--- title: "Steel pipe size for given energy gradient" subject: "Hydraulics" chapter: 1 key: "semester_4/hydraulics:steel-pipe-size-energy-gradient" source: https://www.bctengineer.com/q/semester_4/hydraulics/steel-pipe-size-energy-gradient site: BCT Engineering --- # Steel pipe size for given energy gradient Hydraulics · Chapter 1: Pipe flow - Asked in: 2073 Magh, 8 marks - Video: [PIPE SIZING | LINE SIZING | EXAMPLE | HYDRAULICS | PIPING MANTRA |](https://www.youtube.com/watch?v=JkDnSyBdCLA) — Piping Mantra - Answer checked by Claude Sonnet 5.5 on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_4/hydraulics/steel-pipe-size-energy-gradient - Whole chapter: https://www.bctengineer.com/md/semester_4/hydraulics/chapter-1 ## Question Determine the size of steel pipe required to carry water at 30 l/s if the permissible energy gradient is 0.05. Will the boundary act as smooth or in transition? ## Short answer Trial solution of Darcy-Weisbach equation combined with Colbrook-White gives a required diameter of about 121 mm (commercially 125 mm). The shear Reynolds number u_* k/ν ≈ 5.6 places the boundary in the transition regime. ## Answer **Data.** $Q = 0.03$ m³/s, permissible energy gradient $S = h_f/L = 0.05$. Commercial steel pipe: $k = 0.046$ mm (Moody table). Water at 20 °C, $\nu = 1.0\times10^{-6}$ m²/s. This is a "find the diameter" problem, solved by trial. #### Working equation $$ S = \frac{f}{D}\frac{V^2}{2g},\quad V = \frac{4Q}{\pi D^2} \;\Rightarrow\; S = \frac{8fQ^2}{\pi^2 g D^5} \;\Rightarrow\; D = \left(\frac{8fQ^2}{\pi^2 g S}\right)^{1/5} $$ #### Trials (f from the Colebrook-White equation at each new D) | Trial | f used | D (mm) | |---|---|---| | 0 | 0.02000 | 124.4 | | 1 | 0.01740 | 120.9 | | 2 | 0.01742 | 121.0 | | 3 | 0.01742 | 121.0 | The values converge to $D = 0.1210$ m. Final check: $V = 2.610$ m/s, $Re = 315734$, $k/D = 0.000380$, $f = 0.01742$ (Colebrook), and $S = fV^2/(2gD) = 0.0500$, which matches the required 0.05. **Diameter required:** $D \approx 121$ mm. In practice a 125 mm commercial size would be adopted (it gives $S$ = 0.0424 < 0.05 at $V$ = 2.44 m/s). #### Smooth or transition? $$ u_* = V\sqrt{\frac{f}{8}} = 2.610\sqrt{\frac{0.01742}{8}} = 0.1218\ \text{m/s},\qquad Re_* = \frac{u_*k}{\nu} = \frac{0.1218(0.046\times10^{-3})}{10^{-6}} = 5.60 $$ Since $4 < Re_* < 100$, the boundary acts in the **transition** zone (neither smooth nor fully rough), as the Colebrook-White equation assumes. **Answer:** $D \approx 121$ mm (use 125 mm); the boundary is in the transition regime. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Why must the diameter be found by trial? A. Because Q is unknown B. Because the pipe length is unknown C. Because the energy gradient changes along the pipe D. Because f depends on D (through Re and k/D) while D = (8fQ^2/(pi^2 g S))^(1/5) needs f Answer: D. D needs f, but f from Colebrook-White needs Re and k/D, which both depend on D, so the values are iterated until they converge. (Key point: Working equation for D) 2. In the trials, what is done after each D is computed? A. f is recalculated from Colebrook-White at the new D and D is recomputed until the values agree B. D is doubled until S equals 0.05 C. The velocity is fixed at 1 m/s D. f is kept at 0.02 throughout Answer: A. f changes with D, so each new D gives a new f, and the process stops when D (121 mm) and f (0.01742) stop changing. (Key point: Method of trial) 3. Why is a 125 mm pipe adopted in practice rather than 121 mm? A. Because 121 mm gives a Reynolds number below 2000 B. Because 125 mm is a commercial size and, being slightly larger, gives S = 0.0424, below the permissible 0.05 C. Because a larger pipe gives a higher energy gradient D. Because 125 mm makes the boundary smooth Answer: B. 125 mm is a standard size; its larger diameter gives lower velocity and a gradient 0.0424 < 0.05, so the limit is satisfied. (Key point: Practical pipe size) 4. Re* = u*k/nu is 5.60. What does this show? A. Smooth boundary since Re* is below 100 B. Fully rough boundary since Re* is above 4 C. Laminar flow since Re* is small D. Transition, since 4 < Re* < 100 Answer: D. Smooth is Re* below about 4 and fully rough is above about 100, so 5.60 lies in the transition zone. (Key point: Boundary classification) ## Related in Hydraulics - [Minimum hydrodynamically smooth pipe size](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Asmooth-pipe-minimum-size) — Hydraulics, chapter 1 - [Permissible roughness for minimum energy loss](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Aone-metre-pipe-permissible-roughness) — Hydraulics, chapter 1 - [Water hammer in 1.2 m steel pipe, wall thickness](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Awater-hammer-steel-pipe-thickness-1200mm) — Hydraulics, chapter 4 - [Kerosene flow: smooth, rough or transition](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Akerosene-pipe-rough-or-smooth) — Hydraulics, chapter 1 - [Inclined smooth pipe with wall shear limit](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Asmooth-pipe-inclined-wall-shear-limit) — Hydraulics, chapter 1 - [Velocity profile near smooth boundary](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Aprandtl-smooth-boundary-profile) — Hydraulics, chapter 1 ## Related in other subjects - [Process to determine wire size](https://www.bctengineer.com/question.md?key=semester_1%2Fbasic_electrical_and_electronics_engineering%3Awire-size-determination) — Basic Electrical and Electronics Engineering, chapter 4 - [Timber-steel plate composite member: permissible load](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Atimber-steel-composite-permissible-load) — Strength of Materials, chapter 3 - [Specific energy of 400 t train on 1% up-gradient](https://www.bctengineer.com/question.md?key=semester_7%2Futilization_of_electrical_energy%3Atrain-400t-up-gradient-specific-energy) — Utilization of Electrical Energy, chapter 4 - [Flow between reservoirs (20 cm and 30 cm pipe)](https://www.bctengineer.com/question.md?key=semester_6%2Fhydro_power%3Areservoirs-pipeline-20-30cm) — Hydro Power, chapter 1