Check Valve (G)

Check valve in a gas network

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Libraries:
Simscape / Fluids / Gas / Valves & Orifices / Directional Control Valves

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 P_A is the absolute pressure at port A and P_B is the absolute pressure at port B.

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

• The maximum pressure of the valve, P_Max, 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 P_Atm is the atmospheric pressure specified in the Gas Properties (G) block of the model.

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

• The maximum pressure of the valve, P_Max, 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

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

where:

• P_Ctrl is the control pressure.

• P_Crk is the cracking pressure.

• P_Max 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, $$\widehat{p}$$. 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 $$K_v=0.865C_v$$, 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_v 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.

• p_in is the inlet pressure.

• p_out is the outlet pressure.

• ρ_in is the inlet density.

The expansion factor is

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

where:

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

• x_T 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, B_lam,

$$\dot{m}=C_v{N}_6{Y}_{lam}\sqrt{\frac{{\rho }_{avg}}{p_{avg}(1-B_{lam})}}(p_{in}-p_{out}),$$

where:

$$Y_{lam}=1-\frac{1-B_{lam}}{3F_{\gamma }{x}_T}.$$

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_v by using the relation $$K_v=0.865C_v$$. 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, $$\dot{m}$$, 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_crit is the critical pressure ratio.

• m is the value of the Subsonic index parameter.

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

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

• T_in 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 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_r is the orifice or valve area.

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

• C_d 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, B_lam,

$$\dot{m}=C_d{S}_r\sqrt{\frac{2\gamma }{\gamma -1}{p}_{avg}^{\frac{2-\gamma }{\gamma }}{\rho }_{avg}{B}_{lam}^{\frac{2}{\gamma }}\left[\frac{1-B_{lam}^{\frac{\gamma -1}{\gamma }}}{1-{\left(\frac{S_R}{S}\right)}^2{B}_{lam}^{\frac{2}{\gamma }}}\right]}\left(\frac{p_{in}^{\frac{\gamma -1}{\gamma }}-p_{out}^{\frac{\gamma -1}{\gamma }}}{1-B_{lam}^{\frac{\gamma -1}{\gamma }}}\right).$$

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.

Examples

Antagonistic McKibben Muscle Actuator

This demo shows a muscle actuation based on two air muscle actuators (or McKibben artificial muscles) in antagonistic connection. The air muscle actuators are connected to the opposite sides of a lever. The 4-way directional valve is controlled by an electro-mechanical valve actuator. In the 4-way directional way when the high-pressure path P-A and return line B-T are open, the top air muscle actuator contracts and forces the bottom air muscle actuator on the opposite side to extend. Similarly, as the high-pressure path P-B and return line A-T open, the bottom air muscle actuator starts to contract and forces the top air muscle actuator to extend. The oscillating motions of the muscles are converted into the angular rotation of the output load connected to the mechanical linkage modeled with the slider-cranks.

• Open Model

Ports

Conserving

A — Valve entrance

gas

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

B — Valve entrance

gas

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

Parameters

Opening pressure specification — Choice of pressure measurement to use as valve control signal

Pressure difference of port A relative to port B (default) | Gauge pressure at port A

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.

Cracking pressure differential — Pressure drop required to open valve

0.01 MPa (default) | positive scalar

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.

Maximum opening pressure differential — Pressure drop at which valve is fully open

0.02 MPa (default) | positive scalar

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.

Cracking pressure (gauge) — Minimum pressure at inlet required to open valve

0.01 MPa (default) | positive scalar

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 opening pressure (gauge) — Maximum pressure at inlet

0.02 MPa (default) | positive scalar

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.

Valve parameterization — Parameterization used to specify flow characteristics of orifice

Cv flow coefficient (default) | Kv flow coefficient | Sonic conductance | Orifice area

Method to calculate the mass flow rate.

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

• Kv flow coefficient — The flow coefficient K_v, where $$K_v=0.865C_v$$, 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.

Maximum Cv flow coefficient — Cv coefficient that corresponds to maximally open component

4 (default) | positive scalar

Value of the C_v 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.

xT pressure differential ratio factor at choked flow — Ratio of pressure differentials

0.7 (default) | positive scalar

Ratio between the inlet pressure, p_in, and the outlet pressure, p_out, defined as $$(p_{in}-p_{out})/p_{in}$$ 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 Kv flow coefficient — Kv coefficient that corresponds to maximally open component

3.6 (default) | positive scalar

Maximum value of the K_v 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.

Maximum sonic conductance — Sonic conductance corresponding to maximally open component

12 l/(bar*s) (default) | positive scalar

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.

Critical pressure ratio — Ratio of downstream and upstream pressures at which the flow first chokes

0.3 (default) | positive scalar

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.

Subsonic index — Exponent used to characterize mass flow in subsonic flow regime

0.5 (default) | positive scalar

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.

ISO reference temperature — ISO 8778 reference temperature

293.15 K (default) | scalar

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

You only need to adjust the ISO reference parameter values if you are using sonic conductance values that are obtained at difference reference values.

Dependencies

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

ISO reference density — ISO 8778 reference density

1.185 kg/m^3 (default) | positive scalar

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

You only need to adjust the ISO reference parameter values if you are using sonic conductance values that are obtained at difference reference values.

Dependencies

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

Maximum orifice area — Flow area corresponding to maximally open component

1e-4 m^2 (default) | positive scalar

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.

Discharge coefficient — Discharge coefficient

0.64 (default) | positive scalar in the range of [0,1]

Correction factor that accounts for discharge losses in theoretical flows.

Dependencies

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

Leakage flow fraction — Ratio of flow rates

1e-6 (default) | positive scalar

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

Smoothing factor — Numerical smoothing factor

0.01 (default) | positive scalar in the range [0,1]

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.

Laminar flow pressure ratio — Pressure ratio at which flow transitions between laminar and turbulent regimes

0.999 (default) | positive scalar

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.

Cross-sectional area at ports A and B — Flow area at gas ports

0.01 m^2 (default) | positive scalar

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.

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.

Version History

Introduced in R2018b

R2022b: Model gas systems using improved flow computations

In previous versions, the blocks in the gas library used a conversion factor to implement different orifice parameterizations in terms of sonic conductance. You can use the Valve parameterization parameter to specify the parameterization.

The new parameterizations also include new parameters:

• xT pressure differential ratio factor at choked flow

• Discharge coefficient

• Leakage flow fraction. This parameter replaces the Cv coefficient (USCS) at leakage flow, Kv coefficient (SI) at leakage flow, Sonic conductance at leakage flow, and Restriction area at leakage flow parameters.

To improve clarity:

• The Opening parameterization parameter has been renamed to Opening characteristic.

• The Reference temperature and Reference density parameters have been renamed to ISO reference temperature and ISO reference density, respectively.

• The Cv coefficient (USCS) at maximum flow and Kv coefficient (USCS) at maximum flow parameters have been renamed to Maximum Cv coefficient and Maximum Kv coefficient , respectively.

• The Sonic conductance at maximum flow parameter has been renamed to Maximum sonic conductance.

• The Pressure control specification parameter has been renamed to Opening pressure specification. The options for this parameter have been renamed to Pressure difference of port A relative to port B and Gauge pressure at port A.