Poppet Lift Conical Sharp Edge $—$ Conical poppet lift with sharp edge seat

This is a template model for a conical poppet lift with sharp edge seat.

 Flow Area When the poppet is far from the seat, the flowarea between the two chambers is limited by the area between the seat-side rod and the seat. As the poppet moves nearer, the flowarea is limited by the seat and the poppet. The flow area, $A$, is the area (excluding top and bottom area) of the truncated cone with base radius $r$, top radius $a$, and slant height $s$, as seen in the picture. $A=\mathrm{\pi }\left(a+r\right)s$ $s=X\mathrm{cos}\left(\mathrm{\theta }\right)$ $a=r-s\mathrm{sin}\left(\mathrm{\theta }\right)$ where $X$ is the distance from the seat to the poppet center and $r$ is the seat radius.
 Hydraulic Diameter The hydraulic diameter is $\mathrm{D}=\frac{4A}{P}$, where $P$ is the wetted perimeter and $A$ is the flow area. The wetted perimeter for the poppet lift with sharp edge seat is $P=2\mathrm{\pi }\left(a+r\right)$.
 Chamber Volumes The chamber volumes ${V}_{1}$ (poppet side) and ${V}_{2}$ (seat side) are calculated using the following quantities: The initial volume (when the poppet lift is closed) on both poppet side (V01) and seat side (V02) are known. The seat-side volume is ${V}_{2}=\left\{\begin{array}{cc}{V}_{02}& X\le {X}_{\mathrm{min}}\\ {V}_{02}-\left(X-{X}_{\mathrm{min}}\right){A}_{\mathrm{rod2}}+{V}_{\mathrm{cone1}}+{V}_{\mathrm{cone2}}-{V}_{\mathrm{cone3}}& \mathrm{otherwise}\end{array}$ where cone 1 has base radius $r$ and top radius ${R}_{\mathrm{rod2}}$ cone 2 has base radius $r$ and top radius $a$ cone 3 has base radius $a$ and top radius ${R}_{\mathrm{rod2}}$ and $X-{X}_{\mathrm{min}}$ is the distance the poppet has been lifted from its initial position. The poppet-side volume, ${V}_{1}$, is calculated from ${V}_{1}+{V}_{2}={V}_{\mathrm{01}}+{V}_{\mathrm{02}}-\left(X-{X}_{\mathrm{min}}\right)\left({A}_{\mathrm{rod2}}-{A}_{\mathrm{rod1}}\right)$.
 Implementation This model calculates the volumes on both sides of the poppet lift, the mass flow between the two sides and the force $f$. Pressure is calculated in volume components connected to connector $\mathrm{portA}$ and $\mathrm{portA1}$.
 Equations $\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}=\mathrm{Modelica.Fluid.Utilities.regRoot}\left(\frac{{\left(\sqrt{{k}_{1}^{2}{\mathrm{ν}}^{2}{\mathrm{ρ}}^{2}+\frac{32\mathrm{flowarea}{k}_{2}\left|\mathrm{Δp}\right|\mathrm{ρ}}{\mathrm{π}}}-{k}_{1}\mathrm{ν}\mathrm{ρ}\right)}^{2}\mathrm{max}\left(0,\mathrm{flowarea}\right)\mathrm{π}}{16{k}_{2}^{2}{\mathrm{ρ}}^{2}},\mathrm{regRoot_q}\right)& \mathrm{TransitionType}=1\\ \mathrm{qunsigned}=\mathrm{Cdmax}\mathrm{Modelica.Math.tanh}\left(\frac{2\mathrm{\lambda }}{{\mathrm{λ}}_{c}}\right)\mathrm{flowarea}\mathrm{Modelica.Fluid.Utilities.regRoot}\left(\frac{2\left|\mathrm{Δp}\right|}{\mathrm{ρ}},\mathrm{regRoot_prho}\right)& \mathrm{otherwise}\end{array}\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{VTruncCone1}+\mathrm{VTruncCone2}-\mathrm{VTruncCone3}& \mathrm{otherwise}\end{array}\right\$ $X=\mathrm{min}\left(\mathrm{Xmax},\mathrm{max}\left(\mathrm{Xmin},\mathrm{switch}{s}_{a\left(\mathrm{rel}\right)}+\mathrm{X_0}\right)\right)$ $a=r-X\mathrm{Modelica.Math.cos}\left(\mathrm{\theta }\right)\mathrm{sin}\left(\mathrm{\theta }\right)$ $f=-\left(-\mathrm{\pi }{a}^{2}+\mathrm{Arod_1}\right){p}_{1}+\left(-\mathrm{\pi }{a}^{2}+\mathrm{Arod_2}\right){p}_{2}-\mathrm{fflow}$ $\mathrm{\lambda }=\frac{\mathrm{D}\mathrm{Modelica.Fluid.Utilities.regRoot}\left(\frac{2\left|\mathrm{Δp}\right|}{\mathrm{ρ}},\mathrm{regRoot_prho}\right)}{\mathrm{ν}}$ $\mathrm{lift}=\mathrm{max}\left(0,\mathrm{switch}{s}_{a\left(\mathrm{rel}\right)}-\mathrm{Xmin}\right)$ ${\mathrm{p1}}_{\mathrm{abs}}={p}_{1}+{p}_{\mathrm{atm}\left(\mathrm{oil}\right)}$ ${\mathrm{p2}}_{\mathrm{abs}}={p}_{2}+{p}_{\mathrm{atm}\left(\mathrm{oil}\right)}$ ${V}_{A}=\mathrm{V1}$ ${\mathrm{px}}_{\mathrm{abs}}={\mathrm{p1}}_{\mathrm{abs}}\mathrm{xp}+{\mathrm{p2}}_{\mathrm{abs}}\left(1-\mathrm{xp}\right)$ $\mathrm{qLmin}=60000\mathrm{qunsigned}$ ${s}_{a\left(\mathrm{rel}\right)}={s}_{A}-{s}_{A\left(\mathrm{support}\right)}$ $\mathrm{xp}=\mathrm{Modelon.Math.Smoothing.cubicStep}\left(\mathrm{Δp}+\frac{1}{2}\right)$ $\mathrm{Δp}={p}_{1}-{p}_{2}$ $\mathrm{VTruncCone1}=\mathrm{Hydraulics.Elements.truncConeVolume}\left(\mathrm{_msim_R}=r,\mathrm{_msim_r}=\mathrm{Rrod_2},\mathrm{_msim_h}=\left(r-\mathrm{Rrod_2}\right)\mathrm{Modelica.Math.tan}\left(\mathrm{\theta }\right)\right)$ $\mathrm{VTruncCone2}=\mathrm{Hydraulics.Elements.truncConeVolume}\left(\mathrm{_msim_R}=r,\mathrm{_msim_r}=a,\mathrm{_msim_h}=\left(r-a\right)\mathrm{Modelica.Math.tan}\left(\mathrm{\theta }\right)\right)$ $\mathrm{VTruncCone3}=\mathrm{Hydraulics.Elements.truncConeVolume}\left(\mathrm{_msim_R}=a,\mathrm{_msim_r}=\mathrm{Rrod_2},\mathrm{_msim_h}=\left(a-\mathrm{Rrod_2}\right)\mathrm{Modelica.Math.tan}\left(\mathrm{\theta }\right)\right)$ $\mathrm{cosalpha}=\mathrm{Modelica.Math.cos}\left(\frac{\mathrm{π}}{2}-\mathrm{\theta }\right)$ $\mathrm{flowarea}=\mathrm{flowarea_nom}$ $\mathrm{flowarea_nom}=\mathrm{Hydraulics.Elements.truncConeArea2}\left(\mathrm{_msim_R}=r,\mathrm{_msim_r}=a,\mathrm{_msim_s}=X\mathrm{Modelica.Math.cos}\left(\mathrm{\theta }\right)\right)$ $\mathrm{portA.m_flow}=\mathrm{smooth}\left(0,\mathrm{noEvent}\left(\left\{\begin{array}{cc}\mathrm{ρ}\mathrm{qunsigned}& 0\le \mathrm{Δp}\\ -\mathrm{ρ}\mathrm{qunsigned}& \mathrm{otherwise}\end{array}\right\\right)\right)$ $\mathrm{portA1.V}=\mathrm{V2}$ ${s}_{\mathrm{ab}\left(\mathrm{rel}\right)}=L$ ${s}_{\mathrm{ab}\left(\mathrm{rel}\right)}=\mathrm{switch}\left({s}_{B}-{s}_{A}\right)$ ${s}_{B\left(\mathrm{support}\right)}=L\mathrm{switch}+{s}_{A\left(\mathrm{support}\right)}$ ${p}_{1}={p}_{A}$ ${p}_{2}=\mathrm{portA1.p}$ $\mathrm{V1}+\mathrm{V2}=\mathrm{V01}+\mathrm{V02}-\left(X-\mathrm{Xmin}\right)\left(\mathrm{Arod_2}-\mathrm{Arod_1}\right)$ ${m}_{\mathrm{flow}\left(\mathrm{A1}\right)}+{m}_{\mathrm{flow}\left(A\right)}=0$ ${\partial }_{t}\left(\mathrm{Xmax}\right)=0$

