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## Q. 11.1

Compare the mechanical power requirement to compress 1 kg/s of dry air from 100 kPa and 20°C (ambient conditions) to 800 kPa (absolute pressure) using isothermal, adiabatic, or polytropic (γ = 1.3) compression .

## Verified Solution

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)