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# Ball Valve (G)

Valve with longitudinally translating ball as control element

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

## Description

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

The block supports subsonic flows up to the valve critical pressure, when choking occurs and the velocity downstream of the orifice remains constant. The block does not support supersonic flow.

### Ball Mechanics

The block models the displacement of the ball but not valve opening or closing dynamics. The signal at port L provides the normalized valve position. The overall ball position is the sum of the variable displacement received at port L, L, and its initial Valve lift control offset, L0: $h\left(L\right)=L+{h}_{\text{0}},$ Note that h and L0 are normalized distances between 0 and 1, indicating a fully closed valve and a fully open valve, respectively.

Numerical Smoothing

Numerical smoothing can be applied to soften discontinuities in the simulation when the valve is in the near-open or near-closed position. A 3rd-order polynomial approximates the ball position in these regions, as shown in the two figures below:

Simulated valve position without smoothing

Simulated valve position with smoothing

### Opening Area

The opening area of the valve depends on the Valve seat geometry, which can be either `Sharp-edged` or `Conical`. A Leakage area is defined 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.

Ball Valve Seat Types

The opening area for a sharp-edged seat is:

`$S=2\pi {R}_{\text{O}}OC\left[1-{\left(\frac{{R}_{\text{B}}}{OC}\right)}^{2}\right]+{S}_{\text{Leak}},$`

where:

• R0 is the orifice radius.

• d is the distance between the orifice center and orifice edge.

• RB is the ball radius.

• ALeak is the Leakage area.

The opening area of a conical seat is:

`$S=2\pi {R}_{\text{B}}h\rho \text{sin}\left(\theta \right)+\pi {h}^{2}{\rho }^{2}\text{sin}\left(\theta \right)\text{sin}\left(\frac{\theta }{2}\right)+{S}_{\text{Leak}},$`

where:

• hmax is the maximum ball distance from the valve seat.

• h is the position of the ball.

• θ is the Cone angle.

### Valve Parameterizations

There are four valve parameterization options for calculating the valve mass flow rate:

• Sonic conductance

• Flow coefficient Cv

• Flow factor Kv

• Compute from geometry

Sonic Conductance

The mass flow rate depends on opening area and the sonic conductance of the valve. Sonic conductance is a property of an orifice that characterizes flow transition between subsonic and supersonic regimes. When Valve parameterization is set to ```Sonic conductance```, sonic conductance is treated as linearly proportional to opening area:

`$C\left(S\right)=\frac{S}{{S}_{\text{Max}}}{C}_{\text{Max}},$`

where

• C is the sonic conductance.

• CMax is the Sonic conductance at maximum flow. For ball valves in a physical system, this value is usually stated in the manufacturer's specifications.

• SMax is the maximum valve opening area, calculated from the Orifice diameter and Leakage area.

Flow Coefficients

If sonic conductance is not known, the mass flow rate can be calculated from:

• The flow coefficient, Cv. This coefficient is defined for Imperial System units.

The sonic conductance is computed from the Cv coefficient (USCS) at maximum flow parameter as:

`$C=\left(4×{10}^{-8}{C}_{\text{v}}\right){m}^{3}/\left(sPa\right),$`

• The flow factor, Kv. This coefficient is defined for SI units.

The sonic conductance is computed from the Kv coefficient (SI) at maximum flow parameter as:

`$C=\left(4.758×{10}^{-8}{K}_{\text{v}}\right){m}^{3}/\left(sPa\right),$`

Compute From Geometry

In this parameterization, the sonic conductance is calculated from the geometry of valve opening, based on the formulations for area, A, in Opening Area above. The sonic conductance is calculated from the geometry as:

`$C=0.512\frac{A}{\pi }.$`

### Mass Flow Rate

Continuity

The fluid mass flowing through the valve is conserved:

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

Choked Flow

When the flow is choked, the mass flow rate is a function of the sonic conductance, C, and of the valve inlet pressure and temperature:

`${\stackrel{˙}{m}}_{\text{ch}}=C{\rho }_{\text{0}}{p}_{\text{in}}\sqrt{\frac{{T}_{\text{0}}}{{T}_{\text{in}}}},$`

where:

• ρ0 is the gas Reference density.

• pin is the inlet pressure.

• T0 is the gas Reference temperature.

• Tin is the inlet temperature.

