function calculateFlowRate() {
var pressureDrop = parseFloat(document.getElementById("pressureDrop").value);
var pipeLength = parseFloat(document.getElementById("pipeLength").value);
var pipeDiameter = parseFloat(document.getElementById("pipeDiameter").value);
var fluidViscosity = parseFloat(document.getElementById("fluidViscosity").value);
var pipeRoughness = parseFloat(document.getElementById("pipeRoughness").value);
var resultElement = document.getElementById("result");
resultElement.innerHTML = ""; // Clear previous results
// Input validation
if (isNaN(pressureDrop) || isNaN(pipeLength) || isNaN(pipeDiameter) || isNaN(fluidViscosity) || isNaN(pipeRoughness)) {
resultElement.innerHTML = "Please enter valid numbers for all fields.";
return;
}
if (pipeLength <= 0 || pipeDiameter <= 0 || fluidViscosity <= 0 || pipeRoughness < 0) {
resultElement.innerHTML = "Please enter positive values for length, diameter, and viscosity, and a non-negative value for roughness.";
return;
}
// Calculation using the Darcy-Weisbach equation and an iterative approach for friction factor (Darcy-f)
// This is a simplified approach. In reality, finding the friction factor (f) often requires iteration.
// We'll use the Colebrook equation implicitly or an approximation for the friction factor.
// For simplicity and to avoid complex iteration in a basic JS calculator, we'll use an approximation
// or a direct calculation method suitable for common flow regimes.
// First, estimate the Reynolds number (Re) assuming a turbulent flow and then iterate or use an approximation for 'f'.
// A common starting point for turbulent flow is to assume a friction factor and refine.
// Alternatively, we can use the Swamee-Jain equation which directly calculates flow rate without explicit iteration for 'f'.
// The Swamee-Jain equation for flow rate (Q) is:
// Q = -0.965 * A * sqrt(D * g * L / f) * log( (e / (3.7 * D)) + (1.775 * nu / (D * V)) ) <– This is not directly for Q
// A more direct approach for Q using Swamee-Jain for friction factor:
// Re = (rho * D * V) / nu (rho is density, V is velocity, nu is kinematic viscosity)
// V = Q / A (A is cross-sectional area)
// Re = (rho * D * Q) / (A * nu)
// f = 0.25 / [ log10( (e/3.7D) + (5.74/Re^0.9) ) ]^2 (for turbulent flow, explicit approximation)
// Let's use a simplified method or an explicit equation for flow rate (Q)
// A common approach to solve this iteratively is to guess 'f', calculate Re, then calculate a new 'f' until convergence.
// For a single-step calculation, we can use an explicit approximation of the Colebrook equation or the Swamee-Jain equation.
// Let's use an explicit approximation for friction factor (f) and then Darcy-Weisbach.
// Re = (rho * D * V) / nu
// V = Q / A
// A = pi * (D/2)^2 = pi * D^2 / 4
// Darcy-Weisbach equation: deltaP = f * (L/D) * (rho * V^2 / 2)
// Rearranging for V: V = sqrt( (2 * deltaP * D) / (f * L * rho) )
// Q = V * A = A * sqrt( (2 * deltaP * D) / (f * L * rho) )
// We need fluid density (rho) to proceed. This calculator is missing density.
// Assuming we're dealing with water at room temperature for a common example, rho ~ 1000 kg/m^3.
// Let's add density as an input or use a default. For a more general calculator, density is essential.
// FOR THIS IMPLEMENTATION, we will proceed by making an assumption about density OR
// by using an equation that implicitly handles it or requires kinematic viscosity.
// If we use dynamic viscosity (mu) and density (rho), kinematic viscosity (nu) = mu / rho.
// The inputs provided are dynamic viscosity. So, density is STILL required.
// Let's assume density of water (rho = 1000 kg/m^3) for this calculation.
// If this were a real-world calculator, density would be a crucial input.
var fluidDensity = 1000; // kg/m^3 (Assumption for water)
var pipeArea = Math.PI * Math.pow(pipeDiameter / 2, 2);
var relativeRoughness = pipeRoughness / pipeDiameter;
// We need to solve for flow rate (Q). This typically requires iteration because the friction factor 'f' depends on the Reynolds number (Re), which depends on velocity (V), which depends on Q.
// Let's use an iterative approach to find the friction factor 'f'.
