Valve Plate Orifice (IL)

Variable orifice in an isothermal axial-piston machine

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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. A smoothing function is applied to all calculated angles, but primarily influences the simulation at the extremes of this range.

The normalized transition opening angle is calculated as:

$${\widehat{\gamma }}_{open}=\frac{\gamma }{\left({\gamma }_2-{\gamma }_1\right)}.$$

The Smoothing factor, s, is applied to the normalized angle:

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

The smoothed transition opening angle is:

$${\gamma }_{open,smoothed}={\widehat{\gamma }}_{open,smoothed}\left({\gamma }_2-{\gamma }_1\right).$$

Similarly, the normalized transition closing angle is:

$${\widehat{\gamma }}_{close}=\frac{\gamma }{\left({\gamma }_4-{\gamma }_3\right)}.$$

The Smoothing factor, s, is applied to the normalized angle:

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

The smoothed transition closing angle is:

$${\gamma }_{close,smoothed}={\widehat{\gamma }}_{close,smoothed}\left({\gamma }_4-{\gamma }_3\right)$$

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:

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

Ports

Conserving

A — Liquid port

isothermal liquid

Entry point to the orifice.

B — Liquid port

isothermal liquid

Exit point from the orifice.

Input

G — Cylinder rotation angle, rad

physical signal

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

Parameters

Cylinder block pitch radius — Cylinder pitch radius

0.05 m (default) | positive scalar

Pitch radius of the rotating cylinder.

Cylinder orifice diameter — Diameter of the cylinder slot

2.5e-3 m (default) | positive scalar

Diameter of the cylinder slot.

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.

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.

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.

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.

Model 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.