ANS:B - first power of the roughness.

The losses in open channel flow generally vary as the first power of the roughness.

Explanation:

1. Open Channel Flow:
   • Open channel flow occurs when a fluid flows in a conduit with a free surface exposed to atmospheric pressure, such as rivers, streams, and canals.
2. Losses in Open Channel Flow:
   • Losses in open channel flow are primarily due to friction between the moving fluid and the channel bed and walls, as well as other factors like bends, expansions, contractions, and transitions.
   • The magnitude of these losses depends significantly on the roughness of the channel bed and walls.
3. Variation with Roughness:
   • Empirical studies and equations (such as Manning's formula for uniform flow) show that the losses in open channel flow vary approximately as the first power (linearly) of the roughness height or roughness coefficient nnn.
   • Roughness directly affects the frictional resistance encountered by the flowing fluid, influencing the overall energy losses in the system.
4. Other Options:
   • Inverse of the roughness: This is not generally true; losses do not vary inversely with roughness.
   • Square of the velocity: This relates to kinetic energy and is not directly related to losses due to roughness.
   • Inverse square of hydraulic radius: This describes the resistance coefficient in Manning's equation, but not the variation of losses with roughness.

Conclusion: