# 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.

For more information, see Numerical Smoothing.

### 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:

*S*is the instantaneous valve opening area. The block then smooths this area to remove derivative discontinuities at the limiting valve positions._{open}*D*is the common diameter of the gate and its bore, which are identical. This is the value of the**Orifice diameter**parameter.*S*is the area of overlap between the gate and bore, which the block computes as a function of the gate position,_{C}*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*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.

### 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

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

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 when 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
```

, [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].

### 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 based on geometry`

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

## Ports

### Input

### 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**