(A) The heat conducted through the ice layer of thickness $x$ in time $dt$ is given by $dQ = \frac{KA(T_2 - T_1)}{x} dt$.Here,$T_2 = 0^{\circ}C$ (temp

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(A) The heat conducted through the ice layer of thickness $x$ in time $dt$ is given by $dQ = \frac{KA(T_2 - T_1)}{x} dt$.Here,$T_2 = 0^{\circ}C$ (temperature at the water-ice interface) and $T_1 = -26^{\circ}C$ (outside air temperature).So,$dQ = \frac{KA(0 - (-26))}{x} dt = \frac{26KA}{x} dt$.This heat causes an additional thickness $dx$ of water to freeze. The mass of this new ice layer is $dm = A \cdot dx \cdot \rho$.The heat released during this phase change is $dQ = dm \cdot L = A \cdot dx \cdot \rho \cdot L$.Equating the two expressions for $dQ$:$\frac{26KA}{x} dt = A \cdot dx \cdot \rho \cdot L$.Rearranging to find the rate of increase of thickness $\frac{dx}{dt}$:$\frac{dx}{dt} = \frac{26K}{\rho x L}$.

Class 11 Physics 2 Thermal Properties of Matter: Heat Transfer Heat Conduction and Thermal Conductivity

Medium

$A$ deep rectangular pond of surface area $A$,containing water (density $=\rho$,specific heat capacity $=s$),is located in a region where the outside air temperature is at a steady value of $-26^{\circ}C$. The thickness of the frozen ice layer in this pond at a certain instant is $x$. Taking the thermal conductivity of ice as $K$ and its specific latent heat of fusion as $L$,the rate of increase of the thickness of the ice layer at this instant would be given by:

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A
$26 K / (\rho x L)$
B
$26 K / (\rho x^2 L)$
C
$26 K / (\rho L)$
D
$13 K / (\rho x L)$

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2 Thermal Properties of Matter: Heat Transfer

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Heat Conduction and Thermal Conductivity

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$A$ rod of length $l$ with thermally insulated lateral surface is made of a material whose thermal conductivity $K$ varies as $K = C/T$,where $C$ is a constant. The ends are at temperatures $T_1$ and $T_2$. The heat current density is

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In a steady state,the temperatures at the ends $A$ and $B$ of a $20\,cm$ long rod $AB$ are $100\,^{\circ}C$ and $0\,^{\circ}C$ respectively. The temperature of a point $C$ at a distance of $9\,cm$ from $A$ is....... $^{\circ}C$

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$A$ rod of length $L$ with sides fully insulated is made of a material whose thermal conductivity varies with temperature $T$ as $K = \frac{\alpha}{T}$,where $\alpha$ is a constant. The ends of the rod are kept at temperatures $T_1$ and $T_2$. The temperature $T$ at a distance $x$ from the end kept at $T_1$ is:

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$A$ cylindrical rod has temperatures $T_1$ and $T_2$ at its ends. The rate of flow of heat is $Q_1 \ cal/sec$. If all the linear dimensions are doubled while keeping the temperatures constant,then the new rate of flow of heat $Q_2$ will be:

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Two spheres made of different materials have radii in the ratio $2:1$ and wall thicknesses in the ratio $1:4$. They are completely filled with ice. If the time taken to melt the ice completely in the larger sphere is $25 \ min$ and in the smaller sphere is $16 \ min$,then the ratio of the thermal conductivities of the materials of the larger and smaller spheres is ..........

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$A$ rod of length $l$ with thermally insulated lateral surface is made of a material whose thermal conductivity $K$ varies as $K = C/T$,where $C$ is a constant. The ends are at temperatures $T_1$ and $T_2$. The heat current density is

In a steady state,the temperatures at the ends $A$ and $B$ of a $20\,cm$ long rod $AB$ are $100\,^{\circ}C$ and $0\,^{\circ}C$ respectively. The temperature of a point $C$ at a distance of $9\,cm$ from $A$ is....... $^{\circ}C$

$A$ cylindrical rod has temperatures $T_1$ and $T_2$ at its ends. The rate of flow of heat is $Q_1 \ cal/sec$. If all the linear dimensions are doubled while keeping the temperatures constant,then the new rate of flow of heat $Q_2$ will be:

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