Question

Consider a reversible heat engine with a thermal efficiency of 55%. The engine is reversed and operated as a heat pump between the same temperature reservoirs.
If the heat supplied from the low temperature reservoir is 45 kJ/cycle, then the absolute value of heat rejected is ______ kJ/cycle (answer in integer).

Hint:

First get T_L/T_H from the engine efficiency, then reuse the same ratio for the heat pump's Q_L/Q_H.

Answer 1

Correct Answer: 100

Step 1: Relate efficiency to the reservoir temperatures.
For a reversible engine, thermal efficiency depends only on the reservoir temperatures: \( \eta = 1 - \dfrac{T_L}{T_H} \).
With \( \eta = 0.55 \), this gives \( \dfrac{T_L}{T_H} = 1 - 0.55 = 0.45 \).

Step 2: Use the same temperature ratio for the reversed cycle.
Running the same reversible cycle backward as a heat pump between the same two reservoirs preserves the reversible relation \( \dfrac{Q_L}{Q_H} = \dfrac{T_L}{T_H} \), where \(Q_L\) is heat drawn from the cold reservoir and \(Q_H\) is heat rejected to the hot reservoir.
So \( Q_H = \dfrac{Q_L}{T_L / T_H} = \dfrac{Q_L}{0.45} \), which is the same as \( Q_H = Q_L / (1 - \eta) \).

Final Answer:
Substitute \(Q_L = 45\) kJ/cycle.
\[ Q_H = \frac{45}{0.45} = 100 \ \text{kJ/cycle} \] \[ \boxed{Q_H = 100 \ \text{kJ/cycle}} \]