Sharp-edged variable-area orifice in pneumatic systems

Pneumatic Elements

The Variable Area Pneumatic Orifice block models
the flow rate of an ideal gas through a sharp-edged variable-area
orifice. The area of the orifice is expected to be computed outside
the block and imported via the AR physical signal connection. The **Minimum
area** parameter specifies the minimum orifice area value.
If the input signal falls below this level (for example, turns negative),
the area is saturated to this value.

The flow rate through the orifice is proportional to the orifice area and the pressure differential across the orifice.

$$G={C}_{d}\xb7A\xb7{p}_{i}\sqrt{\frac{2\gamma}{\gamma -1}\xb7\frac{1}{R{T}_{i}}\left[{\left(\frac{{p}_{o}}{{p}_{i}}\right)}^{\frac{2}{\gamma}}-{\left(\frac{{p}_{o}}{{p}_{i}}\right)}^{\frac{\gamma +1}{\gamma}}\right]}$$

where

G | Mass flow rate |

C_{d} | Discharge coefficient, to account for effective loss of area due to orifice shape |

A | Orifice cross-sectional area |

p_{i}, p_{o} | Absolute pressures at the orifice inlet and outlet, respectively.
The inlet and outlet change depending on flow direction. For positive
flow (G > 0), p = _{i}p,
otherwise _{A}p = _{i}p._{B} |

γ | The ratio of specific heats at constant pressure and constant
volume, c_{p} / c_{v} |

R | Specific gas constant |

T | Absolute gas temperature |

The choked flow occurs at the critical pressure ratio defined by

$${\beta}_{cr}=\frac{{p}_{o}}{{p}_{i}}={\left(\frac{2}{\gamma +1}\right)}^{\frac{\gamma}{\gamma -1}}$$

after which the flow rate depends on the inlet pressure only and is computed with the expression

$$G={C}_{d}\xb7A\xb7{p}_{i}\sqrt{\frac{\gamma}{R{T}_{i}}\xb7{\beta}_{cr}{}^{\frac{\gamma +1}{\gamma}}}$$

The square root relationship has infinite gradient at zero flow,
which can present numerical solver difficulties. Therefore, for very
small pressure differences, defined by *p _{o} /
p_{i}* > 0.999, the flow equation
is replaced by a linear flow-pressure relationship

$$G=k{C}_{d}\xb7A\xb7{T}_{i}^{-0.5}\left({p}_{i}-{p}_{o}\right)$$

where *k* is a constant such that the flow
predicted for *p _{o} / p_{i}* is
the same as that predicted by the original flow equation for

The heat flow out of the orifice is assumed equal to the heat flow into the orifice, based on the following considerations:

The orifice is square-edged or sharp-edged, and as such is characterized by an abrupt change of the downstream area. This means that practically all the dynamic pressure is lost in the expansion.

The lost energy appears in the form of internal energy that rises the output temperature and makes it very close to the inlet temperature.

Therefore, *q _{i}* =

The block positive direction is from port A to port B. This means that the flow rate is positive if it flows from A to B.

The gas is ideal.

Specific heats at constant pressure and constant volume,

*c*and_{p}*c*, are constant._{v}The process is adiabatic, that is, there is no heat transfer with the environment.

Gravitational effects can be neglected.

The orifice adds no net heat to the flow.

**Discharge coefficient, Cd**Semi-empirical parameter for orifice capacity characterization. Its value depends on the geometrical properties of the orifice, and usually is provided in textbooks or manufacturer data sheets. The default value is

`0.82`

.**Minimum area**Specifies the minimum orifice area value. If the input signal falls below this level (for example, turns negative), the area is saturated to this value. The default value is

`1e-12`

m^2.

Use the **Variables** tab in the block dialog
box (or the **Variables** section in the block Property
Inspector) to set the priority and initial target values for the block
variables prior to simulation. For more information, see Set Priority and Initial Target for Block Variables.

The block has the following ports:

`A`

Pneumatic conserving port associated with the orifice inlet for positive flow.

`B`

Pneumatic conserving port associated with the orifice outlet for positive flow.

`AR`

Physical signal port that provides the value of the orifice area.

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