When comparing the thermal efficiency of different engine cycles, specifically the Diesel, Otto, and Dual combustion cycles, under the condition of the same maximum pressure and peak temperature, the Diesel cycle demonstrates the highest efficiency.

Engine Cycles Overview

To understand why the Diesel cycle is most efficient under these specific conditions, let's briefly review the characteristics of each cycle:

• Otto Cycle: This cycle is characterized by heat addition occurring at constant volume. It is an ideal cycle for spark-ignition (petrol) engines.
• Diesel Cycle: In this cycle, heat addition occurs at constant pressure. It is an ideal cycle for compression-ignition (diesel) engines.
• Dual Combustion Cycle: This cycle is a combination of both Otto and Diesel cycles. Heat addition occurs partly at constant volume and partly at constant pressure. It aims to represent the actual combustion process in modern high-speed diesel engines more accurately.

Efficiency Comparison Conditions: Same Maximum Pressure and Peak Temperature

The key to this comparison lies in the specified constraints: "same maximum pressure" and "peak temperature". These conditions significantly influence the thermal efficiency of each cycle.

Thermal efficiency ($\eta_{th}$) for any heat engine cycle is generally given by:

$$\eta_{th} = 1 - \frac{Q_{out}}{Q_{in}}$$

where \(Q_{out}\) is the heat rejected and \(Q_{in}\) is the heat added. For higher efficiency, the amount of heat rejected (\(Q_{out}\)) should be as low as possible for a given heat input, or the net work output should be maximized.

Diesel Cycle Efficiency Advantage

Under the constraint of the same maximum pressure and peak temperature, the Diesel cycle is the most efficient due to the following reasons:

• Expansion Ratio: For all three cycles (Otto, Diesel, and Dual) starting from the same initial state and reaching the same maximum pressure and peak temperature, the Diesel cycle will have the largest expansion ratio. This is because heat is added at constant pressure in the Diesel cycle, allowing the volume to increase during heat addition, which means the expansion process starts from a larger volume and continues to expand further compared to the other cycles for the same initial compression ratio.
• Net Work Output: A larger expansion ratio translates to a greater area under the pressure-volume (P-V) diagram during the expansion stroke. This means the Diesel cycle performs more net work output for the same heat input when compared under these specific conditions.
• Heat Rejection: Consequently, because the Diesel cycle has a larger expansion ratio and performs more work, the temperature at the end of the expansion stroke will be lower for the Diesel cycle compared to the Otto or Dual cycles under the same maximum pressure and peak temperature conditions. A lower exhaust temperature implies less heat rejection ($Q_{out}$) to the surroundings for a given amount of heat added.

In essence, with the same maximum pressure and peak temperature, the Diesel cycle is able to extract more work from the heat input by allowing for a greater expansion, which leads to a lower temperature at the point of heat rejection. This results in the highest thermal efficiency among the three cycles under these specific operating conditions.