Conduction Law: Differential Form

Q.

The temperatures of the two outer surfaces of a composite slab, consisting of two materials having coefficients of thermal conductivity K and 2K and thickness x and 4x, respectively, are $T_2$ and $T_1$ ($T_2>T_1$). The rate of heat transfer through the slab, in a steady state is $\left(\frac{A(T_2-T_1)K}{x}\right)f$, with f equal to

1. $\frac{1}{2}$

2. $\frac{2}{3}$

3. $\frac{1}{3}$

4. 1

Q.

Find the rate of heat flow through a cross-section of the tapered conical rod shown in figure $($Temperature of the big end$\left({\theta }_2\right)>$ Temperature of the small end$\left({\theta }_1\right))$, with radius $r_1$ and $r_2$ at the small and big end respectively. Thermal conductivity of the material of the rod is K.

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Q.

Water is densest at 4${}^{\circ }$C; this is the reason lakes do not completely freeze during extreme winters. Say, the atmosphere is at -${\theta }^{\circ }C(\theta >0)$, now the temperature of the lakes water starts to fall, and the denser water from the top sinks, getting the bottom layers up, and cooling them until the temperature reaches 4 ${}^{\circ }$C at the upper surface. Now, further reduction in temperature actually makes the water less dense! Hence the colder water stays on top, starts to freeze, until the top layer of ice reaches -${\theta }^{\circ }C$ and the bottom layer at 0 ${}^{\circ }$ C . And this layer of ice is the only area through which the water below loses heat (by conduction), slowly increasing the thickness of the ice (because ice is a bad conduction), and for a very long time the water at the bottom of the lake remains at 4 ${}^{\circ }$ C! Now, Let the thickness of the layer of ice be $y_1$, at an instant.How much time t, will it take to increase to thickness $y_2$?

$\rho \to$ density of the ice

K $\to$ thermal conductivity of ice.

L $\to$ Latent heat of fusion

$\theta \to$ temperature of surface (as mentioned earlier)

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Q.

Six identical conducting rods are joined as shown in figure. Points A and D are maintained at temperatures 200${}^{\circ }$C and 20 ${}^{\circ }$ C respectively. Find the temperature of junction B.

1. 160 deg C

2. 180 deg C

3. 140 deg C

4. 120 deg C

Q.

On a cold winter day, the atmospheric temperature is -$\theta$ (on Celsius scale) which is below 0${}^{\circ }$C. A cylindrical drum of height h made of a bad conductor is completely filled with water at 0 ${}^{\circ }$ C and is kept outside without any lid. Calculate the time taken for the whole mass of water to freeze. Thermal conductivity of ice is K, $\rho$ is the density and latent heat of fusion is L. Neglect expansion of water on freezing

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Q.

Two different metal rods of the same length have their ends kept at the same temperatures ${\theta }_1$ and ${\theta }_2$ with ${\theta }_2$ > ${\theta }_1$. If $A_1$ and $A_2$ are their cross-sectional areas and $k_1$ and $k_2$ their thermal conductivities, the rate of heat flow in the two rods will be same if

1. $\frac{k_1}{k_2}=\frac{A_1}{A_2}$

2. $\frac{k_1}{k_2}=\frac{A_2}{A_1}$

3. $\frac{k_1}{k_2}=\frac{A_1{\theta }_2}{A_2{\theta }_1}$

4. $\frac{k_1}{k_2}=\frac{A_1{\theta }_1}{A_2{\theta }_2}$

Q. A rod of negligible heat capacity has length 20 cm, area of cross- section $1.0c{m}^2$ and thermal conductivity $200W/m^{\circ }C$. The temperature of one end is maintained at $0^{\circ }C$ and that of the other end is slowly and linearly varied from $0^{\circ }C$ to ${60}^{\circ }C$ in 10 minutes. Assuming no loss of heat through the sides find the total heat transmitted through the rod in these 10 minutes.

1. 2000 J
2. 2200 J
3. 1800 J
4. 2400 J

Q.

The temperatures of the two outer surfaces of a composite slab, consisting of two materials having coefficients of thermal conductivity K and 2K and thickness x and 4x, respectively, are $T_2$ and $T_1$ ($T_2>T_1$). The rate of heat transfer through the slab, in a steady state is $\left(\frac{A(T_2-T_1)K}{x}\right)f$, with f equal to

1. $\frac{2}{3}$

2. $\frac{1}{3}$

3. 1

4. $\frac{1}{2}$

Q. <!--td {border: 1px solid #ccc;}br {mso-data-placement:same-cell;}--> How does the density of water change when : (i) it is heated from 0˚C to 4˚C, (ii) it is heated from 4˚C to 10˚C.