Why is the flow in pipe CD not part of the iteration?

It is outside the loop and carries the fixed 2 m^3/s outflow at D

2073 Chaitra · 8 marks

Head loss and velocity in loop pipes AB, BC, AC, CD

Short answer

Hardy-Cross yields: AB 1.943 m³/s (1.72 m/s, 0.376 m); BC 0.943 m³/s (1.88 m/s, 0.673 m); AC 2.057 m³/s (2.62 m/s, 1.049 m); CD 2 m³/s (2.55 m/s, 0.992 m).

A part of the water distribution network is shown in figure below. If the diameter of pipes AB, BC, AC and CD are 1200 mm, 800 mm, 1000 mm and 1000 mm respectively, calculate the head loss and velocity in pipes AB, BC, AC and CD. Assume length of all the pipes are 100 m and coefficient of friction, f = 0.03.

[Figure: loop A-B-C-D; 4 m $^3$ /s enters at A, 1 m $^3$ /s leaves at B, 1 m $^3$ /s leaves at C, 2 m $^3$ /s leaves at D.]

The 4 m³/s entering at A splits into paths A–B–C and A–C; pipe CD takes 2 m³/s to D. The split is found by Hardy-Cross.

Formulas

Darcy-Weisbach: $h_f = \dfrac{8 f L Q^2}{\pi^2 g D^5} = kQ^2$ , velocity $v = Q/(\pi D^2/4)$ . Hardy-Cross correction for the loop:

\Delta Q = -\frac{\sum kQ|Q|}{2\sum k|Q|}

Flow distribution

Pipe CD carries 2 m³/s (outflow at D) and its flow is fixed. In the triangle A–B–C let AB carry $x$ ; then BC = $x - 1$ (1 m³/s leaves at B) and AC = $4 - x$ . At C: BC + AC = 3 = 1 (outflow at C) + 2 (to D) ✓. Loop condition: $h_{AB} + h_{BC} = h_{AC}$ . Trial: AB = 2, BC = 1, AC = 2.

Constants k for each pipe ( $f = 0.03$ )

Pipe	L (m)	D (m)	k = 8fL/(π²gD⁵)
AB	100	1.2	0.0996
BC	100	0.8	0.7565
AC	100	1.0	0.2479

Hardy-Cross iterations (clockwise positive; flows in m³/s)

Iteration	ΔQ	AB	BC	AC
1	-0.0563	1.9437	0.9437	2.0563
2	-0.0007	1.9430	0.9430	2.0570
3	-0.0000	1.9430	0.9430	2.0570
4	-0.0000	1.9430	0.9430	2.0570

Results

Pipe	Discharge (m³/s)	Velocity (m/s)	Head loss (m)
AB	1.9430	1.718	0.3761
BC	0.9430	1.876	0.6727
AC	2.0570	2.619	1.0488

Pipe CD: $k = 0.2479$ , $Q = 2$ m³/s, $h_f = 0.2479 \times 4 = 0.9915$ m, $v = 2.546$ m/s. Check: AB + BC = 0.3761 + 0.6727 = 1.0488 m = AC = 1.0488 m ✓

Answer: AB: 1.943 m³/s, 1.72 m/s, 0.376 m; BC: 0.943 m³/s, 1.88 m/s, 0.673 m; AC: 2.057 m³/s, 2.62 m/s, 1.049 m; CD: 2 m³/s, 2.55 m/s, 0.992 m.

✓ Answer checked by Claude Opus 5.5 (high effort)

Velocity, head loss, discharge in loop pipes BC, CD, BF, FD · Water Supply Engineering, chapter 7
Velocity, head loss, discharge in pipes BC, CD, BF, FD (0.1 m) · Water Supply Engineering, chapter 7
Hardy-Cross network discharge AB, BC, AD, CD · Water Supply Engineering, chapter 7
Design pipes AB, BC and AC, C = 110 · Water Supply Engineering, chapter 7
Design pipes AB and BC, 20,000 population, f = 0.04 · Water Supply Engineering, chapter 7
Design pipes AB, BC; 160 lpcd, peak factor 3 · Water Supply Engineering, chapter 7

All Water Supply Engineering chapter 7 questions →