Question

(II) Twenty-five kilograms of granite constitute the “hot rock” part of a sauna. Granite (with quartz inclusions) has specific heat capacity Cgranite ≈ 800 J/(kg·C), and emissivity egranite ≈ 0.70. The walls of the sauna are lined with red cedar. Red cedar has specific heat capacity Ccedar ≈ 1500 J/(kg·C), and emissivity ecedar ≈ 0.35. The effective surface area of the rocks is 0.4 m 2 , while that of the interior of the sauna is 20 m 2 . The Stefan–Boltzmann constant is σ ≃ 5 2 3 × 10−8 W/(m 2 ·K 4 ). Outside, it is wintertime. [The granite is insulated from contact with the walls so as to not set them on fire.]

(a) At an initial time, ti , the rock is at an initial temperature of Tg,i = 227 C, while the walls are at Tw = 27 C. The wall temperature remains effectively constant over the timescale of this question.

(i) Compute the rate at which the rocks are radiating heat.

(ii) Compute the rate at which the rocks are absorbing heat radiation from the walls.

(iii) Compute the NET rate at which the rocks are radiating heat.

(b) Roughly how long does it take for the temperature of the granite rocks to reach 225 C? [HINT: Feel free to make reasonable approximations.]

(c) At the instant that the temperature of the granite is 225 C, 250 g of water, at an initial temperature of 40 C, are splashed upon the rocks. The specific heat capacity and latent heat of vaporisation are CH2O = 4186 J/(kg·C) and LVH2O ≃ 2256 kJ/kg, respectively. The water is very quickly [“instantaneously”] heated to 100 C, vaporises, and spreads throughout the sauna as steam at 100 C.

(i) How much energy is absorbed by the water as it undergoes its transformation to steam?

(ii) What is the temperature of the rocks immediately after the water boils away? [Neglect the effects of thermal radiation on this short time-scale.]

Answer 1