Governing equation for transient pipe flow, part 4

From the last part we ended up with these two equations:

These equations are on a form that is suitable for FSI analysis when combining them with the equations of motion for piping systems (longitudinal, rotational and bending vibrations). They could also be used as is for pure hydraulic transients, but the normal starting point is to use pressure head, H, instead of the pressure, P, and velocity, V instead of flow rate Q (even though Q usually is eventually used in the end) and the diameter, D, instead of the radius, R.

Neglecting the Poisson coupling (sturdy piping structure), the last term in equation 3.19 and reorganizing we get:

Here the wave propagation velocity, a, becomes from the outset (for thin walled pipes):

For most cases this is a good approximation, but Wylie & Streeter have elaborated this for lots of different conditions in their books. Based on experience and practicalities however, the wave propagation velocity is a very fuzzy thing to determine. It can hardly be experimentally verified for any real piping system within +- 20%, mostly due to small, and changing, amounts of gas content, gas pockets and so on. Wylie & Streeter writes a lot about that too, and in the MOC (Method of Characteristics) it is used as a parameter that is changed within limits to discretize large systems. Even though it can be calculated theoretically at very high accuracy, that accuracy simply does not exist in real systems. Having a good starting point, is a good idea nonetheless though, because the difference between polymer based pipes and steel pipes can be more than 100% for instance.

What's missing now is the pipe friction. Diving into this, one discover it is super complex material. To this day, no good general transient friction term exists as far as I know. I have written a lot about it in my thesis. For steady state, Darcy-Weisbach and the Moody diagram is what is used. In lack of a better general non steady state model, Darcy-Weisback is also used for transient simulations, even though "everybody" knows this is far from correct. The best we can say about this, is that at least the calculations becomes conservative. Calculating water hammers that are smaller than in real life, does not normally happen. The exceptions are when FSI becomes important and for fluid cavity collapse. If this is not taken into account, then the calculations are usually not conservative anymore.

The Darcy-Weisbach equation:

Here f is the Darcy-Weisbach friction factor we normally obtain from the Moody diagram. The minus sign is there because of integration when positive x is in positive flow direction. This is simply the steady state result of the momentum equation above, and we can "reverse" it back into that equation to achieve exactly that. The final equations for liquid filled transient pipe flow analysis therefore becomes:

The wave propagation velocity, a, is as shown above.