Mechanical Engineering 2021 Paper II 50 marks Compulsory Prove

Q1

(a) Discuss briefly the functional differences between a fan, a blower and a compressor. (10 marks) (b) Prove that shock cannot occur in subsonic flow. (10 marks) (c) Consider a large plane wall of thickness L = 0·4 m, thermal conductivity k = 2·3 W/m°C and surface area A = 20 m². The left side of the wall is maintained at a constant temperature of T₁ = 80°C while the right side loses heat by convection to the surrounding air at Tₐ = 15°C with a heat transfer coefficient of h = 24 W/m²°C. Assuming constant thermal conductivity and no heat generation in the wall, (i) obtain a relation for the variation of temperature in the wall. (ii) evaluate the rate of heat transfer through the wall. (10 marks) (d) The following equation has been proposed for the heat transfer coefficient in natural convection from long vertical cylinders to air at atmospheric pressure: $\bar{h}_c = \frac{536.5(T_s - T_\infty)^{0.33}}{T}$ where T = the film temperature = $\frac{(T_s + T_\infty)}{2}$ and T is in the range 0 to 200°C. The corresponding equation in dimensionless form is $\frac{\bar{h}_c L}{K} = C(Gr Pr)^m$. Compare the two equations to determine the values of C and m such that the second equation will give the same results as the first. Use properties of dry air at 100°C and one atmosphere: K = 0·0307 W/(mk), g = 9·8 m/sec², μ = 21·673×10⁻⁶ NS/m², Cₚ = 1022 J/(kg K). The absolute pressure of one atmosphere = 101,000 N/m². The gas constant R (for air) = 287 J/kg K. Symbols have their usual meaning. (10 marks) (e) Combustion in a diesel engine is assumed to begin at inner dead centre and to be at constant pressure. The air-fuel ratio is 27 : 1, the calorific value of the fuel is 43000 kJ/kg, and the specific heat (at constant volume) of the products of combustion is given by: Cᵥ = (0·71 + 20 × 10⁻⁵ T) kJ/(kg K). R for products = 0·287 kJ/(kg K). If the compression ratio is 15 : 1, and the temperature at the end of compression is 870 K, determine the percentage of stroke at which combustion is completed. (10 marks)

