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What is the gravity flow?Hazen-Williams equationVelocity of water flow in a pipe: an exampleFAQsUse 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:

$v = \mathrm{k} \times C \times R^{0.63} \times S^{0.54}$v=k×C×R0.63×S0.54

where:

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

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

The **roughness coefficient $C$C** depends on the material of the pipe. You can pick a material from the drop-down list of our tool or select the last option to manually input the value of $C$C 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, $R$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:

$R = \frac{A}{P} = \frac{\pi r^2}{2 \pi r} = \frac{r}{2} = \frac{d}{4}$R=PA=2πrπr2=2r=4d

where $r$r is the pipe radius, and $d$d is the pipe diameter. By ticking this pipe flow calculator's `Display more calculated variables`

checkbox, you can view and modify all these parameters (area, perimeter, hydraulic radius).

To calculate the **slope $S$S**, you must divide the drop (height difference between the beginning and endpoints) by the pipe length. 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, $Q$Q,** by multiplying the cross-sectional area of the pipe by the flow speed:

$Q = A \times v$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 endpoints of the pipe is equal to **3 feet**.

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

$\small \quad r = \frac{d}{2} = \frac{0.5}{2} = 0.25 \ \mathrm{ ft}$r=2d=20.5=0.25ft

- Find the cross-sectional area of the pipe.

$\small \quad A = \pi r^2 = \pi \times 0.25^2 \approx 0.1963 \ \mathrm{ft^2}$A=πr2=π×0.252≈0.1963ft2

- Determine the perimeter of the pipe.

$\small \quad P = 2\pi r = 2 \pi \times 0.25 \approx 1.57 \ \mathrm{ft}$P=2πr=2π×0.25≈1.57ft

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

$\small \quad R = \frac{A}{P} = \frac{0.1963}{1.57} \approx 0.125 \ \mathrm{ft}$R=PA=1.570.1963≈0.125ft

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

$\small \quad C = 150$C=150

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

$\small \quad S = \frac{y}{L} = \frac{3}{12} = 0.25$S=Ly=123=0.25

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

$\footnotesize\begin{align*}\quad v &= 1.318 \times C \times R^{0.63} \times S^{0.54} \\[8pt]&= 1.318 \times 150 \times 0.125^{0.63} \times 0.25^{0.54} \\[8pt]& \approx 25.23 \ \frac{\mathrm{ft}}{\mathrm{s}}\end{align*}$v=1.318×C×R0.63×S0.54=1.318×150×0.1250.63×0.250.54≈25.23sft

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

$\begin{align*}\quad Q &= A \times v = 0.1963 \times 25.23 \\[8pt]&= 4.95 \mathrm{\frac{cu \ ft}{s}}\end{align*}$Q=A×v=0.1963×25.23=4.95scuft

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.

### 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 × R ^{0.63} × S^{0.54}**. In this equation,

**k**is either 0.849 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**, this is because the flow rate is directly related to the pipe's cross-sectional area. When the cross-sectional area increases (which happens when the pipe diameter increases), the flow rate increases, too. Similarly, as the diameter decreases, the flow rate decreases.

### 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 faster the gravity-fed flow through a pipe will be.