Spool Valve $—$ Spool valve

This is a model of a spool valve.

 Equations $\left\{\begin{array}{cc}\left\{\mathrm{ds_max}=2\sqrt{\frac{\mathrm{flowareamax}}{\mathrm{\pi }}},\mathrm{\theta }=\mathrm{noEvent}\left(\left\{\begin{array}{cc}\mathrm{Modelica.Math.acos}\left(1-\frac{2\mathrm{s_rel_a}\mathrm{switch}-2\mathrm{x_min}}{\mathrm{ds_max}}\right)& -\frac{2\mathrm{s_rel_a}\mathrm{switch}-2\mathrm{x_min}}{\mathrm{ds_max}}<0<2-\frac{2\mathrm{s_rel_a}\mathrm{switch}-2\mathrm{x_min}}{\mathrm{ds_max}}\\ \left\{\begin{array}{cc}0& \mathrm{s_rel_a}\mathrm{switch}\le \mathrm{x_min}\\ \mathrm{\pi }& \mathrm{otherwise}\end{array}\right\& \mathrm{otherwise}\end{array}\right\\right)\right\}& \mathrm{orificeType}=3\\ \left\{\mathrm{ds_max}=\mathrm{x_max}-\mathrm{x_min},\mathrm{\theta }=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(\sqrt{\mathrm{max}\left(0,{k}_{1}^{2}{\mathrm{\nu }}^{2}{\mathrm{\rho }}^{2}+\frac{32\mathrm{flowarea}{k}_{2}\sqrt{{\mathrm{Δp}}^{2}}\mathrm{\rho }}{\mathrm{\pi }}\right)}-{k}_{1}\mathrm{\nu }\mathrm{\rho }\right)\sqrt{\mathrm{max}\left(0,\mathrm{flowarea}\mathrm{\pi }\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}\sqrt{2}\sqrt{\frac{\sqrt{{\mathrm{Δp}}^{2}}}{\mathrm{\rho }}}& \mathrm{otherwise}\end{array}\right\$ $\left\{\begin{array}{cc}\mathrm{perimeter}=\mathrm{ds_max}\left(\mathrm{\theta }+\mathrm{sin}\left(\mathrm{\theta }\right)\right)& \mathrm{orificeType}=3\\ \mathrm{perimeter}=\mathrm{Modelon.Math.Interpolation.tableInterpolate1b}\left(\mathrm{diam_tab},\mathrm{s_rel_a}\mathrm{switch}\right)& \mathrm{orificeType}=4\\ \mathrm{perimeter}=\mathrm{\pi }\sqrt{\frac{\mathrm{flowarea}}{\mathrm{\pi }}}& \mathrm{otherwise}\end{array}\right\$ $\left\{\begin{array}{cc}\mathrm{flowarea_nom}=\frac{\mathrm{flowareamax}\left(\mathrm{s_rel_a}\mathrm{switch}-\mathrm{x_min}\right)}{\mathrm{x_max}-\mathrm{x_min}}& \mathrm{orificeType}=1\\ \mathrm{flowarea_nom}=\frac{\mathrm{flowareamax}{\left(\mathrm{s_rel_a}\mathrm{switch}-\mathrm{x_min}\right)}^{2}}{{\left(\mathrm{x_max}-\mathrm{x_min}\right)}^{2}}& \mathrm{orificeType}=2\\ \mathrm{flowarea_nom}=\frac{{\mathrm{ds_max}}^{2}\left(\mathrm{\theta }-\frac{\mathrm{sin}\left(2\mathrm{\theta }\right)}{2}\right)}{4}& \mathrm{orificeType}=3\\ \mathrm{flowarea_nom}=\mathrm{Modelon.Math.Interpolation.tableInterpolate1b}\left(\mathrm{area_tab},\mathrm{s_rel_a}\mathrm{switch}\right)& \mathrm{otherwise}\end{array}\right\$ $0=f\mathrm{switch}+{f}_{A}+{f}_{B}$ $0=-f\mathrm{switch}+{f}_{A\left(\mathrm{support}\right)}+{f}_{B\left(\mathrm{support}\right)}$ $\ell =\frac{{V}_{0}}{A-\mathrm{A_rod}}+{s}_{a\left(\mathrm{rel}\right)}\mathrm{switch}$ $\mathrm{\nu }={\mathrm{\nu }}_{\mathrm{oil}}\left({p}_{\mathrm{abs}}={p}_{\mathrm{abs}},T=T,{v}_{\mathrm{air}}={v}_{\mathrm{gas}\left(\mathrm{oil}\right)},{p}_{\mathrm{sat}}={p}_{\mathrm{sat}}\right)$ $\mathrm{\rho }={\mathrm{\rho }}_{\mathrm{oil}}\left({p}_{\mathrm{abs}}={p}_{\mathrm{abs}},T=T,{v}_{\mathrm{air}}={v}_{\mathrm{gas}\left(\mathrm{oil}\right)},{p}_{\mathrm{sat}}={p}_{\mathrm{sat}}\right)$ $\mathrm{D}=2\mathrm{Modelica.Fluid.Utilities.regRoot}\left(\frac{\mathrm{flowarea}}{\mathrm{max}\left(\frac{1}{1000000000},\mathrm{perimeter}\right)}\right)$ $T={T}_{0\left(\mathrm{oil}\right)}+{\mathrm{ΔT}}_{\mathrm{system}}$ $V={V}_{0}+\left(A-\mathrm{A_rod}\right){s}_{a\left(\mathrm{rel}\right)}\mathrm{switch}$ $f=-\left(-A+\mathrm{A_rod}\right)p-\mathrm{fflow}$ $\mathrm{\lambda }=\frac{\mathrm{D}\mathrm{Modelica.Fluid.Utilities.regRoot}\left(\frac{2\left|\mathrm{Δp}\right|}{\mathrm{ρ}}\right)}{\mathrm{ν}}$ $p={p}_{A}$ ${V}_{A}=V$ $\mathrm{qLmin}=60000\mathrm{qunsigned}$ ${s}_{a\left(\mathrm{rel}\right)}={s}_{A}-{s}_{A\left(\mathrm{support}\right)}$ $\mathrm{Δp}={p}_{A}-\mathrm{portA1.p}$ $\mathrm{cosalpha}=\left\{\begin{array}{cc}\frac{159}{100\left(\frac{269}{100}+\frac{\mathrm{s_rel_a}\mathrm{switch}-\mathrm{x_min}}{\mathrm{deltaR}}\right)}+\frac{17}{50}& 0<\mathrm{s_rel_a}\mathrm{switch}-\mathrm{x_min}\\ \frac{467}{500}& \mathrm{otherwise}\end{array}\right\$ $\mathrm{flowarea}=\mathrm{noEvent}\left(\left\{\begin{array}{cc}\mathrm{max}\left(0,\mathrm{flowareamax}\right)& \mathrm{x_min}+\mathrm{ds_max}<\mathrm{s_rel_a}\mathrm{switch}\\ \left\{\begin{array}{cc}\mathrm{flowarea_nom}& \mathrm{x_min}<\mathrm{s_rel_a}\mathrm{switch}<\mathrm{x_min}+\mathrm{ds_max}\\ 0& \mathrm{otherwise}\end{array}\right\& \mathrm{otherwise}\end{array}\right\\right)$ $\mathrm{portA.m_flow}=\mathrm{noEvent}\left(\left\{\begin{array}{cc}\mathrm{\rho }\mathrm{qunsigned}& 0\le \mathrm{Δp}\\ -\mathrm{\rho }\mathrm{qunsigned}& \mathrm{otherwise}\end{array}\right\\right)$ ${s}_{\mathrm{ab}\left(\mathrm{rel}\right)}=L$ ${s}_{\mathrm{ab}\left(\mathrm{rel}\right)}=\mathrm{switch}\left({s}_{B}-{s}_{A}\right)$ ${s}_{B\left(\mathrm{support}\right)}=L\mathrm{switch}+{s}_{A\left(\mathrm{support}\right)}$ ${p}_{\mathrm{abs}}=p+{p}_{\mathrm{atm}\left(\mathrm{oil}\right)}$ ${m}_{\mathrm{flow}\left(\mathrm{A1}\right)}+{m}_{\mathrm{flow}\left(A\right)}=0$

