Head loss due to a sudden enlargement in a pipe (velocities v1 before and v2 after enlargement) is given by:

Introduction / Context:
Energy losses at geometric discontinuities (like sudden enlargement) are major components of minor losses in pipe systems. Correctly recalling the Carnot–Borda formula helps estimate additional head needed and pump sizing.

Given Data / Assumptions:

• Incompressible flow of a liquid (e.g., water).
• Steady flow with a sudden area increase from section 1 to section 2.
• Negligible elevation change between sections.

Concept / Approach:
Downstream of a sudden expansion, the separated eddies dissipate kinetic energy. The loss equals the difference between the incoming kinetic energy and the recoverable kinetic energy associated with the uniform downstream velocity distribution, which leads to a loss proportional to the square of the velocity difference.

Step-by-Step Solution:
Start with energy grade: h_L = (α1 v1^2 − α2 v2^2) / (2 g) − pressure recovery term.For sudden expansion with uniform profiles, the classical result simplifies to h_L = (v1 − v2)^2 / (2 g).This is the Carnot–Borda loss expression.

Verification / Alternative check:
Setting v1 = v2 gives zero loss as expected; a large expansion (v2 ≈ 0) yields h_L ≈ v1^2/(2 g), matching the intuition of losing nearly all incoming kinetic energy.

Why Other Options Are Wrong:
(v1^2 − v2^2)/(2 g) or (v2^2 − v1^2)/(2 g): these are simple kinetic energy differences, not the correct dissipation measure.(v1 + v2)^2/(2 g): unphysical; predicts excessive loss even if v1 ≈ v2.

Common Pitfalls:

• Confusing sudden enlargement with sudden contraction loss formulas.
• Ignoring velocity coefficients (α) where profiles are highly non-uniform.

Final Answer:
(v1 - v2)^2 / (2 g)