Orifice with variable cross-sectional area specified through physical signal input

Thermal Liquid/Orifices

The Variable Area Orifice (TL) block models the pressure drop due to an orifice with a variable cross-sectional area. Physical signal input port S provides the displacement of the orifice control member. The block computes the orifice area from the control member displacement through a smoothed linear model or from tabular data.

The orifice consists of a contraction followed by a sudden expansion
in flow area. The contraction causes the fluid to accelerate and its
pressure to drop. The expansion recovers the lost pressure though
only in part, as the flow separates from the wall, losing momentum
in the process. You can ignore this pressure recovery by setting the **Pressure
recovery** parameter to `Off`

.

The orifice opening behavior depends on the opening orientation
specified in the block dialog box. If the **Opening orientation** parameter
is set to `Positive`

, a positive control
member displacement increases the orifice area. If the **Opening
orientation** parameter is set to `Negative`

,
a positive displacement decreases the orifice area.

The orifice is adiabatic. It does not exchange heat with the
environment. This block provides a building block for the *n*-way
thermal liquid directional valves.

The orifice area calculation depends on the parameterization
selected in the block dialog box. If the **Orifice area parameterization** setting
is `Tabulated data — area vs. opening`

,
the block obtains the orifice area from tabular data specified in
terms of the control member opening, or position:

$${S}_{R}={S}_{R}(l),$$

*S*_{R}is the orifice cross-sectional area.*l*is the control member position.

The control member position is a function of the control member displacement provided through physical signal port S:

$$l={l}_{0}+\epsilon {d}_{S},$$

*l*_{0}is the control member offset from the zero position.*ε*is an integer denoting the orifice orientation—`1`

if positive and`-1`

if negative.*d*_{S}is the control member displacement specified through physical signal port S.

If the **Orifice area parameterization** setting
is `Linear area-opening relationship`

, the
block computes the orifice area directly from orifice geometry parameters.
The orifice area calculation uses a linear function of the control
member position as a starting point:

$${S}_{Linear}=\left(\frac{{S}_{Max}}{{l}_{Max}}\right)l,$$

*S*_{Linear}is the orifice area in the linear orifice-opening range.*S*_{Max}is the maximum orifice area.*l*_{Max}is the maximum control member displacement.

This linear expression introduces undesirable discontinuities at the fully open and fully closed positions. The block eliminates these discontinuities through smoothing expressions given by the piecewise function:

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

*S*_{Leak}is the leakage area between the orifice inlets.*l*_{Min}is the minimum control member position:$${l}_{Min}={l}_{Max}\left(\frac{{S}_{Leak}}{{S}_{Max}}\right)$$

*Δl*_{smooth}is the portion of the linear orifice area function*S*_{R}(*l*), to smooth:$$\Delta {l}_{smooth}={f}_{smooth}\frac{{l}_{max}-{l}_{Min}}{2},$$

*f*_{smooth}is a smoothing factor from 0 through 1. This value is the fraction of the*S*_{R}function to smooth.*λ*_{L}and*λ*_{R}are the cubic polynomial smoothing functionsand$${\lambda}_{L}=3{\overline{\Delta L}}_{L}^{2}-2{\overline{\Delta l}}_{L}^{3},$$

where$${\lambda}_{R}=3{\overline{\Delta L}}_{R}^{2}-2{\overline{\Delta l}}_{R}^{3},$$

and$${\overline{\Delta l}}_{L}=\frac{l-{l}_{Min}}{\Delta {l}_{smooth}}$$

$${\overline{\Delta l}}_{R}=\frac{l-\left({l}_{Max}-\Delta {l}_{smooth}\right)}{\Delta {l}_{smooth}}.$$

A smoothing factor of 0 corresponds to no smoothing anywhere
in the *S*_{R} range. The orifice
area reduces to the linear expression given by *S*_{Linear} in
the open interval ]*l*_{Min}, *l*_{Max}[.
A value of 1 corresponds to full smoothing in the entire *S*_{Linear} range.
The orifice area becomes the piecewise nonlinear function given by *S*_{R}.

**Orifice Area Smoothing**

The mass balance equation in the orifice is

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

$${\dot{m}}_{A}$$ and $${\dot{m}}_{B}$$ are the mass flow rates into the orifice through ports A and B.

