# Pressure Relief Valve (2P)

**Libraries:**

Simscape /
Fluids /
Two-Phase Fluid /
Valves & Orifices /
Pressure Control Valves

## Description

The Pressure Relief Valve (2P) block models a pressure-controlling relief valve in a
two-phase fluid network. The valve opens when the pressure exceeds the set pressure.
Specify the pressure differential that opens the valve in the **Set pressure
specification** parameter, which can be the pressure difference between
ports **A** and **B** or the gauge pressure at port
**A**. When you set **Set pressure control** to
`Controlled`

, the set pressure varies according to the
input signal at port **Ps**.

The **Modeling option** parameter controls the parameterization
options for a valve designed for modeling either vapor or liquid, but does not impact
the fluid properties. The block calculates fluid properties inside the valve from inlet
conditions. There is no heat exchange between the fluid and the environment, and
therefore phase change inside the orifice only occurs due to a pressure drop or a
propagated phase change from another part of the model.

### Pressure Control

The valve opens when the pressure in the valve,
*p _{control}*, exceeds the set pressure,

*p*. The valve is fully open when the control pressure reaches

_{set}*p*+

_{set}*p*, where

_{range}*p*is the value of the

_{range}**Pressure regulation range**parameter.

For the linear parametrizations, the block calculates the opening fraction of the
valve, *λ*. When you set **Set pressure control**
to `Constant`

, *λ* is

$$\lambda =\left(1-{f}_{leak}\right)\frac{\left({p}_{control}-{p}_{set}\right)}{{p}_{range}}+{f}_{leak},$$

where:

*f*is the value of the_{leak}**Leakage flow fraction**parameter.*p*is the control pressure, which depends on the value of the_{control}**Set pressure specification**parameter.When you set

**Set pressure specification**to`Pressure differential`

, the control pressure is*p*._{A}̶ p_{B}When you set

**Set pressure specification**to`Gauge pressure at port A`

, the control pressure is the difference between the pressure at port**A**and atmospheric pressure.*p*is the set pressure, which depends on the value of the_{set}**Set pressure specification**parameter.When you set

**Set pressure specification**to`Pressure differential`

,*p*is the value of the_{set}**Set pressure differential**parameter.When you set

**Set pressure specification**to`Gauge pressure at port A`

,*p*is the value of the_{set}**Set pressure (gauge)**parameter.

When you set **Set pressure control** to
`Controlled`

, the valve opening fraction is

$$\lambda =\left(1-{f}_{leak}\right)\frac{\left({p}_{control}-{p}_{s}\right)}{{p}_{range}}+{f}_{leak},$$

