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Valve with a sliding ball control mechanism

**Library:**Simscape / Fluids / Hydraulics (Isothermal) / Valves / Flow Control Valves

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**

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:

$$h(x)={x}_{0}+\text{}s,$$

where:

*h*is the valve lift.*x*_{0}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:

*A*_{Max}is the maximum opening area.*r*_{O}is the orifice radius.*A*_{Leak}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:

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

where:

*A*is the opening area at a given valve lift value.*r*_{B}is the ball radius.*d*_{OB}(*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:

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

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**

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:

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

where:

*C*_{D}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

*p*_{A}is the pressure at port**A**and*p*_{B}is the pressure at port**B**.*p*_{Cr}is the minimum pressure required for turbulent flow.

The critical pressure *p*_{Cr}
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:

*Re*_{Cr}is the critical Reynolds number.*ν*(*nu*) is the hydraulic fluid dynamic viscosity.*D*_{H}is the orifice hydraulic diameter:$${D}_{H}=\{\begin{array}{ll}{D}_{H}^{Min},\hfill & \text{if}h\le 0\hfill \\ {D}_{H}^{Max}+{D}_{H}^{Min},\hfill & \text{if}h\ge {h}_{Max}\hfill \\ \frac{4A}{l}+{D}_{H}^{Min},\hfill & \text{otherwise}\hfill \end{array},$$

in which:

*D*_{H}^{Min}is the minimum hydraulic diameter, corresponding to the smallest attainable flow area, the leakage flow area.*D*_{H}^{Max}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.

Fluid inertia is assumed to be negligible.