Conduction Law: Differential Form

Q. A metallic spherical shell having inner and outer radii $a$ and $b$ respectively has thermal conductivity $K=\frac{K_0}{r^2}\{a\le r\le b\}$, where $r$ is the distance from the center of the sphere. The inner surface is maintained at temperature ${\theta }_1$ and outer surface is maintained at temperature ${\theta }_2$ . Calculate the rate at which heat flows radially through the material $[{\theta }_2>{\theta }_1]$.

1. $\frac{4\pi K_0({\theta }_1-{\theta }_2)}{b-a}$
2. $\frac{4\pi K_0({\theta }_2-{\theta }_1)}{b-a}$
3. $\frac{4\pi K_0(b-a)}{{\theta }_2-{\theta }_1}$
4. $\frac{8\pi K_0({\theta }_2-{\theta }_1)}{b-a}$

Q.

Two materials A and B of different materials are welded together. Both the materials are having the same cross-sectional area and the same length. Their thermal conductivities are $K_1$ and $K_2$. What is the thermal conductivity of the composite material?

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 metallic spherical shell having inner and outer radii $a$ and $b$ respectively has thermal conductivity $K=\frac{K_0}{r^2}\{a\le r\le b\}$, where $r$ is the distance from the center of the sphere. The inner surface is maintained at temperature ${\theta }_1$ and outer surface is maintained at temperature ${\theta }_2$ . Calculate the rate at which heat flows radially through the material $[{\theta }_2>{\theta }_1]$.

1. $\frac{4\pi K_0({\theta }_1-{\theta }_2)}{b-a}$
2. $\frac{4\pi K_0({\theta }_2-{\theta }_1)}{b-a}$
3. $\frac{4\pi K_0(b-a)}{{\theta }_2-{\theta }_1}$
4. $\frac{8\pi K_0({\theta }_2-{\theta }_1)}{b-a}$

Q. Two rods of same length but made of different material are taken for a study as shown in the figure. Area of cross-section of rod $1$ is $12{\text{cm}}^2$ and that of rod $2$ is $16{\text{cm}}^2$. Thermal conductivity of rod $1$ is $0.0008{\text{cal/cm-s-}}^{\circ }\text{C}$. If rate of loss of heat due to conduction is equal, find the thermal conductivity of rod $2$.

1. $0.0006{\text{cal/cm-s-}}^{\circ }\text{C}$
2. $0.0008{\text{cal/cm-s-}}^{\circ }\text{C}$
3. $0.0010{\text{cal/cm-s-}}^{\circ }\text{C}$
4. $0.0004{\text{cal/cm-s-}}^{\circ }\text{C}$

Q. A refrigerator door is $150\text{cm}$ high, $80\text{cm}$ wide and $6\text{cm}$ thick. If the coefficient of conductivity is $0.0005{\text{cal/cm-s-}}^{\circ }\text{C}$ and the inner and outer surfaces are at $0^{\circ }\text{C}$ and ${30}^{\circ }\text{C}$ respectively, find the heat loss per minute through the door. (in Calories)

1. $3600$
2. $1800$
3. $900$
4. $1200$