हिंदी में प्रश्न पढ़ें

(a) पंखा (फैन), आधमाता (ब्लोअर) और संपीडक के बीच प्रकार्यात्मक अंतरों को संक्षेप में समझाइए। (10 अंक) (b) सिद्ध कीजिए कि अवध्वनिक प्रवाह में प्रघात घटित नहीं हो सकता है। (10 अंक) (c) मोटाई L = 0·4 m, ऊष्मा चालकता k = 2·3 W/m°C और पृष्ठीय क्षेत्रफल A = 20 m² वाली एक बड़ी समतल दीवार पर विचार करें। दीवार के बायीं ओर का तापमान T₁ = 80°C पर स्थिर बनाए रखा जाता है, जबकि दाहिनी ओर Tₐ = 15°C तापमान पर परिवेश की हवा में संवहन द्वारा ऊष्मा-अंतरण गुणांक h = 24 W/m²°C के साथ ऊष्मा का ह्रास होता है। स्थिर ऊष्मा चालकता और दीवार में कोई भी ऊष्मा उत्पादन न होने को मानते हुए: (i) दीवार में तापमान परिवर्तन के लिए एक संबंध प्राप्त करें। (ii) दीवार के माध्यम से ऊष्मा हस्तांतरण की दर का मूल्यांकन करें। (10 अंक) (d) वायुमंडलीय दबाव पर लंबे उर्ध्वाधर सिलिंडर से हवा में प्राकृतिक संवहन के लिए ऊष्मा हस्तांतरण गुणांक का समीकरण $\bar{h}_{c} = \frac{536 \cdot 5(T_{s}-T_{\infty})^{0 \cdot 33}}{T}$ द्वारा प्रस्तावित है। जहाँ T = फिल्म तापमान = $\frac{(T_{s}+T_{\infty})}{2}$ और T, 0 से 200°C की सीमा में है। विमारहित रूप में संगत समीकरण $\frac{\bar{h}_{c} L}{K} = C(Gr Pr)^{m}$ है। दोनों समीकरणों की तुलना करके, दूसरे समीकरण में C और m के उपमानों को निर्धारित करें जो पहले समीकरण के एक सा परिणाम देगा। 100°C और एक वायुमंडलीय दबाव पर शुष्क हवा के निम्न गुणों का उपयोग करें: K = 0·0307 W/(mk), g = 9·8 m/sec², μ = 21·673×10⁻⁶ NS/m², Cₚ = 1022 J/(kg K), एक वायुमंडलीय यथार्थ दाब = 101,000 N/m², वायु के लिए गैस नियतांक R = 287 J/kg K। संकेताक्षरों का सामान्य अर्थ है। (10 अंक) (e) ऐसा माना जाता है कि डीजल इंजन में दहन आंतरिक निश्चल्य स्थिति पर शुरू होता है तथा दहन में दबाव स्थिर रहता है। वायु ईंधन अनुपात 27 : 1, ईंधन का ऊष्मीय मान 43000 kJ/kg, दहन के उत्पादों की विशिष्ट ऊष्मा (अचर आयतन पर) Cᵥ = (0·71 + 20 × 10⁻⁵ T) kJ/(kg K) एवं उत्पादों के लिए R = 0·287 kJ/(kg K) दिया गया है। यदि संपीडन अनुपात 15 : 1 है, और संपीडन के अंत में तापमान 870 K है, तो ज्ञात करें कि कितने प्रतिशत चरण (स्ट्रोक) पर दहन पूरा होता है। (10 अंक)

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Directive word: Prove

This question asks you to prove. The directive word signals the depth of analysis expected, the structure of your answer, and the weight of evidence you must bring.

See our UPSC directive words guide for a full breakdown of how to respond to each command word.

How this answer will be evaluated

Approach

Begin with part (a) using 'discuss' to compare fan/blower/compressor pressure ratios and applications; for (b) 'prove' requires rigorous derivation using Fanno/Rayleigh flow or area-velocity relations showing shock requires supersonic upstream Mach number. Part (c) demands 'solve' for temperature distribution and heat transfer using thermal resistance network. Part (d) involves 'compare' to extract C and m through dimensional analysis matching Gr and Pr exponents. Part (e) requires 'calculate' percentage stroke using constant pressure heat addition with variable specific heat integration. Allocate time proportionally: ~15% each for (a), (b), (d); ~25% each for (c) and (e) due to numerical complexity.

Key points expected

  • Part (a): Fan (<1.1 pressure ratio, low pressure rise), Blower (1.1-2.5 pressure ratio, medium flow), Compressor (>2.5 pressure ratio, high pressure rise); cite applications like HVAC fans, FD/ID blowers in Indian power plants, multi-stage compressors
  • Part (b): Prove using dA/A = (M²-1)/M² * dV/V; for shock, entropy must increase requiring dS>0 which demands supersonic M>1 upstream; or use Prandtl relation M₁*M₂=1 showing M₁>1 needed for real M₂
  • Part (c)(i): Derive T(x) = T₁ - (T₁-T₂)x/L with T₂ found from convection boundary; linear temperature profile in wall
  • Part (c)(ii): Q = (T₁-Tₐ)/(L/kA + 1/hA) = 2934.8 W; thermal resistance network with conduction and convection resistances in series
  • Part (d): Match exponents to find m=0.25 from (T_s-T_∞) exponent 0.33 and Gr∝ΔT; calculate C=0.525 using properties at 100°C with β=1/T_film, ν=μ/ρ
  • Part (e): Integrate CᵥdT from 870K to T₃ using Q=m_f*CV*(A/F+1); find T₃=2154K; use V₃/V₂=T₃/T₂ for constant pressure; percentage stroke = (V₃-V₂)/(V₁-V₂)*100 = 23.8%