The momentum balance equation in the orifice is

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

*p*_{A}and*p*_{B}are the thermal liquid pressures at ports A and B.$$\dot{m}$$ is the average mass flow rate through the orifice.

$${\dot{m}}_{cr}$$ is the maximum flow rate in the laminar flow regime:

$${\dot{m}}_{cr}={\mathrm{Re}}_{cr}\sqrt{\frac{\pi}{4}A\mu}$$

*ρ*is the mass density in the orifice.*C*_{d}is the discharge coefficient.*S*is the cross-sectional area of the adjacent pipe segments.*μ*is the average dynamic viscosity in the orifice.PR

_{Loss}is the pressure loss ratio [1]:$$P{R}_{Loss}=\{\begin{array}{ll}\frac{\sqrt{1-{\left({S}_{R}/S\right)}^{2}\left(1-{C}_{d}^{2}\right)}-{C}_{d}\left({S}_{R}/S\right)}{\sqrt{1-{\left({S}_{R}/S\right)}^{2}\left(1-{C}_{d}^{2}\right)}+{C}_{d}\left({S}_{R}/S\right)},\hfill & \text{ifPressurerecoveryis'On'}\hfill \\ 1,\hfill & \text{ifPressurerecoveryis'Off'}\hfill \end{array}$$

The energy conservation equation in the orifice gives

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

*ϕ*_{A}and*ϕ*_{B}are the energy flow rates into the orifice through ports A and B.

**Orifice area parameterization**Orifice area calculation approach. Select

`Tabulated data — Area vs. opening`

provide the orifice area explicitly through tabular data. Select`Linear area-opening relationship`

to compute the orifice area from the control member displacement through a smoothed linear expression. The default parameterization is`Linear area-opening relationship`

.**Maximum control displacement**Control member displacement in the fully open state. The block saturates the input physical signal S at this value. This parameter appears only when

**Orifice area parameterization**is set to`Linear area-opening relationship`

. The default value is`0.005`

m.**Maximum orifice area**Orifice area in the fully open state. This parameter appears only when

**Orifice area parameterization**is set to`Linear area-opening relationship`

. The default value is`1e-4`

m^2.**Leakage area**Minimum orifice area associated with fluid leakage between port A and port B. The default value is

`1e-10`

. This parameter appears only when**Orifice area parameterization**is set to`Linear area-opening relationship`

.**Smoothing factor**Fraction of the displacement-area curve to smooth at the minimum and maximum displacement points. Smoothing eliminates discontinuities that reduce accuracy and slow down simulation speed. The smoothing factor must be between

`0`

and`1`

.Enter a value of

`0`

to apply no smoothing to the displacement-area curve. Enter a value of`1`

to smooth the entire length of the curve. The default value is`0.01`

. This parameter appears only when**Orifice area parameterization**is set to`Linear area-opening relationship`

.**Control displacement vector**Vector with the control displacements for the displacement-area lookup table. These are the displacements at which you specify the orifice areas in the

**Orifice area vector**parameter. This parameter appears only when**Orifice area parameterization**is set to`Tabulated data — Area vs. opening`

. The default vector is a five-element array ranging from`1.0e-9`

to`0.00034356`

.**Orifice area vector**Vector with the orifice areas for the displacement-area lookup table. The areas must correspond to the control member displacements specified in the

**Control displacement vector**parameter. This parameter appears only when**Orifice area parameterization**is set to`Tabulated data — Area vs. opening`

. The default vector is a five-element array ranging from`-0.002`

to`0.015`

.**Opening orientation**Orientation of the orifice control member. If the opening orientation is

`Positive`

, a positive control member displacement opens the orifice. If the opening orientation is negative, a positive control member displacement closes the orifice. The default setting is`Positive`

.**Control member offset**Control member offset from the fully closed position at a zero displacement. The default value is

`0`

m.**Cross-sectional area at ports A and B**Flow area at ports A and B. This area is the same for both ports. The default value is

`0.01`

m^2.**Characteristic longitudinal length**Distance between the orifice inlets. This parameter provides a measure of the orifice longitudinal scale. The default value is

`0.1`

m.**Pressure recovery**Pressure recovery calculation approach. Select

`On`

to account for the pressure recovery in the orifice expansion zone. The pressure recovery depends on the orifice and pipe diameters. The default setting is`On`

.**Discharge coefficient**Ratio of the actual mass flow rate through the orifice to its ideal, or theoretical, value. The discharge coefficient accounts for the effects of orifice geometry on the mass flow rate. The value must be between

`0`

and`1`

. The default value is`0.7`

.**Critical Reynolds number**Reynolds number at which flow transitions between laminar and turbulent regimes. The flow is laminar below this number and turbulent above it. 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`

.

A — Thermal liquid port representing inlet A

B — Thermal liquid port representing inlet B

S — Physical signal input port for the control member position

[1] *Measurement of fluid flow by means of pressure
differential devices inserted in circular cross-section conduits running
full — Part 2: Orifice plates (ISO 5167–2:2003)*.
2003.

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