Willans line and friction power estimation

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.

 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.