Solve exercise 2.10 with the additional condition that the concrete wall releases heat to air at a temperature ϑ S = ϑ 0 = 20 ◦C, whereby the heat transfer coefficient α = 15.0W/m 2 K is decisive....

Solve exercise 2.10 with the additional condition that the concrete wall releases heat to air at a temperature ϑ_S
= ϑ₀
= 20 ◦C, whereby the heat transfer coefficient α = 15.0W/m²
K is decisive.

exercise 2.10

A very thick concrete wall (λ = 0.80W/K m, = 1950 kg/m³, c = 880 J/kg K) initially has as temperature throughout the wall of ϑ0 = 20 ◦C. It is heated at its surface with a constant heat flux ˙q₀
= 650W/m²
(sunshine). What temperature has the surface reached after 2.0 h? How high is the temperature at this time at a depth of 10 cm?

A sphere made of plastic (λ = 0.35W/K m, c = 2300 J/kg K, = 950 kg/m³) of diameter d = 20 mm, which was heated to ϑ₀
= 110.0 ◦C, is cooled in air at ϑ_S
= 15.0 ◦C; the heat transfer coefficient is α = 8.75W/m²
K .

a) What temperature has the sphere reached after 20 min? Use the approximation equation for small Biot numbers, cf. section 2.3.5.2.

b) Compare the result from a) with the exact solution by calculating the mean temperature ϑ_m
and the temperatures at the surface and the centre of the sphere.