# Pressure Compensator

Valve used to regulate the pressure drop across a hydraulic component

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• Simscape / Fluids / Hydraulics (Isothermal) / Valves / Pressure Control Valves

## Description

The Pressure Compensator block models the flow through a valve that constricts so as to maintain a preset pressure drop between a chosen two hydraulic nodes. The valve has four hydraulic ports, two being flow passages (the inlet, A, and the outlet, B) and two pressure sensors (X and Y). The normally open valve contracts when the pressure drop from X to Y rises above the valve pressure setting. The drop in opening area is a function of the pressure drop—proportional to it in a linear parameterization (the block default) or a general function of it in a tabulated parameterization. The valve serves its purpose until it hits the limit of its pressure regulation range—a point at which the valve is fully closed and the pressure drop can again rise unabated.

### 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 is computed as:

`$S\left(\Delta {p}_{\text{xy}}\right)={S}_{\text{Max}}-k\left(\Delta {p}_{\text{xy}}-\Delta {p}_{\text{Set}}\right),$`

where:

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

• ΔpSet is the value specified in the Valve pressure setting block parameter.

• ΔpXY is the pressure drop from port X to port Y:

`$\Delta {p}_{\text{XY}}={p}_{\text{X}}-{p}_{\text{Y}},$`

where p is the gauge pressure at the port indicated by the subscript (X or Y).

• k is the linear constant of proportionality:

`$k=\frac{{S}_{\text{Max}}-{S}_{\text{Leak}}}{\Delta {p}_{\text{Reg}}},$`

where in turn:

• SLeak is the value specified in the Leakage area block parameter.

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

At and below the valve pressure setting, the opening area is that of a fully open valve:

`$S\left(\Delta {p}_{XY}\le \Delta {p}_{\text{Set}}\right)={S}_{\text{Max}}.$`

At and above a maximum pressure, the opening area is that due to internal leakage alone:

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

where the maximum pressure drop ΔpMax is the sum:

`$\Delta {p}_{\text{Max}}=\Delta {p}_{\text{Set}}+\Delta {p}_{\text{Reg}}.$`

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 opening area is computed as:

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

where SXY 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 valve is 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 Rate

The causes of the pressure losses incurred in the passages of the 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 the valve relative to the theoretical value that it would have in an ideal valve. The flow rate through the valve is defined as:

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

where:

• q is the volumetric flow rate through the valve.

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

• S is the opening area of the valve.

• ΔpAB is the pressure drop from port A to port B.

• 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 ports A and B.

• 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 enter the valve.

Opening through which the flow can exit the valve.

Port at which pressure is measured for the purpose of setting the valve opening area.

Port at which pressure is measured for the purpose of setting the valve opening area.

## 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 its sensor ports. The alternative setting allows for a general, nonlinear relationship to be specified in tabulated form.

Opening area of the valve in the fully open position. The valve is fully open if the pressure drop from port X to port Y is less than the value specified in the Valve pressure setting block parameter.

#### Dependencies

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

Pressure drop from port X to port Y that the valve is to maintain. The valve closes in proportion to the pressure drop if it should exceed the value specified here. If the pressure drop is less than this value, the opening area is that specified in the Maximum passage area block parameter.

#### Dependencies

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

Pressure drop interval over which the valve is designed to operate. This interval stretches from the largest pressure drop at which the valve is fully open to the lowest pressure drop at which the valve is fully closed. Only within this range is the valve capable of maintaining the specified pressure setting.

#### 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.

Opening area of the valve in the fully closed position, when only internal leakage between its ports remains. This parameter serves primarily to ensure that closure of the valve does not cause portions of the thermal liquid 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`.

Vector of pressure differentials from port X to port Y 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 block parameter.

The block uses this data to construct a lookup table by which to determine from the pressure differential the 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 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 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 valve at the start of simulation. The block uses this parameter to calculate the initial rate at which the 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```.