Heat Transfer theory

HEAT TRANSFER

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Published at : Ascent Gate Academy
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TABLE OF CONTENTS

UNIT CHAPTER PAGE NO.
2–9
1 Basic of Heat Transfer 10 – 30
31 – 38
2 Conduction 39 – 51
52 - 55
3 Convection

4 Radiation

5 Heat Exchanger

1. Basics of Heat Transfer

1.1 Thermodynamics and Heat transfer

We all know from experience that a cold canned drink left in a room warms up and a warm
canned drink left in a refrigerator cools down. This is accomplished by the transfer of energy
from the warm medium to the cold one. The energy transfer is always from the higher
temperature medium to the lower temperature one, and the energy transfer stops when the
two mediums reach the same temperature.

You will recall from thermodynamics that energy exists in various forms. In this text we are
primarily interested in heat, which is the form of energy that can be transferred from one
system to another as a result of temperature difference. The science that deals with the
determination of the rates of such energy transfers is heat transfer.

You may be wondering why we need to undertake a detailed study on heat transfer. After
all, we can determine the amount of heat transfer for any system undergoing any process
using a thermodynamic analysis alone. The reason is that thermodynamics is concerned with
the amount of heat transfer as a system undergoes a process from one equilibrium state to
another, and it gives no indication about how long the process will take. A thermodynamic
analysis simply tells us how much heat must be transferred to realize a specified change of
state to satisfy the conservation of energy principle.

In practice we are more concerned about the rate of heat transfer (heat transfer per unit
time) than we are with the amount of it. For example, we can determine the amount of heat
transferred from a thermos bottle as the hot coffee inside cools from 90°C to 80°C by a
thermodynamic analysis alone. But a typical user or designer of a thermos is primarily
interested in how long it will be before the hot coffee inside cools to 80°C, and a
thermodynamic analysis cannot answer this question. Determining the rates of heat transfer to
or from a system and thus the times of cooling or heating, as well as the variation of the
temperature, is the subject of heat transfer (Fig. 1–1).

Fig. 1-1

Thermodynamics deals with equilibrium states and changes from one equilibrium state to
another. Heat transfer, on the other hand, deals with systems that lack thermal equilibrium,
and thus it is a non-equilibrium phenomenon. Therefore, the study of heat transfer cannot be
based on the principles of thermodynamics alone. However, the laws of thermodynamics lay
the frame-work for the science of heat transfer. The first law requires that the rate of energy
transfer into a system be equal to the rate of increase of the energy of that system. The second
law requires that heat be transferred in the direction of decreasing temperature. This is like a
car parked on an inclined road that must go downhill in the direction of decreasing elevation
when its brakes are released. It is also analogous to the electric current flowing in the
direction of decreasing voltage or the fluid flowing in the direction of decreasing total
pressure.

The basic requirement for heat transfer is the presence of a temperature difference. There can
be no net heat transfer between two mediums that are at the same temperature. The
temperature difference is the driving force for heat transfer, just as the voltage difference is
the driving force for electric current flow and pressure difference is the driving force for fluid
flow. The rate of heat transfer in a certain direction depends on the magnitude of the
temperature gradient (the temperature difference per unit length or the rate of change of
temperature) in that direction. The larger the temperature gradient, the higher the rate of heat
transfer.

1.2 Applications

Heat transfer is commonly encountered in engineering systems and other aspects of life, and
one does not need to go very far to see some application areas of heat transfer. In fact, one
does not need to go anywhere. The human body is constantly rejecting heat to its
surroundings, and human comfort is closely tied to the rate of this heat rejection. We try to
control this heat transfer rate by adjusting our clothing to the environmental conditions.

Many ordinary household appliances are designed, in whole or in part, by using the principles
of heat transfer. Some examples include the electric or gas range, the heating and air-
conditioning system, the refrigerator and freezer, the water heater, the iron, and even the
computer, the TV, and the VCR. Of course, energy-efficient homes are designed on the basis
of minimizing heat loss in winter and heat gain in summer. Heat transfer plays a major role in

the design of many other devices, such as car radiators, solar collectors, various components
of power plants, and even spacecraft. The optimal insulation thickness in the walls and roofs
of the houses, on hot water or steam pipes, or on water heaters is again determined on the
basis of a heat transfer analysis with economic consideration (Fig. 1–2)

Fig. 1-2. Some of the applications of Heat Transfer

1.3 Modes of Heat Transfer

Heat can be transferred in three different modes:
1. conduction,
2. convection, and
3. Radiation.

All modes of heat transfer require the existence of a temperature difference, and all modes are
from the high-temperature medium to a lower temperature one. Below we give a brief
description of each mode. A detailed study of these modes is given in later chapters of this
text.

