Question 7.DE.2: In a single-stage gas turbine, gas enters and leaves in axia......
In a single-stage gas turbine, gas enters and leaves in axial direction. The nozzle efflux angle is 68°, the stagnation temperature and stagnation pressure at stage inlet are 800°C and 4 bar, respectively. The exhaust static pressure is 1 bar, total-to-static efficiency is 0.85, and mean blade speed is 480 m/s, determine (1) the work done, (2) the axial velocity which is constant through the stage, (3) the total-to-total efficiency, and (4) the degree of reaction.
Assume γ = 1.33, and C_{pg} = 1.147 kJ/kgK.
(1) The specific work output
W = C_{pg} (T_{01} – T_{03})
= η_{ts}C_{pg}T_{01} [1 – (1/4)^{0.33/1.33}]
= (0.85)(1.147)(1073) [1 – (0.25)^{0.248}] = 304.42 kJ/kg
(2) Since α_1 = 0, α_3 = 0, C_{w1} = 0 and specific work output is given by
W = UC_{w2} \text{or} C_{w2} = \frac{W} {U} = \frac{304.42 \times 1000} {480} = 634.21 m/s
From velocity triangle
\sinα_2 = \frac{C_{w2}} {C_2}
or
C_2 = \frac{C_{w2}} {\sinα_2} = \frac{634.21} {\sin 68°} = 684 m/s
Axial velocity is given by
Ca_2 = 684 cos 68° = 256.23 m/s
(3) Total-to-total efficiency, η_{tt}, is
η_{tt} = \frac{T_{01} – T_{03}} {T_{01} – T^′ _{03}}
= \frac{w_s} {T_{01} – \left(T_3 + \frac{C^2_3}{2C_{pg}}\right)} = \frac{w_s} {\frac{w_s} {η_{ts}} – \frac{C^2_3}{2C_{pg}}}
= \frac{304.42} {\frac{304.42} {0.85} – \frac{(256.23)^2} {2 \times 1147}} = 92.4%
(4) The degree of reaction
Λ = \frac{Ca} {2U} (\tanβ_3 – \tanβ_2)
= \left(\frac{Ca} {2U} \times \frac{U}{Ca}\right) – \left(\frac{Ca}{2U}\tan α_2\right) + \left(\frac{U}{Ca} \times \frac{Ca}{2U}\right)
(from velocity triangle)
L = 1 – \frac{C_a} {2U} \tanα_2 = 1 – \frac{256.23} {(2)(480)} tan 68° = 33.94%