---
title: "Wind turbine output: 6 m blade, 8 and 16 m/s"
subject: "Power Plant Design"
chapter: 1
key: "semester_8/power_plant_design:wind-numerical-6m-8ms"
source: https://www.bctengineer.com/q/semester_8/power_plant_design/wind-numerical-6m-8ms
site: BCT Engineering
---

# Wind turbine output: 6 m blade, 8 and 16 m/s

Power Plant Design · Chapter 1: Energy Sources and Electric Power Generation

- Asked in: 2078 Kartik, 4 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_8/power_plant_design/wind-numerical-6m-8ms
- Whole chapter: https://www.bctengineer.com/md/semester_8/power_plant_design/chapter-1

## Question

Assume that we live in an area slightly above sea level that has an air density of 1.225 kg/m³ and we have installed a 40% efficient wind turbine which has a rotor blade radius of 6 m. Calculate the output power from the turbine at a wind speed of 8 m/s and again at double the velocity of 16 m/s.

## Short answer

A = π(6)² = 113.10 m². At 8 m/s: P = 0.5×1.225×113.10×8³×0.40 ≈ 14.19 kW. At 16 m/s: P = 0.5×1.225×113.10×16³×0.40 ≈ 113.50 kW (8× due to v³).

## Answer

Turbine output power:
$$
P = \frac{1}{2}\rho A v^3 \eta, \qquad A = \pi r^2
$$

#### Given
$\rho = 1.225$ kg/m³, $r = 6$ m, $\eta = 0.40$, $v_1 = 8$ m/s, $v_2 = 16$ m/s.

#### Swept area
$$
A = \pi \times 6^2 = 113.10\ \text{m}^2
$$

#### At $v_1 = 8$ m/s
$$
\begin{aligned}
P_{w1} &= 0.5 \times 1.225 \times 113.10 \times 8^3 \\
&= 0.5 \times 1.225 \times 113.10 \times 512 = 35{,}467\ \text{W} \\
P_1 &= 0.40 \times 35{,}467 = 14{,}187\ \text{W}
\end{aligned}
$$

#### At $v_2 = 16$ m/s
$$
\begin{aligned}
P_{w2} &= 0.5 \times 1.225 \times 113.10 \times 16^3 \\
&= 0.5 \times 1.225 \times 113.10 \times 4096 = 283{,}739\ \text{W} \\
P_2 &= 0.40 \times 283{,}739 = 113{,}495\ \text{W}
\end{aligned}
$$

#### Check
$$
\frac{P_2}{P_1} = \left(\frac{16}{8}\right)^3 = 8, \qquad 8 \times 14.187 = 113.50\ \text{kW}
$$

**Answer:** Output $\approx 14.19$ kW at 8 m/s and $\approx 113.50$ kW at 16 m/s; doubling the wind speed raises the power 8 times.

## Check yourself

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

1. For r = 6 m, the swept area is about
   A. 113.1 m^2
   B. 36 m^2
   C. 18.8 m^2
   D. 226.2 m^2
   Answer: A. A = pi x 36 = 113.10 m^2. (Key point: Swept area)

2. At 8 m/s with 40% efficiency, the output is about
   A. 35.5 kW
   B. 14.2 kW
   C. 113.5 kW
   D. 5.7 kW
   Answer: B. Wind power 35.47 kW x 0.40 = 14.19 kW. (Key point: Output at 8 m/s)

3. When speed is doubled to 16 m/s, the output becomes
   A. 2 times
   B. 4 times
   C. 8 times
   D. 16 times
   Answer: C. P is proportional to v^3, so 2^3 = 8 times (113.5 kW). (Key point: Doubling speed)

4. Why is the output (14.19 kW) less than the power in the wind (35.47 kW)?
   A. Only a fraction eta (0.40) of the wind power is converted
   B. Air density drops at 8 m/s
   C. The area is reduced
   D. Power is divided by v
   Answer: A. The turbine extracts only the fraction equal to its efficiency. (Key point: Wind power vs output)

## Close variants set in other exams

- [Wind turbine output: 6.5 m blade, 9 and 18 m/s](https://www.bctengineer.com/question.md?key=semester_8%2Fpower_plant_design%3Awind-numerical-6-5m-9ms) — Power Plant Design, chapter 1, asked 3×
- [Wind turbine output: 12 m blade, 18 m/s](https://www.bctengineer.com/question.md?key=semester_8%2Fpower_plant_design%3Awind-numerical-12m-18ms) — Power Plant Design, chapter 1, asked 3×

## Related in Power Plant Design

- [Wind turbine merits and output: 5.5 m blade](https://www.bctengineer.com/question.md?key=semester_8%2Fpower_plant_design%3Awind-merits-numerical-5-5m) — Power Plant Design, chapter 1
- [Wind power pros/cons and output: 1.25 m blade](https://www.bctengineer.com/question.md?key=semester_8%2Fpower_plant_design%3Awind-pros-cons-1-25m) — Power Plant Design, chapter 1, asked 2×
- [Wind power pros/cons and output: 1.5 m blade](https://www.bctengineer.com/question.md?key=semester_8%2Fpower_plant_design%3Awind-pros-cons-1-5m) — Power Plant Design, chapter 1, asked 2×

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