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Does the Darcy-Weisbach equation give the change in static pressure or does it give the change in total pressure (static plus dynamic)?

I’m assuming here that the static pressure includes the contribution from elevation change which at least some authors seem to exclude from the definition of static pressure.

I can’t seem to find a reliable answer to this very basic question. Even Frank White’s Fluid Mechanics doesn’t seem to answer this for me.

Also wondering why it’s so rare that authors mention what type of pressure they mean. There are several types of pressure, and it’s not necessarily obvious which of those is being discussed and shown in equations.

Or is it obvious and I’m one of the few who didn’t get the memo?

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The Darcy-Weisbach equation applies to an incompressible fluid in a pipe of uniform cross-section. Hence, the velocity (and therefore the dynamic pressure) is required by conservation of volume to be the same all the way along the pipe. As a result, the change in static pressure and the change in total pressure are the same thing, so the answer is both.

(OK, you may have seen people apply something that looks quite like the Darcy-Weisbach equation to a differentially-thin axial slice of a pipe with a compressible flow in it - but I think what they've done is relegated compressibility effects on the frictional energy loss to second order in the thickness of that differentially-thin slice.)

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Loses due to friction take some of the energy from the fluid, which can generally affect both static as well as dynamic pressure. With imaginary pipe with inlet in reservoir higher than outlet, where you can turn friction on and off, 2 simple extreme cases can be shown:

  • There will be some flow in the pipe due to gravity and the flow will be smaller when you turn on the friction, so in this case, only dynamic pressure is affected.

  • In the second case, you want to keep the flow same even with friction, so when you turn on the friction, you also increase pressure in the reservoir to compensate for friction losses. If you end up with the same flow as when the pipe was without friction, the dynamic pressure was clearly unaffected, but you had to change static pressure at the inlet to achieve this.

You can also ask yourself, what does it actually take for the fluid to flow through the pipe. The answer is probably some pressure difference and gravity. Lets disregard gravity for a moment. If you try to use just static pressure difference, or just dynamic pressure difference, you can find examples where this does not work. Take for example a pipe with decreasing cross section along the length. Dynamic pressure at the inlet might be higher than dynamic pressure at the outlet, so dynamic pressure cannot be the driving force in this case. In case of flow in opposite direction, you might end up with similar problem trying to use static pressure as a driving force. However, in both cases, total pressure will be decreasing along the flow, so the total pressure difference can be used as the driving force. When you add gravity to the mix, the driving force of the flow is basically total pressure difference without the part caused by gravity (hydrostatic).

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