Pressure drop in pipe bends
Δ p = K · ρ/2 · v^{2}
 
Δ p  : Pressure drop of a bend 90º 
K 
: Resistance coefficient (Friction Factor) of a bend 90º 
ρ  : Density 
v  : Average Speed in the pipe 
Re  : Reynolds number Re = ρ · v · d / η 
μ  : Dyn. Viscosity 
d  : Inner diameter of the bend 
The Resistance coefficient in bends is a function of the Renumber, the roughness of the
pipe wall and the bend geometry. A bend affects the flow pattern in front and behind
the bend. The pressure drop of bends in series is lower or equal than the pressure drop calculated
by adding the pressure loss of every single bend. TECCINESS assumes that the inner diameter
of the pipe equals the inner diameter of the bend.
The Resistance coefficient for
Re < 2320 (laminar flow)
is calculated according to [Ghia 1977, page 648]. The graph plotted by Ghia
has been approximated as follows:
K / K,p = 0.026 · Dean ^0.661 +1

Kp  = π/2 · r/d · 64/Re 
 Resistance coefficientof a straight pipe with the length of the bend´s midline 
Dean  = Re · (d/r)^0.5   Deannumber 
r  Radius of the bend´s midline 
The Resistance coefficient for Re > 2320 (transition or turbulent state)
is calculated by a set of equations presented by [Krüger 1970, page 39 ff].
Krüger takes into consideration the bend radius and the wall roughness.
ε /d   2320 < Re < 2 x 10^5  Re > 2 x 10^5 
> 0.001  


< 0.001 
Re > Re* 


Re < Re* 

0  


f  : Darcy Friction Factor of the straight pipe 
f,o  : Darcy Friction Factor of a smooth straight pipe 
ε  : roughness of the pipe wall 
Re* = (2,89/(1+1000 x ε /d))^12 
K = f(Re)
Literature

[Ghia 1977] K.N. Ghia, J.S. Sokhey, Laminar Incompressible Viscous
Flow in Curved Ducts of Regular CrossSections, Transactions of the
ASME Journal of Fluids Engineering, December 1977, page 640 ff 
[Kittredge 1957] C.P. Kittredge, D.S. Rowley, Resistance Coefficients for Laminar
and Turbulent Flow Through OneHalfInch Valves and Fittngs, Transactions of the
American Society of Mechanical Engineering ASME, Volume 79 January 1957, Fig. 6 
[Krüger 1970] H. Krüger, Berechnung strömungstechnischer
Kennwerte von Durchströmteilen für Flüssigkeiten und Gase, Mitteilungen
des Institut für Leichtbau und ökonomische Verwendug von Werkstoffen, IfL
Mitteilungen, Dresden 1970, Beilage 
[Miller 2008] D.S. Miller, Internal Flow Systems, 2nd Edition 2008, Miller Innovations
Bredford UK, page 223 f 
[Ward Smith 1980] A.J. WardSmith, The fluid dynamics of flow in pipes
and ducts, Clarendon Press, Oxford 1980, pages 248268 
[White 1929] C.M. White, Streamline flow through curvd pipes.
Proc. Royal Soc., London, 123 (1929), page 645, cited by [Krüger 1970, page 39] 
