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Combined pressure-relief and pressure-reducing valve in isothermal system

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

The Pressure-Reducing 3-Way Valve (IL) is a combination of a pressure-relief and
pressure-reducing valve. It maintains pressure at the valve outlet, port
**A**, by restricting the inflow area at port
**P** and venting the flow at port **T**.

Valve operation is triggered by comparing the pressure difference between port
**T** and port **A** to a threshold, the set
pressure. When the pressure between **T** and
**A**, *P*_{control},
exceeds this set pressure, *P*_{set,reducing},
port **P** begins to close. A transition pressure range defines the
pressures the valve experiences when both valves at ports **P** and
**T** are closed. When the pressure difference between ports
**A** and **T** exceeds the pressure
transition range, port **T** opens. That is, $${P}_{set,relief}={P}_{set,reducing}+{P}_{range}+{P}_{transition}.$$ The **Pressure regulation range** is specified
for both the pressure-reducing and the pressure-relief valves. The valve parameters,
such as **Leakage Area** and **Maximum Opening
Area**, are the same for all ports.

To simulate pressure relief or pressure reduction with respect to another system element, see Pressure Compensator Valve (IL). To simulate pressure reduction between the valve outlet and atmosphere, see the Pressure-Reducing Valve (IL). To simulate pressure relief with respect to a valve or between the valve outlet and atmospheric pressure, see Pressure Relief Valve (IL).

When *P*_{control},
*P*_{A} –
*P*_{T}, exceeds the threshold pressure,
*P*_{set,reducing}, the valve at port
**P** begins to close. When the **Pressure transition
range** is exceeded, or when
*P*_{control} >
*P*_{set,relief}, the valve at port
**T** begins to open. Both valve closing and opening are
parameterized in two ways:

When

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

, connect a pressure signal to port**Ps**, set the constant**Pressure regulation range**, and set the constant**Pressure transition range**. The pressure-reducing valve begins to close when*P*_{control}is greater than*P*_{set,reducing}and below*P*_{max,reducing}. The relief valve response is triggered when*P*_{control}is greater than*P*_{set,relief}and below*P*_{max,relief}.*P*_{max,relief}is the sum of the**Pressure regulation range**and*P*_{set,relief}.When

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

, valve closing at port**P**is continuously regulated by either a linear or tabular parameterization. Similarly, relief valve opening at port**T**is parameterized linearly or by table lookup. An example of linear parameterization of the reduction valve (solid line) and relief valve (dotted line) is shown below.When the

`Tabulated data`

option is selected,*P*_{set,reducing}and*P*_{max,reducing}are the first and last parameters of the**Pressure differential vector for reducing valve**, respectively, and*P*_{set,relief}and*P*_{max,relief}are the first and last parameters of the**Pressure differential vector for relief valve**, respectively. An example of tabular parameterization of both the reducing and relieving valves are shown below.

Momentum is conserved through the valve:

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

The mass flow rate through the valves is calculated as:

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

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

where:

*C*_{d}is the**Discharge coefficient**.*A*is the instantaneous valve open area between ports**A**and**P**or**A**and**T**, as indicated by the subscript.*A*_{port}is the**Cross-sectional area at ports A, P & T**.$$\overline{\rho}$$ 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, which corresponds to either the
pressure-reducing or pressure relief component of the valve:

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

where *A* is either
*A*_{PA} or
*A*_{AT}, corresponding to the reducing or
relief component of the valve, respectively.

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

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

*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 **Pressure recovery** to
`Off`

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

The opening area *A* is determined by the opening
parameterization (for `Constant`

valves only) of the
reducing valve (**P** to **A**) or relief valve
(**A** to **T**) and the valve opening
dynamics.

Linear parameterization of the valve area for the reducing valve is

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

and for the relief valve is

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

The normalized pressure, $$\widehat{p}$$, is

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

where the set and maximum pressures are the respective reducing or relief valve settings.

At the extremes of the relief and reducing valve pressure ranges, 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 every calculated pressure, 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,PA} and
*A*_{max,PA} are the first and last
parameters of the **Opening area vector reducing valve**,
respectively, and *A*_{leak,AT} and
*A*_{max,AT} are the first and last
parameters of the **Opening area vector for relief valve**,
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 or closing area,
*A*_{dyn}; otherwise,
*A*_{valve} is the steady-state opening
area. This area is specific to the reducing and relief components of the valve,
*A*_{PA} or
*A*_{AT}, respectively. 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** is set to
`Off`

.

Friction between the valve and fluid, the hydraulic force of the fluid on the valve components, and the effect of fluid inertia are neglected.

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