# Variable Overlapping Orifice (IL)

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

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

## Description

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.

### Overlapping Area

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:

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

### Numerically-Smoothed Displacement

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:

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

### Mass Flow Rate Equation

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:

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

## Ports

### Conserving

### Input

## Parameters

## See Also

**Introduced in R2020a**