# Pilot-Operated Check Valve (G)

Check valve with control port in a gas network

**Library:**Simscape / Fluids / Gas / Valves & Orifices / Directional Control Valves

## Description

The Pilot-Operated Check Valve (G) block represents a check valve with an override mechanism that allows backflow when activated. A check valve is an orifice with a unidirectional opening mechanism that prevents backflow.

The **X** port of the block represents a pilot port that serves as an
override mechanism. During normal operation, the pilot port is inactive and the valve
behaves as a check valve. In a check valve, the orifice only opens when the pressure
gradient across the valve drops from inlet to outlet. The orifice opening prevents
backflow, which requires the reverse pressure gradient, and protects the upstream
components of the valve from downstream pressure surges, temperature spikes, and
chemical contamination.

For sufficient values at port **X**, the block pressurizes the pilot port
and forces the control element of the valve off its seat, permitting backflow. The valve
is then open to flow in both directions, with a reverse pressure drop from outlet to
inlet driving the flow upstream. The seat, which lies in the path of the flow,
determines if the valve is open. When it is covered, the flow is cut off and the valve
is closed.

The valve begins opening at the cracking pressure and continues opening until the end of the pressure regulation range. The cracking pressure is the initial resistance, due to friction or spring forces, that the valve must overcome to crack open. Below this threshold, the valve is closed and only leakage flow can pass. Past the end of the pressure regulation range, the valve is fully open and the flow at the maximum value is determined by the instantaneous pressure conditions.

The flow can be laminar or turbulent, and it can reach up to sonic speeds. The maximum velocity happens at the throat of the valve where the flow is narrowest and fastest. The flow chokes and the velocity saturates when a drop in downstream pressure can no longer increase the velocity. Choking occurs when the back-pressure ratio reaches the critical value characteristic of the valve. The block does not capture supersonic flow.

### Control and Other Pressures

The opening of the valve depends on the pilot pressure and the pressure drop from the inlet to the outlet. The pressure that opens the valve is

$${P}_{Ctrl}={k}_{pilot}{P}_{pilot}+{P}_{A}-{P}_{B},$$

where:

*P*is the pilot pressure._{pilot}*k*is the pilot ratio._{pilot}*P*is the gauge pressure at port_{A}**A**.*P*is the gauge pressure at port_{B}**B**.

The block determines the port pressures against absolute zero during simulation.

The pilot pressure can be relative to the inlet, port **A**, or relative
to the environment. You can select the pilot pressure by setting the **Pilot
pressure specification** parameter to ```
Pressure difference
of port X relative to port B
```

or ```
Gauge pressure at
port X
```

.

If you select `Gauge pressure at port X`

, the pilot pressure is

$${P}_{pilot}={P}_{X}-{P}_{atm},$$

where *P _{Atm}* is the
atmospheric pressure specified in the Gas Properties (G) block of
the model and

*P*is the absolute value of the pressure at the pilot port. If you select

_{X}```
Pressure difference of
port X relative to port A
```

, the pilot pressure is$${P}_{pilot}={P}_{X}-{P}_{A},$$

where *P _{A}* is the
absolute value of pressure at the inlet of the valve, port

**A**.

When the **Pilot configuration** parameter is ```
Rigidly
connected pilot spool and poppet
```

, the pilot spool transmits both
positive and negative pilot forces and the block uses
*P _{pilot}* as is in the equation for

*P*. When the

_{Ctrl}**Pilot configuration**parameter is

```
Disconnected pilot spool and
poppet
```

, the pilot spool transmits only positive pilot pressure
forces and *P*is limited to positive values. The block uses

_{pilot}`max(`*P*_{pilot},0)

in
the equation for *P*.

_{Ctrl}### Fraction of Valve Opening

The degree to which the control pressure exceeds the cracking pressure determines how much the valve opens. The fraction of valve opening is

$$\widehat{P}=\frac{{P}_{Ctrl}-{P}_{Crk}}{{P}_{Max}-{P}_{Crk}},$$

where:

*P*is the control pressure._{Ctrl}*P*is the cracking pressure specified by the_{Crk}**Cracking pressure differential**parameter.*P*is the maximum opening pressure specified by the_{Max}**Maximum opening pressure differential**parameter.

The fraction is normalized so that it is `0`

in the fully closed valve and
`1`

in the fully open valve. If the calculation returns a value
outside of these bounds, the block saturates the value to the nearest of the two
limits.

**Numerical Smoothing**

When the **Smoothing factor** parameter is nonzero, the block applies
numerical smoothing to the normalized control pressure, $$\widehat{p}$$. Enabling smoothing helps maintain numerical robustness in
your simulation.

For more information, see Numerical Smoothing.

