Question

According to the Darcy--Weisbach equation, the loss of head for flow through a circular pipe is (\( L \)-length of the pipe, \( D \)-diameter of the pipe, \( V \)-velocity of flow, \( f \)-friction factor)

• A. \( h_f = \frac{f L \sqrt{V}}{2 g D} \)
• B. \( h_f = \frac{f L V}{2 g D} \)
• C. \( h_f = \frac{f L V^2}{2 g D} \)
• D. \( h_f = \frac{f L V^3}{2 g D} \)

Hint:

The Darcy-Weisbach equation \( h_f = \frac{f L V^2}{2 g D} \) calculates frictional head loss, where the \( V^2 \) term reflects the kinetic energy of the flow.

Answer 1

The Correct Option is C

Step 1: Understand the Darcy-Weisbach equation.
The Darcy-Weisbach equation is used to calculate the head loss due to friction in a pipe flow. It accounts for the frictional resistance as fluid flows through a pipe. The equation is commonly expressed as: \[ h_f = f \frac{L}{D} \frac{V^2}{2 g}, \] where:
\( h_f \): Head loss due to friction (in units of length, e.g., meters),
\( f \): Friction factor (dimensionless),
\( L \): Length of the pipe,
\( D \): Diameter of the pipe,
\( V \): Average velocity of the flow,
\( g \): Acceleration due to gravity (\( \approx 9.81 \, \text{m/s}^2 \)).
Step 2: Verify the form of the equation.
The equation can be rewritten for clarity: \[ h_f = \frac{f L V^2}{2 g D}. \] This form matches the structure of the options provided. The head loss is proportional to the square of the velocity (\( V^2 \)), which comes from the kinetic energy term in the derivation of the equation (dynamic pressure \( \frac{1}{2} \rho V^2 \)). Step 3: Evaluate the options.
(1) \( h_f = \frac{f L \sqrt{V}}{2 g D} \): The velocity term should be \( V^2 \), not \( \sqrt{V} \). Incorrect.
(2) \( h_f = \frac{f L V}{2 g D} \): The velocity term should be \( V^2 \), not \( V \). Incorrect.
(3) \( h_f = \frac{f L V^2}{2 g D} \): This matches the Darcy-Weisbach equation exactly. Correct.
(4) \( h_f = \frac{f L V^3}{2 g D} \): The velocity term should be \( V^2 \), not \( V^3 \). Incorrect. Step 4: Select the correct answer.
The Darcy-Weisbach equation for head loss in a circular pipe is \( h_f = \frac{f L V^2}{2 g D} \), matching option (3).