# Pressure-Reducing 3-Way Valve (IL)

Combined pressure-relief and pressure-reducing valve in isothermal system

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• Simscape / Fluids / Isothermal Liquid / Valves & Orifices / Pressure Control Valves

## Description

The Pressure-Reducing 3-Way Valve (IL) is a combination of a pressure-relief and pressure-reducing valve. It maintains pressure at the valve outlet, port A, by restricting the inflow area at port P and venting the flow at port T.

### Valve Functionality

Valve operation is triggered by comparing the pressure difference between port T and port A to a threshold, the set pressure. When the pressure between T and A, Pcontrol, exceeds this set pressure, Pset,reducing, port P begins to close. A transition pressure range defines the pressures the valve experiences when both valves at ports P and T are closed. When the pressure difference between ports A and T exceeds the pressure transition range, port T opens. That is, ${P}_{set,relief}={P}_{set,reducing}+{P}_{range}+{P}_{transition}.$ The Pressure regulation range is specified for both the pressure-reducing and the pressure-relief valves. The valve parameters, such as Leakage Area and Maximum Opening Area, are the same for all ports.

To simulate pressure relief or pressure reduction with respect to another system element, see Pressure Compensator Valve (IL). To simulate pressure reduction between the valve outlet and atmosphere, see the Pressure-Reducing Valve (IL). To simulate pressure relief with respect to a valve or between the valve outlet and atmospheric pressure, see Pressure Relief Valve (IL).

### Pressure Control

When Pcontrol, PAPT, exceeds the threshold pressure, Pset,reducing, the valve at port P begins to close. When the Pressure transition range is exceeded, or when Pcontrol > Pset,relief, the valve at port T begins to open. Both valve closing and opening are parameterized in two ways:

• When Set Pressure control is set to `Controlled`, connect a pressure signal to port Ps, set the constant Pressure regulation range, and set the constant Pressure transition range. The pressure-reducing valve begins to close when Pcontrol is greater than Pset,reducing and below Pmax,reducing. The relief valve response is triggered when Pcontrol is greater than Pset,relief and below Pmax,relief. Pmax,relief is the sum of the Pressure regulation range and Pset,relief.

• When Set Pressure control is set to `Constant`, valve closing at port P is continuously regulated by either a linear or tabular parameterization. Similarly, relief valve opening at port T is parameterized linearly or by table lookup. An example of linear parameterization of the reduction valve (solid line) and relief valve (dotted line) is shown below.

When the `Tabulated data` option is selected, Pset,reducing and Pmax,reducing are the first and last parameters of the Pressure differential vector for reducing valve, respectively, and Pset,relief and Pmax,relief are the first and last parameters of the Pressure differential vector for relief valve, respectively. An example of tabular parameterization of both the reducing and relieving valves are shown below.

### Mass Flow Rate Equation

Momentum is conserved through the valve:

`${\stackrel{˙}{m}}_{A}+{\stackrel{˙}{m}}_{reducing}+{\stackrel{˙}{m}}_{relief}=0.$`

The mass flow rate through the valves is calculated as:

`${\stackrel{˙}{m}}_{reducing}=\frac{{C}_{d}{A}_{PA}\sqrt{2\overline{\rho }}}{\sqrt{P{R}_{loss,reducing}\left(1-{\left(\frac{{A}_{PA}}{{A}_{port}}\right)}^{2}\right)}}\frac{\Delta {p}_{reducing}}{{\left[\Delta {p}_{reducing}{}^{2}+\Delta {p}_{crit,reducing}^{2}\right]}^{1/4}},$`

`${\stackrel{˙}{m}}_{relief}=\frac{{C}_{d}{A}_{AT}\sqrt{2\overline{\rho }}}{\sqrt{P{R}_{loss,relief}\left(1-{\left(\frac{{A}_{AT}}{{A}_{port}}\right)}^{2}\right)}}\frac{\Delta {p}_{relief}}{{\left[\Delta {p}_{relief}{}^{2}+\Delta {p}_{crit,relief}^{2}\right]}^{1/4}},$`

where:

• Cd is the Discharge coefficient.

• A is the instantaneous valve open area between ports Aand P or A and T, as indicated by the subscript.

• Aport is the Cross-sectional area at ports A, P & T.

• $\overline{\rho }$ is the average fluid density.

• Δp is the valve pressure difference: pApB.

