Conduction Law

Q. A body cools in $7$ min from ${60}^{\circ }\text{C}$ to ${40}^{\circ }\text{C}$. What will be its temperature after the next $7$ min ?
The temperature of surroundings is ${10}^{\circ }\text{C}$. [Assume Newton's Law of cooling is applicable]

1. $T={32}^{\circ }\text{C}$
2. $T={28}^{\circ }\text{C}$
3. $T={24}^{\circ }\text{C}$
4. $T={36}^{\circ }\text{C}$

Q. Two rods $\text{A}$ and $\text{B}$ of different materials are welded together as shown in figure. Their thermal conductivities are $K_1$and $K_2.$ The thermal conductivity of the composite rod will be

1. $\frac{K_1+K_2}{2}$
2. $\frac{3(K_1+K_2)}{2}$
3. $K_1+K_2$
4. $2(K_1+K_2)$

Q. If $K_1\text{and}{K}_2$ are the thermal conductivities, $L_1\text{and}{L}_2$ are the lengths and $A_1\text{and}{A}_2$ are the cross sectional areas of steel and copper rods respectively such that $\frac{K_2}{K_1}=9,\frac{A_1}{A_2}=2,\frac{L_1}{L_2}=2$.Then, for the arrangement as shown in the figure, the value of temperature $T$ of the steel-copper junction in the steady state will be

Q. Two ends of a rod of non-uniform area of cross-section are maintained at temperature $T_1$ and $T_2(T_1>T_2)$ as shown in the figure. If $I$ is the heat current through the cross - section of the conductor at distance $x$ from its left face then the variation of $I$ with time $t$ is best represented by

1. 
2. 
3. 
4. 

Q. Two rectangular blocks, having identical dimensions, can be arranged either in configuration I or II as shown in the figure. One of the blocks has thermal conductivity $K$ and the other, $2K$. The temperature difference between the ends along the $x$-axis is the same in both the configuarations. It takes $9\text{s}$ to transport a certain amount of heat from the hot end to the cold end in configuration I. The time to transport the same amount of heat in the configuration II is:-

1. $2.0\text{s}$
2. $3.0\text{s}$
3. $4.5\text{s}$
4. $6.0\text{s}$

Q. A water cooler of storage capacity $120\text{litres}$ can cool water at a constant rate of $P$. In a closed circulation system (as shown schematically in this figure), the water from the cooler is used to cool an external device that constantly generates $3\text{kW}$ of power (thermal load). The temperature of the water fed into the device cannot exceed ${30}^{\circ }\text{C}$ and the entire $120\text{litres}$ of water is initially cooled to ${10}^{\circ }\text{C}$. The entire system is thermally insulated. The minimum value of $P$ for which the device can be operated for 3 hours (taking the specific heat capacity of water $=4.2{\text{kJ kg}}^{-1}{\text{K}}^{-1}$ and the density of water $=1000{\text{kg m}}^{-3})$ is

1. $1600\text{W}$
2. $2067\text{W}$
3. $2533\text{W}$
4. $3933\text{W}$

Q. The heat conduction equation $\frac{dQ}{dt}=-kA\left(\frac{dT}{dx}\right)$ applied for heat transfer through a plane slab presumes:
1. Steady- state one-dimensional conduction.
2. Constant value of thermal conductivity
3. The bounding surfaces are isothermal.
4. Temperature gradient is constant.
of these statements:-

1. $2$ and $4$ are correct
2. $1$, $3$ and $4$ are correct
3. $1,2,$ and $3$ are correct
4. $2,3,$ and $4$ are correct

Q. A rod of length $L$ and uniform cross-sectional area has varying thermal conductivity which changes linearly from $2K$ at end $B$ to $K$ at the other end $A$. The ends $A$ and $B$ of the rod are maintained at constant temperature ${100}^{\circ }C$ and $0^{\circ }C$, respectively. At steady state, the graph of temperature : $T=T\left(x\right)$ where $x=$ distance from end $A$ will be

1. 
2. 
3. 
4. 

Q. A hollow sphere of inner radius $R$ and outer radius $2R$ is made of a material of thermal conductivity $K$. It is surrounded by another hollow sphere of inner radius $2R$ and outer radius $3R$ made of the same material of thermal conductivity $K$. The inside of the smaller sphere is maintained at $0^{\circ }\text{C}$ and the outside of the bigger sphere is at ${100}^{\circ }\text{C}$. The system is in steady state. Find the temperature of the interface.