where *p _{s}* is the value
of the signal at port

**Ps**. If the control pressure exceeds the valve pressure regulation range, the valve opening fraction is 1.

### Liquid Valve

When **Modeling option** is ```
Liquid operating
condition
```

, the block parameterizations depend on the value of the
**Valve parameterization** parameter. The block calculates the
pressure loss and pressure recovery in the same way for all liquid parameterization
options.

The critical pressure difference,
*Δp _{crit}*, is the pressure differential where
the flow transitions between laminar and turbulent flow. For all liquid
parameterizations,

*Δp*is

_{crit}$$\Delta {p}_{crit}=\frac{\left({p}_{A}+{p}_{B}\right)}{2}\left(1-{B}_{lam}\right),$$

where:

*p*and_{A}*p*are the pressure at port_{B}**A**and**B**, respectively.*B*is the value of the_{lam}**Laminar flow pressure ratio**parameter.

The block accounts for pressure loss by using the ratio of the pressure loss across the
whole valve to the pressure drop immediately across the valve restriction area. This
ratio, *PR _{loss}*, is

$$P{R}_{loss}=\frac{\sqrt{1-{\left(\frac{{A}_{valve}}{{A}_{port}}\right)}^{2}\left(1-{C}_{d}^{2}\right)}-{C}_{d}\frac{{A}_{valve}}{{A}_{port}}}{\sqrt{1-{\left(\frac{{A}_{valve}}{{A}_{port}}\right)}^{2}\left(1-{C}_{d}^{2}\right)}+{C}_{d}\frac{{A}_{valve}}{{A}_{port}}},$$

where:

*A*is the value of the_{port}**Cross-sectional area at ports A and B**parameter.*C*is the value of the_{d}**Discharge coefficient**parameter.*A*is the valve area._{valve}

The pressure recovery is the positive pressure change in the valve due to an
increase in area after the orifice hole. If you do not want to capture this increase
in pressure, clear the **Pressure recovery** check box. In this
case, *PR _{loss}* is 1, which reduces the model
complexity. Clear this setting if the orifice hole is quite small relative to the
port area or if the next downstream component is close to the block and any jet does
not have room to dissipate.

**Linear - Nominal Mass Flow Rate vs. Pressure Parameterization**

When you set **Valve Parameterization** to
`Nominal mass flow rate vs. pressure`

, the mass
flow rate through the valve is

$$\dot{m}=\lambda {\dot{m}}_{nom}\left[\sqrt{\frac{{v}_{nom}}{2\Delta {p}_{nom}}}\right]\sqrt{\frac{2}{{v}_{in}}}\frac{\Delta p}{{\left(\Delta {p}^{2}+\Delta {p}_{crit}^{2}\right)}^{0.25}},$$

where:

*Δp*is the pressure drop over the valve,*p*._{A}̶ p_{B}$${\dot{m}}_{nom}$$ is the value of the

**Nominal mass flow rate at maximum valve opening**parameter.*Δp*is the value of the_{nom}**Nominal pressure drop rate at maximum valve opening**parameter.*v*is the nominal inlet specific volume. The block determines this value from the tabulated fluid properties data based on the_{nom}**Nominal inlet specific enthalpy**and**Nominal inlet pressure**parameters.*v*is the inlet specific volume._{in}

**Linear - Area vs. Pressure Parameterization**

When you set **Valve parameterization** to
`Linear - area vs. pressure`

, the valve area is

$${\text{A}}_{valve}=\lambda {A}_{max},$$

where *A _{max}* is the
value of the

**Maximum valve area**parameter.

The mass flow rate is

$$\dot{m}=\frac{{C}_{d}{A}_{valve}\sqrt{\frac{2}{{v}_{in}}}}{\sqrt{P{R}_{loss}\left(1-{\left(\frac{{A}_{valve}}{{A}_{port}}\right)}^{2}\right)}}\frac{\Delta p}{{\left[\Delta {p}^{2}+\Delta {p}_{crit}^{2}\right]}^{1/4}}.$$

When the valve is in a near-open or near-closed position,
you can maintain numerical robustness in your simulation by adjusting the
**Smoothing factor** parameter. If the
**Smoothing factor** parameter is nonzero, the block
smoothly saturates the opening area between
*A _{leak}* and

*A*, where

_{max}*A*. For more information, see Numerical Smoothing.

_{leak}= f_{leak}A_{max}**Tabulated Data - Area vs. Pressure Parameterization**

When you set **Valve Parameterization** to
`Tabulated data - area vs. pressure`

, the block
interpolates the valve area from the **Valve area vector** and
**Opening pressure differential vector** or
**Opening pressure (gauge) vector** parameters.

The block uses the same equation as the ```
Linear - Area vs.
pressure
```

setting to calculate the volumetric flow
rate.

**Fluid Specific Volume Dynamics**

For all parameterizations, the block calculates the fluid specific volume during simulation based on the liquid state.

If the fluid at the valve inlet is a liquid-vapor mixture, the block calculates the specific volume as

$${v}_{in}=\left(1-{x}_{dyn}\right){v}_{liq}+{x}_{dyn}{v}_{vap},$$

where:

*x*is the inlet vapor quality. The block applies a first-order lag to the inlet vapor quality of the mixture._{dyn}*v*is the liquid specific volume of the fluid._{liq}*v*is the vapor specific volume of the fluid._{vap}

If the inlet fluid is liquid or vapor,
*v _{in}* is the respective liquid
or vapor specific volume.

If the inlet vapor quality is a liquid-vapor mixture, the block applies a first-order time lag,

$$\frac{d{x}_{dyn}}{dt}=\frac{{x}_{in}-{x}_{dyn}}{\tau},$$

where:

*x*is the dynamic vapor quality._{dyn}*x*is the current inlet vapor quality._{in}*τ*is the value of the**Inlet phase change time constant**parameter.

If the inlet fluid is a subcooled liquid, *x _{in} =
0*. If the inlet fluid is a superheated vapor,

*x*.

