# Pressure Relief Valve

Pressure control valve maintaining preset pressure in system

## Library

Pressure Control Valves

## Description

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 across the valve.

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:

$A=A\left(p\right)$

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:

$A\left(t=0\right)={A}_{init}$

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

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

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

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

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

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

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

where

 q Flow rate p Pressure differential pA, pB Gauge pressures at the block terminals CD Flow discharge coefficient A Instantaneous orifice passage area A(p) Pressure-dependent orifice passage area Ainit Initial open area of the valve Amax Fully open valve passage area Aleak Closed valve leakage area preg Regulation range pset Valve preset pressure pmax Valve pressure at maximum opening ρ Fluid density ν Fluid kinematic viscosity τ Time constant for the first order response of the valve opening t Time pcr Minimum pressure for turbulent flow Recr Critical Reynolds number DH Valve instantaneous hydraulic diameter

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

## Basic Assumptions and Limitations

• Valve opening is linearly proportional to the pressure differential.

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

## Dialog Box and Parameters

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.

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 12.

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-12m^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.

## 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.

## Examples

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.