91Ó°ÊÓ

The first law of thermodynamics is all about the conservation of energy. In simple terms, it means that energy cannot be created or destroyed, only transformed from one form to another. For a heat engine, which is a device that transforms thermal energy into mechanical work, this law is expressed as: \[ \dot{Q}_H = \dot{Q}_L + \dot{W} \]where:

• \( \dot{Q}_H \) is the heat energy supplied to the engine.
• \( \dot{Q}_L \) is the heat energy rejected by the engine.
• \( \dot{W} \) is the work done by the engine.

Whenever evaluating a heat engine, checking if this equation holds is the first step. If the equation balances, it means the system adheres to the first law, affirming that energy is conserved. For instance, in case (a), with \( \dot{Q}_H = 6 \text{kW} \), \( \dot{Q}_L = 4 \text{kW} \), and \( \dot{W} = 2 \text{kW} \), the energy relationship holds because \( 6 = 4 + 2 \). This indicates that the first law is satisfied.

The second law of thermodynamics is centered on the direction of energy transfers and transformations. It states that processes occur in a direction that increases entropy, or disorder, and that energy has quality as well as quantity. In the context of a heat engine, it implies that not all input energy can be converted into work, as some heat will always be "wasted" to a cold reservoir or environment.According to this law, for a heat engine:\[ \dot{Q}_H \geq \dot{Q}_L + \dot{W} \]This means the heat added to the system must be greater than or equal to the sum of the heat rejected and the work produced, ensuring some energy is unavailable for work. If this condition is not met, the engine is said to violate the second law. For example, in case (b), where \( \dot{Q}_H = 6 \text{kW} \), \( \dot{Q}_L = 0 \text{kW} \), and \( \dot{W} = 6 \text{kW} \), it appears as though all input heat is converted to work with no waste. This scenario constitutes a "perfect" engine, which is impossible in reality, thus violating the second law.

When analyzing a heat engine, it's crucial to check both the first and second laws of thermodynamics to ensure its feasibility. The first law confirms energy conservation, while the second law ensures proper energy transformation direction and accounts for efficiency realism.Considerations during analysis include:

• Energy Balance: Verifying the sum of heat rejected and work matches the heat supplied confirms adherence to the first law.
• Efficiency Check: Calculating the efficiency and checking against theoretical limits ensures the heat engine respects the second law. For example, efficiency, \( \eta \), of an ideal heat engine is calculated as:\[ \eta = \frac{\dot{W}}{\dot{Q}_H} \]

In case (d), \( \dot{Q}_H = 6 \text{kW} \), \( \dot{Q}_L = 6 \text{kW} \), and \( \dot{W} = 0 \text{kW} \), the engine does no work, merely transferring heat completely to the sink, thus aligning with second law's expectations as efficiency becomes effectively zero.