Variables

 Name Value Units Description Modelica ID ${{s}_{\mathrm{rel}}}_{a}$ $m$ Relative position of flange_a wrt support_a s_rel_a ${{s}_{\mathrm{rel}}}_{\mathrm{ab}}$ $m$ Relative position of flange_b wrt flange_a s_rel_ab $f$ $N$ Force acting in positive direction of flange_a f $\mathrm{\lambda }$ Flow coefficient lambda $\mathrm{qunsigned}$ Absolute value of volume flowrate qunsigned $\mathrm{qLmin}$ Volume flowrate in l/min qLmin $X$ $m$ Distance from poppet center/poppet to seat X $\mathrm{lift}$ $m$ 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 ${V}_{1}$ ${m}^{3}$ Volume, poppet side V1 ${V}_{2}$ ${m}^{3}$ 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 $\mathrm{Xmax}$ $m$ Xmax $\mathrm{VTruncCone1}$ ${m}^{3}$ Cone 1 used for calculating chamber volumes VTruncCone1 $\mathrm{VTruncCone2}$ ${m}^{3}$ Cone 2 used for calculating chamber volumes VTruncCone2 $\mathrm{VTruncCone3}$ ${m}^{3}$ Cone 3 used for calculating chamber volumes VTruncCone3

[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 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 reverse $\mathrm{false}$ Reverse the sign convention, see documentation for details reverse $R$ $0.1$ $m$ Poppet radius R $\mathrm{\theta }$ [1] $\mathrm{rad}$ Conical seat/poppet angle theta $r$ $0.05$ $m$ Throat radius r ${R}_{\mathrm{rod}\left(1\right)}$ $m$ Rod radius, poppet side Rrod_1 ${R}_{\mathrm{rod}\left(2\right)}$ $m$ Rod radius, seat side Rrod_2 ${\mathrm{lift}}_{0}$ $0$ $m$ Poppet lift in neutral position lift_0 ${\ell }_{\mathrm{piston}}$ $L$ $m$ Piston length l_piston ${V}_{1}$ [2] ${m}^{3}$ Volume, poppet side, when closed valve V01 ${V}_{2}$ [2] ${m}^{3}$ Volume, seat side, when closed valve V02 ${X}_{\mathrm{min}}$ $0$ $m$ minimum distance between poppet center and seat in basic case, otherwise set as 0 Xmin transition type $1$ Type of transition TransitionType ${k}_{1}$ $10$ Laminar part of flow model k1 ${k}_{2}$ $2.04$ Turbulent part of flow model k2 ${C}_{d\left(\mathrm{max}\right)}$ $0.7$ Discharge coefficient Cdmax ${\mathrm{\lambda }}_{c}$ $14$ Critical flow number lambdac ${X}_{\mathrm{max}\left(\mathrm{start}\right)}$ $r$ XmaxStart ${X}_{\mathrm{max}\left(\mathrm{nom}\right)}$ $0.001{X}_{\mathrm{max}\left(\mathrm{start}\right)}$ XmaxNominal ${\mathrm{ΔT}}_{\mathrm{system}}$ $0$ $K$ Temperature offset from system temperature dT_system

[1] $0.333333333333333\mathrm{\pi }$

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

Constant Parameters

 Name Default Units Description Modelica ID ${A}_{\mathrm{flow}\left(\mathrm{max}\right)}$ [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 ${X}_{0}$ ${\mathrm{lift}}_{0}+{X}_{\mathrm{min}}$ $m$ distance from seat to poppet center X_0

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

[2] $\mathrm{\pi }{R}_{\mathrm{rod}\left(1\right)}^{2}$

[3] $\mathrm{\pi }{R}_{\mathrm{rod}\left(2\right)}^{2}$