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I am working on designing an experiment where I will be creating a wall-jet flow where the wall is a rotating wing. After some thought I realized I need to estimate the pressure losses due to the pipe outlet splitting into multiple branches before opening up to a large tank.

Another question on physics stack exchange gives some good insights on how to calculate the flow rate in a single branch but I am more interested in the head loss from just before the split to after the split. I know that if there was only one branch I would only need to consider the exiting frictional losses, but since there multiple exits I'm not sure how to calculate the major losses due to the split and exits.

Here are my underlying assumptions.

  • The flow through all the branches are incompressible.
  • The flow before the split will already have become fully developed and is laminar.
  • I want the exiting flow from each branch to also be fully developed and laminar.
  • All the branches (main and offshoots) are smooth, and curvature can be neglected so the major head loss in the branches is $f=64/Re$. The major head losses should only come from the split, and the exit.
  • All the branches are of constant diameter $d$, with equal lengths $L$, and the diameter before the split is $D$. Therefore in my diagram $Q_1=Q_2=Q_3=...=Q_N$.
  • The change in potential energy is negligible.
  • The pressure at the inlet and the outlet are the same.

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From my undergraduate textbook (Fluid Mechanics - Fundamentals and Applications by Cengel and Cimbala) in Ch. 8 it describes pipe flows in series and in parallel toward the end of the chapter. In Fig. 8-42,43 $h_L$ is the minor head loss $f\frac{L}{d}\frac{V^2}{2g}$.

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The energy equation for the flow along a streamline starting after the pump through the split and out one of the branches is $$\alpha_{in}\frac{V_{in}^2}{2g}+h_{pump}=\alpha_{out}\frac{V_{out}^2}{2g}+h_L$$

where

  • $\alpha$ is the kinetic energy correction factor (2 for fully developed laminar flow).
  • $V_{in}$ is the fully developed velocity after the pump.
  • $h_{pump}$ is the minor head loss due to the pump, for now treat it as a constant that I will look up from the manufacturer's specs later.
  • $V_{out}$ is the velocity leaving any one of the branches. They should all be the same due to constant cross sectional area and incompressibility.
  • $h_L$ is the sum of the minor and major losses from the piping.

So then the total minor and major head losses should be $$h_L=h_{in}+h_{split}+h_{out}=\frac{64}{\frac{V_{in}D}{\nu}}\frac{L_{in}}{D}\frac{V_{in}^2}{2g}+h_{split}+\left(\frac{64}{\frac{V_{out}d}{\nu}}\frac{L}{d}+\alpha_{out}\right)\frac{V_{out}^2}{2g}$$

where the second term in the $h_{out}$ equation comes from the text I've cited for a pipe exit.

I think I could model $h_{split}$ as multiple sudden contractions, in which case for a single branch. $$h_{split}=\left(1-\frac{d^2}{D^2}\right)^2\frac{V_{out}^2}{2g}$$

Assuming the relationships for parallel flows applies here, how do I change $h_{split}+h_{out}$ to incorporate multiple branches?

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