# Pipe Flow Calculator

Created by Bogna Szyk
Reviewed by Dominik Czernia, PhD and Jack Bowater
Last updated: Oct 14, 2022

Use this pipe flow calculator to analyze 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.

We also suggest checking the orifice flow calculator to see another type of liquid flow!

## 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. 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 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 pipe material. Its roughness causes friction between the sides of the pipe and the water, decreasing the water speed. Also, in the coefficient of discharge calculator, we described the metric related to the theoretical and actual flow rates. Be sure to check it out if you're interested!

## 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:

$v = \mathrm{k} \times C \times R^{0.63} \times S^{0.54}$

where:

• $v$ — Velocity of water flowing in the pipe (in m/s for the metric system and ft/s for the Imperial system);
• $C$ — Roughness coefficient;
• $R$ — Hydraulic radius (in meters or feet depending on the unit system) - check the hydraulic radius calculator to learn more;
• $S$ — Slope of the energy line (frictional head loss per length of pipe). It is unitless, but sometimes expressed in m/m; and
• $\mathrm{k}$ — Conversion factor dependent on the unit system ($\mathrm{k} = 0.849$ for the metric system and $\mathrm{k} = 1.318$ 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 $C$ depends on the material of the pipe. You can pick 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:

$R = \frac{A}{P} = \frac{\pi r^2}{2 \pi r} = \frac{r}{2} = \frac{d}{4}$

where $r$ is the pipe radius, and $d$ is the pipe diameter. In this pipe flow calculator's Advanced mode, you can view and modify all these parameters (area, perimeter, hydraulic radius).

To calculate the slope $S$, you must divide the pipe length by the drop (height difference between the beginning and endpoints). Remember that if the pipe slope is not constant but changes all the time, the actual water flow speed will differ 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:

$Q = A \times v$

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 endpoints of the pipe is equal to 3 feet.

1. Divide the diameter by 2 to find the radius of the pipe.
$\small \quad r = \frac{d}{2} = \frac{0.5}{2} = 0.25 \ \mathrm{ ft}$
1. Find the cross-sectional area of the pipe.
$\small \quad A = \pi r^2 = \pi \times 0.25^2 \approx 0.1963 \ \mathrm{ft^2}$
1. Determine the perimeter of the pipe.
$\small \quad P = 2\pi r = 2 \pi \times 0.25 \approx 1.57 \ \mathrm{ft}$
1. Divide the area by the perimeter to find the hydraulic radius of the pipe.
$\small \quad R = \frac{A}{P} = \frac{0.1963}{1.57} \approx 0.125 \ \mathrm{ft}$
1. Pick "Plastic" from the drop-down list and write down its roughness coefficient.
$\small \quad C = 150$
1. Divide the drop by the length of the pipe to calculate the slope.
$\small \quad S = \frac{y}{L} = \frac{3}{12} = 0.25$
1. Use the Hazen-Williams equation to find the velocity of the gravity flow.
\footnotesize \begin{align*} \quad v &= 1.318 \times C \times R^{0.63} \times S^{0.54} \\[8pt] &= 1.318 \times C \times 0.125^{0.63} \times 0.25^{0.54} \\[8pt] & \approx 25.23 \ \frac{\mathrm{ft}}{\mathrm{s}} \end{align*}
1. Multiply this value with the cross-sectional area of the pipe to find the discharge:
\begin{align*} \quad Q &= A \times v = 0.1963 \times 25.23 \\[8pt] &= 4.95 \mathrm{\frac{cu \ ft}{s}} \end{align*}

That's it! You just found the speed and discharge of a gravity flow. Have you already seen the pipe volume calculator? That's another tool that considers the liquids flowing through pipes.

## FAQ

### How do you calculate gravity flow through a pipe?

First use the Hazen-Williams equation to find the velocity of the fluid: v = k × C × R0.63 × S0.54. In this equation, k is either 0.489 for metric or 1.318 if using imperial units, C is the roughness coefficient of the pipe material, R is the hydraulic radius (cross-sectional area divided by perimeter), and S is the slope of the pipe.

You can then calculate the volume that flows through the pipe per second by multiplying v by the cross-sectional area of the pipe.

### Does flow rate change with pipe diameter?

Yes. The flow rate will increase with pipe diameter squared since it depends on the pipe's cross-sectional area. It then follows that the flow rate will decrease with the square of the pipe diameter.

### How to calculate volume flow rate in a pipe?

You multiply the speed of the liquid flowing through the pipe by the pipe's cross-sectional area.

### What is the roughness coefficient of a plastic pipe?

For a typical plastic pipe, its roughness coefficient is 150. The higher the roughness coefficient, the slower the gravity-fed flow through a pipe will be.

Bogna Szyk
Pipe diameter
in
Material
Plastic
Roughness coefficient
Pipe length
ft
Drop
ft
Flow velocity
ft/s
Flow discharge
cu ft
/s
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