Mechanical Engineering 2024 Paper II 50 marks Compulsory Solve

Q1

(a) A 2 gm quantity of air undergoes the following sequence of quasi-static processes in a piston-cylinder arrangement: (i) An adiabatic expansion in which the volume doubles. (ii) A constant pressure process in which the volume is reduced to its initial value. (iii) A constant volume compression back to the initial state. The air is initially at 150°C and 5 atm. Calculate net work on the air in the sequence of processes. (10 marks) (b) Consider a nozzle of inlet area 'A₁' and outlet area 'A₂'. The velocity is 'V₁' at inlet and 'V₂' at outlet. This nozzle accelerates the incompressible fluid (V₂ > V₁) and decreases the pressure. Can this nozzle in any condition, deaccelerate the fluid? If yes, then justify your answer with the help of continuity, momentum and energy equations. (10 marks) (c) Deduce an expression for the temperature distribution in an infinite long slab of thickness "L" m under one-dimensional steady state heat conduction. The slab uniformly generates heat of q̇ W/m³. One of its surfaces is perfectly insulated and the other surface is maintained at a constant temperature of Tw °C. Also plot the temperature profile clearly mentioning the maximum and minimum temperatures and the location. (10 marks) (d) For special cases of axial flow reaction turbines with degree of reaction in the form R = 1/(k+1), where k is an integer, a special relationship exists between the blade velocity 'u' and fluid inlet velocity or velocity for maximum utilization. Show that this relationship is given by u/V₁ = (k+1)/(2k) cos 'α'. Here 'α' is angle between inlet velocity 'V₁' and blade velocity 'u'. (10 marks) (e) Incompressible fluid having free stream velocity of "u" m/s and temperature of T°C flows over a flat plate maintained at a constant temperature of T_w °C (T ≠ T_w). Flow is within the laminar region. Draw the relative thicknesses of thermal and hydrodynamic boundary layers developed on the flat plate for three fluids having (i) Pr < 1, (ii) Pr = 1 and (iii) Pr > 1. Justify your answer appropriately. (Draw three diagrams for three fluids for better clarity) (10 marks)

हिंदी में प्रश्न पढ़ें

(a) एक पिस्टन-सिलिंडर विन्यास में 2 gm वायु की मात्रा स्थैतिक-कल्प प्रक्रमों के निम्नलिखित अनुक्रम से गुजरती है: (i) एक रूद्धोष्म प्रसार जिसमें आयतन दुगुना हो जाता है। (ii) एक स्थिर दाब प्रक्रम जिसमें आयतन को घटाकर उसके प्रारंभिक मान तक ले आते हैं। (iii) एक स्थिर आयतन संपीडन जो घूमकर अपनी प्रारंभिक अवस्था में आ जाता है। वायु प्रारंभ में 150°C व 5 atm पर है। प्रक्रमों के अनुक्रम में वायु पर किए गए नेट कार्य का परिकलन कीजिए। (10 marks) (b) एक तुंड पर विचार कीजिए जिसका अंतर्गम क्षेत्रफल 'A₁' व निर्गम क्षेत्रफल 'A₂' है। उसके अंतर्गम पर वेग 'V₁' व निर्गम पर 'V₂' है। यह तुंड असंपीड्य तरल (V₂ > V₁) को त्वरित करता है एवं दाब को कम करता है। क्या यह तुंड किसी परिस्थिति में तरल को विवरित कर सकता है? यदि हाँ, तो अपने उत्तर को सांतत्य, संवेग व ऊर्जा समीकरणों की सहायता से उचित सिद्ध कीजिए। (10 marks) (c) एक अपरिमित लम्बाई के "L" m मोटाई के पड़, जो कि एक-विमीय स्थायी दशा ऊष्मा चालन के अन्तर्गत है, के तापमान वितरण के लिए व्यंजक व्युत्पन्न कीजिए। पड़ समान रूप से q̇ W/m³ की ऊष्मा उत्पन्न करता है। उसकी एक सतह पूर्ण रूप से रोधित है तथा दूसरी सतह Tw °C के स्थिर ताप पर अनुरक्षित है। अधिकतम व न्यूनतम तापमान तथा स्थान निर्धारण का स्पष्ट रूप से उल्लेख करते हुए तापमान प्रोफाइल भी बनाइए। (10 marks) (d) अक्षीय प्रवाह प्रतिक्रिया टरबाइनों के विशिष्ट मामलों के लिए प्रतिक्रिया मात्रा R = 1/(k+1) के रूप में है, जहाँ k एक पूर्णांक है। ब्लेड वेग 'u' व तरल अन्तर्गाम वेग, वह वेग जो कि अधिकतम उपयोग के लिए है, के मध्य एक विशिष्ट संबंध स्थापित है। दर्शाइए कि यह संबंध u/V₁ = (k+1)/(2k) cos 'α' द्वारा व्यक्त किया जाता है। यहाँ 'α' अन्तर्गाम वेग V₁ व ब्लेड वेग 'u' के मध्य का कोण है। (10 marks) (e) असंपीड़ीय तरल जो मुक्त प्रवाह वेग "u" m/s व T°C तापमान पर है, एक चपटी पट्टिका के ऊपर से प्रवाहित होता है जबकि पट्टिका T_w °C (T ≠ T_w) के स्थिर तापमान पर रखी गई है। प्रवाह स्तरीय क्षेत्र के भीतर है। तीन तरलों के लिए (i) Pr < 1, (ii) Pr = 1 व (iii) Pr > 1, चपटी पट्टिका पर विकसित हुए ऊष्मीय व द्रवगतिक सीमान्त परतों की अपेक्षिक मोटाइयों का आरेख खींचिए। उपयुक्त रूप से अपने उत्तर का औचित्य बताइए। (बेहतर स्पष्टता के लिए तीन तरलों के लिए तीन आरेख बनाइए।) (10 marks)

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Approach

Solve each sub-part sequentially with clear section headers. For (a), apply first law to each process and sum work terms; for (b), use continuity and Bernoulli to analyze diffuser condition; for (c), set up and solve the heat equation with given boundary conditions; for (d), derive the velocity ratio using stage work and Euler turbine equations; for (e), sketch three comparative boundary layer diagrams with Prandtl number reasoning. Allocate approximately 18-20 minutes per sub-part.

