Mechanical Engineering 2023 Paper II 50 marks Compulsory Solve

Q1

(a) A 50 kg block of iron at 500 K is placed into open atmosphere which is at a temperature of 285 K. The iron block eventually reaches thermal equilibrium with the atmosphere. Assuming an average specific heat of 0·45 kJ/kg-K for iron, determine the (i) entropy change for the iron block and the atmosphere, and (ii) irreversibility. (10 marks) (b) Show that for normal shock in a perfect gas, M*ₓ M*ᵧ = 1. (10 marks) (c) In the axial flow compressor, for 50% reaction, the blading design is sometimes called symmetrical blading. Explain, with proper equations and justification, why it is called so. (10 marks) (d) An industrial furnace (blackbody) is emitting radiation at 2700 °C. Calculate the following: (i) Spectral emissive power at λ = 1·2 μm, (ii) Wavelength at which the emissive power is maximum, (iii) Maximum spectral emissive power, (iv) Total emissive power. Use Planck's distribution law equation given below: $$E_{b\lambda} = \frac{C_1}{\lambda^5 \left[\exp(C_2/\lambda T)-1\right]}$$ where $C_1 = 3.742 \times 10^8$ W-μm⁴/m², $C_2 = 1.438 \times 10^4$ μm-K. Take $\sigma = 5.67 \times 10^{-8}$ W/m²-K⁴. (10 marks) (e) (i) Write down the basic assumptions for LMTD method in case of heat exchanger analysis. (5 marks) (ii) Write down in which case LMTD method and in which case NTU method will be applicable in basic heat exchanger analysis. (5 marks)

हिंदी में प्रश्न पढ़ें

(a) 500 K पर लोहे के 50 kg के एक खंड को खुले वातावरण में रखा जाता है, जिसका तापमान 285 K है। लोह खंड अंततः वायुमंडल के साथ ऊष्मीय साम्यावस्था तक पहुँच जाता है। लोहे के लिए 0·45 kJ/kg-K की औसत विशिष्ट ऊष्मा मानकर (i) लोहे के खंड और वायुमंडल के एन्ट्रॉपी परिवर्तन तथा (ii) अप्रतिक्रम्यता का निर्धारण कीजिए। (10 अंक) (b) दर्शाइए कि एक आदर्श गैस में सामान्य प्रघात के लिए M*ₓ M*ᵧ = 1. (10 अंक) (c) अक्षीय प्रवाह संपीडक में, 50% प्रतिक्रिया हेतु, फलक अभिकल्प को कभी-कभी सममित फलक कहा जाता है। उचित समीकरणों और औचित्य के साथ समझाइए कि इसे ऐसा क्यों कहा जाता है। (10 अंक) (d) 2700 °C पर एक औद्योगिक भट्टी (कृष्णिका) विकिरण उत्सर्जित करती है। निम्नलिखित की गणना कीजिए: (i) λ = 1·2 μm पर स्पेक्ट्रमी उत्सर्जन शक्ति, (ii) तरंगदैर्ध्य, जिस पर उत्सर्जन शक्ति अधिकतम होती है, (iii) अधिकतम स्पेक्ट्रमी उत्सर्जन शक्ति, (iv) कुल उत्सर्जन शक्ति। नीचे दिए गए प्लैंक वितरण नियम समीकरण का उपयोग कीजिए: E_{bλ} = \frac{C_1}{λ^5 [\exp(C_2/λT)-1]} जहाँ, C₁ = 3·742×10⁸ W-μm⁴/m², C₂ = 1·438×10⁴ μm-K. σ = 5·67×10⁻⁸ W/m²-K⁴ लीजिए। (10 अंक) (e) (i) उष्मा विनिमयक विश्लेषण के मामले में एल० एम० टी० डी० विधि के लिए मूल अभिधारणाएँ लिखिए। (5 अंक) (ii) लिखिए कि मूल उष्मा विनिमयक विश्लेषण में किस स्थिति में एल० एम० टी० डी० विधि और किस स्थिति में एन० टी० यू० विधि लागू होगी। (5 अंक)

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Approach

Solve each sub-part systematically with clear section headings. For (a), apply entropy balance for irreversible heat transfer to surroundings; for (b), derive the characteristic Mach number relation using normal shock relations; for (c), prove symmetrical velocity triangles using reaction definition; for (d), apply Planck's law and Wien's displacement law with proper unit conversions; for (e), state LMTD assumptions and compare method applicability. Allocate time proportionally: ~15% each to (a), (b), (c), (d) and ~20% to (e) combined.

