---
title: "Hardy Cross with given constant factors"
subject: "Hydraulics"
chapter: 3
key: "semester_4/hydraulics:hardy-cross-constant-factors-abcd"
source: https://www.bctengineer.com/q/semester_4/hydraulics/hardy-cross-constant-factors-abcd
site: BCT Engineering
---

# Hardy Cross with given constant factors

Hydraulics · Chapter 3: Three reservoirs problem and Pipe networks

- Asked in: 2075 Bhadra, 8 marks
- Video: [Hardy Cross Method](https://www.youtube.com/watch?v=pxCWxGHKo2M) — Kenneth Lamb
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/hydraulics/hardy-cross-constant-factors-abcd
- Whole chapter: https://www.bctengineer.com/md/semester_4/hydraulics/chapter-3

## Question

Using Hardy-Cross method, find the rate of flow in every pipe lines as given below. The constant factor for AB, BC, CD, DA and BD are 1, 2, 1, 2 and 3 respectively. [Figure: network with nodes A, B, C, D and diagonal BD; 100 units enter at A; 25 units leave at B; 75 units leave at C]

## Short answer

After Hardy‑Cross iteration with resistance factors r (AB=1, BC=2, CD=1, DA=2, BD=3), the final discharges are: AB = 58.5 A→B, BC = 31.1 B→C, CD = 43.9 D→C, DA = 41.5 A→D, BD = 2.4 B→D, all in the same flow units.

## Answer

**Data.** Constant factors ($r$ in $h_f = rQ^2$): AB 1, BC 2, CD 1, DA 2, BD 3. 100 units enter at A; 25 leave at B and 75 at C.

**Initial distribution (continuity):** A to B 60, A to D 40; at B (out 25): B to C 20, B to D 15; at D: inflow $40 + 15 = 55$ all goes D to C; at C: $20 + 55 = 75$. 

**Loop I** = A-B-D-A (AB +, BD +, DA $-$ as AD). **Loop II** = B-C-D-B (BC +, CD $-$ as DC, DB $-$ as BD).

#### Hardy-Cross iterations
| Iter | $\sum rQ|Q|$ (I) | $2\sum r|Q|$ (I) | $\Delta Q_I$ | $\sum rQ|Q|$ (II) | $2\sum r|Q|$ (II) | $\Delta Q_{II}$ |
|---|---|---|---|---|---|---|
| 1 | 1075.00 | 370.00 | -2.905 | -2900.00 | 280.00 | 10.357 |
| 2 | -412.90 | 296.24 | 1.394 | -158.93 | 221.14 | 0.719 |
| 3 | -8.07 | 297.50 | 0.027 | -15.38 | 226.63 | 0.068 |

After 10 iterations the corrections are negligible.

#### Final discharges
| Pipe | AB | BC | CD (D to C) | DA (A to D) | BD (B to D) |
|---|---|---|---|---|---|
| Q | 58.52 | 31.15 | 43.85 | 41.48 | 2.37 |

Checks: A: $58.52 + 41.48 = 100.00$; B: $58.52 = 25 + 31.15 + 2.37$; C: $31.15 + 43.85 = 75.00$; D: $41.48 + 2.37 = 43.85$.

**Answer:** AB = 58.5, BC = 31.1, CD = 43.9, DA = 41.5, and BD = 2.4 units (B to D).

## Check yourself

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

1. With 100 units at A and outflows 25 at B and 75 at C, the initial flows are A to B 60, A to D 40, B to C 20, B to D 15. What flows from D to C?
   A. 55
   B. 40
   C. 15
   D. 75
   Answer: A. Node D receives 40 + 15 = 55 and has no external outflow, so all 55 goes to C. (Key point: Initial distribution)

2. In loop II (B-C-D-B), how are DC and BD treated if flows are D to C and B to D?
   A. Both are positive
   B. Only DC is negative
   C. Both are omitted
   D. Both are taken as negative (against the clockwise direction)
   Answer: D. Clockwise B to C to D to B: C to D and D to B are positive, so D to C and B to D are negative. (Key point: Loop definitions)

3. Which node B check holds for the final discharges?
   A. 58.52 + 2.37 = 25 + 31.15
   B. 58.52 = 31.15 + 43.85
   C. 58.52 = 25 + 31.15 + 2.37
   D. 58.52 = 25 + 31.15
   Answer: C. Inflow AB equals the 25 withdrawn, plus BC and BD. (Key point: Final check at B)

## Related in Hydraulics

- [Hardy Cross network with valve on diagonal](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Ahardy-cross-network-six-links-resistance) — Hydraulics, chapter 3
- [Hardy Cross correction derivation and network](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Ahardy-cross-correction-and-kathmandu-network) — Hydraulics, chapter 3
- [Flow distribution in five-pipe network](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Anetwork-five-pipes-flow-distribution) — Hydraulics, chapter 3
- [Verify suggested discharge distribution (Hardy Cross)](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Ahardy-cross-verify-suggested-distribution) — Hydraulics, chapter 3
- [Derive Hardy Cross correction factor](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Ahardy-cross-correction-factor-derivation) — Hydraulics, chapter 3
- [Network: unknown discharges and head losses](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Anetwork-find-unknown-discharges-head-losses) — Hydraulics, chapter 3, asked 2×

## Related in other subjects

- [Hardy-Cross network discharge AB, BC, AD, CD](https://www.bctengineer.com/question.md?key=semester_5%2Fwater_supply_engineering%3Ahardy-cross-as69) — Water Supply Engineering, chapter 7
- [Head loss and velocity in loop pipes AB, BC, AC, CD](https://www.bctengineer.com/question.md?key=semester_5%2Fwater_supply_engineering%3Aloop-c73) — Water Supply Engineering, chapter 7
- [Design pipelines AB, BC and CD, C = 120](https://www.bctengineer.com/question.md?key=semester_5%2Fwater_supply_engineering%3Apd-c74) — Water Supply Engineering, chapter 7
- [Design pipes AB, BC; 160 lpcd, peak factor 3](https://www.bctengineer.com/question.md?key=semester_5%2Fwater_supply_engineering%3Apd-c71) — Water Supply Engineering, chapter 7