Calculate Mass Flow Rate under Choked Flow Conditions
Absolute Pressure in Bar (bar a)
Temperature in Celsius (°C)
Diameter in millimeters (mm)
Typically 0.60 to 0.98
Custom Gas
Air
Nitrogen (N₂)
Oxygen (O₂)
Methane (CH₄)
Helium (He)
Hydrogen (H₂)
Carbon Dioxide (CO₂)
Auto-fills MW and k
Molar Mass in g/mol
Adiabatic Index (Cp/Cv)
Dimensionless (1.0 for Ideal Gas)
Please check your inputs. Pressure and Diameter must be positive.
Calculation Results
Mass Flow Rate
0.000
kg/sec
Mass Flow Rate
0.000
kg/hour
Orifice Area
0.000
mm²
Sonic Velocity
0.000
m/s
Note: Results assume critical (choked) flow conditions are met (P_downstream < P_critical).
Understanding Critical Gas Flow (Choked Flow)
In fluid dynamics, Critical Flow, often referred to as Choked Flow, occurs when a gas flowing through a restriction (such as an orifice plate, nozzle, or valve) reaches the speed of sound (Mach 1) at the narrowest point (the throat). Once this condition is met, lowering the downstream pressure further will not increase the mass flow rate.
This phenomenon is crucial in engineering applications ranging from safety relief valve sizing and pipeline leak estimation to rocket propulsion and flow metering.
The Critical Flow Equation
This calculator uses the standard isentropic flow equations derived from the Ideal Gas Law. The mass flow rate ($\dot{m}$) under choked conditions is calculated as follows:
ṁ = C · A · P₁ · √ [ (k · M) / (Z · R · T₁) ] · [ 2 / (k + 1) ] ^ [ (k + 1) / (2(k – 1)) ]
Where:
ṁ: Mass flow rate (kg/s)
C: Discharge coefficient (dimensionless, typically 0.6 – 0.98)
A: Cross-sectional area of the orifice (m²)
P₁: Upstream absolute pressure (Pa)
T₁: Upstream absolute temperature (K)
k: Specific heat ratio (Cp/Cv, dimensionless)
M: Molecular weight of the gas (kg/mol)
Z: Compressibility factor (dimensionless)
R: Universal Gas Constant (8.314 J/mol·K)
Key Parameters Explained
Discharge Coefficient ($C_d$): This accounts for real-world inefficiencies. An ideal nozzle has a $C_d$ of 1.0. Sharp-edged orifices typically have a $C_d$ around 0.60 to 0.62, while well-rounded nozzles may approach 0.98.
Specific Heat Ratio ($k$ or $\gamma$): A thermodynamic property of the gas. For air, it is approximately 1.40. For monatomic gases like Helium, it is ~1.66, and for more complex molecules like Methane, it is ~1.30.
Compressibility Factor ($Z$): Corrects for deviations from the Ideal Gas Law at high pressures. For most low-pressure applications, Z = 1 is sufficient.
When does Flow become Critical?
Flow becomes choked when the ratio of downstream pressure ($P_2$) to upstream pressure ($P_1$) falls below the critical pressure ratio:
P₂/P₁ ≤ [ 2 / (k + 1) ] ^ [ k / (k – 1) ]
For air ($k=1.4$), this critical ratio is approximately 0.528. If the downstream pressure is less than 52.8% of the upstream pressure, the flow is choked, and this calculator provides the maximum possible flow rate.
Common Applications
Safety Valves: Ensuring relief valves can vent gas fast enough to prevent tank rupture.
Leak Rate Estimation: Calculating how much gas escapes from a puncture in a high-pressure line.
Flow Limiters: Designing orifices to limit the maximum flow to a safe level regardless of downstream pipe rupture.
// Gas Properties Database
var gasData = {
'air': { mw: 28.97, k: 1.40 },
'nitrogen': { mw: 28.01, k: 1.40 },
'oxygen': { mw: 32.00, k: 1.40 },
'methane': { mw: 16.04, k: 1.32 },
'helium': { mw: 4.00, k: 1.66 },
'hydrogen': { mw: 2.02, k: 1.41 },
'co2': { mw: 44.01, k: 1.30 }
};
function updateGasProperties() {
var selection = document.getElementById('gasSelection').value;
if (selection !== 'custom' && gasData[selection]) {
document.getElementById('molecularWeight').value = gasData[selection].mw;
document.getElementById('heatCapacityRatio').value = gasData[selection].k;
}
}
function calculateCriticalFlow() {
// Clear previous errors
document.getElementById('errorMsg').style.display = 'none';
document.getElementById('resultSection').style.display = 'none';
// Get Input Values
var p1_bar = parseFloat(document.getElementById('upstreamPressure').value);
var t1_c = parseFloat(document.getElementById('upstreamTemp').value);
var d_mm = parseFloat(document.getElementById('orificeDiameter').value);
var cd = parseFloat(document.getElementById('dischargeCoeff').value);
var mw_gmol = parseFloat(document.getElementById('molecularWeight').value);
var k = parseFloat(document.getElementById('heatCapacityRatio').value);
var z = parseFloat(document.getElementById('compressibilityZ').value);
// Validation
if (isNaN(p1_bar) || p1_bar <= 0 || isNaN(d_mm) || d_mm <= 0 || isNaN(t1_c)) {
document.getElementById('errorMsg').style.display = 'block';
return;
}
// Unit Conversions
var p1_pa = p1_bar * 100000; // Bar to Pascals
var t1_k = t1_c + 273.15; // Celsius to Kelvin
var d_m = d_mm / 1000; // mm to meters
var area_m2 = Math.PI * Math.pow((d_m / 2), 2); // Area in m^2
var mw_kgmol = mw_gmol / 1000; // g/mol to kg/mol
var R = 8.314462618; // Universal Gas Constant J/(mol·K)
// Calculation Logic for Choked Mass Flow Rate
// Formula: m_dot = Cd * A * P1 * SQRT( (k * M) / (Z * R * T1) ) * (2 / (k+1))^((k+1)/(2*(k-1)))
// 1. Calculate expansion factor term: (2 / (k+1)) ^ ((k+1)/(2(k-1)))
var exponent = (k + 1) / (2 * (k – 1));
var expansionTerm = Math.pow((2 / (k + 1)), exponent);
// 2. Calculate gas properties root term: SQRT( (k * M) / (Z * R * T1) )
var propertiesTerm = Math.sqrt((k * mw_kgmol) / (z * R * t1_k));
// 3. Total Mass Flow (kg/s)
var massFlowKgS = cd * area_m2 * p1_pa * propertiesTerm * expansionTerm;
// 4. Calculate Velocity at throat (Sonic Velocity)
// c = SQRT(k * Z * R * T_throat / M)
// T_throat = T1 * (2 / (k+1))
var t_throat = t1_k * (2 / (k + 1));
var velocity_throat = Math.sqrt((k * z * R * t_throat) / mw_kgmol);
// Display Results
document.getElementById('resMassFlowKgS').innerText = massFlowKgS.toFixed(4);
document.getElementById('resMassFlowKgH').innerText = (massFlowKgS * 3600).toFixed(2);
// Convert Area back to mm^2 for display
document.getElementById('resArea').innerText = (area_m2 * 1000000).toFixed(2);
document.getElementById('resVelocity').innerText = velocity_throat.toFixed(1);
document.getElementById('resultSection').style.display = 'block';
}