The conditions of air at the entry of an axial flow compressor stage are p1 = 100kN/m^2 and Tl = 300 K. The air angles are β1 = 51º, β2 = 10º, a1 = a 3 = 8º. The mean diameter and peripheral speed are 0.5 m and 150 m/s respectively. Mass flow rate through the stage is 30 kg/s; the work done

Question

The conditions of air at the entry of an axial flow compressor stage are [latex] p_{1}=100 \mathrm{kN} / \mathrm{m}^{2} [/latex] and [latex] T_{1}=300 \mathrm{~K} [/latex] . The air angles are [latex] \beta_{1}=51^{\circ}, \beta_{2}=10^{\circ}, \alpha_{1}=\alpha_{3}=8^{\circ} [/latex] .

The mean diameter and peripheral speed are 0.5 m and 150 m/s respectively. Mass flow rate through the stage is 30 kg/s; the work done factor is 0.95 and mechanical efficiency is 90%. Assuming an isentropic stage efficiency of 85%, determine (i) blade height at entry (ii) stage pressure ratio, and (iii) the power required to drive the stage (for air, [latex] R=287 \mathrm{~J} / \mathrm{kg} \mathrm{K}, \quad \gamma=1.4 [/latex] )

Accepted Answer

(i) [latex] ho_{1}=\frac{p_{1}}{R T_{1}}=\frac{100 imes 10^{3}}{287 imes 300}=1.16 \mathrm{~kg} / \mathrm{m}^{3} [/latex]

From Eq. (16.12),

[latex] \frac{U}{V_{f}}= an \alpha_{1}+ an \beta_{1} [/latex]

Hence, [latex] V_{f}=\frac{150}{ an 8^{\circ}+ an 51^{\circ}} [/latex]

[latex] =109.06 \mathrm{~m} / \mathrm{s} [/latex]

[latex] \dot{m}=V_{f} ho_{1}\left(\pi d h_{1} ight) [/latex]

[latex] 30=109.06 imes 1.16 imes \pi imes 0.5 h_{1} [/latex]

which gives [latex] h_{1}=0.15 \mathrm{~m} [/latex]

(ii) From Eq. (16.19)

[latex] \Delta T_{s t}=\frac{\lambda U V_{f}}{c_{p}}\left( an \beta_{1}- an \beta_{2} ight) [/latex]

Again, [latex] c_{p}=\frac{1.4}{(1.4-1)} imes 287=1005 \mathrm{~J} / \mathrm{kg} \mathrm{K} [/latex]

Hence, [latex] \Delta T_{s t}=\frac{0.95 imes 150 imes 190.06}{1005}\left( an 51^{\circ}- an 10^{\circ} ight) [/latex]

[latex] =16.37^{\circ} \mathrm{C} [/latex]

With the help of

Eq. (16.20)

[latex] R_{s}=\left[1+\frac{\eta_{s} \Delta T_{s t}}{T_{1 t}} ight]^{\frac{\gamma}{\gamma-1}} [/latex]

we can write

Pressure ratio, [latex] R_{s}=\left[1+\frac{0.85 imes 16.37}{300} ight]^{\frac{1.4}{0.4}} [/latex]

=1.17

(iii) [latex] P=\frac{\dot{m} w}{\eta_{m}}=\frac{\dot{m} c_{p} \Delta T_{s t}}{\eta_{m}} [/latex]

[latex] =\frac{30 imes 1005 imes 16.37}{0.9 imes 1000} \mathrm{~kW}=548.39 \mathrm{~kW} [/latex]

Question 16.6: The conditions of air at the entry of an axial flow compress...

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