UPSC Mechanical Engineering 2023 — Paper II

(a) A 50 kg block of iron at 500 K is placed into open atmosphere which is at a temperature of 285 K. The iron block eventually reaches thermal equilibrium with the atmosphere. Assuming an average specific heat of 0·45 kJ/kg-K for iron, determine the (i) entropy change for the iron block and the atmosphere, and (ii) irreversibility. (10 marks) (b) Show that for normal shock in a perfect gas, M*ₓ M*ᵧ = 1. (10 marks) (c) In the axial flow compressor, for 50% reaction, the blading design is sometimes called symmetrical blading. Explain, with proper equations and justification, why it is called so. (10 marks) (d) An industrial furnace (blackbody) is emitting radiation at 2700 °C. Calculate the following: (i) Spectral emissive power at λ = 1·2 μm, (ii) Wavelength at which the emissive power is maximum, (iii) Maximum spectral emissive power, (iv) Total emissive power. Use Planck's distribution law equation given below: E_bλ = (C₁)/(λ⁵ [exp(C₂/λ T)-1]) where C₁ = 3.742 × 10⁸ W-μm⁴/m², C₂ = 1.438 × 10⁴ μm-K. Take σ = 5.67 × 10⁻⁸ W/m²-K⁴. (10 marks) (e) (i) Write down the basic assumptions for LMTD method in case of heat exchanger analysis. (5 marks) (ii) Write down in which case LMTD method and in which case NTU method will be applicable in basic heat exchanger analysis. (5 marks)

(a) A 15 m high cylinder with a cross-sectional area of 0.6 m² contains 3 m³ of liquid water at 25 °C on the top of a thin insulated piston of mass 20 kg. Below the piston, argon gas is at 15 °C with a volume of 3 m³, as shown in the figure. Heat is supplied to argon such that the piston rises and pushes the water out over the top edge. Find the (i) work done (kJ) to remove the whole water from the top of the piston and (ii) heat transferred (kJ) to argon during the process. (iii) Plot the process on P-v diagram for argon. Assume atmospheric pressure (P₀) as 101 kPa, Cᵥ and R for argon as 0·312 kJ/kg-K and 0·2081 kJ/kg-K respectively. The specific volume of water at 25 °C is 0·001003 m³/kg. Neglect piston thickness. (20 marks) (b) Air at 100 kPa and 290 K enters a gas turbine cycle with two stages of compression and two stages of expansion. This system uses ideal regenerator, reheater and intercooler. The pressure ratio across each stage is 4. 300 kJ/kg of heat is added in combustion chamber and reheater each. The regenerator increases the air temperature by 20 °C. Draw T-s plot and determine the (i) total heat rejected (kJ/kg), (ii) net work output (kJ/kg) and (iii) thermal efficiency of the system. Assume isentropic operation for all compressors and turbines. Take Cₚ of air = 1·005 kJ/kg-K and γ = 1·4. (20 marks) (c) A convergent-divergent nozzle has a throat area of 250 mm² and an exit area of 500 mm². Air enters the nozzle with a stagnation temperature of 350 K and stagnation pressure of 1 MPa. Determine the maximum flow rate of air through the nozzle and the static pressure, static temperature, Mach number and velocity at the exit from the nozzle. Given γ = 1·4, R = 0·287 kJ/kg-K. Use Gas Table to solve the problem. (10 marks)

(a) Heat is generated in a stainless steel plate (thermal conductivity = 22 W/m-K) of thickness 1 cm, at a uniform rate of 600 MW/m³. The left side of the plate is maintained at 200 °C and the right side is maintained at 100 °C. What will be the (i) temperature distribution across the plate, (ii) location and value of maximum temperature and (iii) heat flux from both sides of the plate and its direction? Assume one-dimensional, steady-state heat conduction. (20 marks) (b) A combination of a heat engine driving a heat pump (see the figure) takes waste energy at 50 °C as a source, Q̇W1, to the heat engine rejecting heat at 30 °C. The remainder, Q̇W2, goes into the heat pump that delivers Q̇H at 150 °C. If the total waste energy is 5 MW, find the rate of energy delivered at the higher temperature. Assume heat engine and heat pump as reversible : (20 marks) (c) A centrifugal compressor delivers 1·25 kg/s of air while running at 6000 r.p.m. The diameters at the inlet and outlet are 0·5 m and 1 m respectively. The power input factor is 1·04, while the slip factor is unity. The power consumed by the compressor is 50 kW. State the type of impeller used, whether forward, radial or backward curved. Draw velocity triangles. Assume no prewhirl at the inlet. (10 marks)

(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)

(a) Show in the form of a table, how the increase in the following variables affects (increase or decrease) the ignition delay period of a compression ignition (CI) engine: (i) Self-ignition temperature, (ii) Cetane number, (iii) Compression ratio, (iv) Intake pressure, (v) Intake temperature, (vi) Air-fuel ratio, (vii) Exhaust gas recirculation. (10 marks) (b) What are the desirable characteristics of an ideal working fluid for vapour power cycle? (10 marks) (c) What is reheat factor of a steam turbine? Derive an expression to show that the reheat factor is always greater than unity. (10 marks) (d) Compare vapour compression and vapour absorption refrigeration systems. (10 marks) (e) Without using psychrometric chart, calculate (i) relative humidity, (ii) humidity ratio, (iii) dew point temperature and (iv) enthalpy of moist air, when DBT is 35 °C and WBT is 23 °C. The barometer reads 755 mm of Hg. Use modified Apjohn equation (take values of pressure in bar): p_v = p'_v - (1.8p(t-t'))/2700 where, p_v = partial pressure of water vapour (w.v.) corresponding to DPT, p'_v = partial pressure of w.v. corresponding to WBT, t = DBT, t' = WBT, p = barometric pressure. Use the properties of water vapour given below: t (°C) | Vapour pressure (bar) ---|--- 10 | 0·012272, 12 | 0·014017, 14 | 0·015977, 16 | 0·018173, 18 | 0·020630, 20 | 0·023373, 22 | 0·026431, 24 | 0·029832, 32 | 0·047552, 34 | 0·053201, 36 | 0·059423. (10 marks)

