Assuming that water ( total amount $Q$) flows down from one river/drain, into multiple rivers/drains, under the action of gravity, as shown below:

enter image description here

How the flow ($Q$) will be divided among different outflows ($Q_i$)?

Assume that we know that

  1. The drain geometry is predetermined—we know the $x,y,z$ coordinates of the start and end node of the river/drain and we know the cross sectional information and length of the river/drain
  2. The velocity is approximated by manning formula, $$V=\frac{k}{n}R_h^\frac{2}{3} S^\frac{1}{2}$$
  3. The flow is incompressible flow, so $Q=VA$


  1. $R_h$ is the hydraulic radius of the river/drain
  2. $S$ is the slope
  3. $k$ conversion between the SI and English units
  4. $n$ the manning coefficient
  5. $A$ is the cross section area of the river/drain

For one, I know that the water flow volume must be conserved


But I don't know what are the other factors that could help us determine $Q_i$.

Edit: After some research, I think that I need use Bernoulli equation to include headloss in this calculation, but I have no idea how to do it, how to proceed?

  • $\begingroup$ Assuming an incompressible fluid the volume flow rate is equal to the area times the velocity. $\endgroup$
    – Farcher
    Commented Feb 28, 2016 at 5:41
  • $\begingroup$ Yes, I know that. Let me update that in the question $\endgroup$
    – Graviton
    Commented Feb 28, 2016 at 7:32
  • $\begingroup$ You need the flow characteristics of the various components. Size, flow shape (circle etc.), length, roughness. You need the characteristics of the fluid. Viscosity, density. You need the boundary conditions. Pressure at least and possibly others depending on the fluid. Then you need to look up the formulas that apply in the current flow regime. $\endgroup$
    – Dan
    Commented Jan 28, 2022 at 13:16
  • $\begingroup$ @Dan, and do you have a formula for that ? $\endgroup$
    – Graviton
    Commented Jan 29, 2022 at 6:04

1 Answer 1


Each drain has a flow coefficient $C_v$ that relates the flow to the pressure drop. The equation for water is $Q=C_v\sqrt {\Delta P}$ where $Q$ is the flow rate and $\Delta P$ is the pressure drop. The pressure at the junction will rise until the total flow out the three drains matches the flow in from the source. The flow down each drain will then be divided so that the inlet pressures of the drains are equal.

  • $\begingroup$ How to calculate the $\Delta P$? Would appreciate if you can furnish a detailed derivation on a 3 outflow drain system $\endgroup$
    – Graviton
    Commented Mar 8, 2016 at 6:25
  • $\begingroup$ What you are hoping to calculate is the $C_v$ for each drain, which I can't help you with. If the outlet of each drain is at the same pressure, then the $\Delta P$ is common and you can solve for $\Delta P$ as $(\frac Q{\sum C_v})^2$ $\endgroup$ Commented Mar 8, 2016 at 14:56
  • $\begingroup$ As the coordinate of each drain is given, can't we use that to calculate the pressure? I just have this vague idea but don't know how to proceed, would you like to show me? $\endgroup$
    – Graviton
    Commented Mar 9, 2016 at 0:27
  • $\begingroup$ I don't know how to use the information you have in point 1 to compute the $C_v$s. Maybe somebody has some software that does that. There should be tables available online for simple shapes, like round pipes. $\endgroup$ Commented Mar 9, 2016 at 0:31
  • $\begingroup$ in that case, would you like to do a derivation based on $C_v$ and what you know? $\endgroup$
    – Graviton
    Commented Mar 9, 2016 at 0:38

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