Question

In design of underground pipeline, if \(v\) is velocity of flow, \(g\) is acceleration due to gravity and friction head loss at pipe entry is \(h_e\), which is the correct relationship

• A. \( h_e = \dfrac{v^2}{2g} \)
• B. \( h_e = 0.5\dfrac{v^2}{2g} \)
• C. \( h_e = 10\dfrac{v^2}{2g} \)
• D. \( h_e = 0.1\dfrac{v^2}{2g} \)

Hint:

Entrance head loss formula: \[ \boxed{ h_e = K_e\frac{v^2}{2g} } \] For ordinary sharp-edged pipe entry: \[ \boxed{ K_e = 0.5 } \]

Answer 1

The Correct Option is B

Concept: Whenever water enters a pipe from a reservoir or open source, some amount of energy is lost because of sudden contraction and turbulence developed at the pipe entrance. This energy loss is known as entry head loss or entrance loss. The general expression for entrance head loss is: \[ h_e = K_e \frac{v^2}{2g} \] where:
• \(h_e\) = entrance head loss
• \(K_e\) = entrance loss coefficient
• \(v\) = velocity of flow
• \(g\) = acceleration due to gravity For ordinary pipe entrance conditions used in underground pipeline design: \[ K_e = 0.5 \] Therefore: \[ h_e = 0.5\frac{v^2}{2g} \]

Step 1: Writing the standard formula for entry loss. The standard entrance loss equation is: \[ h_e = K_e\frac{v^2}{2g} \] This equation represents the energy lost due to turbulence and flow separation at the pipe entrance.

Step 2: Identifying the entrance loss coefficient. For normal sharp-edged pipe entry: \[ K_e = 0.5 \] This value is commonly used in irrigation engineering and underground pipeline hydraulics.

Step 3: Substituting the coefficient value. Substituting \(K_e = 0.5\): \[ h_e = 0.5\frac{v^2}{2g} \] Hence: \[ \boxed{ h_e = 0.5\frac{v^2}{2g} } \]

Step 4: Checking all options carefully. Option (A): \[ h_e = \frac{v^2}{2g} \] This corresponds to \(K_e = 1\), which is not generally used for normal underground pipe entry. Hence: \[ \boxed{ \text{Option (A) is incorrect} } \] Option (B): \[ h_e = 0.5\frac{v^2}{2g} \] This is the standard entrance loss formula. Hence: \[ \boxed{ \text{Option (B) is correct} } \] Option (C): \[ h_e = 10\frac{v^2}{2g} \] This value is extremely high and physically unrealistic for pipe entrance loss. Hence: \[ \boxed{ \text{Option (C) is incorrect} } \] Option (D): \[ h_e = 0.1\frac{v^2}{2g} \] This coefficient is too small for ordinary entrance conditions. Hence: \[ \boxed{ \text{Option (D) is incorrect} } \] Final Conclusion: The correct relationship for entrance head loss in underground pipeline design is: \[ \boxed{ h_e = 0.5\frac{v^2}{2g} } \] Hence the correct answer is: \[ \boxed{ (B) } \]