Answer
a) $ \dot{W}_{elec} = 5.61 \ hp$
b) $Cost= 711.7 \ \$ $
Work Step by Step
Given}
-Energy delivered by heat transfer to the dwelling $\dot{Q}_{out}=40,000 \ Btu/h$
-Coefficient of performance $COP=2.8$
-$Cost = 0.085 \ [\$ / KW \cdot h] $
-Time of operation $ t=2000 \ h $
Required
a) The power input to the cycle $\dot{W}_{elec} \ [hp]$
b) Cost of electricity during the heating season when the
heat pump operates for 2000 hours [\$]
Assumption
-Ideal cycle
Solution}
a)
Coefficient of performance could be defined by.
$$COP= \displaystyle{\frac{\dot{Q}_{out}}{\dot{W}_{elec}}}$$
$\dot{W}_{elec}=\displaystyle{\frac{\dot{Q}_{out}}{COP} \rightarrow \frac{40,000}{2.8}}$
${=14,285.71 \ Btu/h} $
$ \displaystyle{ \times \left( \frac{ 1.055 \left( \frac{KJ}{Btu} \right) }{ 3600 \left( \frac{s}{h} \right) } \right)} $
$\displaystyle{ =4.187 \ KW \times \left( \frac{hp}{0.746 \ Kw} \right) }$
${ =5.61 \ hp }$
b)
Electrical energy could be defined by.
$$W_{elec}=\dot{W}_{elec} \times t $$
$W_{elec} \rightarrow 4.187 \times 2000 = 8873 \ KW \cdot h $
Cost of electricity during the heating season when the heat pump operates for 2000 hours could be defined by.
$$Cost = Cost/Work \times Work$$
$Cost \rightarrow 0.085 \times 8873 = 711.7 \ \$ $
