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Pressure-relief valve in an isothermal system

**Library:**Simscape / Fluids / Isothermal Liquid / Valves & Orifices / Pressure Control Valves

The Pressure Relief Valve (IL) models a pressure-relief valve in an isothermal liquid
network. The valve remains closed when the pressure is less than a specified value. When
this pressure is met or surpassed, the valve opens. This set pressure is either a
threshold pressure differential over the valve, between ports **A** and
**B**, or between port **A** and atmospheric
pressure. For pressure control based on another element in the fluid system, see the
Pressure Compensator
Valve (IL) block.

Two valve control options are available:

When

**Set pressure control**is set to`Controlled`

, connect a pressure signal to port**Ps**and define the constant**Pressure regulation range**. The valve response will be triggered when*P*_{control}, the pressure differential between ports**A**and**B**, is greater than*P*_{set}and below*P*_{max}.*P*_{max}is the sum of*P*_{set}and the pressure regulation range.When

**Set pressure control**is set to`Constant`

, the valve opening is continuously regulated between*P*_{set}and*P*_{max}. There are two options for pressure regulation available in the**Pressure control specification**parameter:*P*_{control}can be the pressure differential between ports**A**and**B**or the pressure differential between port**A**and atmospheric pressure. The opening area is then modeled by either linear or tabular parameterization. When the`Tabulated data`

option is selected,*P*_{set}and*P*_{max}are the first and last parameters of the**Pressure differential vector**, respectively.

Momentum is conserved through the valve:

$${\dot{m}}_{A}+{\dot{m}}_{B}=0.$$

The mass flow rate through the valve is calculated as:

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

where:

*C*_{d}is the**Discharge coefficient**.*A*_{valve}is the instantaneous valve open area.*A*_{port}is the**Cross-sectional area at ports A and B**.$$\overline{\rho}$$ is the average fluid density.

*Δp*is the valve pressure difference*p*_{A}–*p*_{B}.

The critical pressure difference,
*Δp*_{crit}, is the pressure differential
associated with the **Critical Reynolds number**,
*Re*_{crit}, the flow regime transition
point between laminar and turbulent flow:

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

*Pressure loss* describes the reduction of pressure in the
valve due to a decrease in area. *PR*_{loss} is
calculated as:

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

*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 the **Pressure recovery** to
`Off`

. In this case,
*PR*_{loss} is 1.

The opening area *A*_{valve} is determined by
the opening parameterization (for `Constant`

valves only)
and the valve opening dynamics.

Linear parameterization of valve area is:

$${A}_{valve}=\widehat{p}\left({A}_{\mathrm{max}}-{A}_{leak}\right)+{A}_{leak},$$

where the normalized pressure, $$\widehat{p}$$, is

$$\widehat{p}=\frac{{p}_{control}-{p}_{set}}{{p}_{\mathrm{max}}-{p}_{set}}.$$

At the extremes of the valve pressure range, you can maintain numerical robustness
in your simulation by adjusting the block **Smoothing factor**.
With a nonzero smoothing factor, a smoothing function is applied to all calculated
valve pressures, 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:

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

In the `Tabulated data`

parameterization,
*A*_{leak} and
*A*_{max} are the first and last parameters
of the **Opening area vector**, respectively. The smoothed,
normalized pressure is also used when the smoothing factor is nonzero with linear
interpolation and nearest extrapolation.

If **Opening dynamics** are modeled, a lag is introduced to the
flow response to valve opening. *A*_{valve}
becomes the dynamic opening area, *A*_{dyn};
otherwise, *A*_{valve} is the steady-state
opening area. The instantaneous change in dynamic opening area is calculated based
on the **Opening time constant**, *τ*:

$${\dot{p}}_{dyn}=\frac{{p}_{control}-{p}_{dyn}}{\tau}.$$

By default, **Opening dynamics** are turned
`Off`

.

Steady-state dynamics are set by the same parameterization as valve opening, and
are based on the control pressure,
*p*_{control}.
A nonzero **Smoothing factor** can provide additional numerical stability when the orifice is in near-closed or near-open position.

When faults are enabled, the valve open area becomes stuck at a specified value in response to one of these triggers:

Simulation time — Faulting occurs at a specified time.

Simulation behavior — Faulting occurs in response to an external trigger. This exposes port

**Tr**.

Three fault options are available in the **Opening area when faulted** parameter:

`Closed`

— The valve freezes at its smallest value, depending on the**Opening parameterization**:When

**Opening parameterization**is set to`Linear`

, the valve area freezes at the**Leakage area**.When

**Opening parameterization**is set to`Tabulated data`

, the valve area freezes at the first element of the**Opening area vector**.

`Open`

— The valve freezes at its largest value, depending on the**Opening parameterization**:When

**Opening parameterization**is set to`Linear`

, the valve area freezes at the**Maximum opening area**.When

**Orifice parameterization**is set to`Tabulated data`

, the valve area freezes at the last element of the**Opening area vector**.

`Maintain last value`

— The valve area freezes at the valve open area when the trigger occurred.

Due to numerical smoothing at the extremes of the valve area, the
minimum area applied is larger than the **Leakage area**, and the
maximum is smaller than the **Maximum orifice area**, in proportion
to the **Smoothing factor** value.

Once triggered, the valve remains at the faulted area for the rest of the simulation.

Pressure Compensator Valve (IL) | Pressure-Compensated 3-Way Flow Control Valve (IL) | Pressure-Compensated Flow Control Valve (IL) | Check Valve (IL) | Counterbalance Valve (IL)