Question

The power required by a heat pump to maintain a house at 22°C during winter is 3 kW. The rate of heat transfer from the boundary of the house is 4000 kJ/hr per degree temperature difference between inside and outside. The actual COP is 50% of maximum COP. What is the minimum temperature for which the heat pump can meet the heating requirements of the house?

• A. 1.75°C
• B. 1.89°C
• C. 2.04°C
• D. 2.42°C

Answer 1

The correct option is C 2.04°C

$(COP{)}_{HP}=\frac{Q_1}{W}$

$\Rightarrow W=3kW$

and $Q_1=4000(T_2-T_{ambient})kJ/hr$

$Q_1=\frac{4000}{3600}(T_1-T_{ambient})kW$

$(COP{)}_{HP}=\frac{4000}{3600\times 3}(T_1-T_{ambient})$

$(COP{)}_{max}=\frac{T_1}{T_1-T_{ambient}}$

$\Rightarrow (COP{)}_{HP}=0.5\times \frac{T_1}{T_1-T_{ambient}}$

$\Rightarrow \frac{4000}{3600\times 3}(T_1-T_{ambient})=\frac{0.5T_1}{T_1-T_{ambient}}$

$\frac{10}{27}(T_1-T_{ambient}{)}^2=0.5\times T_1$

$\Rightarrow (T_1-T_{ambient}{)}^2=1.35\times T_1=1.35\times 295=398.25$

$T_1-T_{ambient}=19.9562$

$T_{ambient}=295-19.9562=275.0438K={2.0438}^{\circ }C$