The Morgae Flow Rate Calculator is a specialized tool designed to estimate the volumetric flow rate of a fluid through a pipe. It's based on fundamental principles of fluid dynamics, particularly the Darcy-Weisbach equation, which accounts for pressure drop due to friction within the pipe. This calculator helps engineers, technicians, and researchers predict how much fluid will move under specific conditions.
How it Works: The Math Behind the Calculation
The calculation involves several steps, converging on an iterative solution because the friction factor (which influences flow rate) itself depends on the flow regime (laminar or turbulent), which in turn depends on the flow rate.
The core of the calculation relies on the Darcy-Weisbach equation, which relates the pressure drop (ΔP) to the flow rate (Q) and other pipe and fluid properties:
ΔP = f * (L/D) * (ρ * V²/2)
Where:
ΔP is the pressure drop (P1 – P2) in Pascals (Pa).
f is the Darcy friction factor (dimensionless).
L is the pipe length in meters (m).
D is the pipe inner diameter in meters (m).
ρ (rho) is the fluid density in kilograms per cubic meter (kg/m³).
V is the average fluid velocity in meters per second (m/s).
The volumetric flow rate (Q) is related to velocity (V) and the cross-sectional area of the pipe (A) by:
Q = V * A
And the area (A) is calculated as:
A = π * (D/2)²
The challenge lies in determining the friction factor (f). It depends on the Reynolds number (Re) and the relative roughness (ε/D) of the pipe.
Reynolds Number (Re): Re = (ρ * V * D) / μ. This indicates the flow regime. Low Re means laminar flow; high Re means turbulent flow.
Friction Factor (f):
For laminar flow (Re < 2300), f = 64 / Re.
For turbulent flow (Re > 4000), the Colebrook equation is typically used, which is implicit and requires an iterative solution or approximation formulas (like the Swamee-Jain equation). The calculator uses an approximation for f based on Re and relative roughness.
Iterative Process: Since 'f' depends on 'V' (through Re), and 'V' depends on 'f', the calculator must iterate. It starts with an initial guess for 'f' (e.g., from Moody chart approximations for turbulent flow) and calculates 'V' and 'Re'. Then, it recalculates 'f' based on the new 'Re' and roughness, and repeats until 'f' converges.
Input Definitions:
Inlet Pressure (P1): The pressure of the fluid at the beginning of the pipe section being analyzed (in kPa).
Outlet Pressure (P2): The pressure of the fluid at the end of the pipe section being analyzed (in kPa).
Pipe Inner Diameter (D): The internal diameter of the pipe (in mm). This is converted to meters for calculations.
Pipe Length (L): The length of the pipe section being analyzed (in meters).
Fluid Density (ρ): The mass per unit volume of the fluid (in kg/m³).
Fluid Dynamic Viscosity (μ): A measure of the fluid's resistance to flow (in Pa·s).
Pipe Absolute Roughness (ε): The average height of the surface irregularities inside the pipe (in mm). This is converted to meters for calculations.
Use Cases:
Designing and analyzing fluid transport systems (water, oil, gas).
Optimizing pump and pipe sizing.
Troubleshooting flow rate issues in industrial processes.
Estimating energy losses due to friction.
By inputting these parameters, the calculator provides a critical estimate of the flow rate, enabling informed decisions in fluid system design and management.
function calculateMorgaeFlowRate() {
var p1_kPa = parseFloat(document.getElementById("inletPressure").value);
var p2_kPa = parseFloat(document.getElementById("outletPressure").value);
var d_mm = parseFloat(document.getElementById("pipeDiameter").value);
var l_m = parseFloat(document.getElementById("pipeLength").value);
var rho = parseFloat(document.getElementById("fluidDensity").value);
var mu = parseFloat(document.getElementById("fluidViscosity").value);
var epsilon_mm = parseFloat(document.getElementById("roughness").value);
var resultElement = document.getElementById("flowRateResult");
resultElement.textContent = "–"; // Reset result
if (isNaN(p1_kPa) || isNaN(p2_kPa) || isNaN(d_mm) || isNaN(l_m) || isNaN(rho) || isNaN(mu) || isNaN(epsilon_mm)) {
alert("Please enter valid numbers for all fields.");
return;
}
if (p1_kPa <= p2_kPa) {
alert("Inlet pressure (P1) must be greater than outlet pressure (P2) for flow.");
return;
}
if (d_mm <= 0 || l_m <= 0 || rho <= 0 || mu <= 0 || epsilon_mm < 0) {
alert("Physical parameters (diameter, length, density, viscosity) must be positive. Roughness cannot be negative.");
return;
}
// Convert units to SI base units
var p1_Pa = p1_kPa * 1000;
var p2_Pa = p2_kPa * 1000;
var d_m = d_mm / 1000;
var epsilon_m = epsilon_mm / 1000;
var delta_p_Pa = p1_Pa – p2_Pa;
var area_m2 = Math.PI * Math.pow(d_m / 2, 2);
// Initial guess for velocity (can be arbitrary, e.g., 1 m/s)
var v_guess = 1.0;
var iterations = 0;
var maxIterations = 100;
var tolerance = 1e-6;
var v_current = v_guess;
var f_current = 0.02; // Initial guess for friction factor
while (iterations < maxIterations) {
var re = (rho * v_current * d_m) / mu;
var relativeRoughness = epsilon_m / d_m;
var f_next;
if (re < 2300) { // Laminar flow
f_next = 64 / re;
} else { // Turbulent flow – using Swamee-Jain approximation for Colebrook
// Swamee-Jain equation: f = 0.25 / [log10( (epsilon/D)/3.7 + 5.74/Re^0.9 )]^2
f_next = Math.pow(0.772 / Math.log10(relativeRoughness / 3.7 + 5.74 / Math.pow(re, 0.9)), 2);
// Handle cases where re is very large leading to potential log issues or non-convergence with specific approximations.
if (isNaN(f_next) || f_next <= 0) {
// Fallback or simplified approach for extreme conditions
f_next = 0.01; // A very low friction factor assumption
}
}
// Calculate velocity based on Darcy-Weisbach with the current friction factor
// delta_p_Pa = f_current * (l_m/d_m) * (rho * v_current^2 / 2)
// v_current = sqrt( (2 * delta_p_Pa * d_m) / (f_current * l_m * rho) )
var v_calculated = Math.sqrt((2 * delta_p_Pa * d_m) / (f_current * l_m * rho));
// Check for convergence
if (Math.abs(v_calculated – v_current) < tolerance) {
v_current = v_calculated;
break;
}
v_current = v_calculated;
f_current = f_next; // Use the newly calculated friction factor for the next iteration's velocity calculation.
iterations++;
}
if (iterations === maxIterations) {
console.warn("Morgae flow rate calculation did not converge within the maximum number of iterations.");
// Use the last calculated velocity if convergence fails
}
var flowRate_m3s = v_current * area_m2;
resultElement.textContent = flowRate_m3s.toFixed(4); // Display with 4 decimal places
}