Which expression correctly relates Δp/γ to elevations, velocity heads, and head losses in the energy equation?

• A. Δp/γ = (z2 − z1) + (V2^2 − V1^2)/(2g) + h_L
• B. Δp/γ = (z1 − z2) − (V2^2 − V1^2)/(2g) + h_L
• C. Δp/γ = (z1 − z2) + (V2^2 − V1^2)/(2g) − h_L
• D. Δp/γ = (z1 − z2) + (V1^2 − V2^2)/(2g) + h_L

Answer: (C)

The energy equation with a head loss term shows how the pressure head difference relates to changes in elevation and velocity head as flow moves from one section to another. Starting from p1/γ + z1 + V1^2/(2g) = p2/γ + z2 + V2^2/(2g) − h_L, and defining Δp/γ as the downstream minus upstream pressure head difference (p2/γ − p1/γ), you can rearrange to get:

Δp/γ = (z1 − z2) + (V2^2 − V1^2)/(2g) − h_L.

This form makes the signs clear: the elevation difference contributes (z1 − z2), the change in velocity head contributes (V2^2 − V1^2)/(2g), and the head loss subtracts energy, hence −h_L. The other options mix signs of the elevation and velocity terms or treat the head loss with the opposite sign, which would not align with the standard energy balance given this definition of Δp/γ.