# Pressure Reducing 3-Way Valve

Composite valve comprising the functions of pressure reducing and relief valves

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• 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:

• SMax is the value specified in the Maximum passage area block parameter.

• SLeak 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.

• ΔpReg is the value specified in the Valve regulation range block parameter.

• ΔpRdn 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 ΔpSet 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 ΔpRlf 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 ΔpTran 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:

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

where SAT 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:

`$\stackrel{˙}{S}=\frac{S\left(\Delta {p}_{\text{SS}}\right)-S\left(\Delta {p}_{\text{In}}\right)}{\tau },$`

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)..

• CD is the value of the Discharge coefficient block parameter.

• S is the opening area of the valve (SPA or SAT).

• Δp is the pressure drop across the valve (ΔpPA or ΔpAT).

• pCrit 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:

• pAtm is the atmospheric pressure (as defined for the corresponding hydraulic network).

• pAvg is the average of the gauge pressures at the ports (P and A or A and T).

• βCrit 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:

• ReCrit is the value of the Critical Reynolds number block parameter.

• ν is the kinematic viscosity specified for the hydraulic network.

• DH is the instantaneous hydraulic diameter:

`${D}_{\text{H}}=\sqrt{\frac{4S}{\pi }}.$`

## Ports

### Conserving

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Opening through which the flow can exit the valve.

Opening through which the flow can enter the valve.

Opening through which the flow can vent from the valve.

## Parameters

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Method by which to calculate the opening area of the valve. The default setting prescribes a linear relationship between the opening area of the valve and the pressure drop between the outlet and the vent opening. The alternative setting allows for a general, nonlinear relationship to be specified in tabulated form.

Opening area of each valve when fully open. The pressure-reducing and pressure-relief valves are assumed to be identical in size. See the block description for the pressure conditions under which the valves can each be said to be fully open.

#### Dependencies

This parameter is active when the Opening area parameterization block parameter is set to `Linear area-pressure relationship`.

Pressure differential from port A to port T that the composite valve is to maintain. The pressure-reducing valve closes in proportion to the pressure differential if its value should exceed that specified here. The pressure-relief valve opens in proportion to the same if its value should exceed, by an amount given by the Transition pressure block parameter, that needed to fully close the reducing valve.

#### Dependencies

This parameter is active when the Opening area parameterization block parameter is set to `Linear area-pressure relationship`.

Pressure differential interval over which the valves are each designed to operate. This interval spans those pressures over which the opening area changes between its minimum and maximum values. The value specified here applies to both the pressure-reducing and pressure-relief valves.

#### Dependencies

This parameter is active when the Opening area parameterization block parameter is set to `Linear area-pressure relationship`.

Pressure differential in excess of that required to fully close the pressure-reducing valve at which the pressure-relief valve first opens. The greater the transition pressure, the greater the lag between the closing of the pressure-reducing valve and the opening of its pressure-relief counterpart.

#### Dependencies

This parameter is active when the Opening area parameterization block parameter is set to `Linear area-pressure relationship`.

Ratio of the actual flow rate through the valve to the theoretical value that it would have in an ideal valve. This semi-empirical parameter measures the flow allowed through the valve: the greater its value, the greater the flow rate. Refer to the valve data sheet, if available, for this parameter. The value specified here is shared by both pressure-reducing and pressure-relief valves.

Opening area of a valve in the fully closed position, when only internal leakage between its respective ports remains. The leakage area is the same in the pressure-reducing and pressure-relief valves. This parameter serves primarily to ensure that closure of the valve does not cause portions of the hydraulic network to become isolated. The exact value specified here is less important than its being a small number greater than zero.

#### Dependencies

This parameter is active when the Opening area parameterization block parameter is set to `Linear area-pressure relationship`.

Smoothing factor for easing the transition to the fully open or fully closed valve position. The smoothing enhances the numerical robustness of the model, helping to avoid numerical errors due to discontinuities and simulation slowdowns due to zero crossings. Use this parameter to scale the widths of the smoothed regions. The smaller the value, the broader the transition.

#### Dependencies

This parameter is active when the Opening area parameterization block parameter is set to `Linear area-pressure relationship`.

