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# Ball Valve

Hydraulic ball valve

## Library

Flow Control Valves

## Description

The Ball Valve block models a variable orifice created by a spherical ball and a round sharp-edged orifice.

The flow rate through the valve is proportional to the valve opening and to the pressure differential across the valve. The flow rate is determined according to the following equations:

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

$p={p}_{A}-{p}_{B}$

${p}_{cr}=\frac{\rho }{2}{\left(\frac{{\mathrm{Re}}_{cr}\cdot \nu }{{C}_{D}\cdot {D}_{H}}\right)}^{2}$

$h={x}_{0}+x·or$

$D=\sqrt{{\left(\sqrt{{r}_{B}^{2}-{r}_{O}^{2}}+{h}^{2}\right)}^{2}+{r}_{O}^{2}}$

${D}_{H}=\sqrt{\frac{4A\left(h\right)}{\pi }}$

${A}_{\mathrm{max}}=\frac{\pi {d}_{O}^{2}}{4}$

${h}_{\mathrm{max}}={r}_{O}\cdot \left(\sqrt{\frac{{\left(1+\sqrt{1+4\frac{{d}_{B}^{2}}{{d}_{O}^{2}}}\right)}^{2}}{4}-1}-\sqrt{\frac{{d}_{B}^{2}}{{d}_{O}^{2}}-1}\right)$

where

 q Flow rate p Pressure differential pA, pB Gauge pressures at the block terminals CD Flow discharge coefficient A(h) Instantaneous orifice passage area x0 Initial opening x Ball displacement from initial position h Valve opening dO Orifice diameter rO Orifice radius dB Ball diameter rB Ball radius ρ Fluid density ν Fluid kinematic viscosity pcr Minimum pressure for turbulent flow Recr Critical Reynolds number DH Valve instantaneous hydraulic diameter Aleak Closed valve leakage area Amax Maximum valve open area hmax Maximum valve opening

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.

## Basic Assumptions and Limitations

• 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 center and the orifice edge.

## Dialog Box and Parameters

Valve ball diameter

The diameter of the valve ball. It must be greater than the orifice diameter. The default value is 0.01 m.

Orifice diameter

The diameter of the orifice of the valve. The default value is 0.005 m.

Initial opening

The initial opening of the valve. Its value must be nonnegative. The default value is 0.

Flow discharge coefficient

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.

Critical Reynolds number

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.

Leakage area

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.

## Global Parameters

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.

## Ports

The block has the following ports:

A

Hydraulic conserving port associated with the valve inlet.

B

Hydraulic conserving port associated with the valve outlet.

S

Physical signal port to control ball displacement.