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:

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

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, S₀: $$C=\left|S-S_0\right|.$$

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

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

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:

$$\Delta p_{crit}=\frac{\pi \overline{\rho }}{8A_{overlapping}}{\left(\frac{\nu {\mathrm{Re}}_{crit}}{C_d}\right)}^2.$$

Ports

Conserving

A — Liquid port

isothermal liquid

Entry or exit point to the orifice.

B — Liquid port

isothermal liquid

Entry or exit point to the orifice.

Input

S — Sleeve displacement, m

physical signal

Sleeve position with respect to the case, in m.

Parameters

Sleeve hole diameter — Sleeve hole diameter

5e-3 m (default) | positive scalar

Diameter of the hole in the moving element.

Case hole diameter — Case hole diameter

5.5e-3 m (default) | positive scalar

Diameter of hole in the fixed element.

Sleeve position when holes are concentric — Moving element position when hole centers are aligned

0 m (default) | scalar

Moving element position when hole centers are aligned. This parameter is used to the determine instantaneous overlap of the moving and stationary holes during simulation by comparison with the signal received at port S.

Number of interacting pairs — Number of overlapping orifices, in pairs

1 (default) | positive scalar

Number of overlapping orifices, in pairs. The sleeve-case pairs do not need to be aligned in the physical hardware being modeled.

Leakage area — Gap area when in fully closed position

1e-10 m^2 (default) | positive scalar

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

Cross-sectional area at ports A and B — Orifice area at conserving ports

inf m^2 (default) | positive scalar

Areas at the entry and exit ports A and B, which are used in the pressure-flow rate equation that determines the mass flow rate through the orifice.

Discharge coefficient — Discharge coefficient

0.64 (default) | positive scalar

Correction factor that accounts for discharge losses in theoretical flows.

Critical Reynolds number — Upper Reynolds number limit for laminar flow

150 (default) | positive scalar

Upper Reynolds number limit for laminar flow through the orifice.

Smoothing factor — Numerical smoothing factor

0.01 (default) | positive scalar in the range [0,1]

Continuous smoothing factor that introduces a layer of gradual change to the flow response when the valve 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.

Pressure recovery — Whether to account for pressure increase in area expansions

Off (default) | On

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.