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MEASUR-Tools-Suite v1.0.11
The MEASUR Tools Suite is a collection of industrial efficiency calculations written in C++ and with bindings for compilation to WebAssembly.
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This calculator implements the orifice method, which treats a compressed air leak as flow through a sharp-edged circular orifice at choked (sonic) conditions. When supply pressure exceeds approximately twice atmospheric pressure — the typical condition for industrial compressed air systems — the flow through the orifice reaches the speed of sound at the throat and becomes independent of the downstream condition. Under these choked-flow conditions, the mass flow rate is fully determined by the upstream pressure, temperature, and the orifice geometry.
The method applies isentropic compressible-flow theory for dry air (γ = 1.4) using U.S. customary units. The ideal gas law gives local air densities, and the standard isentropic choking relations yield the throat density and sonic velocity. The resulting mass flow per orifice is converted to standard volumetric flow (scfm) and scaled by the number of identical leak points.
The orifice method provides the highest field accuracy of the non-inline-meter survey techniques when the discharge coefficient \(C_d\) is known. For surveys where geometry details are unavailable, \(C_d = 1.0\) serves as a conservative upper bound.
The calculation proceeds through the following steps:
Relevant formulas are documented below.
Conversion of compressed air temperature from °F to °R for thermodynamic calculations.
All gas-dynamics calculations require absolute temperature. The Rankine scale is the absolute temperature scale in U.S. customary units, defined such that 0 °R equals absolute zero. The offset from Fahrenheit to Rankine is 459.67 degrees.
\begin{equation}\label{eq:orifice-method-temperature} T_R = T_F + 459.67 \end{equation}
| \(T_R\) | Air temperature in degrees Rankine \([\unit{ \degreeRankine}]\) |
| \(T_F\) | Compressed air temperature at the leak point \([\unit{ \degreeFahrenheit}]\) |
| \(459.67\) | Fahrenheit-to-Rankine offset \([\unit{ \degreeRankine}]\) |
Conversion of gauge supply pressure to absolute pressure for density calculations.
Pressure gauges read relative to atmospheric pressure (gauge pressure). The ideal gas law requires absolute pressure. Absolute pressure is gauge pressure plus the local atmospheric (barometric) pressure. Standard atmospheric pressure at sea level is 14.7 psia; this default is appropriate for most industrial sites.
\begin{equation}\label{eq:orifice-method-abs-pressure} P_{abs} = P_{atm} + P_{supply} \end{equation}
| \(P_{abs}\) | 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 compressed air supply conditions using the ideal gas law.
The density of dry air at any pressure and temperature follows the ideal gas law. In U.S. customary units, pressure must be in lbf/ft² (multiply psia by 144 to convert from lbf/in² to lbf/ft²). The specific gas constant for dry air is \(R_{air} = 53.34\) ft·lbf/(lbm·°R). This density is used in the isentropic choking relations to find the throat (sonic) density.
\begin{equation}\label{eq:orifice-method-ca-density} \rho_{ca} = \frac{P_{abs} \cdot 144}{R_{air} \cdot T_R} \end{equation}
| \(\rho_{ca}\) | Air density at supply (compressed) conditions \([\unit{ \pound\per\cubicFoot}]\) |
| \(P_{abs}\) | Absolute supply pressure \([\unit{ \psia}]\) |
| \(144\) | Unit conversion: square inches per square foot \([\unit{ \squareInch\per\squareFoot}]\) |
| \(R_{air}\) | Specific gas constant for dry air (53.34 ft·lbf per lbm·°R) \([\unit{ \foot\lbf\per\lbm\per\degreeRankine}]\) |
| \(T_R\) | Air temperature in degrees Rankine \([\unit{ \degreeRankine}]\) |
Air density at standard atmospheric conditions, used to convert mass flow to scfm.
Standard volumetric flow (scfm) is defined at atmospheric pressure. The standard density is computed at the same temperature as the supply density but at atmospheric pressure only. Dividing mass flow (lbm/min) by standard density (lbm/ft³) yields the equivalent volumetric flow at standard conditions (scfm).
\begin{equation}\label{eq:orifice-method-standard-density} \rho_{std} = \frac{P_{atm} \cdot 144}{R_{air} \cdot T_R} \end{equation}
| \(\rho_{std}\) | Air density at standard atmospheric conditions \([\unit{ \pound\per\cubicFoot}]\) |
| \(P_{atm}\) | Local atmospheric pressure \([\unit{ \psia}]\) |
| \(144\) | Unit conversion: square inches per square foot \([\unit{ \squareInch\per\squareFoot}]\) |
| \(R_{air}\) | Specific gas constant for dry air (53.34 ft·lbf per lbm·°R) \([\unit{ \foot\lbf\per\lbm\per\degreeRankine}]\) |
| \(T_R\) | Air temperature in degrees Rankine \([\unit{ \degreeRankine}]\) |
Isentropic density at the choked (sonic) throat of the orifice.
When flow is choked, the throat conditions follow the isentropic relations for an ideal gas. The critical density ratio relates stagnation density (≈ supply density for low approach velocity) to throat density via the ratio of specific heats \(\gamma\). For dry air, \(\gamma = 1.4\).
\begin{equation}\label{eq:orifice-method-sonic-density} \rho^* = \rho_{ca} \cdot \left(\frac{2}{\gamma + 1}\right)^{\frac{1}{\gamma - 1}} \end{equation}
| \(\rho^*\) | 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 (speed-of-sound) air velocity at the choked orifice throat.
