Mechanical Engineering 2022 Paper II 50 marks Compulsory Calculate

Q1

(a) Consider the system shown in Fig. 1(a). The two chambers initially have equal volumes of 28 litres and contain air (C_p = 1·005 kJ/kg-K and C_v = 0·717 kJ/kg-K) and hydrogen (C_p = 14·32 kJ/kg-K and C_v = 10·17 kJ/kg-K), respectively. The chambers are separated by a frictionless piston which is non-heat-conducting. Both the gases are initially at 140 kPa and 40 °C. Heat is added to the air side until the pressure of both the gases reaches 280 kPa. All outside walls of the chambers are insulated except for the surface where heat is added to air. Calculate the final temperature of the air. (10 marks) (b) What is 'choked flow' in a convergent-divergent nozzle? Explain, with diagram, the effect of pressure ratio on exit velocity of compressible gas in a convergent-divergent nozzle. (10 marks) (c) An axial flow compressor with inlet and outlet angles of 40° and 15°, respectively has been designed for 50% reaction. The compressor has a pressure ratio of 6 : 1 and overall isentropic efficiency of 0·80, when inlet static temperature is 41 °C. The blade speed and axial velocity are constant throughout. Assuming a value of 210 m/s for blade speed, find the number of stages required if the work done factor is 0·88 for all the stages. Take Cp = 1·005 kJ/kg-K and γ = 1·4 for air. (10 marks) (d) Hot water is flowing through a pipe made of cast iron having thermal conductivity of 52 W/m-°C, with an average velocity of 1·5 m/s. The inner and outer diameters of the pipe are 3 cm and 3·5 cm, respectively. The pipe passes through a 15 m long section of a basement whose temperature is 15 °C. The temperature of the water drops from 70 °C to 67 °C as it passes through the basement. The heat transfer coefficient on the inner surface of the pipe is 400 W/m²-°C. Determine the combined convection and radiation heat transfer coefficient at the outer surface of the pipe. (10 marks) (e) (i) Define the total and spectral black body emissive powers. How are they related to each other? (5 marks) (ii) Consider two identical bodies, one at 1000 K and the other at 1500 K. Which body emits more radiation in the shorter wavelength region? Which body emits more radiation at a wavelength of 20 μm? (5 marks)

हिंदी में प्रश्न पढ़ें

(a) चित्र 1(a) में दर्शाये गये निकाय पर गौर कीजिये। दोनों कक्ष प्रारम्भ में 28 लीटर के समान आयतन के हैं, जिनमें क्रमशः वायु (C_p = 1·005 kJ/kg-K तथा C_v = 0·717 kJ/kg-K) एवं हाइड्रोजन (C_p = 14·32 kJ/kg-K तथा C_v = 10·17 kJ/kg-K) हैं। ये कक्ष एक घर्षणहीन पिस्टन, जो कि ऊष्मा अचालक है, के द्वारा अलग किये गये हैं। दोनों गैसें आरम्भ में 140 kPa तथा 40 °C पर हैं। वायु की ओर से इस प्रकार ऊष्मा दी जाती है कि दोनों गैसों का दाब 280 kPa तक पहुँच जाये। जहाँ से वायु को ऊष्मा दी जाती है उस सतह को छोड़कर कक्ष की सभी बाहरी दीवारों को रोधित किया गया है। वायु का अन्तिम तापमान ज्ञात कीजिये। (10 अंक) (b) एक अभिसारी-अपसारी नोजल में 'प्रोध प्रवाह' क्या होता है? एक अभिसारी-अपसारी नोजल में संपीड्य गैस के निर्गम वेग पर दाब अनुपात के प्रभाव को आरेख की सहायता से समझाइये। (10 अंक) (c) एक अक्षीय प्रवाह संपीडक, जिसके अंतर्गाम तथा निर्गम कोण क्रमशः: 40° तथा 15° हैं, को 50% प्रतिक्रिया के लिये अभिकल्पित किया गया है। संपीडक का दाब अनुपात 6 : 1 है तथा समग्र समझौती दक्षता 0·80 है, जबकि अंतर्गाम स्थैतिक तापमान 41 °C है। फलक (ब्लेड) चाल तथा अक्षीय वेग आद्योपांत समान हैं। यदि सभी पदों के लिये कृत कार्य गुणक 0·88 हो, तो फलक चाल का मान 210 m/s मानते हुए वांछित पदों की संख्या ज्ञात कीजिये। वायु के लिये Cp = 1·005 kJ/kg-K तथा γ = 1·4 लीजिये। (10 अंक) (d) 52 W/m-°C की ऊष्मीय चालकता वाले ढलवां लोहे से बने एक पाइप में गर्म जल 1·5 m/s के औसत वेग से प्रवाहित हो रहा है। पाइप के आंतरिक तथा बाह्य व्यास क्रमशः: 3 cm तथा 3·5 cm हैं। 15 °C तापमान वाले तलघर के 15 m लम्बे हिस्से से पाइप गुजरता है। पाइप के तलघर से गुजरने से जल का तापमान 70 °C से घटकर 67 °C रह जाता है। पाइप की आंतरिक सतह पर ऊष्मा अंतरण गुणांक 400 W/m²-°C है। पाइप की बाह्य सतह पर संयुक्त संवहन तथा विकिरण ऊष्मा अंतरण गुणांक को निर्धारित कीजिये। (10 अंक) (e) (i) सम्पूर्ण तथा स्पेक्ट्रमी कृष्णिका उत्सर्जक शक्तियों को परिभाषित कीजिये। ये एक-दूसरे से किस प्रकार सम्बन्धित हैं? (5 अंक) (ii) दो समरूप पिंडों पर विचार कीजिये, जिनमें एक 1000 K तथा दूसरा 1500 K पर है। लघु तरंगदैर्घ्य क्षेत्र में कौन-सा पिंड ज्यादा विकिरण उत्सर्जित करता है? 20 μm के तरंगदैर्घ्य पर कौन-सा पिंड ज्यादा विकिरण उत्सर्जित करता है? (5 अंक)