1. ${50}^{\circ }\text{C}$
2. ${100}^{\circ }\text{C}$
3. ${75}^{\circ }\text{C}$
4. ${125}^{\circ }\text{C}$

Q. Three rods of same material, same area of cross-section but different lengths $10\text{cm}$, $20\text{cm}$ and $30\text{cm}$ are connected at a point as shown in the figure. Find the temperature of junction ${}^{\prime }{\text{O}}^{\prime }$.

1. ${19.2}^{\circ }\text{C}$
2. ${16.4}^{\circ }\text{C}$
3. ${11.5}^{\circ }\text{C}$
4. ${22}^{\circ }\text{C}$

Q.

Why does the heat transfer coefficient increase with velocity?

Q. Two identical square rods of metal are welded end to end as shown in figure $\left(a\right)$, $20\text{calories}$ of heat flows through it in $4\text{minutes}$. If the rods are welded as shown in figure $\left(b\right)$, the same amount of heat will flow through the rods in

1. $1\text{minute}$
2. $2\text{minutes}$
3. $4\text{minutes}$
4. $16\text{minutes}$

Q. The dimension of radiation pressure is -

1. $\left[M{L}^1{T}^2\right]$
2. $\left[M{L}^1{T}^{-2}\right]$
3. $\left[M{L}^{-1}{T}^2\right]$
4. $\left[M{L}^{-1}{T}^{-2}\right]$

Q.

A uniform slab of dimension 10 cm $\times$ 10 cm $\times$ 1 cm is kept between two heat reservoirs at temperatures ${10}^{\circ }C$ and ${90}^{\circ }C$. The larger surface areas touch the reservoirs. The thermal conductivity of the material is $0.80W{m}^{-1\circ }{C}^{-1}$ Find the amount of heat flowing through the slab per minute.

Q. If a liquid takes $30\text{s}$ in cooling from ${95}^{\circ }\text{C}$ to ${90}^{\circ }\text{C}$ and $70\text{s}$ in cooling from ${55}^{\circ }\text{C}$ to ${50}^{\circ }\text{C}$ then temperature of room is -

1. ${16.5}^{\circ }\text{C}$
2. ${22.5}^{\circ }\text{C}$
3. ${28.5}^{\circ }\text{C}$
4. ${32.5}^{\circ }\text{C}$

Q. The specific heat of a metal at low temperatures varies according to $S=a{T}^3$ where $a$ is a constant and $T$ is absolute temperature. The heat energy needed to raise the temperature of unit mass of the metal from $T=1K$ to $T=2K$ is

1. $\frac{15a}{4}$
2. $\frac{2a}{3}$
3. $\frac{12a}{5}$
4. $3a$

Q.

Steam at ${120}^{\circ }C$ is continuously passed through a 50 -cm long rubber tube of inner and outer radii 1.0 cm and 1.2 cm. The room temperature is ${30}^{\circ }C$. Calculate the rate of heat flow through the walls of the tube. Thermal conductivity of rubber = $0.15J{s}^{-1}{m}^{-1\circ }{C}^{-1}$.

Q.

In the relation $P=\frac{\alpha }{\beta }{e}^{\frac{\alpha x}{nR\theta }}$, P is power, x is distance, n is number of moles, R is gas constant and $\theta$ is temperature. The dimensional formula of $\beta$ is

1. $\left[M^0{L}^0{T}^0\right]$

2. $\left[M^1{L}^0{T}^1\right]$

3. $\left[M^0{L}^{-1}{T}^1\right]$

4. $\left[M^0{L}^{-1}{T}^{-1}\right]$

Q. Five identical rods are joined as shown in figure. Point A and C are maintained at temperature ${120}^{\circ }$ C and ${20}^{\circ }$ C respectively. The temperature of junction B will be

1. C
2. C
3. C
4. C

Q. A composite wall having three layers of thickness $0.3\text{m}$, $0.2\text{m}$ and $0.1\text{m}$ and of thermal conductivity $0.6,0.4$ and $0.1\text{W/m-K}$ respectively, has a surface area of $1{\text{m}}^2.$ If the inner and outer temperatures of the composite wall are $1840\text{K}$ and $340\text{K}$ respectively, find the rate of heat transfer.

