Pipe Bend - MapleSim Help
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Pipe Bend

Lossy model of a circular pipe

 Description 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.
 Equations 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 ($\mathrm{Model Type}=\mathrm{Idelchik - Circular}$): ${A}_{1}=\left\{\begin{array}{cc}0.9\mathrm{sin}\left(\mathrm{\theta }\right)& \mathrm{\theta }\le 70\\ 0.7+\frac{0.35}{90}\mathrm{\theta }& 100\le \mathrm{\theta }\\ 0& \mathrm{otherwise}\end{array}$ ${A}_{2}=\mathrm{MapleSim.Interpolate1D}\left(\mathrm{data},\frac{{R}_{0}}{{\mathrm{D}}_{h}}\right)$ ${B}_{\mathrm{loc}}={A}_{1}=\left\{\begin{array}{cc}\frac{0.21}{{\left(\frac{{R}_{0}}{{\mathrm{D}}_{h}}\right)}^{0.5}}& 1\le \frac{{R}_{0}}{{\mathrm{D}}_{h}}\\ \frac{0.21}{{\left(\frac{{R}_{0}}{{\mathrm{D}}_{h}}\right)}^{2.5}}& \mathrm{otherwise}\end{array}$ ${c}_{\mathrm{loc}}=1$ ${k}_{\delta }=\left\{\begin{array}{cc}\left\{\begin{array}{cc}1& \mathrm{Re}\le 40000\\ \mathrm{min}\left(1.5,\mathrm{max}\left(1,1+500\frac{\mathrm{\epsilon }}{{\mathrm{D}}_{\mathrm{h_act}}}\right)\right)& \mathrm{otherwise}\end{array}& \frac{{R}_{0}}{{\mathrm{D}}_{{0}_{\mathrm{Bend}}}}\\ \left\{\begin{array}{cc}1& \mathrm{Re}\le 40000\\ \mathrm{min}\left(2,\mathrm{max}\left(1,\frac{{\mathrm{\lambda }}_{\mathrm{tur_roughness}}}{{\mathrm{\lambda }}_{\mathrm{tur_smooth}}}\right)\right)& 40000<\mathrm{Re}<200000\\ \mathrm{min}\left(2,\mathrm{max}\left(1,1+1000\frac{\mathrm{\epsilon }}{{\mathrm{D}}_{\mathrm{h_act}}}\right)\right)& \mathrm{otherwise}\end{array}\end{array}$ Friction coefficient of smooth pipe for ${k}_{\mathrm{Re}}$: ${\mathrm{\lambda }}_{\mathrm{tur_smooth}}=\frac{1}{4}{\left(\frac{1}{\mathrm{log10}\left(\frac{5.74}{{\mathrm{max}\left(\mathrm{Re},1\right)}^{0.9}}\right)}\right)}^{2}$ ${\mathrm{\lambda }}_{\mathrm{tur_roughness}}=\frac{1}{4}{\left(\frac{1}{\mathrm{log10}\left(\frac{\mathrm{\epsilon }}{3.7{\mathrm{D}}_{\mathrm{h_act}}}+\frac{5.74}{{\mathrm{max}\left(\mathrm{Re},1\right)}^{0.9}}\right)}\right)}^{2}$ Correction factor ${k}_{\mathrm{Re}}$ (Reynolds number dependency) ${k}_{\mathrm{Rey}}=\mathrm{MapleSim.Interpolate1D}\left(\mathrm{data},\mathrm{Re}\right)$ Friction resistance is defined with ${\mathrm{zeta}}_{\mathrm{fri}}=\mathrm{\theta }\mathrm{\lambda }\frac{{R}_{0}}{{\mathrm{D}}_{h}}$ Total resistance is defined with ${\mathrm{zeta}}_{\mathrm{act}}={\mathrm{zeta}}_{\mathrm{loc}}+{\mathrm{zeta}}_{\mathrm{fri}}$ $\mathrm{Re}=q\mathrm{D}\frac{{\mathrm{D}}_{h}}{a\mathrm{nu}}\phantom{\rule[-0.0ex]{2.5ex}{0.0ex}}{\mathrm{D}}_{h}=4\frac{A}{U}\phantom{\rule[-0.0ex]{2.5ex}{0.0ex}}A=\mathrm{\pi }\frac{{\mathrm{D}}^{2}}{4}$ ${f}_{L}=64\frac{{f}_{T}}{\mathrm{Re}}\phantom{\rule[-0.0ex]{3.0ex}{0.0ex}}{f}_{T}={f}_{\mathrm{Colebrook}}\left({\mathrm{Re}}_{T},\frac{\mathrm{\epsilon }}{{\mathrm{D}}_{h}}\right)$ $\mathrm{mode}=\left\{\begin{array}{cc}{\mathrm{pos}}_{\mathrm{turbulent}}& {\mathrm{Re}}_{T}<\mathrm{Re}\\ {\mathrm{neg}}_{\mathrm{turbulent}}& {\mathrm{Re}}_{T}<-\mathrm{Re}\\ {\mathrm{pos}}_{\mathrm{mixed}}& {\mathrm{Re}}_{L}<\mathrm{Re}\\ {\mathrm{net}}_{\mathrm{mixed}}& {\mathrm{Re}}_{L}<-\mathrm{Re}\\ \mathrm{laminar}& \mathrm{otherwise}\end{array}$ $\mathrm{\lambda }=\frac{1}{\mathrm{Re}}\left\{\begin{array}{cc}{f}_{\mathrm{Colebrook}}\left(|\mathrm{Re}|,\frac{\mathrm{\epsilon }}{{\mathrm{D}}_{h}}\right)\left|\mathrm{Re}\right|& \left(\mathrm{mode}={\mathrm{pos}}_{\mathrm{turbulent}}\vee \mathrm{mode}={\mathrm{neg}}_{\mathrm{turbulent}}\right)\\ \left({f}_{L}+\frac{{f}_{T}-{f}_{L}}{{\mathrm{Re}}_{T}-{\mathrm{Re}}_{L}}\left(\left|\mathrm{Re}\right|-{\mathrm{Re}}_{L}\right)\right)\left|\mathrm{Re}\right|& \left(\mathrm{mode}={\mathrm{pos}}_{\mathrm{mixed}}\vee \mathrm{mode}={\mathrm{neg}}_{\mathrm{mixed}}\right)\\ 64& \mathrm{otherwise}\end{array}$ ${f}_{\mathrm{Colebrook}}=\left(\mathrm{Re},{\mathrm{\epsilon }}_{\mathrm{D}}\right)\to {\left(1.8{{\mathrm{log}}_{10}\left(\frac{6.9}{\mathrm{Re}}+\left(\frac{{\mathrm{\epsilon }}_{\mathrm{D}}}{3.7}\right)\right)}^{1.11}\right)}^{-2}$ $p={p}_{A}-{p}_{B}=\frac{1}{2}{\mathrm{zeta}}_{\mathrm{total}}\mathrm{\rho }{v}^{2}$ $q={q}_{A}=-{q}_{B}=\mathrm{Re}A\frac{\mathrm{nu}}{{\mathrm{D}}_{h}}$ $v=\frac{q}{A}$ 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. $\mathrm{\lambda }=\frac{\frac{1}{2}\mathrm{Maplesoft.Hydraulics.Restrictions.ColebrookFriction}\left(\left|\mathrm{Re}\right|,{\mathrm{Re}}_{T},\frac{\mathrm{\epsilon }}{{\mathrm{D}}_{h}}\right)\left|\mathrm{Re}\right|\left(1+\mathrm{mode}\right)+\frac{1}{2}\mathrm{Ks}\left(1-\mathrm{mode}\right)}{\mathrm{max}\left(0.1,\mathrm{Re}\right)}$ $\mathrm{mode}=\mathrm{Maplesoft.Hydraulics.Functions.sat}\left(\left|\mathrm{Re}\right|-\frac{1}{2}{\mathrm{Re}}_{L}-\frac{1}{2}{\mathrm{Re}}_{T},\frac{1}{2}\frac{{\mathrm{Re}}_{+1}}{2}{\mathrm{Re}}_{T}\right)$ $p={p}_{A}-{p}_{B}=\frac{1}{2}{\mathrm{zeta}}_{\mathrm{total}}\mathrm{\rho }{v}^{2}$ $q={q}_{A}=\mathrm{\Re }A\frac{\mathrm{\nu }}{{\mathrm{D}}_{h}}$ $v=\frac{q}{A}$

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

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 $\mathrm{Crane}$ Type of Calculation model modelBend $\mathrm{D}$ $0.01$ $m$ Inner diameter D $\mathrm{\epsilon }$ $2.5·{10}^{-5}$ $m$ Height of inner surface roughness epsilon ${R}_{0}$ $0.1$ $m$ Radius of neutral axis R0 $\mathrm{\theta }$ $\frac{\mathrm{\pi }}{6}$ $\mathrm{rad}$ Angle of bend 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 ${A}_{1}$ $\cdot 45$ Coefficient that allows for the effect of bend angle on the local resistance A1 ${A}_{2}$ $2.·{10}^{3}$ Correction factor A2, Idelchik A2 $B$ $\cdot 21$ Correction factor B, Idelchik B ${k}_{\mathrm{Re}}$ $2$ Correction factor k_re, Idelchik K_Rey ${\mathrm{zeta}}_{\mathrm{corr}}$ $1$ Correction

 See Also