Pilot-Operated Check Valve (G)

Check valve with control port in a gas network

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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:

• Ppilot is the pilot pressure.

• kpilot is the pilot ratio.

• PA is the gauge pressure at port A.

• PB is the gauge pressure at port 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 PAtm is the atmospheric pressure specified in the Gas Properties (G) block of the model and PX is the absolute value of the pressure at the pilot port. If you select ```Pressure difference of port X relative to port A```, the pilot pressure is

`${P}_{pilot}={P}_{X}-{P}_{A},$`

where PA 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 Ppilot as is in the equation for PCtrl. When the Pilot configuration parameter is ```Disconnected pilot spool and poppet```, the pilot spool transmits only positive pilot pressure forces and Ppilot is limited to positive values. The block uses `max(Ppilot,0)` in the equation for PCtrl.

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

`$\stackrel{^}{P}=\frac{{P}_{Ctrl}-{P}_{Crk}}{{P}_{Max}-{P}_{Crk}},$`

where:

• PCtrl is the control pressure.

• PCrk is the cracking pressure specified by the Cracking pressure differential parameter.

• PMax is the maximum opening pressure specified by the 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, $\stackrel{^}{p}$. Enabling smoothing helps maintain numerical robustness in your simulation.

Valve Parameterizations

The block behavior depends on the Valve parametrization parameter:

• `Cv flow coefficient` — The flow coefficient Cv 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 Kv, 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, $\stackrel{˙}{m}$, is

`$\stackrel{˙}{m}={C}_{v}{N}_{6}Y\sqrt{\left({p}_{in}-{p}_{out}\right){\rho }_{in}},$`

where:

• Cv is the flow coefficient.

• N6 is a constant equal to 27.3 for mass flow rate in kg/hr, pressure in bar, and density in kg/m3.

• Y is the expansion factor.

• pin is the inlet pressure.

• pout is the outlet pressure.

• ρin is the inlet density.

The expansion factor is

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

where:

• Fγ is the ratio of the isentropic exponent to 1.4.

• xT is the value of the 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, Blam,

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

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

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

When you set Orifice parametrization to ```Kv flow coefficient parameterization```, the block uses these same equations, but replaces Cv with Kv by using the relation ${K}_{v}=0.865{C}_{v}$. For more information on the mass flow equations when the 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, $\stackrel{˙}{m}$, is

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

where:

• C is the sonic conductance.

• Bcrit is the critical pressure ratio.

• m is the value of the Subsonic index parameter.

• Tref is the value of the ISO reference temperature parameter.

• ρref is the value of the ISO reference density parameter.

• Tin is the inlet temperature.

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 Blam,

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

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

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

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, $\stackrel{˙}{m}$, is

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

where:

• Sr is the orifice or valve area.

• S is the value of the Cross-sectional area at ports A and B parameter.

• Cd is the value of the 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, Blam,

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

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

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

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

`${\stackrel{˙}{m}}_{A}+{\stackrel{˙}{m}}_{B}=0,$`

where $\stackrel{˙}{m}$ 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

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Gas conserving port associated with the opening through which the flow must enter the valve during normal operation when the pilot mechanism is disabled. This port can serve as an exit only when the pilot port is pressurized to a sufficient degree.

Gas conserving port associated with the opening through which the flow must exit the valve during normal operation when the pilot mechanism is disabled.

Gas conserving port associated with the opening by which to actuate, by the application of a sufficient pressure, the pilot mechanism that opens the valve for reverse flow.

Parameters

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Method the block uses when the pilot pressure is negative:

• ```Rigidly connected pilot spool and poppet``` — The pilot spool transmits both positive and negative pilot pressure forces, so the block uses ppilot as is.

• `Disconnected pilot spool and poppet` — The pilot spool only transmits positive pilot pressure forces, so the block limits ppilot to positive values only. The block replaces negative values with zero.

