Hazen-Williams Formula for Pipe Friction Loss

1. From Rate of Loss to Total Loss

Let's start by connecting directly to our previous lesson. You learned to look up the friction loss rate from a chart. For example, you might find that for a given flow and pipe size, the loss is 2.8 meters of head per 100 meters of pipe (which can also be written as 2.8%).

To find the total friction loss for a specific pipe segment, you simply use a proportional calculation.

Total Friction Loss = (Friction Loss Rate / 100) × Actual Pipe Length

For example, if the chart indicates a loss of 2.8 meters per 100 meters, and your pipe segment is 65 meters long, the calculation is:

$\text{Total Friction Loss}=\left(\frac{2.8\text{ m}}{100\text{ m}}\right)\times 65\text{ m}=0.028\times 65=1.82\text{ meters of head}$

This means that over the 65-meter run, the water will lose an amount of pressure equivalent to the pressure exerted by a 1.82-meter column of water. We'll talk more about "head" and its relation to pressure (like bars) in a moment.

2. Introducing the Hazen-Williams Formula

While charts are fast and convenient, they are just a snapshot of a more powerful underlying tool. The Hazen-Williams equation is an empirical formula that allows you to calculate friction loss directly.

As you read, the key takeaway is that this formula mathematically connects the four factors that influence friction loss: pipe length, pipe diameter, water flow rate, and the pipe's internal roughness.

3. Deconstructing the Formula

The Hazen-Williams formula looks complex, but it becomes manageable when you understand each part. We will work with the SI (metric) version, as it is standard in most international engineering contexts.

$h_f=10.67\cdot L\cdot {\left(\frac{Q}{C}\right)}^{1.852}\cdot \frac{1}{d^{4.87}}$

This is often written more compactly as:

$h_f=\frac{10.67\cdot L\cdot Q^{1.852}}{C^{1.852}\cdot d^{4.87}}$

Let's break down the variables:

• $h_f$ is the head loss due to friction, measured in meters (m). This is the primary output you are solving for. "Head" is a convenient way to talk about pressure. A head loss of 10 meters means the pressure has dropped by the equivalent of a 10-meter-high column of water, which is approximately 1 bar.
• $L$ is the pipe length in meters (m). This is a direct measurement.
• $Q$ is the flow rate in cubic meters per second (m³/s). This is the trickiest part, as your zone flow rates will likely be in liters per hour (L/h) or cubic meters per hour (m³/h). Unit conversion is mandatory.
• $d$ is the internal pipe diameter in meters (m). Pipe is often sold by its nominal or outer diameter in millimeters, so you must find the internal diameter and convert it to meters.
• $C$ is the Hazen-Williams roughness coefficient. This is a dimensionless number that describes how smooth the inside of the pipe is. A higher C value means a smoother pipe and less friction.

Here are some typical C values for new pipes, which you can also find in the resource LINK:

• PVC, PE (Polyethylene): 140 – 150 (we often use 150 for design)
• New steel or ductile iron: 120 – 140
• Aged or corroded pipe: 100 or lower

The numbers 10.67, 1.852, and 4.87 are constants and exponents derived from empirical measurements that make the formula work for water with these specific units.

4. Worked Example: Calculating Friction Loss

Let's apply the formula to a realistic scenario for a villa garden.

Scenario: You have a lateral pipe line feeding sprinklers in a lawn zone.

• Flow Rate (Q): 2.5 m³/h
• Pipe Material: PVC, with an internal diameter of 27.8 mm
• Pipe Length (L): 45 m
• Roughness (C): 150 (for new PVC)

Step 1: Convert All Units to Match the Formula

• Length (L): 45 m (Already in the correct unit)
• Roughness (C): 150 (No units needed)
• Flow Rate (Q): We need to convert from m³/h to m³/s.
  Since there are 3600 seconds in an hour:
  $Q=\frac{2.5{\text{ m}}^3}{\text{h}}\times \frac{1\text{ h}}{3600\text{ s}}=0.000694{\text{ m}}^3/\text{s}$
• Diameter (d): We need to convert from mm to m.
  $d=27.8\text{ mm}\times \frac{1\text{ m}}{1000\text{ mm}}=0.0278\text{ m}$

Step 2: Plug the Converted Values into the Formula

$h_f=\frac{10.67\cdot 45\cdot (0.000694{)}^{1.852}}{150^{1.852}\cdot (0.0278{)}^{4.87}}$

Let's calculate the terms:

• $L=45$
• $Q^{1.852}=(0.000694{)}^{1.852}\approx 1.05\times 10^{-6}$
• $C^{1.852}=150^{1.852}\approx 12,053$
• $d^{4.87}=(0.0278{)}^{4.87}\approx 1.25\times 10^{-8}$

Now, substitute these back into the equation:

$h_f=\frac{10.67\cdot 45\cdot (1.05\times 10^{-6})}{12,053\cdot (1.25\times 10^{-8})}=\frac{0.000504}{0.000151}\approx 3.34\text{ m}$

Step 3: Interpret the Result

The total friction loss along this 45-meter pipe is 3.34 meters of head.

To convert this to bar, a more familiar pressure unit:
$\text{Pressure Loss (bar)}\approx \frac{\text{Head Loss (m)}}{10}=\frac{3.34}{10}=0.334\text{ bar}$

So, you will lose about a third of a bar of pressure just to friction in this pipe segment.

5. Watching a Pro Do the Math

Now that you've walked through the calculation, let's watch an engineer solve a similar problem. This video is excellent for reinforcing the step-by-step process. Be aware that it uses US Customary units (gallons per minute, feet, inches), so the formula's constant is different (4.73 instead of 10.67), but the methodical approach of converting units and plugging them in is universal.

6. Tying it All Together: Formula, Charts, and Calculators

You now understand the theory. Let's connect it back to the practical tools. The charts you used in the last lesson are simply pre-calculated solutions to the Hazen-Williams equation for standard pipe sizes.

To prove this to yourself, you could take a value from one of those charts (e.g., for 13 GPM in a 1" pipe), convert the units, plug them into the formula, and see if your calculated result for a 100 ft length of pipe matches the PSI loss shown in the chart. (It will!)

For your day-to-day design work, you don't need to manually calculate the formula every time. The goal of this lesson is to understand what the calculation represents. Once you do, you can confidently use professional tools.

Conclusion

You have now moved from being a user of friction loss charts to understanding the engine that drives them. This is a significant step in your journey toward professional irrigation design.

Key Takeaways:

• The Hazen-Williams formula is the industry standard for calculating friction loss for water in pipes.
• The formula is highly sensitive to pipe diameter; a small change in diameter has a large impact on pressure loss.
• Strict unit consistency is essential for an accurate calculation. All inputs (L, Q, d) must be converted to the base units of the formula (meters and m³/s for the SI version).
• Head loss (meters) is a way of expressing pressure loss that can be converted to bars by dividing by approximately 10.
• Online calculators are efficient tools, and you can now use them with confidence, knowing the principles behind their operation.

In our next lesson, "System Pressure and Sizing," we will take the friction loss you just learned to calculate and add it to other pressure losses in the system, such as those from valves, fittings, and changes in elevation. This will allow us to calculate the Total Dynamic Head (TDH), a critical value needed to ensure your system has enough pressure to operate correctly.