A closed cycle ideal gas turbine plant operates between temperature limits of 800°C and 30°C and produces a power of 100 kW. The plant is designed such that there is no need for a regenerator. A fuel of calorific 45000 kJ/kg is used. Calculate the mass flow rate of air through the plant and rate of fuel consumption.
Assume $c_{p}$ = 1 kJ/kg K and γ = 1.4. (GATE, 1998)

Question

A closed cycle ideal gas turbine plant operates between temperature limits of 800°C and 30°C and produces a power of 100 kW. The plant is designed such that there is no need for a regenerator. A fuel of calorific 45000 kJ/kg is used. Calculate the mass flow rate of air through the plant and rate of fuel consumption.
Assume [latex] c_{p}[/latex] = 1 kJ/kg K and γ = 1.4. (GATE, 1998)

Accepted Answer

Given : [latex] T_{1} = 30 + 273 = 303 K ; T_{3} = 800 + 273 = 1073 K ; C = 45000 kJ/kg ; c_{p}[/latex] = 1 kJ/kg K ; γ = 1.4 ; [latex] W_{turbine} – W_{compressor} [/latex] = 100 kW.

[latex]\dot{m}_{a} , \dot{m}_{f} : [/latex]

Since no regenerator is used we can assume the turbine expands the gases upto [latex] T_{4} [/latex] in such a way that the exhaust gas temperature from the turbine is equal to the temperature of air coming out of the compressor i.e., [latex]T_{2} = T_{4} [/latex]

[latex] \frac{p_{2}}{p_{1}} = \frac{p_{3}}{p_{4}} , \frac{p_{2}}{p_{1}} = ( \frac{T_{2}}{T_{1}})^{\frac{γ}{γ - 1}} and \frac{p_{3}}{p_{4}} = ( \frac{T_{3}}{T_{4}})^{\frac{γ}{γ - 1}} [/latex]

∴ [latex] \frac{T_{2}}{T_{1}} = \frac{T_{3}}{T_{4}} = \frac{T_{3}}{T_{2}}[/latex]

(∵ [latex]T_{2} = T_{4} [/latex] ......assumed )

or [latex]T_{2}² = T_{1} T_{3} or T_{2} = \sqrt{T_{1} T_{3}} [/latex]

or [latex]T_{2} = \sqrt{303 × 1073} [/latex] = 570.2 K

Now, [latex] W_{turbine} – W_{compressor} = \dot{m}_{f} [/latex] × C × η

or 100 = [latex] \dot{m}_{f} [/latex] × 45000 × [1 - [latex] \frac{T_{4} - T_{1} }{T_{3} - T_{2} } ] [/latex]

= [latex] \dot{m}_{f} [/latex] × 45000 × [1 - [latex] \frac{570.2 - 303}{1073 - 570.2} ] [/latex]

= [latex] \dot{m}_{f} [/latex] × 21085.9

or [latex] \dot{m}_{f} = \frac{100}{21085.9} = 4.74 × 10^{–3} kg/s.[/latex]

Again, [latex] W_{turbine} – W_{compressor} [/latex] = 100 kW

[latex](\dot{m}_{a} + \dot{m}_{f} ) (T_{3} - T_{4}) - \dot{m}_{a} × 1 × (T_{2} - T_{1}) [/latex] = 100

or [latex](\dot{m}_{a} + 0.00474) (1073 – 570.2) - \dot{m}_{a} [/latex] (570.2 – 303) = 100

or [latex](\dot{m}_{a} + 0.00474) × 502.8 - 267.2 \dot{m}_{a} [/latex] = 100

or [latex] 502.8 \dot{m}_{a} + 2.383 - 267.2 \dot{m}_{a} [/latex] = 100

or 235.5 [latex] \dot{m}_{a} [/latex] = 97.617

∴ [latex] \dot{m}_{a} [/latex] = 0.414 kg/s.

Question 8.37: A closed cycle ideal gas turbine plant operates between temp...

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