# Poppet Valve (G)

**Libraries:**

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

## Description

The Poppet Valve (G) block represents an orifice with a
translating ball that moderates flow through the valve. In the fully closed position,
the ball rests at the perforated seat, and fully blocks the fluid from passing between
ports **A** and **B**. The area between the ball and
seat is the opening area of the valve.

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.

### Ball Mechanics

The block models the displacement of the ball but not the valve opening or closing
dynamics. The signal at port **L** provides the normalized ball
position, *L*. Note that *L* is a normalized
distance between 0 and 1, which indicate a fully closed valve and a fully open
valve, respectively. If the calculation returns a number outside of this range, that
number is set to the nearest bound.

**Numerical Smoothing**

When the **Smoothing factor** parameter is nonzero, the block
applies numerical smoothing to the normalized ball position,
*L*. Enabling smoothing helps maintain numerical robustness
in your simulation.

For more information, see Numerical Smoothing.

### Opening Area

The opening area of the valve depends on the **Valve seat geometry**
parameter, which can be either `Sharp-edged`

or
`Conical`

. The **Leakage flow
fraction** parameter is the ratio of the flow rate of the valve when it
is closed to when it is open. The **Leakage flow fraction** allows
for small contact gaps between the ball and seat in the fully closed position. This
parameter also maintains continuity in the flow for solver performance.

This figure shows the seat types for the poppet valve.

**Sharp-Edged Seat Geometry**

The block calculates the opening area of the valve as

$${S}_{open,sharp-edged}=\pi {r}_{O}\sqrt{{\left({G}_{sharp}+h\right)}^{2}+{r}_{O}^{2}}\left[1-\frac{{r}_{B}^{2}}{{\left({G}_{sharp}+h\right)}^{2}+{r}_{O}^{2}}\right],$$

where:

*h*is the distance between the outer edge of the cylinder and the seat.*r*is the seat orifice radius, which the block calculates from the_{O}**Orifice diameter**parameter.*r*is the radius of the ball, which the block calculates from the_{B}**Ball diameter**parameter.*G*is the geometric parameter: $${G}_{sharp}=\sqrt{{r}_{B}^{2}-{r}_{O}^{2}}.$$_{sharp}

The maximum displacement,
*h _{max}*, bounds the opening area:

$${h}_{\mathrm{max}}=\sqrt{\frac{2{r}_{B}^{2}-{r}_{O}^{2}+{r}_{O}\sqrt{{r}_{O}^{2}+4{r}_{B}^{2}}}{2}}-{G}_{sharp}.$$

For any ball displacement larger than
*h _{max}*,

*S*is the value of the maximum orifice area:

_{open}$${S}_{open}=\frac{\pi}{4}{d}_{O}^{2}.$$

When the signal at port **L** is less than
0, the valve is closed and the **Leakage flow fraction**
parameter determines the mass flow rate.

**Conical Seat Geometry**

The block calculates the opening area of the valve as:

$${S}_{open,conical}={G}_{conical}h+\frac{\pi}{2}\mathrm{sin}\left(\theta \right)\mathrm{sin}\left(\frac{\theta}{2}\right){h}^{2},$$

where:

*h*is the vertical distance between the outer edge of the cylinder and the seat.*θ*is the value of the**Cone angle**parameter.*G*is the geometric parameter, $${G}_{conical}=\pi {r}_{B}\mathrm{sin}\left(\theta \right),$$ where_{conical}*r*is the ball radius._{B}

The maximum displacement,
*h _{max}*, bounds the opening area:

$${h}_{\mathrm{max}}=\frac{\sqrt{{r}_{B}^{2}+\frac{{r}_{O}^{2}}{\mathrm{cos}\left(\frac{\theta}{2}\right)}}-{r}_{B}}{\mathrm{sin}\left(\frac{\theta}{2}\right)}.$$

For any ball displacement larger than
*h _{max}*,

*S*is the value of the maximum orifice area:

_{open}$${S}_{open}=\frac{\pi}{4}{d}_{O}^{2}.$$

When the signal at port **L** is less than
0, the valve is closed and the **Leakage flow fraction**
parameter determines the mass flow rate.

### 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 based on geometry`

— 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.

### Mass Flow Rate

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

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

where:

*C*is the value of the_{v}**Maximum Cv flow coefficient**parameter.*S*is the valve opening area._{open}*S*is the maximum valve area where the valve is fully open._{Max}*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}*,

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

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

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

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

$$\dot{m}=C\frac{{S}_{open}}{{S}_{Max}}{\rho}_{ref}{p}_{in}\sqrt{\frac{{T}_{ref}}{{T}_{in}}}{\left[1-{\left(\frac{\frac{{p}_{out}}{{p}_{in}}-{B}_{crit}}{1-{B}_{crit}}\right)}^{2}\right]}^{m},$$

where:

*C*is the value of the**Maximum sonic conductance**parameter.*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}*,

$$\dot{m}=C\frac{{S}_{open}}{{S}_{Max}}{\rho}_{ref}\sqrt{\frac{{T}_{ref}}{{T}_{avg}}}{\left[1-{\left(\frac{{B}_{lam}-{B}_{crit}}{1-{B}_{crit}}\right)}^{2}\right]}^{m}\left(\frac{{p}_{in}-{p}_{out}}{1-{B}_{lam}}\right).$$

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

$$\dot{m}=C\frac{{S}_{open}}{{S}_{Max}}{\rho}_{ref}{p}_{in}\sqrt{\frac{{T}_{ref}}{{T}_{in}}}.$$

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 based on geometry`

, the mass flow
rate, $$\dot{m}$$, is

$$\dot{m}={C}_{d}{S}_{open}\sqrt{\frac{2\gamma}{\gamma -1}{p}_{in}{\rho}_{in}{\left(\frac{{p}_{out}}{{p}_{in}}\right)}^{\frac{2}{\gamma}}\left[\frac{1-{\left(\frac{{p}_{out}}{{p}_{in}}\right)}^{\frac{\gamma -1}{\gamma}}}{1-{\left(\frac{{S}_{open}}{S}\right)}^{2}{\left(\frac{{p}_{out}}{{p}_{in}}\right)}^{\frac{2}{\gamma}}}\right]},$$

where:

*S*is the valve opening area._{open}*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}*,

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

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

$$\dot{m}={C}_{d}{S}_{open}\sqrt{\frac{2\gamma}{\gamma +1}{p}_{in}{\rho}_{in}\frac{1}{{\left(\frac{\gamma +1}{2}\right)}^{\frac{2}{\gamma -1}}-{\left(\frac{{S}_{open}}{S}\right)}^{2}}}.$$

For more information on the mass flow equations when the **Orifice
parametrization** parameter is ```
Orifice area based on
geometry
```

, see [4].

### 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**.

### 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.

### 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 based on geometry`

parameterization is less accurate for gases that are far from ideal.This block does not model supersonic flow.

## 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 R2018b**