# Pressure Reducing 3-Way Valve

Composite valve comprising the functions of pressure reducing and relief valves

**Library:**Simscape / Fluids / Hydraulics (Isothermal) / Valves / Pressure Control Valves

## Description

The Pressure Reducing 3-Way Valve block models the flow
through a valve that constricts and, if necessary, vents (via a separate line) so as to
maintain a preset pressure differential between its outlet (port **A**)
and its surroundings (**T**, the vent). The valve combines the
functions of two more elementary valves, one a pressure-reducing valve, the other a
pressure-relief valve.

The pressure-reducing valve runs between the inlet (**P**) and the
outlet (**A**). It is normally open but closes as needed to combat
pressure fluctuations upstream of the outlet. The pressure-relief valve runs between the
outlet (**A**) and the vent (**T**). It is normally
closed but opens if the outlet pressure should exceed (by a specified transition amount)
the pressure regulation range of the pressure-reducing valve.

The valve opening areas are functions of the pressure differential from port
**A** to port **T**—either directly proportional
to it or general (tabulated) functions of it. The opening areas can each vary up to a
point—the limit of the respective valve regulation range—beyond which the valve is
saturated at full capacity and unable to counteract any additional pressure
rises.

### Valve Opening

The opening area calculation depends on the valve parameterization selected for
the block: either `Linear area-opening relationship`

or
`Tabulated data - Area vs. pressure`

.

**Linear Parameterization**

If the **Valve parameterization** block parameter is in the
default setting of `Linear area-opening relationship`

,
the opening area of the pressure-reducing valve
(**P**-**A**) is computed as:

$${S}_{\text{PA}}=\frac{{S}_{\text{Max}}+{S}_{\text{Leak}}}{2}-\frac{{S}_{\text{Max}}-{S}_{\text{Leak}}}{2}\text{tanh}\left[\frac{\lambda \left(\Delta {p}_{\text{AT}}-\Delta {p}_{\text{Rdn}}\right)}{\Delta {p}_{\text{Reg}}/2}\right],$$

where the `tanh`

term serves to smooth the
transitions to the fully open and fully closed valve positions. In the equation:

*S*_{Max}is the value specified in the**Maximum passage area**block parameter.*S*_{Leak}is the value specified in the**Leakage area**block parameter.*ƛ*is the value of the**Valve opening adjustment coefficient**block parameter, a measure of the smoothing to apply to the valve transitions. The smaller the value, the smoother the transition.*Δp*_{Reg}is the value specified in the**Valve regulation range**block parameter.*Δp*_{Rdn}is the midpoint of the pressure regulation range of the pressure-reducing valve:$$\Delta {p}_{\text{Rdn}}=\Delta {p}_{\text{Set}}+\frac{\Delta {p}_{\text{Reg}}}{2},$$

with

*Δp*_{Set}being the value of the**Reducing valve pressure setting**block parameter.

The opening area of the pressure-relief valve
(**A**-**T**) is likewise computed as:

$${S}_{\text{AT}}=\frac{{S}_{\text{Max}}+{S}_{\text{Leak}}}{2}+\frac{{S}_{\text{Max}}-{S}_{\text{Leak}}}{2}\text{tanh}\left(\frac{\lambda \left(\Delta {p}_{\text{AT}}-\Delta {p}_{\text{Rlf}}\right)}{\Delta {p}_{\text{Reg}}/2}\right),$$

where *Δp*_{Rlf} is the
midpoint of the pressure regulation range of the pressure-relief valve:

$$\Delta {p}_{\text{Rlf}}=\Delta {p}_{\text{Set}}+\Delta {p}_{\text{Reg}}+\Delta {p}_{\text{Tran}}+\frac{\Delta {p}_{\text{Reg}}}{2},$$

with *Δp*_{Tran} being
the transition pressure differential—that required for the pressure-relief valve
to open after the pressure-reducing valve has fully closed. This value is
obtained from the **Transition pressure** block parameter.