1-3-1. Conduction

Conduction is the transfer of energy from the more energetic particles of a substance to the
adjacent less energetic ones as a result of interactions between the particles. Conduction can
take place in solids, liquids, or gases. In gases and liquids, conduction is due to the collisions
and diffusion of the molecules during their random motion. In solids, it is due to the
combination of vibrations of the molecules in a lattice and the energy transport by free
electrons. A cold canned drink in a warm room, for example, eventually warms up to the
room temperature as a result of heat transfer from the room to the drink through the
aluminum can by conduction.

Mechanism of Conduction

In solids conduction is due to Lattice Vibration and/or movement of free electrons.The
molecules, atoms, electrons which are in immediate vicinity of the flame get heated and start
vibrating about their mean position because there kinetic energy increases (Fig. 1-3)

Fig. 1-3
These more active particles collide with less active molecules lying next to them. During
collision the less active particles also get excited i.e. thermal energy is imparted to them (Fig.
1-4)

Fig. 1-4
This process is repeated for layer after layer of molecules until the end of rod is reached.Each
layer of molecule is at a higher temperature than the preceding one (Fig. 1-5)

Fig. 1-5

Governing equation for conduction -

Governing law for conduction is Fourier's law after J. Fourier, who expressed it first in his
heat transfer text in 1822. According to Fourier’s law the rate of heat conduction through a
plane layer is proportional to the temperature gradient and the heat transfer area, thus

A

Q ∝ −A dT T1 Q
dx T2

x
T1> T2

Here dT/dx is the temperature gradient, which is the slope of the temperature curve on aT-x
diagram (the rate of change of T with x), at location x. The relation above indicates that the
rate of heat conduction in a direction is proportional to the temperature gradient in that
direction. Heat is conducted in the direction of decreasing temperature, and the temperature
gradient becomes negative when temperature decreases with increasing x. The negative sign
in Equation ensures that heat transfer in the positive x direction is a positive quantity.

The constant of proportionality k is the thermal conductivity of the material, which is a
measure of the ability of a material to conduct heat and it is the property of the material. So
the Fourier’s law becomes,

Q = −kA dT
dx

S.I. Units –

Q = Conductive Rate of heat transfer (Watt)

A = Area perpendicular to the direction of heat flow (m2)

T = Absolute temperature (K)

k= Thermal Conductivity (W/m K)

1-3-2. Convection

Convection is the mode of energy transfer between a solid surface and the adjacent liquid or
gas that is in motion, and it involves the combined effects of conduction and fluid motion.
The faster the fluid motion, the greater the convection heat transfer. In the absence of any
bulk fluid motion, heat transfer between a solid surface and the adjacent fluid is by pure

conduction. The presence of bulk motion of the fluid enhances the heat transfer between the
solid surface and the fluid, but it also complicates the determination of heat transfer rates.
Consider the cooling of a hot block by blowing cool air over its top surface (Fig. 1–6).
Energy is first transferred to the air layer adjacent to the block by conduction. This energy is
then carried away from the surface by convection, that is, by the combined effects of
conduction within the air that is due to random motion of air molecules and the bulk or
macroscopic motion of the air that removes the heated air near the surface and replaces it by
the cooler air.

Fig. 1-6
Types of Convection
Convection is called forced convection if the fluid is forced to flow over the surface by
external means such as a fan, pump, or the wind. In contrast, convection is called natural (or
free) convection if the fluid motion is caused by buoyancy forces that are induced by density
differences due to the variation of temperature in the fluid (Fig. 1-7)

Fig. 1-7. The cooling of a boiled egg by forced and natural convection.
For example, in the absence of a fan, heat transfer from the surface of the hot block in Fig. 1–
6 will be by natural convection since any motion in the air in this case will be due to the rise
of the warmer (and thus lighter) air near the surface and the fall of the cooler (and thus
heavier) air to fill its place. Heat transfer between the block and the surrounding air will be by
conduction if the temperature difference between the air and the block is not large enough to
overcome the resistance of air to movement and thus to initiate natural convection currents.
Heat transfer processes that involve change of phase of a fluid are also considered to be
convection because of the fluid motion induced during the process, such as the rise of the
vapor bubbles during boiling or the fall of the liquid droplets during condensation.

Governing equation for Convection

Despite the complexity of convection, the rate of convection heat transfer is observed to be
proportional to the temperature difference, and is conveniently expressed by Newton’s law of
cooling as

Q = hA (Ts − T )

Where, h is the convection heat transfer coefficient in W/m2 · °C.A is the surface area through
which convection heat transfer takes place, Ts Is the surface temperature, and T is the
temperature of the fluid sufficiently far from the surface. Note that at the surface, the fluid
temperature equals the surface temperature of the solid.

