Answer:
a) [tex]\dot W = 0.978\,kW[/tex], b) [tex]I = \left(50000\,\frac{kJ}{h} \right)\cdot \left(\frac{1}{3600}\,\frac{h}{s}\right)\cdot \left(\frac{1}{COP_{real}} \right) - 0.978\,kW[/tex]
Explanation:
a) The ideal Coefficient of Performance for the heat pump is:
[tex]COP_{HP} = \frac{T_{H}}{T_{H}-T_{L}}[/tex]
[tex]COP_{HP} = \frac{298.15\,K}{298.15\,K - 277.15\,K}[/tex]
[tex]COP_{HP} = 14.198[/tex]
The reversible work input is:
[tex]\dot W = \frac{\dot Q_{H}}{COP_{HP}}[/tex]
[tex]\dot W = \left(\frac{50000\,\frac{kJ}{h} }{14.198} \right)\cdot \left(\frac{1}{3600}\,\frac{h}{s} \right)[/tex]
[tex]\dot W = 0.978\,kW[/tex]
b) The irreversibility is given by the difference between real work and ideal work inputs:
[tex]I = \dot W_{real} - \dot W_{ideal}[/tex]
[tex]I = \left(50000\,\frac{kJ}{h} \right)\cdot \left(\frac{1}{3600}\,\frac{h}{s}\right)\cdot \left(\frac{1}{COP_{real}} \right) - 0.978\,kW[/tex]
