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:

For fluid dynamics problems, always identify the flow regime (laminar or turbulent) using the Reynolds number. Memorize the friction factor formula for laminar flow (\(f = 16/Re\)) as it's a frequently tested concept.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
The question asks for the mathematical relationship between the Fanning friction factor (\(f\)) and the Reynolds number (\(Re\)) for a specific flow condition: laminar flow in a smooth circular tube.
Step 2: Key Formula or Approach:
For laminar flow in a circular pipe, the velocity profile is parabolic, and this is described by the Hagen-Poiseuille equation. The friction factor can be derived from the pressure drop given by this equation. The Reynolds number for flow in a pipe is defined as \( Re = \frac{\rho v D}{\mu} \), where \(\rho\) is the fluid density, \(v\) is the average velocity, \(D\) is the pipe diameter, and \(\mu\) is the dynamic viscosity. Laminar flow typically occurs when \(Re<2100\).
Step 3: Detailed Explanation:
The Fanning friction factor (\(f\)) is defined as the ratio of the wall shear stress (\(\tau_w\)) to the kinetic energy per unit volume of the fluid, \( f = \frac{\tau_w}{\frac{1}{2}\rho v^2} \).
From the Hagen-Poiseuille equation for laminar flow, the wall shear stress is given by \( \tau_w = \frac{8 \mu v}{D} \).
Substituting this into the definition of the friction factor: \[ f = \frac{8 \mu v / D}{\frac{1}{2}\rho v^2} = \frac{16 \mu}{\rho v D} \] Recognizing that the Reynolds number is \( Re = \frac{\rho v D}{\mu} \), we can substitute it into the equation for \(f\): \[ f = \frac{16}{Re} \] This relationship is a fundamental result in fluid mechanics for laminar flow in pipes.
It is important to distinguish the Fanning friction factor (\(f\)) from the Darcy-Weisbach friction factor (\(f_D\)), where \(f_D = 4f\). For the Darcy factor, the relationship would be \(f_D = \frac{64}{Re}\). Since \( \frac{64}{Re} \) is not an option, the question is referring to the Fanning friction factor.
Step 4: Final Answer:
The correct relation is \( f = \frac{16}{Re} \).