Valve Plate Orifice (IL)

R2026b

Variable orifice in an isothermal axial-piston machine

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Valve Plate Orifice (IL) block

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Libraries:
Simscape / Fluids / Isothermal Liquid / Pumps & Motors / Auxiliary Components

Description

The Valve Plate Orifice (IL) block models a crescent-shaped opening between moving pistons and a pump in an axial-piston machine. The rotating pistons periodically connect to the pump intake or discharge through the orifice plate. You can connect two valve plate blocks to each cylinder of an axial-piston pump to represent both the pump inlet and outlet slots.

A rotating cylinder, with one crescent slot, is connected to the pump intake at port A and pump discharge at port B. These points are connected to the plate between the Pressure carryover angle and $$\pi$$ radians (180 degrees) from each other. The plate rotation angle is set by the signal at port G. The cylinder position angle, γ, is the sum of the position signal, G, and the initial angle offset, the Phase angle, γ₀:

$$\gamma ={\gamma }_0+G.$$

γ is always between 0 and 2π. For any combined signal and offset larger than 2π rad, γ is maintained at 2π, and for any combined signal and offset lower than 0 rad, and γ is maintained at 0. To change the initial position of the orifice relative to the slot, you can adjust the Phase angle parameter.

Axial-Piston Machine With Five Pistons

The numbers in the diagram indicate the components of an axial-piston machine:

1. Valve plate orifice

2. Rotor

3. Piston

4. Driving shaft

5. Swash plate

Cylinder Angle

The cylinder rotational alignment with the slot is described by the following angles:

• Cylinder angle at rotation onto slot, γ₁:

  $${\gamma }_1=\Psi -\frac{r}{R}$$

• Cylinder angle at complete rotation onto slot, γ₂:

  $${\gamma }_2=\Psi +\frac{r}{R}$$

• Cylinder angle at rotation beyond slot, γ₃:

  $${\gamma }_3=\pi -2\frac{r}{R}$$

• Cylinder angle at complete rotation beyond slot, γ₄:

  $${\gamma }_4=\pi$$

where:

• Ψ is the Pressure carryover angle. This angle represents the average angular distance the piston travels during its pressure transition period from a closed to opened slot.

• r is the half of the Cylinder orifice diameter.

• R is the Cylinder block pitch radius.

Orifice Opening Area

Orifice area calculations during cylinder motion

The transition opening area, which is the opening between the cylinder rotation angles γ₁ and γ₂, is calculated as:

$$A_{{\gamma }_1{\gamma }_2}=S_{opening}+S_{Leak}=r^2\left(2{\beta }_{opening}-\sin (2{\beta }_{opening})\right)+S_{Leak}.$$

The transition closing area, which is the opening between the cylinder rotation angles γ₃ and γ₄, is calculated as:

$$A_{{\gamma }_3{\gamma }_4}=S_{closing}+S_{Leak}=r^2\left(2{\beta }_{closing}-\sin (2{\beta }_{closing})\right)+S_{Leak},$$

where the opening and closing parameters are:

$${\beta }_{opening}={\cos }^{-1}\left(\frac{R}{r}\sin \left(\frac{\left(\psi +\frac{r}{R}\right)-\gamma }{2}\right)\right),$$

and

$${\beta }_{closing}={\cos }^{-1}\left(\frac{R}{r}\sin \left(\frac{\gamma -\left(\pi -2\frac{r}{R}\right)}{2}\right)\right).$$

The area between γ₂ and γ₃ is $$A_{{\gamma }_2{\gamma }_3}=S_{Max}+S_{Leak},$$ and the area between γ₄ and γ₁ is $$A_{{\gamma }_4{\gamma }_1}=S_{Leak}.$$The maximum opening of the orifice is $$S_{Max}=\pi r^2.$$

A nonzero Smoothing factor can provide additional numerical stability when the orifice is in near-closed or near-open position.

Numerically-Smoothed Orifice Angle

At the orifice slot entrance and exit angles, you can maintain numerical robustness in your simulation by adjusting the block Smoothing factor parameter. If the Smoothing factor parameter is nonzero, the block smoothly saturates the transition opening angle between γ₁ and γ₂ and the transition closing angle between γ₃ and γ₄. For more information, see Numerical Smoothing.

Mass Flow Rate Equation

The flow through a valve plate orifice is calculated from the pressure-area relationship:

$$\dot{m}=C_d{A}_{orifice}\sqrt{2\overline{\rho }}\frac{\Delta p}{{\left[\Delta p^2+\Delta p_{crit}^2\right]}^{1/4}},$$

where:

• C_d is the Discharge coefficient.

• A_orifice is the area open to flow.

• $$\overline{\rho }$$ is the average fluid density.

• Δp is the pressure drop over the valve, P_A – P_B.

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:

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

Examples

Hydraulic Axial-Piston Pump with Load-Sensing and Pressure-Limiting Control

A test rig designed to investigate the interaction between an axial-piston pump and a typical control unit simultaneously performing the load-sensing and pressure-limiting functions. To increase the fidelity of the simulation, this example uses a detailed model of the pump that accounts for the interaction between the pistons, swash plate, and valve plate.

• Open Model

Ports

Conserving

A — Liquid port

isothermal liquid

Entry point to the orifice.

Programmatic Use

Port: A

B — Liquid port

isothermal liquid

Exit point from the orifice.

Programmatic Use

Port: B

Input

G — Cylinder rotation angle, rad

physical signal

Cylinder rotation angle in radians, specified as a physical signal..

Programmatic Use

Port: G

Parameters

Cylinder block pitch radius — Cylinder pitch radius

0.05 m (default) | positive scalar

Pitch radius of the rotating cylinder.

Programmatic Use

Parameter: piston_pitch_radius

Cylinder orifice diameter — Diameter of the cylinder slot

2.5e-3 m (default) | positive scalar

Diameter of the cylinder slot.

Programmatic Use

Parameter: orifice_diameter

Pressure carryover angle — Pressure transition distance

0.06 rad (default) | scalar

Average angular distance the piston travels during the pressure transition period from a closed to opening slot.

Programmatic Use

Parameter: press_angle

Phase angle — Initial plate offset angle

0 rad (default) | scalar

Initial plate offset angle. The total angle between the plate and slot is the sum of the Phase angle and the Pressure carryover angle, Ψ.

Programmatic Use

Parameter: phase_angle

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 a 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.

Programmatic Use

Parameter: area_leak

Discharge coefficient — Discharge coefficient

0.64 (default) | positive scalar

Correction factor that accounts for discharge losses in theoretical flows.

Programmatic Use

Parameter: Cd

Critical Reynolds number — Upper Reynolds number limit for laminar flow

150 (default) | positive scalar

Upper Reynolds number limit for laminar flow through the orifice.

Programmatic Use

Parameter: Re_c

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.

Programmatic Use

Parameter: smoothing_factor

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.

Version History

Introduced in R2020a