Answer:
[tex]\dot W_{in} = 49.386\,kW[/tex]
Explanation:
An adiabatic compressor is modelled as follows by using the First Law of Thermodynamics:
[tex]\dot W_{in} + \dot m \cdot c_{p}\cdot (T_{1}-T_{2}) = 0[/tex]
The power consumed by the compressor can be calculated by the following expression:
[tex]\dot W_{in} = \dot m \cdot c_{v}\cdot (T_{2}-T_{1})[/tex]
Let consider that air behaves ideally. The density of air at inlet is:
[tex]P\cdot V = n\cdot R_{u}\cdot T[/tex]
[tex]P\cdot V = \frac{m}{M}\cdot R_{u}\cdot T[/tex]
[tex]\rho = \frac{P\cdot M}{R_{u}\cdot T}[/tex]
[tex]\rho = \frac{(104\,kPa)\cdot (28.02\,\frac{kg}{kmol})}{(8.315\,\frac{kPa\cdot m^{3}}{kmol\cdot K} )\cdot (292\,K)}[/tex]
[tex]\rho = 1.2\,\frac{kg}{m^{3}}[/tex]
The mass flow through compressor is:
[tex]\dot m = \rho \cdot \dot V[/tex]
[tex]\dot m = (1.2\,\frac{kg}{m^{3}})\cdot (0.15\,\frac{m^{3}}{s} )[/tex]
[tex]\dot m = 0.18\,\frac{kg}{s}[/tex]
The work input is:
[tex]\dot W_{in} = (0.18\,\frac{kg}{s} )\cdot (1.005\,\frac{kJ}{kg\cdot K})\cdot (565\,K-292\,K)[/tex]
[tex]\dot W_{in} = 49.386\,kW[/tex]
