# H C VERMA Solutions for Class 12-science Physics Chapter 6 - Heat Transfer

## Chapter 6 - Heat Transfer Exercise 98

A uniform slab of dimension 10cm×10cm×1cm is kept between two heat reservoirs at temperatures 10°C and 90°C The larger surface areas touches the reservoirs. The thermal conductivity of the material is 0.08 W/m-°C Find the amount of heat flowing through the slab per minute.

We know,

A liquid-nitrogen container is made of a 1cm thick thermocoal
sheet having thermal conductivity 0.025 *J/m-s-**°**C* Liquid nitrogen at 80K is kept in
it. A total area of 0.80m^{2} is in contact with the liquid nitrogen.
The atmospheric temperature is 300K. Calculate the rate of heat flow from the
atmosphere to the liquid nitrogen.

We know,

= 440W

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

We know,

Water is boiled in a container having a bottom of surface
area225cm^{2}, thickness 1.0mm and thermal conductivity 50W/m-°C 100g of water is converted into steam per minute in the
steady state after the boiling starts. Assuming that no heat is lost to the
atmosphere, calculate the temperature of the lower surface of the bottom.
Latent heat of vaporization of water = 2.26×10^{6} J/kg.

We know,

Also Rate of conversion =

One end of a steel rod (K=46 J/m-s-°C.) of length of 1.0m is kept in ice at 0°C and the other end is kept in boiling water at 100°C The area of cross - section of the rod is 0.04 cm^{2}.
Assuming no heat loss to the atmosphere, find the mass of the ice melting per
second. Latent heat of fusion of ice = 3.36×10^{6} J/kg.

We know,

An ice box almost completely filled with ice at 0°C is dipped into a large volume of water at 20°C. The box has walls of surface area
2400cm^{2} thickness 2.0mm and thermal conductivity 0.06W/m-°C Calculate the rate at which the ice melts in the box.
Latent heat of fusion of ice = 3.4×10^{5}
J/kg

We know,

Ice melt at rate of m/t which is:

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

We know,

Also

A steel frame (K = 45W/m-°C) of total length 60cm and cross - sectional area 0.20cm^{2}
forms three sides of a square. The free ends are maintained at 20°C and 40°C.
Find the rate of heat flow through a cross - section of the frame.

We know,

Water at 50°C
is filled in a closed cylindrical vessel of height 10cm and cross - sectional
area 10cm^{2}. The walls of the vessel are adiabatic but the flat
parts are made of 1mm thick Aluminium (K=200 J/m-s-°C). Assume that the outside temperature is 20°C. The density of water is 1000 kg/m^{3}, and the
specific heat capacity of water = 4200 J/kg-°C. Estimate the time taken for the temperature to fall by
1.0°C. Make any simplifying
assumptions you need but specify them.

We know,

= 6000

Heat goes out from two surfaces, Hence net heat flow is:

= 12000

Also,

The left end of a copper rod (length = 20cm, area of cross
- section =0.20cm^{2}) is maintained at 20°C and the right end is maintained at 80°C. Neglecting any loss of heat through radiation, find (a)
the temperature at a point 11cm from the left end and (b) the heat current
through the rod. Thermal conductivity of copper = 385 W/m-°C.

(a) We know,

________(1)

Also,

(b) From equation (1) heat flow,

## Chapter 6 - Heat Transfer Exercise 99

The ends of a metre stick are maintained at and 0°C One end of a rod is maintained at 25°C Where should its other end be touched on the metre stick so that there is no heat current in the rod in steady state?

Let the point touched be 0. As no heat is flowing

x is distance between 0°C end and point which is touched.

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

We know,

Figure shows water in a container having 2.0 mm thick
walls made of a material of thermal conductivity 0.50 W/m-°C The container is kept in a melting ice bath at 0°C The total surface area in contact with water is 0.05m^{2}
A wheel is clamped inside the water and is coupled to a block of mass M as
shown in the figure. As the block goes down, the wheel rotates. It is found
that after some time a steady state is reached in which the block goes down
with a constant speed of 10 cm/s and the temperature of the water remains
constant at 1.0°C Find the mass M of the block.
Assume that the heat flows out of the water only through the walls in
contact. Take g=10 m/s^{2}.

