Air enters the compressor of a gas turbine plant operating on Brayton cycle at 1 bar, 27°C. The pressure ratio in the cycle is 6. If WT = 2.5 WC, where WT and WC are the turbine and compressor work respectively, calculate the maximum temperature and the cycle efficiency. (GATE, 1996)

Question

Air enters the compressor of a gas turbine plant operating on Brayton cycle at 1 bar, 27°C. The pressure ratio in the cycle is 6. If [latex] W_{T} = 2.5 W_{C}, where W_{T} and W_{C}[/latex] are the turbine and compressor work respectively, calculate the maximum temperature and the cycle efficiency. (GATE, 1996)

Accepted Answer

Given : [latex]p_{1} = 1 bar ; T_{1}[/latex] = 27 + 273 = 300 K ; [latex] \frac{p_{2}}{p_{1}} = 6 ; W_{T} = 2.5 W_{C} [/latex]

Maximum temperature, [latex]T_{3} [/latex] :

Now, [latex] \frac{T_{2}}{T_{1}} = (\frac{p_{2}}{p_{1}})^{\frac{γ - 1}{γ}} = (6)^{\frac{1.4 - 1}{1.4}} = (6)^{\frac{0.4}{1.4}}[/latex] = 1.668

∴ [latex]T_{2}[/latex] = 300 × 1.668 = 500.4 K

Also, [latex] \frac{T_{3}}{T_{4}} = (\frac{p_{2}}{p_{1}})^{\frac{γ - 1}{γ}} = (6)^{\frac{0.4}{1.4}}[/latex] = 1.668

or [latex]T_{3} = 1.668 T_{4} [/latex]
Now, compressor work
[latex] W_{C} = mc_{p} (T_{2} – T_{1}), [/latex]
and turbine work,
[latex] W_{T} = m c_{p} (T_{3} – T_{4}) [/latex]
Since [latex] W_{T} = 2.5 W_{C}[/latex] ......(Given)
∴ [latex] m c_{p} (T_{3} – T_{4}) = 2.5 × m c_{p} (T_{2} – T_{1}) [/latex]

[latex](T_{3} - \frac{T_{3}}{1.668}) [/latex] = 2.5 (500.4 - 300)

or [latex]T_{3} ( 1 - \frac{1}{1.668}) [/latex] = 501

∴ [latex]T_{3} [/latex] = 1251 K.

Cycle efficiency, [latex]η_{cycle} [/latex] :

[latex]η_{cycle} = \frac{W_{T} - W_{C}}{m c_{p} (T_{3} – T_{2})} = \frac{m c_{p} (T_{3} – T_{4}) - m c_{p} (T_{2} – T_{1}) }{m c_{p} (T_{3} – T_{2})} [/latex]

= [latex] \frac{(1251 - \frac{1251}{1.668} ) - (500.4 - 300)}{(1251 - 500.4) } [/latex]

= [latex] \frac{501 - 200.4}{750.6} [/latex] = 0.40 or 40 % .

Question 8.36: Air enters the compressor of a gas turbine plant operating o...

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