(C) Let $A_1$ and $A_2$ be the cross-sectional areas of the inner cylinder and the outer shell respectively.$A_1 = \pi R^2$$A_2 = \pi (2R)^2 - \pi R^2

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(C) Let $A_1$ and $A_2$ be the cross-sectional areas of the inner cylinder and the outer shell respectively.$A_1 = \pi R^2$$A_2 = \pi (2R)^2 - \pi R^2 = 3\pi R^2$Total area $A = A_1 + A_2 = 4\pi R^2$.Since the two ends are at different temperatures and the system is in a steady state,the heat flow through both parts occurs in parallel.The equivalent thermal resistance $R_{eq}$ is given by $\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2}$.Using $R = \frac{l}{kA}$,we have $\frac{kA}{l} = \frac{k_1 A_1}{l} + \frac{k_2 A_2}{l}$.Substituting the values:$k(4\pi R^2) = k_1(\pi R^2) + k_2(3\pi R^2)$$4k = k_1 + 3k_2$$k = \frac{k_1 + 3k_2}{4}$

Class 11 Physics 10-2.Heat Transfer Thermal Resistance and it's Combination

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$A$ cylinder of radius $R$ made of a material of thermal conductivity $k_1$ is surrounded by a cylindrical shell of inner radius $R$ and outer radius $2R$ made of a material of thermal conductivity $k_2$. The two ends of the combined system are maintained at different temperatures. There is no loss of heat from the cylindrical surface and the system is in steady state. The effective thermal conductivity of the system is

A
$k_1 + k_2$
B
$\frac{k_1 k_2}{k_1 + k_2}$
C
$\frac{1}{4}(k_1 + 3k_2)$
D
$\frac{1}{4}(3k_1 + k_2)$

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10-2.Heat Transfer

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Thermal Resistance and it's Combination

Six identical rods are arranged as shown in the figure. The temperature of the junction $B$ will be......... $^oC$

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Three rods of the same material,same length,and same cross-sectional area are connected as shown in the figure. Find the temperature of their junction in $^oC$.

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For the figure shown,when arcs $ACD$ and $ADB$ are made of the same material,the total heat current carried between $A$ and $B$ is $H$. If $ADB$ is replaced with another material,the total heat current becomes $2H$. If the temperatures at $A$ and $B$ are fixed at $T_1$ and $T_2$,what is the ratio of the new thermal conductivity to the old thermal conductivity of the arc $ADB$?

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Two rods of same length and material transfer a given amount of heat in $12 \, s$ when they are joined end to end. But when they are joined lengthwise (parallel),they will transfer the same amount of heat under the same conditions in .......... $s$.

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$A$ 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 $\,W$)

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Six identical rods are arranged as shown in the figure. The temperature of the junction $B$ will be......... $^oC$

Three rods of the same material,same length,and same cross-sectional area are connected as shown in the figure. Find the temperature of their junction in $^oC$.

Two rods of same length and material transfer a given amount of heat in $12 \, s$ when they are joined end to end. But when they are joined lengthwise (parallel),they will transfer the same amount of heat under the same conditions in .......... $s$.

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