--- title: "Three reservoirs with two parallel pipes" subject: "Hydraulics" chapter: 3 key: "semester_4/hydraulics:three-reservoirs-five-pipes-parallel" source: https://www.bctengineer.com/q/semester_4/hydraulics/three-reservoirs-five-pipes-parallel site: BCT Engineering --- # Three reservoirs with two parallel pipes Hydraulics · Chapter 3: Three reservoirs problem and Pipe networks - Asked in: 2078 Chaitra, 2+2+4 marks - Video: [Fluid Mechanics: Topic 9.2.2 - Example of branched pipe system (three reservoir problem)](https://www.youtube.com/watch?v=l1QRiovdGMw) — CPPMechEngTutorials - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_4/hydraulics/three-reservoirs-five-pipes-parallel - Whole chapter: https://www.bctengineer.com/md/semester_4/hydraulics/chapter-3 ## Question Three reservoirs are connected as shown below, with the connecting pipes having the following properties (length, diameter, friction factor and minor losses) as shown below: | Pipe | L (m) | D (mm) | f | $\Sigma K$ | |---|---|---|---|---| | 1 | 280 | 500 | 0.021 | 3.5 | | 2 | 980 | 300 | 0.022 | 0.0 | | 3 | 270 | 550 | 0.017 | 0.0 | | 4 | 700 | 650 | 0.028 | 3.5 | | 5 | 780 | 750 | 0.030 | 1.5 | The corresponding reservoir elevation are: $Z_A$ = 185 m, $Z_B$ = 105 m and $Z_C$ = 70 m. [Figure: reservoir A connected by pipe 1 to junction D; reservoir B connected to D by pipe 2; pipes 3 and 4 run in parallel from D, then pipe 5 leads to reservoir C]. Calculate: (i) The relationship between the flow rates in the two parallel pipes that form part of the system. (ii) The value of the Hydraulic Grade Line (HGL) at junction D, and (iii) The flow rates in all five pipes for this value. ## Short answer (i) Q₃/Q₄ = √(r₄/r₃) = 1.438, so Q₃ = 1.438 Q₄. (ii) Hydraulic grade line at junction D is 108.95 m. (iii) Pipe flows: Q₁=1.942, Q₂=0.073 (to B), Q₃=1.102, Q₄=0.766, Q₅=1.868 m³/s. ## Answer **Layout assumed from the figure.** A feeds junction D by pipe 1; B joins D by pipe 2; from D two pipes, 3 and 4, run in parallel to a junction E; pipe 5 then runs from E to reservoir C. Minor losses are included through $\sum K$: for each pipe the head loss is $h = \left(\dfrac{fL}{D} + \sum K\right)\dfrac{V^2}{2g} = rQ^2$, with $$ r = \frac{8}{\pi^2gD^4}\left(\frac{fL}{D} + \sum K\right) $$ | Pipe | $fL/D$ | $\sum K$ | r (s²/m⁵) | |---|---|---|---| | 1 | 11.76 | 3.5 | 20.174 | | 2 | 71.87 | 0 | 733.10 | | 3 | 8.345 | 0 | 7.536 | | 4 | 30.15 | 3.5 | 15.578 | | 5 | 31.20 | 1.5 | 8.539 | #### (i) Relation between the flows in the two parallel pipes They have the same head loss, so $r_3Q_3^2 = r_4Q_4^2$: $$ \frac{Q_3}{Q_4} = \sqrt{\frac{r_4}{r_3}} = \sqrt{\frac{15.578}{7.536}} = 1.438 \;\Rightarrow\; Q_3 = 1.438\,Q_4 $$ The equivalent resistance of the pair is given by $\dfrac{1}{\sqrt{r_{par}}} = \dfrac{1}{\sqrt{r_3}} + \dfrac{1}{\sqrt{r_4}}$, so $r_{par} = 2.621$ s²/m⁵, and $Q_5 = Q_3 + Q_4$. #### (ii) HGL at junction D Let $H_D$ be the HGL at D. Pipe 5 is in series with the parallel pair (3, 4), so $H_D - Z_C = (r_{par} + r_5)Q_5^2$. - Pipe 1: $Q_1 = \sqrt{(185 - H_D)/r_1}$ (A to D) - Pipe 2: $Q_2 = \sqrt{|105 - H_D|/r_2}$, towards B if $H_D > 105$, from B if $H_D < 105$ - Pipe 5: $Q_5 = \sqrt{(H_D - 70)/(r_{par} + r_5)}$ - Continuity at D: $Q_1 = Q_2 + Q_5$ (when B is receiving), or $Q_1 + Q_2 = Q_5$ (when B is supplying). Trials ($Q_2 > 0$ means B supplies D, $Q_2 < 0$ means B receives): | $H_D$ (m) | $Q_1$ | $Q_2$ | $Q_5$ | $Q_1 + Q_2 - Q_5$ | |---|---|---|---|---| | 100 | 2.053 | +0.083 | 1.640 | +0.496 | | 105 | 1.991 | +0.000 | 1.771 | +0.220 | | 108 | 1.954 | -0.064 | 1.845 | +0.044 | | 110 | 1.928 | -0.083 | 1.893 | -0.048 | The sum changes sign between 108 and 110 m, and the balance is at $$ H_D = 108.95\ \text{m} $$ Since $H_D > 105$ m, pipe 2 carries water **to** reservoir B. #### (iii) Flow rates | Pipe | Q (m³/s) | Direction | |---|---|---| | 1 | 1.942 | A to D | | 2 | 0.073 | D to B | | 3 | 1.102 | D to E | | 4 | 0.766 | D to E | | 5 | 1.868 | E to C | Pipe 5: $Q_5 = \sqrt{(108.95 - 70)/(2.621 + 8.539)} = 1.868$ m³/s. The HGL at junction E is $70 + r_5Q_5^2 = 99.80$ m, so $Q_3 = \sqrt{(108.95 - 99.80)/7.536} = 1.102$ and $Q_4 = \sqrt{(108.95 - 99.80)/15.578} = 0.766$ m³/s. Continuity check at D: $Q_1 = 1.942 = Q_2 + Q_5 = 0.073 + 1.868$. **Answer:** (i) $Q_3 = 1.438\,Q_4$; (ii) HGL at D $\approx 109.0$ m; (iii) $Q_1 = 1.942$, $Q_2 = 0.073$ (to B), $Q_3 = 1.102$, $Q_4 = 0.766$, $Q_5 = 1.868$ m³/s. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Pipes 3 and 4 are in parallel. What is the relation between their flows? A. Q3 = 0.695 Q4 + Q5 B. Q3 = 1.438 Q4, from r3 Q3^2 = r4 Q4^2 C. Q3 = Q4, because they have the same head loss D. Q4 = 1.438 Q3 Answer: B. Parallel pipes share the same head loss, so Q3/Q4 = sqrt(r4/r3) = sqrt(15.578/7.536) = 1.438. (Key point: Flow relation in parallel pipes) 2. How are minor losses included in the resistance r of a pipe? A. r = 8/(pi^2 g D^4) x (fL/D + sum K) B. r = 8fL/(pi^2 g D^5) and K is added to H_D C. K is ignored because velocity head is small D. r = 8/(pi^2 g D^4) x (fL/D) x sum K Answer: A. The total loss coefficient is (fL/D + sum K), multiplied by V^2/(2g) and expressed through Q. (Key point: Resistance with minor losses) 3. H_D is found to be 108.95 m while Z_B = 105 m. What does this show for pipe 2? A. Water flows from B to D, so B supplies B. There is no flow in pipe 2 C. Pipe 2 must be closed D. Water flows from D to B, so B receives Answer: D. If the junction head exceeds the reservoir level, water flows into that reservoir; continuity is Q1 = Q2 + Q5. (Key point: Direction of flow in pipe 2) 4. Which relation gives the flow in pipe 5 once H_D is known? A. Q5 = Q3 - Q4 B. Q5 = sqrt((185 - H_D)/r5) C. Q5 = sqrt((H_D - 70)/(r_par + r5)) and Q5 = Q3 + Q4 D. Q5 = sqrt((H_D - 70)/r5) only Answer: C. The parallel pair is in series with pipe 5 to reservoir C at 70 m, giving Q5 = 1.868 m^3/s and splitting into Q3 and Q4. (Key point: Flow in pipe 5) ## Related in Hydraulics - [Three reservoirs with 250 mm concrete pipes](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Athree-reservoir-concrete-pipes-flow-rates) — Hydraulics, chapter 3, asked 2× - [Three reservoirs: pipe AEC length and level B](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Athree-reservoir-pipe-aec-length-zb) — Hydraulics, chapter 3 - [Three reservoirs with pump in pipe 2](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Apump-three-reservoir-flow-and-head-j) — Hydraulics, chapter 3 - [Manometer reading in pipe with minor losses](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Amanometer-reading-pipe-minor-losses) — Hydraulics, chapter 2 - [Siphon with 90 mm pipe, number and length](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Asiphon-90mm-pipe-total-length) — Hydraulics, chapter 2 - [Series-parallel pipes with control valve](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Aseries-parallel-valve-equal-discharge) — Hydraulics, chapter 2, asked 2× ## Related in other subjects - [Friction pile length, 300 mm, 250 kN](https://www.bctengineer.com/question.md?key=semester_5%2Ffoundation_engineering%3Afriction-pile-300mm-250kn) — Foundation Engineering, chapter 8 - [Series and parallel pipe network](https://www.bctengineer.com/question.md?key=semester_4%2Ffluid_mechanics_mechanical%3Aseries-parallel-pipe-network-discharge) — Fluid Mechanics, chapter 8 - [Flow between reservoirs (15 cm and 25 cm pipe)](https://www.bctengineer.com/question.md?key=semester_6%2Fhydro_power%3Areservoirs-pipeline-15-25cm) — Hydro Power, chapter 1 - [Friction pile length, 350 mm, 1200 kN](https://www.bctengineer.com/question.md?key=semester_5%2Ffoundation_engineering%3Afriction-pile-350mm-1200kn) — Foundation Engineering, chapter 8