GATE 2024 ME – Question 47

Thermodynamics · Zeroth, first and second laws, work, heat and energy changes · 2 marks · Numerical answer

A heat pump (H.P.) is driven by the work output of a heat engine (H.E.) as shown in the figure. The heat engine extracts 150 kJ of heat from the source at 1000 K. The heat pump absorbs heat from the ambient at 280 K and delivers heat to the room which is maintained at 300 K. Considering the combined system to be ideal, the total amount of heat delivered to the room together by the heat engine and heat pump is _______ kJ (*answer in integer*).

A heat engine takes 150 kJ from a 1000 K source and gives work to a heat pump. Both reject heat to a room at 300 K, and the heat pump also takes heat from a 280 K ambient.

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Correct answer: 1620

Explanation

An ideal engine is a Carnot engine working between 1000 K and the room at 300 K. Its work is $W = 150\left(1 - \frac{300}{1000}\right) = 105$ kJ, and it rejects $150 - 105 = 45$ kJ to the room. The ideal heat pump between 280 K and 300 K has $COP = \frac{300}{300 - 280} = 15$, so it delivers $15 \times 105 = 1575$ kJ to the room. The total is $45 + 1575 = 1620$ kJ.