// We'll start with an initial guess for 'f' (e.g., for fully turbulent flow) and refine it.
var f = 0.02; // Initial guess for friction factor
var maxIterations = 100;
var tolerance = 1e-6;
var iteration = 0;
var V, Re, new_f;
while (iteration < maxIterations) {
// Calculate Reynolds Number using current velocity estimate
// V = Q / A. We need Q or V. Let's try to solve for V first.
// From Darcy-Weisbach: V = sqrt( (2 * pressureDrop * pipeDiameter) / (f * pipeLength * fluidDensity) )
V = Math.sqrt((2 * pressureDrop * pipeDiameter) / (f * pipeLength * fluidDensity));
Re = (fluidDensity * pipeDiameter * V) / fluidViscosity;
// Calculate new friction factor using the Colebrook equation (implicit) or an explicit approximation.
// Using an explicit approximation (e.g., Haaland equation for simplicity)
if (Re < 2300) { // Laminar flow
new_f = 64 / Re;
} else { // Turbulent flow
// Haaland equation (explicit approximation of Colebrook)
new_f = (1 / (-1.8 * Math.log10(Math.pow(relativeRoughness / 3.7, 1.11) + 6.9 / Re)))^2;
// Alternative explicit approximation: Swamee-Jain for friction factor
// new_f = 0.25 / Math.pow(Math.log10(relativeRoughness / 3.7 + 5.74 / Math.pow(Re, 0.9)), 2);
}
if (Math.abs(new_f – f) < tolerance) {
f = new_f; // Converged
break;
}
f = new_f; // Update friction factor for next iteration
iteration++;
}
if (iteration === maxIterations) {
resultElement.innerHTML = "Calculation did not converge. Please check inputs or try again.";
return;
}
// Now calculate the final velocity and flow rate using the converged friction factor 'f'
V = Math.sqrt((2 * pressureDrop * pipeDiameter) / (f * pipeLength * fluidDensity));
var Q = V * pipeArea; // Flow rate in m^3/s
// Convert to more common units if desired (e.g., liters per minute)
var Q_lpm = Q * 60 * 1000;
resultElement.innerHTML = `
Converged Friction Factor (f): ${f.toFixed(5)}
Estimated Velocity (V): ${V.toFixed(3)} m/s
Estimated Reynolds Number (Re): ${Re.toFixed(0)}
Calculated Flow Rate (Q): ${Q.toFixed(5)} m³/s
Calculated Flow Rate (Q): ${Q_lpm.toFixed(2)} L/min
`;
}
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font-family: sans-serif;
border: 1px solid #ccc;
padding: 20px;
border-radius: 8px;
max-width: 500px;
margin: 20px auto;
background-color: #f9f9f9;
}
.calculator-container h2 {
text-align: center;
margin-bottom: 20px;
color: #333;
}
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display: grid;
grid-template-columns: 1fr;
gap: 15px;
margin-bottom: 20px;
}
.input-section label {
font-weight: bold;
color: #555;
}
.input-section input[type="number"] {
width: calc(100% – 22px);
padding: 10px;
border: 1px solid #ccc;
border-radius: 4px;
font-size: 1em;
}
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padding: 12px 20px;
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color: white;
border: none;
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font-size: 1.1em;
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transition: background-color 0.3s ease;
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.input-section button:hover {
background-color: #0056b3;
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margin-top: 20px;
padding-top: 15px;
border-top: 1px solid #eee;
}
.result-section h3 {
color: #333;
margin-bottom: 10px;
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#result {
background-color: #e9ecef;
padding: 15px;
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#result p {
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}
### Understanding Flow Rate Calculation from Pressure Drop
The relationship between pressure drop and flow rate in a fluid system is fundamental in fluid dynamics and engineering. Understanding this relationship allows engineers to design and optimize piping systems, predict fluid velocities, and ensure systems operate within desired parameters. The primary principle governing this is the **Darcy-Weisbach equation**, which relates the head loss (or pressure drop) in a pipe to the flow rate, pipe characteristics, and fluid properties.
**Key Concepts:**
* **Pressure Drop ($\Delta P$):** This is the difference in pressure between two points in a fluid system, typically caused by friction as the fluid flows through pipes, valves, and fittings, or by elevation changes. In this calculator, we are focusing on the pressure drop due to friction along a straight pipe.
* **Flow Rate ($Q$):** This is the volume of fluid that passes through a given cross-section of the pipe per unit of time. It is commonly measured in cubic meters per second (m³/s) or liters per minute (L/min).
