Pipe diameter
Roughness coefficient
Pipe length
Flow velocity
Flow discharge
cu ft

Use this pipe flow calculator to analyse the properties of water flowing in a gravity-fed system. You only need to know the diameter of the pipe, the material it's made of, its length, and the drop in height. We then apply the Hazen-Williams equation for you, which calculates the resulting velocity and discharge. Interested? Read on to discover the formulas we use, and to see an easy-to-follow example calculation.

What is the gravity flow?

The gravity flow of water is when the flow of water in a pipe is caused by the force of gravity. This required the pipe to be inclined, as otherwise the water will not flow downwards. The must also be no external energy (for example, from a pump) used to move the water forward.

Our water 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 material the pipe is made of - its 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.

You can write down this formula as:

v = k * C * R0.63 * S0.54


  • v stands for the velocity of water flowing in the pipe (in m/s for the metric system and ft/s for the Imperial system)
  • C is the roughness coefficient
  • R stands for the hydraulic radius (in meters or feet depending on the unit system)
  • S is the slope of the pipe (unitless)
  • k is a conversion factor dependent on the unit system (k = 0.849 for the metric system and k = 1.318 for the imperial system)

You don't need to know the values of C, R or S in order to use our pipe flow calculator - we calculate them for you!

The roughness coefficient C dependents on the material of the pipe. You can pick a material from a drop-down list or input the value of C manually if you know the roughness coefficient of your flow system. We use the following values:

Material Roughness coefficient
Cast iron 100
Concrete 110
Copper 140
Plastic 150
Steel 120

The hydraulic radius, R, is the proportion between the area and the perimeter of your pipe. If the pipe is circular, you will find it according to the following equation:

R = A / P = πr² / 2πr = r / 2 = d / 4

where r is the pipe radius, and d is the pipe diameter. You can view and modify all these parameters (area, perimeter, hydraulic radius) in the advanced mode of this pipe flow calculator.

To calculate the slope, S,, you need to divide the pipe length by the drop (height difference between the beginning and end points). Remember that if the pipe slope is not constant, but changes all the time, the real water flow speed will be different from the obtained result.

Once you know the velocity of the gravity flow, you can also find the discharge, Q, by multiplying the cross-sectional area of the pipe by the flow speed:

Q = A * v

Make sure to use our flow rate calculator to convert between the discharge (volumetric flow rate) and the mass flow rate.

Velocity of water flow in a pipe: an example

Let's use the pipe flow calculator to determine the velocity and discharge of a plastic pipe, 0.5 feet in diameter. The pipe is 12 feet long, and the difference in height between the beginning and end points of the pipe is equal to 3 feet.

  1. Divide the diameter by 2 to find the radius of the pipe.

    r = d/2 = 0.5 / 2 = 0.25 ft

  2. Find the cross-sectional area of the pipe.

    A = πr² = π * 0.25² ≈ 0.1963 ft²

  3. Determine the perimeter of the pipe.

    P = 2πr = 2π * 0.25 ≈ 1.57 ft

  4. Divide the area by the perimeter to find the hydraulic radius of the pipe.

    R = A/P = 0.1963 / 1.57 ≈ 0.125 ft

  5. Pick "plastic" from the drop-down list and write down its roughness coefficient.

    C = 150

  6. Divide the drop by the length of the pipe to calculate the slope.

    S = y / L = 3 / 12 = 0.25

  7. Use the Hazen-Williams equation to find the velocity of the gravity flow.

v = 1.318 * C * R0.63 * S0.54 = 1.318 * 150 * 0.1250.63 * 0.250.54 = 25.23 ft/s

  1. Multiply this value with the cross-sectional area of the pipe to find the discharge:

    Q = A * v = 0.1963 * 25.23 = 0.1835 cu ft/s

That's it! You just found the speed and discharge of a gravity flow.

Bogna Haponiuk