Modes Heat Transfer

Understanding the Modes of Heat Transfer i.e. Conduction, Convection and Radiation.

Conduction

Rate of Heat Transfer in Steady state and Rate of Heat Absorption in Transient state in Conduction. Thermal Conductivity.

Q1

A slab of unknown material of area 5 m2 and thickness 0.10 m is exposed on the lower surface to steam at 100 oC. A block of ice at 0 oC rests on the upper surface of the slab. If in one hour, 10 kg of ice is melted, calculate the thermal conductivity of the material.
Latent heat of ice = 80 kcal/kg.

Series & Parallel

Combination of Thermal conductors connected in Series and Parallel combination.

Q2

A composite sheet has two layers A and B each made of different material. Both layers have the same thickness. The thermal conductivity of A is 3 times as that of B. In the steady state, the temperature difference across the composite sheet is 36 oC. Find the temperature difference (in oC) across the layer A.


Q3

In one case, two identical rods are joined one after the other and the temperature at one of the ends of this combination is maintained at 100 oC and the other is kept in ice at 0 oC. In the second case, the two rods are placed one on top of the other and has the same temperature difference across its ends. If Q1 and Q2 ( in gram per second ) are the respective rates of melting of ice in the two cases, then the ratio is

Q4

A cylinder of radius R and thermal conductivity k1, is surrounded by a cylindrical shell of inner radius R and outer radius 2R and of thermal conductivity k2.
There is no loss of heat across the cylindrical surface and the system is in steady state. The effective thermal conductivity of the system is

Q5

A room of total wall area 200 m2 is heated by an electric heater to maintain a temperature of 30 oC inside it when the outside temperature is -10 oC. The walls have three layers of different materials. The innermost layer is of wood of thickness 5 cm, the middle is of cement of thickness 3 cm and the outermost layer is brick of thickness 14 cm. The thermal conductivities of wood, cement and brick are 1.25, 1.5 and 1.0 watt cm-1 oC-1 respectively.
Find the power of the electric heater and the temperature at every interface.
Assume that there is no heat loss through the floor and ceiling.

Q6

Three rods of identical cross-sectional area and made from the same metal form the sides of an isosceles triangle ABC, right angled at C. The point A and C are maintained at temperatures T and T respectively. If temperature of point B is TB, the ratio TB / T is

Cylindrical Shell

Conduction of Heat through Cylindrical conductors.

Q7

A thin metal pipe of 1 m length and l cm radius carries steam at 100 oC. This is covered by two layers of insulation. The thermal conductivity of outer layer which is 1 cm thick is 2 x 10-4 cal/cm-oC-sec while that inner layer which is 1 cm thick is 1 x 10-4 cal/cm-oC-sec.
If the outer surface of lagging is kept in ice at 0 oC, find:

Stefan\'\'s Law

Stefan?s Law relates the Power of Thermal Radiation to the Emissivity, Surface Area and Absolute Temperature of material. Discussion on Ideal Black Bodies and Prevost Theory of Exchange.

Q8

A thin rectangular sheet of sides 20 cm and 10 cm is heated in a furnace to 800 K and taken out. How much electric power is needed to maintain the sheet at this temperature, if the temperature of surrounding is
Given that its emissivity is 0.250, Stefan-Boltzmann constant, s = 5.67 x 10-8 W

Q9

A copper sphere is suspended in an vaccum chamber maintained at 300 K. The sphere is maintained at a constant temperature of 600 K by heating it electrically. A total of 400 W of electric power is needed to do so. When surface of copper sphere is completely blackened, 700 W is needed to maintain the same temperature of the sphere. Calculate the emissivity of copper.

Temp of Sun

Calculating the Temperature of Sun from Stefan?s Law of Thermal Radiation

Weins Disp Law

Wein?s Displacement Law relates the Dominant Wavelength of Radiation of a body with its absolute temperature.

Q10

A black body is at a temperature of 2880 K. The energy of radiation emitted by this body between wavelengths 499 nm and 500 nm is U1, between 999 nm and 1000 nm is U2 and between 1499 nm and 1500 nm is U3. The Wiens constant b = 2.88 106 nm K.

Q11

Which of the following is the fm ? T graph for a perfectly black body?




Q12

An ideal black body at room temperature is thrown into a furnace. It is observed that

a) initially it is the darkest body and at later times the brightest
b) it is the darkest body at all times
c) it cannot be distinguished at all times
d) initially it is the darkest body and at later times it cannot be distinguished.

Q13

If the filament of a 100 W bulb has an area 0.25 cm2 and behaves as a perfect black body, find the wavelength and frequency corresponding to the maximum in its energy distribution. Given that Stefans constant is s = 5.67 10-8 J / m2 sec K4.

