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Valve used to vent pressure in excess of a preset value from a hydraulic line

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

The Pressure Relief Valve block models the flow through
a valve that opens to vent so as to maintain a preset pressure drop between inlet (port
**A**) and outlet (port **B**). The normally
closed valve cracks open when the pressure drop from **A** to
**B** exceeds the valve pressure setting. The valve opening area is
computed as a function of the excess in the pressure drop, either a simple linear
expression or a (generally nonlinear) tabulated function. The valve serves its purpose
until its opening area reaches a maximum—a point at the limit of the valve regulation
range, beyond which the pressure drop is again free to rise, unabated.

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

or
`Tabulated data - Area vs. pressure`

.

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

,
the opening area is computed as:

$$S(\Delta {p}_{\text{AB}})={S}_{\text{Leak}}+k\left(\Delta {p}_{\text{AB}}-\Delta {p}_{\text{Set}}\right),$$

where:

*S*_{Max}is the value specified in the**Maximum passage area**block parameter.*Δp*_{Set}is the value specified in the**Valve pressure setting**block parameter.*Δp*_{AB}is the pressure drop from port**A**to port**B**:$$\Delta {p}_{\text{AB}}={p}_{\text{A}}-{p}_{\text{B}},$$

where

*p*is the gauge pressure at the port indicated by the subscript (**A**or**B**).*k*is the linear constant of proportionality:$$k=\frac{{S}_{\text{Max}}-{S}_{\text{Leak}}}{\Delta {p}_{\text{Reg}}},$$

where in turn:

*S*_{Leak}is the value specified in the**Leakage area**block parameter.*Δp*_{Reg}is that specified in the**Valve regulation range**block parameter.

At and below the valve pressure setting, the opening area is that due to internal leakage alone:

$$S(\Delta {p}_{\text{AB}}\le \Delta {p}_{\text{Set}})={S}_{\text{Leak}},$$

At and above a maximum pressure, the opening area is that of a fully open valve:

$$S(\Delta {p}_{\text{AB}}\ge \Delta {p}_{\text{Max}})={S}_{\text{Max}},$$

where the maximum pressure drop
*Δp*_{Max} is the sum:

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

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

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

, the opening
area is computed as:

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

where *S*_{XY} 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**

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:

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

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`

).

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 its being a small number greater than zero. The leakage area is obtained from the block parameter of the same name.

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.*C*_{D}is the value of the**Discharge coefficient**block parameter.*S*is the opening area of the valve.*Δp*_{AB}is the pressure drop from port**A**to port**B**.*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 ports**A**and**B**.*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:$${D}_{\text{H}}=\sqrt{\frac{4S}{\pi}}.$$