The critical pressure difference, Δpcrit, is the pressure differential associated with the Critical Reynolds number, Recrit, the flow regime transition point between laminar and turbulent flow, which corresponds to either the pressure-reducing or pressure relief component of the valve:

`$\Delta {p}_{crit}=\frac{\pi \overline{\rho }}{8A}{\left(\frac{\nu {\mathrm{Re}}_{crit}}{{C}_{d}}\right)}^{2},$`

where A is either APA or AAT, corresponding to the reducing or relief component of the valve, respectively.

Pressure loss describes the reduction of pressure in the valve due to a decrease in area. PRloss is calculated as:

`$P{R}_{loss}=\frac{\sqrt{1-{\left(\frac{A}{{A}_{port}}\right)}^{2}\left(1-{C}_{d}^{2}\right)}-{C}_{d}\frac{A}{{A}_{port}}}{\sqrt{1-{\left(\frac{A}{{A}_{port}}\right)}^{2}\left(1-{C}_{d}^{2}\right)}+{C}_{d}\frac{A}{{A}_{port}}}.$`

Pressure recovery describes the positive pressure change in the valve due to an increase in area. If you do not wish to capture this increase in pressure, set Pressure recovery to `Off`. In this case, PRloss is 1.

The opening area A is determined by the opening parameterization (for `Constant` valves only) of the reducing valve (P to A) or relief valve (A to T) and the valve opening dynamics.

### Opening and Closing Parameterization

Linear parameterization of the valve area for the reducing valve is

`${A}_{valve}={\stackrel{^}{p}}_{reducing}\left({A}_{leak}-{A}_{\mathrm{max}}\right)+{A}_{\mathrm{max}},$`

and for the relief valve is

`${A}_{valve}={\stackrel{^}{p}}_{relief}\left({A}_{\mathrm{max}}-{A}_{leak}\right)+{A}_{leak}.$`

The normalized pressure, $\stackrel{^}{p}$, is

`$\stackrel{^}{p}=\frac{{p}_{control}-{p}_{set}}{{p}_{\mathrm{max}}-{p}_{set}},$`

where the set and maximum pressures are the respective reducing or relief valve settings.

At the extremes of the relief and reducing valve pressure ranges, you can maintain numerical robustness in your simulation by adjusting the block . With a nonzero smoothing factor, a smoothing function is applied to every calculated pressure, but primarily influences the simulation at the extremes of these ranges.

When the Smoothing factor, f, is nonzero, a smoothed, normalized pressure is instead applied to the valve area:

`${\stackrel{^}{p}}_{smoothed}=\frac{1}{2}+\frac{1}{2}\sqrt{{\stackrel{^}{p}}_{}^{2}+{\left(\frac{f}{4}\right)}^{2}}-\frac{1}{2}\sqrt{{\left(\stackrel{^}{p}-1\right)}^{2}+{\left(\frac{f}{4}\right)}^{2}}.$`

In the `Tabulated data` parameterization, Aleak,PA and Amax,PA are the first and last parameters of the Opening area vector reducing valve, respectively, and Aleak,AT and Amax,AT are the first and last parameters of the Opening area vector for relief valve, respectively. The smoothed, normalized pressure is also used when the smoothing factor is nonzero with linear interpolation and nearest extrapolation.

### Opening Dynamics

If Opening dynamics are modeled, a lag is introduced to the flow response to valve opening. Avalve becomes the dynamic opening or closing area, Adyn; otherwise, Avalve is the steady-state opening area. This area is specific to the reducing and relief components of the valve, APA or AAT, respectively. The instantaneous change in dynamic opening area is calculated based on the Opening time constant, τ:

`${\stackrel{˙}{p}}_{dyn}=\frac{{p}_{control}-{p}_{dyn}}{\tau }.$`

By default, Opening dynamics is set to `Off`.

### Assumptions and Limitations

Friction between the valve and fluid, the hydraulic force of the fluid on the valve components, and the effect of fluid inertia are neglected.

## Ports

### Conserving

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Liquid exit port of the valve.

Liquid entry port to the valve.

Liquid relief port of the valve.

### Input

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Varying-signal set pressure for controlled valve operation.

#### Dependencies

To enable this port, set Set pressure control to `Controlled`.

## Parameters

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Valve operation method. A `Constant` valve opens or closes linearly over a fixed pressure regulation range and pressure transition or in accordance with tabulated pressure and opening area data that you provide. A `Controlled` valve opens or closes according to a variable set pressure signal at port Pset over a fixed pressure regulation and pressure transition range. The selected setting applies both to the reducing and relief valve operation.

Method of modeling the valve opening or closing. Valve opening is either parametrized linearly, which correlates the opening area to the provided pressure range, or by a table of values you provide that correlate the valve opening area to pressure differential data.

#### Dependencies

To enable this port, set Set pressure control to `Constant`.

