---
title: "Three reservoirs with pressurised closed tank"
subject: "Hydraulics"
chapter: 3
key: "semester_4/hydraulics:three-reservoir-closed-tank-pressure"
source: https://www.bctengineer.com/q/semester_4/hydraulics/three-reservoir-closed-tank-pressure
site: BCT Engineering
---

# Three reservoirs with pressurised closed tank

Hydraulics · Chapter 3: Three reservoirs problem and Pipe networks

- Asked in: 2075 Baisakh, 8 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-reservoir-closed-tank-pressure
- Whole chapter: https://www.bctengineer.com/md/semester_4/hydraulics/chapter-3

## Question

For the reservoir system shown in figure, determine the flow in each pipe. At C, the pipe discharges into the atmosphere at an elevation of 140.00 m and at Tank B, the top is closed with pressure of 667 kN/m². Take f = 0.02 for all pipes and use following data:

| Pipe | Diameter | Length |
|---|---|---|
| 1 | 15 cm | 800 m |
| 2 | 20 cm | 500 m |
| 3 | 30 cm | 600 m |

[Figure: reservoir A at EL 200.00 m; closed tank B at EL 170.00 m with 667 kN/m² on top; junction J; pipe 1 from A to J, pipe 2 from J to B, pipe 3 from J to C; C discharges to atmosphere at EL 140.00 m]

## Short answer

Tank B’s piezometric head is 170 + 667/9.81 = 238.0 m, making it the highest source. Trial values of the junction head HJ give HJ ≈ 160.0 m, yielding final flows: pipe 1 (A→J) 47.9 L/s, pipe 2 (B→J) 173.7 L/s, pipe 3 (J→C) 221.6 L/s.

## Answer

**Data.** A: water level 200 m. B: closed tank, water level 170 m, surface pressure 667 kN/m². C: outlet at 140 m discharging to the atmosphere. $f = 0.02$ for all pipes. Friction losses only (velocity heads and minor losses neglected). $h_f = rQ^2$, $r = \dfrac{8fL}{\pi^2gD^5}$.

#### Piezometric head of tank B
$$
H_B = 170 + \frac{667}{9.81} = 170 + 67.99 = 237.99\ \text{m}
$$
This is **higher than A** (200 m). So the effective heads are: B = 238.0 m, A = 200 m, C = 140 m. Tank B is the highest source, A is the middle one, and C is the lowest (free outlet).

| Pipe | D (m) | L (m) | r (s²/m⁵) |
|---|---|---|---|
| 1 (A-J) | 0.15 | 800 | 17409.4 |
| 2 (J-B) | 0.20 | 500 | 2582.1 |
| 3 (J-C) | 0.30 | 600 | 408.0 |

#### Direction of flows
At first assume $H_J = 200$ m (A just balanced): $Q_2 = \sqrt{(237.99 - 200)/2582.1} = 0.1213$ m³/s into J, $Q_3 = \sqrt{60/408.0} = 0.3835$ m³/s out. The outflow exceeds the inflow, so A must also supply J, and $H_J < 200$ m. Hence: **B and A both feed J; J feeds C**: $Q_3 = Q_1 + Q_2$.

#### Trial of $H_J$
| $H_J$ (m) | $Q_1$ (A to J) | $Q_2$ (B to J) | $Q_3$ (J to C) | $Q_1 + Q_2 - Q_3$ |
|---|---|---|---|---|
| 200.0 | 0.0000 | 0.1213 | 0.3835 | -0.2622 |
| 180.0 | 0.0339 | 0.1499 | 0.3131 | -0.1293 |
| 165.0 | 0.0448 | 0.1681 | 0.2475 | -0.0346 |
| 160.0 | 0.0479 | 0.1738 | 0.2214 | +0.0003 |

The balance is reached at $H_J = 160.05$ m.

#### Discharges
$$
Q_1 = \sqrt{\frac{200 - 160.05}{17409.4}} = 0.0479\ \text{m}^3/\text{s},\quad Q_2 = \sqrt{\frac{237.99 - 160.05}{2582.1}} = 0.1737\ \text{m}^3/\text{s},\quad Q_3 = \sqrt{\frac{160.05 - 140}{408.0}} = 0.2216\ \text{m}^3/\text{s}
$$
Check: $0.0479 + 0.1737 = 0.2216 = Q_3$.

**Answer:** Pipe 1 (A to J): 47.9 l/s; pipe 2 (B to J): 173.7 l/s; pipe 3 (J to C): 221.6 l/s. Head at J = 160.0 m.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What is the effective head of tank B?
   A. 667/9.81 = 68 m
   B. 170 + 667/9.81 = 237.99 m
   C. 170 m
   D. 170 - 667/9.81 = 102 m
   Answer: B. A closed tank's surface pressure adds p/gamma to its water level, giving about 238 m. (Key point: Piezometric head of closed tank)

2. After adding pressure head, which ordering of effective heads is correct?
   A. B (238) > A (200) > C (140)
   B. A (200) > B (170) > C (140)
   C. A > C > B
   D. B (238) > C > A
   Answer: A. B becomes the highest source, A the middle, and C the lowest. (Key point: Ranking of reservoirs)

3. What are the flow directions at J?
   A. A feeds J; J feeds B and C
   B. B feeds J; J feeds A and C
   C. A and C feed J; J feeds B
   D. B and A feed J; J feeds C
   Answer: D. At H_J = 200 the outflow to C exceeds the inflow from B, so A must also supply J, giving Q3 = Q1 + Q2. (Key point: Direction of flows)

4. Which result is correct?
   A. H_J is about 140 m
   B. H_J is about 238 m
   C. H_J is about 160 m, with Q1 = 47.9, Q2 = 173.7, Q3 = 221.6 l/s
   D. H_J is about 200 m
   Answer: C. At H_J = 160.05 m, 47.9 + 173.7 = 221.6 l/s balances. (Key point: Head at J)

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