The Pipe Bend component models a circular pipe with losses caused by the bending of flow. The pressure drop is computed with the Darcy equation, with the friction factor determined using the Haaland approximation for turbulent flow along with correction factors due to the bend which contributes to the total resistance to the flow inside the pipe.

If $\mathrm{Model\; Type}\=\mathrm{Crane}$:

$K\=\mathrm{MapleSim.Interpolate1D}\left(\mathrm{Crane\_data}\,\frac{R_0}{{\mathrm{D}}_h}\right)$

${\mathrm{zeta}}_{\mathrm{loc}}\=\mathrm{\lambda }K$

${\mathrm{zeta}}_{\mathrm{fri}}\=2\mathrm{\lambda }\frac{\mathrm{\pi }}{4}\frac{R_0}{{\mathrm{D}}_h}$

${\mathrm{zeta}}_{\mathrm{total}}\={\mathrm{zeta}}_{\mathrm{loc}}\+{\mathrm{zeta}}_{\mathrm{fri}}$

otherwise ($\mathrm{Model\; Type}\=\mathrm{Idelchik\; -\; Circular}$):

$A_1\=\{\begin{array}{cc}0.9\sin \left(\mathrm{\theta }\right)& \mathrm{\theta }\le 70\\ 0.7\+\frac{0.35}{90}\mathrm{\theta }& 100\le \mathrm{\theta }\\ 0& \mathrm{otherwise}\end{array}$

$A_2\=\mathrm{MapleSim.Interpolate1D}\left(\mathrm{data}\,\frac{R_0}{{\mathrm{D}}_h}\right)$

$B_{\mathrm{loc}}\=A_1\=\{\begin{array}{cc}\frac{0.21}{{\left(\frac{R_0}{{\mathrm{D}}_h}\right)}^{0.5}}& 1\le \frac{R_0}{{\mathrm{D}}_h}\\ \frac{0.21}{{\left(\frac{R_0}{{\mathrm{D}}_h}\right)}^{2.5}}& \mathrm{otherwise}\end{array}$

$c_{\mathrm{loc}}\=1$

$k_{\delta }\&equals;\{\begin{array}{cc}\{\begin{array}{cc}1& \mathrm{Re}\le 40000\\ \min \left(1.5\&comma;\max \left(1\&comma;1\&plus;500\frac{\mathrm{\epsilon }}{{\mathrm{D}}_{\mathrm{h\_act}}}\right)\right)& \mathrm{otherwise}\end{array}& \frac{R_0}{{\mathrm{D}}_{0_{\mathrm{Bend}}}}\\ \{\begin{array}{cc}1& \mathrm{Re}\le 40000\\ \min \left(2\&comma;\max \left(1\&comma;\frac{{\mathrm{\lambda }}_{\mathrm{tur\_roughness}}}{{\mathrm{\lambda }}_{\mathrm{tur\_smooth}}}\right)\right)& 40000<\mathrm{Re}<200000\\ \min \left(2\&comma;\max \left(1\&comma;1\&plus;1000\frac{\mathrm{\epsilon }}{{\mathrm{D}}_{\mathrm{h\_act}}}\right)\right)& \mathrm{otherwise}\end{array}\end{array}$

Friction coefficient of smooth pipe for $k_{\mathrm{Re}}$:

${\mathrm{\lambda }}_{\mathrm{tur\_smooth}}\&equals;\frac{1}{4}{\left(\frac{1}{\mathrm{log10}\left(\frac{5.74}{{\max \left(\mathrm{Re}\&comma;1\right)}^{0.9}}\right)}\right)}^2$

${\mathrm{\lambda }}_{\mathrm{tur\_roughness}}\&equals;\frac{1}{4}{\left(\frac{1}{\mathrm{log10}\left(\frac{\mathrm{\epsilon }}{3.7{\mathrm{D}}_{\mathrm{h\_act}}}\&plus;\frac{5.74}{{\max \left(\mathrm{Re}\&comma;1\right)}^{0.9}}\right)}\right)}^2$

Correction factor $k_{\mathrm{Re}}$ (Reynolds number dependency)

$k_{\mathrm{Rey}}\&equals;\mathrm{MapleSim.Interpolate1D}\left(\mathrm{data}\&comma;\mathrm{Re}\right)$

Friction resistance is defined with

${\mathrm{zeta}}_{\mathrm{fri}}\&equals;\mathrm{\theta }\mathrm{\lambda }\frac{R_0}{{\mathrm{D}}_h}$

Total resistance is defined with

${\mathrm{zeta}}_{\mathrm{act}}\&equals;{\mathrm{zeta}}_{\mathrm{loc}}\&plus;{\mathrm{zeta}}_{\mathrm{fri}}$

$\mathrm{Re}\&equals;q\mathrm{D}\frac{{\mathrm{D}}_h}{a\mathrm{nu}}{\mathrm{D}}_h\&equals;4\frac{A}{U}A\&equals;\mathrm{\pi }\frac{{\mathrm{D}}^2}{4}$

