Piloted Relief Valve - MapleSim Help

Piloted Relief Valve

Description

The Piloted Relief Valve component is a relief valve with a pilot pressure that can be used to assist in opening or closing the valve.

The valve orifice area is assumed to be a piecewise linear function of the pressure difference between port A and port B.

Based on the orifice area, the pressure vs. flow rate relationship is calculated by the formulation used in the Orifice component.

 Formulation Approaches Two approaches were taken for formulation of the flow equation inside the device. When the boolean parameter $\mathrm{Use constant Cd}$ is true, a constant coefficient of discharge (${C}_{d}$) is used, otherwise a variable coefficient of discharge with maximum value (${C}_{d\left(\mathrm{max}\right)}$) and a critical flow number (${\mathrm{Crit}}_{\mathrm{no}}$) are used.
 Optional Volumes The boolean parameters Use volume A and Use volume B, when true, add optional volumes ${V}_{A}$  and ${V}_{B}$ to ports A and B, respectively. See Port Volumes for details. If two orifices or valves are connected, enabling a volume at the common port reduces the stiffness of the system and improves the solvability.
 Equations $p={p}_{A}-{p}_{B}$ $\mathbf{Orifice Fluid Equations}$ $\left\{\begin{array}{cc}p=\frac{\mathrm{\pi }}{4}\frac{\mathrm{\rho }\mathrm{\nu }q}{{C}_{d}^{2}{A}_{\mathrm{cs}}\sqrt{\mathrm{\pi }{A}_{\mathrm{cs}}}}{\left(\frac{16{q}^{4}}{{\mathrm{\pi }}^{2}{A}_{\mathrm{cs}}^{2}{\mathrm{\nu }}^{4}}+{\mathrm{\Re }}_{\mathrm{Cr}}^{4}\right)}^{\frac{1}{4}}& \mathrm{Use constant Cd}=\mathrm{true}\\ q={C}_{d\left(\mathrm{max}\right)}\mathrm{tanh}\left(4\frac{\sqrt{\frac{{A}_{\mathrm{cs}}}{\mathrm{\pi }}\frac{2\left|p\right|}{\mathrm{\rho }}}}{\mathrm{\nu }{\mathrm{Crit}}_{\mathrm{no}}}\right){A}_{\mathrm{cs}}\sqrt{\frac{2\left|p\right|}{\mathrm{\rho }}}\mathrm{sign}\left(p\right)& \mathrm{otherwise}\end{array}$ $\mathbf{Pilot Equations}$ $\left\{\begin{array}{cc}{A}_{\mathrm{cs}}={A}_{i}={A}_{t}& \mathrm{Exact}\\ \left\{{A}_{\mathrm{cs}}=\mathrm{min}\left({A}_{\mathrm{open}},\mathrm{max}\left({A}_{\mathrm{close}},{A}_{i}\right)\right),{t}_{c}\frac{\mathrm{d}{A}_{i}}{\mathrm{d}t}+{A}_{i}={A}_{t}\right\}& \mathrm{otherwise}\end{array}$ ${A}_{t}=\left\{\begin{array}{cc}{A}_{\mathrm{close}}& {p}_{\mathrm{tot}}\le {p}_{\mathrm{close}}\\ {A}_{\mathrm{close}}+\left({p}_{\mathrm{tot}}-{p}_{\mathrm{close}}\right)\frac{{A}_{\mathrm{open}}-{A}_{\mathrm{close}}}{{p}_{\mathrm{open}}-{p}_{\mathrm{close}}}& {p}_{\mathrm{tot}}<{p}_{\mathrm{open}}\\ {A}_{\mathrm{open}}& \mathrm{otherwise}\end{array}$ ${p}_{\mathrm{tot}}=p+\left\{\begin{array}{cc}{k}_{p}{p}_{C}& \mathrm{dir}=1\\ -{k}_{p}{p}_{C}& \mathrm{otherwise}\end{array}$ ${q}_{C}=0$ $\mathbf{Optional Volume Equations}$ ${V}_{{f}_{A}}=\left\{\begin{array}{cc}\mathrm{Va}\left(1+\frac{{p}_{A}}{\mathrm{El}}\right)& \mathrm{Use volume A}=\mathrm{true}\\ 0& \mathrm{otherwise}\end{array}\phantom{\rule[-0.0ex]{5.0ex}{0.0ex}}{V}_{{f}_{B}}=\left\{\begin{array}{cc}\mathrm{Vb}\left(1+\frac{{p}_{B}}{\mathrm{El}}\right)& \mathrm{Use volume B}=\mathrm{true}\\ 0& \mathrm{otherwise}\end{array}$ $q={q}_{A}-{q}_{{V}_{A}}=-\left({q}_{B}-{q}_{{V}_{B}}\right)$ ${q}_{{V}_{A}}=\left\{\begin{array}{cc}\frac{\mathrm{d}{V}_{{f}_{A}}}{\mathrm{d}t}& \mathrm{Use volume A}=\mathrm{true}\\ 0& \mathrm{otherwise}\end{array}\phantom{\rule[-0.0ex]{4.0ex}{0.0ex}}{q}_{{V}_{B}}=\left\{\begin{array}{cc}\frac{\mathrm{d}{V}_{{f}_{B}}}{\mathrm{d}t}& \mathrm{Use volume B}=\mathrm{true}\\ 0& \mathrm{otherwise}\end{array}$

