(a) What is heat treatment ? What is its purpose ? Explain different heat treatment processes with the help of common…

(a) Heat treatment: purposes and processes on the Fe–Fe₃C diagram (10 marks)

Definition. Heat treatment is the controlled heating of a solid metal to a set temperature, holding it, and cooling it at a set rate, to alter its microstructure and hence its properties without changing its shape.

Purposes. To relieve internal stress, refine grains, soften for machining or cold work, increase hardness and strength, improve toughness and ductility, improve wear resistance of the surface, and homogenise the structure.

Processes. On the Fe–Fe₃C diagram the eutectoid is at 0.8% C and 727 °C (A₁), and A₃ and A_cm are the upper critical lines. Below A₁ steel is ferrite (α, BCC) with cementite (Fe₃C), and above it austenite (γ, FCC).

• Annealing (full): a hypoeutectoid steel is heated 30–50 °C above A₃, held, and furnace cooled. The result is coarse pearlite with ferrite, which is soft and ductile. Spheroidising (just below A₁ for long) gives cementite spheroids in ferrite in high-carbon steel, and process annealing (550–650 °C) recovers cold-worked steel.
• Normalising: heated 30–50 °C above A₃ (A_cm for hypereutectoid) and cooled in still air. Faster cooling gives fine pearlite with ferrite, which is harder and stronger than annealed steel and has a refined, uniform grain.
• Hardening: austenitised at the same temperature, then quenched in water or oil faster than the critical cooling rate. Austenite transforms to martensite, a hard, brittle body-centred tetragonal supersaturated solid solution, with some retained austenite.
• Tempering: quenched steel is reheated below A₁ (150–650 °C), which lets carbon escape as fine carbide. Tempered martensite (ferrite plus cementite) is tough, and higher temperature gives lower hardness and higher toughness.
• Austempering: quenched to 250–400 °C in a salt bath and held until austenite becomes bainite, giving good toughness and little distortion. Martempering quenches to just above the M_s temperature, equalises, and cools in air to form martensite with less cracking.
• Case hardening: carburising (0.8–1.2% C in austenite at 900–950 °C, then quench) gives a hard martensitic case on a tough low-carbon core. Nitriding (500–550 °C, in the ferrite region) forms hard nitrides without quenching, and induction or flame hardening gives a hard skin locally.

(b) Torque transmitted by a thin steel tube by three failure theories (20 marks)

Data. d_m = 100 mm, t = 3 mm, σ_y = 225 MN/m², ν = 0.3, factor of safety 2.25, so the permissible simple tensile stress is σ_y/2.25 = 100 MN/m².

Torsion of a thin tube. The mean enclosed area is A_m = π d_m²/4 = 7853.98 mm². The shear stress is τ = T/(2 A_m t) = T/(2 × 7853.98 × 3) = T/47 123.9 (T in N·mm, τ in N/mm²). Pure shear gives principal stresses σ₁ = +τ, σ₂ = −τ and σ₃ = 0.

(i) Maximum shear stress theory. The maximum shear stress is (σ₁ − σ₂)/2 = τ, and the permissible value is σ_y/(2 × FS) = 50 N/mm². So τ = 50 N/mm², and

T = 50 × 47 123.9 = 2.356 × 10⁶ N·mm = 2.36 kN·m.

(ii) Maximum strain energy theory (Haigh). The strain energy per unit volume is proportional to σ₁² + σ₂² − 2ν σ₁σ₂ and must not exceed its value at the permissible stress, (σ_y/FS)²:

τ² + τ² + 2(0.3) τ² = 2.6 τ² = 100²

τ = 100/√2.6 = 62.02 N/mm², and T = 62.02 × 47 123.9 = 2.923 × 10⁶ N·mm = 2.92 kN·m.

(iii) Maximum shear strain energy (distortion energy) theory. The condition is σ₁² + σ₂² − σ₁σ₂ = (σ_y/FS)², so 3τ² = 100²:

τ = 100/√3 = 57.74 N/mm², and T = 57.74 × 47 123.9 = 2.721 × 10⁶ N·mm = 2.72 kN·m.

Comparison. The order is Tresca (2.36) < von Mises (2.72) < Haigh (2.92 kN·m). The maximum shear stress theory is the most conservative, and it is the usual choice for ductile steel in design.

(c) Flywheel of a two-wheeler engine (20 marks)

(i) Maximum, minimum and mean torque. T = 8000 + 1000 sin 2θ − 2000 cos 2θ can be written as 8000 + R sin(2θ − α), where R = √(1000² + 2000²) = 2236.07 N·m and α = tan⁻¹(2000/1000) = 63.43°.

• T_max = 8000 + 2236.07 = 10 236 N·m, at 2θ − α = 90°, i.e. θ = 76.7°.
• T_min = 8000 − 2236.07 = 5764 N·m, at 2θ − α = 270°, i.e. θ = 166.7°.
• T_mean = 8000 N·m, the constant term, because the sine term averages to zero over the 180° cycle (its 2θ period is 360°).

The curve is a sine wave about the mean line, which is the constant resistance torque. The torque exceeds the resistance (energy accumulates) between θ = 31.7° and 121.7°, and is below it for the rest of the cycle.

(ii) Power developed. ω = 2π × 300/60 = 31.416 rad/s.

P = T_mean ω = 8000 × 31.416 = 251 327 W ≈ 251.3 kW.

(iii) Total percentage fluctuation of speed. The maximum fluctuation of energy is the area of the torque curve above the mean line, between the two crossings where 2θ − α = 0 and 180°:

ΔE = ∫ R sin(2θ − α) dθ = (R/2)[−cos(2θ − α)] from 0 to π = R = 2236.07 J

The moment of inertia is I = m k² = 500 × 0.75² = 281.25 kg·m². Using ΔE = I ω² C_s:

C_s = ΔE/(I ω²) = 2236.07/(281.25 × 31.416²) = 2236.07/277 583 = 0.008055

The total fluctuation of speed is 0.806%.

(iv) With k = 0.60 m. I = 500 × 0.6² = 180 kg·m², and ΔE is unchanged.

C_s = 2236.07/(180 × 986.96) = 0.012587, that is 1.259%.

Yes, the fluctuation increases, by 1.259 − 0.806 = 0.453 percentage points. This is a rise of 56.25% of the original value, since C_s varies as 1/k² and (0.75/0.60)² = 1.5625.