Pressure control valve maintaining preset pressure in system

Pressure Control Valves

The Pressure Relief Valve block represents a
hydraulic pressure relief valve as a data-sheet-based model. The following
figure shows the typical dependency between the valve passage area * A* and
the pressure differential

`p`

The valve remains closed while pressure at the valve inlet is lower than the valve preset pressure. When the preset pressure is reached, the valve control member (spool, ball, poppet, etc.) is forced off its seat, thus creating a passage between the inlet and outlet. Some fluid is diverted to a tank through this orifice, thus reducing the pressure at the inlet. If this flow rate is not enough and pressure continues to rise, the area is further increased until the control member reaches its maximum. At this moment, the maximum flow rate is passing through the valve. The value of a maximum flow rate and the pressure increase over the preset level to pass this flow rate are generally provided in the catalogs. The pressure increase over the preset level is frequently referred to as valve steady state error, or regulation range. The valve maximum area and regulation range are the key parameters of the block.

In addition to the maximum area, the leakage area is also required to characterize the valve. The main purpose of the parameter is not to account for possible leakage, even though this is also important, but 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 failure of computation. Therefore, the parameter value must be greater than zero.

By default, the block does not include valve opening dynamics, and the valve sets its opening area directly as a function of pressure:

$$A=A(p)$$

Adding valve opening dynamics provides continuous behavior that
is more physically realistic, and is particularly helpful in situations
with rapid valve opening and closing. The pressure-dependent orifice
passage area *A(p)* in the block equations then becomes
the steady-state area, and the instantaneous orifice passage area
in the flow equation is determined as follows:

$$A(t=0)={A}_{init}$$

$$\frac{dA}{dt}=\frac{A(p)-A}{\tau}$$

In either case, the flow rate through the valve is determined according to the following equations:

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

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

$$A(p)=\{\begin{array}{ll}{A}_{leak}\hfill & \text{for}p={p}_{set}\hfill \\ {A}_{leak}+k\xb7\left(p-{p}_{set}\right)\hfill & \text{for}{p}_{set}p{p}_{\mathrm{max}}\hfill \\ {A}_{\mathrm{max}}\hfill & \text{for}p={p}_{\mathrm{max}}\hfill \end{array}$$

$$k=\frac{{A}_{\mathrm{max}}-{A}_{leak}}{{p}_{reg}}$$

where

q | Flow rate |

p | Pressure differential |

p_{A}, p_{B} | Gauge pressures at the block terminals |

C_{D} | Flow discharge coefficient |

A | Instantaneous orifice passage area |

A(p) | Pressure-dependent orifice passage area |

A_{init} | Initial open area of the valve |

A_{max} | Fully open valve passage area |

A_{leak} | Closed valve leakage area |

p_{reg} | Regulation range |

p_{set} | Valve preset pressure |

p_{max} | Valve pressure at maximum opening |

ρ | Fluid density |

τ | Time constant for the first order response of the valve opening |

t | Time |

p_{cr} | Minimum pressure for turbulent flow |

The minimum pressure for turbulent flow, *p*_{cr},
is calculated according to the laminar transition specification method:

By pressure ratio — The transition from laminar to turbulent regime is defined by the following equations:

*p*_{cr}= (*p*_{avg}+*p*_{atm})(1 –*B*_{lam})*p*_{avg}= (*p*_{A}+*p*_{B})/2where

*p*_{avg}Average pressure between the block terminals *p*_{atm}Atmospheric pressure, 101325 Pa *B*_{lam}Pressure ratio at the transition between laminar and turbulent regimes ( **Laminar flow pressure ratio**parameter value)By Reynolds number — The transition from laminar to turbulent regime is defined by the following equations:

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

$${D}_{H}=\sqrt{\frac{4A}{\pi}}$$

where

*D*_{H}Valve instantaneous hydraulic diameter *ν*Fluid kinematic viscosity *Re*_{cr}Critical Reynolds number ( **Critical Reynolds number**parameter value)

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

Valve opening is linearly proportional to the pressure differential.

No loading on the valve, such as inertia, friction, spring, and so on, is considered.

**Maximum passage area**Valve passage maximum cross-sectional area. The default value is

`1e-4`

m^2.**Valve pressure setting**Preset pressure level, at which the orifice of the valve starts to open. The default value is

`50e5`

Pa.**Valve regulation range**Pressure increase over the preset level needed to fully open the valve. MathWorks recommends using values less than 0.2 of the

**Valve pressure setting**parameter value. The default value is`5e5`

Pa.**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

`0.7`

.**Laminar transition specification**Select how the block transitions between the laminar and turbulent regimes:

`Pressure ratio`

— The transition from laminar to turbulent regime is smooth and depends on the value of the**Laminar flow pressure ratio**parameter. This method provides better simulation robustness.`Reynolds number`

— The transition from laminar to turbulent regime is assumed to take place when the Reynolds number reaches the value specified by the**Critical Reynolds number**parameter.

**Laminar flow pressure ratio**Pressure ratio at which the flow transitions between laminar and turbulent regimes. The default value is

`0.999`

. This parameter is visible only if the**Laminar transition specification**parameter is set to`Pressure ratio`

.**Critical Reynolds number**The maximum Reynolds number for laminar flow. 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

`12`

. This parameter is visible only if the**Laminar transition specification**parameter is set to`Reynolds number`

.**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. The parameter value must be greater than 0. The default value is

`1e-12`

m^2.**Opening dynamics**Select one of the following options:

`Do not include valve opening dynamics`

— The valve sets its orifice passage area directly as a function of pressure. If the area changes instantaneously, so does the flow equation. This is the default.`Include valve opening dynamics`

— Provide continuous behavior that is more physically realistic, by adding a first-order lag during valve opening and closing. Use this option in hydraulic simulations with the local solver for real-time simulation. This option is also helpful if you are interested in valve opening dynamics in variable step simulations.

**Opening time constant**The time constant for the first order response of the valve opening. This parameter is available only if

**Opening dynamics**is set to`Include valve opening dynamics`

. The default value is`0.1`

s.**Initial area**The initial opening area of the valve. This parameter is available only if

**Opening dynamics**is set to`Include valve opening dynamics`

. 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.

The block has the following ports:

`A`

Hydraulic conserving port associated with the valve inlet.

`B`

Hydraulic conserving port associated with the valve outlet.

The Closed-Loop Pump Control with Flexible Drive Shaft example illustrates the use of the Pressure Relief Valve block in hydraulic systems. The valve is set to 75e5 Pa and starts diverting fluid to tank as soon as the pressure at its inlet reaches this value.

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