We know,

On a winter day when the atmospheric temperature drops
to -10°C, ice forms on the surface of a lake. (a) Calculate the
rate of increase of thickness of the ice when 10cm of ice is already formed.
(b) Calculate the total time taken in forming 10cm of ice. Assume that the
temperature of the entire water reaches 0°C before the ice starts forming. Density of water = 1000
kg/m^{3} latent heat of fusion of ice = 3.36×10^{5} J/kg and thermal conductivity of ice = 1.7
W/m-°C Neglect the expansion of water
on freezing.

We know,

Rearranging

(As Q = mL)

)

(b) Now, we consider a small ship of ice.

And,

Or

Or

Integrating

Consider the situation of the previous problem. Assume that the temperature of the water at the bottom of the lake remains constant at 4°C as the ice forms on the surface (the heat required to maintain the temperature of the bottom layer may come from the bed of the lake). The depth of the lake is 1.0m. Show that the thickness of the ice formed attains a steady state maximum value. Find this value. The thermal conductivity of water = 0.50 W/m-°C Take other relevant data from the previous problem.

Heat conducted at point above which ice cannot be formed is same from both levels. Thus,

Three rods of lengths 20cm each and area of cross -
section 1cm^{2} are joined to form a triangle ABC. The conductivities
of the rods are K_{AC} = 50 J/m-s-°C, K_{BC} = 200 J/m-s-°C and K_{AC} = 400 J/m-s- °C. Then junctions A, B and C are maintained at 40°C, 80°C
and 80°C respectively. Find the rate of
heat flowing through the rods AB, AC and BC.

(a) We know

(b)

(c)

A semicircular rod is joined at its end to a straight rod of the same material and the same cross - sectional area. The straight rod forms a diameter of the other rod. The junctions are maintained at different temperatures. Find the ratio of the heat transferred through a cross - section of the semicircular rod to the heat transferred through a cross - section of the straight rod in a given time.

We know,

And

Taking ratio we get

A metal rod of cross - sectional area 10cm^{2} is
being heated at one end. At one time, the temperature gradient is 5.0/cm at cross - section A and is 2.5°C /cm at cross - section B. Calculate the rate at which
the temperature is increasing in the part AB = 0.40J/°C, thermal conductivity of the material of the rod = 200
W/m-°C. Neglect any loss of heat to the
atmosphere.

We know,

And

Net

Also,

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

We know,

A hole of radius r_{1} is made centrally in a
uniform circular disc of thickness *d* and radius
r_{2}. The inner surface (a cylinder of length d and radius r_{1})
is maintained at a temperature θ_{1} and the outer surface (a cylinder
of length *d* and radius r_{2}) is
maintained at a temperature θ_{2}(θ_{1 }> θ_{2}). The thermal conductivity of the
material of the disc is K. Calculate the heat flowing per unit time through
the disc.

We know,

Now let

A hollow tube has a length *L,*
inner radius R_{1} and the outer radius R_{2}. The material has a thermal
conductivity K. Find the heat flowing through the walls of the tube if (a)
the flat ends are maintained at temperature T_{1} and T_{2 }(T_{1 }> T_{2}) and the inside of the tube is
maintained at temperature and the outside is
maintained at
T_{2}.

We know,

Cylinders are concentric and heat flow through them is:

Rearranging and integrating

A composite slab is prepared by pasting two plates of
thicknesses L_{1} and L_{2} and thermal conductivity K_{1} and K_{2}. The slabs have equal cross -
sectional area. Find the equivalent conductivity of the slab.

In this case thermal conductivities are in series with each other. Thus,

Solving we get

Figure shows a copper rod joined to a steel rod. The rods have equal length and equal cross - sectional area. The free end of the copper rod is kept at 0°C and that of the steel rod is kept at 100°C Find the temperature at the junction of the rods. Conductivity of copper = 390 W/m - °C and that of steel = 46 W/m - °C.

Both elements are in series with each other. Thus,

An aluminium rod and a copper rod of equal length 1.0m and
cross - sectional area 1cm^{2} are welded together as shown in
figure. One end is kept at a temperature of 20°C and the other at 60°C. Calculate the amount of heat taken out per second from
the hot end. Thermal conductivity of aluminium = 200 W/m - °C and of copper = 390 W/m -°C.