Variables

 Name Units Description Modelica ID $p$ $\mathrm{Pa}$ Pressure drop from A to B p ${p}_{X}$ $\mathrm{Pa}$ Pressure at port X pX $q$ $\frac{{m}^{3}}{s}$ Flow rate from A to B q ${q}_{X}$ $\frac{{m}^{3}}{s}$ Flow into port X qX ${q}_{\mathrm{VX}}$ $\frac{{m}^{3}}{s}$ Flow rate into port X's optional volume qVX ${V}_{{f}_{X}}$ ${m}^{3}$ Effective volume at port X VfX

Connections

 Name Description Modelica ID $\mathrm{portA}$ Hydraulic port on left side portA $\mathrm{portB}$ Hydraulic port on right side portB $\mathrm{portC}$ Hydraulic port on the bottom (no volume, used as pilot) portC

Parameters

General Parameters

 Name Default Units Description Modelica ID ${p}_{\mathrm{close}}$ $1.9·{10}^{7}$ $\mathrm{Pa}$ Pressure at which valve is fully closed (A = Aclose) pclose ${p}_{\mathrm{open}}$ $2.05·{10}^{7}$ $\mathrm{Pa}$ Pressure at which valve is fully open (A = Aopen) popen ${A}_{\mathrm{close}}$ $1·{10}^{-12}$ ${m}^{2}$ Valve area when closed (leakage) Aclose ${A}_{\mathrm{open}}$ $1·{10}^{-5}$ ${m}^{2}$ Valve area when fully open Aopen Pilot action $1$ Select whether the pilot operates to open or close the valve dir ${k}_{\mathrm{pilot}}$ $3$ Pilot ratio kp $\mathrm{Exact}$ $\mathrm{false}$ When true (checked) a first-order dynamics is used for the valve area Exact ${t}_{c}$ $\frac{1}{10}$ $s$ Time constant tc $\mathrm{Use constant Cd}$ $\mathrm{true}$ True (checked) means a constant coefficient of discharge is implemented, otherwise a variable ${C}_{d}$ is used in flow calculation UseConstantCd ${C}_{d}$ $0.7$ Flow-discharge coefficient; used when $\mathrm{Use constant Cd}$ is true Cd ${\mathrm{\Re }}_{\mathrm{Cr}}$ $12$ Reynolds number at critical flow; used when $\mathrm{Use constant Cd}$ is true ReCr ${C}_{d\left(\mathrm{max}\right)}$ $0.7$ Maximum flow-discharge coefficient; used when $\mathrm{Use constant Cd}$ is false Cd_max ${\mathrm{Crit}}_{\mathrm{no}}$ $1000$ Critical flow number; used when $\mathrm{Use constant Cd}$ is false Crit_no

Optional Volumes

 Name Default Units Description Modelica ID $\mathrm{Use volume A}$ $\mathrm{false}$ True (checked) means a hydraulic volume chamber is added to portA useVolumeA ${V}_{A}$ $1·{10}^{-6}$ ${m}^{3}$ Volume of chamber A Va $\mathrm{Use volume B}$ $\mathrm{false}$ True (checked) means a hydraulic volume chamber is added to portB useVolumeB ${V}_{B}$ $1·{10}^{-6}$ ${m}^{3}$ Volume of chamber B Vb

 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