# Variable Orifice

Generic hydraulic variable orifice

## Library

Orifices

## Description

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 flow rate is computed according to the following equations:

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

$$\Delta p={p}_{\text{A}}-{p}_{\text{B}},$$

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

For the first parameterization, the opening area (*A*) is a piecewise
function of the control member position (*h*). The area saturates at its leakage
value when the control member is in the fully closed position
(*h*_{min}). It saturates at its maximum value when the
control member is in the fully open position
(*h*_{max}).

$$A(h)=\{\begin{array}{ll}{A}_{\text{leak}}\hfill & h\le {h}_{\text{min}}\hfill \\ \frac{{A}_{\text{max}}}{{h}_{\text{max}}}h,\hfill & h>0\hfill \\ {A}_{\text{max}},\hfill & h\ge {h}_{\text{max}}\hfill \end{array}$$

The minimum control member position is calculated as:

$${h}_{\text{min}}=\frac{{h}_{\text{max}}}{{A}_{\text{max}}}{A}_{\text{leak}}$$

For the second parameterization, the opening area is a tabulated function of control member displacement. As with the linear parameterization, the area saturates at its leakage value when the control member is in the fully closed position, and it saturates at its maximum value when the control member is in the fully open position. Between the closed and open positions:

$$A=f(h)$$

The table summarizes the parameters used in the equations.

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 |

A_{leak} | Closed orifice leakage area |

p_{cr} | Minimum pressure for turbulent flow |

The minimum pressure for turbulent flow, *p*_{cr}, is
calculated according to the laminar transition specification method:

By pressure ratio — The transition from laminar to turbulent regime is defined by the following equations:

*p*_{cr}= (*p*_{avg}+*p*_{atm})(1 –*B*_{lam})*p*_{avg}= (*p*_{A}+*p*_{B})/2where

*p*_{avg}Average pressure between the block terminals *p*_{atm}Atmospheric pressure, 101325 Pa *B*_{lam}Pressure ratio at the transition between laminar and turbulent regimes ( **Laminar flow pressure ratio**parameter value)By Reynolds number — The transition from laminar to turbulent regime is defined by the following equations:

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

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

where

*D*_{H}Instantaneous orifice hydraulic diameter *ν*Fluid kinematic viscosity *Re*_{cr}Critical Reynolds number ( **Critical Reynolds number**parameter value)

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.

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 $$\Delta p={p}_{\text{A}}-{p}_{\text{B}},$$. Positive signal at the physical signal port `S`

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

## Basic Assumptions and Limitations

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.

## Parameters

**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 two 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 two 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`

.**Orifice opening vector, s**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 smooth 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.**Orifice area vector**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`

.**Pressure differential vector, dp**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 smooth 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`

.**Volumetric flow rate table, q(s,dp)**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`

— Select this option to get the best performance.`Smooth`

— Select this option to produce a continuous curve (`By area vs. opening table`

) or surface (`By pressure-flow characteristic`

) with continuous first-order derivatives.

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:

`Linear`

— Select this option to produce a curve or surface with continuous first-order derivatives in the extrapolation region and at the boundary with the interpolation region.`Nearest`

— Select this option to produce an extrapolation that does not go above the highest point in the data or below the lowest point in the data.

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`

.**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. The parameter value must be greater than 0. The default value is

`1e-12`

m^2.**Laminar transition specification**Select how the block transitions between the laminar and turbulent regimes:

`Pressure ratio`

— The transition from laminar to turbulent regime is smooth and depends on the value of the**Laminar flow pressure ratio**parameter. This method provides better simulation robustness.`Reynolds number`

— The transition from laminar to turbulent regime is assumed to take place when the Reynolds number reaches the value specified by the**Critical Reynolds number**parameter.

**Laminar flow pressure ratio**Pressure ratio at which the flow transitions between laminar and turbulent regimes. The default value is

`0.999`

. This parameter is visible only if the**Laminar transition specification**parameter is set to`Pressure ratio`

.**Critical Reynolds number**The maximum Reynolds number for laminar flow. 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`

. This parameter is visible only if the**Laminar transition specification**parameter is set to`Reynolds number`

.

**Volumetric flow rate**Flow rate specified as the fluid volume per unit time at time zero. The value entered serves as a guide during model compilation. The actual flow rate can differ if needed to satisfy all model constraints. Set the

**Priority**level to`High`

to prioritize the specified value during model assembly.**Pressure drop**Pressure change from port A to port B at time zero. The value entered serves as a guide during model compilation. The actual flow rate can differ if needed to satisfy all model constraints. Set the

**Priority**level to`High`

to prioritize the specified value during model assembly.

## Global Parameters

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.

## Ports

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**.

## Examples

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

## Extended Capabilities

## Version History

**Introduced in R2006a**