Pressure Compensator

Description

The Pressure Compensator component models a hydraulic reducing valve as a sharp-edged orifice with the orifice area dependent on the pressure across the pilot ports C and D.

The area varies linearly from ${A}_{\mathrm{open}}$ to ${A}_{\mathrm{close}}$ as the pressure varies from ${p}_{\mathrm{open}}$ to ${p}_{\mathrm{close}}$, and remains at the endpoints for pressures outside this range.

Based on the orifice area, the pressure vs flow rate relationships are derived by the formulation used in the Orifice component.

 Formulation Approaches One of two approaches can be selected for modeling the flow in 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 $p={p}_{A}-{p}_{B}$ ${p}_{\mathrm{CD}}={p}_{C}-{p}_{\mathrm{D}}$ $\mathbf{Orifice Fluid Equations}$ $\left\{\begin{array}{cc}p=\frac{\mathrm{\pi }}{4}\frac{\mathrm{\rho }\mathrm{\nu }q}{{C}_{d}^{2}{A}_{\mathrm{cs}}\sqrt{\mathrm{\pi }{A}_{\mathrm{cs}}}}{\left(\frac{16{q}^{4}}{{\mathrm{\pi }}^{2}{A}_{\mathrm{cs}}^{2}{\mathrm{\nu }}^{4}}+{\mathrm{Re}}_{\mathrm{Cr}}^{4}\right)}^{\frac{1}{4}}& \mathrm{Use constant Cd}=\mathrm{true}\\ q={C}_{d\left(\mathrm{max}\right)}\mathrm{tanh}\left(4\frac{\sqrt{\frac{{A}_{\mathrm{cs}}}{\mathrm{\pi }}\frac{2\left|p\right|}{\mathrm{\rho }}}}{\mathrm{\nu }{\mathrm{Crit}}_{\mathrm{no}}}\right){A}_{\mathrm{cs}}\sqrt{\frac{2\left|p\right|}{\mathrm{\rho }}}\mathrm{sign}\left(p\right)& \mathrm{otherwise}\end{array}$ $\mathbf{Pilot Equations}$ ${p}_{\mathrm{tot}}={k}_{C}{p}_{C}-{k}_{\mathrm{D}}{p}_{\mathrm{D}}$ $\mathbf{Orifice Area Equations}$ $\left\{\begin{array}{cc}{A}_{\mathrm{cs}}={A}_{i}={A}_{t}& \mathrm{Exact}\\ \left\{{A}_{\mathrm{cs}}=\mathrm{min}\left({A}_{\mathrm{open}},\mathrm{max}\left({A}_{\mathrm{close}},{A}_{i}\right)\right),{t}_{c}\frac{\mathrm{d}{A}_{i}}{\mathrm{d}t}+{A}_{i}={A}_{t}\right\}& \mathrm{otherwise}\end{array}$ ${A}_{t}=\left\{\begin{array}{cc}{A}_{\mathrm{open}}& {p}_{\mathrm{tot}}<{p}_{\mathrm{contract}}\\ {A}_{\mathrm{open}}-\left({A}_{\mathrm{open}}-{A}_{\mathrm{close}}\right)\mathrm{SmoothTrans}\left(S,\frac{{p}_{\mathrm{tot}}-{p}_{\mathrm{contract}}}{{p}_{\mathrm{close}}-{p}_{\mathrm{contract}}}\right)& {p}_{\mathrm{tot}}<{p}_{\mathrm{close}}\\ {A}_{\mathrm{close}}& \mathrm{otherwise}\end{array}$ $S=\left\{\begin{array}{cc}\mathrm{smoothness}& \mathrm{smoothTransition}\\ 0& \mathrm{otherwise}\end{array}$ $\mathbf{Optional Volume Equations}$ ${V}_{{f}_{A}}=\left\{\begin{array}{cc}\mathrm{Va}\left(1+\frac{{p}_{A}}{\mathrm{El}}\right)& \mathrm{Use volume A}=\mathrm{true}\\ 0& \mathrm{otherwise}\end{array}\phantom{\rule[-0.0ex]{5.0ex}{0.0ex}}{V}_{{f}_{B}}=\left\{\begin{array}{cc}\mathrm{Vb}\left(1+\frac{{p}_{B}}{\mathrm{El}}\right)& \mathrm{Use volume B}=\mathrm{true}\\ 0& \mathrm{otherwise}\end{array}$ $q={q}_{A}-{q}_{{V}_{A}}=-\left({q}_{B}-{q}_{{V}_{B}}\right)$ ${q}_{{V}_{A}}=\left\{\begin{array}{cc}\frac{\mathrm{d}{V}_{{f}_{A}}}{\mathrm{d}t}& \mathrm{Use volume A}=\mathrm{true}\\ 0& \mathrm{otherwise}\end{array}\phantom{\rule[-0.0ex]{4.0ex}{0.0ex}}{q}_{{V}_{B}}=\left\{\begin{array}{cc}\frac{\mathrm{d}{V}_{{f}_{B}}}{\mathrm{d}t}& \mathrm{Use volume B}=\mathrm{true}\\ 0& \mathrm{otherwise}\end{array}$

Variables

 Name Units Description Modelica ID ${p}_{\mathrm{CD}}$ $\mathrm{Pa}$ Pressure drop from C to D pCD $p$ $\mathrm{Pa}$ 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 ${A}_{\mathrm{cs}}$ ${m}^{2}$ Cross-sectional area of orifice Acs ${A}_{i}$ ${m}^{2}$ Filtered interpolated area At ${A}_{t}$ ${m}^{2}$ Interpolated area At ${q}_{{V}_{X}}$ $\frac{{m}^{3}}{s}$ 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 $\mathrm{portA}$ Upstream hydraulic port portA $\mathrm{portB}$ Downstream hydraulic port portB $\mathrm{portC}$ Hydraulic pilot port portC $\mathrm{portD}$ Hydraulic pilot port portD

Parameters

General

 Name Default Units Description Modelica ID ${p}_{\mathrm{close}}$ $2.1·{10}^{7}$ $\mathrm{Pa}$ Pressure at which valve is fully open (A = Aclose) pclose ${p}_{\mathrm{open}}$ $1.9·{10}^{7}$ $\mathrm{Pa}$ Pressure at which valve is fully closed (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 $\mathrm{Exact}$ $\mathrm{false}$ When false (not checked) first-order dynamics are used for the valve area Exact ${t}_{c}$ $0.1$ $s$ Time constant tc $\mathrm{Smooth Transition}$ $\mathrm{false}$ True (checked) means enable the smoothness factor smoothTransition $\mathrm{smoothness}$ $0.5$ Smoothness factor (0: sharpest, 1: smoothest); used when $\mathrm{Smooth Transition}$ is enabled smoothness ${k}_{C}$ $1$ Pilot ratio at port C kpC ${k}_{\mathrm{D}}$ $1$ Pilot ratio at port D kpD

Orifice

 Name Default Units Description Modelica ID $\mathrm{Use constant Cd}$ $\mathrm{true}$ True (checked) means a constant coefficient of discharge is implemented, otherwise a variable ${C}_{d}$ is used in flow calculations 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