---
title: "Penstock thickness and economic diameter"
subject: "Hydro Power"
chapter: 5
key: "semester_6/hydro_power:penstock-thickness-economic-diameter"
source: https://www.bctengineer.com/q/semester_6/hydro_power/penstock-thickness-economic-diameter
site: BCT Engineering
---

# Penstock thickness and economic diameter

Hydro Power · Chapter 5: Components of Hydropower System

- Asked in: 2080 Chaitra, 4 marks
- Video: [Penstock Thickness and Economic Diameter Calculation | Formulae Explained #Thickness #economic](https://www.youtube.com/watch?v=4UqCfkXvVI8) — Hydropower
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/hydro_power/penstock-thickness-economic-diameter
- Whole chapter: https://www.bctengineer.com/md/semester_6/hydro_power/chapter-5

## Question

How do you calculate thickness and economic diameter of penstock? Discuss with mathematical equations and derivations as required.

## Short answer

Penstock thickness t = pD/(2 σ η_j) + corrosion allowance, with design pressure p including water hammer. Economic diameter minimizes total annual cost C = k1 D^2 + k2/D^5, giving optimum when 2C1 = 5C2; empirical formulas like USBR D = 0.176(P/H)^0.466 provide preliminary estimates.

## Answer

#### Thickness of penstock
A penstock is a thin cylinder under internal pressure. Consider a unit length of pipe of internal diameter $D$ cut along a diameter. The bursting force due to pressure $p$ is resisted by hoop tension in the two walls:

$$
\begin{aligned}
p \times D \times 1 &= 2\,\sigma\, t \times 1 \times \eta_j \\
\Rightarrow\ t &= \frac{p\,D}{2\,\sigma\,\eta_j} + t_c
\end{aligned}
$$

where $p = \gamma_w (H + h_{wh})$ is the design pressure (static head plus water hammer rise, typically 15–30%), $\sigma$ = allowable (design) stress of steel, $\eta_j$ = joint efficiency (0.85–0.95), $t_c$ = corrosion allowance (1.5–3 mm). A minimum thickness for handling is also used, e.g. $t_{min} = \dfrac{D + 500}{400}$ mm ($D$ in mm), not less than 6 mm.

#### Economic diameter of penstock
The economic diameter gives minimum **total annual cost** = annual cost of the penstock + annual value of energy lost by friction.

1. Pipe weight $\propto \pi D t$, and $t \propto pD$, so the annual cost of the penstock is
$$
C_1 = k_1 D^2
$$
2. Head loss (Darcy):
$$
h_f = \frac{f L V^2}{2 g D} = \frac{8 f L Q^2}{\pi^2 g D^5}
$$
Power lost $= 9.81\,Q\,h_f\,\eta$; annual value of lost energy
$$
C_2 = \frac{k_2}{D^5}
$$
3. Total annual cost and minimum:
$$
\begin{aligned}
C &= k_1 D^2 + \frac{k_2}{D^5} \\
\frac{dC}{dD} &= 2k_1 D - \frac{5k_2}{D^6} = 0 \\
D_{eco} &= \left(\frac{5k_2}{2k_1}\right)^{1/7}
\end{aligned}
$$

At the optimum, $2C_1 = 5C_2$, i.e. annual pipe cost is 2.5 times the annual cost of energy lost.

4. **Graphical method:** compute $C_1$ and $C_2$ for several trial diameters, plot them and their sum; the minimum of the total curve gives $D_{eco}$.

5. **Empirical formulas** for preliminary design ($P$ in kW, $H$ in m, $D$ in m):
- USBR (Bier): $D = 0.176\,(P/H)^{0.466}$
- Sarkaria: $D = 0.71\,P^{0.43}/H^{0.65}$
- Fahlbusch: $D = 0.52\,H^{-0.17}(P/H)^{0.43}$

## Check yourself

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

1. The penstock plate thickness from hoop tension is
   A. t = p D / (2 sigma eta_j) plus corrosion allowance
   B. t = p D / (sigma eta_j)
   C. t = 2 p D / (sigma eta_j)
   D. t = sigma D / (2 p)
   Answer: A. Pressure on the diameter is resisted by two walls: pD = 2 sigma t eta_j. (Key point: Thickness formula)

2. The design pressure p used for thickness includes
   A. static head only
   B. static head plus water hammer rise (typically 15-30%)
   C. water hammer only
   D. atmospheric pressure
   Answer: B. p = gamma_w (H + h_wh). (Key point: Design pressure)

3. In the economic diameter derivation, the annual pipe cost and energy-loss cost vary as
   A. C1 = k1 / D^2 and C2 = k2 D^5
   B. C1 = k1 D^5 and C2 = k2 / D^2
   C. C1 = k1 D^2 and C2 = k2 / D^5
   D. both vary as D^2
   Answer: C. Weight is proportional to D t with t proportional to D; Darcy head loss varies as 1/D^5. (Key point: Cost components)

4. At the optimum diameter, the relation between the costs is
   A. C1 = C2
   B. 5 C1 = 2 C2
   C. C1 = 2 C2
   D. 2 C1 = 5 C2
   Answer: D. Setting dC/dD = 0 gives 2 k1 D^2 = 5 k2/D^5, i.e. 2C1 = 5C2. (Key point: Optimum condition)

## Related in Hydro Power

- [Economic diameter of penstock](https://www.bctengineer.com/question.md?key=semester_6%2Fhydro_power%3Apenstock-economic-diameter) — Hydro Power, chapter 5, asked 5×
- [Economic penstock diameter by empirical formulas](https://www.bctengineer.com/question.md?key=semester_6%2Fhydro_power%3Apenstock-economic-diameter-formulas) — Hydro Power, 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
- [Penstock plate thickness (1.5 m, 250 m head)](https://www.bctengineer.com/question.md?key=semester_6%2Fhydro_power%3Apenstock-thickness-1-5m) — Hydro Power, chapter 5
- [Penstock thickness (1.3 m, 20 kg/cm²)](https://www.bctengineer.com/question.md?key=semester_6%2Fhydro_power%3Apenstock-thickness-1-3m) — Hydro Power, chapter 5
- [Penstock thickness (1.20 m, 17.5 kg/cm²)](https://www.bctengineer.com/question.md?key=semester_6%2Fhydro_power%3Apenstock-thickness-1-2m) — Hydro Power, chapter 5

## Related in other subjects

- [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
- [Economic diameter factors; penstock wall thickness (3 m)](https://www.bctengineer.com/question.md?key=semester_7%2Fhydropower_engineering%3Aeconomic-diameter-and-wall-thickness-factors) — 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
- [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