orifice_method Namespace Reference

Compressed air leak flow estimation using orifice pressure and geometry. More...

Classes
struct	Input
	Input parameters for the orifice method compressed air leak calculation. More...
struct	Result
	Result of the orifice method leak flow rate and annual consumption calculation. More...

Functions
Result	calculate (const Input &input)
	Estimates compressed air leak flow rate and annual consumption using the orifice method.

Detailed Description

The orifice method estimates the air loss from a compressed air leak by treating the leak path as a sharp-edged orifice with known diameter and discharge coefficient. When the supply pressure exceeds approximately twice the atmospheric pressure, flow through the orifice is choked (sonic), and the mass flow rate depends on the upstream (supply) pressure and temperature rather than the downstream condition.

The calculation proceeds through a series of isentropic gas-dynamics steps:

1. Convert air temperature from °F to °R for thermodynamic calculations.
2. Compute air density at supply conditions and at standard conditions using the ideal gas law.
3. Apply the isentropic choking relations to find sonic density and velocity at the orifice throat.
4. Compute the mass flow rate per orifice from throat conditions.
5. Convert to volumetric flow (scfm) using standard-condition density and scale by the number of orifices.
6. Scale by annual operating time to estimate yearly air consumption.

This method provides the highest field accuracy of the leak-survey techniques that do not require an inline flow meter, provided the discharge coefficient is known or can be estimated from the orifice geometry.

See also

Orifice Method Calculator

Function Documentation

calculate()

Result orifice_method::calculate

(

const Input &

input

)

Applies isentropic choked-flow theory to estimate the mass flow rate of air escaping through a small orifice at sonic conditions. The flow is assumed choked when the supply absolute pressure exceeds twice atmospheric pressure, which is the typical case for industrial compressed air systems operating above 14.7 psia (i.e., above ~15 psig gauge).

All densities use the ideal gas law: \(\rho = P \cdot 144 / (R_{air} \cdot T_R)\) where the factor 144 converts pressure from psia (lbf/in²) to lbf/ft², \(R_{air} = 53.34\) ft·lbf/(lbm·°R) is the specific gas constant for dry air, and \(T_R\) is absolute temperature in degrees Rankine.

The isentropic choking relations use \(\gamma = 1.4\) for dry air.

Temperature Conversion:

\begin{equation}\label{eq:orifice-method-temperature} T_R = T_F + 459.67 \end{equation}

where:

\(T_R\)	Air temperature in degrees Rankine \([\unit{ \degreeRankine}]\)
\(T_F\)	Compressed air temperature at the leak point \([\unit{ \degreeFahrenheit}]\)
\(459.67\)	Fahrenheit-to-Rankine offset \([\unit{ \degreeRankine}]\)

Absolute Supply Pressure:

\begin{equation}\label{eq:orifice-method-abs-pressure} P_{abs} = P_{atm} + P_{supply} \end{equation}

where:

\(P_{abs}\)	Absolute supply pressure \([\unit{ \psia}]\)
\(P_{atm}\)	Local atmospheric (barometric) pressure \([\unit{ \psia}]\)
\(P_{supply}\)	Compressed air supply pressure (gauge) \([\unit{ \psig}]\)

Air Density at Supply Conditions:

\begin{equation}\label{eq:orifice-method-ca-density} \rho_{ca} = \frac{P_{abs} \cdot 144}{R_{air} \cdot T_R} \end{equation}

where:

\(\rho_{ca}\)	Air density at supply (compressed) conditions \([\unit{ \pound\per\cubicFoot}]\)
\(P_{abs}\)	Absolute supply pressure \([\unit{ \psia}]\)
\(144\)	Unit conversion factor (in² per ft²) \([\unit{ \squareInch\per\squareFoot}]\)
\(R_{air}\)	Specific gas constant for dry air (53.34) \([\unit{ \unitless}]\)
\(T_R\)	Air temperature in degrees Rankine \([\unit{ \degreeRankine}]\)

Air Density at Standard (Atmospheric) Conditions:

\begin{equation}\label{eq:orifice-method-standard-density} \rho_{std} = \frac{P_{atm} \cdot 144}{R_{air} \cdot T_R} \end{equation}

where:

\(\rho_{std}\)	Air density at standard atmospheric conditions \([\unit{ \pound\per\cubicFoot}]\)
\(P_{atm}\)	Local atmospheric pressure \([\unit{ \psia}]\)
\(144\)	Unit conversion factor (in² per ft²) \([\unit{ \squareInch\per\squareFoot}]\)
\(R_{air}\)	Specific gas constant for dry air (53.34) \([\unit{ \unitless}]\)
\(T_R\)	Air temperature in degrees Rankine \([\unit{ \degreeRankine}]\)

