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Variable orifice in an isothermal axial-piston machine

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

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**,
*γ*_{0}:

$$\gamma ={\gamma}_{0}+G.$$

*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:

Valve plate orifice

Rotor

Piston

Driving shaft

Swash plate

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

Cylinder angle at rotation onto slot,

*γ*_{1}:$${\gamma}_{1}=\Psi -\frac{r}{R}$$

Cylinder angle at complete rotation onto slot,

*γ*_{2}:$${\gamma}_{2}=\Psi +\frac{r}{R}$$

Cylinder angle at rotation beyond slot,

*γ*_{3}:$${\gamma}_{3}=\pi -2\frac{r}{R}$$

Cylinder angle at complete rotation beyond slot,

*γ*_{4}:$${\gamma}_{4}=\pi $$

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 area calculations during cylinder motion**

The transition opening area, which is the opening between the cylinder rotation
angles *γ*_{1} and
*γ*_{2}, is calculated as:

$${A}_{{\gamma}_{1}{\gamma}_{2}}={S}_{opening}+{S}_{Leak}={r}^{2}\left(2{\beta}_{opening}-\mathrm{sin}(2{\beta}_{opening})\right)+{S}_{Leak}.$$

The transition closing area, which is the opening between the cylinder rotation
angles *γ*_{3} and
*γ*_{4}, is calculated as:

$${A}_{{\gamma}_{3}{\gamma}_{4}}={S}_{closing}+{S}_{Leak}={r}^{2}\left(2{\beta}_{closing}-\mathrm{sin}(2{\beta}_{closing})\right)+{S}_{Leak},$$

where the opening and closing parameters are:

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

and

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

The area between *γ*_{2} and
*γ*_{3} is $${A}_{{\gamma}_{2}{\gamma}_{3}}={S}_{Max}+{S}_{Leak},$$ and the area between *γ*_{4}
and *γ*_{1} is $${A}_{{\gamma}_{4}{\gamma}_{1}}={S}_{Leak}.$$The maximum opening of the orifice is $${S}_{Max}=\pi {r}^{2}.$$

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:

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