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The head loss in a pipe of diameter d carrying oil at a flow rate Q over a distance l is h. The pipe is replaced by another with half the diameter, all other things remaining the same. The head loss in this case will be
  • a)
    0.5h    
  • b)
    2.0h
  • c)
    8.0h    
  • d)
    32.0h
Correct answer is option 'D'. Can you explain this answer?
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Head Loss in a Pipe

Head loss is the loss of pressure in a pipeline due to the friction between the fluid and the walls of the pipe. It is a crucial factor in the design and operation of pipelines.

Head Loss Equation

The head loss in a pipe can be calculated using the Darcy-Weisbach equation:

h = f (L / d) (V² / 2g)

where h is the head loss, f is the friction factor, L is the length of the pipe, d is the diameter of the pipe, V is the velocity of the fluid, and g is the acceleration due to gravity.

Effect of Diameter on Head Loss

The head loss in a pipe is directly proportional to the length of the pipe and the square of the velocity of the fluid, and inversely proportional to the diameter of the pipe and the friction factor.

When the diameter of the pipe is reduced by half, the velocity of the fluid will double to maintain the same flow rate. This will result in a four-fold increase in the head loss due to the quadratic relationship between velocity and head loss.

Therefore, the head loss in the new pipe with half the diameter will be 32 times the head loss in the original pipe.

Answer: Option D (32.0h)
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