$f_L\&equals;64\frac{f_T}{\mathrm{Re}}{f}_T\&equals;f_{\mathrm{Colebrook}}\left({\mathrm{Re}}_T\&comma;\frac{\mathrm{\epsilon }}{{\mathrm{D}}_h}\right)$

$\mathrm{mode}\&equals;\{\begin{array}{cc}{\mathrm{pos}}_{\mathrm{turbulent}}& {\mathrm{Re}}_T<\mathrm{Re}\\ {\mathrm{neg}}_{\mathrm{turbulent}}& {\mathrm{Re}}_T<-\mathrm{Re}\\ {\mathrm{pos}}_{\mathrm{mixed}}& {\mathrm{Re}}_L<\mathrm{Re}\\ {\mathrm{net}}_{\mathrm{mixed}}& {\mathrm{Re}}_L<-\mathrm{Re}\\ \mathrm{laminar}& \mathrm{otherwise}\end{array}$

$\mathrm{\lambda }\&equals;\frac{1}{\mathrm{Re}}\{\begin{array}{cc}{f}_{\mathrm{Colebrook}}\left(\&verbar;\mathrm{Re}\&verbar;\&comma;\frac{\mathrm{\epsilon }}{{\mathrm{D}}_h}\right)\left|\mathrm{Re}\right|& \left(\mathrm{mode}\&equals;{\mathrm{pos}}_{\mathrm{turbulent}}\vee \mathrm{mode}\&equals;{\mathrm{neg}}_{\mathrm{turbulent}}\right)\\ \left(f_L\&plus;\frac{f_T-f_L}{{\mathrm{Re}}_T-{\mathrm{Re}}_L}\left(\left|\mathrm{Re}\right|-{\mathrm{Re}}_L\right)\right)\left|\mathrm{Re}\right|& \left(\mathrm{mode}\&equals;{\mathrm{pos}}_{\mathrm{mixed}}\vee \mathrm{mode}\&equals;{\mathrm{neg}}_{\mathrm{mixed}}\right)\\ 64& \mathrm{otherwise}\end{array}$

$f_{\mathrm{Colebrook}}\&equals;\left(\mathrm{Re}\&comma;{\mathrm{\epsilon }}_{\mathrm{D}}\right)\to {\left(1.8{{\log }_{10}\left(\frac{6.9}{\mathrm{Re}}\&plus;\left(\frac{{\mathrm{\epsilon }}_{\mathrm{D}}}{3.7}\right)\right)}^{1.11}\right)}^{\mathrm{-2}}$

$p\&equals;p_A-p_B\&equals;\frac{1}{2}{\mathrm{zeta}}_{\mathrm{total}}\mathrm{\rho }{v}^2$

$q\&equals;q_A\&equals;-q_B\&equals;\mathrm{Re}A\frac{\mathrm{nu}}{{\mathrm{D}}_h}$

$v\&equals;\frac{q}{A}$

References

[1] : Crane : Flow of Fluids Through Valves, Fittings, and Pipes, Crane LTD, Technical Paper No. 410M

[2] : Idelchik,I.E.: Handbook of hydraulic resistance, Jaico Publishing House, Mumbai, 3rd edition, 2006.

[3] : Swamee P.K., Jain A.K. (1976): Explicit equations for pipe-flow problems, Proc. ASCE, J.Hydraul. Div., 102 (HY5), pp. 657-664.

$\mathrm{\lambda }\&equals;\frac{\frac{1}{2}\mathrm{Maplesoft.Hydraulics.Restrictions.ColebrookFriction}\left(\left|\mathrm{Re}\right|\&comma;{\mathrm{Re}}_T\&comma;\frac{\mathrm{\epsilon }}{{\mathrm{D}}_h}\right)\left|\mathrm{Re}\right|\left(1\&plus;\mathrm{mode}\right)\&plus;\frac{1}{2}\mathrm{Ks}\left(1-\mathrm{mode}\right)}{\max \left(0.1\&comma;\mathrm{Re}\right)}$

$\mathrm{mode}\&equals;\mathrm{Maplesoft.Hydraulics.Functions.sat}\left(\left|\mathrm{Re}\right|-\frac{1}{2}{\mathrm{Re}}_L-\frac{1}{2}{\mathrm{Re}}_T\&comma;\frac{1}{2}\frac{{\mathrm{Re}}_{\mathrm{+1}}}{2}{\mathrm{Re}}_T\right)$

$p\&equals;p_A-p_B\&equals;\frac{1}{2}{\mathrm{zeta}}_{\mathrm{total}}\mathrm{\rho }{v}^2$

$q\&equals;q_A\&equals;\mathrm{\Re }A\frac{\mathrm{\nu }}{{\mathrm{D}}_h}$

$v\&equals;\frac{q}{A}$