GATE 2023 AG – Question 65
Copper and steel balls of diameters d₁,d₂ start at 200°C and both reach120°C after the same time in 30°C air. Bi<.1. Copper density8950 kg/m³,cp383 J/(kg·°C),k386 W/(m·°C);steel7800,460,36 in the same units. Find d₁/d₂,to three decimals.
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Correct answer: MTA
Explanation
The official grading is **marks to all (MTA)**. With lumped cooling and a common convection coefficient,$(T-T_\infty)/(T_i-T_\infty)=\exp[-6ht/(\rho c_pd)]$. Equal times and temperature ratios give $\rho_1c_{p1}d_1=\rho_2c_{p2}d_2$,so $$d_1/d_2=\frac{7800(460)}{8950(383)}=1.046704\ldots\approx\mathbf{1.047}.$$ Conductivities only support the small-Biot assumption. If convection coefficients differ, the ratio also depends on h₁/h₂; the source does not specify that equality, which is why the conditional solution is distinguished from official MTA grading.