GATE 2021 BT – Question 62
Milk in a stainless-steel inner tube (inner diameter 40 mm) is heated from 10°C to 85°C by steam condensing at 120°C. Heat transfer is 146200 kcal/h and U=750 kcal/(h m² °C). Total tube length (one decimal place m) is _______.
Practise this question in The GATE Grind →
Show answer and explanation
Correct answer: 23.58 to 23.82
Explanation
**Heat-transfer equation:** $Q=U\,A\,\Delta T_{lm}$, with $A=\pi d\,L$ for the tube.
**Step 1: log-mean temperature difference.** The steam is at a constant 120 °C, and the milk goes from 10 °C to 85 °C:
$$\Delta T_1=120-10=110,\qquad\Delta T_2=120-85=35 .$$
$$\Delta T_{lm}=\frac{110-35}{\ln(110/35)}=\frac{75}{1.1451}=65.5\ ^\circ\text{C}.$$
**Step 2: length.** With $Q=146\,200$ kcal/h, $U=750$ kcal h⁻¹ m⁻² °C⁻¹ and $d=0.040$ m:
$$L=\frac{Q}{U\,\pi d\,\Delta T_{lm}}=\frac{146\,200}{750\times\pi\times0.040\times65.5}=\frac{146\,200}{6174}=\mathbf{23.7\ m}.$$