Choice of pressure measurement to use as valve control signal. The block uses this setting to determine when the valve should begin to open. In the default setting, ```Pressure difference of port X relative to port A```, the opening pressure of the valve is expressed as a pressure drop from the pilot port, X, to the inlet, A. In the alternative setting, ```Gauge pressure at port X```, it is expressed as a gauge inlet pressure measured against the environment pressure.

Effective pressure differential at which the valve begins to open. This differential is the sum of the pressure drop from inlet to outlet with the product of the pilot pressure and the pilot ratio. The pilot pressure used depends on the setting of the Pilot pressure specification parameter.

Effective pressure differential at which the valve is fully open. This differential is the sum of the pressure drop from inlet to outlet with the product of the pilot pressure and the pilot ratio. The pilot pressure used depends on the setting of the Pilot pressure specification parameter.

Ratio of the pilot port area to the inlet port area.

Method to calculate the mass flow rate.

• `Cv flow coefficient` — The flow coefficient Cv determines the block parameterization.

• `Kv flow coefficient` — The flow coefficient Kv, where ${K}_{v}=0.865{C}_{v}$, determines the block parameterization.

• `Sonic conductance` — The sonic conductance of the resistive element at steady state determines the block parameterization.

• `Orifice area` — The size of the flow restriction determines the block parametrization.

Value of the Cv flow coefficient when the restriction area available for flow is at a maximum. This parameter measures the ease with which the gas traverses the resistive element when driven by a pressure differential.

Dependencies

To enable this parameter, set Valve parameterization to ```Cv flow coefficient```.

Ratio between the inlet pressure, pin, and the outlet pressure, pout, defined as $\left({p}_{in}-{p}_{out}\right)/{p}_{in}$ where choking first occurs. If you do not have this value, look it up in table 2 in ISA-75.01.01 [3]. Otherwise, the default value of 0.7 is reasonable for many valves.

Dependencies

To enable this parameter, set Valve parameterization to ```Cv flow coefficient``` or ```Kv flow coefficient```.

Maximum value of the Kv flow coefficient when the restriction area available for flow is at a maximum. This parameter measures the ease with which the gas traverses the resistive element when driven by a pressure differential.

Dependencies

To enable this parameter, set Valve parameterization to ```Kv flow coefficient```.

Value of the sonic conductance when the cross-sectional area available for flow is at a maximum.

Dependencies

To enable this parameter, set Valve parameterization to ```Sonic conductance```.

Pressure ratio at which flow first begins to choke and the flow velocity reaches its maximum, given by the local speed of sound. The pressure ratio is the outlet pressure divided by inlet pressure.

Dependencies

To enable this parameter, set Valve parameterization to ```Sonic conductance```.

Empirical value used to more accurately calculate the mass flow rate in the subsonic flow regime.

Dependencies

To enable this parameter, set Valve parameterization to ```Sonic conductance```.

Temperature at standard reference atmosphere, defined as 293.15 K in ISO 8778.

Dependencies

To enable this parameter, set Valve parameterization to ```Sonic conductance```.

Density at standard reference atmosphere, defined as 1.185 kg/m3 in ISO 8778.

Dependencies

To enable this parameter, set Valve parameterization to ```Sonic conductance```.

Cross-sectional area of the orifice opening when the cross-sectional area available for flow is at a maximum.

Dependencies

To enable this parameter, set Valve parameterization to ```Orifice area```.

Correction factor that accounts for discharge losses in theoretical flows.

Dependencies

To enable this parameter, set Valve parameterization to ```Orifice area```.

Ratio of the flow rate of the orifice when it is closed to when it is open.

Continuous smoothing factor that introduces a layer of gradual change to the flow response when the orifice is in near-open or near-closed positions. Set this parameter to a nonzero value less than one to increase the stability of your simulation in these regimes.

Pressure ratio at which flow transitions between laminar and turbulent flow regimes. The pressure ratio is the outlet pressure divided by inlet pressure. Typical values range from `0.995` to `0.999`.

Area normal to the flow path at each port. The ports are equal in size. The value of this parameter should match the inlet area of the components to which the resistive element connects.

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.

Version History

Introduced in R2018b

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