Subsonic Turbulent Flow

When the flow is in the turbulent, subsonic regime, the mass flow rate is:

`${\stackrel{˙}{m}}_{\text{tur}}=C{\rho }_{\text{0}}{p}_{\text{in}}\sqrt{\frac{{T}_{\text{0}}}{{T}_{\text{in}}}}{\left[1-{\left(\frac{{p}_{\text{r}}-{b}_{\text{cr}}}{1-{b}_{\text{cr}}}\right)}^{2}\right]}^{m},$`

where:

• pr is the ratio between the inlet and outlet pressures:${P}_{\text{r}}=\frac{{p}_{\text{out}}}{{p}_{\text{in}}}$.

• bcr is the Critical pressure ratio for choked flow. When Valve parameterization is set to `Cv coefficient (USCS)` or `Kv coefficient (SI)`, bcr is 0.3.

• m is the Subsonic index, an empirical constant that characterizes subsonic flows. When Valve parameterization is set to `Cv coefficient (USCS)` or `Kv coefficient (SI)`, m is 0.5.

Subsonic Laminar Flow

When the flow is in the laminar subsonic regime, the mass flow rate is:

`${\stackrel{˙}{m}}_{\text{lam}}=C{\rho }_{\text{0}}{p}_{\text{in}}\left[\frac{1-{p}_{\text{r}}}{1-{b}_{\text{lam}}}\right]\sqrt{\frac{{T}_{\text{0}}}{{T}_{\text{in}}}}{\left[1-{\left(\frac{{b}_{\text{lam}}-{b}_{\text{cr}}}{1-{b}_{\text{cr}}}\right)}^{2}\right]}^{m}$`

where blam, the Laminar flow pressure ratio, is the pressure ratio associated with the flow transition from laminar to turbulent regime.

### Energy Balance

The valve is adiabatic:

`${\varphi }_{\text{A}}+{\varphi }_{\text{B}}=0,$`

where ϕ is the energy flow rate. The sign convention is positive for energy flows into the valve.

## Ports

### Conserving

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Entry or exit point to the valve.

Entry or exit point to the valve.

### Input

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Normalized ball displacement. The ball position, which does not include any initial valve offset 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.

Initial distance between the ball and seat. The instantaneous ball position is calculated during simulation as the sum of this offset and the control signal specified at port L. The valve is partially open in its normal position when the offset is a fraction between `0` and `1`.

Valve opening parameterization. The mass flow rate through the valve is calculated using the sonic conductance provided in ```Sonic conductance```, or is calculated from valve geometry in `Compute from geometry` or flow coefficients Cv in ```Cv coefficient (USCS)```, Kv in ```Kv coefficient (SI)```.

Valve characteristic, influenced by opening area and inlet-outlet pressure ratio, at the maximum flow rate through the valve. Sonic conductance is defined as the ratio of the mass flow rate through the valve to the product of the pressure and density upstream of the valve inlet. This parameter is often referred to in the literature as the C-value. This is the value generally reported by manufacturers in technical data sheets.

#### Dependencies

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

Ratio of downstream to upstream pressures that leads to choked flow in the valve. This parameter is often referred to in the literature as the b-value.

#### Dependencies

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

Empirical value that characterizes subsonic flows. This parameter is sometimes referred to as the m-index. Components with fixed paths, such as a valve, have a subsonic index of approximately `0.5`.

#### Dependencies

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

Flow coefficient of the fully open valve, formulated for US customary units. This value is generally reported by manufacturers on technical data sheets.

#### Dependencies

To enable this parameter, set Valve parameterization to ```Cv coefficient (USCS)```.

Flow coefficient of the fully open valve, formulated for SI units. This value is generally reported by manufacturers on technical data sheets.

#### Dependencies

To enable this parameter, set Valve parameterization to ```Kv coefficient (SI)```.

Sum of all gaps when the valve is in fully closed position. Any area smaller than this value is saturated to the specified leakage area. This contributes to numerical stability by maintaining continuity in the flow.

#### Dependencies

To enable this parameter, set Valve parameterization to ```Opening area```.

Area normal to the flow path at the valve ports. The ports are assumed to be the same in size.

Downstream-to-upstream pressure ratio when the flow regime transitions from laminar to turbulent. Typical values range from `0.995` to `0.999`.

Temperature at standard reference atmosphere, defined as 293.15 K in ISO 8778.

Density at standard reference atmosphere, defined as 1.185 kg/m3 in ISO 8778.

Introduces a layer of gradual change to the flow response when the valve is in near-open or near-closed positions. Set this value to a nonzero value less than one to increase the stability of your simulation in these regimes.

## Version History

Introduced in R2018b