# Gate Valve (G)

Gate valve in a gas network

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

## Description

The Gate Valve (G) block represents an orifice with a translating gate, or sluice, as a flow control mechanism. The gate is circular and must slide perpendicular to the flow due to the constrains of the groove of its seat. The seat of the valve is annular. The flow passes through the bore, which is sized to match the gate. The overlap of the gate and the bore determines the opening area of the valve.

This image shows a gate valve with a conical seat.

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.

Gate valves generally open quickly. Gate valves are most sensitive to gate displacement near the closed position, where a small displacement translates into a disproportionately large change in opening area. Consequently, gate valves have too high a gain in that region to effectively throttle or modulate flow. You can use this block as a binary switch to open and close gas circuits.

### Gate Mechanics

In a real valve, the gate connects by a gear mechanism to a handle. When the handle is turned from a fully closed position, the gate rises from the bore and progressively opens the valve up to a maximum. Hard stops keep the disk from breaching its minimum and maximum positions.

The block captures the motion of the disk but not the detail of its mechanics. You specify the motion as a normalized displacement at port L. The input physical signal carries the fraction of the instantaneous displacement over its value in the fully open valve.

If you want to model the action of the handle and hard stop, use Simscape mechanical blocks to capture the displacement signal. However, in many cases, it suffices to know what displacement to impart to the disk and you do not need to model the mechanics of the valve.

### Gate Position

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

Numerical Smoothing

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

### Opening Area

The opening area of the valve is the area of the bore adjusted for the instantaneous overlap of the gate

`${S}_{open}=\frac{p{D}^{2}}{4}-{S}_{C},$`

where:

• Sopen is the instantaneous valve opening area. The block then smooths this area to remove derivative discontinuities at the limiting valve positions.

• D is the common diameter of the gate and its bore, which are identical. This is the value of the Orifice diameter parameter.

• SC is the area of overlap between the gate and bore, which the block computes as a function of the gate position, L:

`${S}_{C}=\frac{{D}^{2}}{2}\text{acos}\left(L\right)-\frac{LD}{2}\sqrt{{D}^{2}-{L}^{2}{D}^{2}}.$`

The figure shows a front view of the valve when maximally closed, partially open, and fully open, from left to right. The figure also shows the parameters and variables in the opening area calculation.

### 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.

### 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

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

where:

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

• Sopen is the valve opening area.

• SMax is the maximum valve area when 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```, [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].

### 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 based on geometry``` parameterization is less accurate for gases that are far from ideal.

• This block does not model supersonic flow.

## Ports

### Input

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Input port through which to specify the relative displacement of the gate as a fraction of its maximum. This input specifies the gate position, which the block uses to compute the mass flow rate through the valve. This position should range between `0` for a maximally closed valve to `1` for a fully open valve.

### Conserving

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Gas conserving port associated with the opening through which the flow enters or exits the valve. The direction of flow depends on the pressure differential established across the valve. The block allows both forward and backward directions.

Gas conserving port associated with the opening through which the flow enters or exits the valve. The direction of flow depends on the pressure differential established across the valve. The block allows both forward and backward directions.

## Parameters

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Diameter of the bore of the valve and of the gate that controls the opening area. The orifice is assumed to be constant in cross section throughout its length from one port to the other.

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.

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```.

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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