Question

The relationship between fanning friction factor (f) and Reynolds number (N$_{Re}$) in laminar flow is

• A. \( f = \frac{0.079}{N_{Re}^{0.2}} \)
• B. \( f = \frac{0.316}{N_{Re}^{0.2}} \)
• C. \( f = \frac{16}{N_{Re}} \)
• D. \( f = \frac{64}{N_{Re}} \)

Hint:

Always check whether a question asks for the Fanning or Darcy friction factor.
Fanning friction factor: \( f = 16/N_{Re} \)
Darcy/Blasius friction factor: \( f_D = 64/N_{Re} \)
Remembering this distinction will help you avoid simple calculation mistakes.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
The question asks for the mathematical relationship between the Fanning friction factor and the Reynolds number for laminar flow inside a circular pipe.
This is a classic relation in fluid dynamics.

Step 2: Key Formula or Approach:
In laminar flow (where Reynolds number \( N_{Re} \le 2100 \)), the pressure drop can be determined analytically using the Hagen-Poiseuille equation:
\[ \Delta P = \frac{32 \cdot \mu \cdot L \cdot v}{D^2} \]
We can also express the pressure drop in terms of the Fanning friction factor \( f \):
\[ \Delta P = \frac{4 \cdot f \cdot L \cdot \rho \cdot v^2}{2 \cdot D} \]

Step 3: Detailed Explanation:

• Equate the two expressions for the pressure drop:
\[ \frac{4 \cdot f \cdot L \cdot \rho \cdot v^2}{2 \cdot D} = \frac{32 \cdot \mu \cdot L \cdot v}{D^2} \]

• Simplify the equation by canceling common terms (\( L, v, D \)):
\[ \frac{2 \cdot f \cdot \rho \cdot v}{1} = \frac{32 \cdot \mu}{D} \]

• Rearrange the terms to solve for the Fanning friction factor \( f \):
\[ f = \frac{16 \cdot \mu}{\rho \cdot v \cdot D} \]

• Since the Reynolds number is defined as \( N_{Re} = \frac{\rho \cdot v \cdot D}{\mu} \), we can substitute it into the expression:
\[ f = \frac{16}{N_{Re}} \]

• For comparison, the Darcy-Weisbach friction factor \( f_D \) is defined as \( 4f \), which gives \( f_D = \frac{64}{N_{Re}} \).


Step 4: Final Answer:
The Fanning friction factor in laminar flow is related to the Reynolds number by \( f = \frac{16}{N_{Re}} \).