Valve with longitudinally translating ball as control element

**Library:**Simscape / Fluids / Gas / Valves & Orifices / Flow Control Valves

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.

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**, *L*_{0}: $$h(L)=L+{h}_{\text{0}},$$ Note that *h* and
*L*_{0} are normalized distances between 0
and 1, indicating a fully closed valve and a fully open valve, respectively.

Numerical smooothing 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**

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:

*R*_{0}is the orifice radius.*d*is the distance between the orifice center and orifice edge.*R*_{B}is the ball radius.*A*_{Leak}is the**Leakage area**.

The opening area of a conical seat is:

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

where:

*h*_{max}is the maximum ball distance from the valve seat.*h*is the position of the ball.*θ*is the**Cone angle**.

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

Sonic conductance

Flow coefficient

*C*_{v}Flow factor

*K*_{v}Compute from geometry

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:

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

where

*C*is the sonic conductance.*C*_{Max}is the**Sonic conductance at maximum flow**. For ball valves in a physical system, this value is usually stated in the manufacturer's specifications.*S*_{Max}is the maximum valve opening area, calculated from the**Orifice diameter**and**Leakage area**.

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

The flow coefficient,

*C*_{v}. This coefficient is defined for Imperial System units.The sonic conductance is computed from the

**Cv coefficient (USCS) at maximum flow**parameter as:$$C=\left(4\times {10}^{-8}{C}_{\text{v}}\right){m}^{3}/(sPa),$$

The flow factor,

*K*_{v}. This coefficient is defined for SI units.The sonic conductance is computed from the

**Kv coefficient (SI) at maximum flow**parameter as:$$C=\left(4.758\times {10}^{-8}{K}_{\text{v}}\right){m}^{3}/(sPa),$$

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:

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

.

The fluid mass flowing through the valve is conserved:

$${\dot{m}}_{A}+{\dot{m}}_{B}=0,$$

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:

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

where:

*ρ*_{0}is the gas**Reference density**.*p*_{in}is the inlet pressure.*T*_{0}is the gas**Reference temperature**.*T*_{in}is the inlet temperature.

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

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

where:

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

*b*_{cr}is the**Critical pressure ratio**for choked flow. When**Valve parameterization**is set to`Cv coefficient (USCS)`

or`Kv coefficient (SI)`

,*b*_{cr}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.

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

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

where *b*_{lam}, the
**Laminar flow pressure ratio**, is the pressure ratio
associated with the flow transition from laminar to turbulent regime.

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.