What is Gravity Flow?
The gravity flow of water is when the flow of water in a pipe is caused by the force of gravity. The flow will happen as long as there is an altitude difference between the source water (upstream source) and the discharge point. There must also be no external energy (for example, from a pump) used to move the water forward.
Our pipe flow calculator takes into consideration the particular case of gravity flow, where the water flows in a closed pipe. Its velocity is influenced not only by the inclination and size of the pipe but also by the pipe material. Roughness causes friction between the sides of the pipe and the water, decreasing the water speed.
Hazen-Williams Equation
The Hazen-Williams equation is an empirically derived formula that describes the velocity of water in a gravity flow. Remember that the Hazen-Williams equation is valid only for water — applying it for any other fluid will give you inaccurate results. It also doesn't take into account the temperature of the water, and is only accurate for the 40–75 °F (4–25 °C) range.
The formula is written as:
v = k × C × R0.63 × S0.54
Where:
- v — Velocity of water flowing in the pipe (m/s for metric; ft/s for imperial)
- C — Hazen-Williams roughness coefficient (dimensionless)
- R — Hydraulic radius = D/4 for a full circular pipe (m or ft)
- S — Slope of the energy line = height drop / pipe length (dimensionless)
- k — Unit conversion factor: 0.849 for metric, 1.318 for imperial
Roughness Coefficient C
The roughness coefficient C depends on the material of the pipe:
| Material | C (Hazen-Williams) |
|---|---|
| Cast Iron | 100 |
| Concrete | 110 |
| Steel | 120 |
| Copper | 140 |
| Plastic (PVC) | 150 |
A higher C value means a smoother pipe with less friction and a higher flow velocity.
Hydraulic Radius
The hydraulic radius R is the ratio of the cross-sectional area of the flow to the wetted perimeter. For a full circular pipe of diameter D:
R = A / P = (π × D² / 4) / (π × D) = D / 4
Discharge (Flow Rate)
Once the velocity v is known, the volumetric flow rate (discharge) Q is calculated as:
Q = v × A = v × π × (D/2)²
The result is expressed in m³/s and L/s for the metric system, or in ft³/s and gallons per minute (gal/min) for the imperial system.
Velocity of Water Flow in a Pipe — Example
Suppose a plastic PVC pipe (C = 150) has a diameter of 200 mm, a length of 500 m, and an elevation drop of 5 m. Let's find the flow velocity using the metric system.
- Slope: S = 5 / 500 = 0.010
- Diameter in meters: D = 0.200 m
- Hydraulic radius: R = D / 4 = 0.050 m
- Velocity: v = 0.849 × 150 × 0.0500.63 × 0.0100.54 ≈ 1.34 m/s
- Area: A = π × (0.10)² ≈ 0.03142 m²
- Discharge: Q = 1.34 × 0.03142 ≈ 0.0421 m³/s = 42.1 L/s
FAQs
What does the Hazen-Williams roughness coefficient C mean?
C quantifies the friction caused by the inner surface of the pipe. The higher the C value, the smoother the pipe and the faster the water flows. New plastic pipes have C ≈ 150, while old cast iron pipes can have C as low as 80.
Is the Hazen-Williams equation valid for all fluids?
No. The Hazen-Williams equation was developed specifically for water and is only accurate for water temperatures between 4 °C and 25 °C (40–75 °F). For other fluids or temperatures, use the Darcy-Weisbach equation instead.
What is hydraulic radius?
The hydraulic radius is the effective size of a cross-section for flow calculations. For a full circular pipe, it equals diameter divided by 4. It accounts for the relationship between the flow area and the wetted perimeter.
What unit system should I use?
Use Metric if you work with SI units (mm, m). Use American (Imperial) if you work with inches and feet. The calculator automatically applies the correct Hazen-Williams constant (k = 0.849 for metric, k = 1.318 for imperial).