Variables

 Name Value Units Description Modelica ID ${{s}_{\mathrm{rel}}}_{a}$ $m$ Relative position of 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 ${p}_{\mathrm{sat}}$ [1] $\mathrm{Pa}$ Gas saturation pressure p_sat $\mathrm{\lambda }$ Flow coefficient lambda $\mathrm{qunsigned}$ Absolute value of volume flowrate qunsigned $\mathrm{qLmin}$ Volume flowrate in l/min qLmin $\mathrm{cosalpha}$ Defines direction of flow forces cosalpha $\mathrm{fflow}$ $N$ Static flow forces fflow $\mathrm{flowarea}$ ${m}^{2}$ Effective orifice area flowarea ${\mathrm{flowarea}}_{\mathrm{nom}}$ ${m}^{2}$ Orifice area (no saturation) flowarea_nom ${\mathrm{ds}}_{\mathrm{max}}$ $m$ Displacement to move from A=0+ to A=Amax ds_max $\ell$ $m$ Chamber length length $\mathrm{D}$ $m$ Hydraulic diameter D $\mathrm{perimeter}$ $m$ Wetted perimeter perimeter $V$ ${m}^{3}$ Chamber volume V $T$ $K$ Temperature T $\mathrm{Δp}$ $\mathrm{Pa}$ Pressure difference over the valve dp $p$ $\mathrm{Pa}$ Gauge pressure in the chamber p ${p}_{\mathrm{abs}}$ $\mathrm{Pa}$ Absolute pressure, used for all property calls p_abs $\mathrm{\rho }$ $\frac{\mathrm{kg}}{{m}^{3}}$ Fluid density in the chamber rho $\mathrm{\nu }$ $\frac{{m}^{2}}{s}$ Fluid viscosity in the chamber nu $\mathrm{qout}$ [2] qout

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

[2] $\frac{\mathrm{portA}.\mathrm{m_flow}}{\mathrm{oil}.\mathrm{density}\left({p}_{\mathrm{abs}}={p}_{\mathrm{abs}},T=T,{v}_{\mathrm{air}}=\mathrm{oil.v_gas},{p}_{\mathrm{sat}}={p}_{\mathrm{sat}}\right)}$

Connections

 Name Description Modelica ID $\mathrm{supportA}$ supportA $\mathrm{supportB}$ supportB $\mathrm{flangeA}$ flangeA $\mathrm{flangeB}$ flangeB $\mathrm{oil}$ 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 ${V}_{0}$ $0$ ${m}^{3}$ Dead volume V0 $A$ ${m}^{2}$ Spool area A ${A}_{\mathrm{rod}}$ ${m}^{2}$ Rod area A_rod $\mathrm{deltaR}$ ${10}^{-5}$ $m$ Radial clearance > 0 deltaR $\mathrm{TransitionType}$ $1$ 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

Orifice Parameters

 Name Default Units Description Modelica ID $\mathrm{orificeType}$ $1$ orificeType $\mathrm{flowareamax}$ ${m}^{2}$ Maximum flow area flowareamax ${x}_{\mathrm{min}}$ $m$ Position>x_min => valve opens x_min ${x}_{\mathrm{max}}$ $m$ Position>x_max => valve fully open x_max ${\mathrm{area}}_{\mathrm{tab}}$ $\left\{\left\{0\right\},\left\{0.1\right\},\left\{0.2\right\},\left\{0.3\right\}\right\}$ Table for orifice area area_tab ${\mathrm{diam}}_{\mathrm{tab}}$ $\left\{\left\{0\right\},\left\{0.1\right\},\left\{0.2\right\},\left\{0.3\right\}\right\}$ Table for orifice hydraulic diameter diam_tab