Orifice with Pressure-Dependent Area

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Orifice with Pressure-Dependent Area

Orifice with table-based pressure-dependent opening area

Description

The Orifice with Pressure-Dependent Area (or Orifice Pressure Dependent Area) component component models a sharp-edged orifice whose orifice varies with the pressure across the device.

The orifice area is computed as a function of the pressure difference between port A and port B from a data table containing values of pressure difference vs the corresponding orifice area. Based on the orifice area, the pressure vs flow rate relationships are derived by the formulation used in the Orifice component.

If $A (- dp) = - A (dp)$ , then the symmetric parameter can be assigned true and only the first quadrant of the $A$ vs $dp$ curve needs to be defined in the table.

	Formulation Approaches
	Two approaches were taken for formulation of the flow equation inside 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.

Pressure vs Area

For this component, the orifice area must be specified as a function of the pressure difference in the form of a table. The following parameters allows users to specify the necessary data for this calculation.

Data Source: This parameter allows users to specify the nature of the source file for the Pressure vs. Area table. The available options are file, attachment, and inline.

file: Use this option to specify a source file (an Excel or a CSV file) located on the hard drive. Use the $A (dP)$ parameter to browse to and/or select the data file. The data file will not be attached to the MapleSim model.

attachment: Use this option to specify a data file (an Excel or a CSV file) that is attached to the MapleSim model. To be used as an attachment, data files must be attached to the MapleSim model and be categorized under Data Sets. The attached files in a MapleSim model can be viewed in the Attachments pane under the Data Sets section. For more information, see Attaching a File to a Model. If the required file is not already attached to the MapleSim model, use the $A (dP)$ parameter to attach a new file.

Note:

If the file option is used, subsequent changes to the source file will be incorporated into MapleSim simulations

If the attachment option is used, subsequent changes to the source file will not affect the version attached to the MapleSim model. To ensure updates to the source files are incorporated in the simulation, the file must be reattached.

The Data Generation App can be used to generate table data from mathematical functions using Maple code. For more information about MapleSim apps, see Opening MapleSim Apps and Templates.

inline: Use this option to directly specify the pressure vs. area relationship in the form of a table. In the $A (dP)$ parameter, enter an $m \times 2$ matrix. The first column must contain values of pressure difference and the second column must contain corresponding values of the area.

Note: To change the dimensions of table, right-click (Control-click for Mac®) the parameter field and select Edit Matrix Dimensions. In the Matrix Dimensions dialog, enter values for the number or rows and columns and then click OK.

Skip rows: This option allows users to specify the number of top rows to be ignored during the calculations. This option can be useful when the Excel file contains header information.

Smooth: The smooth option specifies how the data table is interpolated.

Pressure difference unit: This option specifies the unit of pressure used in the supplied data.

Area unit: This option specifies the unit of area used in the supplied data.

	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

$A_{i} = Interpolate1D (A (dP), ({\begin{array}{c} |p| & symmetric \\ p & otherwise \end{array}), smoothness, skipRows)$

$A_{cs} = \max (minArea, \frac{A_{i}}{areaScale})$

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

$Orifice Fluid Equations$

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

$Optional Volume Equations$

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

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

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

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 A to B	q
$q_{VX}$	$\frac{m^{3}}{s}$	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
$portA$	Upstream hydraulic port	portA
$portB$	Downstream hydraulic port	portB

Parameters

General

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 calculation	UseConstantCd
$C_{d}$	$0.7$		Flow-discharge coefficient; used when $Use constant Cd$ is true	Cd
$ℜ_{Cr}$	$12$		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

Area vs. Pressure

Name	Default	Description	Modelica ID
$symmetric$	$true$	True (checked) means the data for positive pressure difference is used for both directions	symmetric
Data source	inline	Data source	datasourcemode
$A (dP)$	[1]	Data set	table
Skip rows	$0$	Number of rows to skip	skipRows
Smooth	linear	Smoothness of table interpolation	smoothness
Pressure difference unit	$kPa$	Unit of pressure used in the data	pressUnit
Area unit	${cm}^{2}$	Unit of area used in the data	areaUnit

[1] $[0, 0 &semi; 100, 1 &semi; 100, 1]$

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 Fluid 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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