Calculating the mass flow rate of air based on pressure differential is a critical task in pneumatic engineering, HVAC system design, and fluid dynamics. This calculation determines how much air (in terms of mass) passes through a restriction, such as an orifice plate, nozzle, or valve, given a specific pressure drop.
The Physics of Air Flow
Air is a compressible fluid, meaning its density changes with pressure and temperature. When air flows through a restriction, potential energy (pressure) is converted into kinetic energy (velocity). The relationship between the pressure drop across the restriction and the flow rate is governed by Bernoulli's principle, adjusted for compressibility and friction losses via the Discharge Coefficient (Cd).
The Formula
For sub-sonic (unchoked) flow, this calculator uses the standard orifice flow equation derived from the ideal gas law and fluid mechanics principles:
ṁ = Cd × A × √ ( 2 × ρ × (P₁ – P₂) )
Where:
ṁ (m_dot): Mass flow rate (kg/s)
Cd: Discharge coefficient (typically ~0.61 for sharp-edged orifices)
A: Cross-sectional area of the orifice (m²)
ρ (rho): Density of air at upstream conditions (kg/m³)
P₁: Upstream absolute pressure (Pa)
P₂: Downstream absolute pressure (Pa)
Key Parameters Explained
Upstream Pressure (P₁): The absolute pressure of the air before it enters the restriction. Higher pressure generally increases density and flow potential.
Pressure Differential (ΔP): The difference between P₁ and P₂. This provides the driving force for the flow.
Temperature: Air temperature inversely affects density. Warmer air is less dense, which impacts the mass flow rate for a given volume.
Choked Flow: If the downstream pressure drops below approximately 52.8% of the upstream pressure (for air), the flow reaches the speed of sound (Mach 1). At this point, lowering the downstream pressure further does not increase the mass flow rate. This state is called "Choked Flow". This calculator detects this condition.
Common Applications
Engineers use these calculations for sizing pneumatic valves, calibrating flow meters, estimating leaks in compressed air systems, and designing engine intake manifolds.
function calculateAirFlow() {
// Clear previous results/errors
var errorBox = document.getElementById('errorBox');
var resultBox = document.getElementById('resultBox');
errorBox.style.display = 'none';
resultBox.style.display = 'none';
// 1. Get Inputs
var p1Input = document.getElementById('upstreamPressure').value;
var p2Input = document.getElementById('downstreamPressure').value;
var tempInput = document.getElementById('airTemp').value;
var diaInput = document.getElementById('orificeDiameter').value;
var cdInput = document.getElementById('dischargeCoeff').value;
// 2. Units
var pUnit = document.getElementById('pressureUnit').value;
var tUnit = document.getElementById('tempUnit').value;
var dUnit = document.getElementById('diameterUnit').value;
// 3. Validation
if (p1Input === "" || p2Input === "" || tempInput === "" || diaInput === "" || cdInput === "") {
errorBox.innerText = "Please fill in all fields.";
errorBox.style.display = 'block';
return;
}
var p1 = parseFloat(p1Input);
var p2 = parseFloat(p2Input);
var t = parseFloat(tempInput);
var d = parseFloat(diaInput);
var cd = parseFloat(cdInput);
if (isNaN(p1) || isNaN(p2) || isNaN(t) || isNaN(d) || isNaN(cd)) {
errorBox.innerText = "Please enter valid numbers.";
errorBox.style.display = 'block';
return;
}
if (d <= 0 || cd <= 0 || p1 = p1_pa) {
errorBox.innerText = "Upstream pressure (P1) must be greater than Downstream pressure (P2).";
errorBox.style.display = 'block';
return;
}
// Temperature to Kelvin
var t_k = 0;
if (tUnit === "c") {
t_k = t + 273.15;
} else if (tUnit === "f") {
t_k = (t – 32) * 5/9 + 273.15;
} else {
t_k = t;
}
// Diameter to Meters
var d_m = 0;
if (dUnit === "mm") {
d_m = d / 1000;
} else if (dUnit === "in") {
d_m = d * 0.0254;
} else if (dUnit === "cm") {
d_m = d / 100;
}
// 5. Physics Constants
var R_air = 287.05; // Specific gas constant for dry air J/(kg·K)
var k_air = 1.4; // Heat capacity ratio (adiabatic index)
// 6. Calculation Logic
// Calculate Density at Upstream (Ideal Gas Law: rho = P / RT)
var rho = p1_pa / (R_air * t_k);
// Calculate Orifice Area (A = pi * r^2)
var area = Math.PI * Math.pow((d_m / 2), 2);
// Check for Choked Flow
// Critical pressure ratio for air = (2/(k+1))^(k/(k-1)) ≈ 0.528
var criticalRatio = 0.528;
var actualRatio = p2_pa / p1_pa;
var isChoked = actualRatio 0
// Simplified Y ≈ 1 – (1/k) * (dP/P1) * (some factor).
// For this simpler tool, we stick to the widely used hydraulic approximation:
// m = Cd * A * sqrt(2 * rho * dP)
// Note: This overestimates slightly as dP increases, but is standard for basic tools.
var dP = p1_pa – p2_pa;
massFlow = cd * area * Math.sqrt(2 * rho * dP);
}
// 7. Result Conversions
var massFlowKgHr = massFlow * 3600;
// SCFM Calculation
// Standard conditions: 1 atm (101325 Pa), 20°C (293.15 K) -> Density_std approx 1.204 kg/m3
// Sometimes SCFM uses 15C or 0C. Let's use 20C/1atm (NIST normal).
var rho_std = 1.2041;
var volFlowStd_m3s = massFlow / rho_std;
var scfm = volFlowStd_m3s * 2118.88; // m3/s to cfm
// 8. Display Results
document.getElementById('resKgS').innerText = massFlow.toFixed(4) + " kg/s";
document.getElementById('resKgHr').innerText = massFlowKgHr.toFixed(2) + " kg/h";
document.getElementById('resSCFM').innerText = scfm.toFixed(2) + " SCFM";
document.getElementById('resDensity').innerText = rho.toFixed(3) + " kg/m³";
document.getElementById('resDrop').innerText = (p1_pa – p2_pa).toFixed(0) + " Pa";
document.getElementById('resCondition').innerText = flowConditionText;
resultBox.style.display = 'block';
}