Mechanical Engineering 2022 Paper II 50 marks Prove

Q3

(a) (i) How do the specific work output and efficiency vary with pressure ratio in a gas turbine? (ii) Prove that the efficiency of a gas turbine corresponding to the maximum work done in a Brayton cycle is given by the relation $$\eta_{w \max} = 1 - \frac{1}{\sqrt{t}}$$ where $t$ is the ratio of the maximum and minimum temperatures. (20 marks) (b) A solar collector, as shown in Fig. 3(b) below, having dimensions as 1 m wide and 5 m long, has constant spacing of 3 cm between the glass cover and the collector plate. Air enters the collector at 30 °C and at a rate of 0·15 m³/s through the 1 m wide edge and flows along the 5 m long passageway. If the average temperatures of the glass cover and the collector plate are 20 °C and 60 °C, respectively, determine (i) the net rate of heat transfer to the air in the collector and (ii) the temperature rise of air as it flows through the collector. Fig. 3(b) The properties of air at 1 atm and an estimated average temperature of 35 °C may be taken as : ρ = 1·145 kg/m³, k = 0·02625 W/m-°C, ν = 1·655×10⁻⁵ m²/s, Cₚ = 1007 J/kg-°C, Pr = 0·7268 (20 marks) (c) A windshield of a car, having dimensions as 0·6 m high and 1·8 m long, is electrically heated and is subjected to parallel winds at 1 atm, 0 °C and 80 km/hr. The electrical power consumption is observed to be 50 W, when the exposed surface temperature of the windshield is 4 °C. Disregarding the radiation and heat transfer from the inner surface and using the momentum heat transfer analogy, determine the drag force the wind exerts on the windshield. The properties of air at 0 °C and 1 atm may be taken as : ρ = 1·292 kg/m³, Cₚ = 1·006 kJ/kg-K, Pr = 0·7362 (10 marks)

हिंदी में प्रश्न पढ़ें

(a) (i) एक गैस टरबाइन में विशिष्ट उत्पादित कार्य तथा दक्षता, दाब अनुपात के साथ किस प्रकार बदलते हैं? (ii) सिद्ध कीजिए कि एक गैस टरबाइन की दक्षता, एक ब्रेटन चक्र के लिये अधिकतम कृत कार्य के तदनुसार, निम्न प्रकार से सम्बन्धित है $$\eta_{w \max} = 1 - \frac{1}{\sqrt{t}}$$ जहाँ $t$ अधिकतम और न्यूनतम तापमानों का अनुपात है। (20 अंक) (b) नीचे चित्र 3(b) में दिखाये गये एक सौर संग्राहक, जिसकी विमायें 1 m चौड़ी तथा 5 m लम्बी हैं, में शीशे के आवरण तथा संग्राहक पट्टिका के बीच 3 cm का समान अन्तराल है। संग्राहक में, 30 °C पर 0·15 m³/s की दर से वायु 1 m चौड़े किनारे से प्रविष्ट होती है तथा 5 m लम्बे गलियारे में एक छोर से दूसरे तक प्रवाहित होती है। यदि शीशों के आवरण तथा संग्राहक पट्टिका के औसत तापमान क्रमशः: 20 °C तथा 60 °C हों, तो निर्धारित कीजिये (i) संग्राहक में, वायु में, कुल ऊष्मा संचरण दर और (ii) संग्राहक में प्रवाहित होने पर वायु की तापमान वृद्धि। 1 atm तथा अनुमानित औसत तापमान 35 °C पर वायु के गुणधर्म निम्न प्रकार लिये जा सकते हैं : ρ = 1·145 kg/m³, k = 0·02625 W/m-°C, ν = 1·655×10⁻⁵ m²/s, Cₚ = 1007 J/kg-°C, Pr = 0·7268 (20 अंक) (c) एक कार के एक हवारोधी शीशा, जिसकी विमायें 0·6 m ऊँची तथा 1·8 m लम्बी हैं, को वैद्युतीय रूप से गर्म किया जाता है तथा यह 1 atm, 0 °C तथा 80 km/hr की समानान्तर हवाओं के अधीन है। वैद्युत शक्ति की खपत 50 W देखी गई, जबकि हवारोधी शीशे की उजागर सतह का तापमान 4 °C है। अन्दर की सतह से होने वाले ऊष्मा अन्तरण और विकिरण की उपेक्षा करते हुए तथा संवेग ऊष्मा अन्तरण सादृश्य को प्रयोग में लेते हुए, हवारोधी शीशे पर हवा द्वारा लगाये जाने वाले विकर्ष बल को निर्धारित कीजिये। 1 atm तथा 0 °C पर वायु के गुणधर्म निम्न प्रकार से लिये जा सकते हैं : ρ = 1·292 kg/m³, Cₚ = 1·006 kJ/kg-K, Pr = 0·7362 (10 अंक)