Since rod of the both elements are joined in parallel. Thus,

Also,

## Chapter 6 - Heat Transfer Exercise 100

Figure shows an aluminium rod joined to a copper rod. Each
of the rods has a length of 20cm and area of cross - section 0.20cm^{2}.
The junction is maintained at a constant temperature 40°C and the two ends are maintained at 80°C Calculate the amount of heat taken out from the cold
junction in one minute after the steady state is reached. The conductivities
are = 200 W/m - °C and K_{cu} = 400 W/m - °C.

Net heat per second is given as:

+

Heat drawn per minute is

Consider the situation shown in figure. The frame is made of the same material and has a uniform cross - sectional area everywhere. Calculate the amount of heat flowing per second through a cross - section of the bent part if the total heat taken out per second from the end at 100°C is 130 J.

According to Question

Also,

And,

Taking ratio, we get

= =

________(2)

Substituting equation (2) in (1) we get

Suppose the bent part of the frame of the previous problem has a thermal conductivity of 780 J/m-s-°C whereas it is 390 J/m-s-°C for the straight part. Calculate the ratio of the rate of heat flow through the bent part to the rate of heat flow through the straight part.

For bent part,

For straight part,

Dividing (1) and (2) we get,

A room has a window fitted with a single 1.0m × 2.0m glass of thickness 2mm. (a) Calculate the rate of heat flow through the closed window when the temperature inside the room is 32°C and that outside is 40°C (b) The glass is now replaced by two glasspanes, each having a thickness of 1mm and separated by a distance of 1mm. Calculate the rate of heat flow under the same conditions of temperature. Thermal conductivity of window glass = 1.0 J/m-s-°C and that of air = 0.025 J/m-s-°C

(a) We know,

=

(b) Resistance by glass is,

And resistance by air is

We know,

The two rods shown in figure have identical geometrical dimensions. They are in contact with two heat baths at temperatures 100°C and 0°C The temperature of the junction is 70°C Find the temperature of the junction if the rods are interchanged.

For case I:

And

Here

For case II:

And

Here

The three rods shown in figure have identical geometrical dimensions. Heat flows from the hot end at a rate of 40W in the arrangement (a). Find the rates of heat flow when the rods are joined as in arrangement (b) and in (c). Thermal conductivities of aluminium and copper are 200 W/m -°C and 400 W/m -°C respectively.

Net Resistance is:

Also,

(b) Now,

=

=

Also,

(c)

Also,

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.

According to Question,

Seven rods A, B, C, D, E, F and G are joined as shown in
figure. All the rods have equal cross - sectional area A and the length l.
The thermal conductivities of the rods are K_{A}= K_{C}= K_{0},
K_{B}= K_{D}= 2K_{0}, K_{E}=
3K_{0}, K_{F}= 4K_{0} and K_{G}= 5K_{0}.The rod E is kept at a constant
temperature T_{1} and the rod G is kept at a constant temperature T_{2}(T_{2
}> T_{1}) (a) Show that the
rod F has a uniform temperature (b) Find the rate
of heat flow from the source which maintains the temperature.

In bar F, temperature at both ends is same:

As

Find the rate of heat flow through a cross - section of
the rod shown in figure (θ_{2} > θ_{1}) Thermal conductivity of the
material of the rod is K.

We Know,

And

Simplifying and differentiating w.r.t 'x'

- 0

Also,

Integrating

## Chapter 6 - Heat Transfer Exercise 101

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

We Know,

Also,

Solving we get

A hollow metallic sphere of radius 20cm surrounds a concentric metallic sphere of radius 5cm. The space between the two spheres is filled with a nonmetallic material. The inner and outer spheres are maintained at 50°C and 10°C respectively and it is found that 100 J of heat passes from the inner sphere to the outer sphere per second. Find the thermal conductivity of the material between the spheres.

For small strip having thickness 'dr', we have,

Rearranging and integrating we get,

Solving we get

Figure shows two adiabatic vessels, each containing a mass m of water at different temperatures. The ends of a metal rod of length L, area of cross - section A and thermal conductivity K, are inserted in the water as shown in the figure. Find the time taken for the difference between the temperatures in the vessels to become half of the original value. The specific heat capacity of water is S. Neglect the heat capacity of the rod and the container and any loss of heat to the atmosphere.

