(a) A single-stage impulse turbine rotor has a mean blade ring diameter of 500 mm and rotates at a speed of 10000 r.p.m. The…

(a) Single-stage impulse turbine

u = πDN/60 = π × 0.5 × 10000/60 = 261.8 m/s. Inlet: Cw₁ = 900 cos20° = 845.7 m/s, Cf₁ = 900 sin20° = 307.8 m/s. Blade inlet angle for shockless entry: tanβ₁ = Cf₁/(Cw₁ − u) = 307.8/583.9 = 0.527 ⟹ β₁ = 27.8° (= β₂, equiangular blades). Cr₁ = Cf₁/sinβ₁ = 660 m/s, Cr₂ = 0.85 × 660 = 561 m/s. Outlet: Cf₂ = 561 sin27.8° = 261.6 m/s, Cw₂ = 561 cos27.8° − 261.8 = 234.5 m/s (opposite to blade motion). ṁ = 750/3600 = 0.2083 kg/s, ΣCw = 845.7 + 234.5 = 1080.2 m/s.

(i) Diagram power P = ṁ u ΣCw = 0.2083 × 261.8 × 1080.2 = 58.9 kW. (ii) Diagram efficiency η_d = 2uΣCw/C₁² = 2 × 261.8 × 1080.2/900² = 69.8%. (iii) Axial thrust F_a = ṁ(Cf₁ − Cf₂) = 0.2083 × 46.2 = 9.6 N. (iv) Friction loss = ṁ(Cr₁² − Cr₂²)/2 = 0.2083 × (660² − 561²)/2 = 12.6 kW.

(b)(i) Effect of impeller blade shape

The exit blade angle β₂ fixes the exit whirl Cw₂ and hence the ideal work u₂Cw₂ and pressure rise.

• Backward-curved (β₂ below 90°): Cw₂ less than u₂, and Cw₂ falls as flow rises, so the pressure ratio falls steadily with mass flow. This negative slope gives stable operation, a wide range and the highest efficiency, though pressure ratio per stage is lower.
• Radial (β₂ = 90°): Cw₂ ≈ u₂ (less slip), the ideal head is independent of flow, and the curve is nearly flat. Radial blades are strong, simple to make and suit high tip speeds, so most aircraft and high-speed compressors use them.
• Forward-curved (β₂ above 90°): Cw₂ greater than u₂, giving the largest theoretical pressure ratio, but it rises with flow, so operation is unstable and prone to surge, with high exit kinetic energy that the diffuser recovers poorly. They are rarely used.

(b)(ii) Surging and choking

Surging: at low flow, on the rising (positive-slope) part of the characteristic, the pressure the compressor can deliver drops below the downstream pressure. Flow momentarily reverses, the delivery pressure falls, forward flow resumes and the cycle repeats. The result is violent pressure and flow oscillation, noise, vibration, overheating and possible blade damage. It is avoided by operating to the right of the surge line, with blow-off or recirculation valves, variable inlet guide vanes or diffuser vanes.

Choking: at high flow the velocity at the impeller eye, throat or diffuser reaches sonic speed. Mass flow then cannot increase at that speed however low the back pressure, efficiency and pressure ratio fall steeply, and the speed lines become vertical. It sets the maximum capacity and is delayed by larger flow passages and suitable inlet design.

(c) Shell-and-tube heat exchanger (2 shell passes, 4 tube passes)

Heat duty from glycol: Q = (4500/3600) × 2742 × (140 − 80) = 205.65 kW. Water flow: ṁ_w = 205650/(4175 × 50) = 0.985 kg/s (≈ 3546 kg/h).

C_h = 1.25 × 2742 = 3427.5 W/K = C_min, C_c = 0.985 × 4175 = 4113 W/K, Cr = 0.833. ε = Q/[C_min(140 − 35)] = 205650/(3427.5 × 105) = 0.571. From the two-shell-pass ε–NTU relation (or chart), NTU ≈ 1.27. A = NTU × C_min/U = 1.27 × 3427.5/850 ≈ 5.1 m². (Check by LMTD: ΔT_lm,cf = 49.8 °C, giving F ≈ 0.95, consistent.)