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Directive word: Prove

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Approach

Prove the Brayton cycle efficiency relation in part (a) using calculus-based optimization; for (b) and (c), solve the numerical problems using appropriate heat transfer correlations and momentum-heat transfer analogy. Allocate approximately 40% time to part (a) due to its derivation-heavy 20 marks, 35% to part (b) for complex convection calculations, and 25% to part (c) for the Reynolds analogy application. Structure as: (a) theoretical derivation with T-s diagram, (b) step-wise Nusselt number calculation and energy balance, (c) Stanton number and drag coefficient linkage.

Key points expected

  • Part (a)(i): Specific work output increases with pressure ratio to a maximum then decreases; efficiency increases monotonically with pressure ratio for ideal Brayton cycle
  • Part (a)(ii): Derivation of η_wmax = 1 - 1/√t by differentiating net work w.r.t. pressure ratio, setting to zero, and substituting optimal pressure ratio r_p,opt = √t
  • Part (b): Hydraulic diameter calculation (D_h = 2b = 0.06 m), Reynolds number determination, Nusselt number using appropriate correlation (laminar/turbulent), heat transfer coefficient, and net heat transfer to air
  • Part (b): Temperature rise from energy balance Q = ṁC_pΔT, with mass flow rate from ρ and volumetric flow rate
  • Part (c): Application of Reynolds/Colburn analogy (St = Cf/2 × Pr^(-2/3)), calculation of Stanton number from heat transfer data, determination of friction coefficient and drag force
  • Part (c): Recognition that electrical power equals convective heat loss q = hA(T_s - T_∞) for establishing heat transfer coefficient

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Concept correctness20%10Correctly identifies Brayton cycle optimization condition (∂w/∂r_p = 0); recognizes part (b) as mixed convection in a channel and selects appropriate correlation; applies momentum-heat transfer analogy correctly in part (c) with proper Prandtl number exponent.Gets the final η_wmax formula correct but weak justification for optimization; uses Dittus-Boelter or similar correlation in (b) without checking flow regime; applies analogy but with incorrect Pr exponent.Confuses Brayton with Otto or Diesel cycle; treats (b) as external flow over flat plate; fails to recognize analogy applicability in (c).
Numerical accuracy20%10All calculations precise: Re ≈ 1845 (laminar) or appropriate turbulent value in (b), Nu ≈ 3.66 or 7.54 for fully developed flow, h ≈ 2.9 W/m²°C, Q ≈ 580 W, ΔT ≈ 3.8°C; part (c) St ≈ 0.00144, Cf ≈ 0.0039, Drag ≈ 0.45 N.Correct methodology but arithmetic slips (e.g., hydraulic diameter error as 0.03 m instead of 0.06 m); final answers within 15% of correct values.Order of magnitude errors (Re off by 10², missing factor of 2 in D_h); completely wrong final answers; unit conversion errors (kJ vs J in C_p).
Diagram quality15%7.5Clear T-s diagram for Brayton cycle with isobars, temperatures T_min, T_max labelled, and optimal pressure ratio indicated; schematic of solar collector showing flow direction and temperature profile; control volume for windshield with heat and momentum fluxes.T-s diagram present but missing isobars or temperature labels; collector schematic generic without dimension labels; no diagram for part (c).No diagrams; or incorrect cycle (p-v instead of T-s); diagrams without labels or values.
Step-by-step derivation25%12.5Complete derivation in (a): w_net = c_p[T_max(1-1/r_p^((γ-1)/γ)) - T_min(r_p^((γ-1)/γ)-1)], differentiation w.r.t. r_p, simplification to r_p,opt = (T_max/T_min)^(γ/2(γ-1)) = √t^(γ/(γ-1)), substitution back; in (b) explicit Nu correlation selection with justification; in (c) clear analogy statement St·Pr^(2/3) = Cf/2.Jumps from w_net expression to final result without showing differentiation; uses correlation without stating name or conditions; states analogy without derivation.Final formulas stated without derivation; no working shown for numerical parts; missing critical steps like hydraulic diameter or Reynolds number calculation.
Practical interpretation20%10Discusses why maximum work condition differs from maximum efficiency in gas turbines (ISRO, DRDO applications); comments on solar collector thermal efficiency and pressure drop trade-off; relates windshield drag to vehicle fuel economy at high speeds.Brief mention that maximum work and efficiency conditions are different; states that collector provides useful heat gain; notes drag affects vehicle performance.No interpretation; purely mathematical treatment; irrelevant discussion of unrelated applications.

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