Mechanical Engineering 2024 Paper II 50 marks Solve

Q3

(a) A shell and tube heat exchanger used in a thermal power plant is designed to condense 500 kg/s of saturated steam entering the condenser at 20 kPa to saturated water. Cooling water enters the heat exchanger at 35°C and leaves at 45°C while flowing through copper tubes of 50 mm diameter with negligible thickness. Overall heat transfer coefficient is estimated to be 1500 W/m²K. Find the following for the heat exchanger: (i) Total water flow rate required. (ii) Number of tubes required if water velocity = 1·0 m/s in the tube. (iii) Length of each tube. (iv) Total length of the tubes. Take the following property values: Cₚ of water = 4·2 kJ/kg.K Density of water = 1000 kg/m³ For Steam: T saturation = 60°C h_fg = 2600 kJ/kg (20 marks) (b) Explain in detail the differences between a centrifugal pump and a reciprocating pump. Explain the term slip with reference to reciprocating pump. Can slip be negative in a reciprocating pump? If yes, then when? (20 marks) (c) A steady-flow compressor is used to compress air from 1 atm, 25°C to 10 atm in an adiabatic process. The first-law efficiency for the process is 90%. Calculate the irreversibility for the process and the second-law efficiency. Take T₀ = 15°C. (10 marks)

हिंदी में प्रश्न पढ़ें

(a) एक कोश और नलिका उष्मा विनियामित्र जिसका उपयोग एक ऊष्मीय शक्ति संयंत्र में किया जाता है, उसका डिज़ाइन 20 kPa पर 500 kg/s संतृप्त भाप को जो संयंत्र में प्रवेश कर रही है, उसे संतृप्त जल में संघनित करने का है। शीतल जल उष्मा विनियामित्र की ताँबे की नलिकाओं से जो 50 mm व्यास तथा नगन्य मोटाई की है, 35°C पर प्रवेश कर 45°C पर बाहर निकलता है। संपूर्ण अनुमानित उष्मा अंतरण गुणांक 1500 W/m²K है। उष्मा विनियामित्र के लिए निम्नलिखित को ज्ञात कीजिए: (i) कुल आवश्यक जल प्रवाह दर। (ii) आवश्यक नलिकाओं की संख्या यदि नलिका में जल का वेग = 1·0 m/s हो। (iii) प्रत्येक नलिका की लंबाई। (iv) नलिकाओं की कुल लंबाई। निम्नलिखित गुणों के मान लीजिए: जल का Cₚ = 4·2 kJ/kg.K जल का घनत्व = 1000 kg/m³ भाप के लिए: Tसंतृप्त = 60°C h_fg = 2600 kJ/kg (20 अंक) (b) एक अपकेन्द्री पंप व एक प्रत्यागामी पंप में अंतरों को सविस्तार समझाइए। प्रत्यागामी पंप के संदर्भ में स्लिप पद को समझाइए। क्या स्लिप प्रत्यागामी पंप में ऋणात्मक हो सकती है? यदि हाँ, तो कब? (20 अंक) (c) एक अपरिवर्ती-प्रवाह संपीडक का प्रयोग, 1 atm, 25°C की वायु को एक रुद्धोष्म प्रक्रम में 10 atm तक संपीडित करने के लिए किया जाता है। प्रक्रम के लिए प्रथम नियम दक्षता 90% है। प्रक्रम के लिए अपरिवर्तनीयता एवं द्वितीय नियम दक्षता की गणना कीजिए। T₀ = 15°C लीजिए। (10 अंक)

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Approach

Solve the numerical parts systematically: for (a) apply energy balance to find cooling water flow, then use continuity and LMTD to determine tube geometry; for (b) construct a comparative table highlighting operating principles, flow characteristics, and applications of both pump types, then explain slip with negative slip conditions; for (c) apply first and second law analysis using isentropic efficiency and exergy destruction concepts. Allocate approximately 40% time to part (a) given its 20 marks and four sub-parts, 35% to part (b) for detailed explanation, and 25% to part (c) for thermodynamic calculations.