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Concept correctness20%10Correctly distinguishes fan/blower/compressor by pressure ratio ranges; rigorously proves shock impossibility in subsonic flow using entropy increase requirement or Prandtl relation; sets up correct thermal resistance network for composite wall; properly identifies Gr∝ΔT and Pr independence in natural convection correlation; applies correct constant pressure combustion model with variable Cᵥ integrationCorrect pressure ratio ranges stated but applications generic; shock proof attempted but misses entropy argument or assumes result; thermal resistance concept correct but boundary condition errors; dimensional analysis attempted but m value incorrect; combustion uses constant Cᵥ approximation or wrong heat addition modelConfuses fan/blower/compressor definitions; states shock cannot occur without proof or uses incorrect governing equations; treats wall as convection-only or uses parallel resistances; fails to identify Gr and Pr in dimensional analysis; applies constant volume combustion or ignores variable specific heat
Numerical accuracy20%10Part (c): Q=2934.8 W (or ~2935 W), T₂=29.4°C; Part (d): C≈0.525, m=0.25 with full property calculations including ρ=P/RT, β=0.00268 K⁻¹, ν=2.3×10⁻⁵ m²/s, α=3.28×10⁻⁵ m²/s; Part (e): T₃≈2150-2160 K, percentage stroke 23-24%; all unit conversions correct (kJ to J where needed)Correct order of magnitude for Q but arithmetic slip; C within ±20% due to property evaluation errors; T₃ within ±100 K; percentage stroke 20-28% with correct methodologyQ wrong by factor >2 (e.g., omits area or thickness); C and m not matching dimensional analysis; T₃ calculation fails due to integration error or constant Cᵥ assumption; percentage stroke calculation uses wrong volume ratio or ignores compression ratio
Diagram quality15%7.5Part (a): Clear schematic showing pressure rise vs flow rate curves for all three devices; Part (b): T-s diagram showing shock process with entropy increase direction; Part (c): Thermal resistance network diagram with temperatures and resistances labelled; Part (e): Diesel cycle P-V or T-S diagram showing constant pressure heat addition from TDCAt least two relevant diagrams drawn with correct qualitative features but missing labels or values; or only text descriptions where diagrams expectedNo diagrams despite clear need (especially for thermal resistance and cycle); or completely wrong diagrams (e.g., Otto cycle for diesel, isothermal process shown)
Step-by-step derivation25%12.5Part (b): Full derivation from continuity, momentum, energy equations leading to Prandtl relation or entropy condition; Part (c): Explicit integration of Fourier's law with boundary conditions; Part (d): Complete dimensional analysis showing Gr=gβΔTL³/ν², Pr=ν/α, exponent matching step-by-step; Part (e): Clear integration of Cᵥ from 870K to T₃, expansion work calculation, volume ratio derivationKey steps shown but some skipped (e.g., jumps to Prandtl relation without derivation, assumes linear profile in (c) without integration); dimensional analysis mostly correct but algebraic steps condensedFinal formulae stated without derivation; no integration shown in (e); dimensional analysis missing or incorrect; essentially a list of equations with answers
Practical interpretation20%10Part (a): Cites Indian context—axial fans in cooling towers, centrifugal blowers in NTPC thermal plants, multi-stage compressors in process industries; Part (c): Discusses insulation effectiveness, critical thickness concept; Part (d): Notes validity range (0-200°C) and applicability to chimney/natural draft systems; Part (e): Relates percentage stroke to combustion duration, diesel knock considerations, engine efficiency implicationsGeneric applications mentioned without Indian context; some physical insight but not linked to engineering design or operational implicationsNo interpretation; pure mathematical exercise; or incorrect physical interpretation (e.g., suggests shock desirable, misinterprets heat flow direction)

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