Pressure differential between port T and port A. When this set pressure differential is surpassed, the valve at port P begins to close. The closing is parametrized linearly or by lookup table as defined in the Opening parameterization.

Operational pressure range of the reducing valve. The pressure regulation range lies between the Set pressure differential and the maximum valve operating pressure. At the end of the Pressure regulation range, the pressure-reducing valve is closed and the Pressure transition range begins.

#### Dependencies

To enable this parameter, set

• Set pressure control to `Controlled`, or

• Set pressure control to `Constant` and Opening parameterization to `Linear`.

Pressure range of the 3-way valve. This parameter defines the pressure range, which begins at the end of the Pressure regulation range, over which both ports P and T are closed. Below this range, the reducing valve at port P is open to flow, and above this range, the relief valve at port T opens.

#### Dependencies

To enable this parameter, set either:

• Set pressure control to `Controlled`

• Set pressure control to `Constant` and Opening parameterization to `Linear`

Cross-sectional area of the valve (P-A or A-T) in its fully-open position.

#### Dependencies

To enable this parameter, set either:

• Set pressure control to `Controlled`

• Set pressure control to `Constant` and Opening parameterization to `Linear`

Sum of all gaps when the valve is in fully closed position. Any area smaller than this value is saturated to the specified leakage area. This contributes to numerical stability by maintaining continuity in the flow.

#### Dependencies

To enable this parameter, set either:

• Set pressure control to `Controlled`

• Set pressure control to `Constant` and Opening parameterization to `Linear`

Vector of pressure differential values for the tabular parameterization of opening area. The vector elements correspond one-to-one to the values in the Opening area vector reducing valve parameter. Pressure differential vector values are listed in ascending order and must have the same number of elements as the Opening area vector reducing valve parameter. Linear interpolation is employed between table data points.

#### Dependencies

To enable this parameter, set Set pressure control to `Constant` and Opening parameterization to `Tabulated data`.

Vector of opening area values for the tabular parameterization of opening area. The vector elements must correspond one-to-one to the values in the Pressure differential vector for reducing valve parameter. Areas are listed in descending order. Linear interpolation is employed between table data points.

#### Dependencies

To enable this parameter, set Set pressure control to `Constant` and Opening parameterization to `Tabulated data`.

Vector of pressure differential values for the tabular parameterization of opening area. The vector elements correspond one-to-one to the values in the Opening area vector relief valve parameter. Pressure differential vector values are listed in ascending order and must have the same number of elements as the Opening area vector relief valve parameter. Linear interpolation is employed between table data points.

#### Dependencies

To enable this parameter, set Set pressure control to `Constant` and Opening parameterization to `Tabulated data`.

Vector of opening area values for the tabular parameterization of opening area. The vector elements must correspond one-to-one to the values in the Pressure differential vector for relief valve parameter. Areas are listed in ascending order. Linear interpolation is employed between table data points.

#### Dependencies

To enable this parameter, set Set pressure control to `Constant` and Opening parameterization to `Tabulated data`.

Cross-sectional area at the entry and exit ports A, P, and T. This area is used in the pressure-flow rate equation that determines the mass flow rate through the orifice.

Correction factor accounting for discharge losses in theoretical flows. The default discharge coefficient for a valve in Simscape™ Fluids™ is 0.64.

Upper Reynolds number limit for laminar flow through the valve.

Continuous smoothing factor that introduces a layer of gradual change to the flow response when the valve is in near-open or near-closed positions. Set this value to a nonzero value less than one to increase the stability of your simulation in these regimes.

Accounts for pressure increase when fluid flows from a region of smaller cross-sectional area to a region of larger cross-sectional area. This increase in pressure is not captured when Pressure recovery is set to `Off`.

Accounts for transient effects to the fluid system due to the valve opening. Setting Opening dynamics to `On` approximates the opening conditions by introducing a first-order lag in the flow response. The Opening time constant also impacts the modeled opening dynamics.

Initial cross-sectional area at port P at the time of dynamic opening. This value is used to calculate the instantaneous opening area for the dynamic opening between P and A at the following time step.

#### Dependencies

To enable this parameter, set Opening dynamics to `On`.

Initial cross-sectional area at port T of opening at the time of dynamic opening. This value is used to calculate the instantaneous opening area for the dynamic opening between A and T at the following time step.

#### Dependencies

To enable this parameter, set Opening dynamics to `On`.

Constant that captures the time required for the fluid to reach steady-state when opening or closing the valve from one position to another. This parameter impacts the modeled opening dynamics.

#### Dependencies

To enable this parameter, set Opening dynamics to `On`.

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