Question

A copper sphere of diameter 5 cm is initially at temperature 200°C.It cools in air by convection and radiation. The temperature T of the sphere is governed by the equation

\rho C V_{0} \frac{\mathrm{d} T}{\mathrm{~d} t}=-\left[\varepsilon \sigma\left(T^{4}-T_{\infty}^{4}\right)+h\left(T-T_{\infty}\right)\right] A where p is the density of copper, C its specific heat, V, the volume of the sphere, t the time, ɛ a property of the surface known as emissivity, o a constant known as the Stefan-Boltzmann constant, T the ambient temperature, A the surface area of the sphere, and h the convective heat transfer coefficient. The initial condition is as follows: At t = 0:1 = 200°C Using Heun's method, without iteration, solve this differential equa-tion to find the temperature variation with time, until the temperature drops below 50°C. Use the following values: p = 9000 kg/m³ C = 400 J/kg K & = 0.5 o = 5.67 x 10-8 W/m² K4 T, =25°C h=15 W/m² K Employ time steps of 0.5 and1.0 min, and compare the results obtained in the two cases.

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