If the radius and length of a copper rod are both doubled,the rate of flow of heat along the rod increases ....... times.

$A$ metallic rod of cross-sectional area $9.0 \, cm^2$ and length $0.54 \, m$,with the surface insulated to prevent heat loss,has one end immersed in boiling water and the other in an ice-water mixture. The heat conducted through the rod melts the ice at the rate of $1 \, g$ for every $33 \, s$. The thermal conductivity of the rod is ....... $W m^{-1} K^{-1}$.

Two sheets of thickness $d$ and $3d$ are touching each other. The temperature just outside the thinner sheet side is $A$,and on the side of the thicker sheet is $C$. The interface temperature is $B$. If $A, B$,and $C$ are in arithmetic progression,the ratio of the thermal conductivity of the thinner sheet to the thicker sheet is:

Three rods of equal length and cross-sectional area with thermal conductivities $K, 2K$,and $3K$ are joined as shown in the figure. The temperatures of their outer ends are $110\ ^oC, 20\ ^oC$,and $0\ ^oC$ respectively. Find the temperature of the junction in $^oC$.

The length of a metal bar is $20 \ cm$ and the area of cross-section is $4 \times 10^{-4} \ m^2$. If one end of the rod is kept in ice at $0^{\circ} C$ and the other end is kept in steam at $100^{\circ} C$,the mass of ice melted in one minute is $5 \ g$. The thermal conductivity of the metal in $W \ m^{-1} \ K^{-1}$ is (Latent heat of fusion $= 80 \ cal/g$):

For a spherical shell with inner and outer radii $r_1$ and $r_2$ respectively,the formula for the heat flow rate is = ......... where $T_1$ and $T_2$ are the temperatures of the inner and outer surfaces of the shell $(T_1 > T_2)$. The thermal conductivity of the material of the shell is $k$.