(a) A shell and tube type condenser is employed in a steam power plant to handle 35000 kg/h of dry and saturated steam at 50 °C. The cooling water enters the condenser at 15 °C and leaves at 25 °C. The tubes are of 22·5 mm inside diameter and 25 mm o

Question

(a) A shell and tube type condenser is employed in a steam power plant to handle 35000 kg/h of dry and saturated steam at 50 °C. The cooling water enters the condenser at 15 °C and leaves at 25 °C. The tubes are of 22·5 mm inside diameter and 25 mm outside diameter. The water flows through the tubes at an average velocity of 2 m/s. The heat transfer coefficient on steam side is 5001 W/m²-K.

Calculate the following :

(i) The mass flow rate of water (in kg/s)

(ii) The heat transfer surface area (based on U₀)

(iii) The number of tubes required for water flow

(iv) The number of tube passes in condenser if the length of each tube pass should not be more than 2·5 m

Assume that condensate coming out from the condenser is saturated water and resistance of tube wall is negligible. For waterside heat transfer coefficient, use the correlation Nu = 0·023 Re⁰·⁸ Pr⁰·⁴. Take latent heat of steam as 2374 kJ/kg. The properties of water at mean bulk temperature of 20 °C are as follows :

Pr = 6·98
ρ = 998·9 kg/m³
Cₚ = 4·180 kJ/kg-K
ν = 1·0006 × 10⁻⁶ m²/s
kƒ = 0·59859 W/m-K

Assume fully developed flow through tubes and a single shell is used. (20 marks)

(b) A stationary gas turbine plant operates on a Brayton cycle and delivers 20 MW to an electric generator. The maximum temperature is 1200 K and the minimum temperature is 290 K. The minimum pressure is 95 kPa and the maximum pressure is 380 kPa. If the isentropic efficiencies of the turbine and compressor are 0·85 and 0·8, respectively, find—

(i) the mass flow rate of air to the compressor;

(ii) the volume flow rate of air to the compressor;

(iii) the fraction of turbine work output needed to drive the compressor;

(iv) the cycle efficiency.

If a regenerator of 75% effectiveness is added to the plant, what would be the changes in the cycle efficiency and net work output?

Assume Cp and Cv for air as 1·005 kJ/kg-K and 0·718 kJ/kg-K, respectively. (20 marks)

(c) The heat transfer rate due to free convection from a vertical surface, 1 m high and 0·6 m wide, to quiescent air that is 20 K colder than the surface is known. What is the ratio of the heat transfer rate for that situation to the rate corresponding to a vertical surface, 0·6 m high and 1 m wide, when the quiescent air is 20 K warmer than the surface? Neglect heat transfer by radiation and any influence of temperature on the relevant thermophysical properties of air.

The correlation between Nusselt number and Rayleigh number is given as
NūL = 0·10 RaL^0·25. (10 marks)

UPSC Answer Check · Accepted Answer

Calculate all numerical sub-parts systematically: spend ~40% time on part (a) condenser calculations (mass flow, LMTD, overall U, tube count, passes), ~35% on part (b) gas turbine cycle with regeneration analysis, and ~25% on part (c) free convection scaling. Begin each part with stated assumptions, show formulae with substituted values, and present final answers with proper units. Include T-s diagrams for (b) and note physical implications.
- Part (a)(i): Water mass flow rate = 9.92 kg/s from energy balance Q = m_steam × h_fg = m_water × Cp × ΔT
- Part (a)(ii): Overall U₀ = 1/(1/h₀ + r₀/rᵢ × 1/hᵢ) with hᵢ from Dittus-Boelter correlation; A₀ = Q/(U₀ × LMTD) where LMTD = 14.43°C
- Part (a)(iii)-(iv): Tube count from continuity equation; tube passes determined from length constraint L_total/(N_tubes × L_max_pass)
- Part (b): Compressor and turbine work with isentropic efficiencies; T₂ = 430.9 K, T₄ = 766.3 K; mass flow from W_net = 20 MW
- Part (b) with regenerator: Effectiveness ε = 0.75 gives T₅ = T₂ + ε(T₄-T₂); revised efficiency and unchanged work output
- Part (c): Heat transfer ratio = (L₁/L₂)^0.25 × (W₁/W₂) = (1/0.6)^0.25 × (0.6/1) = 0.6 × 1.136 = 0.682 for cold vs hot surface orientation
- All parts: Proper unit conversions (kg/h to kg/s, kW to W), significant figures, and physical reasonableness checks

