Bernoulli's equation represents the principle of conservation of energy in fluid dynamics. It relates pressure, velocity, and elevation within a moving fluid. This calculator uses the Bernoulli principle, specifically applied to a Venturi effect scenario (a pipe with a constriction), to determine the volumetric flow rate based on pressure differential.
The Formula
Assuming the pipe is horizontal ($h_1 = h_2$) and the fluid is incompressible, we can derive the flow rate ($Q$) using the following formula:
Q = A₂ × √ [ 2(P₁ – P₂) / (ρ(1 – (A₂/A₁)²)) ]
Where:
Q: Volumetric Flow Rate (m³/s)
A₁: Cross-sectional area of the inlet pipe
A₂: Cross-sectional area of the throat (constriction)
P₁: Static pressure at the inlet (Pa)
P₂: Static pressure at the throat (Pa)
ρ (rho): Density of the fluid (kg/m³)
How It Works
When a fluid flows through a constricted section of a pipe (the throat), its velocity increases to satisfy the continuity equation ($A_1v_1 = A_2v_2$). According to Bernoulli's principle, this increase in velocity is accompanied by a decrease in static pressure.
By measuring the pressure difference ($P_1 – P_2$) between the wide section and the narrow section, and knowing the geometry of the pipe (diameters) and fluid properties (density), we can calculate exactly how much fluid is passing through the system.
Important Considerations
This calculator assumes ideal conditions:
Incompressible Flow: The fluid density remains constant (liquids like water).
Inviscid Flow: Friction losses are negligible.
Horizontal Flow: No change in height/elevation between the two measurement points.
Steady Flow: Fluid properties do not change with time.
In real-world engineering, a "discharge coefficient" ($C_d$) is often applied to the final result to account for energy losses due to viscosity and turbulence, typically ranging from 0.90 to 0.99 for Venturi meters.
function calculateFlow() {
// Clear previous error messages
var errorDiv = document.getElementById("error-message");
var resultDiv = document.getElementById("results");
errorDiv.style.display = "none";
errorDiv.innerHTML = "";
// Get Input Values
var d1 = parseFloat(document.getElementById("d1").value); // Inlet Diameter in mm
var d2 = parseFloat(document.getElementById("d2").value); // Throat Diameter in mm
var p1 = parseFloat(document.getElementById("p1").value); // Inlet Pressure in Pa
var p2 = parseFloat(document.getElementById("p2").value); // Throat Pressure in Pa
var rho = parseFloat(document.getElementById("rho").value); // Density in kg/m3
// Validation
if (isNaN(d1) || isNaN(d2) || isNaN(p1) || isNaN(p2) || isNaN(rho)) {
errorDiv.innerHTML = "Please enter valid numeric values for all fields.";
errorDiv.style.display = "block";
resultDiv.style.display = "none";
return;
}
if (d1 <= 0 || d2 <= 0 || rho = d1) {
errorDiv.innerHTML = "Throat diameter (D₂) must be smaller than inlet diameter (D₁) to create a Venturi effect.";
errorDiv.style.display = "block";
resultDiv.style.display = "none";
return;
}
if (p2 >= p1) {
errorDiv.innerHTML = "Throat pressure (P₂) usually must be lower than inlet pressure (P₁) in a horizontal Venturi setup.";
errorDiv.style.display = "block";
resultDiv.style.display = "none";
return;
}
// Convert units
// Convert mm to meters
var d1_m = d1 / 1000;
var d2_m = d2 / 1000;
// Calculate Areas (A = pi * r^2 = pi * (d/2)^2 = (pi * d^2)/4)
var A1 = (Math.PI * Math.pow(d1_m, 2)) / 4;
var A2 = (Math.PI * Math.pow(d2_m, 2)) / 4;
// Pressure difference
var deltaP = p1 – p2;
// Bernoulli Calculation for Q (Volumetric Flow Rate)
// Q = A2 * sqrt( (2 * deltaP) / (rho * (1 – (A2/A1)^2)) )
var areaRatioSq = Math.pow((A2 / A1), 2);
var denominator = rho * (1 – areaRatioSq);
if (denominator <= 0) {
errorDiv.innerHTML = "Calculation Error: Invalid area ratio.";
errorDiv.style.display = "block";
resultDiv.style.display = "none";
return;
}
var velocity2 = Math.sqrt((2 * deltaP) / denominator);
// Calculate Q
var Q = A2 * velocity2;
// Calculate v1 (Continuity: Q = A1 * v1)
var velocity1 = Q / A1;
// Convert Q to Liters/min for better readability
// 1 m3/s = 60000 L/min
var Q_lpm = Q * 60000;
// Display Results
document.getElementById("result-flow-rate").innerText = Q.toFixed(5) + " m³/s";
document.getElementById("result-lpm").innerText = Q_lpm.toFixed(2) + " L/min";
document.getElementById("result-v1").innerText = velocity1.toFixed(2) + " m/s";
document.getElementById("result-v2").innerText = velocity2.toFixed(2) + " m/s";
resultDiv.style.display = "block";
}