To solve this problem, we need to apply the principles of thermodynamics, particularly focusing on the operation of a reversible heat engine. We have two heat sources at different temperatures, and we need to determine how much heat is supplied by each source and the efficiency of the engine. Let's break this down step by step.
Understanding the Engine's Operation
A reversible engine operates between two heat reservoirs, absorbing heat from the hot reservoirs and rejecting some heat to a cold reservoir. In this case, we have:
- Hot source 1 at 900 K
- Hot source 2 at 600 K
- Cold sink at 300 K
Given Data
We know the following:
- Work output (W) = 100 kW = 100 kJ/min (since 1 kW = 60 kJ/min)
- Heat rejected (Q_out) = 3600 kJ/min
Applying the First Law of Thermodynamics
The first law of thermodynamics states that the energy input to the system (heat absorbed) minus the energy output (work done and heat rejected) equals the change in internal energy. For a steady-state engine, we can simplify this to:
Q_in = W + Q_out
Calculating Total Heat Supplied
Substituting the known values:
Q_in = 100 kJ/min + 3600 kJ/min = 3700 kJ/min
Determining the Efficiency of the Engine
The efficiency (η) of a heat engine is defined as the ratio of the work output to the heat input:
η = W / Q_in
Substituting the values we have:
η = 100 kJ/min / 3700 kJ/min = 0.0270 or 2.70%
Heat Supplied by Each Source
Now, we need to find out how much heat is supplied by each source. For a reversible engine, the heat absorbed from each source can be determined using the Carnot efficiency formula:
η_Carnot = 1 - (T_c / T_h)
Where T_c is the temperature of the cold sink and T_h is the temperature of the hot source. We will calculate the efficiency for both sources:
For the 900 K Source
η_Carnot (900 K) = 1 - (300 K / 900 K) = 1 - 1/3 = 2/3
Heat absorbed from the 900 K source (Q_1):
Q_1 = η_Carnot * Q_in = (2/3) * 3700 kJ/min = 2466.67 kJ/min
For the 600 K Source
η_Carnot (600 K) = 1 - (300 K / 600 K) = 1 - 1/2 = 1/2
Heat absorbed from the 600 K source (Q_2):
Q_2 = η_Carnot * Q_in = (1/2) * 3700 kJ/min = 1850 kJ/min
Final Results
In summary, the heat supplied by each source per minute is:
- From the 900 K source: approximately 2466.67 kJ/min
- From the 600 K source: approximately 1850 kJ/min
The efficiency of the engine is approximately 2.70%. This low efficiency indicates that a significant amount of energy is lost as waste heat, which is typical for engines operating between such temperature differences.
