Which formula correctly expresses the head loss across a sudden expansion in terms of V1, V2, and g?

• A. h_L = (V1 - V2)^2 /(2g)
• B. h_L = (V1^2 - V2^2)/(2g)
• C. h_L = (V1 + V2)^2 /(2g)
• D. h_L = (V1 - V2)/(2g)

Answer: (C)

When flow moves from a smaller to a larger cross-section, the velocity drops from V1 to V2 and some energy is dissipated as turbulence and mixing, appearing as head loss. The standard way to express this loss uses the decrease in kinetic energy across the expansion. From the continuity relation V2 = (A1/A2) V1, the head loss can be written as h_L = (V1 − V2)^2 / (2g). This form shows the loss depends on how much the velocity decreases: the larger the drop from V1 to V2, the greater the head loss, and it also connects directly to the upstream velocity via h_L = V1^2 (1 − A1/A2)^2 / (2g).

A quick check helps intuition: if the downstream area doubles, V2 = V1/2, so h_L = (V1/2)^2 / (2g) = V1^2/(8g), which matches the idea that head loss scales with the square of the change in velocity.

Using the sum of velocities, (V1 + V2)^2 /(2g), would imply energy changes in a way that does not reflect the actual loss from the velocity drop, so it does not describe the head loss for a sudden expansion.