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One-way switching valve in an isothermal liquid system

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

The Shuttle Valve (IL) block models a one-way pressure control valve or switching
component in an isothermal liquid network. Flow through the valve travels from port
**A** or port **A1** to port
**B**. When the pressure differential between
**A** and **A1**,
*p*_{AA1}, is above a specified threshold
pressure, the path between ports **A** and **B** is
open to flow and the port at **A1** closes. The path has been fully
changed when the pressure reaches the **Pressure at which A-B is fully open and
A1-B is fully closed**. When *p*_{AA1}
falls below the threshold pressure, the flow inlet switches to port
**A1**.

Mass is conserved through the valve:

$${\dot{m}}_{A}+{\dot{m}}_{A1}+{\dot{m}}_{B}=0.$$

There is no flow between ports **A** and
**A1**.

The mass flow rate through the valve is calculated as:

$$\dot{m}=\frac{{C}_{d}{A}_{valve}\sqrt{2\overline{\rho}}}{\sqrt{P{R}_{loss}\left(1-{\left(\frac{{A}_{valve}}{{A}_{port}}\right)}^{2}\right)}}\frac{\Delta p}{{\left[\Delta {p}^{2}+\Delta {p}_{crit}^{2}\right]}^{1/4}},$$

where:

*C*_{d}is the**Discharge coefficient**.*A*_{valve}is the valve open area, either between ports**A**and**B**or ports**A1**and**B**.*A*_{port}is the**Cross-sectional area at ports A and B**.$$\overline{\rho}$$ is the average fluid density.

*Δp*is the valve pressure difference. Depending on the flow path through the valve, this is either*p*_{A}–*p*_{B},*p*_{A1}–*p*_{B}, or the normalized pressure when switching between the two inlet ports, $$\widehat{p}$$, defined below.

The critical pressure difference,
*Δp*_{crit}, is the pressure differential
associated with the **Critical Reynolds number**,
*Re*_{crit}, and is also dependent on the
flow path through the valve:

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

The critical Reynolds number is the flow regime transition point between laminar and turbulent flow.

*Pressure loss* describes the reduction of pressure in the
valve due to a decrease in area, and can change if different valve flow paths have
different areas. *PR*_{loss} is calculated as:

$$P{R}_{loss}=\frac{\sqrt{1-{\left(\frac{{A}_{valve}}{{A}_{port}}\right)}^{2}\left(1-{C}_{d}^{2}\right)}-{C}_{d}\frac{{A}_{valve}}{{A}_{port}}}{\sqrt{1-{\left(\frac{{A}_{valve}}{{A}_{port}}\right)}^{2}\left(1-{C}_{d}^{2}\right)}+{C}_{d}\frac{{A}_{valve}}{{A}_{port}}}.$$

*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 the **Pressure recovery** to
`Off`

. In this case,
*PR*_{loss} is 1.

The opening area, *A*_{valve}, is also
impacted by the valve opening dynamics.

The linear parameterization of the valve area depends on the flow path through the
valve. The dynamic area is based on a normally open path between ports
**A** and **B**:

$${A}_{valve,AB}=\widehat{p}\left({A}_{\mathrm{max}}-{A}_{leak}\right)+{A}_{leak}.$$

The normalized pressure, $$\widehat{p}$$, is

$$\widehat{p}=\frac{{p}_{AA1}-{p}_{open,A1B}}{{p}_{open,AB}-{p}_{open,A1B}},$$

when *p*_{AA1} is between
the parameters **Pressure at which A-B is fully closed and A1-B is fully
open** and **Pressure at which A-B is fully open and A1-B is
fully closed**. If *p*_{AA1} is
below **Pressure at which A-B is fully closed and A1-B is fully
open**, $$\widehat{p}$$ is 0. If *p*_{AA1} is above
**Pressure at which A-B is fully open and A1-B is fully
closed**, $$\widehat{p}$$ is 1.

When the valve inlet switches from port **A** to port
**A1**, the valve opening area is:

$${A}_{valve,A1B}={A}_{\mathrm{max}}+{A}_{leak}-{A}_{valve,AB}.$$

If opening dynamics are modeled, a lag is introduced to the flow response to the
modeled control pressure. *p*_{control} becomes
the dynamic control pressure, *p*_{dyn};
otherwise, *p*_{control} is the steady-state
pressure. The instantaneous change in dynamic control pressure is calculated based
on the **Opening time constant**, *τ*:

$${\dot{p}}_{dyn,AB}=\frac{{p}_{control}-{p}_{dyn,AB}}{\tau}.$$

By default, **Opening dynamics** is set to
`Off`

.

At the extremes of the control pressure range, you can maintain numerical
robustness in your simulation by adjusting the block **Smoothing
factor**. A smoothing function is applied to every calculated control
pressure, but primarily influences the simulation at the extremes of this range.

The **Smoothing factor**, *s*, is applied to the
normalized pressure, $$\widehat{p}$$:

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

and the smoothed pressure is:

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