1. $150\text{W}$
2. $150\text{W}$
3. $75\text{W}$
4. $750\text{W}$

Q. A heat flux of $4000\text{J/s}$ is to be passed through a copper rod of length $10\text{cm}$ and area of cross section $100{\text{cm}}^2$. The thermal conductivity of copper is $400{\text{W/m}}^{\circ }C$. The two ends of the rod must be kept at temperature difference of

1. ${100}^{\circ }C$
2. $1^{\circ }C$
3. ${10}^{\circ }C$
4. ${1000}^{\circ }C$

Q.

Two plates of the same area and the same thickness having thermal conductivities $k_1$ and $k_2$ are placed one on top of the other. The top and bottom faces of the composite plate are maintained at different constant temperatures. The thermal conductivity of the composite plate will be

1. $\frac{k_1{k}_2}{(k_1+k_2)}$

2. $(k_1+k_2)$

3. $\frac{2k_1{k}_2}{(k_1+k_2)}$

4. $\frac{1}{2}(k_1+k_2)$

Q.

A pitcher with 1 -mm thick porous walls contains 10 kg of water. Water comes to its outer surface and evaporates at the rate of $0.1g{s}^{-1}$. The surface are aof the pitcher (one side) = 200 $c{m}^2$. The room temperature = ${42}^{\circ }$, latent heat of vaporization = $2.27\times {10}^{6}Jk{g}^{-1}$, and the thermal conductivity of the porous walls = $0.80J{s}^{-1}{m}^{-1\circ }{C}^{-1}$. Calculate the temperature of water in the pitcher when it attains a constant value.

Q.

Four identical rods AB, CD, CF and DE are joined as shown in figure. The length, cross -sectional area and thermal conductivity of each rod are l, A and K respectively. The ends A, E and F are maintained at temperatures $T_1,T_2$ and $T_3$ respectively. Assuming no loss of heat to the atmosphere, find the temperature at B.

Q.

The normal body-temperature of a person is ${97}^{\circ }F$ Calculate the rate at which heat is flowing out of his body through the clothes assuming the following values. Room temperature = ${47}^{\circ }F$, surface of the body under clothes = $1.6m^2$, conductivity of the cloth = $0.04J{s}^{-1}{m}^{-1\circ }{C}^{-1}$, thickness of the cloth = 0.5 cm.

Q. Four rods of same material but different radii $r$ and length $l$ are used to connect two reservoirs of heat at different temperatures. Which one will conduct heat fastest?

1. $r=2\text{cm},l=0.5\text{m}$
2. $r=1\text{cm},l=0.5\text{m}$
3. $r=1\text{cm},l=1\text{m}$
4. $r=2\text{cm},l=2\text{m}$

Q. Which of the following options is incorrect in reference to Fourier's law of heat conduction?

1. Rate of heat flow is proportional to the cross-sectional area of rod
2. Rate of heat flow is proportional is the temperature difference between the ends of the rod
3. Rate of heat flow is inversely proportional to thickness of the rod
4. Rate of heat flow is proportional to the coefficient of thermal conductivity of material of the rod

Q.

A cubical box of volume $216c{m}^3$ is made up of 0.1 cm thick wood. The inside is heated electrically by a 100 W heater. It is found that the temperature difference between the inside and the outside surface is $5^{\circ }C$ in steady state. Assuming that the entire electrical energy spent appears as heat, find the thermal conductivity of the material of the box.

Q. Two walls of thickness $d_1$ and $d_2$, thermal conductivities $K_1$ and $K_2$ are in contact. In the steady state, if the temperature at the outer surfaces are $T_1$ and $T_2$, the temperature at the contact of the walls will be

1. $\frac{K_1{T}_1+K_2{T}_2}{d_1+d_2}$
2. $\frac{K_1{T}_1{d}_2+K_2{T}_2{d}_1}{K_1{d}_2+K_2{d}_1}$
3. $\frac{(K_1{d}_1+K_2{d}_2)T_1{T}_2}{T_1+T_2}$
4. $\frac{K_1{d}_1{T}_1+K_2{d}_2{T}_2}{K_1{d}_1+K_2{d}_2}$

Q. The ends of two rods of different materials with their thermal conductivities, radii of cross-sections and lengths all are in the ratio $1:2$ are maintained at the same temperature difference. If the rate of flow of heat in the longer rod is $4\text{cal/sec}$, then that in the shorter rod in $\text{cal/sec}$ will be

1. $1$
2. $8$
3. $16$
4. $2$