# Pressure Reducing 3-Way Valve

Pressure reducing/relieving valve

## Library

Pressure Control Valves

## Description

The Pressure Reducing 3-Way Valve block represents a hydraulic 3-way valve that is also known as a pressure reducing/relieving valve. The valve reduces inlet pressure to a preset value, similar to a conventional pressure reducing valve, but, additionally, starts working as a pressure-relief valve if the pressure continues to rise.

Initially, orifice P-A is fully open. It remains fully open while the outlet pressure at port A is lower than the valve preset pressure. When the preset pressure is reached, the valve control member is forced off its stop and starts closing the orifice, thus trying to maintain outlet pressure at the preset level. Any further increase in the outlet pressure causes the control member to close the orifice even more, until the orifice is fully closed. The pressure increase needed to close the valve is referred to as regulation range, and is generally provided in the catalogs, along with the valve maximum area. In a conventional pressure reducing valve, outlet pressure is no longer under control after orifice P-A is closed. The pressure reducing 3-way valve provides an additional relieving function by diverting some flow from the outlet to a tank through an additional orifice A-T.

The valve has the following area-pressure differential relationship:

Ared = Amed – (AmaxAmed)· tanh(k · (ppred_med) / (pred_maxpred_med))

Arel = Amed + (AmaxAmed)· tanh(k · (pprel_med) / (prel_maxprel_med))

Amed = (Amax + Aleak) / 2

pred_max = pset + preg

pred_med = (pset + preg) / 2

prel_set = pset + preg + ptr

prel_max = prel_set + preg

prel_med = (prel_set + preg) / 2

where

 Amax Maximum opening area of both the reducing and relieving valves Ared Pressure reducing (orifice P-A) valve opening area Arel Pressure-relief (orifice A-T) valve opening area Aleak Leakage area (the area that remains open even after the orifice is completely closed) p Pressure drop across the valve, pA – pT pset Preset pressure differential preg Regulation range ptr Transition pressure (the pressure increment above the pressure of the fully closed reducing valve and setting pressure of the pressure-relief valve) k Valve opening adjustment coefficient

The distinctive feature of a pressure reducing 3-way valve is a sharp change of orifice openings. To avoid computational problems, the openings are approximated using the hyperbolic tangent function. The following figure shows an example of the relationship between the opening area and pressure.

In the example, the setting pressure is set to 20e5 Pa, and the regulation range is set to 1e5 Pa. The valve opening adjustment coefficient is set to 2. The higher the value of the coefficient, the closer the transition is to a linear relationship. The transition is close to the experimental data at k in the range of 3 to 4.

For both orifices, a small leakage area is assumed to exist even after the orifice is completely closed. Physically, it represents a possible clearance in the closed valve, but 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 failure of computation.

After the areas have been determined, the block computes the flow rate for both orifices according to the following equations:

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

where

 q Flow rate p Pressure differential pA, pB, pR Gauge pressures at the block terminals CD Flow discharge coefficient A Instantaneous orifice passage area ρ Fluid density pcr Minimum pressure for turbulent flow

The minimum pressure for turbulent flow, pcr, 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:

pcr = (pavg + patm)(1 – Blam)

pavg = (pA + pB)/2

where

 pavg Average pressure between the block terminals patm Atmospheric pressure, 101325 Pa Blam 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

 DH Instantaneous orifice hydraulic diameter ν Fluid kinematic viscosity Recr Critical Reynolds number (Critical Reynolds number parameter value)

By default, the block does not include valve opening dynamics. Adding valve opening dynamics provides continuous behavior that is particularly helpful in situations with rapid valve opening and closing. The instantaneous orifice passage area A in the flow equations above is then determined as follows:

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

$\frac{d{A}_{red_dyn}}{dt}=\frac{{A}_{red}-{A}_{red_dyn}}{\tau }$

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

$\frac{d{A}_{rel_dyn}}{dt}=\frac{{A}_{rel}-{A}_{rel_dyn}}{\tau }$

where

 Ared_dyn Instantaneous open area for pressure reducing valve (orifice P-A) with opening dynamics Arel_dyn Instantaneous open area for pressure-relief valve (orifice A-T) with opening dynamics Ared_init Initial open area for pressure reducing valve (orifice P-A) Arel_init Initial open area for pressure-relief valve (orifice A-T) τ Time constant for the first order response of the valve opening t Time

Connections P, A, and T are the conserving hydraulic ports associated with the valve inlet, outlet, and return, respectively. The block positive direction is from port P to port A and from port A to port T.

## Assumptions and Limitations

Fluid inertia, valve inertia, friction, and hydraulic forces are not taken into account.

## Parameters

Valve maximum area

The area of a fully opened orifice. Both orifices are assumed to have the same maximum area. The parameter value must be greater than zero. The default value is 1e-4 m^2.

Reducing valve pressure setting

Preset pressure level, at which the orifice P-A of the valve starts to close. The default value is 6e5 Pa.

Valve regulation range

Pressure increase over the preset level needed to fully close the pressure reducing valve. The lower the value of the range, the higher the valve sensitivity. The default value is 0.3e5 Pa.

Transition pressure

Pressure increment above the pressure of the fully closed reducing valve needed to reach the pressure at which the pressure-relief valve starts opening. The transition pressure must be greater than or equal to zero. The default value is 2e5 Pa.

Valve flow discharge coefficient

Semi-empirical parameter for valve capacity characterization. The value depends on the geometrical properties of the orifice, and usually is provided in textbooks or manufacturer data sheets. The default value is 0.6.

Valve 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 Valve 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 Valve critical Reynolds number parameter.

Valve 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 Valve laminar transition specification parameter is set to Pressure ratio.

Valve 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 Valve laminar transition specification parameter is set to Reynolds number.

Valve 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-9 m^2.

The coefficient controls how close the hyperbolic tangent function approximates the linear relationship between the orifice area and control pressure. See the block description for more information. The default value is 1.

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 reducing valve area

The initial opening area of the reducing valve. This parameter is available only if Opening dynamics is set to Include valve opening dynamics. The default value is 1e-9 m^2.

Initial relief valve area

The initial opening area of the relief valve. This parameter is available only if Opening dynamics is set to Include valve opening dynamics. The default value is 1e-9 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:

P

Hydraulic conserving port associated with the valve inlet.

A

Hydraulic conserving port associated with the valve outlet.

T

Hydraulic conserving port that connects with the tank.

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