# Variable Orifice Between Round Holes

Hydraulic variable orifice at intersection of two holes

## Library

Orifices

## Description

The Variable Orifice Between Round Holes block models a variable orifice created by two interacting round holes. These holes can have different diameters. One hole is located in the sleeve, while the other is drilled in the case, with the sleeve sliding along the case. Such a configuration is frequently seen in cartridge valves, as shown in this 3-way valve schematic.

The block can contain multiple identical interacting pairs of holes. The following schematic shows the calculation diagram for one such pair of round holes, where

s | Sleeve displacement from initial position |

c | Distance between hole centers |

d_{s} | Sleeve hole diameter |

d_{c} | Case hole diameter |

The flow rate through the orifice is proportional to the orifice area and to the pressure differential across the orifice, according to these equations:

$$A=\{\begin{array}{ll}{A}_{leak}\hfill & \text{forc}\ge (r+R)\text{}\hfill \\ \begin{array}{l}{A}_{leak}+{r}^{2}\cdot a\mathrm{cos}\left(\frac{{c}^{2}+{r}^{2}-{R}^{2}}{2cr}\right)+{R}^{2}\cdot a\mathrm{cos}\left(\frac{{c}^{2}-{r}^{2}+{R}^{2}}{2cr}\right)\\ \text{\hspace{1em}}-\frac{1}{2}\sqrt{\left(-c+r+R\right)\left(c+R-r\right)\left(c+r-R\right)\left(c+r+R\right)}\end{array}\hfill & \text{forc}(r+R)\hfill \end{array}$$

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

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

where

q | Flow rate |

p | Pressure differential |

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

A | Instantaneous passage area between orifices |

c | Instantaneous distance between hole centers |

r | Smaller hole radius |

R | Larger hole radius |

z | Number of hole pairs |

C_{D} | Flow discharge coefficient |

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:

$${p}_{cr}=\frac{\pi}{4A}{\left(\frac{{\mathrm{Re}}_{cr}}{{C}_{D}}\right)}^{2}\cdot \rho \cdot \frac{{\nu}^{2}}{2}$$

where

*ν*Fluid kinematic viscosity *Re*_{cr}Critical Reynolds number ( **Critical Reynolds number**parameter value)

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. Positive signal at port S moves the sleeve in the positive direction.

## Basic Assumptions and Limitations

Inertial effects are not taken into account.

## Parameters

**Sleeve hole diameter**Diameter of the holes drilled in the sleeve. The default value is

`0.005`

m.**Case hole diameter**Diameter of the holes drilled in the case. The default value is

`0.0054`

m.**Number of interacting pairs**Number of interacting hole pairs. The default value is

`1`

.**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`

.**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`

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

`1e-12`

m^2.

## 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 that provides the instantaneous value of the distance between the hole centers.

## Extended Capabilities

## Version History

**Introduced in R2014b**