Use our pipe flow calculator to determine the velocity and flow rate of water that flows by gravity. This tool employs the gravitational form of the Hazen–Williams equation to calculate velocity in a pipe.

This empirical equation, exclusively applicable to water, allows calculating the velocity or the head loss of a gravity flow. If you'd like to learn more about the Hazen–Williams equation's parameters and how to calculate water flow rate and velocity for yourself, we invite you to continue reading.

Gravity flow

The flow in a piping system is largely determined by the available energy and the losses in the pipes. This energy could be provided by gravity or/and by a pump.

In the case of gravity-fed systems, only gravity is used to transport water (or other fluid) from a source to a final application. The only energy available in these systems is supplied by the difference in heights between the source, usually an atmospheric tank placed at a higher altitude and the system's lowest altitude point. In these cases, the flow of a fluid is referred to as gravity flow.

Hazen–Williams equation to calculate velocity in a pipe

The Hazen–Williams equation is an empirical formula used to calculate water's velocity in a gravity-fed system. In contrast to Darcy–Weisbach's equation, Hazen–Williams has the advantage that it doesn't require an iterative calculation or guessing the friction factor or Reynolds' number.

This equation only applies to water, and it calculates the velocity of the water by relating the geometric properties of the pipe and the slope of the energy line. The Hazen–Williams equation or pipe velocity equation is given by:

v=kCR0.63S0.54\footnotesize v = k \cdot C \cdot R^{0.63}\cdot S^{0.54}

where:

  • vv - Velocity of water in m/s or ft/s;
  • kk - Conversion factor dependent on the unit system (k = 0.849 for the metric system and k = 1.318 for the imperial system);
  • CC - Pipe roughness coefficient for a given material;
  • RR - Hydraulic radius in meters or feet; and
  • SS - Slope of the energy line (frictional head loss per length of pipe). It's unitless, but sometimes expressed in m/m.

The pipe roughness coefficient (C) is dependent on the material. Below you can find the values for this coefficient for different materials:

Material

Roughness coefficient

Cast iron

100

Concrete

110

Copper

140

Plastic (PVC)

150

Steel

120

These same values are included in the pipe flow calculator. By changing the Material variable of the calculator, you can see different values for the pipe Roughness coefficient.

Notice that the Hazen–Williams equation has some constraints (besides water only), making the results relatively accurate only for:

  • Pipe sizes between 50 mm (~2") and 1850 mm (~73");
  • Velocities below 3 m/s;
  • Reynolds' number over 105 (turbulent flow); and
  • Water temperature should be between 40 °F and 75 °F (5 °C and 25 °C), ideally 16 °F (60.8 °C).

Viscosity and density of water are affected by temperature. You can take a look at our density calculator to find out more.

How to calculate the water flow rate and water's velocity – An example

A numerical example is the best way to understand how to use the Hazen–Williams equation to determine the velocity and flow of water. Assume we need to calculate the velocity and flow rate in a system with the following characteristics:

A pipe made of steel, with a diameter of 2.5 inches, with a length of 18 feet, and a difference in heights of 3 feet. Let's do the math:

  1. For the Hazen–Williams equation, we need the values of conversion factor kk, roughness coefficient CC, hydraulic radius RR, and the slope of the energy line SS.

  2. The conversion factor for imperial units is:

k=1.318\qquad \footnotesize k = 1.318
  1. In the case of steel, the roughness coefficient CC is 120.

  2. To calculate the hydraulic radius RR, divide the cross-sectional area (A=πr2) (A = \pi \cdot r^2) of the pipe by the wetted perimeter (P=2πr)(P = 2 \cdot \pi \cdot r):

R=AP=πr22πr=r2=0.0521 ft\qquad \begin{aligned} \footnotesize R &= \cfrac{A}{P} = \cfrac{ \pi \cdot r^2}{2 \cdot \pi \cdot r} \\ \footnotesize &= \cfrac{r}{2} = 0.0521\ \text{ft}\\ \end{aligned}
  1. The last variable we require is the slope of the energy line SS. To obtain it, dive the vertical head loss hfh_f by the total length of the pipe LL:
S=hfL=3 ft18 ft=0.16667 ft/ft\qquad \begin{aligned} \footnotesize S &= \cfrac{h_f}{L} \\ \footnotesize &= \cfrac{3 \ \text{ft}}{18 \ \text{ft}} = 0.16667\ \text{ft/ft}\\ \end{aligned}
  1. Now that we have all of the necessary numbers, we can substitute them into the Hazen–Williams equation to calculate velocity in the pipes:
v=kCR0.63S0.54=1.318120(0.0521 ft)0.630.16670.54=9.34 ft/s\qquad \begin{aligned} \footnotesize v &= k \cdot C \cdot R^{0.63}\cdot S^{0.54} \\ \footnotesize &= 1.318 \cdot 120 \cdot (0.0521\ \text{ft})^{0.63}\cdot 0.1667^{0.54} \\ \footnotesize &= 9.34 \ \text{ft/s} \end{aligned}
  1. Finally, knowing the velocity vv, we can calculate the flow rate QQ by multiplying it by the pipe's cross-sectional area AA:
Q=vA=9.34 ft/s 0.0341 ft2=0.3184 ft3/s\qquad \begin{aligned} \footnotesize Q &= v \cdot A = 9.34 \ \text{ft/s} \ \cdot 0.0341 \ \text{ft}^2\\ \footnotesize &= 0.3184 \ \text{ft}^3/\text{s} \end{aligned}

🙋 In the Advanced mode of the pipe flow calculator, you can find or input other parameters such as area, perimeter, hydraulic radius, and the slope.

Gabriela Diaz
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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