Question 9.2: A house has a winter heating requirement of 30 kJ.s^−1 and a...
A house has a winter heating requirement of 30 kJ⋅s^{−1} and a summer cooling requirement of 60 kJ⋅s^{−1}. Consider a heat pump installation to maintain the house temperature at 20°C in winter and 25°C in summer. This requires circulation of the refrigerant through interior exchanger coils at 30°C in winter and 5°C in summer. Underground coils provide the heat source in winter and the heat sink in summer. For a year-round ground temperature of 15°C, the heat-transfer characteristics of the coils necessitate refrigerant temperatures of 10°C in winter and 25°C in summer. What are the minimum power requirements for winter heating and summer cooling?
The minimum power requirements are provided by a Carnot heat pump. For winter heating, the house coils are at the higher-temperature level T_H , and the heat requirement is QH = 30 kJ⋅s^{−1} . Application of Eq. (5.4) gives:
\frac{-Q_C}{T_C}=\frac{Q_H}{T_H} (5.4)
Q_C=-Q_H \frac{T_C}{T_H}=30\left(\frac{10+273.15}{30+273.15}\right)=28.02 \mathrm{~kJ} \cdot \mathrm{s}^{-1}
This is the heat absorbed in the ground coils. By Eq. (9.1),
W=-\left(Q_C+Q_H\right) (9.1)
W=-Q_H-Q_C=30-28.02=1.98 \mathrm{~kJ} \cdot \mathrm{s}^{-1}
Thus the power requirement is 1.98 kW.
For summer cooling, Q_C=60 \mathrm{~kJ} \cdot \mathrm{s}^{-1} , and the house coils are at the lowertemperature level T_C . Combining Eqs. (9.2) and (9.3) and solving for W:
\omega \equiv \frac{\text { heat absorbed at the lower temperature }}{\text { net work }}=\frac{Q_C}{W} (9.2)
\omega=\frac{T_C}{T_H-T_C} (9.3)
W=Q_C \frac{T_H-T_C}{T_C}=60\left(\frac{25-5}{5+273.15}\right)=4.31 \mathrm{~kJ} \cdot \mathrm{s}^{-1}
The power requirement here is therefore 4.31 kW. Actual power requirements for practical heat pumps are likely to be more than twice this lower limit.