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Variable-area spool orifice in an isothermal system

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

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.

The force on the spool is calculated as:

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

where:

$${\dot{m}}_{A}$$ 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**.

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**, *A*_{leak}, 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.

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:

*n*_{0}is the number of holes.*d*_{0}is the diameter of the holes.*A*_{leak}is $${A}_{leak}=c{d}_{0}{n}_{0}.$$*ΔS*is the control member travel distance,*ε(S - S*, where_{min})*S*is the_{min}**Control member position at closed orifice**.

The maximum open area is:

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

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 *ΔS*_{max} is the
slot orifice **Spool travel between closed and open orifice**
distance.

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

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

At the extremes of the orifice opening range, you can maintain numerical
robustness in your simulation by adjusting the block **Smoothing
factor**. 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:

$$\Delta \widehat{S}=\frac{\Delta S}{\Delta {S}_{\mathrm{max}}},$$

where *ΔS _{max}* 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:

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

The smoothed opening is:

$$\Delta {S}_{smoothed}=\Delta {\widehat{S}}_{smoothed}\Delta {S}_{\mathrm{max}}.$$

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

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

where:

*C*_{d}is the**Discharge coefficient**.*A*is the**Cross-sectional area at ports A and B**.$$\overline{\rho}$$ is the average fluid density.

*A*_{orifice}, 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*A*_{max}.The orifice opening is less than or equal to the minimum opening distance. The orifice area is then

*A*_{leak}.

The critical pressure difference,
*Δp*_{crit}, is the pressure differential
associated with the **Critical Reynolds number**,
*Re*_{crit}, 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. *PR*_{loss}
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,
*PR*_{loss} is 1.

Spool Orifice Flow Force (IL) | Variable Overlapping Orifice (IL) | Annular Leakage (IL) | Orifice (IL)