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Determine the temperature in oC of the low temperature for the heat pump.

A Carnot heat engine which operates between temperature levels of 927°C and 33°C rejects 30 KJ to the low temperature sink. The heat pump receives 270 KJ of heat from a low temperature reservoir and rejects it to the surroundings at 33°C.

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Given: $T_1 = 1200K, T_2 = T4 = 306K, Q_2 = 30kJ, Q_3 = 270kJ$

(Note: Here T2 and T4 represent the same temperature reservoir, but we use different notation so as to not confuse between the Engine sink and Heat pump sink)

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For Heat Engine, the Heat transfer is given by the ratio,

$\frac{Q_1}{Q_2} = \frac{T_1}{TT_2}$

$\frac{Q_1}{30} = \frac{1200}{306}$

$Q_1 = 177.647kJ$

Now, For Energy Balance of Heat Engine,

$W =Q_1 - Q_2$

W = 117.647 - 30 = 87.647kJ

Now, For Energy Balance of Heat Pump,

$Q_4 = W + Q_3$

$Q_4 = 87.647 + 270$

$Q_4 = 357.647kJ$

For Heat Pump, the Heat transfer is given by the ratio,

$\frac{Q_4}{Q_3} = \frac{T_4}{T_3}$

$\frac{357.647}{270} = \frac{306}{T_3}$

$T_3 = 231K = -42^oC$

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