# Pressure Relief Valve (G)

Pressure relief valve in a gas network

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Simscape / Fluids / Gas / Valves & Orifices / Pressure Control Valves

## Description

The Pressure Relief Valve (G) block represents an orifice that opens with inlet pressure to keep the pressure from reaching extreme levels. At normal pressures, the valve is closed and only releases leakage flow. Above a specified pressure setting, the valve begins to open, which allows the gas to vent from the region of high pressure. The opening area increases with inlet pressure up to a maximum value, at which point the valve is fully open and pressure rises unabated.

The valve responds to inlet pressure relative to the outlet or inlet pressure relative to the environment. The chosen pressure measurement is the control pressure of the valve. The Control pressure specification parameter determines which of the measurements the block uses during simulation.

The relationship between the opening area and the pressure drop depends on the parameterization of the valve. That relationship can take the form of a linear analytical expression or a tabulated function.

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.

### Valve Opening Fraction

The inlet pressure is the valve control signal. The more the inlet pressure rises over the pressure setting of the valve, the greater the opening area becomes. The amount that the control pressure exceeds the pressure setting determines how much the valve opens. When the Opening characteristic parameter is `Linear`, the block calculates the valve opening fraction to scale the flow characteristic parameters. The valve opening fraction is

`$\stackrel{^}{p}=\frac{{P}_{Ctl}-{P}_{Set}}{\Delta P},$`

where:

• PCtl is the control pressure.

When Opening pressure specification is ```Pressure difference of port A relative to port B```,

`${P}_{\text{Ctl}}={P}_{\text{A}}-{P}_{\text{B}},$`

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

When Opening pressure specification is `Gauge pressure at port A`,

`${P}_{Ctl}={P}_{A}-{P}_{Atm},$`

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

• PSet is the set pressure.

When Opening pressure specification is ```Pressure difference of port A relative to port B```, PSet, is the value of the Set pressure differential parameter

When Opening pressure specification is `Gauge pressure at port A`, PSet is the value of the Set pressure (gauge) parameter

• ΔP is the value of the Pressure regulation range parameter.

Numerical Smoothing

When the parameter is `Linear` and 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.

### Opening Characteristics

The flow characteristic relates the opening of the valve to the input that produces it. The opening is expressed as a sonic conductance, flow coefficient, or restriction area, determined by the Valve parameterization parameter.

The flow characteristic is normally given at steady state, with the inlet at a constant, carefully controlled pressure. The flow characteristic depends only on the valve and can be linear or nonlinear. To capture the flow characteristics, use the Opening characteristic parameter:

• `Linear` — The measure of flow capacity is a linear function of the valve opening fraction. As the opening fraction rises from `0` to `1`, the measure of flow capacity scales from the specified minimum to the specified maximum.

• `Tabulated` — The measure of flow capacity is a general function, which can be linear or nonlinear, of the orifice opening fraction. The function is specified in tabulated form, with the independent variable specified by the Opening pressure differential vector or the Opening pressure (gauge) vector.

### 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 inlet pressure.

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.

Method the block uses to calculate the opening area of the valve. The linear setting treats the opening area as a linear function of the orifice opening fraction. The tabulated setting allows for a general, nonlinear relationship you can specify in tabulated form.

Pressure drop, from inlet to outlet, required to open the valve. This value marks the beginning of the pressure regulation range, over which the valve progressively opens in an effort to contain a pressure rise. The opening area of the valve depends on the difference between the actual pressure drop to the value specified here.

#### Dependencies

To enable this parameter, set Opening pressure specification to ```Pressure difference at port A relative to port B``` and Opening characteristic to `Linear`.

Valve operational range. The valve begins to open at the set pressure value, and is fully open at pmax, the end of the pressure regulation range, where pmax = pset + prange.

#### Dependencies

To enable this parameter, set Opening characteristic to `Linear`.

Inlet gauge pressure required to open the valve. This parameter marks the beginning of the pressure regulation range, over which the valve progressively opens in an effort to contain a pressure rise The opening area of the valve depends on the difference between the actual inlet pressure to the value specified here.

