# Pressure Relief Valve (G)

Pressure relief valve in a gas network

**Libraries:**

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.

### Control and Other Pressures

The pressure at the inlet is the control signal of the valve. The more the inlet pressure
rises over the pressure setting of the valve, the greater the opening area becomes.
The inlet pressure measurement can be relative to the outlet or the environment. 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
```

. The term *control pressure* refers to both
measurements.

**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

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

where

*P*is the absolute pressure at port_{A}**A**and*P*is the absolute pressure at port_{B}**B**.The pressure setting,

*P*, is the value of the_{Set}**Set pressure differential**parameter when the**Opening characteristic**parameter is`Linear`

. When the**Opening characteristic**parameter is`Tabulated`

,*P*is the first element of the_{Set}**Opening pressure differential vector**parameter.When the

**Opening characteristic**parameter is`Linear`

and the valve is fully open, the maximum pressure of the valve,*P*, is_{Max}$${P}_{Max}={P}_{Set}+\Delta P,$$

where

*ΔP*is the pressure regulation range of the valve. When the**Opening characteristic**parameter is`Tabulated`

,*P*is the last element of the_{Max}**Opening pressure differential vector**parameter.

**Gauge Pressure at Port A**

When the **Opening pressure specification** parameter is
`Gauge pressure at port A`

:

The control pressure is

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

where

*P*is the atmospheric pressure specified in the Gas Properties (G) block of the model._{Atm}The pressure setting,

*P*, is the value of the_{Set}**Set pressure (gauge)**parameter when the**Opening characteristic**parameter is`Linear`

. When the**Opening characteristic**parameter is`Tabulated`

,*P*is the first element of the_{Set}**Opening pressure (gauge) vector**parameter.When the

**Opening characteristic**parameter is`Linear`

and the valve is fully open, the maximum pressure of the valve,*P*, is_{Max}$${P}_{Max}={P}_{Set}+\Delta P,$$

where

*ΔP*is the pressure regulation range of the valve. When the**Opening characteristic**parameter is`Tabulated`

,*P*is the last element of the_{Max}**Opening pressure (gauge) vector**parameter.

### Fraction of Valve Opening

The degree to which 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

$$\widehat{p}=\frac{{P}_{Ctl}-{P}_{Set}}{\Delta P},$$

where:

*P*is the control pressure._{Ctl}*P*is the pressure specified by the_{Set}**Set pressure differential**or**Set pressure (gauge)**parameters.*ΔP*is the pressure regulation range specified by the**Pressure regulation range**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 **Opening characteristic** parameter is
`Linear`

and 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.

### 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, $$\dot{m}$$, 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**