Change

Sudden or gradual cross sectional area change in pipe with contraction and enlargement

 Description The Change component models sudden or gradual changes in the diameter of a pipe and captures both contraction and enlargement within the component.
 Equations The ratio of diameters is calculated as $\beta =\left\{\begin{array}{cc}\frac{{d}_{b}}{{d}_{a}}& {d}_{a}>{d}_{b}\\ \frac{{d}_{a}}{{d}_{b}}& \mathrm{otherwise}\end{array}$ If ${d}_{a}>{d}_{b}$     Loss coefficient of contraction at inlet: ${K}_{c}=\left\{\begin{array}{cc}0.8\mathrm{sin}\left(\frac{\mathrm{\theta }}{2}\right)\frac{1-{\mathrm{\beta }}^{2}}{{\mathrm{\beta }}^{4}}& \mathrm{\theta }\le \frac{45}{180}\mathrm{\pi }\\ 0.5\left(1-{\mathrm{\beta }}^{2}\right)\frac{\sqrt{\mathrm{sin}\left(\frac{\mathrm{\theta }}{2}\right)}}{{\mathrm{\beta }}^{4}}& \mathrm{otherwise}\end{array}$     Loss coefficient of enlargement at outlet: ${K}_{e}=\left\{\begin{array}{cc}2.6\mathrm{sin}\left(\frac{\mathrm{\theta }}{2}\right)\frac{{\left(1-{\mathrm{\beta }}^{2}\right)}^{2}}{{\mathrm{\beta }}^{4}}& \mathrm{\theta }\le \frac{45}{180}\mathrm{\pi }\\ \frac{{\left(1-{\mathrm{\beta }}^{2}\right)}^{2}}{{\mathrm{\beta }}^{4}}& \mathrm{otherwise}\end{array}$ otherwise, ${d}_{a}\le {d}_{b}$     Loss coefficient of contraction at inlet: ${K}_{{}_{c}}=\left\{\begin{array}{cc}0.8\mathrm{sin}\left(\frac{\mathrm{\theta }}{2}\right)\left(-{\mathrm{\beta }}^{2}+1\right)& \mathrm{\theta }\le \frac{45}{180}\mathrm{\pi }\\ 0.5\left(-{\mathrm{\beta }}^{2}+1\right)\sqrt{\mathrm{sin}\left(\frac{\mathrm{\theta }}{2}\right)}& \mathrm{otherwise}\end{array}$     Loss coefficient of enlargement at outlet: ${K}_{{}_{e}}=\left\{\begin{array}{cc}2.6\mathrm{sin}\left(\frac{\mathrm{\theta }}{2}\right){\left(-{\mathrm{\beta }}^{2}+1\right)}^{2}& \mathrm{\theta }\le \frac{45}{180}\mathrm{\pi }\\ {\left(-{\mathrm{\beta }}^{2}+1\right)}^{2}& \mathrm{otherwise}\end{array}$ The total loss coefficient is defined by using the linear approximation for the transition between contraction and enlargement: $K=\left\{\begin{array}{cc}{K}_{e}& \mathrm{dp}<-{\mathrm{dp}}_{{}_{\mathrm{transition}}}\\ \\ \frac{{K}_{{}_{c}}-{K}_{{}_{e}}}{2{\mathrm{dp}}_{{}_{\mathrm{transition}}}}\mathrm{dp}+\frac{{K}_{{}_{c}}+{K}_{{}_{e}}}{2}& ,-{\mathrm{dp}}_{{}_{\mathrm{transition}}}\le \mathrm{dp}\le {\mathrm{dp}}_{{}_{\mathrm{transition}}}\\ {K}_{{}_{c}}& \mathrm{otherwise}\end{array}$ $p={p}_{A}-{p}_{B}=\frac{\mathrm{\pi }}{4}\mathrm{\rho }\mathrm{\nu }q\frac{{\left(16\frac{{q}^{4}}{{\mathrm{\pi }}^{2}{A}_{\mathrm{cs}}^{2}{\mathrm{\nu }}^{4}}+{\mathrm{Re}}_{\mathrm{Cr}}^{4}\right)}^{\frac{1}{4}}}{{C}_{d}^{2}{A}_{\mathrm{cs}}\sqrt{\mathrm{\pi }{A}_{\mathrm{cs}}}}$ $q={q}_{A}=-{q}_{B}$ ${v}_{a}=4\frac{q}{{d}_{a}^{2}}\mathrm{\pi }\phantom{\rule[-0.0ex]{4.0ex}{0.0ex}}{v}_{b}=4\frac{q}{{d}_{b}^{2}}\mathrm{\pi }$ References [1] : Flow of Fluids Through Valves, Fittings, and Pipes, Crane Valves North America, Technical Paper No. 410M. 1979, p A-26

Variables

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

Connections

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

Parameters

General

 Name Default Units Description Modelica ID ${d}_{a}$ $0.05$ $m$ Diameter, port_a side d_a ${d}_{b}$ $0.02$ $m$ Diameter, port_b side d_b $\mathrm{\theta }$ $\frac{\mathrm{\pi }}{9}$ $\mathrm{rad}$ Angle of Contraction/Enlargement theta ${\mathrm{Re}}_{\mathrm{Cr}}$ $12$ Reynolds number at critical flow ReCr ${\mathrm{dp}}_{\mathrm{transition}}$ $10$ $\mathrm{Pa}$ Pressure difference for transition zone $\left|\mathrm{dp}\right|<{\mathrm{dp}}_{\mathrm{transition}}$ p_transition

Fluid Parameters

The following parameters, used in the equations, are properties of the Hydraulic System Properties component used in the model.

 Name Units Description Modelica ID $\mathrm{\nu }$ $\frac{{m}^{2}}{s}$ Kinematic viscosity of fluid nu $\mathrm{\rho }$ $\frac{\mathrm{kg}}{{m}^{3}}$ Density of fluid rho $\mathrm{El}$ $\mathrm{Pa}$ Bulk modulus of fluid El