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Orifice created by open segments with variable overlap in an isothermal system

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

The Variable Overlapping Orifice (IL) block models flow through round holes with varying overlapping areas, such as a moving sleeve within a fixed case. The overlapping holes can have different diameters, but additional holes along the spool or sleeve have the same diameter.

The flow rate depends on the variable open area created by overlapping holes in the sleeve and casing. This instantaneous opening area is calculated as:

$$\begin{array}{l}{A}_{overlap}={r}^{2}\left[{\mathrm{cos}}^{-1}\left(\frac{{C}^{2}+{r}^{2}-{R}^{2}}{2Cr}\right)-\left(\frac{{C}^{2}+{r}^{2}-{R}^{2}}{2Cr}\right)\sqrt{1-{\left(\frac{{C}^{2}+{r}^{2}-{R}^{2}}{2Cr}\right)}^{2}}\right]\\ +{R}^{2}\left[{\mathrm{cos}}^{-1}\left(\frac{{C}^{2}-{r}^{2}+{R}^{2}}{2CR}\right)-\left(\frac{{C}^{2}-{r}^{2}+{R}^{2}}{2CR}\right)\sqrt{1-{\left(\frac{{C}^{2}-{r}^{2}+{R}^{2}}{2CR}\right)}^{2}}\right],\end{array}$$

where:

*r*is the diameter of the smaller hole.*R*is the diameter of the larger hole.*C*is the absolute distance between the hole centers, calculated from the physical signal at port**S**, the instantaneous sleeve position, and the**Sleeve position when holes are concentric**,*S*_{0}: $$C=\left|S-{S}_{0}\right|.$$

If the holes on the sleeve and casing have the same diameter, the overlap area becomes:

$${A}_{overlap}=2{r}^{2}\left[{\mathrm{cos}}^{-1}\left(\frac{C}{2r}\right)-\left(\frac{C}{2r}\right)\sqrt{1-{\left(\frac{C}{2r}\right)}^{2}}\right].$$

You can maintain numerical robustness in your simulation by adjusting the block
**Smoothing factor** at the nearest and farthest points
between hole centers. A smoothing function is applied to every calculated distance,
but primarily influences the simulation at the extremes of this range.

The normalized hole center distance is:

$$\widehat{C}=\frac{\left(C-R+r\right)}{\left(2r\right)},$$

The **Smoothing factor**, *s*, is applied to the
normalized opening:

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

The smoothed opening is:

$${C}_{smoothed}={\widehat{C}}_{smoothed}(2r)+\left(R-r\right).$$

The flow through an orifice pair is calculated from the pressure-area relationship:

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

where:

*C*_{d}is the**Discharge coefficient**.*A*_{orifice}is the area open to flow, $${A}_{orifice}={A}_{overlap}+{A}_{leak}.$$*A*is the**Cross-sectional area at ports A and B**.$$\overline{\rho}$$ is the average fluid density.

*Pressure loss* describes the reduction of pressure in the
valve due to a decrease in area. The pressure loss term,
*PR*_{loss} is calculated as:

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

*Pressure recovery* describes the positive
pressure change in the valve 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.

The critical pressure difference,
*Δp*_{crit}, is the pressure differential
associated with the **Critical Reynolds number**,
*Re*_{crit}, the flow regime transition
point between laminar and turbulent flow:

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