How is the total head loss in a pipe with both major and minor losses computed?

• A. h_L = h_f - Σ K_i (V^2/(2g))
• B. h_L = h_f + Σ K_i (V^2/(2g))
• C. h_L = Σ K_i (V^2/(2g))
• D. h_L = h_f × Σ K_i

Answer: (B)

Total head loss is the sum of all energy-dissipation sources in the flow: the frictional loss along the pipe length (major loss) and the losses caused by fittings, valves, bends, and other disturbances (minor losses). The friction loss is represented by h_f, while each minor loss is h_L,i = K_i (V^2/(2g)), where K_i is the loss coefficient for that component and V^2/(2g) is the velocity head. Therefore, the total head loss is h_L = h_f + Σ K_i (V^2/(2g)). This can also be written as h_L = (V^2/(2g)) [f (L/D) + Σ K_i], showing that both types of losses contribute additively to the total head loss. The units are consistent since V^2/(2g) is a head (length) and the K’s are dimensionless.