Relative efficiency of an engine is defined as the ratio of:

Explanation:

The relative efficiency of an engine is defined as the ratio of the actual thermal efficiency to the air-standard efficiency. This definition helps in understanding how well the engine performs in comparison to the ideal cycle upon which it is based. Let's delve deeper into what this means and why option 2 is the correct answer.

Thermal Efficiency: Thermal efficiency is a measure of the efficiency of a heat engine, and it is defined as the ratio of the work output to the heat input. Mathematically, it is expressed as:

Thermal Efficiency = (Work Output) / (Heat Input)

For actual engines, the thermal efficiency is influenced by various factors such as friction, heat losses, and other real-world inefficiencies. This actual thermal efficiency is what we measure in practice.

Air-Standard Efficiency: The air-standard efficiency is a theoretical efficiency based on the idealized air-standard cycle. This cycle assumes ideal conditions, such as no friction, perfect combustion, and no heat losses. It provides a benchmark or a reference point for comparison.

The air-standard efficiency can be calculated for different types of cycles, such as the Otto cycle, Diesel cycle, or Brayton cycle, depending on the type of engine. These cycles assume that the working fluid behaves as an ideal gas and that the processes occur in a perfectly reversible manner.

Relative Efficiency: The relative efficiency of an engine is a measure of how the actual performance of the engine compares to the idealized performance as predicted by the air-standard cycle. It is defined as:

Relative Efficiency = (Actual Thermal Efficiency) / (Air-Standard Efficiency)

This ratio helps engineers and designers understand the performance gap between the real engine and the idealized cycle, providing insights into potential areas for improvement.

Why Option 2 is Correct:

Option 2 correctly identifies the relative efficiency as the ratio of the actual thermal efficiency to the air-standard efficiency. This definition aligns with the principles of comparing real-world engine performance to the theoretical ideal performance. By understanding this ratio, one can evaluate how closely an engine approaches the theoretical maximum efficiency under given operating conditions.

Now, let's briefly analyze the other options to understand why they are not correct:

Option 1: Mechanical Efficiency to Volumetric Efficiency

Mechanical efficiency refers to the ratio of the brake power (useful power) of the engine to the indicated power (power developed inside the cylinders). Volumetric efficiency, on the other hand, measures how effectively an engine fills its cylinders with air during the intake process. While both are important metrics for engine performance, their ratio does not define relative efficiency. Therefore, this option is incorrect.

Option 3: Brake Thermal Efficiency to Indicated Thermal Efficiency

Brake thermal efficiency is the ratio of the brake power to the heat input, while indicated thermal efficiency is the ratio of the indicated power to the heat input. Although both relate to thermal efficiency, their ratio does not represent relative efficiency. Relative efficiency specifically compares actual thermal efficiency to air-standard efficiency, making this option incorrect.

Option 4: Actual Thermal Efficiency to Carnot Efficiency

Carnot efficiency is the maximum theoretical efficiency that a heat engine can achieve, based on the Carnot cycle. While comparing actual thermal efficiency to Carnot efficiency can provide insights into performance, relative efficiency is specifically defined as the ratio of actual thermal efficiency to air-standard efficiency, not Carnot efficiency. Hence, this option is incorrect.

In summary, the correct definition of relative efficiency is the ratio of actual thermal efficiency to air-standard efficiency, making option 2 the correct choice. This ratio helps in evaluating how well an engine performs compared to the idealized cycle, providing valuable insights for improving engine design and performance.

Important Information:

To further understand the concept of relative efficiency and why the other options are incorrect, let's explore some fundamental principles and definitions related to engine performance:

Mechanical Efficiency: This is a measure of how effectively an engine converts the power developed inside the cylinders (indicated power) into useful power at the crankshaft (brake power). It accounts for losses due to friction and other mechanical inefficiencies. The formula for mechanical efficiency is:

Mechanical Efficiency = (Brake Power) / (Indicated Power)

Volumetric Efficiency: This is a measure of the effectiveness of the engine's intake process. It is defined as the ratio of the actual volume of air-fuel mixture drawn into the cylinder to the theoretical volume that could be drawn in, based on the cylinder's displacement. The formula is:

Volumetric Efficiency = (Actual Volume of Air-Fuel Mixture) / (Theoretical Volume)

Brake Thermal Efficiency: This efficiency measures how effectively the engine converts the heat input from fuel into useful power at the crankshaft. It is defined as:

Brake Thermal Efficiency = (Brake Power) / (Heat Input)

Indicated Thermal Efficiency: This efficiency measures how effectively the engine converts the heat input from fuel into power developed inside the cylinders. It is defined as:

Indicated Thermal Efficiency = (Indicated Power) / (Heat Input)

Carnot Efficiency: This is the maximum theoretical efficiency that a heat engine operating between two temperature limits can achieve, based on the Carnot cycle. It is defined as:

Carnot Efficiency = 1 - (T_cold / T_hot)

Where T_cold is the temperature of the cold reservoir, and T_hot is the temperature of the hot reservoir.

Understanding these definitions and principles is crucial for analyzing engine performance and identifying areas for improvement. By comparing actual thermal efficiency to air-standard efficiency, engineers can gain insights into how closely an engine approaches its theoretical maximum performance, guiding efforts to enhance efficiency and reduce losses.