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Cartridge Valve

Slip-in Cartridge Valve

Description

The Cartridge Valve component has a pilot pressure that can be used to assist in opening or closing the valve.

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

The pressure at port C, in addition to the force of a preloaded spring, act to close the valve while the pressure at port A and B act to open the valve.

The effect of inertia or friction for the poppet is not considered.

Based on the orifice area, the pressure vs. flow rate relationship is calculated 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.

Pilot Ratio

The pilot ratio for the cartridge valve is considered to be

$k_{pilot} = \frac{A_{C}}{A_{A}} = 1 + \frac{A_{B}}{A_{A}}$

As implied above, $1 \leq k_{pilot}$ .

The figure below shows an approximation of the effective areas for pressure at port A, B, and C. The effective area for port Y is the same as that for port C.

For example, if the area ratio of $A_{A}$ to $A_{B}$ for a cartridge valve was 1:1 then $k_{pilot} = 2$ .

Equations

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

$Orifice Fluid Equations$

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

$Pilot Equations$

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

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

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

$p_{tot} = p_{A} + (k_{pilot} - 1) p_{B} - k_{pilot} p_{C} - k_{pilot} p_{Y}$

$p_{Y} = {\begin{array}{c} portY.p & Additional Pressure Port \\ 0 & otherwise \end{array}$

$q_{C} = 0 q_{Y} = 0$

$Optional Volume Equations$

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

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

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

Variables

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

$X \in \{A, B, C, Y\}$

Connections

Name	Description	Modelica ID
$portA$	Hydraulic port on bottom	portA
$portB$	Hydraulic port on right side	portB
$portC$	Hydraulic port on top (no volume, no flow, used as pilot)	portC
$portY$	Optional hydraulic port (no volume, no flow, used as additional pilot)	portY

Parameters

General

Name	Default	Units	Description	Modelica ID
$p_{close}$	$4 \cdot 10^{5}$	$Pa$	Pressure at which valve is fully closed $(A = A_{close})$	pclose
$p_{open}$	$5 \cdot 10^{5}$	$Pa$	Pressure at which valve is fully open $(A = A_{open})$	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
$k_{pilot}$	$2$		Pilot ratio	kp
$Exact$	$false$		When false (not checked) a first-order dynamics is used for the valve area	Exact
$t_{c}$	$0.1$	$s$	Time constant	tc
$Additional Pressure Port$	$false$		True enables optional port Y	AdditionalPressurePort
$Smooth Transition$	$false$		True (checked) means enable the smoothness factor	smoothTransition
$smoothness$	$0.5$		Smoothness factor (0: sharpest, 1: smoothest); used when $Smooth Transition$ is enabled	smoothness

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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