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Pressure control valve for maintaining reduced pressure in fluid network portion

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

The Pressure Reducing Valve (TL) block represents a valve for maintaining a reduced pressure in portion of a fluid network. The valve stays fully open when the pressure at port B is lower than the valve set pressure. At the set pressure, the valve control member moves to reduce the flow rate through the valve. The valve opening area continues to decrease with increasing pressure until only leakage flow remains.

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**

The mass conservation equation in the valve is

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

where:

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

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

The momentum conservation equation in the valve is

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

where:

*p*_{A}and*p*_{B}are the pressures at port A and port B.$$\dot{m}$$ is the mass flow rate.

$${\dot{m}}_{cr}$$ is the critical mass flow rate.

*ρ*_{Avg}is the average liquid density.*C*_{d}is the discharge coefficient.*S*_{R}is the valve opening area.*S*is the valve inlet area.*PR*_{Loss}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

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

where:

*S*_{Leak}is the valve leakage area.*S*_{Linear}is the linear valve opening area:$${S}_{Linear}=\left(\frac{{S}_{Leak}-{S}_{Max}}{{p}_{Max}-{p}_{set}}\right)\left({p}_{control}-{p}_{set}\right)+{S}_{Max}$$

*S*_{Max}is the maximum valve opening area.*p*_{control}is the valve control pressure:$${p}_{control}={p}_{B}$$

*p*_{set}is the valve set pressure:$${p}_{set}={p}_{set,gauge}+{p}_{Atm}$$

*p*_{Min}is the minimum pressure.*p*_{Max}is the maximum pressure:$${p}_{max}={p}_{set,gauge}+{p}_{range}+{p}_{Atm}$$

*Δp*is the portion of the pressure range to smooth.*λ*_{L}and*λ*_{R}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

$${\overline{p}}_{R}=\frac{{p}_{B}-\left({p}_{max}-\Delta {p}_{smooth}\right)}{{p}_{max}-\left({p}_{max}-\Delta {p}_{smooth}\right)}$$

The critical mass flow rate is

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

The energy conservation equation in the valve is

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

where:

*ϕ*_{A}is the energy flow rate into the valve through port A.*ϕ*_{B}is the energy flow rate into the valve through port B.

A — Thermal liquid conserving port representing valve inlet A

B — Thermal liquid conserving port representing valve inlet B

**Valve set pressure (gauge)**Minimum gauge pressure at port B required to actuate the valve. A pressure rise above the set pressure causes the valve to gradually close until only leakage flow remains. The default value is

`0.1`

MPa.**Pressure regulation range**Difference between the maximum and set pressures at port B. The valve begins to close at the set pressure. It is fully closed 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.**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`

.

**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`

.