Question

A closed system undergoes a process in which it interacts with the surroundings maintained at \(T_{0} = 300\text{ K}\) and \(P_{0} = 100\text{ kPa}\). During the process:
\(\bullet\) The system receives 1000 kJ of heat from a reservoir at 1200 K.
\(\bullet\) The system does 500 kJ of boundary work on the surroundings.
\(\bullet\) There is no change in kinetic or potential energy, and the process ends at the dead state (equilibrium with surroundings).
Which of the following statements is most correct?

• A. The maximum useful work (availability) from the given heat input is 750 kJ, and the irreversibility is 750 kJ.
• B. The maximum useful work (availability) from the given heat input is 500 kJ, and the irreversibility is 250 kJ.
• C. The maximum useful work (availability) from the given heat input is 750 kJ, and the irreversibility is 500 kJ.
• D. The maximum useful work (availability) from the given heat input is 750 kJ, and the irreversibility is 250 kJ.

Hint:

First calculate $W_{\text{max,useful}}$ using the Carnot factor.
Since $1000 \times (1 - 300/1200) = 750\text{ kJ}$, this eliminates option B.
Then, the difference between this theoretical limit and the actual work done ($750 - 500$) gives the irreversibility of $250\text{ kJ}$, pointing directly to option D.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
The question asks for the maximum useful work (availability/exergy) of the heat input and the irreversibility of a closed system process.

Step 2: Key Formula or Approach:
The maximum useful work (\(W_{\text{max,useful}}\)) that can be obtained from a heat transfer \(Q\) from a reservoir at temperature \(T\) is given by:
\[ W_{\text{max,useful}} = Q \left( 1 - \frac{T_{0}}{T} \right) \] The irreversibility (\(I\)) of the process is the difference between this maximum useful work and the actual useful work (\(W_{\text{actual,useful}}\)) performed:
\[ I = W_{\text{max,useful}} - W_{\text{actual,useful}} \]

Step 3: Detailed Explanation:

• Identify the given parameters:
Heat input, \(Q = 1000\text{ kJ}\).
Reservoir temperature, \(T = 1200\text{ K}\).
Surroundings temperature, \(T_{0} = 300\text{ K}\).
Actual work done, \(W_{\text{actual}} = 500\text{ kJ}\).

• Calculate the maximum useful work (availability) of the heat input:
\[ W_{\text{max,useful}} = 1000 \left( 1 - \frac{300}{1200} \right) \] \[ W_{\text{max,useful}} = 1000 \left( 1 - 0.25 \right) = 1000 \times 0.75 = 750\text{ kJ} \]
• Calculate the actual useful work done.
Since the process ends at the dead state, the boundary work of \(500\text{ kJ}\) is the actual useful work performed by the system:
\[ W_{\text{actual,useful}} = 500\text{ kJ} \]
• Calculate the irreversibility of the process:
\[ I = W_{\text{max,useful}} - W_{\text{actual,useful}} = 750 - 500 = 250\text{ kJ} \]

Step 4: Final Answer:
The maximum useful work is \(750\text{ kJ}\) and the irreversibility is \(250\text{ kJ}\).