---
title: "Willans line and friction power estimation"
subject: "Internal Combustion Engines"
chapter: 8
key: "semester_6/internal_combustion_engines:willans-line-friction-power-methods"
source: https://www.bctengineer.com/q/semester_6/internal_combustion_engines/willans-line-friction-power-methods
site: BCT Engineering
---

# Willans line and friction power estimation

Internal Combustion Engines · Chapter 8: Engine performance and testing of engines

- Asked in: year not given, 6 marks
- Video: [MOTORING TEST & WILLIAM'S LINE METHOD FOR MEASURING FRICTION POWER](https://www.youtube.com/watch?v=AaGAOH6N7UU) — Professor kartik
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/internal_combustion_engines/willans-line-friction-power-methods
- Whole chapter: https://www.bctengineer.com/md/semester_6/internal_combustion_engines/chapter-8

## Question

Describe the methods of finding the friction power of an engine. The following results were obtained from a constant-speed test on a diesel engine: brake power 10, 20, 30 and 40 kW with fuel consumption 4.5, 6.6, 8.7 and 10.8 kg/h. Use the Willans line method to find the friction power and the mechanical efficiency at 30 kW and at 40 kW.

## Short answer

Methods: Willans line (extrapolate fuel vs BP to zero fuel), Morse test, motoring, retardation, indicator diagram. From given data, FP =11.4 kW; ηm =72.4% at 30 kW, 77.8% at40 kW.

## Answer

#### Methods of finding friction power
1. **Willans line:** plot fuel consumption against brake power at constant speed and extrapolate back to zero fuel; the intercept on the negative BP axis is the friction power. Suitable for CI engines, where the line is straight to about 70% load.
2. **Morse test:** for multi-cylinder engines; cylinders cut out one by one (details in another answer).
3. **Motoring test:** the engine is driven by an electric motor (fuel off) at the test speed; the power taken equals FP. Gives hot FP if done soon after a run.
4. **Retardation test:** the engine is run at no load at rated speed; the ignition (or fuel) is cut and the speed fall is timed, once with the engine alone and once with a known extra load. The deceleration times give FP.
5. **From indicator diagram:** $FP = IP - BP$ with IP found by an engine indicator.

#### Willans line from the data
The values lie on a straight line.

| BP (kW) | 10 | 20 | 30 | 40 |
|---|---|---|---|---|
| Fuel (kg/h) | 4.5 | 6.6 | 8.7 | 10.8 |

```text
 fuel
 kg/h |             . 40
  8.7 |         . 30
  6.6 |     . 20
  4.5 | . 10
  2.4 |.   (BP = 0)
  0.0 +---+--------------- BP
    -11.4 0
```

$$
\text{Slope} = \frac{10.8 - 4.5}{40 - 10} = 0.21\ \text{kg/h per kW}
$$

Fuel at zero BP $= 4.5 - 0.21 \times 10 = 2.4$ kg/h. The line meets zero fuel when

$$
BP = -\frac{2.4}{0.21} = -11.43\ \text{kW}
$$

so $FP = 11.43$ kW (assumed constant at the speed).

#### Mechanical efficiency
$$
\begin{aligned}
\eta_m(30\ \text{kW}) &= \frac{30}{30 + 11.43} = 0.724 \\
\eta_m(40\ \text{kW}) &= \frac{40}{40 + 11.43} = 0.778
\end{aligned}
$$

**Answer:** FP = 11.4 kW; $\eta_m$ = 72.4% at 30 kW and 77.8% at 40 kW.

## Check yourself

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

1. How is friction power found from the Willans line?
   A. Extrapolate fuel vs BP back to zero fuel; the negative BP intercept is FP
   B. Read fuel at the highest BP
   C. Take the slope as FP
   D. Plot SFC vs speed
   Answer: A. The line is extended to zero fuel consumption; the intercept on the negative BP axis equals the friction power. (Key point: Willans line method)

2. What is the friction power from the data?
   A. 11.4 kW
   B. 2.4 kW
   C. 0.21 kW
   D. 24 kW
   Answer: A. Slope 0.21 kg/h per kW, fuel at zero BP is 2.4 kg/h, so BP at zero fuel = -2.4/0.21 = -11.43 kW. (Key point: Willans calculation)

3. What is the mechanical efficiency at 40 kW?
   A. 77.8%
   B. 72.4%
   C. 100%
   D. 88.9%
   Answer: A. eta_m = 40/(40 + 11.43) = 0.778. (Key point: Mechanical efficiency)

4. What is measured in the motoring test?
   A. Fuel flow at no load
   B. Power taken by an electric motor to drive the engine with fuel off, equal to FP
   C. Deceleration time only
   D. The indicator diagram area
   Answer: B. The engine is driven by a motor at test speed with fuel off, and the power taken equals friction power. (Key point: Motoring test)

## Related in Internal Combustion Engines

- [Morse test for friction power](https://www.bctengineer.com/question.md?key=semester_6%2Finternal_combustion_engines%3Amorse-test-four-cylinder-numerical) — Internal Combustion Engines, chapter 8
- [Efficiency ratio and specific fuel consumption](https://www.bctengineer.com/question.md?key=semester_6%2Finternal_combustion_engines%3Arelative-efficiency-sfc-numerical) — Internal Combustion Engines, chapter 8
- [Heat balance sheet of a diesel engine](https://www.bctengineer.com/question.md?key=semester_6%2Finternal_combustion_engines%3Adiesel-engine-heat-balance-numerical) — Internal Combustion Engines, chapter 8
- [IP, BP and efficiencies of a petrol engine](https://www.bctengineer.com/question.md?key=semester_6%2Finternal_combustion_engines%3Aindicated-brake-power-efficiency-numerical) — Internal Combustion Engines, chapter 8
- [Engine performance parameters and efficiencies](https://www.bctengineer.com/question.md?key=semester_6%2Finternal_combustion_engines%3Aengine-performance-parameters-definitions) — Internal Combustion Engines, chapter 8
- [Engine displacement and mean piston speed](https://www.bctengineer.com/question.md?key=semester_6%2Finternal_combustion_engines%3Aengine-displacement-piston-speed-numerical) — Internal Combustion Engines, chapter 2

## Related in other subjects

- [Diesel engine 75 kW: fuel consumption and efficiencies](https://www.bctengineer.com/question.md?key=semester_7%2Fpower_plant_equipment%3Adiesel-75kw-fuel-consumption) — Power Plant Equipment, chapter 5
- [Two-stroke motoring test with specific fuel consumption](https://www.bctengineer.com/question.md?key=semester_7%2Fpower_plant_equipment%3Amotoring-test-sfc) — Power Plant Equipment, chapter 5, asked 6×
- [Engine test: fuel measured in litres](https://www.bctengineer.com/question.md?key=semester_7%2Fpower_plant_equipment%3Aengine-test-fuel-litres) — Power Plant Equipment, chapter 5, asked 2×
- [Engine test: four-cylinder, 85% mechanical efficiency](https://www.bctengineer.com/question.md?key=semester_7%2Fpower_plant_equipment%3Aengine-test-85-percent) — Power Plant Equipment, chapter 5, asked 5×