Rate of Cooling

Newton?s Law of Cooling

Q14

A body cools down from 60 oC to 55 oC in 50 sec. Temperature of surrounding is 45 oC. Calculate a) the time taken by the same body to cool by another 5 oC b) temperature of body after 50 sec when it was at 55 oC


Q15

The graph shown in the following figure, shows fall of temperature (T) of two bodies x and y, having the same surface area, with time (t) due to emission of radiation. Find the correct relation between emissive power (E) and absorptive power (A)

Q16

Two solid copper spheres of radii R1 = 15 cm and R2 = 20 cm are both at a temperature of 60 oC. If the temperature of surrounding is 50 oC, then find
a) the ratio of their initial heat loss per second from their surfaces
b) the ratio of their initial rates of cooling

1

A semi-circular rod and a straight rod of the same material and of same cross-sectional area are joined as shown in the figure. The points A and B are maintained at different temperatures. The ratio of rate of heat transfer through the semi-circular rod to that of the straight rod is

2

A refrigerator is 10 cm thick and has a total area of 5 m2. Inside temperature of refrigerator is maintained at 10 oC while the outside temperature is 40 oC. Output power of its cooling system is equal to the rate at which heat is drained from it. If the efficiency of its cooling system is 30%, find the input power required to maintain the refrigerator at constant temperature.
Thermal conductivity of material of refrigerator is k = 0.05 W/mK.

3

Three rods made of the same material and having the same cross-section have been joined as shown in the figure. Each rod is of the same length. The left and right end are kept at 0 oC and 90 oC respectively. The temperature of the junction of the three rods will be

4

A double pane window is used for insulating a room thermally from outside. It consists of two glass sheets each of area 1 m2 and thickness 0.01 m separated by a 0.05 m of air space. In the steady state, the room glass interface and the glass outdoor interface are at constant temperature of 27 oC and 0 oC respectively.
Calculate the rate of heat flow through the window.
Also find the temperature at other interfaces.
Given thermal conductivities of glass and air as 0.8 Wm-1K-1 and 0.08 Wm-1K-1 respectively.

5

Three rods of same dimensions are arranged as shown in the figure. They have thermal conductivities k1, k2, and k3. The points P and Q are maintained at different temperatures for the heat to flow at the same rate along PRQ and PQ. Which of following options is correct?

6

A metal rod AB of length l has its one end in ice at 0 oC and the other in water at 100 oC. If a point on the rod is maintained at 400 oC, then it is found that the amount of water evaporating per unit time is equal to the amount of ice melting per unit time. Find the distance of this point from ice in terms of the fraction of length of rod.
Latent heat of vaporization of water is 540 cal/g and the latent heat of fusion of ice is 80 cal/g. Neglect any heat loss to the surrounding.

7

A closed cubical box made of perfectly insulating material has walls of thickness 10 cm and the only way for heat to enter or leave the box is through two solid, cylindrical, metallic plugs, each of cross-sectional area 10 cm2 and thickness 10 cm fixed in the opposite walls of the box as shown in figure. Outer surface A is kept at 100 oC while the outer surface B of the outer plug is maintained at 0 oC. The thermal conductivity of material of the plug is 1 cal / sec - cm - oC . A source of heat is enclosed inside the box. Find the equilibrium temperatures of the inner surface of the box assuming that it is the same at all point on the inner surface.

8

Three rods AB, BC and BD having thermal conductivities in the ratio 1 : 2 : 3 and lengths in the ratios 2 : 3 : 4 are joined as shown in figure. The ends A, C and D are at temperatures A, C, and D respectively.
Find the temperature of the junction B. Assume steady state.

9

The intensity of radiation emitted by one star has its maximum value at a wavelength of 500 nm and that emitted by another star has maximum value at 300 nm. If these stars behave as black bodies, the ratio of their surface temperatures is

a) 1/2
b) 3/5
c) 9/25
d) 5/3

If the power of radiation of first star is 81 times as that of second star, the ratio of their radii will be

a) 3/5
b) 5/3
c) 9/25
d) 25/9

10

Two bodies have thermal emissivities of 0.01 and 0.81 respectively. The outer surface areas of the two bodies are the same. If they emit total radiant energy at the same rate, find the ratio of their temperatures.

11

A cubical black body of side l radiates 450 W power at T Kel. If the side of cube was halved and the temperature doubled, the power radiated in watt would be

(a) 225
(b) 450
(c) 900
(d) 1800

12

Earth receives 1600 Wm-2 of solar power. If all the solar energy falling on a lens of area 4.18 m2 is focused on a block of ice of mass 5 kg, the time taken to melt the ice will be

(a) 2 min 10 sec
(b) 3 min 10 sec
(c) 4 min 10 sec
(d) 5 min 10 sec
Latent heat of fusion of ice = 80 kcal/kg

13

Initially a black body at absolute temperature T is kept inside a closed chamber at absolute temperature To. Now the chamber is slightly opened to allow sun rays to enter. It is observed that temperatures T and To remains constant. Which of the following statement is/are true?
(a)The rate of emission of energy from the black body remains the same.
(b)The rate of emission of energy from the black body increases.
(c)The rate of absorption of energy by the black body increases.
(d)The energy radiated by the black body equals the energy absorbed by it.



