# Check Valve (G)

Check valve in a gas network

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## Description

The Check Valve (G) block represents an orifice with a unidirectional opening mechanism that prevents unwanted backflow. The opening mechanism responds to pressure, and opens the orifice when the pressure gradient falls from the inlet at port A to the outlet at port B. Check valves protect components upstream against pressure surges, temperature spikes, and chemical contamination stemming from points downstream.

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 pressure to which the valve responds is the control pressure. By default, the control pressure is the drop from inlet to outlet. This setting ensures that the valve closes if the direction of flow should reverse.

You can also specify the control pressure as the gauge pressure at the inlet. Use this setting if you know that the inlet will always be at a higher pressure than the outlet. For example, when the inlet connects to a pressure source, such as a pump.

You can select the control pressure by setting the Opening pressure specification parameter to Pressure difference of port A relative to port B or Gauge pressure at port A.

Pressure difference of port A relative to port B

When the Opening pressure specification parameter is Pressure difference of port A relative to port B:

• The control pressure is:

${P}_{Ctl}={P}_{A}-{P}_{B},$

where PA is the absolute pressure at port A and PB is the absolute pressure at port B.

• The cracking pressure, PCrk, is the value of the Cracking pressure differential parameter.

• The maximum pressure of the valve, PMax, where the valve is fully open, is the value of the Maximum opening pressure differential parameter.

Gauge pressure at port A

When the Opening pressure specification parameter is Gauge pressure at port A:

• The control pressure is

${P}_{Ctrl}={P}_{A}-{P}_{Atm},$

where PAtm is the atmospheric pressure specified in the Gas Properties (G) block of the model.

• The cracking pressure, PCrk, is the value of the Cracking pressure (gauge) parameter.

• The maximum pressure of the valve, PMax, where the valve is fully open, is the value of the Maximum opening pressure (gauge) parameter.

### 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.

• PMax is the maximum opening pressure.

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.

Gas conserving port associated with the opening through which the flow must exit the valve.

## Parameters

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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 A relative to port B, the opening pressure of the valve is expressed as a pressure drop from inlet to outlet. In the alternative setting, Gauge pressure at port A, it is expressed as a gauge pressure.

Minimum pressure drop, from inlet to outlet, required to open the valve. This value marks the beginning of the pressure differential range of the valve, over which it progressively opens to allow for increased flow.

#### Dependencies

To enable this parameter, set Opening pressure specification to Pressure difference of port A relative to port B.

Pressure drop, from inlet to outlet, at which the valve is fully open. This value marks the end of the pressure differential range of the valve, over which the same progressively opens to allow for increased flow.

#### Dependencies

To enable this parameter, set Opening pressure specification to Pressure difference of port A relative to port B.

Minimum gauge pressure at the inlet (port A) required to open the valve. This value marks the beginning of the pressure range of the valve, over which the same progressively opens to allow for increased flow.

#### Dependencies

To enable this parameter, set Opening pressure specification to Gauge pressure at port A.

Maximum pressure at inlet port A, at which the valve is fully open. This value marks the end of the pressure differential range of the valve, over which the same progressively opens to allow for increased flow.

#### Dependencies

To enable this parameter, set Opening pressure specification to Gauge pressure at port A.

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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