Mechanical Engineering 2025 Paper II 50 marks Calculate

Q3

(a) A long cylindrical rod of radius 10 cm consists of a nuclear reacting material (k = 0·5 W/m-K) generating 24000 W/m³ uniformly throughout its volume. The rod is encapsulated within another cylinder whose outer radius is 20 cm and that has a thermal conductivity of 4 W/m-K. The outer surface is surrounded by a fluid at 100 °C, and the convection coefficient between the surface and the fluid is 20 W/m²-K. Find the temperatures of the interface between the two cylinders, at the outer surface and the maximum temperature under steady-state condition. (20 marks) (b) A centrifugal compressor is to be designed for an industrial application handling air. The inlet stagnation conditions are P₀₁ = 1·1 bar and T₀₁ = 295 K. The air enters the eye of the impeller axially, without any prewhirl. The axial velocity is uniform throughout the eye and is equal to 143 m/s. The eye tip and root diameters are 0·30 m and 0·15 m, respectively. Calculate the mass flow rate of air. The overall diameter of the impeller is 0·50 m. The power input factor is 1·04 and the slip factor is 0·9. The rotational speed of the compressor is 290 revolutions/second. The isentropic efficiency of the compressor (based on total head) is 0·78. The radial velocity at the impeller tip is 143 m/s. Assume that 'half the total losses' occurs in the impeller. Determine the pressure ratio and the power required to drive the compressor. Also, determine the axial depth of the impeller channels at the periphery of the impeller. Draw the T-s diagram showing the variations of both static and stagnation pressures and temperatures in the impeller and the diffuser. For air, γ = 1·4, Cp = 1·005 kJ/kg-K. (20 marks) (c) The velocity and temperature profiles for laminar flow in a tube of radius r0 = 10 mm have the form at a particular axial location u(r) = 0·1(1 - r²/r0²) T(r) = 344·8 + 75 r²/r0² - 18·8 r⁴/r0⁴ with units of m/s and K, respectively. Determine the corresponding value of the mean (or bulk) temperature Tm at this axial position. (10 marks)

हिंदी में प्रश्न पढ़ें

(a) 10 cm त्रिज्या वाली एक लंबी बेलनाकार छड़ एक नाभिकीय प्रतिक्रिया सामग्री (k = 0·5 W/m-K) से बनी हुई है, जो अपने पूरे आयतन में समान रूप से 24000 W/m³ का उत्पादन कर रही है। छड़ को एक अन्य बेलनाकार संरचना के अंदर समाहित किया गया है, जिसकी बाहरी त्रिज्या 20 cm है तथा तापीय चालकता 4 W/m-K है। बाहरी सतह एक तरल द्वारा 100 °C पर धिरी हुई है तथा सतह और तरल के बीच संवहन गुणांक (कन्वेक्शन कोएफिशिएंट) 20 W/m²-K है। दोनों बेलनों के बीच सम्पर्क सतह (इंटरफेस) के तापमान, बाहरी सतह पर तापमान और स्थिर अवस्था की स्थिति में अधिकतम तापमान ज्ञात कीजिये। (20) (b) एक औद्योगिक उपयोग के लिये वायु को संपीडित करने हेतु एक अपकेन्द्री संपीडक (सेंट्रीफ्यूगल कंप्रेसर) डिजाइन किया जाना है। अन्तर्गाम (इनलेट) पर ठहराव (स्टेगनेशन) की स्थितियाँ P₀₁ = 1·1 bar तथा T₀₁ = 295 K हैं। वायु इम्पेलर के नेत्र में किसी भी पूर्व-घूर्णन (प्रीव्हर्ल) के बिना अक्षीय रूप से प्रवेश करती है। अक्षीय वेग सम्पूर्ण नेत्र में समान है और यह 143 m/s के बराबर है। नेत्र की टिप और जड़ के व्यास क्रमशः 0·30 m और 0·15 m हैं। वायु की द्रव्यमान प्रवाह दर की गणना कीजिये। इम्पेलर का कुल व्यास 0·50 m है। पावर निवेश (इनपुट) फैक्टर 1·04 तथा स्लिप फैक्टर 0·9 है। कंप्रेसर की घूर्णन गति 290 चक्कर प्रति सेकंड है। कंप्रेसर की आइसेन्ट्रॉपिक दक्षता (कुल शीर्ष पर आधारित) 0·78 है। इम्पेलर के सिरे (टिप) पर त्रिज्य (रेडियल) वेग 143 m/s है। यह मानिये कि 'कुल हानियों का आधा हिस्सा' इम्पेलर में होता है। कंप्रेसर को चलाने के लिये आवश्यक दाब अनुपात (प्रेशर रेशियो) तथा शक्ति निर्धारित कीजिये। साथ ही इम्पेलर की परिधि पर इम्पेलर चैनलों की अक्षीय गहराई (एक्सियल डेप्थ) को भी निर्धारित कीजिये। इम्पेलर और डिफ्यूजर में स्थैतिक (स्टैटिक) तथा ठहराव (स्टेगनेशन) दाबों और तापमानों में होने वाले परिवर्तन को दर्शाते हुए T-s आरेख निर्मित कीजिये। वायु के लिये γ = 1·4, Cₚ = 1·005 kJ/kg-K. (20) (c) एक निश्चित अक्षीय स्थिति पर त्रिज्या r0 = 10 mm वाली एक नली में स्तरीय (लेमिनर) प्रवाह के लिये वेग और तापमान प्रोफाइल u(r) = 0·1(1 - r²/r0²) T(r) = 344·8 + 75 r²/r0² - 18·8 r⁴/r0⁴ हैं, जिनकी इकाइयाँ क्रमशः m/s और K हैं। इस अक्षीय स्थिति पर औसत (या बल्क) तापमान Tm का संबंधित मान निर्धारित कीजिये। (10)

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Approach

Calculate systematically across all three sub-parts: for (a) apply cylindrical heat conduction with internal heat generation and composite wall resistance; for (b) solve centrifugal compressor thermodynamics with velocity triangles, slip factor, and efficiency corrections; for (c) integrate velocity-weighted temperature for bulk mean temperature. Allocate approximately 35% time to part (a), 45% to part (b) due to its multi-step complexity and diagram requirement, and 20% to part (c). Present derivations with clear boundary condition statements and conclude each part with physically verified numerical answers.

