Generic hydraulic variable orifice

Orifices

The block represents a variable orifice of any type as a data-sheet-based model. Depending on data listed in the manufacturer's catalogs or data sheets for your particular orifice, you can choose one of the following model parameterization options:

`By maximum area and opening`

— Use this option if the data sheet provides only the orifice maximum area and the control member maximum stroke.`By area vs. opening table`

— Use this option if the catalog or data sheet provides a table of the orifice passage area based on the control member displacement.`A=A(h)`

`By pressure-flow characteristic`

— Use this option if the catalog or data sheet provides a two-dimensional table of the pressure-flow characteristics.`q=q(p,h)`

In the first case, the passage area is assumed to be linearly
dependent on the control member displacement, that is, the orifice
is assumed to be closed at the initial position of the control member
(zero displacement), and the maximum opening takes place at the maximum
displacement. In the second case, the passage area is determined by
one-dimensional interpolation from the table * A=A(h)*.
In both cases, a small leakage area is assumed to exist even after
the orifice is completely closed. Physically, it represents a possible
clearance in the closed valve, but the main purpose of the parameter
is to maintain numerical integrity of the circuit by preventing a
portion of the system from getting isolated after the valve is completely
closed. An isolated or "hanging" part of the system
could affect computational efficiency and even cause failure of computation.

In the first and second cases, the model accounts for the laminar
and turbulent flow regimes by monitoring the Reynolds number (* Re*)
and comparing its value with the critical Reynolds number (

`Re`_{cr}

$$q={C}_{D}\cdot A(h)\sqrt{\frac{2}{\rho}}\cdot \frac{p}{{\left({p}^{2}+{p}_{cr}^{2}\right)}^{1/4}}$$

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

$${p}_{cr}=\frac{\rho}{2}{\left(\frac{{\mathrm{Re}}_{cr}\cdot \nu}{{C}_{D}\cdot {D}_{H}}\right)}^{2}$$

$$h={x}_{0}+x\xb7or$$

$$A(h)=\{\begin{array}{ll}h\xb7{A}_{\mathrm{max}}/{h}_{\mathrm{max}}+{A}_{leak}\hfill & \text{for}h0\hfill \\ {A}_{leak}\hfill & \text{for}h=0\text{}\hfill \end{array}$$

$${D}_{H}=\sqrt{\frac{4A(h)}{\pi}}$$

where

q | Flow rate |

p | Pressure differential |

p_{A}, p_{B} | Gauge pressures at the block terminals |

C_{D} | Flow discharge coefficient |

A(h) | Instantaneous orifice passage area |

A_{max} | Orifice maximum area |

h_{max} | Control member maximum displacement |

x_{0} | Initial opening |

x | Control member displacement from initial position |

h | Orifice opening |

or | Orifice orientation indicator. The variable assumes +1 value if the control member displacement in the globally assigned positive direction opens the orifice, and –1 if positive motion decreases the opening. |

ρ | Fluid density |

ν | Fluid kinematic viscosity |

p_{cr} | Minimum pressure for turbulent flow |

Re_{cr} | Critical Reynolds number |

D_{H} | Instantaneous orifice hydraulic diameter |

A_{leak} | Closed orifice leakage area |

In the third case, when an orifice is defined by its pressure-flow characteristics, the flow rate is determined by two-dimensional interpolation. In this case, neither flow regime nor leakage flow rate is taken into account, because these features are assumed to be introduced through the tabulated data. Pressure-flow characteristics are specified with three data sets: array of orifice openings, array of pressure differentials across the orifice, and matrix of flow rate values. Each value of a flow rate corresponds to a specific combination of an opening and pressure differential. In other words, characteristics must be presented as the Cartesian mesh, i.e., the function values must be specified at vertices of a rectangular array. The argument arrays (openings and pressure differentials) must be strictly increasing. The vertices can be nonuniformly spaced. You have a choice of three interpolation methods and two extrapolation methods.