**Opening area in the Linear area-opening
relationship parameterization**

**Tabulated Parameterization**

If the **Valve parameterization** block parameter is set to
`Tabulated data - Area vs. pressure`

, the valve
opening areas are computed as:

$$S=S(\Delta {p}_{\text{AT}}),$$

where *S*_{AT} is a tabulated function
constructed from the **Pressure drop vector** and
**Opening area vector** block parameters. The function is
based on linear interpolation (for points within the data range) and
nearest-neighbor extrapolation (for points outside the data range). The leakage
and maximum opening areas are the minimum and maximum values of the
**Valve opening area vector** block parameter.

**Opening area in the Tabulated data - Area vs.
pressure parameterization**

**Opening Dynamics**

By default, the valve opening dynamics are ignored. The valves are each
assumed to respond instantaneously to changes in the pressure drop, without time
lag between the onset of a pressure disturbance and the increased valve opening
that the disturbance produces. If such time lags are of consequence in a model,
you can capture them by setting the **Opening dynamics** block
parameter to `Include valve opening dynamics`

. The
valves then open each at a rate given by the expression:

$$\dot{S}=\frac{S(\Delta {p}_{\text{SS}})-S(\Delta {p}_{\text{In}})}{\tau},$$

where *τ* is a measure of the time needed
for the instantaneous opening area (subscript `In`

) to reach a
new steady-state value (subscript `SS`

).

**Leakage Area**

The primary purpose of the leakage area of a closed valve is to ensure that at no time does a portion of the hydraulic network become isolated from the remainder of the model. Such isolated portions reduce the numerical robustness of the model and can slow down simulation or cause it to fail. Leakage is generally present in minuscule amounts in real valves but in a model its exact value is less important than it being a small number greater than zero. The leakage area is obtained from the block parameter of the same name.

### Valve Flow Rates

The causes of the pressure losses incurred in the passages of the composite valve are ignored in the block. Whatever their natures—sudden area changes, flow passage contortions—only their cumulative effect is considered during simulation. This effect is captured in the block by the discharge coefficient, a measure of the flow rate through a valve relative to the theoretical value that it would have in the ideal case. The flow rates through each of the elementary valves are defined as:

$$q={C}_{\text{D}}S\sqrt{\frac{2}{\rho}}\frac{\Delta p}{{\left[{\left(\Delta p\right)}^{2}+{p}_{\text{Crit}}^{2}\right]}^{1/4}},$$

where:

*q*is the volumetric flow rate through the valve (**P**-**A**or**A**-**T**)..*C*_{D}is the value of the**Discharge coefficient**block parameter.*S*is the opening area of the valve (*S*_{PA}or*S*_{AT}).*Δp*is the pressure drop across the valve (*Δp*_{PA}or*Δp*_{AT}).*p*_{Crit}is the pressure differential at which the flow shifts between the laminar and turbulent flow regimes.

The calculation of the critical pressure depends on the setting of the
**Laminar transition specification** block parameter. If this
parameter is in the default setting of `By pressure ratio`

:

$${p}_{\text{Crit}}=\left({p}_{\text{Atm}}+{p}_{\text{Avg}}\right)\left(1-{\beta}_{\text{Crit}}\right),$$

where:

*p*_{Atm}is the atmospheric pressure (as defined for the corresponding hydraulic network).*p*_{Avg}is the average of the gauge pressures at the ports (**P**and**A**or**A**and**T**).*β*_{Crit}is the value of the**Laminar flow pressure ratio**block parameter.

If the **Laminar transition specification** block parameter is
instead set to `By Reynolds number`

:

$${p}_{\text{Crit}}=\frac{\rho}{2}{\left(\frac{{\text{Re}}_{\text{Crit}}\nu}{{C}_{\text{D}}{D}_{\text{H}}}\right)}^{2},$$

where:

*Re*_{Crit}is the value of the**Critical Reynolds number**block parameter.*ν*is the kinematic viscosity specified for the hydraulic network.*D*_{H}is the instantaneous hydraulic diameter:$${D}_{\text{H}}=\sqrt{\frac{4S}{\pi}}.$$

## Ports

### Conserving

## Parameters

## Extended Capabilities

**Introduced in R2016a**