Pressure drop Straight Piping    

Pressure drop Straight Piping



Pressure drop of the viscous flow through straight pipes



Δ p = f · l/d · ρ/2 · v2       (Darcy-Weisbach formula)

dimensioned variables (metric as example)
Δ p[kg/(m s²) = 10-5 bar = 10-2mbar]   : Pressure drop of the straight pipe
l[m]: Length
d[m]: Internal Diameter
ρ[kg/m³]: Density
v[m/s]: Average speed in the pipe
η[kg/(m s) = Pas]: Dyn. Viscosity
ε [m]: Roughness
dimensionless values
Re[-]: Reynolds number Re = ρ v d / η
f[-] : Darcy Friction Factor


For Re < 2320 (laminar flow) the Darcy Friction Factor is calculated according to Hagen-Poiseuille:

f = 64 / Re         Hagen Poiseuille / Couette Hagen Poiseuille and Couette Flow


For Re > 2320 (turbulent flow) the Darcy Friction Factor is calculated according to [Colebrook 1939, page 137]:

   

For Re => ∞ (completely turbulent flow) the equation of Colebrook converges to the equation of [Prandtl 1933, page 110] according to Kármán:





and for ε/d => 0 (smooth pipe) to the equation of [Prandtl 1933, page 111]:



Laminar flow is possible for Re < 8000 [VDI 1984, page Lb1]. In the critical transition zone (2320 <= Re < 8000) the value of the Darcy Friction Factor might be overestimated by the means of calculation presented above.

The Moody-diagram [Moody 1944, page 672] is a successful illustration of the equations presented above.

Moody 10-10^6  Moody 100-10^8
Moody
Re: 10-106
  Moody
Re: 100-108

With 1/f^0,5 = Re/200 · ε /d   and    1/f^0,5 = 2log(r/ε) + 1,74   [Moody 1944, page 676] draws a line between the transition zone and the completely turbulent pipe flow.





Literature
[Colebrook 1939] C.F. Colebrook, Turbulent Flow in Pipes, with particular reference to the Transition Region between the Smooth and Rough Pipe Laws, Journal of The Institution of Civil Engineers, Volume 11, 1939
[Moody 1944] L.F.Moody, Friction Factors of Pipe Flow, Transactions of the ASME, Volume 66, 1944
[Prandtl 1933] L. Prandtl, Neuere Ergebnisse der Turbulenzforschung, Zeitschrift des Vereins Deutscher Ingenieure, Band 77, 1933
[VDI 1984] VDI-Wärmeatlas 4. Auflage 1984



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