Poppet Lift $—$ Ball poppet lift with sharp edge seat

This is a model of a poppet lift.

 Equations $\left\{\begin{array}{cc}X=\mathrm{pre}\left(\mathrm{Xmax}\right)& \mathrm{pre}\left(\mathrm{Xmax}\right)<\mathrm{s_rel_a}\mathrm{switch}+\mathrm{X_0}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{and}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{pre}\left(\mathrm{maX}\right)=\mathrm{true}\\ X=\mathrm{max}\left(\mathrm{Xmin},\mathrm{s_rel_a}\mathrm{switch}+\mathrm{X_0}\right)& \mathrm{otherwise}\end{array}\right\$ $\left\{\begin{array}{cc}\mathrm{fflow}=-\frac{2\mathrm{noEvent}\left(\left|\mathrm{Δp}\right|\right)\mathrm{flowarea}\mathrm{cosalpha}}{\sqrt{{k}_{2}}}& \mathrm{TransitionType}=1\\ \mathrm{fflow}=-2\mathrm{noEvent}\left(\left|\mathrm{Δp}\right|\right)\mathrm{Cdmax}\mathrm{flowarea}\mathrm{cosalpha}& \mathrm{otherwise}\end{array}\right\$ $\left\{\begin{array}{cc}\mathrm{qunsigned}=\frac{\left(\mathrm{Modelica.Fluid.Utilities.regRoot}\left(\mathrm{max}\left(0,{k}_{1}^{2}{\mathrm{\nu }}^{2}{\mathrm{\rho }}^{2}+\frac{32\mathrm{flowarea}{k}_{2}\mathrm{Modelica.Fluid.Utilities.regRoot}\left({\mathrm{Δp}}^{2}\right)\mathrm{\rho }}{\mathrm{\pi }}\right)\right)-{k}_{1}\mathrm{\nu }\mathrm{\rho }\right)\mathrm{Modelica.Fluid.Utilities.regRoot}\left(\mathrm{max}\left(0,\mathrm{flowarea}\mathrm{\pi }\right)\right)}{4{k}_{2}\mathrm{\rho }}& \mathrm{TransitionType}=1\\ \mathrm{qunsigned}=\mathrm{Cdmax}\mathrm{Modelica.Math.tanh}\left(\frac{2\mathrm{\lambda }}{{\mathrm{\lambda }}_{c}}\right)\mathrm{flowarea}\mathrm{Modelica.Fluid.Utilities.regRoot}\left(\frac{2\mathrm{Modelica.Fluid.Utilities.regRoot}\left({\mathrm{Δp}}^{2}\right)}{\mathrm{\rho }}\right)& \mathrm{otherwise}\end{array}\right\$ $\mathrm{msim/ALGORITHM}\left(\mathrm{msim/WHEN}\left(\mathrm{flowareamax}<\mathrm{flowarea},\left\{\mathrm{msim/ASSIGN}\left(\mathrm{Xmax},\mathrm{pre}\left(X\right)\right),\mathrm{msim/ASSIGN}\left(\mathrm{maX},\mathrm{true}\right)\right\},\mathrm{flowarea}<\mathrm{flowareamax},\mathrm{msim/ASSIGN}\left(\mathrm{maX},\mathrm{false}\right)\right)\right)$ $0=f\mathrm{switch}+{f}_{A}+{f}_{B}$ $0=-f\mathrm{switch}+{f}_{A\left(\mathrm{support}\right)}+{f}_{B\left(\mathrm{support}\right)}$ $\mathrm{\nu }={\mathrm{\nu }}_{\mathrm{oil}}\left({p}_{\mathrm{abs}}={\mathrm{px}}_{\mathrm{abs}},T=T,{v}_{\mathrm{air}}={v}_{\mathrm{gas}\left(\mathrm{oil}\right)},{p}_{\mathrm{sat}}={p}_{\mathrm{sat}}\right)$ $\mathrm{\rho }={\mathrm{\rho }}_{\mathrm{oil}}\left({p}_{\mathrm{abs}}={\mathrm{px}}_{\mathrm{abs}},T=T,{v}_{\mathrm{air}}={v}_{\mathrm{gas}\left(\mathrm{oil}\right)},{p}_{\mathrm{sat}}={p}_{\mathrm{sat}}\right)$ $\mathrm{D}=\frac{4\mathrm{flowarea}}{\mathrm{\pi }\left(2a+2R\right)}$ $T={T}_{0\left(\mathrm{oil}\right)}+{\mathrm{ΔT}}_{\mathrm{system}}$ $\mathrm{V2}=\left\{\begin{array}{cc}\mathrm{V02}& X\le \mathrm{Xmin}\\ \mathrm{V02}-\left(X-\mathrm{Xmin}\right)\mathrm{Arod_2}+\mathrm{Hydraulics.Elements.Templates.sphericalCapVolume}\left(\mathrm{_msim_R}=R,\mathrm{_msim_a}=r\right)+\mathrm{Vcone}-\mathrm{Vslice}& \mathrm{otherwise}\end{array}\right\$ $\mathrm{Vcone}=\frac{1}{3}\mathrm{\pi }{r}^{2}X$ $\mathrm{Vslice}=\mathrm{Hydraulics.Elements.Templates.sphericalCapVolume}\left(\mathrm{_msim_R}=R,\mathrm{_msim_a}=a\right)+\frac{1}{3}\mathrm{\pi }{a}^{2}\sqrt{{R}^{2}-{a}^{2}}$ $a=\frac{R}{\sqrt{1+\frac{{X}^{2}}{{r}^{2}}}}$ $f=-\left(-\mathrm{\pi }{a}^{2}+\mathrm{Arod_1}\right){p}_{1}+\left(-\mathrm{\pi }{a}^{2}+\mathrm{Arod_2}\right){p}_{2}-\mathrm{fflow}$ $\mathrm{\lambda }=\frac{\mathrm{D}\mathrm{Modelica.Fluid.Utilities.regRoot}\left(\frac{2\left|\mathrm{Δp}\right|}{\mathrm{ρ}}\right)}{\mathrm{ν}}$ $\mathrm{lift}=X-\mathrm{Xmin}$ ${\mathrm{p1}}_{\mathrm{abs}}={p}_{1}+{p}_{\mathrm{atm}\left(\mathrm{oil}\right)}$ ${\mathrm{p2}}_{\mathrm{abs}}={p}_{2}+{p}_{\mathrm{atm}\left(\mathrm{oil}\right)}$ ${V}_{A}=\mathrm{V1}$ ${\mathrm{px}}_{\mathrm{abs}}={\mathrm{p1}}_{\mathrm{abs}}\mathrm{xp}+{\mathrm{p2}}_{\mathrm{abs}}\left(1-\mathrm{xp}\right)$ $\mathrm{qLmin}=60000\mathrm{qunsigned}$ ${s}_{a\left(\mathrm{rel}\right)}={s}_{A}-{s}_{A\left(\mathrm{support}\right)}$ $\mathrm{xp}=\mathrm{Modelon.Math.Smoothing.cubicStep}\left(\mathrm{Δp}+\frac{1}{2}\right)$ $\mathrm{Δp}={p}_{1}-{p}_{2}$ $\mathrm{cosalpha}=\frac{a}{R}$ $\mathrm{flowarea}=\mathrm{flowarea_nom}$ $\mathrm{flowarea_nom}=\mathrm{Hydraulics.Elements.Templates.truncConeArea}\left(\mathrm{_msim_R}=r,\mathrm{_msim_X}=X-\sqrt{{R}^{2}-{a}^{2}},\mathrm{_msim_h}=X\right)$ $\mathrm{portA.m_flow}=\mathrm{noEvent}\left(\left\{\begin{array}{cc}\mathrm{\rho }\mathrm{qunsigned}& 0\le \mathrm{Δp}\\ -\mathrm{\rho }\mathrm{qunsigned}& \mathrm{otherwise}\end{array}\right\\right)$ $\mathrm{portA1.V}=\mathrm{V2}$ ${s}_{\mathrm{ab}\left(\mathrm{rel}\right)}=L$ ${s}_{\mathrm{ab}\left(\mathrm{rel}\right)}=\mathrm{switch}\left({s}_{B}-{s}_{A}\right)$ ${s}_{B\left(\mathrm{support}\right)}=L\mathrm{switch}+{s}_{A\left(\mathrm{support}\right)}$ ${p}_{1}={p}_{A}$ ${p}_{2}=\mathrm{portA1.p}$ $\mathrm{V1}+\mathrm{V2}=\mathrm{V01}+\mathrm{V02}-\left(X-\mathrm{Xmin}\right)\left(\mathrm{Arod_2}-\mathrm{Arod_1}\right)$ ${m}_{\mathrm{flow}\left(\mathrm{A1}\right)}+{m}_{\mathrm{flow}\left(A\right)}=0$

