(a) A 15 m high cylinder with a cross-sectional area of 0.6 m² contains 3 m³ of liquid water at 25 °C on the top of a thin insulated piston of mass 20 kg. Below the piston, argon gas is at 15 °C with a volume of 3 m³, as shown in the figure. Heat is

Question

(a) A 15 m high cylinder with a cross-sectional area of 0.6 m² contains 3 m³ of liquid water at 25 °C on the top of a thin insulated piston of mass 20 kg. Below the piston, argon gas is at 15 °C with a volume of 3 m³, as shown in the figure. Heat is supplied to argon such that the piston rises and pushes the water out over the top edge. Find the (i) work done (kJ) to remove the whole water from the top of the piston and (ii) heat transferred (kJ) to argon during the process. (iii) Plot the process on P-v diagram for argon. Assume atmospheric pressure (P₀) as 101 kPa, Cᵥ and R for argon as 0·312 kJ/kg-K and 0·2081 kJ/kg-K respectively. The specific volume of water at 25 °C is 0·001003 m³/kg. Neglect piston thickness. (20 marks)

(b) Air at 100 kPa and 290 K enters a gas turbine cycle with two stages of compression and two stages of expansion. This system uses ideal regenerator, reheater and intercooler. The pressure ratio across each stage is 4. 300 kJ/kg of heat is added in combustion chamber and reheater each. The regenerator increases the air temperature by 20 °C. Draw T-s plot and determine the (i) total heat rejected (kJ/kg), (ii) net work output (kJ/kg) and (iii) thermal efficiency of the system. Assume isentropic operation for all compressors and turbines. Take Cₚ of air = 1·005 kJ/kg-K and γ = 1·4. (20 marks)

(c) A convergent-divergent nozzle has a throat area of 250 mm² and an exit area of 500 mm². Air enters the nozzle with a stagnation temperature of 350 K and stagnation pressure of 1 MPa. Determine the maximum flow rate of air through the nozzle and the static pressure, static temperature, Mach number and velocity at the exit from the nozzle. Given γ = 1·4, R = 0·287 kJ/kg-K. Use Gas Table to solve the problem. (10 marks)

UPSC Answer Check · Accepted Answer

Solve all three parts systematically, allocating approximately 40% time to part (a) as it carries the highest marks (20), and 30% each to parts (b) and (c). For (a), establish equilibrium states of argon gas with varying water head; for (b), construct the complete T-s diagram for the modified Brayton cycle with regeneration, reheating and intercooling; for (c), use isentropic flow tables to determine choked conditions and exit properties. Present derivations clearly with state points labelled on all diagrams.
- Part (a): Initial argon pressure = P₀ + (m_piston·g)/A + (ρ_water·g·h_water)/A = 101 + 0.327 + 49.05 ≈ 150.4 kPa; mass of argon calculated using ideal gas law
- Part (a): Work done = ∫P_ext dV = area under P-V curve = P_avg·ΔV + mgh term for water lifting; final answer ~90-95 kJ
- Part (a): Heat transfer using First Law Q = ΔU + W = m·Cv·(T₂-T₁) + W; temperature rise from ideal gas law with variable pressure
- Part (b): T-s diagram shows 6 state points with two isentropic compressions (intercooled), two isentropic expansions (reheated), regenerator heat exchange, and heat addition/rejection
- Part (b): With pressure ratio 4 per stage, overall ratio = 16; regenerator effectiveness implied by 20°C temperature rise; net work and efficiency calculated from enthalpy changes
- Part (c): Check if nozzle choked: A_exit/A* = 2.0 > 1, so subsonic or supersonic solution possible; from gas tables at γ=1.4, A/A*=2.0 gives Mach ~0.3 (subsonic) or ~2.2 (supersonic)
- Part (c): Maximum flow occurs when throat is choked (M=1 at throat); use choked flow equation with stagnation conditions; mass flow rate ~0.45-0.50 kg/s
- Part (c): For supersonic exit solution: static pressure ~0.1 MPa, temperature ~175 K, velocity ~450 m/s; subsonic alternative also acceptable if justified

Dimension	Weight	Max marks	Excellent	Average	Poor
Concept correctness	20%	10	Correctly applies: (a) variable external pressure work with hydrostatic head, First Law for closed system; (b) ideal regenerator, reheater, intercooler concepts in modified Brayton cycle; (c) choked flow condition, area-Mach number relation from gas dynamics. Identifies that part (a) requires integration due to varying water head.	Uses correct basic equations but treats (a) as constant pressure work or misses regenerator effectiveness interpretation; (c) applies isentropic relations but confuses subsonic/supersonic branches.	Fundamental errors: treats argon as incompressible, confuses open/closed systems in (a); omits intercooler/reheater in cycle analysis; applies Bernoulli instead of compressible flow relations in (c).
Numerical accuracy	20%	10	All calculations to 3-4 significant figures: (a) W≈92 kJ, Q≈180-190 kJ; (b) η_thermal≈45-50%, net work≈250-280 kJ/kg; (c) ṁ_max≈0.47 kg/s, exit Mach≈2.2 with consistent property values. Unit conversions (mm² to m², kPa to Pa) handled correctly.	Correct methodology but arithmetic slips (e.g., factor of 10 in area conversion, wrong specific heat value); final answers within 10% of expected.	Major calculation errors: wrong gas constant used, temperature in Celsius not Kelvin, choked flow condition not recognized leading to impossible mass flows.
Diagram quality	20%	10	Three clear diagrams: (a) P-V for argon showing curved process line (polytropic-like) with initial and final states; (b) T-s with all 6 states, constant pressure lines, equal entropy intervals for isentropic processes, regenerator shown as horizontal line; (c) nozzle sketch with throat, exit, Mach number variation. All axes labelled, values marked, flow direction indicated.	Diagrams present but missing key labels or incorrect process directions; T-s shows compression/expansion but missing regeneration detail; P-V for (a) drawn as straight line.	Missing or seriously flawed diagrams: no T-s diagram for cycle, P-V axes unlabelled, no nozzle sketch, or diagrams contradict written solution.
Step-by-step derivation	20%	10	Explicit derivation for (a): force balance → P_argon expression → work integral ∫(P₀ + mg/A + ρg(V₀+A·h)/A)dV; state table with P,V,T for argon at start, intermediate, end. For (b): all state temperatures derived from isentropic relations T₂=T₁·(r_p)^((γ-1)/γ). For (c): table lookup or interpolation method shown for A/A*=2.0.	Key equations stated but some steps skipped (e.g., jumps to final work formula without showing integral); state table partially complete; gas table mentioned but not shown.	No derivations shown; final answers stated without intermediate steps; no state tables; appears to copy memorized solutions without adaptation.
Practical interpretation	20%	10	Discusses engineering significance: (a) why piston mass matters for gas storage systems; (b) why regeneration, reheating and intercooling are used in gas turbines (e.g., Indian power plants, aircraft engines); compares with simple cycle efficiency. (c) nozzle applications in rocket propulsion, steam turbines; explains why convergent-divergent needed for supersonic flow.	Brief mention of practical relevance without elaboration; states that efficiency improves but no explanation of why; mentions rockets without connecting to nozzle design.	No interpretation; purely mathematical treatment; misses opportunity to discuss real-world relevance of all three problems.

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