The speed of sound at the throat temperature follows from isentropic relations. The throat temperature is lower than the supply temperature by the factor \(2/(\gamma+1)\). Substituting into the sound-speed formula and using the gravitational conversion constant \(g_c = 32.2\) lbm·ft/(lbf·s²) to reconcile U.S. customary units gives the expression below. Note: the supply temperature \(T_R\) is used directly because the stagnation-to-throat temperature ratio \(2/(\gamma+1)\) is already absorbed into the leading coefficient.
\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}
| \(V^*\) | 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 ft·lbf per lbm·°R) \([\unit{ \foot\lbf\per\lbm\per\degreeRankine}]\) |
| \(T_R\) | Air temperature in degrees Rankine \([\unit{ \degreeRankine}]\) |
| \(g_c\) | Gravitational conversion constant (32.2 lbm·ft per lbf·s²) \([\unit{ \lbm\foot\per\lbf\per\second\squared}]\) |
Mass flow rate of compressed air escaping through one orifice at sonic conditions.
The mass flow rate is computed from the one-dimensional choked-flow continuity equation: mass flow = density × area × velocity. The orifice area is in ft², converted from the diameter in inches by dividing \(\pi d^2/4\) by 144. The discharge coefficient \(C_d\) accounts for the vena contracta (the narrowing of the stream below the geometric orifice area) and friction losses. The factor 60 converts the per-second velocity to per-minute for consistent units.
\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}
| \(\dot{m}\) | 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: square inches per square foot \([\unit{ \squareInch\per\squareFoot}]\) |
| \(V^*\) | 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 at standard atmospheric conditions (scfm).
Dividing the mass flow rate (lbm/min) by the standard-condition density (lbm/ft³) gives the volumetric flow rate referenced to standard conditions (scfm). This conversion allows direct comparison of leak rates at different supply pressures and temperatures, since all values are normalized to the same reference state.
\begin{equation}\label{eq:orifice-method-volume-flow} Q_{scfm} = \frac{\dot{m}}{\rho_{std}} \end{equation}
| \(Q_{scfm}\) | Volumetric flow rate per orifice at standard conditions \([\unit{ \scfm}]\) |
| \(\dot{m}\) | Mass flow rate per orifice \([\unit{ \pound\per\minute}]\) |
| \(\rho_{std}\) | Air density at standard atmospheric conditions \([\unit{ \pound\per\cubicFoot}]\) |
Total compressed air leak flow rate summed across all orifice leak points.
When multiple identical leak points are present (e.g., several identical fittings of the same worn type), the total leak rate is the per-orifice flow rate multiplied by the number of orifices. This aggregation is appropriate when all orifices share the same supply pressure, temperature, diameter, and discharge coefficient.
\begin{equation}\label{eq:orifice-method-total-leak} Q_{leak} = Q_{scfm} \cdot n \end{equation}
| \(Q_{leak}\) | Total volumetric leak flow rate for all orifices \([\unit{ \scfm}]\) |
| \(Q_{scfm}\) | Volumetric flow rate per orifice \([\unit{ \scfm}]\) |
| \(n\) | Number of identical orifice leak points \([\unit{ \unitless}]\) |
Estimated annual compressed air loss from all orifice leak points.
Annual consumption is the total instantaneous leak flow rate scaled by the number of operating hours per year, with a conversion from minutes to hours (factor of 60) and from standard cubic feet to kiloscf (factor of 1000). This gives the projected yearly air loss in kiloscf, a practical unit for annual energy assessment.
\begin{equation}\label{eq:orifice-method-annual-consumption} C_{annual} = \frac{Q_{leak} \cdot t_{op} \cdot 60}{1000} \end{equation}
| \(C_{annual}\) | Estimated annual compressed air loss \([\unit{ \kscf}]\) |
| \(Q_{leak}\) | Total volumetric 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 \([\unit{ \unitless}]\) |
Modules | |
| Orifice Method Temperature Conversion Formula | |
| Conversion of compressed air temperature from °F to °R for thermodynamic calculations. | |
| Orifice Method Absolute Supply Pressure Formula | |
| Conversion of gauge supply pressure to absolute pressure for density calculations. | |
| Orifice Method Supply-Condition Density Formula | |
| Air density at compressed air supply conditions using the ideal gas law. | |
| Orifice Method Standard-Condition Density Formula | |
| Air density at standard atmospheric conditions, used to convert mass flow to scfm. | |
| Orifice Method Sonic Throat Density Formula | |
| Isentropic density at the choked (sonic) throat of the orifice. | |
| Orifice Method Sonic Velocity Formula | |
| Sonic (speed-of-sound) air velocity at the choked orifice throat. | |
| Orifice Method Mass Flow Rate Formula | |
| Mass flow rate of compressed air escaping through one orifice at sonic conditions. | |
| Orifice Method Volumetric Flow Rate Formula | |
| Volumetric flow rate per orifice at standard atmospheric conditions (scfm). | |
| Orifice Method Total Leak Rate Formula | |
| Total compressed air leak flow rate summed across all orifice leak points. | |
| Orifice Method Annual Consumption Formula | |
| Estimated annual compressed air loss from all orifice leak points. | |
Files | |
| file | orifice_method.h |
| Declarations for the orifice method compressed air leak flow rate estimation. | |
Namespaces | |
| namespace | orifice_method |
| Compressed air leak flow estimation using orifice pressure and geometry. | |
Classes | |
| struct | orifice_method::Input |
| Input parameters for the orifice method compressed air leak calculation. More... | |
| struct | orifice_method::Result |
| Result of the orifice method leak flow rate and annual consumption calculation. More... | |
1.9.8