Variable Area Pneumatic Orifice

Sharp-edged variable-area orifice in pneumatic systems

Library

Pneumatic Elements

Description

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.

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

where

 G Mass flow rate Cd Discharge coefficient, to account for effective loss of area due to orifice shape A Orifice cross-sectional area pi, po Absolute pressures at the orifice inlet and outlet, respectively. The inlet and outlet change depending on flow direction. For positive flow (G > 0), pi = pA, otherwise pi = pB. γ The ratio of specific heats at constant pressure and constant volume, cp / cv 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

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

The square root relationship has infinite gradient at zero flow, which can present numerical solver difficulties. Therefore, for very small pressure differences, defined by po / pi > 0.999, the flow equation is replaced by a linear flow-pressure relationship

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

where k is a constant such that the flow predicted for po / pi is the same as that predicted by the original flow equation for po / pi = 0.999.

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, qi = qo, where qi and qo are the input and output heat flows, respectively.

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.

Basic Assumptions and Limitations

• The gas is ideal.

• Specific heats at constant pressure and constant volume, cp and cv, are constant.

• 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.

Dialog Box and Parameters

Parameters Tab

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.

Variables Tab

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.

Ports

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.