# Ball Valve

Valve with a sliding ball control mechanism

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• Simscape / Fluids / Hydraulics (Isothermal) / Valves / Flow Control Valves

## Description

The Ball Valve block models an orifice with a variable opening area controlled by a sliding ball mechanism. The opening area changes with the relative position of the ball—the valve lift. A displacement toward the orifice decreases flow while a displacement away increases flow. The interface between the orifice and the ball—the valve seat—can be `Sharp-edged`, shown left in the figure, or `Conical`, shown right.

Ball Valve Seat Types

### Opening Area

The valve lift is a function of the displacement signal specified through port S. The two can, but generally do not, have the same value. The valve lift differs from the displacement whenever the Ball displacement offset parameter is nonzero:

`$h\left(x\right)={x}_{0}+\text{​}s,$`

where:

• h is the valve lift.

• x0 is the ball displacement offset.

• s is the ball displacement (relative to the specified offset).

The valve is fully closed when the valve lift is equal to zero or less. It is fully open when the valve lift reaches or exceeds a (geometry-dependent) value sufficient to completely clear the orifice. A fully closed valve has an opening area equal to the specified Leakage area parameter while a fully open valve has the maximum possible opening area. Adjusting for internal leakage:

`${A}_{Max}=\pi {r}_{O}^{2}+{A}_{Leak},$`

where:

• AMax is the maximum opening area.

• rO is the orifice radius.

• ALeak is the internal leakage area between the ports.

At intermediate values of the valve lift, the opening area depends on the valve seat geometry. If the Valve seat specification parameter is set to `Sharp-Edged`, the opening area as a function of valve lift is:

`$A\left(h\right)=\pi {r}_{O}\left(1-{\left[\frac{{r}_{B}}{{d}_{OB}}\right]}^{2}\right){d}_{OB}\left(h\right),$`

where:

• A is the opening area at a given valve lift value.

• rB is the ball radius.

• dOB(h) is the distance from the center of the ball (point O in the figure) to the edge of the orifice (point B). This distance is a function of the valve lift (h).

If the Valve seat specification parameter is set to `Conical`, the opening area becomes:

`$A\left(h\right)=\pi \text{\hspace{0.17em}}h\text{\hspace{0.17em}}cos\left(\theta \right)\text{​}\text{\hspace{0.17em}}\text{sin(}\theta \right)\cdot \left(2{r}_{B}+\text{​}h\mathrm{sin}\left(\theta \right)\right),$`

where θ is the angle between the conical surface and the orifice centerline. The geometrical parameters and variables used in the equations are shown in the figure.

Valve Geometries

### Flow Rate

The volumetric flow rate through the valve is a function of the opening area, A(h), and of the pressure differential between the valve ports:

`$q={C}_{D}A\left(h\right)\sqrt{\frac{2}{\rho }}\frac{\Delta p}{{\left({\left(\Delta p\right)}^{2}+{p}_{Cr}^{2}\right)}^{1/4}},$`

where:

• CD is the flow discharge coefficient.

• ρ is the hydraulic fluid density.

• Δp is the pressure differential between the valve ports, defined as:

`$\Delta p={p}_{A}-{p}_{B},$`

where pA is the pressure at port A and pB is the pressure at port B.

• pCr is the minimum pressure required for turbulent flow.

The critical pressure pCr is computed from the critical Reynolds number as:

`${p}_{Cr}=\frac{\rho }{2}{\left(\frac{R{e}_{Cr}\nu }{{C}_{D}{D}_{H}}\right)}^{2},$`

where:

• ReCr is the critical Reynolds number.

• ν (nu) is the hydraulic fluid dynamic viscosity.

• DH is the orifice hydraulic diameter:

in which:

• DHMin is the minimum hydraulic diameter, corresponding to the smallest attainable flow area, the leakage flow area.

• DHMax is the maximum hydraulic diameter, corresponding to the largest attainable flow area, that of the valve in the fully open position.

• l is the wetted length of the valve perimeter—which can, but need not, be that of a circle.

### Assumptions

Fluid inertia is assumed to be negligible.

## Ports

### Input

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Physical signal input port for the ball displacement.

### Conserving

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Hydraulic (isothermal liquid) conserving port associated with the valve inlet.

Hydraulic (isothermal liquid) conserving port associated with the valve outlet.

## Parameters

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Parameters Tab

Geometry of the valve seat. The choice of geometry impacts the opening characteristics of the valve. See the block description for the valve opening calculations.

Angle between the sloping edges of the seat in a transverse cross-sectional view.

Diameter of the ball control element. This diameter must be greater than that specified for the orifice.

Diameter of the valve orifice. This diameter must be smaller than that specified for the ball control element.

Neutral position of the ball control element. The neutral position is that to which the ball returns when the displacement signal is zero. This parameter must be greater than or equal to zero.

Ratio of the actual and theoretical flow rates through the valve. This parameter depends on the geometrical properties of the valve. Values are usually provided in textbooks and manufacturer data sheets.

Total area of internal leaks in the completely closed state. The purpose of this parameter is to maintain the numerical integrity of the fluid network by preventing a portion of that network from becoming isolated when the valve is completely closed.

Select the parameter to base the laminar-turbulent transition on. Options include:

• `Pressure ratio` — Flow transitions between laminar and turbulent at the pressure ratio specified in the Laminar flow pressure ratio parameter. Use this option for the smoothest and most numerically robust flow transitions.

• `Reynolds number` — The transition occurs at the Reynolds number specified in the Critical Reynolds number parameter. Flow transitions are more abrupt and can cause simulation issues at near-zero flow rates.

Pressure ratio at which the flow transitions between the laminar and turbulent regimes. The pressure ratio is the fraction of the outlet pressure over the inlet pressure.

#### Dependencies

This parameter is active when the Laminar transition specification parameter is set to ```Pressure ratio```.

Maximum Reynolds number for laminar flow. This parameter depends on the orifice geometrical profile. You can find recommendations on the parameter value in hydraulics textbooks. The default value, `12`, corresponds to a round orifice in thin material with sharp edges.

#### Dependencies

This parameter is active when the Laminar transition specification parameter is set to ```Reynolds number```.

Variables Tab

Volumetric flow rate through the valve at time zero. Simscape™ software uses this parameter to guide the initial configuration of the component and model. Initial variables that conflict with each other or are incompatible with the model may be ignored. Set the Priority column to `High` to prioritize this variable over other, low-priority, variables.

Pressure drop from port A to port B at time zero. The pressure drop is positive if pressure is greater at port A than at port B. Simscape software uses this parameter to guide the initial configuration of the component and model. Initial variables that conflict with each other or are incompatible with the model may be ignored. Set the Priority column to `High` to prioritize this variable over other, low-priority, variables.

## Version History

Introduced in R2006a