line branches into two lines at a 90 degree T-Junction and I can't find anything helpful.

Lets just assume water at room temperature in steady-state conditions.

It seems that pressure losses will have an effect on proportion of distribution. If the two paths have similar designs, ie; the same energy losses, the flow will split roughly 50:50. However, if one path has more energy losses than the other, then < 50% of fluid will go down that path. So there is a relationship between the proportion of pressure losses in the line to the proportion of flow down that path. What is this relationship? Is there a numerical model to determine this?

Also, intuitively, at a 90 degree T-junction, momentum/inertia should have an effect. Obviously the fluid would prefer to continue on straight rather than take a 90 degree turn, especially at high speeds. Do we consider this when determining flow division? Is there a numerical model to help me determine it's effect on how much flow will go down a given path?

Any answers or resources guiding me in the right direction, or numerical models to help me determine flow distribution at a T junction would be greatly appreciated!

Thank you in advance.

  • $\begingroup$ More flow in the direction of least losses. $\endgroup$
    – Solar Mike
    Mar 4, 2021 at 6:46

1 Answer 1


In flow problems, it's generally a good idea to work backwards in realtion to the flow: Find where each branch is ending and the pressure at this point (height, atmospheric pressure, $p_{out 1}, p_{out 2}$ ). Then, find the relation between pressure loss and flowrate for each branch ($\Delta p_1(Q_1), \Delta p_2(Q_2)$).

You know the the pressure at the junction is the same for both flows ($p_{in}$), and that $p_{in}-\Delta p = p_{out}$. So it follows: $p_{out 1}+\Delta p_1(Q_1)=p_{out 2}+\Delta p_2(Q_2)$. Knowing $Q_{total}=Q_1+Q_2$ you can find the flows.

You won't solve this analytically, only numerically.

As for your second question, consider this (admittedly a bit cryptic) diagram:

enter image description here

(from here)

The red dashed line is the $\zeta$ value for flow straight through the pipe, the lines above are for branching flows (at different sizes of the off-branch). You see that the $\zeta$ value is always higher for branching flows, so your hunch in effect correct.

  • $\begingroup$ Hi thank yo so much for your answer, exactly what I'm looking for! I really appreciate it! I just don't fully grasp the diagram. So if we know the ratio of areas, and we want to find the flow rate ratio, how do we find the ζ value? Or is it meant to be used in such a way that you choose a desired volume rate and find the ζ value you need? Sorry, I'm not well versed in hydraulic resistance. I would be ever more thankful if you could help me with this! :) $\endgroup$
    – Lamijops
    Mar 5, 2021 at 1:48
  • $\begingroup$ working with the diagram, I would guess a starting value V3/V2, find the according zeta values and adjust my calculation, reiterate a few times. You could also approximate the T as an albow (with V3) and a straight pipe run with V1. As you can see, when the flow divides 50/50 the zeta value for the branch is approximately double that of the straight part.. $\endgroup$
    – mart
    Mar 5, 2021 at 7:13
  • $\begingroup$ When you need formulas (to implement in spreadsheet), find a recent edition of Idelchiks handbook of hydraulik resistance. $\endgroup$
    – mart
    Mar 5, 2021 at 7:14
  • 1
    $\begingroup$ Just to be perverse. Remember with small branches they are/can be effected by venturi effect. $\endgroup$
    – Brad
    Mar 5, 2021 at 11:43
  • $\begingroup$ lol could have safed me the trouble of writing this answer and instead linked to this: en.wikipedia.org/wiki/Pipe_network_analysis $\endgroup$
    – mart
    Mar 9, 2021 at 10:06

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