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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) 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,

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

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

$T_3 = 231K = -42^oC$