The block positive direction is from port A to port B. This
means that the flow rate is positive if it flows from A to B and the
pressure differential is determined as $$p={p}_{A}-{p}_{B}$$. Positive signal at the physical
signal port `S`

opens or closes the orifice depending
on the value of the orifice orientation indicator.

Fluid inertia is not taken into account.

For orifices specified by pressure-flow characteristics (the third parameterization option), the model does not explicitly account for the flow regime or leakage flow rate, because the tabulated data is assumed to account for these characteristics.

**Model parameterization**Select one of the following methods for specifying the orifice:

`By maximum area and opening`

— Provide values for the maximum orifice area and the maximum orifice opening. The passage area is linearly dependent on the control member displacement, that is, the orifice is closed at the initial position of the control member (zero displacement), and the maximum opening takes place at the maximum displacement. This is the default method.`By area vs. opening table`

— Provide tabulated data of orifice openings and corresponding orifice areas. The passage area is determined by one-dimensional table lookup. You have a choice of three interpolation methods and two extrapolation methods.`By pressure-flow characteristic`

— Provide tabulated data of orifice openings, pressure differentials, and corresponding flow rates. The flow rate is determined by two-dimensional table lookup. You have a choice of three interpolation methods and two extrapolation methods.

**Orifice maximum area**Specify the area of a fully opened orifice. The parameter value must be greater than zero. The default value is

`5e-5`

m^2. This parameter is used if**Model parameterization**is set to`By maximum area and opening`

.**Orifice maximum opening**Specify the maximum displacement of the control member. The parameter value must be greater than zero. The default value is

`5e-4`

m. This parameter is used if**Model parameterization**is set to`By maximum area and opening`

.**Tabulated orifice openings**Specify the vector of input values for orifice openings as a one-dimensional array. The input values vector must be strictly increasing. The values can be nonuniformly spaced. The minimum number of values depends on the interpolation method: you must provide at least two values for linear interpolation, at least three values for cubic or spline interpolation. The default values, in meters, are

`[-0.002 0 0.002 0.005 0.015]`

. If**Model parameterization**is set to`By area vs. opening table`

, the**Tabulated orifice openings**values will be used together with**Tabulated orifice area**values for one-dimensional table lookup. If**Model parameterization**is set to`By pressure-flow characteristic`

, the**Tabulated orifice openings**values will be used together with**Tabulated pressure differentials**and**Tabulated flow rates**for two-dimensional table lookup.**Tabulated orifice area**Specify the vector of orifice areas as a one-dimensional array. The vector must be of the same size as the orifice openings vector. All the values must be positive. The default values, in m^2, are

`[1e-09 2.0352e-07 4.0736e-05 0.00011438 0.00034356]`

. This parameter is used if**Model parameterization**is set to`By area vs. opening table`

.**Tabulated pressure differentials**Specify the pressure differential vector as a one-dimensional array. The vector must be strictly increasing. The values can be nonuniformly spaced. The minimum number of values depends on the interpolation method: you must provide at least two values for linear interpolation, at least three values for cubic or spline interpolation. The default values, in Pa, are

`[-1e+07 -5e+06 -2e+06 2e+06 5e+06 1e+07]`

. This parameter is used if**Model parameterization**is set to`By pressure-flow characteristic`

.**Tabulated flow rates**Specify the flow rates as an

`m`

-by-`n`

matrix, where`m`

is the number of orifice openings and`n`

is the number of pressure differentials. Each value in the matrix specifies flow rate taking place at a specific combination of orifice opening and pressure differential. The matrix size must match the dimensions defined by the input vectors. The default values, in m^3/s, are:This parameter is used if[-1e-07 -7.0711e-08 -4.4721e-08 4.4721e-08 7.0711e-08 1e-07; -2.0352e-05 -1.4391e-05 -9.1017e-06 9.1017e-06 1.4391e-05 2.0352e-05; -0.0040736 -0.0028805 -0.0018218 0.0018218 0.0028805 0.0040736; -0.011438 -0.0080879 -0.0051152 0.0051152 0.0080879 0.011438; -0.034356 -0.024293 -0.015364 0.015364 0.024293 0.034356;]

