I've been studying Fluid Mechanics recently and on the book Fundamentals of Fluid Mechanics 7e, Munson et al. on page 456, the book states

Another common multiple pipe system contains pipes in parallel ... However, by writing the energy equation between points A and B it is found that head losses experienced by any fluid particle traveling between these two locations is the same, independent of the path taken.

Can someone please clarify this for me. How is it possible that two pipes different in diameter have the same losses?

I thought,

$$h_L = f\frac{l}{d_i}\frac{v^2}{2g}$$

Edit 1:

I'm sorry that I wasn't clear enough. Here's a diagram I sketched on the problem at hand. As you can see the pipes are parallel, and the flow is unidirectional, that is from A to B.

enter image description here

However they all have different diameters, but are equal in length.

Note: Regarding the name and nomenclature. These aren't heat losses, they're actually called head losses.

The equation is called the major head loss during a viscous flow in a pipe. According to this equation, the losses experienced by the fluid are proportional to the velocity squared and to a roughness factor f. The rest is also self-explanatory I believe. What I can't seem to grasp is how this is equal to the three pipes in this configuration.

  • $\begingroup$ 1. Can you clarify the pipe configuration a bit more? Coaxial, just touching, or separated. 2. Co-flowing or opposite? 3. I don't get your equation. $\endgroup$ – Bernhard Jun 9 '14 at 5:21
  • $\begingroup$ Heat loss or head loss makes a HUGE difference ;) No wonder I was confused. $\endgroup$ – Bernhard Jun 9 '14 at 11:43

The head loss is the requiring pressure to create a given flow. The head loss will be the same for the tree pipes (if we neglect potential difference due to gravity and pipe height) since it is set by pressure difference between tank A and B. But flows through the tree pipes will be different. For a given head loss, at constant friction factor, flow will be greater for larger pipe diameter.


The equation you cited is called the Darcy–Weisbach equation. It is a phenomenological law (such as Ohm Law) where the friction factor (Darcy factor is dependent of the flow). It can be assessed using Moody diagram (friction factor versus Reynolds number) or dimensionless formula.

To overcome your difficulty you must realize that:

  • Your pressure difference is set by the fluid column height difference between A and B;
  • All your pipes are subject to the same pressure difference as long as they are not too far from each other and equilibrium level is high before pipe height.
  • Your head loss for each pipe will equals the pressure difference between tanks because there is nothing else in serie;
  • Flow through pipes are different because of their Darcy friction factor and diameters are different.
  • $\begingroup$ Can you please explain what do you mean by neglecting pipe geometry? $\endgroup$ – gezibash Jun 9 '14 at 15:09
  • $\begingroup$ I mean we can assume that their respective heights are not too different and therefore there is no significant pressure differences in each inlet due to different $\rho g z$ term. $\endgroup$ – jlandercy Jun 9 '14 at 15:13

Diameter factor that you have mentioned in head loss equation will be compensated by the flow rate (Q) or Velocity(V) in the numerator. In Parallel pipes, flow rate is divided among the branches based on the area of the pipe.


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