Key points expected

  • Part (a)(i): Energy balance Q = m_steam × h_fg = m_water × C_p × ΔT_water → m_water = 500 × 2600 / (4.2 × 10) = 30,952 kg/s
  • Part (a)(ii): Continuity equation m_water = ρ × A × v × N → N = m_water / (ρ × πd²/4 × v) = 30,952 / (1000 × π × 0.05²/4 × 1) ≈ 15,746 tubes
  • Part (a)(iii)-(iv): LMTD calculation for condenser (ΔT₁ = 60-35 = 25°C, ΔT₂ = 60-45 = 15°C), Q = U × A × LMTD → A = Q/(U×LMTD), then L = A/(N×πd)
  • Part (b): Centrifugal vs reciprocating comparison—continuous vs intermittent flow, high vs low discharge, smooth vs pulsating delivery, suitable for high vs low viscosity fluids; slip = (Q_theoretical - Q_actual)/Q_theoretical; negative slip occurs with air vessel or when delivery valve opens before suction valve closes
  • Part (c): Isentropic work from T₂s/T₁ = (P₂/P₁)^((γ-1)/γ), actual work = w_isen/0.90, actual T₂ from energy balance, irreversibility = T₀(s₂-s₁), second-law efficiency = w_isen/w_actual or exergy out/exergy in
  • Proper unit conversions throughout: kJ/kg to J/kg, kPa to Pa where needed, temperature in Kelvin for thermodynamic calculations
  • Physical interpretation: condenser design implications for NTPC plants, pump selection criteria for water supply vs high-pressure applications, compressor irreversibility sources

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Concept correctness22%11Correctly applies LMTD method for condenser (noting one fluid is at constant temperature); distinguishes between theoretical and actual discharge in slip definition; properly uses isentropic relations with variable specific heats or ideal gas assumption stated; identifies that negative slip requires air vessel or specific valve timing in reciprocating pumps.Uses LMTD correctly but may confuse arithmetic vs logarithmic mean; defines slip correctly but explanation of negative slip is vague; applies isentropic efficiency but may use constant specific heat without stating assumption.Uses arithmetic mean temperature difference instead of LMTD; confuses slip with cavitation or defines it incorrectly; applies isothermal relations for adiabatic compression; fundamental misunderstanding of pump operating principles.
Numerical accuracy24%12All numerical values correct: (i) m_water ≈ 30,952 kg/s, (ii) N ≈ 15,746 tubes, (iii) L ≈ 4.37 m per tube, (iv) total length ≈ 68,800 m; part (c) irreversibility and second-law efficiency calculated with proper T₀ = 288 K; consistent significant figures; intermediate values shown.Correct methodology but minor arithmetic errors (e.g., LMTD calculation off by 1-2°C, tube count error due to rounding, final answers within 5-10% of correct value); part (c) correct approach but may use T₀ = 15°C directly in Celsius.Major calculation errors (order of magnitude wrong for tube count or water flow rate); incorrect formula application (e.g., using h_fg for water, using C_p for steam); part (c) completely wrong efficiency interpretation or irreversibility calculation.
Diagram quality14%7Clear temperature profile diagram for part (a) showing steam at 60°C constant, water heating from 35°C to 45°C with parallel/counterflow indication; schematic of shell-and-tube heat exchanger with proper labelling; T-s diagram for part (c) compression showing actual vs isentropic paths, irreversibility represented as area or exergy destruction; pump comparison diagrams if relevant.Basic temperature profile sketched but lacks proper labelling or scale; schematic of heat exchanger present but missing key components (baffles, tube sheet); part (c) has T-s diagram but may not clearly distinguish actual vs isentropic processes.No diagrams provided where clearly needed; or diagrams drawn without any relevance to the problem (e.g., wrong thermodynamic cycle, incorrect pump types illustrated).
Step-by-step derivation22%11Explicit energy balance equation stated for part (a)(i); continuity equation with substitution shown for (a)(ii); LMTD formula derived or stated with substitution for (a)(iii)-(iv); for part (b), structured comparison with clear headings; for part (c), first law applied to find actual exit state, then entropy generation calculated, then irreversibility and second-law efficiency derived systematically.Key equations stated but some substitution steps skipped; final answers correct but working condensed; part (c) may jump from given efficiency to final answer without showing entropy calculation.No derivation shown—only final answers stated; or incorrect formulae used throughout with no logical flow between steps; part (c) completely omits irreversibility calculation steps.
Practical interpretation18%9Interprets condenser tube count and length in context of NTPC thermal plant design constraints; discusses why such large cooling water flow requires coastal or riverine location; explains when centrifugal pumps are preferred (municipal water supply, irrigation) vs reciprocating (high pressure, metering); identifies sources of irreversibility in real compressors (friction, heat transfer, non-ideal gas behavior) and implications for energy efficiency in process industries.Brief mention of practical applications without specific industry context; states that large tube count implies large heat exchanger but no discussion of implications; identifies that reciprocating pumps suit high pressure but without clear examples.No practical interpretation provided; treats all parts as purely academic exercises with no connection to engineering practice or Indian power sector context.

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