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Hydraulic ball valve with conical seat
The Ball Valve with Conical Seat block models a valve created by a spherical ball and a conical seat.
The valve is characterized by the ball diameter, cone angle, and orifice diameter. The flow rate through the valve is proportional to the ball displacement and pressure differential. If passage area in the ball-cone contact exceeds the area of the orifice, the latter is assumed as the valve passage area. The flow rate is determined according to the following equations:
$$q={C}_{D}\cdot A(h)\sqrt{\frac{2}{\rho}}\cdot \frac{p}{{\left({p}^{2}+{p}_{cr}^{2}\right)}^{1/4}}$$
$$p={p}_{A}-{p}_{B}$$
$${p}_{cr}=\frac{\rho}{2}{\left(\frac{{\mathrm{Re}}_{cr}\cdot \nu}{{C}_{D}\cdot {D}_{H}}\right)}^{2}$$
$$h={x}_{0}+x\xb7or$$
$$A(h)=\{\begin{array}{ll}{A}_{leak}\hfill & \text{for}h=0\hfill \\ \pi \xb7\mathrm{cos}\frac{\theta}{2}\xb7\mathrm{sin}\frac{\theta}{2}\xb7h\left(D+\mathrm{sin}\frac{\theta}{2}\xb7h\right)\hfill & \text{for}0h{h}_{\mathrm{max}}\hfill \\ {A}_{\mathrm{max}}+{A}_{leak}\hfill & \text{for}h={h}_{\mathrm{max}}\hfill \end{array}$$
$${D}_{H}=\sqrt{\frac{4A(h)}{\pi}}$$
$${A}_{\mathrm{max}}=\frac{\pi {d}_{O}^{2}}{4}$$
where
q | Flow rate |
p | Pressure differential |
p_{A}, p_{B} | Gauge pressures at the block terminals |
C_{D} | Flow discharge coefficient |
A(h) | Instantaneous orifice passage area |
x_{0} | Initial opening |
x | Ball displacement from initial position |
h | Valve opening |
d_{O} | Orifice diameter |
D | Ball diameter |
Θ | Cone angle of the valve seat |
ρ | Fluid density |
ν | Fluid kinematic viscosity |
p_{cr} | Minimum pressure for turbulent flow |
Re_{cr} | Critical Reynolds number |
D_{H} | Valve instantaneous hydraulic diameter |
A_{leak} | Closed valve leakage area |
A_{max} | Maximum valve open area |
The block positive direction is from port A to port B. This means that the flow rate is positive if it flows from A to B and the pressure differential is determined as $$p={p}_{A}-{p}_{B}$$. Positive signal at the physical signal port S opens the valve.
Fluid inertia is not taken into account.
The flow passage area is assumed to be equal to the side surface of the frustum of the cone located between the ball and the conical seat.
The diameter of the valve ball. It must be greater than the orifice diameter. The default value is 0.01 m.
The diameter of the orifice of the valve. The default value is 0.005 m.
The cone angle of the valve seat. The default value is 120 degrees.
The initial opening of the valve. Its value must be nonnegative. The default value is 0.
Semi-empirical parameter for valve capacity characterization. Its value depends on the geometrical properties of the orifice, and usually is provided in textbooks or manufacturer data sheets. The default value is 0.65.
The maximum Reynolds number for laminar flow. The transition from laminar to turbulent regime is assumed to take place when the Reynolds number reaches this value. The value of the parameter depends on the orifice geometrical profile. You can find recommendations on the parameter value in hydraulics textbooks. The default value is 10.
The total area of possible leaks in the completely closed valve. The main purpose of the parameter is to maintain numerical integrity of the circuit by preventing a portion of the system from getting isolated after the valve is completely closed. An isolated or "hanging" part of the system could affect computational efficiency and even cause simulation to fail. Therefore, MathWorks recommends that you do not set this parameter to 0. The default value is 1e-12 m^2.
Parameters determined by the type of working fluid:
Fluid density
Fluid kinematic viscosity
Use the Hydraulic Fluid block or the Custom Hydraulic Fluid block to specify the fluid properties.