# Spool Orifice (IL)

Variable-area spool orifice in an isothermal system

• Library:
• Simscape / Fluids / Isothermal Liquid / Valves & Orifices / Orifices

• ## Description

The Spool Orifice (IL) block models a variable-area orifice between a spool and a sleeve with holes. The sleeve holes can be either a series of round or rectangular holes. The flow rate is based on the total opening area between the sleeve, holes, and spool, which extends or retracts according to the signal received at port S. Multiple Spool Orifice (IL) blocks can be connected for multiple sets of holes along a spool-sleeve pair.

If the spool displacement in your system is supplied by an external source or custom block and you would like the axial flow force to be transmitted to the system, you can use the Spool Orifice Flow Force (IL) block, which applies the same equations for force as the Spool Orifice (IL) block.

### Flow Force

The force on the spool is calculated as:

`$F=\frac{-{\stackrel{˙}{m}}_{A}^{2}}{\rho A}\mathrm{cos}\left(\alpha \right)\epsilon ,$`

where:

• ${\stackrel{˙}{m}}_{A}$ is the mass flow rate at port A.

• ρ is the fluid density.

• A is the orifice open area, which is determined by the spool position and orifice parameterization.

• α is the jet angle, which is calculated from an approximation of the Von Mises formula:

`${\alpha }_{jet}=0.3663+0.8373\left(1-{e}^{\frac{-h}{1.848c}}\right),$`

where c is the Radial clearance, and h is the orifice opening.

• ε is the Orifice orientation, which indicates orifice opening that is associated with a positive or negative signal at S.

### Orifice Opening Area

For variable orifices, setting Orifice orientation to `Positive spool displacement opens the orifice` indicates that the orifice opens when the control member extends, while `Negative spool displacement opens the orifice` indicates that the orifice opens when the control member retracts.

The Leakage area, Aleak, is considered a small area open to flow when the orifice is closed, which maintains numerical continuity. Additionally, a nonzero Smoothing factor can provide increased numerical stability when the orifice is in near-closed or near-open position.

Round Holes

Setting Orifice parameterization to ```Round holes``` evenly distributes a user-defined number of holes along the sleeve perimeter with the equal diameters and centers aligned in the same plane.  The open area is calculated as:

`${A}_{orifice}={n}_{0}\frac{{d}_{0}^{2}}{8}\left(\theta -\mathrm{sin}\left(\theta \right)\right)+{A}_{leak}.$`

The orifice opening angle, θ, is set by the control signal at S:

`$\theta =2{\mathrm{cos}}^{-1}\left(\theta -\frac{2\Delta S}{{d}_{0}}\right),$`

where:

• n0 is the number of holes.

• d0 is the diameter of the holes.

• Aleak is ${A}_{leak}=c{d}_{0}{n}_{0}.$

• ΔS is the control member travel distance, ε(S - Smin), where Smin is the Control member position at closed orifice.

The maximum open area is:

`${A}_{\mathrm{max}}=\frac{\pi }{4}{d}_{0}^{2}{n}_{0}+{A}_{leak}.$`

Rectangular Slot

Setting Orifice parameterization to ```Rectangular slot``` models one rectangular slot in the tube sleeve. For an orifice with a slot in a rectangular sleeve, the open area is

`${A}_{orifice}=w\Delta S+{A}_{leak},$`

where w is the orifice width.

The maximum opening distance between the sleeve and case is:

`${A}_{\mathrm{max}}=w\Delta {S}_{\mathrm{max}}+{A}_{leak}.$`

where ΔSmax is the slot orifice Spool travel between closed and open orifice distance.

At the minimum orifice opening area, the leakage area is:

`${A}_{leak}=cw.$`

### Numerically-Smoothed Displacement

At the extremes of the orifice opening range, you can maintain numerical robustness in your simulation by adjusting the block . A smoothing function is applied to every calculated displacement, but primarily influences the simulation at the extremes of this range.

The normalized orifice opening is:

`$\Delta \stackrel{^}{S}=\frac{\Delta S}{\Delta {S}_{\mathrm{max}}},$`

where ΔSmax is the:

• Diameter of round holes, when Orifice parameterization is set to ```Round holes```.

• Spool travel between closed and open orifice, when Orifice parameterization is set to `Rectangular slot`.