#### Dependencies

To enable this parameter, set Opening pressure specification to ```Gauge pressure at port A``` and Opening characteristic to `Linear`.

Vector of control pressures at which to specify the chosen measure of valve opening—sonic conductance, flow coefficient, or opening area. This vector must be equal in size to those containing the valve opening data. The vector elements must be positive and increase monotonically from left to right. These pressure values are the pressure drop from inlet to outlet.

The first element of this parameter is the pressure setting of the valve, at which the valve begins to open. The last element is the maximum pressure, at which the valve is fully open. The difference between the two is the pressure regulation range of the valve.

#### Dependencies

To enable this parameter, set Opening pressure specification to ```Pressure difference at port A relative to port B``` and Opening characteristic to `Tabulated`.

Vector of control pressures at which to specify the chosen measure of valve opening—sonic conductance, flow coefficient, or opening area. This vector must be equal in size to those containing the valve opening data. The vector elements must be positive and increase monotonically in value from left to right. These pressures are the gauge pressure at the inlet.

The first element of this parameter is the pressure setting of the valve, at which the valve begins to open. The last element is the maximum pressure, at which the valve is fully open. The difference between the two is the pressure regulation range of the valve.

#### Dependencies

To enable this parameter, set Opening pressure specification to ```Gauge pressure at port A``` and Opening characteristic to `Tabulated`.

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`, and Opening characteristic to `Linear`.

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```.

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

#### Dependencies

To enable this parameter, set Opening characteristic to `Linear`.

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.

#### Dependencies

To enable this parameter, set Opening characteristic to `Linear`.

Vector of Cv flow coefficients. Each coefficient corresponds to a value in the Opening fraction vector parameter. This parameter measures the ease with which the gas traverses the resistive element when driven by a pressure differential. The flow coefficients must increase monotonically from left to right, with greater opening fractions representing greater flow coefficients. The size of the vector must be the same as the Opening fraction vector.

#### Dependencies

To enable this parameter, set Valve parameterization to `Cv flow coefficient`, and Opening characteristic to `Tabulated`.

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`, and Opening characteristic to `Linear`.

Vector of Kv flow coefficients. Each coefficient corresponds to a value in the Opening fraction vector parameter. This parameter measures the ease with which the gas traverses the resistive element when driven by a pressure differential. The flow coefficients must increase monotonically from left to right, with greater opening fractions representing greater flow coefficients. The size of the vector must be the same as the Opening fraction vector parameter.

#### Dependencies

To enable this parameter, set Valve parameterization to `Kv flow coefficient`, and Opening characteristic to `Tabulated`.

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`, and Opening characteristic to `Linear`.

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```, and Opening characteristic to `Linear`.

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.

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```.

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```.

Vector of sonic conductances inside the resistive element. Each conductance corresponds to a value in the Opening fraction vector parameter. The sonic conductances must increase monotonically from left to right, with greater opening fractions representing greater sonic conductances. The size of the vector must be the same as the Opening fraction vector parameter.

#### Dependencies

To enable this parameter, set Valve parameterization to `Sonic conductance`, and Opening characteristic to `Tabulated`.

Vector of critical pressure ratios at which the flow first chokes, with each critical pressure ratio corresponding to a value in the Opening fraction vector parameter. The critical pressure ratio is the fraction of downstream-to-upstream pressures at which the flow velocity reaches the local speed of sound. The size of the vector must be the same as the Opening fraction vector parameter.

#### Dependencies

To enable this parameter, set Valve parameterization to `Sonic conductance`, and Opening characteristic to `Tabulated`.

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`, and Opening characteristic to `Linear`.

Correction factor that accounts for discharge losses in theoretical flows.

#### Dependencies

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

Vector of cross-sectional areas of the orifice opening. The values in this vector correspond one-to-one with the elements in the Opening fraction vector parameter. The first element of this vector is the orifice leakage area and the last element is the maximum orifice area.

#### Dependencies

To enable this parameter, set Valve parameterization to ```Orifice area```, and Opening characteristic to `Tabulated`.

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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