Question

For laminar fluid flow through a smooth circular tube, the relation between friction factor (\(f\)) and Reynolds number (\(Re\)) is

• A. \( f = \frac{16}{Re} \)
• B. \( f = \frac{24}{Re} \)
• C. \( f = \frac{16}{\sqrt{Re}} \)
• D. \( f = \frac{24}{\sqrt{Re}} \)

Hint:

Be aware of the two friction factors: Darcy (\(64/Re\)) and Fanning (\(16/Re\)). Usually, exam options will only include one of these correct forms, making the choice clear. If you see \(16/Re\) as an option for laminar pipe flow, it is almost certainly the intended correct answer.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
This question asks for the relationship that defines the friction factor for laminar flow in a pipe. It's important to know that there are two common definitions for the friction factor, which differ by a factor of four.
Step 2: Key Formula or Approach:
The two main friction factors are: Darcy Friction Factor (\(f_D\)): Used in the Darcy-Weisbach equation \(\Delta P = f_D \frac{L}{D} \frac{\rho v^2}{2}\). For laminar flow (\(Re<2100\)), it is defined as: \[ f_D = \frac{64}{Re} \] Fanning Friction Factor (\(f_F\)): Defined as \(f_F = \frac{\tau_w}{\frac{1}{2}\rho v^2}\), where \(\tau_w\) is the wall shear stress. It is related to the Darcy factor by \(f_F = f_D / 4\). For laminar flow, it is defined as: \[ f_F = \frac{16}{Re} \] In many engineering contexts, particularly in chemical engineering, the symbol \(f\) without a subscript refers to the Fanning friction factor. Given the options, \(\frac{16}{Re}\) is available, which corresponds to the Fanning friction factor.
Step 3: Detailed Explanation:
The flow is specified as laminar through a smooth circular tube. For this condition, the Hagen-Poiseuille equation can be used to derive the friction factor. The derivation shows a direct inverse relationship between the friction factor and the Reynolds number.
- Option (A) \(f = \frac{16}{Re}\) is the correct formula for the Fanning friction factor.
- If the question intended the Darcy friction factor, the correct answer would be \(f = \frac{64}{Re}\), which is not an option.
- Options (C) and (D) with \(\sqrt{Re}\) are incorrect for laminar flow and are more related to expressions for turbulent flow over flat plates or in pipes under certain assumptions.
Step 4: Final Answer:
Assuming \(f\) represents the Fanning friction factor, the correct relation is \(f = \frac{16}{Re}\).
Step 5: Why This is Correct:
This is a standard, fundamental formula in fluid mechanics for laminar flow in pipes. The choice of the Fanning friction factor convention is confirmed by the available options.