# Variable Area Orifice (TL)

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

## Library

Thermal Liquid/Orifices

## Description

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 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.

### Orifice Area

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:

`${S}_{R}={S}_{R}\left(l\right),$`
where:

• SR 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:

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

• l0 is the control member offset from the zero position.

• ε is an integer denoting the orifice orientation—`1` if positive and `-1` if negative.

• dS 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:

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

• SLinear is the orifice area in the linear orifice-opening range.

• SMax is the maximum orifice area.

• lMax 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:

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

• SLeak is the leakage area between the orifice inlets.

• lMin is the minimum control member position:

`${l}_{Min}={l}_{Max}\left(\frac{{S}_{Leak}}{{S}_{Max}}\right)$`

• Δlsmooth is the portion of the linear orifice area function SR(l), to smooth:

`$\Delta {l}_{smooth}={f}_{smooth}\frac{{l}_{max}-{l}_{Min}}{2},$`

• fsmooth is a smoothing factor from 0 through 1. This value is the fraction of the SR function to smooth.

• λL and λR are the cubic polynomial smoothing functions

`${\lambda }_{L}=3{\overline{\Delta L}}_{L}^{2}-2{\overline{\Delta l}}_{L}^{3},$`
and
`${\lambda }_{R}=3{\overline{\Delta L}}_{R}^{2}-2{\overline{\Delta l}}_{R}^{3},$`
where
`${\overline{\Delta l}}_{L}=\frac{l-{l}_{Min}}{\Delta {l}_{smooth}}$`
and
`${\overline{\Delta l}}_{R}=\frac{l-\left({l}_{Max}-\Delta {l}_{smooth}\right)}{\Delta {l}_{smooth}}.$`

A smoothing factor of 0 corresponds to no smoothing anywhere in the SR range. The orifice area reduces to the linear expression given by SLinear in the open interval ]lMin, lMax[. A value of 1 corresponds to full smoothing in the entire SLinear range. The orifice area becomes the piecewise nonlinear function given by SR.

Orifice Area Smoothing

### Mass Balance

The mass balance equation in the orifice is

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

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

### Momentum Balance

The momentum balance equation in the orifice is

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

• pA and pB are the thermal liquid pressures at ports A and B.

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

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

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

• ρ is the mass density in the orifice.

• Cd is the discharge coefficient.

• S is the cross-sectional area of the adjacent pipe segments.

• μ is the average dynamic viscosity in the orifice.

• PRLoss is the pressure loss ratio [1]:

### Energy Balance

The energy conservation equation in the orifice gives

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

• ϕA and ϕB are the energy flow rates into the orifice through ports A and B.

## Parameters

### Parameters Tab

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 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`.

### 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 port representing inlet A

• B — Thermal liquid port representing inlet B

• S — Physical signal input port for the control member position

## References

[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.