Mechanical Engineering UPSC PYQ 2024

(a) Air flows through a 5 cm diameter pipe with inlet velocity 70 m/s, temperature 80 °C and pressure 1 MPa. For a pipe length of 25 m, assuming adiabatic flow with mean friction factor 0·005, determine the exit temperature, pressure and Mach number using the attached Fanno table. (b) Saturated liquid refrigerant at –7 °C flows through a horizontal copper tube (inside diameter 25 mm, thickness 2·5 mm, length 10 m) exposed to air at 20 °C. For a mass flow rate of 0·0012 kg/s and latent heat of evaporation 400 kJ/kg, find the exit dryness fraction. Property values of air at 280 K: ρ = 1·271 kg/m³, k = 0·0246 W/mK, ν = 1·4 × 10⁻⁵ m²/s, Pr = 0·717. Use the correlation Nu_f = (0·48)[Gr·Pr]^0·25 and neglect temperature drop in the tube wall and tube thermal resistance. (c) (i) Explain how Stefan–Boltzmann law is obtained from Planck’s law. Compute the total emissive power of a black sphere of 5 cm diameter maintained at 500 K, taking σ = 5·67 × 10⁻⁸ W/m²K⁴. Derive Stefan–Boltzmann law from Planck’s law and find total emissive power of a 5 cm black sphere at 500 K. [5M] <!--qid:MAINS_2024_Mechanical_Engineering-II_Q2-->

(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. (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. (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. (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 the angle between inlet velocity V1 and blade velocity u. (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 Tw °C (T ≠ Tw). 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 the three fluids for better clarity.)2a:["$","s <!--qid:MAINS_2024_Mechanical_Engineering-II_Q1-->