Elbow Bend

Lossy model of a Elbow

 Description The Elbow Bend component models a circular pipe with losses caused by the sharp 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.
 Equations If $\mathrm{Model Type}=\mathrm{Crane - Miter bend}$: $K=\mathrm{Interpolate1D}\left(\mathrm{Crane_data},\mathrm{\theta }\right)$ ${\mathrm{zeta}}_{\mathrm{loc}}=\mathrm{\lambda }K$ ${\mathrm{zeta}}_{\mathrm{fri}}=2\frac{L}{{\mathrm{D}}_{h}}\mathrm{\lambda }$ ${\mathrm{zeta}}_{\mathrm{total}}={\mathrm{zeta}}_{\mathrm{loc}}+{\mathrm{zeta}}_{\mathrm{fri}}$  From [1] $L=2{\mathrm{D}}_{h}$ otherwise ($\mathrm{Model Type}=\mathrm{Idelchik - Circular}$ $\left[\frac{{R}_{0}}{{\mathrm{D}}_{h}}<3.0,0<\mathrm{\theta }<\mathrm{\pi },\frac{L}{{\mathrm{D}}_{h}}\ge 10\right]$ ): ${A}_{1}=\mathrm{Interpolate1D}\left(\mathrm{data},\mathrm{\theta }\right)$ ${c}_{\mathrm{loc}}=1$ $\mathrm{k__δ}=\left\{\begin{array}{cc}1& \mathrm{Re}\le 4{10}^{4}\\ \mathrm{min}\left(1.5,\mathrm{max}\left(1,1+500\frac{\mathrm{\epsilon }}{{\mathrm{D}}_{{}_{{h}_{\mathrm{act}}}}}\right)\right)& \mathrm{otherwise}\end{array}$ ${K}_{\mathrm{Rey}}=\mathrm{Interpolate1D}\left(\mathrm{data},\mathrm{Re}\right)$ ${\mathrm{zeta}}_{\mathrm{total}}={\mathrm{zeta}}_{\mathrm{fri}}$ $\mathrm{Re}=q\mathrm{D}\frac{{\mathrm{D}}_{h}}{a\mathrm{nu}}\phantom{\rule[-0.0ex]{2.5ex}{0.0ex}}{\mathrm{D}}_{h}=4\frac{A}{U}\phantom{\rule[-0.0ex]{2.5ex}{0.0ex}}A=\mathrm{\pi }\frac{{\mathrm{D}}^{2}}{4}$ ${f}_{L}=64\frac{{f}_{T}}{\mathrm{Re}}\phantom{\rule[-0.0ex]{3.0ex}{0.0ex}}{f}_{T}={f}_{\mathrm{Colebrook}}\left({\mathrm{Re}}_{T},\frac{\mathrm{\epsilon }}{{\mathrm{D}}_{h}}\right)$ $\mathrm{mode}=\left\{\begin{array}{cc}{\mathrm{pos}}_{\mathrm{turbulent}}& {\mathrm{Re}}_{T}<\mathrm{Re}\\ {\mathrm{neg}}_{\mathrm{turbulent}}& {\mathrm{Re}}_{T}<-\mathrm{Re}\\ {\mathrm{pos}}_{\mathrm{mixed}}& {\mathrm{Re}}_{L}<\mathrm{Re}\\ {\mathrm{net}}_{\mathrm{mixed}}& {\mathrm{Re}}_{L}<-\mathrm{Re}\\ \mathrm{laminar}& \mathrm{otherwise}\end{array}$ $\mathrm{\lambda }=\frac{1}{\mathrm{Re}}\left\{\begin{array}{cc}{f}_{\mathrm{Colebrook}}\left(|\mathrm{Re}|,\frac{\mathrm{\epsilon }}{{\mathrm{D}}_{h}}\right)\left|\mathrm{Re}\right|& \left(\mathrm{mode}={\mathrm{pos}}_{\mathrm{turbulent}}\vee \mathrm{mode}={\mathrm{neg}}_{\mathrm{turbulent}}\right)\\ \left({f}_{L}+\frac{{f}_{T}-{f}_{L}}{{\mathrm{Re}}_{T}-{\mathrm{Re}}_{L}}\left(\left|\mathrm{Re}\right|-{\mathrm{Re}}_{L}\right)\right)\left|\mathrm{Re}\right|& \left(\mathrm{mode}={\mathrm{pos}}_{\mathrm{mixed}}\vee \mathrm{mode}={\mathrm{neg}}_{\mathrm{mixed}}\right)\\ 64& \mathrm{otherwise}\end{array}$ ${f}_{\mathrm{Colebrook}}=\left(\mathrm{Re},{\mathrm{\epsilon }}_{\mathrm{D}}\right)\to {\left(1.8{{\mathrm{log}}_{10}\left(\frac{6.9}{\mathrm{Re}}+\left(\frac{{\mathrm{\epsilon }}_{\mathrm{D}}}{3.7}\right)\right)}^{1.11}\right)}^{-2}$ $p={p}_{A}-{p}_{B}=\frac{1}{2}{\mathrm{zeta}}_{\mathrm{total}}\mathrm{\rho }{v}^{2}$ $q={q}_{A}=-{q}_{B}=\mathrm{Re}A\frac{\mathrm{nu}}{{\mathrm{D}}_{h}}$ $v=\frac{q}{A}$ 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.

Variables

 Name Value Units Description Modelica ID $p$ $\mathrm{Pa}$ Pressure drop from A to B p $q$ $\frac{{m}^{3}}{s}$ Flow rate from port A to port B q $v$ $\frac{m}{s}$ v

Connections

 Name Description Modelica ID $\mathrm{portA}$ Upstream hydraulic portA $\mathrm{portB}$ Downstream hydraulic port portB

Parameters

 Name Default Units Description Modelica ID Model Type Crane - Miter bend Type of calculation model TypeofModel $\mathrm{D}$ $0.01$ $m$ Inner diameter D $\mathrm{\epsilon }$ $2.5·{10}^{-5}$ $m$ Absolute roughness of pipe, with a default for a smooth steel pipe epsilon $\mathrm{\theta }$ $\frac{\mathrm{\pi }}{6}$ $\mathrm{rad}$ Angle of Elbow theta ${\mathrm{Re}}_{L}$ $2·{10}^{3}$ Reynolds number at transition to laminar flow ReL ${\mathrm{Re}}_{T}$ $4·{10}^{3}$ Reynolds number at transition to turbulent flow ReT Apply Coefficients $\mathrm{false}$ Override ${A}_{1}$ $0.45$ Coefficient that allows for the effect of bend angle on the local resistance A1 ${K}_{\mathrm{Re}}$ $1$ Correction factor K_rey, Idelchik K_Rey