(a) The power output of a six-cylinder, four-stroke CI engine is absorbed by a hydraulic dynamometer for which the law is P = WN/20000, where P is the power in kW, W is the brake load in newton and N is the engine speed in r.p.m. The following observations are made during a test on the engine: Bore = 100 mm; Stroke = 110 mm; Brake load = 540 N; Engine speed = 2500 r.p.m.; C/H ratio of the fuel (by mass) = 83/17; Ambient pressure = 1·0 bar; Ambient temperature = 27 °C; Time taken for 100 cc of fuel consumption = 18 s; Fuel density = 780 kg/m³; Calorific value of the fuel = 45 MJ/kg; Mass flow rate of atmospheric air consumed by the engine = 5·301126 kg/min. Calculate the bmep, bsfc, brake thermal efficiency, volumetric efficiency and the percentage of excess air. Given, R_air = 0·287 kJ/kg-K. (20 marks) (b) (i) Draw typical velocity triangles of a stage of a reaction turbine, clearly showing the various velocities. (ii) Derive an expression to show that the optimum value of ρ, the blade-to-steam speed ratio for a Parsons reaction turbine is given by ρ = cos α, where α is the inlet angle of the fixed blades. (iii) Also, show that the maximum efficiency of the Parsons reaction turbine is given by η_maximum = (2 cos² α)/(1 + cos² α). (iv) Draw the velocity triangles of the Parsons reaction turbine operating at ρ_optimum. (20 marks) (c) Explain briefly the various methods of air-conditioning duct design. (10 marks)

(a) A convergent-divergent nozzle receives steam at 5 bar, 250 °C and expands it isentropically into a space at 1 bar. Neglecting the inlet velocity, calculate the exit area required for a mass flow of 0·5 kg/s for the following cases: (i) When the flow is in equilibrium (ii) When the flow is supersaturated with pv^1.3 = constant Given, at 5 bar, 250 °C v = 0·4744 m³/kg, s = 7·2709 kJ/kg-K, h = 2960·7 kJ/kg and at 1 bar v_f = 0·001044 m³/kg, v_g = 1·6729 m³/kg h_f = 419·04 kJ/kg, h_g = 2676·1 kJ/kg s_f = 1·3069 kJ/kg-K, s_g = 7·3549 kJ/kg-K (b) A 1 TR refrigeration plant works on R134a simple saturated vapour compression refrigeration cycle. The evaporator and condenser temperatures are –10 °C and 44 °C respectively. Determine the (i) mass flow rate of the refrigerant, (ii) compressor power, (iii) volumetric cooling capacity and (iv) COP. Also, calculate the (v) increase in the specific compressor work due to superheat horn and (vi) throttling loss, in comparison to reversed Carnot cycle operating between the same temperature limits. Consider the entry to compressor as saturated vapour and saturated liquid refrigerant is leaving the condenser in the reversed Carnot cycle. The properties of R134a are given in the table: Take specific heat of vapour refrigerant as 1·26 kJ/kg-K. (c) Explain how the following characteristics of the lubricating oils affect the operation of an internal combustion (IC) engine: (i) Viscosity (ii) Viscosity index (iii) Pour point (iv) Flash point and fire point

(a) An SI engine working on the Otto cycle has cylinder bore of 210 mm and stroke length of 240 mm. The clearance volume is 1550 cc. The pressure and temperature at the beginning of compression are 1 bar and 17 °C respectively. The maximum pressure of the cycle is 50 bar. Determine the pressure and temperature at the salient points in the cycle, the air-standard efficiency, the work done and the mean effective pressure. Show the cycle on P-v and T-s diagrams. Evaluate the fuel consumption in kg/kWh, if the calorific value of the fuel is 40 MJ/kg. Take Cp and Cv of air as 1·005 kJ/kg-K and 0·718 kJ/kg-K respectively. (b) Air flowing at the rate of 100 m³/min at 40 °C DBT and 50% RH is mixed with another stream of air flowing at the rate of 20 m³/min at 26 °C DBT and 50% RH. The mixture flows over a cooling coil whose ADP temperature is 10 °C and bypass factor is 0·2. Find the DBT and RH of air leaving the coil. If this air is supplied to an air-conditioned room, where DBT of 26 °C and RH of 50% are maintained, then calculate the (i) room sensible heat factor and (ii) coil cooling capacity in tons of refrigeration. Draw a schematic diagram of the system and show all the processes on a skeleton psychrometric chart. Psychrometric chart is given. (c) A process industry employs a medium pressure boiler to produce steam. The mass flow rate of fuel consumed is 0·847 kg/s and c.v. of the fuel is 44 MJ/kg. For efficient combustion, 16 kg of air per kg of fuel is required, for which a draught of 30 mm of the water column is required at the base of the chimney. The flue gases leave the boiler at 350 °C. The average temperature of gases in the stack may be taken as 300 °C. The atmosphere is at 20 °C. Assuming the velocity of gases at the stack exit to be negligible, determine the height of the stack and the diameter at its base. Also, calculate the mass flow rate of the gases. Take Patmosphere = 101·3 kPa, Rair = Rgases = 0·287 kJ/kg-K, g = 9·81 m/s², ρwater = 1000 kg/m³.