Poppet Lift Cone $—$ Ball poppet lift with conical seat

This is a template model for a poppet lift with conical 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 flow area is limited by the seat and the poppet. The flow area is the area of a truncated cone with base radius $s$, top radius $a$, and slant height $h$, $A=\mathrm{\pi }\left(s+a\right)h$ $s=a+X{\mathrm{cos}\left(\mathrm{\theta }\right)}^{2}$ $h=X\mathrm{cos}\left(\mathrm{\theta }\right)$ with $a$ constant.  See the following figure.
 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 conical seat is $P=2\mathrm{\pi }\left(2a+X{\mathrm{cos}\left(\mathrm{\theta }\right)}^{2}\right)$
 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 (${V}_{\mathrm{01}}$) and seat side (${V}_{\mathrm{02}}$) are known. The seat side volume is ${V}_{2}=\left\{\begin{array}{cc}{V}_{\mathrm{02}}& X\le {X}_{\mathrm{min}}\\ {V}_{\mathrm{02}}-X{A}_{\mathrm{rod}\left(2\right)}+{V}_{\mathrm{cap}}\left(R,a\right)+{V}_{\mathrm{cone}}-{V}_{\mathrm{slice}}& \mathrm{otherwise}\end{array}$ where $X-{X}_{\mathrm{min}}$ is the distance the poppet has been lifted from its initial position. ${V}_{1}$ is calculated from the relationship ${V}_{1}+{V}_{2}={V}_{\mathrm{01}}+{V}_{\mathrm{02}}-\left(X-{X}_{\mathrm{min}}\right)\left({A}_{\mathrm{rod}\left(2\right)}-{A}_{\mathrm{rod}\left(1\right)}\right)$.
 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\$ $0=f\mathrm{switch}+{f}_{A}+{f}_{B}$ $0=-f\mathrm{switch}+{f}_{A\left(\mathrm{support}\right)}+{f}_{B\left(\mathrm{support}\right)}$ $\mathrm{\nu }={\mathrm{\nu }}_{\mathrm{oil}}\left({p}_{\mathrm{abs}}={\mathrm{px}}_{\mathrm{abs}},T=T,{v}_{\mathrm{air}}={v}_{\mathrm{gas}\left(\mathrm{oil}\right)},{p}_{\mathrm{sat}}={p}_{\mathrm{sat}}\right)$ $\mathrm{\rho }={\mathrm{\rho }}_{\mathrm{oil}}\left({p}_{\mathrm{abs}}={\mathrm{px}}_{\mathrm{abs}},T=T,{v}_{\mathrm{air}}={v}_{\mathrm{gas}\left(\mathrm{oil}\right)},{p}_{\mathrm{sat}}={p}_{\mathrm{sat}}\right)$ $\mathrm{D}=\frac{4\mathrm{flowarea}}{\mathrm{π}\left(4a+2X{\mathrm{Modelica.Math.cos}\left(\mathrm{\theta }\right)}^{2}\right)}$ $T={T}_{0\left(\mathrm{oil}\right)}+{\mathrm{ΔT}}_{\mathrm{system}}$ $\mathrm{V2}=\left\{\begin{array}{cc}\mathrm{V02}& X\le \mathrm{Xmin}\\ -X\mathrm{Arod_2}+\mathrm{V02}+\mathrm{Hydraulics.Elements.sphericalCapVolume}\left(\mathrm{_msim_R}=R,\mathrm{_msim_a}=a\right)+\mathrm{Vcone}-\mathrm{Vslice}& \mathrm{otherwise}\end{array}\right\$ $\mathrm{Vcone}=\frac{\mathrm{π}{a}^{2}\left(X+\frac{a}{\mathrm{Modelica.Math.tan}\left(\mathrm{\theta }\right)}\right)}{3}$ $\mathrm{Vslice}=\mathrm{Hydraulics.Elements.sphericalCapVolume}\left(\mathrm{_msim_R}=R,\mathrm{_msim_a}=\mathrm{a2}\right)+\frac{1}{3}\mathrm{\pi }{\mathrm{a2}}^{2}\sqrt{{R}^{2}-{\mathrm{a2}}^{2}}$ $X=\mathrm{min}\left(\mathrm{Xmax},\mathrm{max}\left(\mathrm{Xmin},\mathrm{switch}{s}_{a\left(\mathrm{rel}\right)}+\mathrm{X_0}\right)\right)$ $a=R\mathrm{sin}\left(\mathrm{\theta }\right)$ $\mathrm{a2}=\frac{Ra}{\sqrt{{a}^{2}+{\left(X+\sqrt{{R}^{2}-{a}^{2}}\right)}^{2}}}$ $f=-\left(-\mathrm{\pi }{a}^{2}+\mathrm{Arod_1}\right){p}_{1}+\left(-\mathrm{\pi }{a}^{2}+\mathrm{Arod_2}\right){p}_{2}-\mathrm{fflow}$ $\mathrm{\lambda }=\frac{\mathrm{D}\mathrm{Modelica.Fluid.Utilities.regRoot}\left(\frac{2\left|\mathrm{Δp}\right|}{\mathrm{ρ}},\mathrm{regRoot_prho}\right)}{\mathrm{ν}}$ $\mathrm{lift}=\mathrm{max}\left(0,\mathrm{switch}{s}_{a\left(\mathrm{rel}\right)}-\mathrm{Xmin}\right)$ ${\mathrm{p1}}_{\mathrm{abs}}={p}_{1}+{p}_{\mathrm{atm}\left(\mathrm{oil}\right)}$ ${\mathrm{p2}}_{\mathrm{abs}}={p}_{2}+{p}_{\mathrm{atm}\left(\mathrm{oil}\right)}$ ${V}_{A}=\mathrm{V1}$ ${\mathrm{px}}_{\mathrm{abs}}={\mathrm{p1}}_{\mathrm{abs}}\mathrm{xp}+{\mathrm{p2}}_{\mathrm{abs}}\left(1-\mathrm{xp}\right)$ $\mathrm{qLmin}=60000\mathrm{qunsigned}$ ${s}_{a\left(\mathrm{rel}\right)}={s}_{A}-{s}_{A\left(\mathrm{support}\right)}$ $\mathrm{xp}=\mathrm{Modelon.Math.Smoothing.cubicStep}\left(\mathrm{Δp}+\frac{1}{2}\right)$ $\mathrm{Δp}={p}_{1}-{p}_{2}$ $\mathrm{cosalpha}=\frac{a}{R}$ $\mathrm{flowarea}=\mathrm{flowarea_nom}$ $\mathrm{flowarea_nom}=\mathrm{Hydraulics.Elements.truncConeArea2}\left(\mathrm{_msim_R}=a+X{\mathrm{Modelica.Math.cos}\left(\mathrm{\theta }\right)}^{2},\mathrm{_msim_r}=a,\mathrm{_msim_s}=X\mathrm{Modelica.Math.cos}\left(\mathrm{\theta }\right)\right)$ $\mathrm{portA.m_flow}=\mathrm{smooth}\left(0,\mathrm{noEvent}\left(\left\{\begin{array}{cc}\mathrm{ρ}\mathrm{qunsigned}& 0\le \mathrm{Δp}\\ -\mathrm{ρ}\mathrm{qunsigned}& \mathrm{otherwise}\end{array}\right\\right)\right)$ $\mathrm{portA1.V}=\mathrm{V2}$ ${s}_{\mathrm{ab}\left(\mathrm{rel}\right)}=L$ ${s}_{\mathrm{ab}\left(\mathrm{rel}\right)}=\mathrm{switch}\left({s}_{B}-{s}_{A}\right)$ ${s}_{B\left(\mathrm{support}\right)}=L\mathrm{switch}+{s}_{A\left(\mathrm{support}\right)}$ ${p}_{1}={p}_{A}$ ${p}_{2}=\mathrm{portA1.p}$ $\mathrm{V1}+\mathrm{V2}=\mathrm{V01}+\mathrm{V02}-\left(X-\mathrm{Xmin}\right)\left(\mathrm{Arod_2}-\mathrm{Arod_1}\right)$ ${m}_{\mathrm{flow}\left(\mathrm{A1}\right)}+{m}_{\mathrm{flow}\left(A\right)}=0$ ${\partial }_{t}\left(\mathrm{Xmax}\right)=0$

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 $\mathrm{Vcone}$ ${m}^{3}$ Volume of cone used to calculate chamber volumes Vcone $\mathrm{Vslice}$ ${m}^{3}$ Volume of intersection ball-cone Vslice $\mathrm{a2}$ $m$ Radius of cone used to calculate chamber volumes a2

[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] $0.25\mathrm{\pi }$

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