Total Head Loss from a Reservoir-Fed Pipe Test: 04-BS-7, May 2016, Q5

Question 5: Total Head Loss from a Reservoir-Fed Pipe Test (5 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given.

Quantity	Value
Pipe inlet elevation z₁	6 m
Reservoir surface above the inlet	3 m (⇒ surface at z = 9 m)
Pipe outlet elevation z₂ (open to atmosphere)	4 m
Pipe diameter D	50 mm
Pipe length L	120 m
Flow rate Q	0.006 m³/s

Find. The total head loss h_L in the system.

Fig. Q5 — reservoir at +9 m feeds a 50 mm, 120 m pipe discharging at +4 m; the HGL (red) drops 4.52 m of total head loss below the reservoir level.

Approach. Apply the steady-flow energy equation between the reservoir free surface (V ≈ 0, p = 0 gauge) and the open pipe outlet (p = 0 gauge, V = Q/A), with the smooth entrance meaning all losses are lumped into a single h_L term.

Exit velocity. $$A = \frac{\pi}{4}(0.050)^2 = 1.9635\times10^{-3}\ \text{m}^2, \qquad V = \frac{Q}{A} = \frac{0.006}{1.9635\times10^{-3}} = 3.056\ \text{m/s}$$
Energy equation, reservoir surface (z=9 m) to pipe outlet (z=4 m). $$0 + 9 + 0 = 0 + 4 + \frac{V^2}{2g} + h_L$$
Solve for h_L. $$h_L = (9-4) - \frac{V^2}{2g} = 5 - \frac{3.056^2}{2(9.81)} = 5 - 0.476 = \boxed{4.52\ \text{m}}$$

Quantity	Result
Exit velocity V	3.06 m/s
Total head loss h_L	4.52 m

Topic: Energy (Bernoulli + losses) equation for a reservoir-fed pipeline.

Key relations: p/ρg + z + V²/2g = const − h_L; exit kinetic energy V²/2g is NOT itself a "loss" — it is energy leaving with the jet.

Why this works. With a smooth (loss-free) entrance and an open free discharge, the ONLY places energy can go missing between the reservoir surface and the jet are friction and any fittings along the 120 m of pipe, so a single energy-balance equation isolates that total loss directly from the measured flow rate.

Common pitfall. Forgetting to include the 3 m of reservoir depth above the pipe connection when finding the total driving elevation (it is 9 m to the free surface, not 6 m to the connection) understates h_L by exactly 3 m.

Source: White, Fluid Mechanics, Ch. 3 (energy equation), Ch. 6 (pipe flow head loss).

Question 5 of 13: Total Head Loss from a Reservoir-Fed Pipe Test

Question 5: Total Head Loss from a Reservoir-Fed Pipe Test (5 marks)