Variables

 Name Value Units Description Modelica ID ${{s}_{\mathrm{rel}}}_{a}$ $m$ Relative position of flangeA wrt supportA s_rel_a ${{s}_{\mathrm{rel}}}_{\mathrm{ab}}$ $m$ Relative position of flangeB wrt flangeA s_rel_ab $f$ $N$ Force acting in positive direction of flangeA f $\mathrm{\lambda }$ Flow coefficient lambda $\mathrm{qunsigned}$ Absolute value of volume flowrate qunsigned $\mathrm{qLmin}$ Volume flowrate in l/min qLmin $X$ Distance from poppet center to seat X $\mathrm{lift}$ Poppet lift lift $\mathrm{fflow}$ $N$ Static flow forces fflow $\mathrm{cosalpha}$ Defines direction of flow forces cosalpha $\mathrm{flowarea}$ ${m}^{2}$ Effective orifice area flowarea ${\mathrm{flowarea}}_{\mathrm{nom}}$ ${m}^{2}$ Orifice area (no saturation) flowarea_nom $\mathrm{D}$ $m$ Hydraulic diameter D $\mathrm{V1}$ ${m}^{3}$ Volume, poppet side V1 $\mathrm{V2}$ ${m}^{3}$ Volume, seat side V2 $T$ $K$ Temperature T $\mathrm{Δp}$ $\mathrm{Pa}$ Pressure difference over the valve dp ${p}_{1}$ $\mathrm{Pa}$ Pressure, poppet side p1 ${p}_{2}$ $\mathrm{Pa}$ Pressure, seat side p2 $\mathrm{\rho }$ $\frac{\mathrm{kg}}{{m}^{3}}$ Fluid density in the chamber rho $\mathrm{\nu }$ $\frac{{m}^{2}}{s}$ Fluid viscosity in the chamber nu ${p}_{\mathrm{sat}}$ [1] $\mathrm{Pa}$ Gas saturation pressure p_sat ${\mathrm{p1}}_{\mathrm{abs}}$ $\mathrm{Pa}$ Absolute pressure, used for all property calls p1_abs ${\mathrm{p2}}_{\mathrm{abs}}$ $\mathrm{Pa}$ Absolute pressure, used for all property calls p2_abs

[1] $\mathrm{oil.gasSaturationPressure}\left(T=T,{v}_{\mathrm{gas}}={\mathrm{oil.v}}_{\mathrm{gas}}\right)$

Connections

 Name Description Modelica ID $\mathrm{supportA}$ supportA $\mathrm{supportB}$ supportB $\mathrm{flangeA}$ flangeA $\mathrm{flangeB}$ flangeB $\mathrm{oil}$ Fluid property model oil $\mathrm{portA}$ portA $\mathrm{portA1}$ portA1

Parameters

General Parameters

 Name Default Units Description Modelica ID $L$ $m$ Element length L $\mathrm{reverse}$ $\mathrm{false}$ Reverse the sign convention, see documentation for details reverse $\mathrm{switch}$ $\left\{\begin{array}{cc}-1& \mathrm{reverse}=\mathrm{true}\\ 1& \mathrm{otherwise}\end{array}\right\$ -1 if reverse, else 1 switch $R$ $0.1$ $m$ Poppet radius R $r$ $0.05$ $m$ Throat radius r ${\mathrm{Rrod}}_{1}$ $m$ Rod radius, poppet side Rrod_1 ${\mathrm{Rrod}}_{2}$ $m$ Rod radius, seat side Rrod_2 ${\mathrm{lift}}_{0}$ $0$ $m$ Poppet lift in neutral position lift_0 $\mathrm{V01}$ [1] ${m}^{3}$ Volume, poppet side, when closed valve V01 $\mathrm{V02}$ [1] ${m}^{3}$ Volume, seat side, when closed valve V02 $\mathrm{TransitionType}$ $1$ Type of transition TransitionType ${k}_{1}$ $10$ Laminar part of flow model k1 ${k}_{2}$ $2.04$ Turbulent part of flow model k2 $\mathrm{Cdmax}$ $0.7$ Discharge coefficient Cdmax ${\mathrm{\lambda }}_{c}$ $14$ Critical flow number lambdac ${\mathrm{ΔT}}_{\mathrm{system}}$ $0$ $K$ Temperature offset from system temperature dT_system

[1] $2.0\mathrm{\pi }L{R}^{2}$

Constant Parameters

 Name Default Units Description Modelica ID $\mathrm{Xmin}$ $\sqrt{{R}^{2}-{r}^{2}}$ minimum distance between ball center and seat Xmin $\mathrm{flowareamax}$ [1] ${m}^{2}$ Maximum flow area flowareamax ${\mathrm{Arod}}_{1}$ [2] ${m}^{2}$ Rod area, poppet side Arod_1 ${\mathrm{Arod}}_{2}$ [3] ${m}^{2}$ Rod area, seat side Arod_2 ${X}_{0}$ $\mathrm{lift}$0+Xmin_ $m$ distance from seat to poppet center X_0

[1] $\mathrm{\pi }{r}^{2}-\mathrm{Arod_2}$

[2] $\mathrm{\pi }{\mathrm{Rrod_1}}^{2}$

[3] $\mathrm{\pi }{\mathrm{Rrod_2}}^{2}$