Question 11.1: Compare the mechanical power requirement to compress 1 kg/s ...

Using Eq. (11.3) with P_{i }= 100 kPa, P_{o} = 800 kPa, T_{i} = 293 K, R_{a} = 287 J/(kg.K); the mechanical power requirement for the isothermal compression of 1.0 kg/s can be estimated:

\dot{W}_{m}=\dot{m}_{a}.R_{a}.T_{i}.Ln(\frac {P_{o}}{P_{i}})               (11.3)

\dot{W}_{m}=(1.0 kg/s).(287 J/kg .K).(293 K).Ln(\frac {800kPa}{100kPa}) =174.86KW

The mechanical power for the adiabatic compression is calculated using Eq. (11.4) with γ = k = 1.4 :

\dot{W}_{m}=\frac {\dot{m}_{a}.R_{a}.T_{i}.γ}{γ-1}[(\frac {P_{o}}{P_{i}})^{\frac {γ-1}{γ}} -1]               (11.4)

\dot{W}_{m}=\frac {(1.0 kg/s).(287 J/kg .K).(293 K).(1.4)}{1.4-1}[(\frac {800kPa}{100kPa})^{\frac {1.4-1}{1.4}}-1]=238.82kW

For polytropic compression, the mechanical power requirement can be determined using Eq.(11.4) again but with γ = k = 1.3 :

\dot{W}_{m}=\frac {(1.0 kg/s).(287 J/kg .K).(293 K).(1.3)}{1.3-1}[(\frac {800kPa}{100kPa})^{\frac {1.3-1}{1.3}}-1]=224.42kW

As expected, the isothermal compression requires less mechanical power and the adiabatic compression requires more power input than any of the three compression types. Note that the polytropic compression which is more representative of an actual compression process has mechanical power requirements between those of the isothermal and adiabatic compressions.

It is interesting to note that the mechanical power input per unit of  airflow rate (i.e., kW/L/s) can be calculated for the three types of compression by estimating the density of air at the inlet conditions. The ideal gas equation Eq. (11.2) can be used to determine the density of the air (with Z_{a,I} = 1). For T_{i} = 293 K and P_{i} = 100 kPa, the density of air 1.189 kg/m3. Thus, the mechanical input for the three compression types is 0.21 kW/L/s for an isothermal compression, 0.27 kW/L/s for a polytropic compression, and 0.29 kW/L/s for an adiabatic compression.

P_{i} = ρ_{i} .Z_{a,i} .R_{a} .T_{i}             (11.2)