Atmospheric air is at 20°C, 100 kPa with zero velocity. An adiabatic reversible compressor takes atmospheric air in through a pipe with cross-sectional area of 0.1 m² at a rate of 1 kg/s. It is compressed up to a measured stagnation pressure of 500 kPa and leaves through a pipe with cross-sectional

Question

Atmospheric air is at 20°C, 100 kPa with zero velocity. An adiabatic reversible compressor takes atmospheric air in through a pipe with cross-sectional area of 0.1 m² at a rate of 1 kg/s. It is compressed up to a measured stagnation pressure of 500 kPa and leaves through a pipe with cross-sectional area of 0.01 m². What are the required compressor work and the air velocity, static pressure, and temperature in the exit pipeline?

Accepted Answer

C.V. compressor out to standing air and exit to stagnation point.

[latex]\begin{aligned}& \dot{ m }h_{ o 1}+\dot{ W }_{ c }=\dot{ m }\left( h + \mathbf{V} ^2 / 2 ight)_{ ex }=\dot{ mh }_{ o , ex } \& \dot{ ms }_{ o 1}=\dot{ ms }_{ o , ex } \Rightarrow P _{ r , o , ex }= P _{ r , o 1} \& \left( P _{ st , ex } P _{ o 1} ight)=1.028(500 / 100)=5.14 \& \quad \Rightarrow T _{ o , ex }=463 K , \quad h _{ o , ex }=465.38\, kJ / kg , \quad h _{ o 1}=209.45 \,kJ / kg\end{aligned}[/latex]

[latex]\begin{aligned}& \dot{ W }_{ c }=\dot{ m }\left( h _{ o , ex }- h _{ o 1} ight)=1(465.38-209.45)= 2 5 5 . 9\, k W \& P _{ ex }= P _{ o , ex }\left( T _{ ex } / T _{ o , ex } ight)^{ k /( k -1)} \quad T _{ ex }= T _{ o , ex }- V _{ ex }^2 / 2 C _{ p } \& \dot{ m }=1 \,kg / s =( ho AV )_{ ex }= P _{ ex } AV _{ ex } / RT _{ ex }\end{aligned}[/latex]

Now select 1 unknown amongst [latex]P_{e x}, T_{e x}, V_{e x}[/latex] and write the continuity eq. [latex]\dot{ m }[/latex] and solve the nonlinear equation. Say, use [latex]T _{ ex }[/latex] then

[latex]\begin{aligned}& \mathbf{V} _{e x}=\sqrt{2 C_p\left(T_{o, e x}-T_{e x} ight)} \& \dot{m}=1 \,kg / s =P_{o, e x}\left(T_{e x} / T_{o, e x} ight)^{k / k-1} A \sqrt{2 C_p\left(T_{o, e x}-T_{e x} ight)} / R T_{e x}\end{aligned}[/latex]

solve for [latex]T _{ ex } / T _{ o , ex }[/latex] (close to 1)

[latex]T _{ ex }=462.6 \,K \Rightarrow \mathbf{V} _{ ex }=28.3 \,m / s , \quad P _{ ex }=498.6\, kPa[/latex]

Question 17.CSGP.82: Atmospheric air is at 20°C, 100 kPa with zero velocity. An a......

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