Short Answer:

The air-standard efficiency of the Diesel cycle is the ratio of useful work done by the engine to the total heat energy supplied, assuming air as the working medium and all processes are reversible. It depends on the compression ratio (r), cut-off ratio (β), and specific heat ratio (γ) of air. The formula is:

The efficiency of the Diesel cycle increases with higher compression ratio but decreases with an increase in the cut-off ratio.

Detailed Explanation :

Air-standard efficiency of the Diesel cycle

The Diesel cycle is a thermodynamic cycle that represents the working of a compression-ignition (CI) engine, commonly known as a Diesel engine. In this cycle, the working fluid is assumed to be air, and all processes are reversible and ideal. The air-standard efficiency provides a theoretical measure of how efficiently the Diesel cycle converts heat energy into useful mechanical work.

The Diesel cycle differs from the Otto cycle mainly in the process of heat addition. In the Otto cycle, heat is added at constant volume, whereas in the Diesel cycle, heat is added at constant pressure.

1. Processes of the Diesel Cycle

The Diesel cycle consists of four thermodynamic processes:

1. Process 1–2: Isentropic Compression
   • Air is compressed adiabatically (without heat exchange).
   • During this process, the pressure and temperature of air increase significantly due to compression.
   • The relation between temperature and volume is given as:

where  is the compression ratio.

1. Process 2–3: Constant Pressure Heat Addition
   • At the end of compression, fuel is injected into the hot air.
   • The fuel burns and adds heat at constant pressure, causing the air to expand.
   • The volume increases from  to , and the ratio of these volumes is called the cut-off ratio (β):

1. Process 3–4: Isentropic Expansion (Power Stroke)
   • The high-pressure gas expands adiabatically, pushing the piston downward and producing work.
   • The temperature and pressure drop as the volume increases.
   • The relation between temperature and volume is:

1. Process 4–1: Constant Volume Heat Rejection
   • At the end of expansion, heat is rejected at constant volume.
   • This completes the thermodynamic cycle and prepares the system for the next cycle.

2. Derivation of Air-standard Efficiency

The air-standard efficiency (η) of the Diesel cycle is defined as:

Where,

•  = heat supplied during constant pressure process (2–3)
•  = heat rejected during constant volume process (4–1)

For constant pressure heat addition (2–3):

For constant volume heat rejection (4–1):

Substituting into the efficiency equation:

Since :

Using temperature relations derived from the isentropic processes:

 

After substituting and simplifying, the final expression becomes:

This is the air-standard efficiency of the Diesel cycle.

3. Factors Affecting Air-standard Efficiency

1. Compression Ratio (r):
   • Higher compression ratios lead to higher efficiency.
   • Diesel engines typically operate at compression ratios between 14:1 and 22:1, higher than petrol engines.
   • This allows them to achieve greater efficiency than Otto cycle engines.
2. Cut-off Ratio (β):
   • The cut-off ratio represents how much the volume increases during combustion.
   • A larger β means longer fuel burning and lower efficiency.
   • Hence, to maximize efficiency, β should be kept as small as possible.
3. Specific Heat Ratio (γ):
   • The ratio  depends on the gas properties.
   • For air, γ ≈ 1.4.
   • A higher γ leads to higher efficiency since the gas expands more effectively during the power stroke.

4. Comparison with Otto Cycle Efficiency

• The Diesel cycle is less efficient than the Otto cycle for the same compression ratio because part of the heat addition occurs at constant pressure.
• However, Diesel engines operate at much higher compression ratios, so their actual efficiency can be equal to or even higher than that of Otto engines.
• Also, Diesel engines are more fuel-efficient because of their leaner air-fuel mixture and absence of throttle losses.

5. Practical Example

For a Diesel engine with:

• Compression ratio ,
• Cut-off ratio ,
• Specific heat ratio ,

The air-standard efficiency can be calculated as:

Thus, the efficiency is about 56%, which is higher than that of typical petrol engines (around 45%).

Conclusion

The air-standard efficiency of the Diesel cycle defines the ideal performance of a compression-ignition engine, assuming perfect conditions. It depends mainly on the compression ratio, cut-off ratio, and specific heat ratio. The formula,

shows that efficiency increases with higher compression ratio and decreases with higher cut-off ratio. Though real engines have losses, this efficiency equation is very useful for engine analysis and design improvements.