Key points expected

  • (a) Entropy change of iron: ΔS_iron = mc ln(T2/T1) = 50×0.45×ln(285/500) = -12.52 kJ/K; heat transfer Q = mcΔT = 4837.5 kJ; ΔS_atm = Q/T_atm = +16.97 kJ/K; S_gen = +4.45 kJ/K; Irreversibility I = T0×S_gen = 285×4.45 = 1268.25 kJ
  • (b) Characteristic Mach number M* = V/a* where a* is critical speed of sound; using M*² = [(γ+1)M²]/[2+(γ-1)M²]; apply across normal shock using M_y² = [2+(γ-1)M_x²]/[2γM_x²-(γ-1)]; algebraic manipulation yields M*_x × M*_y = 1
  • (c) 50% reaction R = (Δh_rotor)/(Δh_stage) = 0.5 implies equal enthalpy rise in rotor and stator; from velocity triangles, R = 0.5 gives V1 = V3 and V2 = V4 (mirror symmetry); blade shapes are mirror images → symmetrical blading
  • (d) T = 2973 K; (i) E_bλ at 1.2 μm using Planck's law = 2.42×10¹⁴ W/m²·μm; (ii) λ_max = 2898/2973 = 0.974 μm (Wien's law); (iii) E_bλ,max using Planck's law at λ_max = 4.10×10¹⁴ W/m²·μm; (iv) E_b = σT⁴ = 3.72×10⁶ W/m²
  • (e)(i) LMTD assumptions: steady-state, constant overall heat transfer coefficient U, constant specific heats, no phase change, negligible heat loss to surroundings, counter-flow or parallel-flow configuration
  • (e)(ii) LMTD method: suitable when inlet/outlet temperatures are known, design problem; NTU method: suitable when effectiveness is required, rating problem, or when outlet temperatures are unknown

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Concept correctness20%10Correctly identifies (a) as externally irreversible process with finite temperature difference; (b) uses characteristic Mach number definition properly; (c) links reaction degree to velocity triangle symmetry; (d) converts 2700°C to Kelvin and applies Planck's law with correct constants; (e) distinguishes LMTD design vs NTU rating problems accurately.Correct final formulas but weak conceptual linkage between parts; minor errors in identifying reversible vs irreversible processes or mixing up design/rating problem definitions.Fundamental misconceptions: treats (a) as reversible process, confuses characteristic Mach number with local Mach number, misunderstands reaction definition, uses Celsius in Planck's law directly, or cannot distinguish LMTD from NTU applicability.
Numerical accuracy20%10All numerical values accurate to 2-3 significant figures: (a) ΔS_iron = -12.52 kJ/K, ΔS_atm = +16.97 kJ/K, I = 1268 kJ; (d) λ_max = 0.974 μm, E_b = 3.72×10⁶ W/m²; proper handling of exponential terms in Planck's law; unit consistency throughout (kJ, kW, μm, K).Correct methodology but arithmetic slips (e.g., sign errors in entropy, factor of 10 in Planck's law constants, or °C used instead of K in Wien's law); final answers within 10% of correct values.Major calculation errors: wrong heat transfer in (a), algebraic errors in (b) derivation, incorrect temperature conversion in (d), or orders of magnitude wrong in spectral calculations; missing units or inconsistent unit systems.
Diagram quality15%7.5Clear T-s diagram for (a) showing entropy generation; velocity triangles for (c) drawn with inlet/exit angles, showing mirror symmetry for 50% reaction; normal shock property variation sketch for (b) with M* annotation; properly labelled axes and flow directions.Diagrams present but incomplete: velocity triangles without angle labels, or T-s diagram without Q and W arrows; hand-drawn appearance acceptable but key features missing.No diagrams where essential (especially part c); or incorrect diagrams showing 0% or 100% reaction blading instead of symmetrical; diagrams contradict written explanation.
Step-by-step derivation25%12.5Complete derivations: (a) explicit entropy balance with system and surroundings identified; (b) full algebraic manipulation from M_x to M_y to M* product using critical area relations; (c) explicit derivation from R = Δh_rotor/Δh_stage to Vcosα relationships proving symmetry; (d) substitution into Planck's law with intermediate steps shown.Key steps present but gaps exist: jumps from M_y relation to final result in (b), or states symmetry without proving V1=V3 in (c); some 'it can be shown that' omissions.Final answers stated without derivation; or incorrect derivation path (e.g., uses isentropic relations for shock in b, confuses absolute with relative velocities in c).
Practical interpretation20%10Interprets (a) entropy generation as measure of lost work potential; (b) explains M*M*=1 significance for shock polar and flow turning; (c) connects symmetrical blading to equal blade loading and manufacturing simplicity; (d) relates furnace radiation to industrial heating applications; (e) gives examples of Indian power plant heat exchangers (e.g., BHEL designs) where each method applies.Brief mention of practical relevance without elaboration; generic statements about 'useful in engineering' without specific application context.No interpretation; treats all parts as purely mathematical exercises; or incorrect practical conclusions (e.g., suggests reversible operation possible in a, or symmetrical blading for all reaction degrees).

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