14

A heat sensor of area As is at a distance d from a source at temperature T. If the temperature of source is doubled, how far should the sensor be moved from the source so that the power received by it remains the same.



15

Find the relation between the temperature of a planet and its distance from the Sun. Assume both Sun and planet behave like black bodies.

16

Two identical spheres A and B are suspended in an air chamber which is maintained at a temperature of 50 oC. Find the ratio of the heat lost per second from the surface of the spheres if

a) A and B are at temperatures 60 oC and 55 oC, respectively.

b) A and B are at temperatures 250 oC and 200 oC, respectively.

17

A cube and a sphere of equal edge and radius, made of the same substance are allowed to cool under identical conditions. Determine which of the two will cool at a faster rate.

18

Two metallic spheres S1 and S2 are made of the same material and have identical surface finish. The mass of S1 is eight times that of S2. Both the spheres are heated to the same temperature and put in the same room having lower temperature but are thermally insulated from each other. The ratio of the initial rate of cooling of S1 to that of S2 is

19

A solid cube of edges 1 cm is suspended in an evacuated enclosure. Its temperature is found to fall from 100 oC to 99 oC in 100 sec. Another solid cube of edges 2 cm, with similar surface nature, is suspended in a similar manner. The time required for this cube to cool from 100 oC to 99 oC will be approximately
(a) 25 sec
(b) 50 sec
(c) 200 sec
(d) 400 sec


20

There are two thin spheres A and B of the same material and same thickness. They emit like black bodies. Radius of A is double that of B and both have same temperature T. When A and B are kept in a room of temperature To ((a) 2:1
(b) 1:1
(c) 4:1
(d) 8:1
(Assume negligible heat exchange between A and B)

21

A sphere of radius R m has its outer surface painted black. Find the time required for the sphere to cool down from T1 K to T2 K.
Given, Density of copper, r and its specific heat C.
Temperature of environment is negligible as compared to T1 and T2, so the rate of absortion can be neglected.

22

A hot body placed in air is cooled down according to Newtons Law of cooling, the rate of decrease of temperature being B times the temperature difference from the surrounding. Find the time in which the body will lose half the maximum heat it can lose.

23

A cylindrical rod of heat capacity 250 J/K is in a room with temperature 27 oC. It is heated internally by heater supplying power at 1 kW. Temperature of rod finally becomes 77 oC and does not rise further. Find:
a) the initial rate of increase in temperature of rod
b) the rate of emission of radiant heat in steady state

If the heater is switched off, find
d) the rate of fall in temperature of the cylinder just when the heater is turned off
e) rate of fall in temperature when its temperature is 52 oC
f) the net amount of heat lost till this point
g) maximum net amount of heat lost by the cylinder

24

One end of a rod of length L cross-sectional area A is kept in a furnace at temperature T1. The other end of rod is kept at a temperature T2. The thermal conductivity of the material of the rod is k and emissivity of the rod is e. It is given that T2 = Ts + DT, where DT << Ts, Ts is the temperature of the surroundings. If DT ? ( T1 - T2 ), find the proportionality constant.
Consider that heat is lost only by radiation at the end where the temperature of the rod is T2.

25

Three rods of material x and three rods of material y are connected as shown in fig. All the rods are of identical length and cross-sectional area. If the end A is maintained at 100 oC and the junction E at 0 oC, calculate the temperature of the junctions B, C and D. The thermal conductivity of x is 3 times as that of y.

26

A tapered rod of length l and thermal conductivity k has its end radii r1 and r2 maintained at temperature T1 and T2 respectively. Calculate the rate of flow of heat through the rod.


27

A rod of length l and cross section A is initially at room temperature.
Its one end is put in furnace and the other is maintained at the room temperature.
a) Find heat absorbed by the rod before steady state is achieved.

b) If the maximum rate at which furnace can supply heat is Pmax, find the minimum time taken to achieve steady state.

28

A cylinder of length 1m and area of cross section 0.1 m2 is placed coaxially on a thin metal disc of mass 1 kg and of the same cross section. The upper face of the cylinder is maintained at a constant temperature of 400 K and the initial temperature of the disc is 200 K.
If thermal conductivity of the material of the cylinder is 10 Watt/m-K and the specific heat of the material of the disc is 800 J/kg-K, how long will it take for the temperature of the disc to increase to 300 K?
Assume, and system to be thermally insulated except for the upper face of the cylinder.

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