---
title: "Economic diameter factors; penstock wall thickness (3 m)"
subject: "Hydropower Engineering"
chapter: 6
key: "semester_7/hydropower_engineering:economic-diameter-and-wall-thickness-factors"
source: https://www.bctengineer.com/q/semester_7/hydropower_engineering/economic-diameter-and-wall-thickness-factors
site: BCT Engineering
---

# Economic diameter factors; penstock wall thickness (3 m)

Hydropower Engineering · Chapter 6: Water Conveyance Structures

- Asked in: 2075 Ashwin, 2+3 marks
- Video: [Penstock Pipe Numerical || Determine Economical Diameter || Hydropower Engineering](https://www.youtube.com/watch?v=pcmr7R09-_4) — OPengineer
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_7/hydropower_engineering/economic-diameter-and-wall-thickness-factors
- Whole chapter: https://www.bctengineer.com/md/semester_7/hydropower_engineering/chapter-6

## Question

Discuss the various factors which govern the determination of economic diameter of a penstock. Find the wall thickness of a penstock pipe if the internal diameter is 3.0 m which supplies water from a head of 220 m with a possibility of increase in pressure up to 40% due to transient condition. Take $\sigma_{st} = 1400$ kg/cm² and efficiency of joint = 0.95.

## Short answer

Economic diameter depends on discharge, head, steel and fabrication cost, annual charges, value of lost energy, friction loss, allowable velocity, and water hammer. For D=3 m, design head 308 m corresponding to 30.8 kg/cm2, allowable stress 1400 kg/cm2 and joint efficiency 0.95, wall thickness is 34.7 mm.

## Answer

#### Factors governing economic diameter
The economic diameter is found by balancing the annual cost of the penstock against the annual value of the energy lost in friction. It depends on:

- **Discharge and head:** a larger $Q$ or a lower head needs a larger diameter, because the same loss is a larger share of the head.
- **Cost of steel and fabrication:** cost of plates, welding, transport, erection, painting and anchor blocks; the pipe cost varies roughly with $d^2$ (weight $\propto d \cdot t$ and $t \propto d$).
- **Annual charges:** interest, depreciation, maintenance and insurance, as a percentage of the penstock cost.
- **Value of energy lost:** tariff or unit cost of power, and the plant load factor; a high value of power favours a bigger pipe.
- **Friction loss:** pipe roughness (friction factor), length and velocity; the loss varies as $Q^2/d^5$.
- **Turbine and overall efficiency:** the loss of power is $\gamma Q h_f \eta$.
- **Permissible velocity:** limits for erosion (silt), water hammer and noise; usually 3 to 6 m/s (up to 8 m/s for high heads).
- **Other factors:** surge/water-hammer pressures, bends and fittings, transport limits on plate size, and the number of penstocks.

#### Wall thickness of the penstock
Design pressure head $= 1.4 \times 220 = 308$ m of water. Since 10 m of water $\approx 1\ \text{kg/cm}^2$,

$$
p = 30.8\ \text{kg/cm}^2
$$

For a thin-walled pipe (hoop stress), with $D = 300$ cm, $\sigma_{st} = 1400\ \text{kg/cm}^2$, $\eta_j = 0.95$:

$$
t = \frac{pD}{2\sigma_{st}\eta_j} = \frac{30.8 \times 300}{2 \times 1400 \times 0.95} = 3.47\ \text{cm}
$$

**Answer: thickness $\approx 34.7$ mm; in practice provide about 36 to 37 mm (34.7 mm plus a 1 to 2 mm corrosion allowance).**

## Check yourself

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

1. What principle underlies the economic diameter?
   A. Balance annual penstock cost against annual value of energy lost in friction
   B. Maximise flow velocity
   C. Minimise wall thickness
   D. Match the diameter to the tunnel
   Answer: A. Total annual cost is minimised at the economic diameter. (Key point: Principle)

2. How does friction head loss vary with Q and d?
   A. As Q / d^2
   B. As Q^2 / d^5
   C. As d^5 / Q^2
   D. As Q^2 d^2
   Answer: B. hf = f L v^2/(2gd) with v proportional to Q/d^2 gives hf proportional to Q^2/d^5. (Key point: Friction loss variation)

3. What design pressure head is used for the 220 m penstock with 40% transient rise?
   A. 220 m, or 22 kg/cm2
   B. 440 m
   C. 1.4 x 220 = 308 m, or 30.8 kg/cm2
   D. 154 m
   Answer: C. Design head includes the 40% rise: 308 m of water = 30.8 kg/cm2. (Key point: Design pressure)

4. Which formula gives the wall thickness?
   A. t = 2 p D / (sigma eta_j)
   B. t = p D sigma / 2
   C. t = p / (2 sigma D) giving 0.2 cm
   D. t = p D / (2 sigma eta_j) = 3.47 cm
   Answer: D. Hoop stress in a thin cylinder gives t = pD/(2 sigma eta_j) = 34.7 mm. (Key point: Thickness formula)

## Related in Hydropower Engineering

- [Economic diameter; penstock thickness (1.25 m)](https://www.bctengineer.com/question.md?key=semester_7%2Fhydropower_engineering%3Apenstock-economic-diameter-thickness-1-25m) — Hydropower Engineering, chapter 6
- [Most economic penstock diameter (cost numerical)](https://www.bctengineer.com/question.md?key=semester_7%2Fhydropower_engineering%3Apenstock-economic-diameter-cost-numerical) — Hydropower Engineering, chapter 6
- [Economic diameter of penstock by graphical method](https://www.bctengineer.com/question.md?key=semester_7%2Fhydropower_engineering%3Aeconomic-diameter-penstock-graphical) — Hydropower Engineering, chapter 6
- [Economic diameter of penstock (analysis)](https://www.bctengineer.com/question.md?key=semester_7%2Fhydropower_engineering%3Aeconomic-diameter-penstock-mathematical) — Hydropower Engineering, chapter 6, asked 2×
- [Penstock thickness with gate closure time](https://www.bctengineer.com/question.md?key=semester_7%2Fhydropower_engineering%3Apenstock-thickness-water-hammer-600m) — Hydropower Engineering, chapter 6

## Related in other subjects

- [Water pipe thickness from pressure head](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Awater-pipe-thickness-pressure-head) — Strength of Materials, chapter 5
- [Surge tank objectives and economic penstock diameter factors](https://www.bctengineer.com/question.md?key=semester_6%2Fhydro_power%3Asurge-tank-objectives-economic-diameter) — Hydro Power, chapter 5
- [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
- [Penstock thickness and economic diameter](https://www.bctengineer.com/question.md?key=semester_6%2Fhydro_power%3Apenstock-thickness-economic-diameter) — Hydro Power, chapter 5