Key points expected

  • (a) Process 1-2: T₂ = T₁(V₁/V₂)^(γ-1) = 423.15×(0.5)^0.4 = 323.6 K; W₁₂ = mR(T₁-T₂)/(γ-1); Process 2-3: W₂₃ = p₂(V₃-V₂) = mR(T₃-T₂); Process 3-1: W₃₁ = 0; Net work = W₁₂ + W₂₃
  • (b) Nozzle can act as diffuser when A₂ > A₁ and pressure gradient reverses; continuity: ρA₁V₁ = ρA₂V₂; momentum: dp/dx = -ρV(dV/dx); energy: p₁/ρ + V₁²/2 = p₂/ρ + V₂²/2; deceleration occurs when subsonic flow in diverging section
  • (c) Governing equation: d²T/dx² + q̇/k = 0; BCs: dT/dx = 0 at x = 0 (insulated), T = T_w at x = L; Solution: T(x) = T_w + (q̇/2k)(L² - x²); T_max = T_w + q̇L²/2k at x = 0; T_min = T_w at x = L; parabolic profile
  • (d) Degree of reaction R = Δh_rotor/Δh_stage = 1/(k+1); Utilization factor ε = 2u(V₁cosα - u)/(V₁² + u² - 2uV₁cosα); For maximum utilization, dε/du = 0 yields u/V₁ = (k+1)cosα/(2k)
  • (e) δ/δ_t = Pr^(1/3) for laminar flow; (i) Pr < 1: thermal boundary layer thicker than hydrodynamic (δ_t > δ), e.g., liquid metals; (ii) Pr = 1: δ_t = δ, gases; (iii) Pr > 1: δ_t < δ, e.g., water, oil; three clearly labelled comparative sketches

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Concept correctness20%10Correctly identifies: (a) cyclic process with three distinct thermodynamic processes; (b) diffuser operation via area-velocity relationship for subsonic flow; (c) heat generation term in conduction equation with mixed boundary conditions; (d) stage reaction definition and utilization optimization; (e) Prandtl number effect on relative boundary layer thicknesses.Identifies most processes correctly but confuses (a) work sign conventions, or (b) supersonic/subsonic area-velocity relations, or (d) reaction degree definition.Fundamental errors: treats (a) as isothermal, or (b) claims nozzle can never decelerate, or (c) omits heat generation term, or (e) reverses Pr < 1 and Pr > 1 relationships.
Numerical accuracy20%10(a) Correct mass calculation (m = 2×10⁻³ kg), T₂ ≈ 323.6 K, p₂ ≈ 1.91 atm, W₁₂ ≈ 71.8 J, W₂₃ ≈ -57.4 J, net W ≈ 14.4 J; all unit conversions correct (°C to K, atm to Pa); (d) algebraic simplification leads to exact form u/V₁ = (k+1)cosα/(2k).Correct methodology but arithmetic slips in (a) such as using γ = 1.3 instead of 1.4, or sign errors in net work; (d) correct approach but algebraic error in final simplification.Major numerical errors: uses mass in grams directly, or wrong gas constant, or incorrect temperature ratios; (d) fails to reach required expression.
Diagram quality20%10(a) Clear p-V diagram with three processes labelled 1-2-3-1, areas indicating work; (c) Parabolic temperature profile with T_max at insulated surface, T_w at cooled surface, axes labelled; (e) Three distinct boundary layer sketches showing δ vs δ_t for Pr < 1, Pr = 1, Pr > 1 with relative thickness clearly depicted.Diagrams present but missing labels or key features: (a) missing arrow directions; (c) linear instead of parabolic profile; (e) only qualitative without relative thickness indication.Missing critical diagrams especially for (c) and (e); or (a) shows T-s instead of p-V; diagrams unlabelled or incorrect shapes.
Step-by-step derivation20%10(a) Explicit first law application per process with state property calculations; (b) Full development of continuity, momentum (Euler), and energy equations with clear reasoning for deceleration condition; (c) Complete integration of d²T/dx² = -q̇/k with both boundary conditions applied; (d) Systematic differentiation of utilization factor and algebraic manipulation to required form.Derivations present but skips key steps: (b) states Bernoulli without development; (c) writes final solution without showing integration constants; (d) quotes optimum condition without derivation.No derivations shown: only final answers stated; or fundamental errors in differential equation setup such as wrong sign on heat generation term.
Practical interpretation20%10(a) Comments on positive net work indicating heat engine cycle possibility; (b) Identifies diffuser applications in HVAC, aircraft intakes, and subsonic wind tunnels; (c) Relates to nuclear fuel plates or electronic cooling with one insulated boundary; (d) Connects to 50% reaction turbine (k=1, Parsons turbine) and other special cases; (e) Cites practical fluids: mercury/NaK for Pr<<1, air for Pr≈1, water/oil for Pr>1.Brief mention of applications without elaboration; or generic statements without specific engineering context for each sub-part.No practical interpretation provided; treats all sub-parts as purely mathematical exercises with no engineering relevance mentioned.

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