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Pressure Compensator

Piloted spring-loaded 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 $A_{open}$ to $A_{close}$ as the pressure varies from $p_{open}$ to $p_{close}$ , 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 $Use constant Cd$ is true, a constant coefficient of discharge ( $C_{d}$ ) is used, otherwise a variable coefficient of discharge with maximum value ( $C_{d (\max)}$ ) and a critical flow number ( ${Crit}_{no}$ ) 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

$p = p_{A} - p_{B}$

$p_{CD} = p_{C} - p_{D}$

$Orifice Fluid Equations$

${\begin{array}{c} p = \frac{π}{4} \frac{ρ ν q}{C_{d}^{2} A_{cs} \sqrt{π A_{cs}}} {(\frac{16 q^{4}}{π^{2} A_{cs}^{2} ν^{4}} + {Re}_{Cr}^{4})}^{\frac{1}{4}} & Use constant Cd = true \\ q = C_{d (\max)} \tanh (4 \frac{\sqrt{\frac{A_{cs}}{π} \frac{2 |p|}{ρ}}}{ν {Crit}_{no}}) A_{cs} \sqrt{\frac{2 |p|}{ρ}} sign (p) & otherwise \end{array}$

$Pilot Equations$

$p_{tot} = k_{C} p_{C} - k_{D} p_{D}$

$Orifice Area Equations$

${\begin{array}{c} A_{cs} = A_{i} = A_{t} & Exact \\ \{A_{cs} = \min (A_{open}, \max (A_{close}, A_{i})), t_{c} \frac{d A_{i}}{d t} + A_{i} = A_{t}\} & otherwise \end{array}$

$A_{t} = {\begin{array}{c} A_{open} & p_{tot} < p_{contract} \\ A_{open} - (A_{open} - A_{close}) SmoothTrans (S, \frac{p_{tot} - p_{contract}}{p_{close} - p_{contract}}) & p_{tot} < p_{close} \\ A_{close} & otherwise \end{array}$

$S = {\begin{array}{c} smoothness & smoothTransition \\ 0 & otherwise \end{array}$

$Optional Volume Equations$

$V_{f_{A}} = {\begin{array}{c} Va (1 + \frac{p_{A}}{El}) & Use volume A = true \\ 0 & otherwise \end{array} V_{f_{B}} = {\begin{array}{c} Vb (1 + \frac{p_{B}}{El}) & Use volume B = true \\ 0 & otherwise \end{array}$

$q = q_{A} - q_{V_{A}} = - (q_{B} - q_{V_{B}})$

$q_{V_{A}} = {\begin{array}{c} \frac{d V_{f_{A}}}{d t} & Use volume A = true \\ 0 & otherwise \end{array} q_{V_{B}} = {\begin{array}{c} \frac{d V_{f_{B}}}{d t} & Use volume B = true \\ 0 & otherwise \end{array}$

Variables

Name	Units	Description	Modelica ID
$p_{CD}$	$Pa$	Pressure drop from C to D	pCD
$p$	$Pa$	Pressure drop from A to B	p
$p_{X}$	$Pa$	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
$A_{cs}$	$m^{2}$	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
$portA$	Upstream hydraulic port	portA
$portB$	Downstream hydraulic port	portB
$portC$	Hydraulic pilot port	portC
$portD$	Hydraulic pilot port	portD

Parameters

General

Name	Default	Units	Description	Modelica ID
$p_{close}$	$2.1 \cdot 10^{7}$	$Pa$	Pressure at which valve is fully open (A = Aclose)	pclose
$p_{open}$	$1.9 \cdot 10^{7}$	$Pa$	Pressure at which valve is fully closed (A = Aopen)	popen
$A_{close}$	$1 \cdot 10^{−12}$	$m^{2}$	Valve area when closed (leakage)	Aclose
$A_{open}$	$1 \cdot 10^{−5}$	$m^{2}$	Valve area when fully open	Aopen
$Exact$	$false$		When false (not checked) first-order dynamics are used for the valve area	Exact
$t_{c}$	$0.1$	$s$	Time constant	tc
$Smooth Transition$	$false$		True (checked) means enable the smoothness factor	smoothTransition
$smoothness$	$0.5$		Smoothness factor (0: sharpest, 1: smoothest); used when $Smooth Transition$ is enabled	smoothness
$k_{C}$	$1$		Pilot ratio at port C	kpC
$k_{D}$	$1$		Pilot ratio at port D	kpD

Orifice

Name	Default	Units	Description	Modelica ID
$Use constant Cd$	$true$		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 $Use constant Cd$ is true	Cd
${Re}_{Cr}$	$12$		Reynolds number at critical flow; used when $Use constant Cd$ is true	ReCr
$C_{d (\max)}$	$0.7$		Maximum flow-discharge coefficient; used when $Use constant Cd$ is false	Cd_max
${Crit}_{no}$	$1000$		Critical flow number; used when $Use constant Cd$ is false	Crit_no

Optional Volumes

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

For more information see Port Volumes.

Fluid Parameters

The following parameters, used in the equations, are properties of the Hydraulic System Properties component used in the model.

Name	Units	Description	Modelica ID
$ν$	$\frac{m^{2}}{s}$	Kinematic viscosity of fluid	nu
$ρ$	$\frac{kg}{m^{3}}$	Density of fluid	rho
$El$	$Pa$	Bulk modulus of fluid	El

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