### Valve Parameterizations

The block behavior depends on the **Valve parametrization**
parameter:

`Cv flow coefficient`

— The flow coefficient*C*_{v}determines the block parameterization. The flow coefficient measures the ease with which a gas can flow when driven by a certain pressure differential.`Kv flow coefficient`

— The flow coefficient*K*_{v}, where $${K}_{v}=0.865{C}_{v}$$, determines the block parameterization. The flow coefficient measures the ease with which a gas can flow when driven by a certain pressure differential.`Sonic conductance`

— The sonic conductance of the resistive element at steady state determines the block parameterization. The sonic conductance measures the ease with which a gas can flow when*choked*, which is a condition in which the flow velocity is at the local speed of sound. Choking occurs when the ratio between downstream and upstream pressures reaches a critical value known as the*critical pressure ratio*.`Orifice area`

— The size of the flow restriction determines the block parametrization.

The block scales the specified flow capacity by the fraction of valve opening. As the
fraction of valve opening rises from `0`

to `1`

,
the measure of flow capacity scales from its specified minimum to its specified
maximum.

### Momentum Balance

The block equations depend on the **Orifice parametrization** parameter.
When you set **Orifice parametrization** to ```
Cv
flow coefficient parameterization
```

, the mass flow rate, $$\dot{m}$$, is

$$\dot{m}={C}_{v}{N}_{6}Y\sqrt{({p}_{in}-{p}_{out}){\rho}_{in}},$$

where:

*C*is the flow coefficient._{v}*N*is a constant equal to 27.3 for mass flow rate in kg/hr, pressure in bar, and density in kg/m_{6}^{3}.*Y*is the expansion factor.*p*is the inlet pressure._{in}*p*is the outlet pressure._{out}*ρ*is the inlet density._{in}

The expansion factor is

$$Y=1-\frac{{p}_{in}-{p}_{out}}{3{p}_{in}{F}_{\gamma}{x}_{T}},$$

where:

*F*is the ratio of the isentropic exponent to 1.4._{γ}*x*is the value of the_{T}**xT pressure differential ratio factor at choked flow**parameter.

The block smoothly transitions to a linearized form of the equation when the
pressure ratio, $${p}_{out}/{p}_{in}$$, rises above the value of the **Laminar flow pressure
ratio** parameter,
*B _{lam}*,

$$\dot{m}={C}_{v}{N}_{6}{Y}_{lam}\sqrt{\frac{{\rho}_{avg}}{{p}_{avg}(1-{B}_{lam})}}({p}_{in}-{p}_{out}),$$

where:

$${Y}_{lam}=1-\frac{1-{B}_{lam}}{3{F}_{\gamma}{x}_{T}}.$$

When the pressure ratio, $${p}_{out}/{p}_{in}$$, falls below $$1-{F}_{\gamma}{x}_{T}$$, the orifice becomes choked and the block switches to the equation

$$\dot{m}=\frac{2}{3}{C}_{v}{N}_{6}\sqrt{{F}_{\gamma}{x}_{T}{p}_{in}{\rho}_{in}}.$$

When you set **Orifice parametrization** to ```
Kv flow
coefficient parameterization
```

, the block uses these same
equations, but replaces *C _{v}* with

*K*by using the relation $${K}_{v}=0.865{C}_{v}$$. For more information on the mass flow equations when the

_{v}**Orifice parametrization**parameter is

```
Kv
flow coefficient parameterization
```

or ```
Cv flow
coefficient parameterization
```

, see [2][3].When you set **Orifice parametrization** to
`Sonic conductance parameterization`

, the mass flow
rate, $$\dot{m}$$, is

$$\dot{m}=C{\rho}_{ref}{p}_{in}\sqrt{\frac{{T}_{ref}}{{T}_{in}}}{\left[1-{\left(\frac{\frac{{p}_{out}}{{p}_{in}}-{B}_{crit}}{1-{B}_{crit}}\right)}^{2}\right]}^{m},$$

where:

*C*is the sonic conductance.*B*is the critical pressure ratio._{crit}*m*is the value of the**Subsonic index**parameter.*T*is the value of the_{ref}**ISO reference temperature**parameter.*ρ*is the value of the_{ref}**ISO reference density**parameter.*T*is the inlet temperature._{in}

The block smoothly transitions to a linearized form of the equation when the
pressure ratio, $${p}_{out}/{p}_{in}$$, rises above the value of the **Laminar flow pressure
ratio** parameter
*B _{lam}*,

$$\dot{m}=C{\rho}_{ref}\sqrt{\frac{{T}_{ref}}{{T}_{avg}}}{\left[1-{\left(\frac{{B}_{lam}-{B}_{crit}}{1-{B}_{crit}}\right)}^{2}\right]}^{m}\left(\frac{{p}_{in}-{p}_{out}}{1-{B}_{lam}}\right).$$

When the pressure ratio, $${p}_{out}/{p}_{in}$$, falls below the critical pressure ratio,
*B _{crit}*, the orifice becomes
choked and the block switches to the equation

$$\dot{m}=C{\rho}_{ref}{p}_{in}\sqrt{\frac{{T}_{ref}}{{T}_{in}}}.$$

For more information on the mass flow equations when the **Orifice
parametrization** parameter is ```
Sonic conductance
parameterization
```

, see [1].

