--- title: "Three reservoirs: pipe AEC length and level B" subject: "Hydraulics" chapter: 3 key: "semester_4/hydraulics:three-reservoir-pipe-aec-length-zb" source: https://www.bctengineer.com/q/semester_4/hydraulics/three-reservoir-pipe-aec-length-zb site: BCT Engineering --- # Three reservoirs: pipe AEC length and level B Hydraulics · Chapter 3: Three reservoirs problem and Pipe networks - Asked in: 2082 Kartik, 8 marks - Video: [Unit:2 | Pipe Networking | Numerical from Three Reservoirs| Types:1 | Prashant YT | BE Civil |](https://www.youtube.com/watch?v=pIXbQd4ZiiM) — Prashant YT - Page: https://www.bctengineer.com/q/semester_4/hydraulics/three-reservoir-pipe-aec-length-zb - Whole chapter: https://www.bctengineer.com/md/semester_4/hydraulics/chapter-3 ## Question For the following three reservoir problem, calculate length of pipe AEC and elevation of reservoir B. [Figure: reservoir A at elevation 40 m feeds two parallel pipes AEC and AFC to junction C; pipe BC joins reservoir B (elevation $Z_B$ = ?) to C; pipe CD runs from C to a free outlet D at EL = 10 m; datum below] | Pipe | L (m) | D (m) | f | |---|---|---|---| | AEC | ? | 0.60 | 0.03 | | AFC | 1500 | 0.75 | 0.03 | | BC | 1200 | 0.50 | 0.04 | | CD | 3000 | 0.40 | 0.02 | ## Short answer Assuming equal discharge in parallel pipes, L_AEC = L_AFC (D_AEC/D_AFC)⁵ = 491.5 m. Total resistance A–D yields Q_CD ≈ 0.248 m³/s; with zero flow in BC, Z_B ≈ 39.8 m. ## Answer **Data missing from the question.** The figure gives only the levels (A = 40 m, outlet D = 10 m) and the pipe table; no discharge is given, and two quantities are unknown ($L_{AEC}$ and $Z_B$). As the paper says "assume suitable data", the following standard conditions are assumed: 1. The two parallel pipes AEC and AFC (which together join A to C) carry **equal discharges**. 2. Reservoir B neither supplies nor takes water (no flow in BC), so its level equals the piezometric head at C. If B supplies a discharge $Q_B$ the level would be $Z_B = H_C + r_{BC}Q_B^2$. Friction losses only; $h_f = rQ^2$ with $r = \dfrac{8fL}{\pi^2gD^5}$. #### Length of pipe AEC Parallel pipes have the same head loss: $\dfrac{f_1L_1Q_1^2}{D_1^5} = \dfrac{f_2L_2Q_2^2}{D_2^5}$. With $f$ equal and $Q_1 = Q_2$: $$ L_{AEC} = L_{AFC}\left(\frac{D_{AEC}}{D_{AFC}}\right)^5 = 1500\left(\frac{0.60}{0.75}\right)^5 = 1500(0.32768) = 491.5\ \text{m} $$ (General case: $L_{AEC} = L_{AFC}\left(\dfrac{D_{AEC}}{D_{AFC}}\right)^5\left(\dfrac{Q_{AFC}}{Q_{AEC}}\right)^2$.) #### Discharge from A to D Resistance of AFC: $r_{AFC} = 15.67$ s²/m⁵. Since AEC has the same resistance (equal $L/D^5$ ratio and flow), the pair in parallel has $r_{par} = r_{AFC}/4 = 3.917$ s²/m⁵. Resistance of CD: $r_{CD} = \dfrac{8(0.02)(3000)}{\pi^2(9.81)(0.4)^5} = 484.1$ s²/m⁵. Between the levels of A and D: $$ 40 - 10 = (r_{par} + r_{CD})Q^2 \;\Rightarrow\; Q = \sqrt{\frac{30}{3.917 + 484.1}} = 0.2479\ \text{m}^3/\text{s} $$ Each parallel pipe carries 0.1240 m³/s. #### Elevation of reservoir B (no flow in BC) Head loss in the parallel pair $= r_{par}Q^2 = 0.24$ m; in CD $= r_{CD}Q^2 = 29.76$ m. The piezometric head at the junction C is the head at D plus the loss in CD: $$ H_C = 10 + 29.76 = 39.76\ \text{m} $$ $$ Z_B = H_C = 39.76\ \text{m} $$ **Answer:** $L_{AEC} \approx 491.5$ m (about 492 m) and $Z_B \approx 39.8$ m, with $Q \approx 0.248$ m³/s in CD under the stated assumptions. (For a B that supplies water, $Z_B$ is higher than this by $r_{BC}Q_B^2$, with $r_{BC} = 126.9$ s²/m⁵.) ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What is the key property of parallel pipes AEC and AFC? A. They have the same head loss between A and C B. They have the same discharge always C. They have the same length D. They have the same velocity Answer: A. Parallel pipes connect the same two nodes, so their head losses are equal. (Key point: Parallel pipe head loss) 2. With equal f and equal Q, which formula gives L_AEC? A. L_AFC (D_AEC/D_AFC)^5 B. L_AFC (D_AEC/D_AFC)^2 C. L_AFC (D_AFC/D_AEC)^5 D. L_AFC Answer: A. f L Q^2/D^5 equal gives L proportional to D^5, so 1500 x 0.8^5 = 491.5 m. (Key point: Length of AEC) 3. What is assumed to find Z_B? A. No flow in BC, so Z_B equals piezometric head at C B. B supplies 100 l/s C. Z_B = 40 m D. B is empty Answer: A. If no flow occurs in BC, B's level equals H_C. (Key point: Assumption on B) 4. How is H_C found? A. H_C = 10 + head loss in CD = 39.76 m B. H_C = 40 m C. H_C = 40 - loss in CD D. H_C = 10 m Answer: A. C is above outlet D by the friction loss in CD, 29.76 m. 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