A cylindrical block of length $0.4 m$ and area of cross-section $0.04 m^{2}$ is placed coaxially on a thin metal disc of mass $0.4 k g$ and of the same cross-section. The upper face of the cylinder is maintained at a constant temperature of $400 K$ and the initial temperature of disc is $300 K$ . If the thermal conductivity of the material of disc is $600 J / k g - K$ . How long will it take for the temperature of the disc of increase to $350 K$ ? Assume, for purpose of calculation, the thermal conductivity of the disc to be very high and the system to be thermally insulated, except for the upper face of the cylinder.

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Solution

In steady state, fundamental equation of heat conduction is
$H = \frac{d Q}{d t} = \frac{k A (T_{1} - T_{2})}{l}$
Here temperature of upper face of cylinder is $400 K$ and that of lower face is changing $\frac{d Q}{d t} = k A \frac{(400 - T)}{l}$
where $T$ is intermediate temperature of lower face of cylinder $d Q$ is heat absorbed by disc in time $d t$ .
Also $d Q = m c d T$
where $m =$ mass of disc $= 0.4 k g, c =$ specific heat of disc
$= 600 J / k g . K$
$\frac{m C d T}{d t} = k A \frac{(400 - T)}{l}$
$\Rightarrow \int_{0}^{t} d t = \frac{m c l}{k A} \int_{300}^{350} \frac{d T}{400 - T}$
$t = \frac{m c l}{k A} {[\frac{l n (400 - T)}{- 1}]}_{300}^{350} = \frac{m c l}{k A} l n (\frac{400 - 300}{400 - 350})$
Substituting the value of $m, c, l, k$ and $A$ we get
$t = 166.3 s e c .$

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