The convection heat transfer coefficient h is not a property of the fluid. It is an
experimentally determined parameter whose value depends on all the variables influencing
convection such as the surface geometry, the nature of fluid motion, the properties of the
fluid, and the bulk fluid velocity.

1-3-3. Radiation

Radiation is the energy emitted by matter in the form of electromagnetic waves (or photons)
as a result of the changes in the electronic configurations of the atoms or molecules. Unlike
conduction and convection, the transfer of energy by radiation does not require the presence
of an intervening medium. In fact, energy transfer by radiation is fastest (at the speed of light)
and it suffers no attenuation in a vacuum. This is how the energy of the sun reaches the earth.

In heat transfer studies we are interested in thermal radiation, which is the form of radiation
emitted by bodies because of their temperature. It differs from other forms of electromagnetic
radiation such as x-rays, gamma rays, microwaves, radio waves, and television waves that are
not related to temperature. All bodies at a temperature above absolute zero emit thermal
radiation.

Radiation is a volumetric phenomenon, and all solids, liquids, and gases emit, absorb, or
transmit radiation to varying degrees. However, radiation is usually considered to be a
surface phenomenon for solids that are opaque to thermal radiation such as metals, wood, and
rocks since the radiation emitted by the interior regions of such material can never reach the
surface, and the radiation incident on such bodies is usually absorbed within a few microns
from the surface.

Important points about Radiation
• Every substance above absolute zero (0 k) emits radiation. These radiations are in the
form of electromagnetic waves.

• These waves travel with the speed of light (3×108 m/s).

• Radiation emitted over wavelength of 0.1 µm to 100 µm is known as thermal
radiation.

• Unlike Conduction and Convection, no material medium is required for radiation.

Governing law for Thermal Radiation

Thermal Radiation is governed by Stefan- Boltzmann law and the maximum rate of radiation
that can be emitted from a surface at an absolute temperature T(in K) is given by

E = σAT 4

E = Rate of Radiative Heat Transfer (W)

σ = Stefan- Boltzmann constant (5.67×10-8) W/m2 K4

2. Conduction

Heat transfer has direction as well as magnitude. The rate of heat conduction in a specified
direction is proportional to the temperature gradient, which is the change in temperature per
unit length in that direction. Heat conduction in a medium, in general, is three-dimensional
and time dependent. That is, T T(x, y, z, t) and the temperature in a medium varies with
position as well as time. Heat conduction in a medium is said to be steady when the
temperature does not vary with time, and unsteady or transient when it does. Heat conduction
in a medium is said to be one-dimensional when conduction is significant in one dimension
only and negligible in the other two dimensions, two-dimensional when conduction in the
third dimension is negligible, and three-dimensional when conduction in all dimensions is
significant.

Thermal Conductivity (k)

We have seen that different materials store heat differently, and we have defined the property
specific heat Cp as a measure of a material’s ability to store thermal energy. For example, Cp=
4.18 kJ/kg · °C for water and Cp= 0.45 kJ/kg · °C for iron at room temperature, which
indicates that water can store almost 10 times the energy that iron can per unit mass.
Likewise, the thermal conductivity k is a measure of a material’s ability to conduct heat. For
example, k= 0.608 W/m · °C for water and k = 80.2 W/m · °C for iron at room temperature,
which indicates that iron conducts heat more than 100 times faster than water can. Thus we
say that water is a poor heat conductor relative to iron, although water is an excellent medium
to store thermal energy.
The thermal conductivities of materials vary over a wide range, as shown in Fig. 2-1. The
thermal conductivities of gases such as air vary by a factor of 104 from those of pure metals
such as copper. Note that pure crystals and metals have the highest thermal conductivities, and
gases and insulating materials the lowest.

Fig. 2-1
Temperature is a measure of the kinetic energies of the particles such as the molecules or atoms
of a substance. In a liquid or gas, the kinetic energy of the molecules is due to their random

translational motion as well as their vibrational and rotational motions. When two molecules
possessing different kinetic energies collide, part of the kinetic energy of the more energetic
(higher-temperature) molecule is transferred to the less energetic (lower- temperature)
molecule, much the same as when two elastic balls of the same mass at different velocities
collide, part of the kinetic energy of the faster ball is transferred to the slower one. The higher
the temperature, the faster the molecules move and the higher the number of such collisions,
and the better the heat transfer.

The kinetic theory of gases predicts and the experiments confirm that the thermal
conductivity of gases is proportional to the square root of the absolute temperature T, and
inversely proportional to the square root of the molar mass M. Therefore, the thermal
conductivity of a gas increases with increasing temperature and decreasing molar mass. So it
is not surprising that the thermal conductivity of helium (M 4) is much higher than those of
air (M 29) and argon (M 40).

