Valve with a sliding ball control mechanism

**Library:**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:

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

*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:

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

*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:

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

*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:

$$A(h)=\pi \text{\hspace{0.17em}}h\text{\hspace{0.17em}}cos(\theta )\text{}\text{\hspace{0.17em}}\text{sin(}\theta )\text{\hspace{0.17em}}\left({d}_{OB}(h)+\text{}h\mathrm{sin}(\theta )\right),$$

**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:

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

*C*_{D}is the flow discharge coefficient.*ρ*is the hydraulic fluid density.*Δp*is the pressure differential between the valve ports, defined as:where$$\Delta p={p}_{A}-{p}_{B},$$

*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:

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

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

*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.

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