Vector of pressure drops from port P to port A at which to specify the opening area of the valve. The vector elements must increase monotonically from left to right. This order is important when specifying the Opening area vector for reducing valve block parameter.

The block uses this data to construct a lookup table by which to determine from the pressure differential the reducing-valve opening area. Data is handled with linear interpolation (within the tabulated data range) and nearest-neighbor extrapolation (outside of the range).

#### Dependencies

This parameter is active when the Opening area parameterization block parameter is set to ```Tabulated data - Area vs. pressure```.

Vector of opening areas corresponding to the breakpoints defined in the Pressure differential vector for reducing valve block parameter. The vector elements must decrease monotonically from left to right (with increasing pressure). For best results, avoid regions of flattened slope.

The block uses this data to construct a lookup table by which to determine from the pressure differential the reducing-valve opening area. Data is handled with linear interpolation (within the tabulated data range) and nearest-neighbor extrapolation (outside of the range).

#### Dependencies

This parameter is active when the Opening area parameterization block parameter is set to ```Tabulated data - Area vs. pressure```.

Vector of pressure drops from port A to port T at which to specify the opening area of the valve. The vector elements must increase monotonically from left to right. This order is important when specifying the Opening area vector for relief valve block parameter.

The block uses this data to construct a lookup table by which to determine from the pressure differential the reducing-valve opening area. Data is handled with linear interpolation (within the tabulated data range) and nearest-neighbor extrapolation (outside of the range).

#### Dependencies

This parameter is active when the Opening area parameterization block parameter is set to ```Tabulated data - Area vs. pressure```.

Vector of opening areas corresponding to the breakpoints defined in the Pressure differential vector for relief valve block parameter. The vector elements must decrease monotonically from left to right (with increasing pressure). For best results, avoid regions of flattened slope.

The block uses this data to construct a lookup table by which to determine from the pressure differential the reducing-valve opening area. Data is handled with linear interpolation (within the tabulated data range) and nearest-neighbor extrapolation (outside of the range).

#### Dependencies

This parameter is active when the Opening area parameterization block parameter is set to ```Tabulated data - Area vs. pressure```.

Parameter in terms of which to specify the boundary between the laminar and turbulent flow regimes. The pressure ratio of the default parameterization is defined as the gauge pressure at the outlet divided by the same at the inlet.

Pressure ratio at which the flow is assumed to transition between laminar and turbulent regimes. The pressure ratio is defined as the gauge pressure at the outlet divided by the same at the inlet. The transition is assumed to be smooth and centered on this value.

#### Dependencies

This parameter is active when the Laminar transition specification block parameter is set to `Pressure ratio`.

Reynolds number at which the flow is assumed to transition between laminar and turbulent regimes.

#### Dependencies

This parameter is active when the Laminar transition specification block parameter is set to `Reynolds number`.

Choice of whether to capture the opening dynamics of the valve. Selecting `Include valve opening dynamics` causes the valve to open gradually, so as to approach its new steady-state area over a small time span. The characteristic time for such transitions is given in the Opening time constant block parameter.

Setting this parameter to the alternative ```Do not include valve opening dynamics``` is equivalent to specifying a value of `0` for the opening time constant. The opening area is in this case assumed to reach its new steady-state value instantaneously.

Include opening dynamics to more accurately capture the behavior of a real valve. For best real-time simulation performance, use with a local solver or disable valve opening dynamics altogether.

Measure of the time taken by the valve to transition from its current opening area to a new steady-state value. The block uses this parameter to calculate the rate at which a valve is opening and from it the instantaneous opening area at the next time step.

#### Dependencies

This parameter is active when the Opening dynamics block parameter is set to ```Include valve opening dynamics```.

Area normal to the direction of flow within the pressure-reducing valve at the start of simulation. The block uses this parameter to calculate the initial rate at which the pressure-reducing valve is opening and from it the instantaneous opening area at the next time step.

#### Dependencies

This parameter is active when the Opening dynamics block parameter is set to ```Include valve opening dynamics```.

Area normal to the direction of flow within the pressure-relief valve at the start of simulation. The block uses this parameter to calculate the initial rate at which the pressure-relief valve is opening and from it the instantaneous opening area at the next time step.

#### Dependencies

This parameter is active when the Opening dynamics block parameter is set to ```Include valve opening dynamics```.