Question

The head loss (\( K_L \)) associated with the flow entry of water to an internal passage depends on the shape of the entry. The following figure shows three different types of flow entry into a pipe. Which one of the following relationships correctly represents the head loss associated with the three different flow entries?
\includegraphics[width=0.75\linewidth]{imager7.png}

• A. \( (K_L)_a > (K_L)_b > (K_L)_c \)
• B. \( (K_L)_b > (K_L)_a > (K_L)_c \)
• C. \( (K_L)_b \leq (K_L)_a = (K_L)_c \)
• D. \( (K_L)_b < (K_L)_a < (K_L)_c \)

Hint:

When analyzing head loss due to flow entry, the more abrupt the change in direction or shape, the greater the turbulence and the higher the head loss.

Answer 1

The Correct Option is B

Head loss \( K_L \) in pipe flow is influenced by the geometry of the flow entry. A sharp or abrupt entry tends to cause higher turbulence, resulting in greater head loss.

Step 1: Analyze the shapes of the flow entries.
- Flow entry (a) represents a relatively smooth, sharp entry into the pipe. This entry creates minimal turbulence and hence, lower head loss compared to the others.
- Flow entry (b) involves a more abrupt change in direction, resulting in more turbulence and, thus, higher head loss.
- Flow entry (c) represents a smoother curve, resulting in the least turbulence and, therefore, the least head loss.

Step 2: Conclusion.
From this analysis, we conclude that \( (K_L)_b > (K_L)_a > (K_L)_c \), as the abrupt change in flow direction at entry (b) causes the highest turbulence and thus the greatest head loss. The correct answer is (B).

Final Answer: (B) \( (K_L)_b > (K_L)_a > (K_L)_c \)