Gas Turbine Blades (Power Turbine): 25-Nav-B5 Marine Control Systems, May 2015, Q5

Question 5: Gas Turbine Blades (Power Turbine) (10 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. A 50 %-reaction free-turbine first stage on a twin-shaft industrial gas turbine.

Given data (per single gas turbine)
Stator angles α₀/α₁	30° / 60°
Rotor angles β₁/β₂	30° / 60°
Tip / root diameter	1500 mm / 1050 mm
Speed	3000 rev/min
Exhaust c_p	1.148 kJ/kg°C
Exhaust flow / power (half of combined)	139 kg/s / 30.43 MW

Find. Mean blade speed, the velocity triangle, gas velocities, power by two methods and the discrepancy.

Figure 4. First-stage inlet triangle for the 50 %-reaction power turbine (symmetric: α₁ = β₂, β₁ = α₀).

Approach. Take the mean-diameter blade speed, exploit the symmetry of a 50 %-reaction stage to fix the axial velocity from $U=C_a(\tan\alpha_1-\tan\beta_1)$, then compute the Euler work and compare with the enthalpy-drop power.

Mean blade speed. $D_m=\tfrac{1.5+1.05}{2}=1.275\ \text{m}$, $U=\dfrac{\pi D_m N}{60}=\dfrac{\pi\cdot1.275\cdot3000}{60}=200.3\ \text{m/s}$.
Axial velocity (50 % reaction). $C_a=\dfrac{U}{\tan60^{\circ}-\tan30^{\circ}}=173.4\ \text{m/s}$, so $C_1=C_a/\cos60^{\circ}=346.9$ and $W_1=C_a/\cos30^{\circ}=200.3\ \text{m/s}$.
Stage work (velocity triangles). $\Delta C_w=C_a(\tan60^{\circ}-\tan30^{\circ})$ gives $w_{stage}=U\,\Delta C_w=40.1\ \text{kJ/kg}$; for three like stages $w=120.3\ \text{kJ/kg}$.
Power from gas velocities. With $\dot m=278/2=139\ \text{kg/s}$, $P_d=139\cdot120.3=16.7\ \text{MW}$.
Power from temperature drop. $P_e=\dot m c_p\Delta T=139\cdot1.148\cdot(682-483)=31.8\ \text{MW}$.
Comparison. The specified per-turbine output is $60.86/2=30.4\ \text{MW}$, matching the enthalpy method (31.8 MW) but well above the velocity-triangle figure — $\boxed{P_e\approx P_{spec}\gg P_d}$.

Check: a single 50 %-reaction stage carrying the entire enthalpy drop would need an absolute velocity above Mach 1, which is impossible; the power turbine must therefore be multi-staged. Three like stages are assumed here, which keeps the velocities subsonic and reconciles the temperature-drop power with the specified output.

The point of parts (d)–(f) is the deliberate discrepancy: the velocity-triangle calculation for the assumed number of stages under-predicts the power because the real machine extracts the full enthalpy drop from inlet (682°C) to exhaust (483°C), whereas the idealised triangles capture only the whirl change at the assumed stage loading. The enthalpy-drop method, which does not depend on the blade geometry, agrees with the nameplate 30.4 MW and is the reliable figure.

Question 5 results (per single gas turbine)
Quantity	Value
(a) Mean blade speed	200.3 m/s
(c) C₁ / W₁ / C_a	346.9 / 200.3 / 173.4 m/s
(d) Power from velocities (3 stages)	16.7 MW
(e) Power from ΔT	31.8 MW
(f) Specified output	30.4 MW

Concept & Theory

Topic: 50 %-reaction axial stage and the two ways to compute power

Key relations: Blade speed $U=\pi D_m N/60$; symmetry $\alpha_1=\beta_2,\ \beta_1=\alpha_0$; $U=C_a(\tan\alpha_1-\tan\beta_1)$; Euler $w=U\Delta C_w$; enthalpy $P=\dot m c_p\Delta T$.

Why this works: In a 50 %-reaction stage the stator and rotor blade rows are mirror images, so the inlet and exit triangles are congruent and the axial velocity follows directly from the blade angles and the blade speed. The Euler (velocity) power depends on how many stages share the whirl change, while the enthalpy-drop power depends only on the measured temperatures — comparing them exposes the assumed stage count.

Common pitfall: Trying to put the whole enthalpy drop into one stage (giving a supersonic C1) instead of recognising the machine must be multi-staged.

Source: Cohen, Rogers & Saravanamuttoo, Gas Turbine Theory, 6th ed., Ch. 7 (axial turbines).

Question 5 of 8: Gas Turbine Blades (Power Turbine)

Question 5: Gas Turbine Blades (Power Turbine) (10 marks)