Mechanical Engineering — UPSC Mains Optional PYQ (2023–2025)

(a) A low carbon steel stock of thickness 25 mm is to be rolled in two stages. In the first stage, the reduction is to be from 25 mm to 15 mm and in the second stage from 15 mm to 5 mm. Determine the minimum diameter of the rolls for the two stages if the maximum angle of bite is 35° for the first stage and 25° for the second stage. Also, calculate the required coefficient of friction in both the stages. (b) In an arc welding process of steel with a potential of 15 V, current of 150 A and travel speed of 5 mm s⁻¹, the cross-sectional area of joint is 15 mm². If heat required to melt the steel is 10 J mm⁻³ and heat-transfer efficiency is 0·75, calculate the melting efficiency. (c) In an automobile manufacturing industry the demand for a specific part was 250 in April, 100 in May and 200 in June. The forecast for April was 150. Calculate the forecast for the month of July with a smoothing constant of 0·25 using first-order exponential smoothing. (d) Five different products are manufactured in a mixed-model production line. The time required for each task (in seconds) is : Product-1 Product-2 Product-3 Product-4 Product-5 Task A 5 5 – 5 5 B 6 6 6 – 6 C 6 – 6 6 – D 7 7 – – 7 E – 4 4 4 – Each product requires a set of tasks. Calculate the total number of work-stations for this mixed-model assembly line if the cycle time is 15 s. (e) For the given dimensions of mated parts, determine the allowance, hole tolerance and shaft tolerance using the basic hole system. Hole : 57·50 mm, 57·52 mm Shaft : 57·47 mm, 57·45 mm

(a-i) If the power source characteristic in a Metal Inert Arc welding process is Vp = 38 – I⁄60 and the arc characteristic is Va = 3La + 27, calculate the change in arc power when arc length changes from 1 mm to 3 mm. Also find the maximum arc length sustainable when source current limit is 300 A. (a-ii) An EDM process cuts a 6 mm deep cavity in high-carbon steel using (I) copper–tungsten and (II) copper electrodes. Taking wear ratios 9 : 1 and 3 : 1 respectively, determine the required spindle movement. (b-i) In ECM a 15 mm diameter, 125 mm deep hole is to be drilled in high-carbon steel. With 45 A current and 15 % NaCl electrolyte, find the machining time. Take valency of Fe = 2, atomic weight 56, density of steel 7·8 g cm⁻³. (b-ii) Using ECM to machine Nimonic-75 alloy (composition table given) and the lowest valency of dissolution for each element (data table given), compute the material-removal rate when current is 1050 A. (c) With a line diagram explain the different flow patterns used in plant layouts. State the conditions that must be satisfied by an ideal flow pattern.

(a) In an orthogonal machining process of medium-carbon steel using an HSS tool with 9° rake angle, the following were observed: feed 0·25 mm rev⁻¹, cutting speed 250 m min⁻¹, depth of cut 1·5 mm, chip-thickness ratio 0·30, vertical cutting force 1100 N, horizontal cutting force 600 N. Calculate (i) shear force along shear plane, (ii) normal force on shear plane, (iii) friction force along rake surface, (iv) normal force along rake surface, (v) friction angle, (vi) work done in shear, (vii) work done in friction. (b) Annual demand for an item is 2400 units, unit cost ₹ 6. A 5 % discount is offered for orders of 500 units or more. Ordering cost is ₹ 32 per order and inventory carrying cost is 16 %. Determine whether accepting the discount is economical.

(a) What are the various computer languages used for NC machines? Explain the merits and limitations of each language. (b) A small manufacturing unit produces bolts and the weight of each bolt is measured. The target weight for a bolt is 60 g. A sample of 5 bolts is taken every day for 30 days. The weight (in grams) for the first 5 days is given below: Day Sample 1 Sample 2 Sample 3 Sample 4 Sample 5 1 59·8 60·2 60·0 60·1 59·9 2 60·1 60·3 60·2 60·0 59·8 3 59·7 60·0 60·1 59·9 60·2 4 60·0 59·8 60·2 60·1 59·9 5 60·2 60·3 60·1 60·0 59·9 Calculate the control limits for the X̄ (X-bar) chart. The constant A₂ for a sample size of 5 is 0·577. (c) A certain component can be manufactured either by welding or by forging process. The factory has an order for 5,00,000 units. The costs involved for both methods of manufacturing are as follows: Welding Forging Fixed cost ₹ 15,000 ₹ 94,000 Variable cost per unit ₹ 5 ₹ 4·25 (i) Which is the most economical method of manufacturing the component? Draw the indicative graph also for BEP. (ii) What will be the loss if a wrong choice is made?

(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.)

(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.

A medium carbon steel cylindrical rod is being machined under orthogonal cutting condition with an HSS cutting tool having rake angle as 12°. While machining, following data were recorded: Vertical component of cutting force = 1600 N Horizontal component of cutting force = 1250 N Chip thickness ratio = 0·25 Calculate the following for the above-mentioned machining condition: (i) Normal force on the rake face (ii) Friction force along the rake face (iii) Resultant cutting force (iv) Coefficient of friction at chip–tool interface (v) Normal force on the shear plane (vi) Shear force along the shear plane

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. Show that for normal shock in a perfect gas, M′x M′y = 1. 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. 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 : Eλb = C1 / [λ5 (exp(C2 / λT) – 1)] where C1 = 3·742 × 108 W · µm4 / m2, C2 = 1·438 × 104 µm-K Take σ = 5·67 × 10-8 W / m2 · K4. Write down three basic assumptions for LMTD method in case of heat exchanger analysis. Write down in which case LMTD method and in which case NTU method will be applicable in basic heat exchanger analysis.

A manufacturing company wants to arrange work-centres A, B, C and D so as to minimize inter-departmental parts handling costs. The flow of parts and existing work-centres layout are shown below: A B C D A – 400 500 50 B 300 – 200 0 C 0 0 – 700 D 0 0 0 – (Parts moved between work-centres) Existing layout with distances: A B C D (30 m between successive centres) Suggest a modified layout.

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 a 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. 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 a 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. 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.

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. 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.