Hazen–Williams Equation

What Is the Hazen–Williams Equation?

The Hazen–Williams equation is an empirical hydraulic formula used to estimate friction losses in water flowing through pressurized pipes.

It was developed in 1902 by American engineers Allen Hazen and Gardner Stewart Williams based on experimental testing of water flow in pipes. Unlike the Darcy–Weisbach equation, which is derived from fundamental fluid mechanics principles, the Hazen–Williams equation is based purely on observed data.

It simplifies pipe flow calculations by:

• Avoiding the need to calculate Reynolds number
• Avoiding friction factor charts (like the Moody diagram)
• Eliminating viscosity from the formula
• Using a single roughness coefficient (C-value)

Because of this simplicity, it became the standard design equation for water distribution systems.

What Is Head Loss?

Head loss is the reduction in total hydraulic energy of water as it flows through a pipe.

In simple terms:

Head loss = energy lost due to friction.

When water moves through a pipe, it rubs against:

• The internal pipe wall
• Pipe fittings (elbows, valves, tees)
• Surface irregularities
• Corrosion or scale buildup

This friction converts mechanical energy into heat (very small amount), reducing pressure as the water moves downstream.

Head loss is usually measured in:

• Meters (SI units)
• Feet (US customary units)

What Causes Head Loss?

Head loss increases when:

• Pipe length increases
• Flow rate increases
• Pipe diameter decreases
• Pipe roughness increases
• The pipe ages or corrodes

The Hazen–Williams equation quantifies exactly how these factors interact.

Notably:

• Flow rate has an exponent of ~1.85 → small increases in flow cause large increases in head loss.
• Diameter has an exponent of ~4.87 → increasing pipe diameter dramatically reduces head loss.

This is why pipe sizing is so critical in hydraulic design.

Why Is Head Loss Important?

Head loss directly affects system performance.

Reduced Pressure at Fixtures

Excessive head loss causes:

• Low pressure at taps
• Weak shower flow
• Poor irrigation performance
• Inadequate fire sprinkler discharge

Increased Pumping Costs

If head loss is high:

• Pumps must work harder
• Energy consumption increases
• Operating costs rise
• Equipment wears faster

Over decades, undersized pipes can cost far more in energy than they saved in installation cost.

Fire Protection System Failure Risk

In fire suppression systems, insufficient pressure due to friction loss can lead to:

• Inadequate sprinkler discharge density
• Code compliance issues
• Increased life safety risk

This is why Hazen–Williams is widely used in fire system hydraulic calculations.

What Is the Equation Good For?

The Hazen–Williams equation is especially useful for:

• Designing municipal water distribution systems
• Sizing pipes for residential and commercial buildings
• Estimating pump head requirements
• Checking pressure drop in long pipelines
• Irrigation system design
• Fire sprinkler hydraulic calculations

It is implemented in hydraulic modeling software such as:

• EPANET
• WaterCAD

These programs use the equation to simulate large pipe networks and predict pressure behavior throughout entire cities.

Practical Engineering Insight

The Hazen–Williams equation is popular not because it is perfect, but because it is practical.

It allows engineers to:

• Quickly estimate pressure loss
• Compare pipe diameters
• Optimize design cost vs performance
• Make conservative assumptions using C-values

For typical water distribution temperatures and turbulent flow conditions, its accuracy is more than sufficient for engineering design.

Simple Conceptual Explanation

Think of water in a pipe like cars on a highway:

• A wider highway (larger diameter) → smoother flow → less resistance
• A rough road surface (low C-value) → more resistance
• More cars (higher flow rate) → traffic buildup → higher energy loss
• Longer distance → more total resistance

The Hazen–Williams equation mathematically describes this behavior.

Conditions for Use

The equation is valid when:

• Fluid is water
• Temperature is approximately 5°C – 25°C
• Flow is fully turbulent
• Pipe is flowing full
• Diameter ≥ 50 mm (2 inches)

Not recommended for:

• Oils or gases
• Laminar flow
• Very small diameter tubes

Comparison with Darcy–Weisbach

Feature	Hazen–Williams	Darcy–Weisbach
Basis	Empirical	Theoretical
Fluids	Water only	Any fluid
Accuracy	Moderate	High
Complexity	Simple	More complex

The Darcy–Weisbach equation remains more universal, but Hazen–Williams is preferred in water distribution engineering for simplicity.

Hazen–Williams Formula (SI Units)

$$h_f = 10.67 \times \frac{L}{C^{1.852} \times D^{4.87}} \times Q^{1.852}$$

Where:

Symbol	Description	Unit
$h_f$	Head loss	m
$L$	Pipe length	m
$C$	Roughness coefficient	–
$D$	Pipe diameter	m
$Q$	Flow rate	m³/s

Formula Variants (Rearranged Forms)

Solve for Length (L)

$$L = \frac{h_f \times C^{1.852} \times D^{4.87}}{10.67 \times Q^{1.852}}$$​

Solve for Flow Rate (Q)

$$Q = \left( \frac{h_f \times C^{1.852} \times D^{4.87}}{10.67 \times L} \right)^{\frac{1}{1.852}}$$​

Solve for Diameter (D)

$$D = \left( \frac{10.67 \times L \times Q^{1.852}}{h_f \times C^{1.852}} \right)^{\frac{1}{4.87}}$$​

C-Values

Higher C = smoother pipe.

Pipe Material	Typical C-Value
PVC / Plastic	140–150
HDPE	140
Copper	130–140
Ductile Iron (new)	130
Steel (new)	120
Cast Iron (new)	120
Cast Iron (old)	80–100
Concrete (new, lined)	120–140
Concrete (unlined)	100–120
Asbestos Cement	140
Galvanized Steel	110–120

For general municipal design, engineers often assume:

C = 130

Unit Conversion – mm to m

In engineering practice, pipe diameters are usually given in millimeters.

To convert:$$\text{Diameter (m)} = \frac{\text{Diameter (mm)}}{1000}$$Diameter (m)=1000Diameter (mm)​

Example:

150 mm pipe:$$150 / 1000 = 0.15 \, m$$150/1000=0.15m

Always convert mm to meters before using the equation in SI format.

Example Calculation

Given:

• L = 200 m
• D = 150 mm → 0.15 m
• Q = 0.02 m³/s
• C = 130

Substitute into formula to find head loss:$$h_f = 10.67 \times \frac{200}{130^{1.852} \times 0.15^{4.87}} \times 0.02^{1.852}$$

Result ≈ 2–3 meters of head loss

Hazen–Williams Calculator

US Formula Used:

$$h_f = \frac{4.52 \times L \times Q^{1.85}}{C^{1.85} \times D^{4.87}}$$​

Where:

• $L$ = feet
• $D$ = inches
• $Q$ = gallons per minute (gpm)
• $h_f$​ = feet of head loss

• EPANET – Official EPA Water Distribution Modeling Software
• WaterCAD (via Bentley OpenFlows Water)
• EPANET GitHub – Open Water Analytics Community