Find the required air-fuel ratio in a gas turbine whose turbine and compressor efficiencies are 85% and 80%, respectively. Maximum cycle temperature is 875°C. The working fluid can be taken as air (cp = 1.0 kJ/kg K, γ = 1.4) which enters the compressor at 1 bar and 27°C. The pressure ratio is 4.

Question

Find the required air-fuel ratio in a gas turbine whose turbine and compressor efficiencies are 85% and 80%, respectively. Maximum cycle temperature is 875°C. The working fluid can be taken as air ([latex] c_{p} [/latex] = 1.0 kJ/kg K, γ = 1.4) which enters the compressor at 1 bar and 27°C. The pressure ratio is 4. The fuel used has calorific value of 42000 kJ/kg. There is a loss of 10% of calorific value in the combustion chamber. (GATE, 1998)

Accepted Answer

Given : [latex] η_{turbine} [/latex] = 85 % ; [latex]η_{compressor}[/latex] = 80% ; [latex]T_{3} = 273 + 875 = 1148 K ; T_{1} = 27 + 273 = 300 K ; c_{p} = 1.0 kJ/kg K ; γ = 1.4 ; p_{1} = 1 bar; p_{2} [/latex] = 4 bar (since pressure ratio is 4) ; C = 42000 kJ/kg K, [latex] η_{cc}[/latex] = 90% (since loss in the combustion chamber is 10%)
For isentropic compression 1–2, we have

[latex] \frac{T_{2}}{T_{1}} = (\frac{p_{2}}{p_{1}})^{\frac{γ - 1}{γ}} = (4)^{\frac{1.4 - 1}{1.4}} =[/latex] 1.486

∴ [latex]T_{2}[/latex] = 300 × 1.486 = 445.8 K

[latex] η_{compressor} = \frac{T_{2} - T_{1}}{T_{2}′ - T_{1}}[/latex]

or 0.8 = [latex] \frac{445.8 - 300}{T_{2}′ - 300}[/latex]

or [latex]T_{2}′ = \frac{445.8 - 300}{0.8} [/latex] + 300 = 482.2 K

Now, heat supplied by the fuel = heat taken by the burning gases
0.9 × [latex] m_{f} × C = (m_{a} + m_{f} ) × c_{p} × ( T_{3} - T_{2}′)[/latex]

∴ C = [latex] (\frac{m_{a} + m_{f} }{m_{f}}) × \frac{ c_{p}( T_{3} - T_{2}′)}{0.9} = (\frac{m_{a} }{m_{f}} + 1) × \frac{c_{p} ( T_{3} - T_{2}′)}{0.9} [/latex]

or 42000 = [latex] (\frac{m_{a} }{m_{f}} + 1) × \frac{1.00 (1148 - 482.27)}{0.9} = 739.78 (\frac{m_{a} }{m_{f}} + 1) [/latex]

∴ [latex] \frac{m_{a} }{m_{f}} = \frac{42000}{739.78} [/latex] - 1 = 55.77 say 56

∴ A/F ratio = 56 : 1.

Question 8.39: Find the required air-fuel ratio in a gas turbine whose turb...

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