In a straight pipe with major and minor losses, which expression represents the total head loss?

• A. h_L = h_f + Σ K_i (V^2/(2g)); h_f = f (L/D) (V^2/(2g)); K_i are minor loss coefficients.
• B. h_L = Q^2 /(2gA^2).
• C. h_L = p/(ρg).
• D. h_L = f (P) (V^2/(2g)).

Answer: (A)

In pipe flow, total head loss accumulates from two kinds of losses: friction along the pipe (major losses) and losses at fittings or other local disturbances (minor losses). The total head loss is written as h_L = h_f + Σ K_i (V^2/(2g)). Here, h_f is the major (friction) loss, given by h_f = f (L/D) (V^2/(2g)), where f is the Darcy–Weisbach friction factor, L is the pipe length, D is the diameter, and V is the mean flow velocity. The term Σ K_i (V^2/(2g)) represents all minor losses, with K_i being dimensionless loss coefficients for each fitting or component; each contributes a head loss equal to K_i times the velocity head V^2/(2g).

This form makes sense because every loss scales with the velocity head, so you can sum the friction loss and all local losses in a consistent way. The other expressions don’t represent total head loss: Q^2/(2gA^2) is just the velocity head V^2/(2g), not a loss; p/(ρg) is the pressure head, not a loss; and f (P) (V^2/(2g)) mixes concepts incorrectly and doesn’t reflect the standard separation into major and minor losses.