# Pressure Relief Valve (TL)

Pressure control valve for maintaining preset pressure in fluid network

## Library

Thermal Liquid/Valves/Pressure Control Valves

## Description

The Pressure Relief Valve (TL) block represents a valve for maintaining a preset pressure in a fluid network. The valve remains closed until the pressure at port A reaches the valve set pressure. A pressure rise above the set pressure causes the valve to gradually open, allowing the fluid network to relieve excess pressure.

A smoothing function allows the valve opening area to change smoothly between the fully closed and fully open positions. The smoothing function does this by removing the abrupt opening area changes at the zero and maximum ball positions. The figure shows the effect of smoothing on the valve opening area curve.

Opening-Area Curve Smoothing

### Mass Balance

The mass conservation equation in the valve is

`${\stackrel{˙}{m}}_{A}+{\stackrel{˙}{m}}_{B}=0,$`
where:

• ${\stackrel{˙}{m}}_{A}$ is the mass flow rate into the valve through port A.

• ${\stackrel{˙}{m}}_{B}$ is the mass flow rate into the valve through port B.

### Momentum Balance

The momentum conservation equation in the valve is

`${p}_{A}-{p}_{B}=\frac{\stackrel{˙}{m}\sqrt{{\stackrel{˙}{m}}^{2}+{\stackrel{˙}{m}}_{cr}^{2}}}{2{\rho }_{Avg}{C}_{d}^{2}{S}^{2}}\left[1-{\left(\frac{{S}_{R}}{S}\right)}^{2}\right]P{R}_{Loss},$`
where:

• pA and pB are the pressures at port A and port B.

• $\stackrel{˙}{m}$ is the mass flow rate.

• ${\stackrel{˙}{m}}_{cr}$ is the critical mass flow rate.

• ρAvg is the average liquid density.

• Cd is the discharge coefficient.

• SR is the valve opening area.

• S is the valve inlet area.

• PRLoss is the pressure ratio:

`$P{R}_{Loss}=\frac{\sqrt{1-{\left({S}_{R}/S\right)}^{2}\left(1-{C}_{d}^{2}\right)}-{C}_{d}\left({S}_{R}/S\right)}{\sqrt{1-{\left({S}_{R}/S\right)}^{2}\left(1-{C}_{d}^{2}\right)}+{C}_{d}\left({S}_{R}/S\right)}.$`

The valve opening area is computed as

`${S}_{R}=\left\{\begin{array}{ll}{S}_{Leak},\hfill & {p}_{control}\le {p}_{set}\hfill \\ {S}_{Leak}\left(1-{\lambda }_{L}\right)+{S}_{Linear}{\lambda }_{L},\hfill & {p}_{control}<{p}_{Min}+\Delta {p}_{smooth}\hfill \\ {S}_{Linear},\hfill & {p}_{control}\le {p}_{Max}-\Delta {p}_{smooth}\hfill \\ {S}_{Linear}\left(1-{\lambda }_{R}\right)+{S}_{Max}{\lambda }_{R},\hfill & {p}_{control}<{p}_{Max}\hfill \\ {S}_{Max},\hfill & {p}_{control}\ge {p}_{Max}\hfill \end{array},$`
where:

• SLeak is the valve leakage area.

• SLinear is the linear valve opening area:

`${S}_{Linear}=\left(\frac{{S}_{Max}-{S}_{Leak}}{{p}_{Max}-{p}_{set}}\right)\left({p}_{control}-{p}_{set}\right)+{S}_{Leak}$`

• SMax is the maximum valve opening area.

• pcontrol is the valve control pressure:

• pset is the valve set pressure:

• pMin is the minimum pressure.

• pMax is the maximum pressure:

• Δp is the portion of the pressure range to smooth.