When you set **Orifice parametrization** to
`Orifice area parameterization`

, the mass flow
rate, $$\dot{m}$$, is

$$\dot{m}={C}_{d}{S}_{r}\sqrt{\frac{2\gamma}{\gamma -1}{p}_{in}{\rho}_{in}{\left(\frac{{p}_{out}}{{p}_{in}}\right)}^{\frac{2}{\gamma}}\left[\frac{1-{\left(\frac{{p}_{out}}{{p}_{in}}\right)}^{\frac{\gamma -1}{\gamma}}}{1-{\left(\frac{{S}_{R}}{S}\right)}^{2}{\left(\frac{{p}_{out}}{{p}_{in}}\right)}^{\frac{2}{\gamma}}}\right]},$$

where:

*S*is the orifice or valve area._{r}*S*is the value of the**Cross-sectional area at ports A and B**parameter.*C*is the value of the_{d}**Discharge coefficient**parameter.*γ*is the isentropic exponent.

The block smoothly transitions to a linearized form of the equation when the
pressure ratio, $${p}_{out}/{p}_{in}$$, rises above the value of the **Laminar flow pressure
ratio** parameter,
*B _{lam}*,

$$\dot{m}={C}_{d}{S}_{r}\sqrt{\frac{2\gamma}{\gamma -1}{p}_{avg}^{\frac{2-\gamma}{\gamma}}{\rho}_{avg}{B}_{lam}^{\frac{2}{\gamma}}\left[\frac{1-\text{\hspace{0.17em}}{B}_{lam}^{\frac{\gamma -1}{\gamma}}}{1-{\left(\frac{{S}_{R}}{S}\right)}^{2}{B}_{lam}^{\frac{2}{\gamma}}}\right]}\left(\frac{{p}_{in}^{\frac{\gamma -1}{\gamma}}-{p}_{out}^{\frac{\gamma -1}{\gamma}}}{1-{B}_{lam}^{\frac{\gamma -1}{\gamma}}}\right).$$

When the pressure ratio, $${p}_{out}/{p}_{in}$$, falls below$${\left(\frac{2}{\gamma +1}\right)}^{\frac{\gamma}{\gamma -1}}$$ , the orifice becomes choked and the block switches to the equation

$$\dot{m}={C}_{d}{S}_{R}\sqrt{\frac{2\gamma}{\gamma +1}{p}_{in}{\rho}_{in}\frac{1}{{\left(\frac{\gamma +1}{2}\right)}^{\frac{2}{\gamma -1}}-{\left(\frac{{S}_{R}}{S}\right)}^{2}}}.$$

For more information on the mass flow equations when the **Orifice
parametrization** parameter is ```
Orifice area
parameterization
```

, see [4].

### Mass Balance

The block assumes the volume and mass of fluid inside the valve is very small and ignores these values. As a result, no amount of fluid can accumulate in the valve. By the principle of conservation of mass, the mass flow rate into the valve through one port equals that out of the valve through the other port

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

where $$\dot{m}$$ is defined as the mass flow rate into the valve through the port
indicated by the **A** or **B** subscript.

### Energy Balance

The resistive element of the block is an adiabatic component. No heat exchange can
occur between the fluid and the wall that surrounds it. No work is done on or by the
fluid as it traverses from inlet to outlet. Energy can flow only by advection,
through ports **A** and **B**. By the principle of conservation of energy, the sum of the port
energy flows is always equal to zero

$${\varphi}_{\text{A}}+{\varphi}_{\text{B}}=0,$$

where *ϕ* is the energy flow rate into the valve through ports
**A** or **B**.

### Assumptions and Limitations

The

`Sonic conductance`

setting of the**Valve parameterization**parameter is for pneumatic applications. If you use this setting for gases other than air, you may need to scale the sonic conductance by the square root of the specific gravity.The equation for the

`Orifice area`

parameterization is less accurate for gases that are far from ideal.This block does not model supersonic flow.

## Ports

### Conserving

## Parameters

## References

[1] ISO 6358-3. "Pneumatic fluid power – Determination of flow-rate characteristics of components using compressible fluids – Part 3: Method for calculating steady-state flow rate characteristics of systems". 2014.

[2] IEC 60534-2-3. "Industrial-process control valves – Part 2-3: Flow capacity – Test procedures". 2015.

[3] ANSI/ISA-75.01.01. "Industrial-Process Control Valves – Part 2-1: Flow capacity – Sizing equations for fluid flow underinstalled conditions". 2012.

[4] P. Beater. *Pneumatic
Drives*. Springer-Verlag Berlin Heidelberg. 2007.

## Extended Capabilities

## Version History

**Introduced in R2018b**