The temperatures of the two outer surfaces of a composite slab are $T_2$ and $T_1$ $(T_2 > T_1)$. The thermal conductivities of the materials are $K$ and $2K$,and their thicknesses are $x$ and $4x$,respectively. In the steady state,the rate of heat flow through the slab is given by $f \left( \frac{A(T_2 - T_1)K}{x} \right)$. Find the value of $f$.

$A$ window used to thermally insulate a room from outside consists of two parallel glass sheets each of area $2.6 \,m^2$ and thickness $1 \,cm$ separated by $5 \,cm$ thick stagnant air. In the steady state, the room-glass interface is at $18^{\circ} C$ and the glass-outdoor interface is at $-2^{\circ} C$. If the thermal conductivities of glass and air are respectively $0.8 \,Wm^{-1} K^{-1}$ and $0.08 \,Wm^{-1} K^{-1}$, the rate of flow of heat through the window is (in $\,W$)

Two metal plates of equal thickness have thermal conductivities $K_1$ and $K_2$ respectively. They are joined together to form a single plate. The equivalent thermal conductivity of this composite plate is ..........

Two identical metal rods are welded end-to-end as shown in figure $(1)$. It takes $4 \ min$ for $20 \ cal$ of heat to flow through them. If these rods are now welded side-by-side as shown in figure $(2)$,the time taken for the same amount of heat to flow through them is .......... $\min$.

Five metal rods of the same cross-sectional area $A$ and same length $l$ are connected as shown in the figure. If the thermal conductivities of copper and steel are $K_1$ and $K_2$ respectively,then the resultant thermal resistance between $A$ and $C$ is:

$A$ partition wall has two layers $A$ and $B$ in contact,each made of a different material. They have the same thickness but the thermal conductivity of layer $A$ is twice that of layer $B$. If the steady state temperature difference across the wall is $60 \ K$,then the corresponding difference across the layer $A$ is ....... $K$.