# Poppet Valve (G)

Poppet valve in a gas network

Since R2018b

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.

### 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.

• rO is the seat orifice radius, which the block calculates from the Orifice diameter parameter.

• rB is the radius of the ball, which the block calculates from the Ball diameter parameter.

• Gsharp is the geometric parameter: ${G}_{sharp}=\sqrt{{r}_{B}^{2}-{r}_{O}^{2}}.$

The maximum displacement, hmax, 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 hmax, Sopen is the value of the maximum orifice area:

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

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.

• Gconical is the geometric parameter, ${G}_{conical}=\pi {r}_{B}\mathrm{sin}\left(\theta \right),$ where rB is the ball radius.

The maximum displacement, hmax, 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 hmax, Sopen is the value of the maximum orifice area:

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

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 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 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, $\stackrel{˙}{m}$, is

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

where:

• Cv is the value of the Maximum Cv flow coefficient parameter.

• Sopen is the valve opening area.

• SMax is the maximum valve area where the valve is fully open.

• 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,

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

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

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

When you set Orifice parametrization to ```Kv flow coefficient parameterization```, the block uses these same equations, but replaces 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

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

where:

• C is the value of the Maximum sonic conductance parameter.

• 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,

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

When the pressure ratio, ${p}_{out}/{p}_{in}$, falls below the critical pressure ratio, Bcrit, the orifice becomes choked and the block switches to the equation

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

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, $\stackrel{˙}{m}$, is

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

where:

• Sopen is the valve opening 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,

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

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

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

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

`${\stackrel{˙}{m}}_{A}+{\stackrel{˙}{m}}_{B}=0,$`

where $\stackrel{˙}{m}$ is defined as the mass flow rate into the valve through the port indicated by the A or B subscript.

### 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

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Gas conserving port associated with the opening through which the flow enters or exits the valve.

Gas conserving port associated with the opening through which the flow enters or exits the valve.

### Input

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Normalized ball displacement. The ball position is normalized by the maximum opening distance. A value of 0 indicates a fully closed valve and a value of 1 indicates a fully open valve.

## Parameters

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Geometry of the seat of the ball. This parameter determines the opening area of the valve.

Angle formed by the slope of the conical seat against its center line.

#### Dependencies

To enable this parameter, set Valve seat specification to `Conical`.

Diameter of the ball control element.

Diameter of the valve constant orifice. For a conical geometry, the diameter the root of the seat.

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 based on geometry` — The size of the flow restriction determines the block parametrization.

Correction factor that accounts for discharge losses in theoretical flows.

#### Dependencies

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

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```.

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```.

Maximum 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```.

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```.

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```.

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.

#### 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.

#### Dependencies

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

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

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.

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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