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Air turbine 1.5 kg/s: heat loss and diameters

Short answer

Applying the steady-flow energy equation, with Ẇ = 180 kW, gives a heat loss of 16.5 kW. The inlet diameter is 62.2 mm and the exhaust diameter is 165 mm.

Air flows at a rate of 1.5 kg/s through a turbine, entering at 500 kPa, 150°C and with a velocity of 120 m/s and leaving at 100 kPa, 25°C and with a velocity of 60 m/s. Power produced by the turbine is 180 MW. Determine: a) Heat loss from the turbine and b) Diameters of inlet and exhaust pipe [Take R = 287 J/kgK, Cp = 1005 J/kgK]

Given: air, ṁ = 1.5 kg/s; inlet: 500 kPa, 150 °C (423.15 K), 120 m/s; exit: 100 kPa, 25 °C (298.15 K), 60 m/s; R = 0.287, c_p = 1.005 kJ/kg·K.

Note on the data: the printed power of 180 MW is evidently a misprint for 180 kW: the total energy drop of the air (enthalpy plus kinetic energy) is only about 197 kW, so 180 MW is impossible. The solution uses Ẇ = 180 kW.

a) Heat loss

Steady-flow energy equation (ΔPE = 0):
Q̇ = Ẇ + ṁ [ c_p (T₂ − T₁) + (V₂² − V₁²)/2 ]

c_p (T₂ − T₁) = 1.005 × (25 − 150) = −125.63 kJ/kg
(V₂² − V₁²)/2 = (60² − 120²)/2 = −5400 J/kg = −5.4 kJ/kg

Q̇ = 180 + 1.5 × (−125.63 − 5.4) = 180 − 196.5 = −16.5 kW

So heat is lost from the turbine at about 16.5 kW.

b) Pipe diameters

v₁ = RT₁/P₁ = 0.287 × 423.15/500 = 0.2429 m³/kg
A₁ = ṁ v₁/V₁ = 1.5 × 0.2429/120 = 3.036 × 10⁻³ m²
d₁ = √(4A₁/π) = 0.0622 m (62.2 mm)

v₂ = RT₂/P₂ = 0.287 × 298.15/100 = 0.8557 m³/kg
A₂ = ṁ v₂/V₂ = 1.5 × 0.8557/60 = 0.02139 m²
d₂ = √(4A₂/π) = 0.165 m (165 mm)

Answer: heat loss 16.5 kW; d_inlet = 62.2 mm; d_exhaust = 165 mm.

✓ Answer checked by Claude Opus 5.5 (high effort)

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