Why are velocities in hundreds of m/s?

5 m^3/s is forced through only 0.1 m diameter pipes

The friction factor is too small

k in h_f = kQ^2 depends on:

f, L and D (k = 8fL/(pi^2 g D^5))

How is the final answer checked?

Head loss along B-C-D equals head loss along B-F-D

Total head loss is zero in each pipe

2072 Chaitra · 8 marks

Velocity, head loss, discharge in pipes BC, CD, BF, FD (0.1 m)

Short answer

Hardy-Cross solution gives: BC 2.809 m³/s (357.7 m/s, hf=195605 m); CD 1.809 m³/s (230.3 m/s, 162256 m); BF 2.191 m³/s (279.0 m/s, 356948 m); FD 0.192 m³/s (24.4 m/s, 913 m).

Determine the velocity, head loss and discharge in the pipes BC, CD, BF and FD. The length of pipe BC, CD, BF and FD are 100 m, 200 m, 300 m and 100 m respectively and dia of all pipes are 0.1 m and Darcy's coefficient of friction, f = 0.03.

[Figure: 5 m $^3$ /s enters at A and reaches B; the loop B-C-D-F; 1 m $^3$ /s leaves at C, 2 m $^3$ /s leaves at F, and 2 m $^3$ /s leaves from D towards E.]

The 5 m³/s entering at A reaches B and splits between the paths B–C–D and B–F–D. All pipes are only 0.1 m in diameter, so the numbers are not realistic (velocities of hundreds of m/s), but the method is the same.

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

BC = $x$ , CD = $x - 1$ (1 m³/s leaves at C), BF = $5 - x$ and FD = $3 - x$ (2 m³/s leaves at F). At D: CD + FD = 2 ✓. Trial: BC = 3, CD = 2, BF = 2, FD = 0 (a very small value 0.001 is used so that the correction can be calculated).

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

Pipe	L (m)	D (m)	k = 8fL/(π²gD⁵)
BC	100	0.1	24788.0572
CD	200	0.1	49576.1143
BF	300	0.1	74364.1715
FD	100	0.1	24788.0572

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

Iteration	ΔQ	BC	CD	BF	FD
1	-0.1923	2.8077	1.8077	2.1923	0.1933
2	+0.0014	2.8091	1.8091	2.1909	0.1919
3	+0.0000	2.8091	1.8091	2.1909	0.1919
4	+0.0000	2.8091	1.8091	2.1909	0.1919

Results

Pipe	Discharge (m³/s)	Velocity (m/s)	Head loss (m)
BC	2.8091	357.667	195604.8111
CD	1.8091	230.343	162256.3632
BF	2.1909	278.953	356948.4203
FD	0.1919	24.432	912.7540

Check: upper path = 357861.2 m; lower path = 357861.2 m ✓

Answer: BC = 2.809 m³/s (v = 357.7 m/s, h_f = 195605 m); CD = 1.809 m³/s (230.3 m/s, 162256 m); BF = 2.191 m³/s (279.0 m/s, 356948 m); FD = 0.192 m³/s (24.4 m/s, 913 m). The very large velocities and head losses show that the data (5 m³/s through 100 mm pipes) is not physically practical; the pipe sizes must be increased in a real design.

✓ 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
Head loss and velocity in loop pipes AB, BC, AC, CD · Water Supply Engineering, chapter 7

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