_{in}= 1### Vapor Valve

When **Modeling option** is ```
Vapor operating
condition
```

, the block behavior depends on the **Valve
parameterization** and **Opening characteristic**
parameters.

The flow rate in the valve depends on the **Opening
characteristic** parameter:

`Linear`

— The block scales the measure of flow capacity by*λ*to account for the valve opening area.`Tabulated`

— The block interpolates the measure of flow capacity from either the**Cv flow coefficient vector**,**Kv flow coefficient vector**, or**Orifice area vector**parameters. This function uses a one-dimensional lookup table.

**Cv Flow Coefficient Parameterization**

When you set **Valve parametrization** to ```
Cv
flow coefficient
```

, the mass flow rate is

$$\dot{m}={C}_{v}{N}_{6}Y\sqrt{\frac{({p}_{in}-{p}_{out})}{{v}_{in}}},$$

where:

*C*is the flow coefficient._{v}*N*is a constant equal to 27.3 when mass flow rate is in kg/hr, pressure is in bar, and density is in kg/m_{6}^{3}.*Y*is the expansion factor.*p*is the inlet pressure._{in}*p*is the outlet pressure._{out}*v*is the inlet specific volume._{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}*,

$$\dot{m}={C}_{v}{N}_{6}{Y}_{lam}\sqrt{\frac{1}{{p}_{avg}(1-{B}_{lam}){v}_{avg}}}({p}_{in}-{p}_{out}),$$

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 valve becomes choked and the block uses the equation

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

**Kv Flow Coefficient Parameterization**

When you set **Valve parametrization** to ```
Kv flow
coefficient
```

, the block uses the same equations as the
`Cv flow coefficient`

parametrization, but replaces
*C _{v}* with

*K*using the relation $${K}_{v}=0.865{C}_{v}$$.

_{v}**Valve Area Parameterization**

When you set **Valve parametrization** to
`Orifice area`

, the mass flow rate is

$$\dot{m}={C}_{d}{A}_{valve}\sqrt{\frac{2\gamma}{\gamma -1}{p}_{in}\frac{1}{{v}_{in}}{\left(\frac{{p}_{out}}{{p}_{in}}\right)}^{\frac{2}{\gamma}}\left[\frac{1-{\left(\frac{{p}_{out}}{{p}_{in}}\right)}^{\frac{\gamma -1}{\gamma}}}{1-{\left(\frac{{A}_{valve}}{{A}_{port}}\right)}^{2}{\left(\frac{{p}_{out}}{{p}_{in}}\right)}^{\frac{2}{\gamma}}}\right]},$$

where:

*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}*,

$$\dot{m}={C}_{d}{A}_{valve}\sqrt{\frac{2\gamma}{\gamma -1}{p}_{avg}^{\frac{2-\gamma}{\gamma}}\frac{1}{{v}_{avg}}{B}_{lam}^{\frac{2}{\gamma}}\left[\frac{1-\text{\hspace{0.17em}}{B}_{lam}^{\frac{\gamma -1}{\gamma}}}{1-{\left(\frac{{A}_{valve}}{{A}_{port}}\right)}^{2}{B}_{lam}^{\frac{2}{\gamma}}}\right]}\left(\frac{{p}_{in}^{\frac{\gamma -1}{\gamma}}-{p}_{out}^{\frac{\gamma -1}{\gamma}}}{1-{B}_{lam}^{\frac{\gamma -1}{\gamma}}}\right).$$

When the pressure ratio, $${p}_{out}/{p}_{in}$$, falls below$${\left(\frac{2}{\gamma +1}\right)}^{\frac{\gamma}{\gamma -1}}$$ , the valve becomes choked and the block uses the equation

$$\dot{m}={C}_{d}{A}_{valve}\sqrt{\frac{2\gamma}{\gamma +1}{p}_{in}\frac{1}{{v}_{in}}\frac{1}{{\left(\frac{\gamma +1}{2}\right)}^{\frac{2}{\gamma -1}}-{\left(\frac{{A}_{valve}}{{A}_{port}}\right)}^{2}}}.$$

### Mass Balance

Mass is conserved in the valve,

$${\dot{m}}_{A}+{\dot{m}}_{B}=0,$$

where:

$${\dot{m}}_{A}$$ is the mass flow rate at port

**A**.$${\dot{m}}_{B}$$ is the mass flow rate at port

**B**.

### Energy Balance

Energy is conserved in the valve,

$${\Phi}_{A}+{\Phi}_{B}=0,$$

where:

*Φ*is the energy flow at port_{A}**A**.*Φ*is the energy flow at port_{B}**B**.

### Assumptions and Limitations

There is no heat exchange between the valve and the environment.

When

**Modeling option**is`Liquid operating condition`

, the results may not be accurate outside of the subcooled liquid region. When**Modeling option**is`Vapor operating condition`

, the results may not be accurate outside of the superheated vapor region. To model a valve in a liquid-vapor mixture, set**Modeling option**to`Liquid operating condition`

.

## Ports

### Conserving

### Input

## 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 R2021a**