Question

One end of $25cm$ long lagged metal bar is in steam and the other is in contact with ice. If $12g$ of ice melt per minute in the steady state, what is the thermal conductivity of the metal? (Cross section of the bar is $25c{m}^2$ and the latent heat of fusion of ice is $80kcal/kg.\left(67.2W/mK\right)$)

Answer 1

Given: One end of 25 cm long lagged metal bar is in steam and the other is in contact with ice. If 12 g of ice melt per minute in the steady state,

To find the thermal conductivity of the metal

Solution:

As per the given condition,

Length of the rod, $x=25cm=0.25m$

Area of cross section,$A=25c{m}^2=25\times {10}^{–4}{m}^2$

Mass of the ice, $m_i=12g=12\times {10}^{–3}kg$

latent heat of fusion of ice, $L=80kcal/kg$

Temperature difference between two ends of the rod $(T_1–T_2)=(100–0)=100$

time, $t=1min=60s$

If $Q$ is the amount of heat required to melt the ice, then

$Q=m_{i}L⟹Q=12\times {10}^{–3}\times 80⟹Q=960\times {10}^{-3}kcal=960\times {10}^{-3}\times 4184=4016.64J$

We know,

$Q=\frac{KAt(T_1-T_2)}{x}⟹K=\frac{Q\times x}{At(T_1-T_2)}⟹K=\frac{4016.64\times 0.25}{25\times {10}^{–4}\times 60\times 100}⟹K=0.669\times {10}^2=66.9W/mK$

is the thermal conductivity of the metal