Heat transfer-Radiation

Question

A metal cylinder of 5 kg is heated electrically by a 20 W heater in a room 20^{o}C. The cylinder temperature rises uniformly to 30^{o}C in 5 minute and finally becomes constant at 45^{o}C. Assuming that the rate of heat loss is proportional to the excess temperature over the surroundings.

Moderate

Solution

$\begin{array}{l}{\mathrm{T}}_{\text{room}}={20}^{\circ}\mathrm{C}\\ {\mathrm{m}}_{\text{cylinder}}=5\mathrm{kg}\\ \mathrm{P}=20\mathrm{W}\end{array}$

Since temperature becomes constant at 45°C

$\Rightarrow $Power lost at 45°C = 20 W

From Newton's law of cooling

$\begin{array}{l}\therefore \phantom{\rule{1em}{0ex}}{\left|\frac{dQ}{\mathrm{dt}}\right|}_{\mathrm{T}={45}^{\circ}\mathrm{C}}=20=\mathrm{K}(45-20)\\ \Rightarrow \phantom{\rule{1em}{0ex}}\mathrm{K}=\frac{4}{5}\end{array}$

Since temperature rises uniformly to 30°C thus, at 25°C,

$\frac{dQ}{\mathrm{dt}}=\mathrm{k}(25-20)=4W$

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