Calculate fluid velocity and volumetric flow rate using Bernoulli's principle.
Fluid Properties
Section 1 (Inlet)
Section 2 (Outlet/Throat)
Flow Velocity at Inlet (v₁)–
Flow Velocity at Outlet (v₂)–
Volumetric Flow Rate (Q)–
Flow Rate in Liters–
Pressure Difference (ΔP)–
Understanding the Bernoulli Equation
The Bernoulli equation is a cornerstone of fluid dynamics, representing the principle of conservation of energy for flowing fluids. It states that for an incompressible, non-viscous fluid flowing in a steady streamline, the sum of pressure energy, kinetic energy, and potential energy per unit volume remains constant.
P₁ + 0.5·ρ·v₁² + ρ·g·h₁ = P₂ + 0.5·ρ·v₂² + ρ·g·h₂
Where:
P = Static Pressure (Pa)
ρ (rho) = Fluid Density (kg/m³)
v = Flow Velocity (m/s)
g = Acceleration due to gravity (9.81 m/s²)
h = Elevation height (m)
How This Calculator Works
This tool utilizes the Continuity Equation alongside Bernoulli's Principle to solve for the flow rate (Q). The Continuity Equation for an incompressible fluid states that the mass flow rate must remain constant:
A₁·v₁ = A₂·v₂ = Q
By substituting v₁ in the Bernoulli equation with (A₂/A₁)·v₂, we can solve for the velocity at the second section (v₂) based on the pressure difference, height difference, and the ratio of the areas (or diameters).
Calculation Steps
Calculate Areas: The cross-sectional areas A₁ and A₂ are calculated from the input diameters.
Calculate Velocity (v₂): The derived formula determines the velocity at the constriction (or outlet).
v₂ = √ [ 2·((P₁ – P₂) + ρ·g·(h₁ – h₂)) / (ρ·(1 – (A₂/A₁)²)) ]
Calculate Flow Rate (Q): Once velocity is known, Q is calculated as Q = A₂ · v₂.
Real-World Example
Consider a horizontal Venturi meter (h₁ = h₂) measuring water flow:
Density (ρ): 1000 kg/m³
Inlet Diameter: 100 mm
Throat Diameter: 50 mm
Pressure Drop (P₁ – P₂): 50,000 Pa
Using the calculator, you would find that the velocity at the throat increases significantly to balance the drop in pressure, resulting in a specific flow rate determined by the constriction ratio.
Applications
This calculation is vital in various engineering fields, including:
Venturi Meters: Measuring flow rate in pipes by creating a constriction and measuring pressure drop.
Aerodynamics: Analyzing airflow over wings (airfoils).
Hydraulics: Designing pipe systems, nozzles, and siphons.
Carburetors: Mixing air and fuel in internal combustion engines.
function calculateBernoulli() {
// Get Inputs
var density = parseFloat(document.getElementById('density').value);
var gravity = parseFloat(document.getElementById('gravity').value);
var d1_mm = parseFloat(document.getElementById('d1').value);
var p1 = parseFloat(document.getElementById('p1').value);
var h1 = parseFloat(document.getElementById('h1').value);
var d2_mm = parseFloat(document.getElementById('d2').value);
var p2 = parseFloat(document.getElementById('p2').value);
var h2 = parseFloat(document.getElementById('h2').value);
var errorDiv = document.getElementById('errorMsg');
var resultBox = document.getElementById('resultBox');
// Reset display
errorDiv.style.display = 'none';
resultBox.classList.remove('active');
// Validation
if (isNaN(density) || isNaN(d1_mm) || isNaN(p1) || isNaN(h1) || isNaN(d2_mm) || isNaN(p2) || isNaN(h2)) {
errorDiv.innerHTML = "Please fill in all fields with valid numbers.";
errorDiv.style.display = 'block';
return;
}
if (density <= 0 || d1_mm <= 0 || d2_mm A2, but the formula works mathematically
// as long as the term under the square root is positive.
// However, standard use implies D1 > D2.
// Calculate differences
var deltaP = p1 – p2; // Pa
var deltaH = h1 – h2; // m
// Bernoulli Derived Formula for v2:
// v2 = sqrt( (2 * ( (P1-P2) + rho*g*(h1-h2) ) ) / (rho * (1 – (A2/A1)^2 )) )
var numerator = 2 * (deltaP + (density * gravity * deltaH));
var areaRatioSq = Math.pow((a2 / a1), 2);
var denominator = density * (1 – areaRatioSq);
if (Math.abs(1 – areaRatioSq) < 0.000001) {
errorDiv.innerHTML = "Diameters cannot be identical (causes division by zero in this specific derivation).";
errorDiv.style.display = 'block';
return;
}
var rootTerm = numerator / denominator;
if (rootTerm v1 = v2 * (A2/A1)
var v1 = v2 * (a2 / a1);
// Calculate Flow Rate Q = A2 * v2
var Q = a2 * v2;
// Conversions for display
var Q_liters_min = Q * 60000; // m3/s to L/min
// Display Results
document.getElementById('resV1').innerHTML = v1.toFixed(4) + " m/s";
document.getElementById('resV2').innerHTML = v2.toFixed(4) + " m/s";
document.getElementById('resQ').innerHTML = Q.toFixed(6) + " m³/s";
document.getElementById('resQLiters').innerHTML = Q_liters_min.toFixed(2) + " L/min";
document.getElementById('resDeltaP').innerHTML = deltaP.toFixed(2) + " Pa";
resultBox.classList.add('active');
}