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Approach

Calculate numerical solutions for parts (a), (c), and (d) while explaining theoretical concepts with diagrams for parts (b) and (e). Allocate approximately 25% time each to (a), (c), and (d) as they involve multi-step calculations; 15% to (b) for the choked flow diagram and explanation; and 10% to (e) for definitions and Wien's displacement law application. Begin each numerical part with stated assumptions and end with unit verification.

Key points expected

  • Part (a): Apply first law to adiabatic hydrogen compression (γ_H2 = 1.407) to find T_H2_final, then use piston equilibrium and ideal gas law to find T_air_final ≈ 586-590 K
  • Part (b): Define choked flow as Mach 1 at throat with maximum mass flow; sketch P-V diagram showing over-expanded, design, and under-expanded nozzle conditions with exit velocity trends
  • Part (c): Use 50% reaction condition (α1 = β2, α2 = β1) to find flow angles, compute stage temperature rise from degree of reaction and velocity triangles, then determine stages n ≈ 8-9
  • Part (d): Apply thermal resistance network (convection-inner, conduction-pipe, convection+radiation-outer) using LMTD for water temperature drop, solve for h_outer ≈ 12-15 W/m²°C
  • Part (e)(i): Define E_bλ = C1λ⁻⁵/[exp(C2/λT)-1] and Eb = σT⁴ with Planck's law integration; state Stefan-Boltzmann constant σ = 5.67×10⁻⁸ W/m²K⁴
  • Part (e)(ii): Apply Wien's law (λ_maxT = 2898 μm·K) to show 1500 K body emits more at shorter wavelengths; use Planck's distribution to compare emission at 20 μm

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Concept correctness20%10Correctly applies: (a) adiabatic process for H2 and piston equilibrium; (b) choked flow definition with critical pressure ratio; (c) 50% reaction velocity triangles and stage loading; (d) composite thermal resistance with LMTD; (e) Planck's law and Wien's displacement law with proper spectral integrationUses correct major formulas but confuses: adiabatic vs isothermal in (a), or reaction degree definition in (c), or LMTD vs arithmetic mean in (d); partial understanding of choked flowFundamental errors: treats piston as fixed volume for both gases, confuses convergent-only with C-D nozzle choking, uses wrong reaction degree formula, applies simple cylindrical conduction without convection, or states incorrect Stefan-Boltzmann relation
Numerical accuracy20%10All calculations precise: (a) T_air ≈ 587 K; (c) stages = 8-9 with exact temperature rise per stage ≈ 24-25 K; (d) h_outer ≈ 13-14 W/m²°C; (e)(ii) correct ratio comparisons using Planck's law values; unit conversions (litres to m³, °C to K) flawlessFinal answers within 5-10% of correct values; minor arithmetic errors in intermediate steps (e.g., velocity triangle trigonometry, log mean calculation) but method soundOrder-of-magnitude errors (e.g., stages calculated as 80 instead of 8, temperature in °C not K for gas calculations, missing π in circumference calculations); inconsistent units throughout
Diagram quality20%10Clear labeled diagram for (b): convergent-divergent nozzle with throat, exit, pressure variation along length, and three operating regimes (pe < pb, pe = pb, pe > pb) with velocity vectors; schematic for (a) showing insulated piston and heat addition; velocity triangles for (c) with blade angles markedBasic C-D nozzle sketch without pressure distribution curves or regime labels; velocity triangles drawn but angles not marked or incorrectly labeledNo diagram for (b) despite explicit instruction; confusing or unlabeled sketches; omits velocity triangles for 50% reaction in (c); thermal resistance network missing for (d)
Step-by-step derivation20%10Complete derivations: (a) γ calculation, adiabatic relation T2/T1 = (P2/P1)^((γ-1)/γ), volume ratio from piston movement; (c) tan(αm) = (tanα1 + tanα2)/2 for 50% reaction, stage temperature rise ΔT0 = λUCa(tanα1 - tanα2)/Cp; (d) 1/U = 1/hi + ln(r2/r1)/(2πkL) + 1/ho with Q = ṁCpΔT = UAΔT_LMTDJumps to final formulas with minimal derivation; shows some intermediate steps but skips key substitutions (e.g., directly uses stage efficiency without relating to polytropic index)No working shown—only final answers stated; or incorrect formula substitution without explanation (e.g., uses isothermal relations for adiabatic process, confuses static and stagnation temperatures)
Practical interpretation20%10Physical insights: (a) notes hydrogen acts as spring, limiting air temperature rise; (b) relates choked flow to rocket nozzle design (ISRO PSLV/Vikram engines); (c) discusses surge margin and blade loading limits for axial compressors (jet engines, NTPC gas turbines); (d) comments on pipe insulation economics; (e) relates to solar thermal collectors and pyrometryBrief mention of applications (e.g., 'used in turbines') without specific Indian/industrial context; generic statements about efficiencyNo physical interpretation; treats all parts as pure mathematics; fails to recognize that (a) involves coupled thermodynamic systems or that (b) explains why nozzle flow becomes independent of downstream pressure

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