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Shuttle Valve

Shuttle valve with pressure bias

Description

The Shuttle Valve component allows flow out of port C from the either port A or port B, depending which has the higher pressure.

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 $Use volume A$ , $Use volume B$ , and $Use volume C$ when true, add optional volumes $V_{A}$ , $V_{B}$ , and $V_{C}$ to ports A, B, and C, 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

$p_{AB} = p_{A} - p_{B} P_{BC} = p_{B} - p_{C}$

$q_{AA} = q_{A} - q_{V_{A}} q_{BB} = q_{B} - q_{V_{B}} q_{CC} = q_{C} - q_{V_{C}}$

$q_{AA} + q_{BB} + qCC = 0$

$Orifice Fluid Equations$

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

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

${\begin{array}{c} \{A_{{cs}_{1}} = A_{i} = A_{t}\} & Exact \\ \{A_{{cs}_{1}} = \min (A_{open}, \max (A_{close}, A_{i})), t_{c} \frac{d A_{i}}{d t} + A_{i} = A_{t}\} & otherwise \end{array}$

$A_{{cs}_{2}} = A_{open} + A_{close} - A_{{cs}_{1}}$

$A_{t} = \frac{1}{2} (A_{open} - A_{close}) (1 + \tanh (eps \cdot (p_{AB} - p_{open})))$

$Optional Volume Equations$

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

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

$V_{f_{C}} = {\begin{array}{c} Vb (1 + \frac{p_{C}}{El}) & Use volume C = true \\ 0 & otherwise \end{array} q_{V_{C}} = {\begin{array}{c} \frac{d V_{f_{C}}}{d t} & Use volume C = true \\ 0 & otherwise \end{array}$

Variables

Name	Units	Description	Modelica ID
$p_{X}$	$Pa$	Pressure at port X	pX
$p_{XY}$	$Pa$	Pressure drop from X to Y	pXY
$q_{XX}$	$\frac{m^{3}}{s}$	Flow rate into port X	qXX
$q_{V_{X}}$	$\frac{m^{3}}{s}$	Flow rate into port X's optional volume	qVX
$V_{f_{X}}$	$m^{3}$	Effective volume at port X	VfX
$A_{{cs}_{1}}$	$m^{2}$	Cross-sectional area from A to B	Acs[1]
$A_{{cs}_{2}}$	$m^{2}$	Cross-sectional area from B to C	Acs[2]
$A_{i}$	$m^{2}$	Filtered interpolated area	Ai
$A_{t}$	$m^{2}$	Interpolated area	At
$u_{1}$	$\frac{m}{s}$	Fluid velocity from A to B	u[1]
$u_{2}$	$\frac{m}{s}$	Fluid velocity from B to C	u[2]

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

Connections

Name	Description	Modelica ID
$portA$	Hydraulic port of left inlet	portA
$portB$	Hydraulic port of right inlet	portB
$portC$	Hydraulic port of bottom outlet	portC

Parameters

General

Name	Default	Units	Description	Modelica ID
$p_{open}$	$1 \cdot 10^{4}$	$Pa$	Pressure difference to make one side fully open	popen
$A_{close}$	$1 \cdot 10^{−12}$	$m^{2}$	Orifice area when closed (leakage)	Aclose
$A_{open}$	$1 \cdot 10^{−5}$	$m^{2}$	Orifice area when fully open	Aopen
$eps$	$0.2$	$\frac{1}{Pa}$	Positive factor that effects transition between ports A and B	eps
$Exact$	$false$		True (checked) means first-order dynamics are used for the valve area	Exact
$t_{c}$	$0.1$	$s$	Time constant	tc

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
Use volume C	$false$		True (checked) means a hydraulic volume chamber is added to portC	useVolumeC
$V_{C}$	$1 \cdot 10^{−6}$	$m^{3}$	Volume of chamber C	Vc

For more information see Port Volumes.

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