91Ó°ÊÓ

Thermal efficiency is a measure of how well an engine converts heat into work. In the case of a Carnot engine, which is an idealized engine, the thermal efficiency (\( \eta \)) indicates the fraction of heat energy absorbed from the high-temperature reservoir that is converted into mechanical work. This efficiency can be expressed mathematically as:
\[ \eta = \frac{W}{Q_h} \]where \( W \) is the work done by the engine, and \( Q_h \) is the heat absorbed from the high-temperature reservoir.
For the Carnot engine in this example, the work done per cycle is 215 J, and the heat absorbed is 550 J. Hence, the thermal efficiency is approximately 39.09%. This means that about 39% of the heat energy absorbed is used to perform mechanical work, while the remaining energy is lost to the low-temperature reservoir.
The efficiency depends on the temperatures of the two reservoirs. The higher the efficiency, the more effective the engine is in converting heat into work.

The First Law of Thermodynamics is a fundamental principle that describes the conservation of energy. It states that energy cannot be created or destroyed, only transformed from one form to another. In terms of thermodynamic processes, this principle can be expressed as:

• \( Q = \Delta E + W \)

where \( Q \) is the heat added to the system, \( \Delta E \) is the change in internal energy, and \( W \) is the work done by the system.
For a Carnot engine, which is a cyclic process, the change in internal energy over a complete cycle is zero (\( \Delta E = 0 \)). Therefore, the First Law simplifies to:
\[ Q_h = Q_c + W \]where \( Q_c \) is the heat rejected to the low-temperature reservoir. This equation confirms that the energy taken in as heat from the high-temperature reservoir (\( Q_h \)) is equal to the sum of the work done and the heat released.
In this particular exercise, the Carnot engine absorbs 550 J of heat and releases 335 J to the surrounding. The difference between these two amounts, which is 215 J, is the work done by the engine as mechanical work during each cycle.

The low-temperature reservoir is critical in the operation of a Carnot engine. It is the thermal sink where the engine releases heat that cannot be converted into work. This reservoir plays a fundamental role in defining the efficiency of the Carnot cycle.
The efficiency of the Carnot engine depends on the temperature difference between the high-temperature and low-temperature reservoirs. For maximum efficiency, the temperature of the low-temperature reservoir (\( T_c \)) should be as low as possible relative to the high-temperature reservoir (\( T_h \)). The relationship between temperatures and efficiency is given by:
\[ \eta = 1 - \frac{T_c}{T_h} \]In the given example, with a high-temperature reservoir at 620 K and a calculated efficiency of approximately 39.09%, the low-temperature reservoir's temperature can be determined to be 378 K.
This temperature differential allows the engine to transfer heat through the cycle and perform work. Maintaining effective temperature management is crucial in real-world applications to ensure the engine operates with high efficiency.