Calculate Flow Rate Given Pressure and Pipe Diameter

Flow Rate Calculator (Darcy-Weisbach Equation)

Calculating the flow rate of a fluid through a pipe is a fundamental task in many engineering disciplines, including mechanical, civil, and chemical engineering. The flow rate, often expressed in units like liters per minute (LPM) or cubic meters per second (m³/s), is influenced by several factors, most notably the pressure difference driving the flow and the characteristics of the pipe itself. This calculator utilizes the Darcy-Weisbach equation, a widely accepted method for determining pressure loss due to friction in pipe flow.

The Darcy-Weisbach Equation

The Darcy-Weisbach equation is expressed as:

$h_f = f_D \frac{L}{D} \frac{V^2}{2g}$

Where:

• $h_f$ is the head loss due to friction (in meters of fluid).
• $f_D$ is the Darcy friction factor (dimensionless).
• $L$ is the length of the pipe (in meters).
• $D$ is the hydraulic diameter of the pipe (in meters). For a circular pipe, this is the inner diameter.
• $V$ is the average velocity of the fluid (in meters per second).
• $g$ is the acceleration due to gravity (approximately 9.81 m/s²).

To use this equation for flow rate (Q), we need to relate velocity to flow rate: $Q = V \times A$, where $A$ is the cross-sectional area of the pipe ($A = \frac{\pi D^2}{4}$). Thus, $V = \frac{Q}{A}$.

Furthermore, pressure drop ($\Delta P$) is related to head loss ($h_f$) by $\Delta P = \rho g h_f$, where $\rho$ is the fluid density.

The Darcy friction factor ($f_D$) is the most complex term, as it depends on the Reynolds number (Re) and the relative roughness of the pipe. The Reynolds number is given by $Re = \frac{\rho V D}{\mu}$, where $\mu$ is the dynamic viscosity of the fluid. For turbulent flow (typically Re > 4000), the friction factor can be approximated using the Colebrook equation or its explicit approximations like the Swamee-Jain equation. This calculator simplifies this by iteratively solving for $f_D$ or using an approximation.

How the Calculator Works

This calculator takes the following inputs to determine the flow rate:

• Pressure Drop (Pa): The difference in pressure between the two ends of the pipe.
• Pipe Inner Diameter (m): The internal diameter of the pipe.
• Pipe Length (m): The total length of the pipe.
• Fluid Dynamic Viscosity (Pa·s): A measure of the fluid's resistance to flow.
• Fluid Density (kg/m³): The mass per unit volume of the fluid.
• Pipe Roughness (m): A measure of the internal surface roughness of the pipe.

The calculator will first calculate the Reynolds number and then iteratively determine the Darcy friction factor ($f_D$). Once $f_D$ is found, it calculates the velocity ($V$) using a rearranged form of the Darcy-Weisbach equation and finally computes the flow rate ($Q = V \times A$).

Example Calculation

Let's consider pumping water through a pipe:

• Pressure Drop: 50,000 Pa
• Pipe Inner Diameter: 0.025 meters (2.5 cm)
• Pipe Length: 50 meters
• Fluid Dynamic Viscosity (water @ 20°C): 0.001 Pa·s
• Fluid Density (water @ 20°C): 998 kg/m³
• Pipe Roughness (commercial steel): 0.000045 meters

Plugging these values into the calculator should provide an estimated flow rate, which will likely be in cubic meters per second (m³/s) and can be converted to more intuitive units like liters per minute (LPM).

Error:

Warning:

${flowRateM3s.toFixed(6)} m³/s

${flowRateLpm.toFixed(2)} LPM