* **Friction Factor ($f$):** A dimensionless number that quantifies the frictional resistance to flow in a pipe. It depends on the flow regime (laminar or turbulent) and the relative roughness of the pipe.
* **Reynolds Number ($Re$):** A dimensionless number that helps predict flow patterns. It is the ratio of inertial forces to viscous forces.
* $Re 4000$: Turbulent flow (chaotic, irregular)
* **Pipe Roughness ($\epsilon$):** The absolute roughness of the inner surface of the pipe, measured in meters. This accounts for the texture of the pipe material.
* **Relative Roughness ($\epsilon/D$):** The ratio of the pipe's absolute roughness to its inner diameter.
* **Fluid Viscosity ($\mu$):** A measure of a fluid's resistance to flow. Dynamic viscosity is used here (Pa·s).
* **Fluid Density ($\rho$):** The mass per unit volume of the fluid (kg/m³).
**The Darcy-Weisbach Equation:**
The Darcy-Weisbach equation is the cornerstone for calculating head loss (or pressure drop) due to friction in a pipe:
$h_f = f \frac{L}{D} \frac{V^2}{2g}$
Where:
* $h_f$ is the head loss due to friction (in meters of fluid column)
* $f$ is the Darcy friction factor
* $L$ is the pipe length (m)
* $D$ is the pipe inner diameter (m)
* $V$ is the average flow velocity (m/s)
* $g$ is the acceleration due to gravity (approximately 9.81 m/s²)
To relate this to pressure drop ($\Delta P$), we use $\Delta P = \rho g h_f$. Thus:
$\Delta P = f \frac{L}{D} \frac{\rho V^2}{2}$
**The Challenge:**
The difficulty in calculating flow rate directly from pressure drop lies in the interdependence of these variables. The friction factor ($f$) is not constant; it depends on the Reynolds number ($Re$), which in turn depends on the flow velocity ($V$). Since velocity ($V$) is directly related to flow rate ($Q$) ($V = Q/A$, where $A$ is the pipe's cross-sectional area), solving for $Q$ from $\Delta P$ requires an iterative approach.
**How the Calculator Works:**
1. **Inputs:** The calculator takes the pressure drop ($\Delta P$), pipe length ($L$), pipe inner diameter ($D$), fluid dynamic viscosity ($\mu$), and pipe absolute roughness ($\epsilon$) as inputs.
2. **Assumptions:** For simplicity, a fluid density ($\rho$) is assumed (e.g., 1000 kg/m³ for water). In a more advanced calculator, density would also be an input.
3. **Iterative Calculation:**
* An initial guess for the friction factor ($f$) is made.
* Using this $f$, an initial velocity ($V$) is calculated from the rearranged Darcy-Weisbach equation.
* The Reynolds number ($Re$) is calculated using this $V$.
* A new friction factor ($f_{new}$) is calculated using an explicit approximation of the Colebrook equation (like the Haaland or Swamee-Jain equation), which accounts for $Re$ and relative roughness ($\epsilon/D$).
* This process repeats, using the new $f_{new}$ to calculate a new $V$, then a new $Re$, and subsequently a new $f$, until the friction factor converges (i.e., the change between successive iterations is very small).
4. **Final Calculation:** Once a stable friction factor is found, the final velocity ($V$) and flow rate ($Q = V \times A$) are calculated.
5. **Output:** The results are displayed, including the converged friction factor, estimated velocity, Reynolds number, and the calculated flow rate in both m³/s and L/min.
**Example:**
Let's consider a scenario:
* **Pressure Drop ($\Delta P$):** 50,000 Pascals (Pa)
* **Pipe Length ($L$):** 100 meters (m)
* **Pipe Inner Diameter ($D$):** 0.1 meters (m) (i.e., 10 cm)
* **Fluid Viscosity ($\mu$):** 0.001 Pa·s (typical for water)
* **Pipe Roughness ($\epsilon$):** 0.000002 meters (m) (e.g., smooth steel pipe)
Plugging these values into the calculator would provide an estimated flow rate, taking into account the complex interplay between friction, velocity, and pipe characteristics. The calculator's iterative process would determine the appropriate friction factor for the calculated flow regime.
This calculator is a powerful tool for estimating flow rates when pressure drop is known, essential for many hydraulic and pneumatic systems.