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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.

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