We know,

Rise in temperature is :

And Fall in temperature is :

Also, Final temperatures are given as :

And

Now, net temperature is

=

Rearranging and integrating we get

Two bodies of masses and specific heat
capacities s_{1} and s_{2} are connected by a rod of length
l, cross-sectional area A, thermal conductivity K and negligible heat
capacity. The whole system is thermally insulated. At time t = 0, the
temperature of the first body is and the temperature of the second body is T_{2} (T_{2}> T_{1}) Find the
temperature difference between the two bodies at time t.

We know,

Rise in temperature is :

And Fall in temperature is :

Final temperature are given as:

And

And also,

Solving we get

Integrating we get

An amount n (in moles) of a monatomic gas at an initial temperature is enclosed in a cylindrical vessel fitted with a light piston. The surrounding air has a temperature and the atmospheric pressure is . Heat may be conducted between the surrounding and the gas through the bottom of the cylinder. The bottom has a surface area A, thickness and thermal conductivity K. Assuming all changes to be slow, find the distance moved by the piston in time t.

We know,

Also

Integrating and solving we get

And

Assume that the total surface area of a human body is 1.6cm^{2}
and that it radiates like an ideal radiator. Calculate the amount of energy
radiated per second by the body if the body temperature is 37°C. Stefan constant.

Energy radiated is given as:

Calculate the amount of heat radiated per second by a body
of surface area 12cm^{2} kept in thermal equilibrium in a room at
temperature 20°C. The emissivity of the surface =
0.80 and .

We know,

A solid aluminium sphere and a solid copper sphere of twice the radius are heated to the same temperature and are allowed to cool under identical surrounding temperatures. Assume that the emissivity of both the spheres is the same. Find the ratio of (a) the rate of heat loss from the aluminium sphere to the rate of heat loss from the copper sphere and (b) the rate of fall of temperature of the aluminium sphere to the rate of fall of temperature of the copper sphere. The specific heat capacity of aluminium = 900 J/kg -°C and that of copper = 390 J/kg - °C. The density of copper = 3.4 times the density of aluminium.

(a) Heat loss is the energy radiated

Ratio is 1:4

(b) Emissivity is same in both cases

Ratio is 2.9:1

A 100 W bulb has tungsten filament of length 1.0m and
radius 4×10^{-5}m. The emissivity
of the filament is 0.8 and . Calculate the temperature of the filament when the bulb
is operating at correct wattage.

We know,

A spherical ball of surface area absorbs any radiation that falls on it. It is suspended in a closed box maintained at 57°C. (a) Find the amount of radiation falling on the ball per second. (b) Find the net rate of heat flow to or from the ball at an instant when its temperature is 200°C. Stefan constant.

(a) We know,

(b) Also,

A spherical tungsten piece of radius 1.0cm is suspended in an evacuated chamber maintained at 300K. The piece is maintained at 1000K by heating it electrically. Find the rate at which the electrical energy must be supplied. The emissivity of tungsten is 0.30 and the Stefan constant .

We know

A cubical block of mass 1.0 kg and edge 5.0 cm is heated to 227°C It is kept in an evacuated chamber maintained at 27°C Assuming that the block emits radiation like a blackbody, find the rate at which the temperature of the block will decrease. Specific heat capacity of the material of the block is 400 J/kg - K.

Let us assume that cube is a black body

Thus,

A copper sphere is suspended in an evacuated chamber maintained at 300K. The sphere is maintained at a constant temperature of 500K by heating it electrically. A total of 210W of electric power is needed to do it. When the surface of the copper sphere is completely blackened, 700W is needed to maintain the same temperature of the sphere. Calculate the emissivity of copper.

We know

In case of anybody:

And in case of black body:

Dividing equation (1) and (2) we get

A spherical ball A of surface area 20cm^{2} is
kept at the centre of a hollow spherical shell B of area 80cm^{2}.
The surface of A and the inner surface of B emit as blackbodies. Assume that
the thermal conductivity of the material of B is very poor and that of A is
very high and that the air between A and B has been pumped out. The heat
capacities of A and B are 42 J/°C
and 82 J/°C respectively. Initially, the
temperature of A is 100 and that of B is 20°C. Find the rate of change of temperature of A and that of
B at this instant. Explain the effects of the assumptions listed in the
problem.

