The Darcy-Weisbach Equation: Mastering Pipe Pressure Drop

Introduction to Pressure Drop

When fluid flows through a pipe, it experiences resistance due to shear stress at the pipe wall (friction) and internal viscous forces. This resistance causes a loss of mechanical energy, observed as a drop in pressure along the length of the pipe.

The most accurate and universally accepted method for calculating this frictional head loss in full-flowing pipes is the Darcy-Weisbach equation.

The Darcy-Weisbach Equation

The equation expresses the frictional head loss ($h_f$) as a function of the pipe dimensions, fluid velocity, and a friction factor:

Where:

• $h_f$ = Frictional head loss (meters of fluid)
• $f_D$ = Darcy friction factor (dimensionless)
• $L$ = Length of the pipe (m)
• $D$ = Internal diameter of the pipe (m)
• $v$ = Average fluid velocity (m/s)
• $g$ = Acceleration due to gravity (9.81 $m/s^2$)

Head Loss vs. Pressure Drop

The equation above yields head loss, which is energy per unit weight of fluid. Process engineers usually need the actual pressure drop ($\Delta P$). To convert head loss to pressure drop, multiply by the fluid density and gravity:

Where:

• $\Delta P$ = Pressure drop (Pascals, Pa)
• $\rho$ = Fluid density ($kg/m^3$)

The Friction Factor ($f_D$) Controversy: Darcy vs. Fanning

It is critical to know which friction factor you are using. There are two common definitions in chemical and mechanical engineering:

1. Darcy Friction Factor ($f_D$): Used in the standard Darcy-Weisbach equation (common in mechanical/civil engineering).
2. Fanning Friction Factor ($f_f$): Based on wall shear stress (common in chemical engineering textbooks like Perry's Handbook).

The relationship is simple but causes countless errors if mixed up:

If you use the Fanning friction factor, the pressure drop equation becomes: $\Delta P = 4 f_f \frac{L}{D} \frac{\rho v^2}{2}$. Always verify which friction factor a software tool or chart is providing!

How to Find the Darcy Friction Factor

The value of $f_D$ depends entirely on the flow regime (determined by the Reynolds number, $Re$):

• Laminar Flow ($Re < 2000$): The friction factor is independent of pipe roughness.
  $f_D = \frac{64}{Re}$
• Turbulent Flow ($Re > 4000$): The friction factor depends on both $Re$ and the relative roughness of the pipe ($\varepsilon/D$). It is found using the Moody Diagram or the Colebrook-White equation.

Worked Example

Problem: Calculate the pressure drop in 500 meters of 6-inch Schedule 40 carbon steel pipe (ID = 154.1 mm, roughness $\varepsilon = 0.045$ mm) carrying water at 20°C ($\rho = 998 \ kg/m^3$, $\mu = 1.002$ cP) at a flow rate of $100 \ m^3/h$.

Step 1: Calculate Velocity

Step 2: Calculate Reynolds Number

Step 3: Determine Friction Factor

Relative roughness: $\varepsilon / D = 0.045 / 154.1 = 0.00029$.
Using the Colebrook equation (or a Moody chart) for $Re = 228,700$ and $\varepsilon/D = 0.00029$, we find: $$ f_D \approx 0.0175 $$

Step 4: Calculate Pressure Drop

The pressure drop through the 500 meters of piping is approximately 0.63 bar.

Try the Calculator

Use the Friction Factor Calculator on this page to instantly calculate the Darcy and Fanning friction factors for your specific pipe material and flow conditions, bypassing the complex Colebrook iteration.