Mechanical Engineering 2023 Paper II 50 marks Derive

Q4

(a) Considering an ideal, isentropic gas flow through a nozzle, show that choking will occur at Mach number (M) = 1. (20 marks) (b) Water at 30 °C enters a 1·5 cm diameter horizontal tube with a velocity of 1 m/s. The tube wall is maintained at a constant temperature of 90 °C. Calculate the length of the tube if the exit water temperature is 65 °C. One may assume that the flow is turbulent, fully developed and the internal surface of the tube is smooth. The properties of water are given: Thermal conductivity (k) = 0·656 W/m-K Density (ρ) = 984·4 kg/m³ Kinematic viscosity (ν) = 0·497×10⁻⁶ m²/s Specific heat (Cp) = 4178 J/kg-K Prandtl number (Pr) = 3·12 Friction factor (F) = 0·079 (Reynolds number)⁻⁰·²⁵ Reynolds number ≡ ReD Average Nusselt number (NuD) = [(F/2)(ReD - 1000)Pr] / [1 + 12·7(F/2)¹/²(Pr²/³ - 1)] Also, calculate the water temperature at the middle of the tube and pressure drop across the tube. (20 marks) (c) An evacuated 150 L tank is connected to a line flowing air (constant specific heat) at room temperature 25 °C and 8 MPa pressure. The valve is opened, allowing air to flow into the tank until the pressure inside is 6 MPa. At this point, the valve is closed. The filling process occurs rapidly and is essentially adiabatic. The tank is then placed in storage, where it eventually returns to room temperature. What is the final pressure inside the tank? (10 marks)

हिंदी में प्रश्न पढ़ें

(a) एक नोजल के माध्यम से होने वाले एक आदर्श, समएन्ट्रॉपी गैस प्रवाह पर विचार करते हुए दर्शाइए कि मैक संख्या (M) = 1 पर प्रोधन होगा। (20 अंक) (b) 30 °C पर पानी 1 m/s के वेग से 1·5 cm व्यास वाली एक क्षैतिज नलिका में प्रवेश करता है। नलिका की दीवार को 90 °C के स्थिर तापमान पर बनाए रखा जाता है। यदि निकास पानी का तापमान 65 °C है, तो नलिका की लम्बाई की गणना कीजिए। कोई यह मान सकता है कि प्रवाह विषमुख है, पूरी तरह से विकसित है और नलिका का आंतरिक पृष्ठ चिकना है। पानी के गुणधर्म दिए गए हैं: ऊष्मा चालकता (k) = 0·656 W/m-K घनत्व (ρ) = 984·4 kg/m³ श्यानता (ν) = 0·497×10⁻⁶ m²/s विशिष्ट ऊष्मा (Cₚ) = 4178 J/kg-K प्रांडल संख्या (Pr) = 3·12 घर्षण गुणक (F) = 0·079 (रेनॉल्ड्स संख्या)⁻⁰·²⁵ रेनॉल्ड्स संख्या ≡ Reᴅ औसत नुसेल्ट संख्या (Nuᴅ) = [(F/2)(Reᴅ - 1000)Pr] / [1 + 12·7(F/2)¹/²(Pr²/³ - 1)] इसके अलावा, नलिका के मध्य में पानी के तापमान और नलिका में दाब-पात की गणना कीजिए। (20 अंक) (c) एक खाली की गई 150 L की टंकी कमरे के तापमान 25 °C और 8 MPa दाब पर बहने वाली वायु (स्थिर विशिष्ट ऊष्मा) की एक लाइन से जुड़ी है। वाल्व खोला जाता है, जिससे वायु को टंकी में तब तक प्रवाहित होने दिया जाता है, जब तक कि अंदर का दाब 6 MPa न हो जाए। इस क्षण पर वाल्व बंद किया जाता है। यह भरने की प्रक्रिया तेजी से होती है और मूलतः रूद्धोष्म है। फिर टंकी को भंडारण में रखा जाता है, जहाँ वह अंततः कमरे के तापमान पर वापस आ जाती है। टंकी के अंदर अंतिम दाब क्या है? (10 अंक)

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Directive word: Derive

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Approach

Derive the choking condition in part (a) by starting from isentropic relations and showing that mass flow rate maximizes at M=1. For part (b), calculate Reynolds number first, then apply the given Nusselt number correlation to find heat transfer coefficient, use the LMTD or energy balance to find tube length, then compute midpoint temperature and pressure drop. For part (c), apply unsteady filling analysis with adiabatic assumption first, then isochoric cooling to find final pressure. Allocate approximately 40% time to (a), 40% to (b), and 20% to (c) based on marks distribution.

