Determining the flow rate of a fluid through a pipe based on the pressure difference between two points is a fundamental task in fluid mechanics, civil engineering, and hydraulic system design. This calculator uses the Hagen-Poiseuille equation to estimate the volumetric flow rate given the pressure drop, pipe dimensions, and fluid viscosity.
Understanding the Physics: Pressure vs. Flow
Fluid flow in a pipe is driven by a pressure gradient. In simple terms, fluid moves from areas of high pressure to areas of low pressure. The rate at which the fluid moves (Flow Rate) depends heavily on the resistance offered by the pipe.
The resistance is influenced by three main factors:
Pipe Diameter: This is the most critical factor. Resistance decreases drastically as the pipe gets wider.
Pipe Length: Longer pipes create more friction, reducing flow rate.
Viscosity: "Thicker" fluids (like oil) flow slower than "thin" fluids (like water) under the same pressure.
The Calculation Formula
This calculator relies on the Hagen-Poiseuille Law, which describes laminar flow of an incompressible Newtonian fluid through a long cylindrical pipe of constant cross-section.
Q = (π · r⁴ · ΔP) / (8 · η · L)
Where:
Q = Volumetric flow rate (m³/s)
r = Internal radius of the pipe (m)
ΔP = Pressure difference/drop (Pa)
η (eta) = Dynamic viscosity (Pa·s)
L = Length of the pipe (m)
π = Pi (approx. 3.14159)
Why is Pipe Diameter so Important?
Notice the r⁴ term in the equation above. This indicates that the flow rate is proportional to the fourth power of the radius. This is a massive geometric relationship.
Example: If you double the diameter of a pipe (e.g., going from 1 inch to 2 inches) while keeping pressure and length constant, the flow rate increases by a factor of 16 (2⁴ = 16). This explains why even small increases in pipe size can drastically improve hydraulic performance.
Typical Viscosity Values
Viscosity is a measure of a fluid's resistance to flow. It changes significantly with temperature.
Fluid
Dynamic Viscosity (Pa·s)
Water (20°C)
0.00100
Water (25°C)
0.00089
Motor Oil (SAE 30)
0.20000
Honey
10.0000
Air
0.000018
Limitations of This Calculator
While the Hagen-Poiseuille equation is excellent for understanding the relationships between variables, it specifically applies to laminar flow (smooth, orderly flow). In industrial applications involving water at high speeds, flow often becomes turbulent.
In turbulent flow regimes, the relationship between pressure and flow becomes non-linear (often involving the square root of pressure), and the friction factor becomes dependent on the pipe roughness. However, for low-velocity systems, capillary tubes, viscous fluids (like oil), or initial estimations, this calculator provides an accurate scientific baseline.