Question

When the flow of a fluid through a circular pipe, the friction factor is

• A. \( f = \frac{16}{N_{\text{Re}}} \)
• B. \( f = \frac{24}{N_{\text{Re}}} \)
• C. \( f = 0.079 N_{\text{Re}}^{-1/2} \)
• D. \( f = 0.079 N_{\text{Re}}^{-1/4} \)

Hint:

The Fanning friction factor for laminar flow in a circular pipe is \( f = \frac{16}{N_{\text{Re}}} \), while the Darcy friction factor is \( f = \frac{64}{N_{\text{Re}}} \). Be mindful of which definition is used.

Answer 1

The Correct Option is A

Step 1: Understand the friction factor in pipe flow.
The friction factor \( f \) (often the Darcy friction factor) relates the pressure drop due to friction in a pipe to the flow velocity, pipe diameter, and fluid properties. It depends on the flow regime, which is determined by the Reynolds number (\( N_{\text{Re}} = \frac{\rho v D}{\mu} \), where \( \rho \) is density, \( v \) is velocity, \( D \) is pipe diameter, and \( \mu \) is viscosity).
Step 2: Determine the flow regime and corresponding friction factor.
The problem does not specify the flow regime, but we can infer it from the correct answer. There are two main regimes for pipe flow:
Laminar flow (\( N_{\text{Re}}<2000 \)): The flow is smooth and orderly. For a circular pipe, the friction factor is derived from the Hagen-Poiseuille equation: \[ \Delta P = \frac{32 \mu L v}{D^2}, \] where \( \Delta P \) is the pressure drop, \( L \) is the pipe length. The Darcy friction factor is defined as: \[ \Delta P = f \frac{L}{D} \frac{\rho v^2}{2}. \] Equating the two expressions: \[ f \frac{L}{D} \frac{\rho v^2}{2} = \frac{32 \mu L v}{D^2}, \] \[ f = \frac{32 \mu v}{D \rho v^2} \cdot \frac{2D}{L} \cdot \frac{L}{D} = \frac{64 \mu}{\rho v D} = \frac{64}{N_{\text{Re}}}. \] Thus, for laminar flow, \( f = \frac{64}{N_{\text{Re}}} \).
Turbulent flow (\( N_{\text{Re}}>4000 \)): The friction factor depends on the pipe roughness and Reynolds number. For smooth pipes, empirical correlations like the Blasius formula are used: \[ f = 0.316 N_{\text{Re}}^{-1/4} \quad (\text{for } 4000<N_{\text{Re}}<10^5), \] or a simplified form: \[ f \approx 0.079 N_{\text{Re}}^{-1/4}. \] Another correlation for rough pipes or different \( N_{\text{Re}} \) ranges might use \( f \propto N_{\text{Re}}^{-1/2} \), but this is less common for smooth pipes.
Step 3: Match the correct answer to the flow regime.
The correct answer is \( f = \frac{16}{N_{\text{Re}}} \). Note that: \[ \frac{64}{N_{\text{Re}}} \text{ (Darcy friction factor)} = 4 \times \frac{16}{N_{\text{Re}}}. \] The factor of 4 difference arises because the problem likely uses the Fanning friction factor, which is defined differently: \[ f_{\text{Fanning}} = \frac{f_{\text{Darcy}}}{4}. \] So: \[ f_{\text{Fanning}} = \frac{64}{4 N_{\text{Re}}} = \frac{16}{N_{\text{Re}}}, \] which matches option (1). This suggests the problem refers to laminar flow using the Fanning friction factor.
Step 4: Evaluate the options.
(1) \( f = \frac{16}{N_{\text{Re}}} \): Correct, as this is the Fanning friction factor for laminar flow in a circular pipe. Correct.
(2) \( f = \frac{24}{N_{\text{Re}}} \): Incorrect, as this does not correspond to a standard friction factor for circular pipes (it might apply to non-circular ducts). Incorrect.
(3) \( f = 0.079 N_{\text{Re}}^{-1/2} \): Incorrect, as this form is not standard; turbulent flow typically uses \( N_{\text{Re}}^{-1/4} \). Incorrect.
(4) \( f = 0.079 N_{\text{Re}}^{-1/4} \): Incorrect, as this applies to turbulent flow (Blasius correlation), not laminar flow. Incorrect.

Step 5: Select the correct answer.
For laminar flow in a circular pipe, the friction factor (Fanning) is \( f = \frac{16}{N_{\text{Re}}} \), matching option (1).