Dimension	Weight	Max marks	Excellent	Average	Poor
Concept correctness	20%	10	Correctly applies: (a) LMTD method for condenser with phase change, Dittus-Boelter correlation for turbulent flow; (b) Brayton cycle with non-isentropic compression/expansion, regenerator effectiveness definition; (c) Grashof/Prandtl dependence in free convection, Nu ∝ Ra^0.25 scaling. Identifies that LMTD uses ΔT₁ = T_sat - T_c,out and ΔT₂ = T_sat - T_c,in.	Uses correct basic formulae but misapplies LMTD (uses arithmetic mean) or confuses regenerator effectiveness with efficiency; gets Ra scaling direction correct but exponent handling sloppy.	Confuses condenser with evaporator LMTD; treats turbine/compressor as isentropic without efficiency correction; applies forced convection correlation for free convection in (c).
Numerical accuracy	20%	10	All calculations to 3-4 significant figures: (a)(i) 9.92 kg/s, (a)(ii) hᵢ ≈ 8700 W/m²K, U₀ ≈ 3200 W/m²K, A₀ ≈ 195 m²; (a)(iii) ~115 tubes; (a)(iv) ~4 passes; (b) ṁ ≈ 108 kg/s, η_cycle ≈ 28.5% without regenerator, ~35% with; (c) ratio 0.682. Cross-checks energy balances.	Correct methodology but arithmetic slips in exponent calculations (Re^0.8) or unit conversions (kJ to J); final answers within 10% of correct values.	Order-of-magnitude errors (e.g., mass flow in g/s, area in cm²); ignores diameter ratio in U₀ calculation; uses Cp for steam instead of latent heat; ratio in (c) inverted or >1.
Diagram quality	20%	10	Clear T-s diagram for Brayton cycle in (b) showing: 1-2 actual compression (dashed) vs isentropic, 2-3 heat addition, 3-4 actual expansion vs isentropic, 4-1 heat rejection; with regenerator showing 2-5 and 4-6 heat exchange. States and sketches condenser temperature profile in (a) showing constant T_steam = 50°C vs rising water temperature.	T-s diagram drawn but processes not clearly labelled or isentropic/actual paths not distinguished; no temperature profile for condenser.	No diagrams; or PV diagram instead of T-s for gas turbine; completely missing visual representation of temperature profiles or cycle processes.
Step-by-step derivation	20%	10	Explicit derivation chain: (a) Q → m_water → Re → Pr → Nu → hᵢ → U₀ → LMTD → A₀ → N_tubes → L_total → passes; (b) isentropic T₂s, T₄s → actual T₂, T₄ → w_C, w_T → ṁ from W_net → back work ratio → η; (c) Gr = gβΔTL³/ν² → Ra → Nu → h → q = hAΔT → ratio. All formulae stated before substitution.	Some steps combined or skipped (e.g., jumps to hᵢ value without showing Re, Pr calculation); final answers correct but working condensed.	Bare answers with no derivation; or incorrect formula sequence (e.g., calculates area before finding U); missing critical steps like isentropic temperature calculation.
Practical interpretation	20%	10	Comments on: (a) typical condenser design velocities (1-3 m/s), tube material selection for power plants (admiralty brass), fouling factor omission justification; (b) why regenerator improves efficiency (reduces Q_in) but not work, practical effectiveness limits (0.6-0.8), back work ratio significance for plant startup; (c) orientation effect on electronic cooling, chimney design implications. References NTPC/Indian power plant standards where relevant.	Brief comment on one or two aspects (e.g., regenerator saves fuel) but no depth on design constraints or operational implications.	No physical interpretation; treats as pure mathematics; or irrelevant commentary not tied to calculated results.

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