Explanation:

Understanding the Problem:

The problem describes a heat engine that receives heat from a source at a high temperature of 1500 K at a rate of 600 kJ/s and rejects the waste heat to a sink at a low temperature of 300 K. The power output of the engine is given as 400 kW. We need to calculate the second-law efficiency of the heat engine.

Second Law of Thermodynamics:

The second law of thermodynamics states that no heat engine can have a higher efficiency than a Carnot cycle operating between the same two temperatures. The Carnot cycle is a theoretical cycle that operates between two reservoirs at different temperatures. It consists of four processes: two isothermal processes and two adiabatic processes.

Carnot Engine:

The Carnot cycle is the most efficient heat engine possible. The efficiency of a Carnot engine is given by the formula:

η = 1 - (Tc/Th)

where η is the efficiency, Tc is the temperature of the cold reservoir, and Th is the temperature of the hot reservoir.

Calculating the Second-Law Efficiency:

To calculate the second-law efficiency of the heat engine, we first need to calculate the temperature of the hot reservoir (Th) and the temperature of the cold reservoir (Tc).

Given:

Th = 1500 K
Tc = 300 K
Qin = 600 kJ/s
Pout = 400 kW

We can calculate the heat input (Qin) using the formula:

Qin = Pout/η

where η is the efficiency of the heat engine.

Substituting the values, we get:

Qin = 400,000/η

600,000 = 400,000/η

η = 400,000/600,000

η = 0.67

Therefore, the second-law efficiency of the heat engine is 0.67 or 67%. This means that the heat engine is able to convert 67% of the heat input into useful work, while the remaining 33% is lost to the surroundings.