**Model parameterization**is set to`By pressure-flow characteristic`

.**Interpolation method**Select one of the following interpolation methods for approximating the output value when the input value is between two consecutive grid points:

`Linear`

— For one-dimensional table lookup (`By area vs. opening table`

), uses a linear interpolation function. For two-dimensional table lookup (`By pressure-flow characteristic`

), uses a bilinear interpolation algorithm, which is an extension of linear interpolation for functions in two variables.`Cubic`

— For one-dimensional table lookup (`By area vs. opening table`

), uses the Piecewise Cubic Hermite Interpolation Polynomial (PCHIP). For two-dimensional table lookup (`By pressure-flow characteristic`

), uses the bicubic interpolation algorithm.`Spline`

— For one-dimensional table lookup (`By area vs. opening table`

), uses the cubic spline interpolation algorithm. For two-dimensional table lookup (`By pressure-flow characteristic`

), uses the bicubic spline interpolation algorithm.

For more information on interpolation algorithms, see the PS Lookup Table (1D) and PS Lookup Table (2D) block reference pages.

**Extrapolation method**Select one of the following extrapolation methods for determining the output value when the input value is outside the range specified in the argument list:

`From last 2 points`

— Extrapolates using the linear method (regardless of the interpolation method specified), based on the last two output values at the appropriate end of the range. That is, the block uses the first and second specified output values if the input value is below the specified range, and the two last specified output values if the input value is above the specified range.`From last point`

— Uses the last specified output value at the appropriate end of the range. That is, the block uses the last specified output value for all input values greater than the last specified input argument, and the first specified output value for all input values less than the first specified input argument.

For more information on extrapolation algorithms, see the PS Lookup Table (1D) and PS Lookup Table (2D) block reference pages.

**Orifice orientation**The parameter is introduced to specify the effect of the orifice control member motion on the valve opening. The parameter can be set to one of two options:

`Opens in positive direction`

or`Opens in negative direction`

. The value`Opens in positive direction`

specifies an orifice whose control member opens the valve when it is shifted in the globally assigned positive direction. The parameter is extremely useful for building a multi-orifice valve with all the orifices being controlled by the same spool. The default value is`Opens in positive direction`

.**Flow discharge coefficient**Semi-empirical parameter for orifice capacity characterization. Its value depends on the geometrical properties of the orifice, and usually is provided in textbooks or manufacturer data sheets. The default value is

`0.7`

.**Initial opening**Orifice initial opening. The parameter can be positive (underlapped orifice), negative (overlapped orifice), or equal to zero for zero lap configuration. The value of initial opening does not depend on the orifice orientation. The default value is

`0`

.**Critical Reynolds number**The maximum Reynolds number for laminar flow. The transition from laminar to turbulent regime is assumed to take place when the Reynolds number reaches this value. The value of the parameter depends on the orifice geometrical profile. You can find recommendations on the parameter value in hydraulics textbooks. The default value is

`12`

.**Leakage area**The total area of possible leaks in the completely closed orifice. The main purpose of the parameter is to maintain numerical integrity of the circuit by preventing a portion of the system from getting isolated after the valve is completely closed. An isolated or "hanging" part of the system could affect computational efficiency and even cause simulation to fail. Therefore, MathWorks recommends that you do not set this parameter to 0. The default value is

`1e-12`

m^2.

Parameters determined by the type of working fluid:

**Fluid density****Fluid kinematic viscosity**

Use the Hydraulic Fluid block or the Custom Hydraulic Fluid block to specify the fluid properties.

The block has the following ports:

`A`

Hydraulic conserving port associated with the orifice inlet.

`B`

Hydraulic conserving port associated with the orifice outlet.

`S`

Physical signal port to control spool displacement.

The flow rate is positive if fluid flows from port `A`

to
port `B`

. Positive signal at the physical signal
port `S`

opens or closes the orifice depending on
the value of the parameter **Orifice orientation**.

The Hydraulic Flapper-Nozzle Amplifier example illustrates the use of the Variable Orifice block in hydraulic systems.

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