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Sharp-edged orifice in pneumatic systems
The Constant Area Pneumatic Orifice block models the flow rate of an ideal gas through a sharp-edged orifice.
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_{A}, otherwise 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 p_{o} / p_{i} = 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, q_{i} = q_{o}, where q_{i} and q_{o} 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.
The gas is ideal.
Specific heats at constant pressure and constant volume, c_{p} and c_{v}, 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.
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.
Specify the orifice cross-sectional area. The default value is 1e-5 m^2.
Use the Variables tab 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.