Key points expected

  • Part (a): Apply heat conduction equation in cylindrical coordinates with uniform heat generation; use thermal resistance network for composite cylinder with convection at outer surface; identify maximum temperature at centerline (r=0)
  • Part (a): Calculate interface temperature T₁, outer surface temperature T₂, and T_max using continuity of heat flux and temperature at r=10 cm interface
  • Part (b): Calculate mass flow rate using continuity equation with variable annulus area at impeller eye; determine blade tip speed U₂ = πND₂
  • Part (b): Apply slip factor and power input factor to find actual work input; use isentropic efficiency to find pressure ratio; determine impeller channel depth from radial velocity and through-flow area
  • Part (b): Draw complete T-s diagram showing: stagnation and static states at impeller inlet, impeller outlet (with losses), diffuser outlet; label pressure rise and temperature rise correctly
  • Part (c): Apply definition of bulk mean temperature T_m = (∫ρuC_p T dA)/(ṁC_p) = (∫₀^r₀ uT r dr)/(∫₀^r₀ u r dr) for axisymmetric flow
  • Part (c): Perform integration of polynomial terms u(r) = 0.1(1-r²/r₀²) and T(r) = 344.8 + 75r²/r₀² - 18.8r⁴/r₀⁴ to obtain numerical T_m

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Concept correctness22%11Correctly applies cylindrical heat conduction with generation for (a), recognizing T_max at center and proper interface matching; for (b) correctly uses Euler turbomachinery equation with slip factor, distinguishes between isentropic and actual work, and applies efficiency definition properly; for (c) correctly defines and applies bulk temperature integral formulationUses correct basic formulas but makes minor errors: treats (a) as plane wall or misses generation term in outer cylinder; for (b) confuses static and stagnation quantities or misapplies slip factor; for (c) uses arithmetic mean instead of bulk meanFundamental conceptual errors: uses Fourier's law without generation term for (a), treats compressor as axial flow or ignores slip factor entirely for (b), integrates T(r) alone without velocity weighting for (c)
Numerical accuracy24%12Part (a): T_interface ≈ 140-145°C, T_outer ≈ 115-120°C, T_max ≈ 160-165°C; Part (b): mass flow ≈ 2.3-2.5 kg/s, pressure ratio ≈ 2.8-3.2, power ≈ 280-320 kW, channel depth ≈ 8-10 mm; Part (c): T_m ≈ 360-365 K; all calculations show proper unit conversions and 3-4 significant figuresCorrect methodology but arithmetic slips: one part has >10% error, or consistent unit errors (cm vs m, bar vs Pa), or slip factor applied incorrectly leading to plausible but wrong powerMultiple order-of-magnitude errors; uses radius as diameter; temperature in °C where K required; final answers physically impossible (e.g., T_max below ambient, pressure ratio < 1)
Diagram quality16%8Clear T-s diagram for part (b) with: two distinct curves (static and stagnation), proper slope (γ/(γ-1)), labeled states 01, 1, 2, 02, 3, 03 showing impeller and diffuser processes; vertical distance showing losses; isentropic and actual compression paths distinguished; all temperatures and pressures marked with numerical values or clear trendsT-s diagram drawn but missing key elements: only stagnation states shown, or static/stagnation distinction unclear, or losses not indicated, or missing diffuser sectionNo diagram provided for part (b); or incorrect diagram (P-v instead of T-s, or open cycle shown); or part (a) diagram drawn unnecessarily while (b) diagram missing
Step-by-step derivation22%11Part (a): Governing equation 1/r d/dr(r dT/dr) + q'''/k = 0 stated, integrated twice with constants found from boundary conditions; thermal resistance R_total = R_cylinder2 + R_conv shown; Part (b): Velocity triangles sketched, U₂ calculated, work input derived from w = σψU₂², efficiency relation derived; Part (c): Integral set up with dA = 2πr dr, substitution shown, limits 0 to r₀Key formulas stated but derivation compressed: jumps from differential equation to solution without integration constants; or skips velocity triangle and quotes Euler equation directly; or presents final integral without setupNo derivations shown: only final formulas quoted with numbers substituted; or incorrect governing equations stated; or algebraic manipulation errors in integration
Practical interpretation16%8For (a): comments on nuclear fuel rod thermal limits, cladding purpose, and critical heat flux considerations; for (b): discusses industrial compressor applications (e.g., BHEL gas turbine packages), surge margin, and material stress at high tip speeds; for (c): relates bulk temperature to heat exchanger design and Nusselt number correlation validityBrief mention of practical relevance: states that temperatures must stay below material limits, or notes that pressure ratio affects plant efficiency, without specific industrial contextNo interpretation provided; or irrelevant commentary (discusses beam bending in a heat transfer problem); or physically incorrect statements about applications

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