Key points expected

  • Part (a): Derivation showing d(m_dot)/dM = 0 leads to M=1; area-Mach relation A/A* = (1/M)[(2/(γ+1))(1+(γ-1)M²/2)]^((γ+1)/(2(γ-1)))
  • Part (a): Physical explanation that sonic throat is the minimum area where upstream information cannot propagate
  • Part (b): Re_D = VD/ν = 1×0.015/(0.497×10⁻⁶) ≈ 30181 (turbulent, valid for correlation)
  • Part (b): Application of Gnielinski correlation (given) to find h, then energy balance Q = m_dot×Cp×(T_out-T_in) = h×A_s×LMTD to solve for length L
  • Part (b): Midpoint temperature found from exponential temperature profile or iterative energy balance; pressure drop from ΔP = f(L/D)(ρV²/2)
  • Part (c): Adiabatic filling gives T_tank = γT_line for ideal gas with constant specific heats; final pressure after cooling p_f = p_tank×(T_room/T_tank)
  • Part (c): Final pressure calculation yielding approximately 4.5 MPa (or exact value based on γ=1.4)

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Concept correctness20%10Correctly identifies that choking occurs when Mach number reaches 1 at the throat where dA/dM changes sign; recognizes that part (b) requires simultaneous solution of energy balance with the given Nusselt correlation; understands that part (c) involves two distinct thermodynamic processes (adiabatic filling followed by isochoric cooling).Identifies M=1 for choking but explanation lacks rigor; applies correlations correctly in (b) but may confuse LMTD with arithmetic mean; recognizes two stages in (c) but makes errors in temperature/pressure relations.Confuses choking with normal shock or other phenomena; applies Dittus-Boelter instead of given correlation in (b); treats filling as isothermal or confuses mass and volume in (c).
Numerical accuracy20%10Re_D = 30181, f = 0.0079, Nu_D ≈ 180-190, h ≈ 8000 W/m²K, L ≈ 2.5-3 m, midpoint T ≈ 48-52°C, ΔP ≈ 15-20 kPa; part (c) final pressure ≈ 4.5 MPa with correct intermediate steps.Correct order of magnitude for all quantities but 10-20% error in one or two final values due to arithmetic slips; correct methodology but calculator errors.Order-of-magnitude errors (e.g., L in cm instead of m, h in W instead of kW); wrong correlation used leading to Nu off by factor of 10; final pressure in (c) exceeding supply pressure or below atmospheric.
Diagram quality15%7.5Clear T-s diagram for part (a) showing isentropic expansion with stagnation and sonic states labelled; schematic of tube flow in (b) with inlet/outlet conditions and thermal boundary layer indicated; p-V or T-s diagram for part (c) showing adiabatic filling path and isochoric cooling.One relevant diagram present (typically for part a) with correct qualitative features but missing labels or scales; other parts described textually only.No diagrams despite the question's conceptual nature; or incorrect diagrams (e.g., showing shock in converging nozzle, or confusing filling process with steady flow).
Step-by-step derivation25%12.5Part (a): Full derivation from isentropic relations, mass flow rate expression, differentiation to show maximum at M=1; Part (b): Sequential calculation of Re, f, Nu, h, then energy balance with LMTD derivation, explicit solve for L; Part (c): Control volume analysis for unsteady filling with uniform state assumption, derivation of T_tank = γT_line/(1+(γ-1)M_i/2M_t) simplified for evacuated initial state.Key steps shown but some shortcuts taken (e.g., stating d(m_dot)/dM=0 without full expansion); energy balance correct but LMTD formula stated without temperature substitution shown; part (c) uses correct final formula but derivation abbreviated.Final answers stated without derivation; or fundamental errors in derivation (e.g., treating unsteady filling as steady flow, or using Bernoulli for compressible flow in part a).
Practical interpretation20%10Discusses implications: part (a) explains why rocket nozzles use converging-diverging design (ISRO PSLV/Vikram engines); part (b) comments on whether computed length is practical, effect of fouling on smooth tube assumption, and whether water property variation is significant; part (c) relates to compressed air energy storage systems and safety considerations for high-pressure tanks.Brief mention of practical relevance for one part (typically nozzle applications); or generic statements about heat exchanger design without specific connection to computed values.No physical interpretation; purely mathematical treatment; or incorrect physical statements (e.g., claiming choking can be eliminated by higher pressure ratio).

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