When the Smoothing factor, f, is nonzero, a smoothing function is applied to the normalized opening:

`$\Delta {\stackrel{^}{S}}_{smoothed}=\frac{1}{2}+\frac{1}{2}\sqrt{{\left(\Delta \stackrel{^}{S}\right)}_{}^{2}+{\left(\frac{f}{4}\right)}^{2}}-\frac{1}{2}\sqrt{{\left(\Delta \stackrel{^}{S}-1\right)}^{2}+{\left(\frac{f}{4}\right)}^{2}}.$`

The smoothed opening is:

`$\Delta {S}_{smoothed}=\Delta {\stackrel{^}{S}}_{smoothed}\Delta {S}_{\mathrm{max}}.$`

### The Mass Flow Rate Equation

The flow through a spool orifice is calculated by the pressure-flow rate equation:

`$\stackrel{˙}{m}=\frac{{C}_{d}{A}_{orifice}\sqrt{2\overline{\rho }}}{\sqrt{P{R}_{loss}\left(1-{\left(\frac{{A}_{orifice}}{A}\right)}^{2}\right)}}\frac{\Delta p}{{\left[\Delta {p}^{2}+\Delta {p}_{crit}^{2}\right]}^{1/4}},$`

where:

• Cd is the Discharge coefficient.

• A is the Cross-sectional area at ports A and B.

• $\overline{\rho }$ is the average fluid density.

• Aorifice, is the orifice open area, unless:

• The opening is larger than or equal to the area at the Spool travel between closed and open orifice distance. The orifice area is then Amax.

• The orifice opening is less than or equal to the minimum opening distance. The orifice area is then Aleak.

The critical pressure difference, Δpcrit, is the pressure differential associated with the Critical Reynolds number, Recrit, which is the point of transition between laminar and turbulent flow in the fluid:

`$\Delta {p}_{crit}=\frac{\pi \overline{\rho }}{8{A}_{orifice}}{\left(\frac{\nu {\mathrm{Re}}_{crit}}{{C}_{d}}\right)}^{2}.$`

Pressure loss describes the reduction of pressure in the orifice due to a decrease in area. PRloss is calculated as:

`$P{R}_{loss}=\frac{\sqrt{1-{\left(\frac{{A}_{orifice}}{{A}_{port}}\right)}^{2}\left(1-{C}_{d}^{2}\right)}-{C}_{d}\frac{{A}_{orifice}}{{A}_{port}}}{\sqrt{1-{\left(\frac{{A}_{orifice}}{{A}_{port}}\right)}^{2}\left(1-{C}_{d}^{2}\right)}+{C}_{d}\frac{{A}_{orifice}}{{A}_{port}}}.$`

Pressure recovery describes the positive pressure change in the orifice due to an increase in area. If you do not wish to capture this increase in pressure, set Pressure recovery to `Off`. In this case, PRloss is 1.

## Ports

### Conserving

expand all

Entry or exit port of the liquid to or from the orifice.

Entry or exit port of the liquid to or from the orifice.

### Input

expand all

Control member displacement for a variable-area orifice, in m.

### Output

expand all

Axial flow force, in N.

## Parameters

expand all

Hole geometry in the sleeve. The round holes are spaced evenly about the cross-sectional circumference. There is only one hole in the ```Rectangular slot``` setting.

Diameter of all holes on the sleeve.

#### Dependencies

To enable this parameter, set Orifice parametrization to ```Round holes```.

Number of holes along the sleeve circumference.

#### Dependencies

To enable this parameter, set Orifice parametrization to ```Round holes```.

Rectangular slot width.

#### Dependencies

To enable this parameter, set Orifice parametrization to ```Rectangular slot```.

Maximum distance of the control member travel. This value provides an upper limit to calculations so that simulations do not return unphysical values.

#### Dependencies

To enable this parameter, set Orifice parametrization to ```Rectangular slot```.

Whether to model the axial hydraulic force on the spool. When this parameter is set to `On`, port F is enabled and outputs the axial force as a physical signal, in N.

Radial distance between the spool and the sleeve.

#### Dependencies

To enable this parameter, set Flow force effect to `On`.

Sum of all gaps when the valve is in the fully closed position. Any area smaller than this value is saturated to the specified leakage area. This contributes to numerical stability by maintaining continuity in the flow.

Spool offset when the orifice is fully open. A positive, nonzero value indicates a partially closed orifice. A negative, nonzero value indicates an overlapped orifice that remains open for an initial displacement set by the physical signal at port .

Cross-sectional area at the entry and exit ports A and B. This area is used in the pressure-flow rate equation that determines the mass flow rate through the orifice.

Direction of the area change for variable orifices. A positive opening orientation indicates an increase in the orifice opening. A negative orientation indicates a decrease in the orifice opening. The magnitude is always positive.

Correction factor that accounts for discharge losses in theoretical flows.

Upper Reynolds number limit for laminar flow through the orifice.

Continuous smoothing factor that introduces a layer of gradual change to the flow response when the orifice is in near-open or near-closed positions. Set this value to a nonzero value less than one to increase the stability of your simulation in these regimes.

Whether to account for pressure increase when fluid flows from a region of smaller cross-sectional area to a region of larger cross-sectional area. 