Main Content

Axial fluid force on spool orifice in an isothermal liquid system

**Library:**Simscape / Fluids / Valve Actuators & Forces

The Spool Orifice Flow Force (IL) block models the hydraulic axial force on a spool orifice.
It receives the spool position as a physical signal at port **S**. You
can also model the flow through a spool orifice with round or rectangular holes. A
positive force acts to close the orifice.

If you would like to model the spool and axial force in one block, use the Spool Orifice (IL) block. The axial
force is output as a physical signal at port **F**.

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 opening orientation, which indicates orifice opening that is associated with a positive or negative signal at**S**.

The orifice opening is based on the open area created by the displaced spool:

$$\Delta S=\left(S-{S}_{\mathrm{min}}\right)\epsilon ,$$

where:

*S*_{min}is the**Spool position at closed orifice**.*S*is the displacement signal at port**S**.

If *Δs* falls below 0, the orifice leakage area is
used. If *ΔS* is greater than the **Spool travel between
closed and open orifice**, the maximum orifice area is used.

Setting **Orifice geometry** to ```
Round
holes
```

evenly distributes a user-defined number of holes along
the sleeve perimeter that have equal diameters and centers aligned in the same
plane.

The open area is calculated as:

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

where:

*n*_{0}is the number of holes.*d*_{0}is the diameter of the holes.*θ*is the orifice opening angle:$$\theta ={\mathrm{cos}}^{-1}\left(1-2\frac{\Delta S}{{d}_{0}}\right).$$

If

*θ*is greater than*2π*,*θ*remains at*2π*.*A*_{leak}is the**Leakage area**.

The maximum open area is:

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

Setting **Orifice geometry** 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
**Spool travel between closed and open orifice**.

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:

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

[1] Manring, N.
*Hydraulic Control Systems*. John Wiley & Sons,
2005.

[2] Merritt, H.
*Hydraulic Control Systems*. Wiley, 1967.