Sonic Density at Orifice Throat (Isentropic Choked Flow):

\begin{equation}\label{eq:orifice-method-sonic-density} \rho^* = \rho_{ca} \cdot \left(\frac{2}{\gamma + 1}\right)^{\frac{1}{\gamma - 1}} \end{equation}

where:

\(\rho^*\)	Air density at the isentropic sonic throat \([\unit{ \pound\per\cubicFoot}]\)
\(\rho_{ca}\)	Air density at supply conditions \([\unit{ \pound\per\cubicFoot}]\)
\(\gamma\)	Ratio of specific heats for dry air (1.4) \([\unit{ \unitless}]\)

Sonic Velocity at Orifice Throat:

\begin{equation}\label{eq:orifice-method-sonic-velocity} V^* = \sqrt{\frac{2\gamma}{\gamma + 1} \cdot R_{air} \cdot T_R \cdot g_c} \end{equation}

where:

\(V^*\)	Sonic air velocity at the orifice throat \([\unit{ \foot\per\second}]\)
\(\gamma\)	Ratio of specific heats for dry air (1.4) \([\unit{ \unitless}]\)
\(R_{air}\)	Specific gas constant for dry air (53.34) \([\unit{ \unitless}]\)
\(T_R\)	Air temperature in degrees Rankine \([\unit{ \degreeRankine}]\)
\(g_c\)	Gravitational conversion constant (32.2) \([\unit{ \unitless}]\)

Mass Flow Rate per Orifice:

\begin{equation}\label{eq:orifice-method-mass-flow} \dot{m} = \rho^* \cdot \frac{\pi d^2}{4 \cdot 144} \cdot V^* \cdot 60 \cdot C_d \end{equation}

where:

\(\dot{m}\)	Mass flow rate of leaked air through one orifice \([\unit{ \pound\per\minute}]\)
\(\rho^*\)	Air density at the sonic throat \([\unit{ \pound\per\cubicFoot}]\)
\(d\)	Orifice (leak opening) diameter \([\unit{ \inch}]\)
\(144\)	Unit conversion factor (in² per ft²) \([\unit{ \squareInch\per\squareFoot}]\)
\(V^*\)	Sonic velocity at the orifice throat \([\unit{ \foot\per\second}]\)
\(60\)	Seconds per minute conversion \([\unit{ \second\per\minute}]\)
\(C_d\)	Orifice discharge coefficient \([\unit{ \unitless}]\)

Volumetric Flow Rate per Orifice:

\begin{equation}\label{eq:orifice-method-volume-flow} Q_{scfm} = \frac{\dot{m}}{\rho_{std}} \end{equation}

where:

\(Q_{scfm}\)	Volumetric flow rate of leaked air through one orifice \([\unit{ \scfm}]\)
\(\dot{m}\)	Mass flow rate per orifice \([\unit{ \pound\per\minute}]\)
\(\rho_{std}\)	Air density at standard atmospheric conditions \([\unit{ \pound\per\cubicFoot}]\)

Total Leak Rate:

\begin{equation}\label{eq:orifice-method-total-leak} Q_{leak} = Q_{scfm} \cdot n \end{equation}

where:

\(Q_{leak}\)	Total volumetric leak flow rate through all orifices \([\unit{ \scfm}]\)
\(Q_{scfm}\)	Volumetric flow rate per orifice \([\unit{ \scfm}]\)
\(n\)	Number of identical orifice leak points \([\unit{ \unitless}]\)

Annual Consumption:

\begin{equation}\label{eq:orifice-method-annual-consumption} C_{annual} = \frac{Q_{leak} \cdot t_{op} \cdot 60}{1000} \end{equation}

where:

\(C_{annual}\)	Estimated annual compressed air loss \([\unit{ \kscf}]\)
\(Q_{leak}\)	Total leak flow rate \([\unit{ \scfm}]\)
\(t_{op}\)	Annual system operating time \([\unit{ \hour}]\)
\(60\)	Minutes per hour conversion \([\unit{ \minute\per\hour}]\)
\(1000\)	Standard cubic feet per kiloscf conversion \([\unit{ \unitless}]\)

Note

Supply pressure must be in gauge (psig); the calculation converts to absolute pressure internally. Air temperature must be in degrees Fahrenheit; conversion to Rankine is performed internally. The discharge coefficient \(C_d\) for a sharp-edged orifice is typically 0.61; for a well-rounded nozzle it approaches 1.0. For most leak-survey work, \(C_d = 1.0\) is used as a conservative upper bound.

Parameters

[in]

input

Input

Returns

Result containing all intermediate and final flow values.