function calculateBernoulliFlow() {
// Clear previous results/errors
document.getElementById("resultSection").style.display = "none";
document.getElementById("errorMsg").style.display = "none";
document.getElementById("errorMsg").innerHTML = "";
// Get Input Values
var rho = parseFloat(document.getElementById("fluidDensity").value);
var d1_mm = parseFloat(document.getElementById("diameter1").value);
var p1 = parseFloat(document.getElementById("pressure1").value);
var h1 = parseFloat(document.getElementById("height1").value);
var d2_mm = parseFloat(document.getElementById("diameter2").value);
var p2 = parseFloat(document.getElementById("pressure2").value);
var h2 = parseFloat(document.getElementById("height2").value);
// Validation
if (isNaN(rho) || isNaN(d1_mm) || isNaN(p1) || isNaN(h1) || isNaN(d2_mm) || isNaN(p2) || isNaN(h2)) {
document.getElementById("errorMsg").innerHTML = "Please fill in all fields with valid numbers.";
document.getElementById("errorMsg").style.display = "block";
return;
}
if (rho <= 0 || d1_mm <= 0 || d2_mm = A1) {
// While physically possible in expanding pipes, this calculator uses the Venturi format
// where generally section 2 is the throat (smaller). However, Bernoulli works either way.
// But the formula typically assumes we are solving for flow based on a constriction.
// We will proceed, but note that the term (1 – (A2/A1)^2) will be negative if A2 > A1,
// requiring the pressure difference to flip.
}
// Bernoulli with Continuity Equation Substitution
// P1 + 0.5*rho*v1^2 + rho*g*h1 = P2 + 0.5*rho*v2^2 + rho*g*h2
// v1 = v2 * (A2/A1)
// Rearranging for v2:
// v2 = sqrt( 2 * ( (P1 – P2) + rho*g*(h1 – h2) ) / ( rho * (1 – (A2/A1)^2) ) )
var pressureDiff = p1 – p2;
var heightDiffTerm = rho * g * (h1 – h2);
var numerator = 2 * (pressureDiff + heightDiffTerm);
var areaRatioSq = Math.pow((A2 / A1), 2);
var denominator = rho * (1 – areaRatioSq);
// Check for physical impossibility (complex numbers)
var termUnderRoot = numerator / denominator;
if (termUnderRoot < 0) {
document.getElementById("errorMsg").innerHTML = "Calculation Error: The input values result in a physically impossible scenario (negative square root).Ensure Pressure 1 is sufficient to overcome the height difference and velocity changes, or check your diameters.";
document.getElementById("errorMsg").style.display = "block";
return;
}
if (denominator === 0) {
document.getElementById("errorMsg").innerHTML = "Diameters cannot be identical while solving this specific rearrangement. D1 and D2 must differ.";
document.getElementById("errorMsg").style.display = "block";
return;
}
var v2 = Math.sqrt(termUnderRoot);
var v1 = v2 * (A2 / A1);
var Q = A2 * v2; // Flow rate in m³/s
// Formatting Results
var Q_liters_min = Q * 60000; // Convert m³/s to L/min
document.getElementById("resFlowRate").innerHTML = Q.toExponential(4) + " m³/s";
document.getElementById("resFlowLpm").innerHTML = Q_liters_min.toFixed(2) + " L/min";
document.getElementById("resVel1").innerHTML = v1.toFixed(2) + " m/s";
document.getElementById("resVel2").innerHTML = v2.toFixed(2) + " m/s";
document.getElementById("resAreaRatio").innerHTML = (A2/A1).toFixed(4);
document.getElementById("resultSection").style.display = "block";
}
Understanding the Bernoulli Flow Rate Calculator
The Bernoulli Flow Rate Calculator is an essential engineering tool designed to determine the fluid flow rate through a pipe with varying cross-sections, such as a Venturi meter, nozzle, or orifice plate. By applying Bernoulli's principle alongside the Continuity Equation, this tool calculates the velocity and volumetric flow rate based on pressure and height differences between two points.
How It Works
Bernoulli's principle states that for a non-viscous, incompressible fluid in steady flow, the sum of pressure energy, kinetic energy, and potential energy per unit volume remains constant at any point along a streamline. This calculator combines this principle with the mass continuity equation ($A_1v_1 = A_2v_2$) to solve for the unknown flow velocity.
P₁ / P₂: Static Pressure at point 1 (inlet) and point 2 (outlet/throat), measured in Pascals (Pa).
D₁ / D₂: Pipe diameter at the respective sections. The calculator uses these to determine the Cross-Sectional Area (A).
h₁ / h₂: Elevation height relative to a reference plane. This accounts for the hydrostatic pressure change due to gravity.
ρ (Rho): Fluid density (e.g., Water ≈ 1000 kg/m³, Air ≈ 1.225 kg/m³).
Q: Volumetric Flow Rate, typically expressed in cubic meters per second (m³/s) or liters per minute (L/min).
Applications in Engineering
This calculation is widely used in fluid mechanics for:
Venturi Meters: Measuring flow rate by creating a constriction in the pipe and measuring the pressure drop.
Orifice Plates: Calculating flow based on the pressure differential across a plate with a hole.
Pitot Tubes: Determining fluid velocity in aerodynamics and hydrodynamics.
Plumbing Systems: Analyzing pressure changes in reducing pipe sections.
Important Assumptions
To maintain accuracy, this calculator assumes the fluid is incompressible (density is constant) and inviscid (friction losses are negligible). In real-world piping scenarios, engineers often apply a "Discharge Coefficient" ($C_d$) to the final flow rate to account for energy losses due to friction and turbulence.