Fluid Flow Rate Calculator
This calculator estimates the volumetric flow rate of a fluid through a pipe, considering the pressure difference and the pipe's diameter. It utilizes a simplified form of the Hagen-Poiseuille equation, assuming laminar flow and an incompressible Newtonian fluid.
Understanding Flow Rate, Pressure, and Diameter
Volumetric Flow Rate (Q): This is the volume of fluid that passes through a given cross-sectional area per unit of time. It's typically measured in cubic meters per second (m³/s) or liters per minute (L/min). A higher flow rate means more fluid is moving.
Pressure Difference (ΔP): This is the difference in pressure between two points in a system. For fluid to flow, there must be a pressure gradient. A larger pressure difference generally leads to a higher flow rate, as it provides a stronger driving force for the fluid. It's measured in Pascals (Pa).
Pipe Diameter (D): The inner diameter of the pipe significantly impacts flow. A larger diameter offers less resistance to flow, allowing for a higher flow rate under the same pressure conditions. A smaller diameter constricts the flow, reducing the flow rate. Measured in meters (m).
Pipe Length (L): Longer pipes introduce more friction, which resists flow. Therefore, for the same pressure difference and diameter, a longer pipe will result in a lower flow rate. Measured in meters (m).
Fluid Dynamic Viscosity (μ): Viscosity is a measure of a fluid's resistance to flow. Thicker fluids (like honey) have higher viscosity than thinner fluids (like water). Higher viscosity increases the resistance to flow, thus decreasing the flow rate. Measured in Pascal-seconds (Pa·s).
The relationship between these factors can be approximated by the Hagen-Poiseuille equation for laminar flow in a cylindrical pipe: $Q = \frac{\pi R^4 \Delta P}{8 \mu L}$ where R is the pipe radius ($R = D/2$). Substituting R: $Q = \frac{\pi (D/2)^4 \Delta P}{8 \mu L} = \frac{\pi D^4 \Delta P}{128 \mu L}$
Example Calculation
Let's consider water (dynamic viscosity approximately 0.001 Pa·s at 20°C) flowing through a pipe with an inner diameter of 0.05 meters (5 cm) and a length of 10 meters. If the pressure difference across the pipe is 500 Pascals:
- Pressure Difference (ΔP): 500 Pa
- Pipe Inner Diameter (D): 0.05 m
- Pipe Length (L): 10 m
- Fluid Dynamic Viscosity (μ): 0.001 Pa·s
Using the formula: $Q = \frac{\pi \times (0.05 \text{ m})^4 \times 500 \text{ Pa}}{128 \times 0.001 \text{ Pa·s} \times 10 \text{ m}}$ $Q \approx \frac{3.14159 \times 0.00000625 \times 500}{128 \times 0.001 \times 10}$ $Q \approx \frac{0.009817}{1.28} \approx 0.00767 \text{ m³/s}$