K_{A} is high so it will
conduct all heat while K_{B} is low so it will so it will act as poor
conductor.

Also

Similarly

A cylindrical rod of length 50 cm and cross - sectional area is fitted between a large ice chamber at 0°C and an evacuated chamber maintained at 27°C as shown in figure. Only small portions of the rod are inside the chambers and the rest is thermally insulated from the surrounding. The cross - section going into the evacuated chamber is blackened so that it completely absorbs any radiation falling on it. The temperature of the blackened end is 17°C when steady state is reached. Stefan constant . Find the thermal conductivity of the material of the rod.

We know,

Also,

From Equation (1) and (2)

## Chapter 6 - Heat Transfer Exercise 102

One end of a rod of length 20cm is inserted in a furnace at 800K. The sides of the rod are covered with an insulating material and the other end emits radiation like a blackbody. The temperature of this end is 750K in the steady state. The temperature of the surrounding air is 300K. Assuming radiation to be the only important mode of energy transfer between the surrounding and the open end of the rod, find the thermal conductivity of the rod. Stefan constant .

We know,

Also,

A calorimeter of negligible heat capacity contains 100cc
of water at 40°C. The water cools to 35°C in 5 minutes. The water is now replaced by K-oil of
equal volume at 40°C Find the time taken for the
temperature to become 35°C
under similar conditions. Specific heat capacities of water and K-oil are
4200 J/kg - K and 2100 J/kg - K respectively. Density of K-oil = 800 kg/m^{3}.

In case of water,

In case of Kerosene,

Equating (1) and (2) we get

A body cools down from 50°C to 45°C in 5 minutes and to 40°C in another 8 minutes. Find the temperature of the surrounding.

I st CASE:

Average temperature is given as:

And Average temperature that is different from surroundings is:

Now,

Rate of fall of temperature =

And also Rate of fall of temperature =

__________(1)

II nd CASE:

Average temperature is given as:

And Average temperature that is different from surroundings is:

Now,

Rate of fall of temperature =

And also Rate of fall of temperature =

A calorimeter contains 50g of water at 50°C. The temperature falls to 40°C in 10 minutes. When the calorimeter contains 100g of water at 50°C, it takes 18 minutes for the temperature to become 45°C Find the water equivalent of the calorimeter.

We know,

Rate of heat flow is given as:

And

A metal ball of mass 1 kg is heated by means of a 20W heater in a room at 20°C The temperature of the ball becomes steady at 50°C. (a) Find the rate of loss of heat to the surrounding when the ball is at 50°C. (b) Assuming Newton's law of cooling, calculating the rate of loss of heat to the surrounding when the ball is at 30°C (c) Assume that the temperature of the ball rises uniformly from 20°C to 30°C in 5 minutes. Find the total loss of heat to the surrounding during this period. (d) Calculate the specific heat capacity of the metal.

(a) We know,

Also,

P = H (As in steady state there is no loss of heat)

(b) We know,

(By Newton's law of cooling)

Now again,

(c) Here net heat absorbed is:

A metal block of heat capacity 80 J/°C placed in a room at 20°C is heated electrically. The heater is switched off when the temperature reaches 30°C The temperature of the block rises at the rate of 2°C/s just after the heater is switched on and falls at the rate of 0.2°C/s just after the heater is switched off. Assume Newton's law of cooling to hold. (a) Find the power of the heater. (b) Find the power radiated by the block just after the heater is switched off. (c) Find the power radiated by the block when the temperature of the block is 25°C represents the average value in the heating process, find the time for which the heater was kept on.

(a) Power of heater is given as:

(b) Power Radiated is given as:

(c) We know,

And

A hot body placed in a surrounding of temperature θ_{0} obeys Newton's law of cooling . Its temperature at t = 0 is θ_{1}. The specific heat capacity of
the body is *s* and its mass is *m*. Find (a) The maximum heat that the body can lose and
(b) the time starting from t = 0 in which it will lose 90% of this maximum heat.

(a) Maximum heat loss by body is given as:

(b) If 90% of max. heat is lost then, temperature decrease is:

Temperature at

Also

Integrating , we get

(From (1))

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