(a)(i) 3 kg of air is compressed in a reversible steady flow polytropic process from 100 kPa, 40°C to 1000 kPa. During this process the law of compression followed is PV^1.25 = C. Determine the shaft work, heat transferred and the change in entropy.

Question

(a)(i) 3 kg of air is compressed in a reversible steady flow polytropic process from 100 kPa, 40°C to 1000 kPa. During this process the law of compression followed is PV^1.25 = C. Determine the shaft work, heat transferred and the change in entropy. Assume for air C_v = 0.717 kJ/kg K and R = 0.287 kJ/kg K. (ii) Distinguish between pdv work and -vdp work. (20 marks)
(b) Calculate the displacement thickness and momentum thickness of a laminar boundary layer, in terms of the nominal boundary layer thickness δ, for the following velocity distribution: u/U_0 = sin(π/2 y/δ) (20 marks)
(c) An ideal gas turbine engine operates with air as the working fluid at a pressure ratio 18 : 1 and a maximum temperature of 700°C. The air enters the compressor at 100 kPa and 20°C. Determine the thermal efficiency, the heat addition and the temperature of exhaust air. For air take C_p = 1.0035 kJ/kg K and γ = 1.4. (10 marks)

UPSC Answer Check · Accepted Answer

Calculate numerical solutions for all three sub-parts with systematic derivations. For (a)(i), apply polytropic relations for steady flow; for (a)(ii), distinguish flow work from shaft work using control volume analysis. For (b), integrate the sine velocity profile to obtain displacement and momentum thickness. For (c), apply Brayton cycle analysis with given pressure ratio and temperature limits. Allocate approximately 40% time to part (a) combined, 35% to part (b), and 25% to part (c) based on mark distribution.
- Part (a)(i): T2 = 40*(1000/100)^(0.25/1.25) = 40*10^0.2 = 63.4 K → T2 = 313.4 K; W_shaft = n/(n-1)*mR(T1-T2) = 5*3*0.287*(313-40) = -1176 kJ; Q = W*(γ-n)/(γ-1) = -1176*(1.4-1.25)/0.4 = -441 kJ; ΔS = m[Cp*ln(T2/T1) - R*ln(P2/P1)]
- Part (a)(ii): pdv work is boundary work for closed systems (quasi-static process); -vdp is flow work or shaft work in steady flow (Euler work equation); relate via steady flow energy equation
- Part (b): δ* = ∫[0,δ](1 - u/U0)dy = δ[1 - 2/π] = 0.363δ; θ = ∫[0,δ](u/U0)(1 - u/U0)dy = δ[2/π - 1/2] = 0.137δ; show integration steps with substitution
- Part (c): T2/T1 = (18)^(0.4/1.4) = 2.58; T2 = 293*2.58 = 756 K; T4 = T3/2.58 = 973/2.58 = 377 K; η = 1 - 1/(r_p)^((γ-1)/γ) = 1 - 1/2.58 = 61.2%; Q_add = Cp(T3-T2) = 1.0035*(973-756) = 217.8 kJ/kg
- All parts: State assumptions clearly, show unit consistency, and verify results against physical expectations

Dimension	Weight	Max marks	Excellent	Average	Poor
Concept correctness	20%	10	Correctly applies polytropic relations for steady flow in (a)(i); distinguishes closed system vs control volume work in (a)(ii); uses proper boundary layer integral definitions in (b); applies ideal Brayton cycle with correct isentropic relations in (c)	Uses correct formulas but with minor conceptual gaps (e.g., confuses n and γ in polytropic relations, or misidentifies flow work direction)	Applies isentropic relations to polytropic process, or confuses displacement thickness with momentum thickness, or uses Otto cycle for gas turbine
Numerical accuracy	20%	10	All numerical values accurate: (a)(i) W ≈ -1176 kJ, Q ≈ -441 kJ, ΔS ≈ -0.92 kJ/K; (b) δ* = 0.363δ, θ = 0.137δ; (c) η = 61.2%, Q_add ≈ 218 kJ/kg, T_exhaust ≈ 377 K; 3-4 significant figures maintained	Correct methodology but arithmetic errors in 1-2 parts (e.g., temperature ratio error, or integration constant mishandling)	Order-of-magnitude errors, wrong temperature units (K vs °C), or completely incorrect final answers without verification
Diagram quality	15%	7.5	P-v diagram for polytropic compression in (a) with areas representing work; T-s diagram showing entropy change; velocity profile sketch for (b) with δ, δ*, θ marked; T-s or P-v diagram for Brayton cycle in (c) with all states labelled	Diagrams present but missing key labels or incorrect process curves (e.g., isothermal instead of polytropic)	No diagrams provided where essential for explanation, or diagrams contradict the analysis
Step-by-step derivation	25%	12.5	Complete derivations: polytropic work from ∫-vdp, entropy change from Tds equations; integration by parts for boundary layer thicknesses with substitution steps; Brayton efficiency derived from net work/heat input with all intermediate temperatures shown	Key steps shown but skips algebraic manipulations or assumes standard results without derivation	Final formulas stated without derivation, or incorrect derivation with missing steps
Practical interpretation	20%	10	Interprets negative work and heat in (a) as compression requiring input; explains why δ* > θ for laminar profiles; relates 61% efficiency to real gas turbine limitations (material temperature limits, compressor efficiency); mentions Indian applications (GTs in NTPC plants, HAL engine development)	Brief physical interpretation of signs and magnitudes but no engineering context or real-world limitations discussed	Purely mathematical treatment with no physical insight or application context

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