Thermal Efficiency of a Reversible Engine

A reversible engine is an idealized engine that operates in a cycle consisting of two isothermal and two isobaric processes. The engine uses an ideal gas as the working substance and assumes that the specific heat at constant pressure (CP) and the specific heat at constant volume (CV) are constant throughout the process. The thermal efficiency of a reversible engine can be calculated using the following formula:

Thermal Efficiency = 1 - (T2/T1)

Where T1 is the temperature at the beginning of the isothermal expansion (hot reservoir) and T2 is the temperature at the end of the isothermal compression (cold reservoir).

Explanation:

1. Reversible Engine Cycle: The reversible engine operates in a cycle consisting of two isothermal and two isobaric processes. During the isothermal expansion, the working substance absorbs heat from the hot reservoir at a constant temperature (T1) and expands, doing work on the surroundings. This process is reversible, meaning that the temperature of the working substance remains constant throughout the expansion.

2. Isothermal Expansion: During the isothermal expansion, the working substance absorbs heat (Q1) from the hot reservoir and expands from an initial volume (V1) to a final volume (V2). Since the process is isothermal, the temperature remains constant at T1. The work done by the gas during this process is given by the equation:

W1 = nRT1 * ln(V2/V1)

Where n is the number of moles of the gas, R is the ideal gas constant, and ln represents the natural logarithm.

3. Isobaric Compression: After the isothermal expansion, the working substance undergoes an isobaric compression. During this process, the gas is compressed at a constant pressure (P2) and the temperature decreases from T1 to T2. The heat (Q2) released during this process is given by:

Q2 = nCP(T1 - T2)

Where CP is the specific heat at constant pressure.

4. Isothermal Compression: The working substance then undergoes an isothermal compression, during which it rejects heat (Q3) to the cold reservoir at a constant temperature (T2) and decreases in volume from V2 to V1. The work done on the gas during this process is given by:

W2 = nRT2 * ln(V1/V2)

5. Isobaric Expansion: Finally, the working substance undergoes an isobaric expansion, during which it absorbs heat (Q4) from the surroundings and increases in volume from V1 to V2. The heat absorbed during this process is given by:

Q4 = nCV(T2 - T1)

Where CV is the specific heat at constant volume.

6. Thermal Efficiency: The thermal efficiency of the reversible engine can be calculated using the formula:

Thermal Efficiency = (Q1 - Q3)/(Q1) = 1 - (Q3/Q1) = 1 - (T2/T1)

This formula is derived from the first law of thermodynamics, which states that the net work done by the