Willans' Line Method for Measuring Friction Power (FP) in Engines

 Willans' Line Method is an experimental technique used to estimate friction power (FP) and mechanical efficiency (ηₘ) in internal combustion engines. It involves plotting fuel consumption rate against brake power (BP) at different loads and extrapolating to find friction losses.

1. Concept & Theory

• Willans' Line is a straight-line approximation of the relationship between:

  • Fuel consumption rate (ṁ_f) (x-axis)

  • Brake power (BP) (y-axis)

• The intercept of this line on the BP axis gives Friction Power (FP).

Key Assumptions:

1. At zero BP (engine motoring), all input energy is used to overcome friction.

2. The relationship between fuel consumption and BP is linear under constant speed.

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2. Procedure to Determine FP

1. Run the engine at constant speed (N) under varying loads.

2. Measure:

   • Brake Power (BP) (using a dynamometer).

   • Fuel consumption rate (ṁ_f) (using a fuel flow meter).

3. Plot ṁ_f vs. BP and draw the best-fit line (Willans' Line).

4. Extrapolate the line to BP = 0 → The intercept gives FP.

Graphical Representation:

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Brake Power (BP)
  ^
  |      / Willans' Line (ṁ_f vs BP)
  |     /
  |    /
  |   /
  |  /
  | / 
  |/___________> Fuel Consumption (ṁ_f)
  |   |
  | FP (Intercept at BP=0)
  |

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3. Mathematical Formulation

The Willans' Line equation is:

$$��=�\cdot {\dot{�}}_{�}-��$$

Where:

• $�$ = Slope (represents engine's fuel-to-power conversion efficiency)

• $��$ = Friction Power (obtained from the intercept)

Steps to Calculate FP:

1. Record BP and ṁ_f at different loads (e.g., 25%, 50%, 75%, 100%).

2. Perform linear regression to find the best-fit line:

   $$��=�\cdot {\dot{�}}_{�}+�$$

3. The intercept $�=-��$, so:

   $$��=\mathrm{\mid }�\mathrm{\mid }$$

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4. Example Calculation

Given:

Load (%)	BP (kW)	Fuel Consumption (kg/h)
25	20	5.0
50	40	9.5
75	60	14.2
100	80	19.0

Step 1: Plot ṁ_f vs. BP

• Perform linear regression (e.g., using Excel or manual calculation).

Step 2: Find Line Equation
Assume best-fit line is:

$$��=4.5\cdot {\dot{�}}_{�}-2.5$$

• Slope (m) = 4.5 kW·h/kg

• Intercept (C) = -2.5 kW

Step 3: Determine FP

$$��=\mathrm{\mid }�\mathrm{\mid }=2.5\text{kW}$$

Step 4: Calculate Mechanical Efficiency (ηₘ) at Full Load

$$��=��+��=80+2.5=82.5\text{kW}$$$${�}_{�}=\frac{��}{��}=\frac{80}{82.5}=97\mathrm{\%}$$

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5. Advantages & Limitations

Advantages:

✅ Simple and inexpensive (no need for complex instruments).
✅ Works well for diesel engines (linearity holds better).
✅ Helps estimate mechanical efficiency without disassembly.

Limitations:

❌ Assumes linearity (may not hold for all engines, especially petrol).
❌ Requires constant speed testing.
❌ Less accurate at very low loads.

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6. Comparison with Other Methods

Method	Measures FP By	Accuracy	Complexity
Willans' Line	Extrapolating fuel vs. BP	Moderate	Low
Morse Test	Cutting off cylinders	High	Medium
Motoring Test	Driving engine externally	High	High

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Conclusion

• Willans' Line is a simple, cost-effective way to estimate FP and ηₘ.

• Best suited for diesel engines under steady-speed conditions.

• For higher accuracy, combine with other methods (e.g., Morse Test)