• λL and λR are the cubic polynomial smoothing functions

`${\lambda }_{L}=3{\overline{p}}_{L}^{2}-2{\overline{p}}_{L}^{3}$`
and
`${\lambda }_{R}=3{\overline{p}}_{R}^{2}-2{\overline{p}}_{R}^{3}$`
where:
`${\overline{p}}_{L}=\frac{{p}_{control}-{p}_{set}}{\left({p}_{set}+\Delta {p}_{smooth}\right)-{p}_{set}}$`
and
`${\overline{p}}_{R}=\frac{{p}_{control}-\left({p}_{max}-\Delta {p}_{smooth}\right)}{{p}_{max}-\left({p}_{max}-\Delta {p}_{smooth}\right)}$`

The critical mass flow rate is

`${\stackrel{˙}{m}}_{cr}={\mathrm{Re}}_{cr}{\mu }_{Avg}\sqrt{\frac{\pi }{4}{S}_{R}}.$`

### Energy Balance

The energy conservation equation in the valve is

`${\varphi }_{A}+{\varphi }_{B}=0,$`
where:

• ϕA is the energy flow rate into the valve through port A.

• ϕB is the energy flow rate into the valve through port B.

## Parameters

### Parameters Tab

Pressure control specification

Specification method for the valve set pressure parameter. Options include `Pressure at port A` and `Pressure differential`.

Valve set pressure (gauge)

Minimum gauge pressure at port A required to open the valve. A pressure rise above the set pressure causes the valve to gradually open until it reaches the fully open state. This parameter is active only when the Pressure control specification parameter is set to `Pressure at port A`. The default value is `0.1` MPa.

Valve set pressure differential

Minimum pressure differential between ports A and B required to open the valve. A pressure differential rise above this value causes the valve to gradually open until it reaches the fully open state. This parameter is active only when the Pressure control specification parameter is set to ```Pressure differential```. The default value is `0` MPa.

Pressure regulation range

Difference between the maximum and set pressures at port A. The valve begins to open at the set pressure. It is fully open at the maximum pressure. The default value is `0.01` MPa.

Maximum opening area

Flow cross-sectional area in the fully open state. This state corresponds to pressures lower than the set pressure. The default value is `1e-4` MPa.

Leakage area

Aggregate area of all fluid leaks in the valve. The leakage area helps to prevent numerical issues due to isolated fluid network sections. For numerical robustness, set this parameter to a nonzero value. The default value is `1e-12`.

Smoothing factor

Fraction of the opening-area curve, expressed as a fraction from 0 to 1, to smooth. The block replaces the discontinuities in the opening area curve with smooth transitions that span the specified fraction of the curve. The default value is `0.01`.

A smoothing factor of 0 corresponds to a linear function that is discontinuous at the set and maximum-area pressures. A smoothing factor of 1 corresponds to a nonlinear function that changes continuously throughout the entire function domain.

A smoothing factor between 0 and 1 corresponds to a continuous piece-wise function with smooth nonlinear transitions at the set and maximum-area pressures and linear segments elsewhere.

Opening-Area Curve Smoothing

Cross-sectional area at ports A and B

Flow area at the valve inlets. The inlets are assumed equal in size. The default value is `0.01` m^2.

Characteristic longitudinal length

Approximate length of the valve. This parameter provides a measure of the longitudinal scale of the valve. The default value is `0.1` m^2.

Discharge coefficient

Semi-empirical parameter commonly used as a measure of valve performance. The discharge coefficient is defined as the ratio of the actual mass flow rate through the valve to its theoretical value.

The block uses this parameter to account for the effects of valve geometry on mass flow rates. Textbooks and valve data sheets are common sources of discharge coefficient values. By definition, all values must be greater than 0 and smaller than 1. The default value is `0.7`.

Critical Reynolds number

Reynolds number corresponding to the transition between laminar and turbulent flow regimes. The flow through the valve is assumed laminar below this value and turbulent above it. The appropriate values to use depend on the specific valve geometry. The default value is `12`.

### Variables Tab

Mass flow rate into port A

Mass flow rate into the component through port A at the start of simulation. The default value is ```1 kg/s```.

## Ports

• A — Thermal liquid conserving port representing valve inlet A

• B — Thermal liquid conserving port representing valve inlet B

## See Also

#### Introduced in R2016a

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