The mechanism of heat conduction in a liquid is complicated by the fact that the molecules are
more closely spaced, and they exert a stronger intermolecular force field. The thermal
conductivities of liquids usually lie between those of solids and gases. The thermal
conductivity of a substance is normally highest in the solid phase and lowest in the gas phase.
Unlike gases, the thermal conductivities of most liquids decrease with increasing
temperature, with water being a notable exception. Like gases, the conductivity of liquids
decreases with increasing molar mass. Liquid metals such as mercury and sodium have high
thermal conductivities and are very suitable for use in applications where a high heat transfer
rate to a liquid is desired, as in nuclear power plants

In solids, heat conduction is due to two effects: the lattice vibrational waves induced by the
vibrational motions of the molecules positioned at relatively fixed positions in a periodic
manner called a lattice, and the energy trans-ported via the free flow of electrons in the solid
(Fig. 2-2). The thermal conductivity of a solid is obtained by adding the lattice and electronic
components. The relatively high thermal conductivities of pure metals are primarily due to
the electronic component. The lattice component of thermal conductivity strongly depends
on the way the molecules are arranged. For example, diamond, which is a highly ordered
crystalline solid, has the highest known thermal conductivity at room temperature.

Fig. 2-2. The mechanisms of heat conduction in different phases of a substance.

Unlike metals, which are good electrical and heat conductors, crystallinesolids such as
diamond and semiconductors such as silicon are good heat con-ductors but poor electrical
conductors. As a result, such materials find wide-spread use in the electronics industry.
Despite their higher price, diamond heat sinks are used in the cooling of sensitive electronic
components because of the excellent thermal conductivity of diamond. Silicon oils and
gaskets are commonly used in the packaging of electronic components because they provide
both good thermal contact and good electrical insulation.

Pure metals have high thermal conductivities, and one would think that metal alloys should
also have high conductivities. One would expect an alloy made of two metals of thermal
conductivities k1 and k2 to have a conductivity k between k1 and k2. But this turns out not to
be the case. The thermal conductivity of an alloy of two metals is usually much lower than
that of either metal, as shown in Table 1–2. Even small amounts in a pure metal of “foreign”
molecules that are good conductors themselves seriously disrupt the flow of heat in that
metal. For example, the thermal conductivity of steel containing just 1 percent of chrome is
62 W/m · °C, while the thermal conductivities of iron and chromium are 83 and 95 W/m · °C,
respectively.

Thermal Diffusivity (α)

The product ρCp, which is frequently encountered in heat transfer analysis, is called the heat
capacity of a material. Both the specific heat Cp and the heat capacity ρCp represent the heat
storage capability of a material. But Cp expresses it per unit mass whereas ρCp expresses it
per unit volume, as can be noticed from their units J/kg · °C and J/m3 · °C, respectively.

Another material property that appears in the transient heat conduction analysis is the thermal
diffusivity, which represents how fast heat diffuses through a material and is defined asRatio
of heat conducted and heat stored

α = k m2
ρ c sec

Note that the thermal conductivity k represents how well a material conducts heat, and the
heat capacity Cp represents how much energy a material stores per unit volume. Therefore,
the thermal diffusivity of a material can be viewed as the ratio of the heat conducted through
the material to the heat stored per unit volume. A material that has a high thermal
conductivity or a low heat capacity will obviously have a large thermal diffusivity. The larger
the thermal diffusivity, the faster the propagation of heat into the medium. Asmall value of
thermal diffusivity means that heat is mostly absorbed by the material and a small amount of
heat will be conducted further.

2-1. GENERAL HEAT CONDUCTION EQUATION

2-1-1. Rectangular Coordinates

Consider a small rectangular element of length x, width y, and height z, as shown in
Figure 2–3. Assume the density of the body is ρ and the specific heat is C. An energy balance
on this element during a small time interval t can be expressed as

⎛ Rate of heat ⎞ ⎛ Rate of heat ⎞ Right of heat ⎞ ⎛ Rate of change ⎞
⎜ conduction ⎟⎛ ⎜ ⎟
⎜ ⎟ ⎜ at x + x+x, ⎟ ⎜ ⎟ ⎜ of the energy ⎟
⎜ conduction at ⎟ − ⎜ + +y, and z + ⎟ + ⎜ generation ⎟ = ⎟
⎝⎜ x, y and z ⎠⎟ ⎜ ⎟ ⎝⎜ inside the element ⎟⎠ ⎜ content of ⎟
⎝ ⎠ ⎜ ⎠
y +z ⎝ the element

Or

Q x + Q y + Qz − Qx ++x −Q y ++y −Qz ++z +Gelement = +
Element
+t

Where, G or Q g= Heat generated in the element. (For per unit volume, g or qg will be used)

……………..2‐1
Above equation is thegeneral heat conduction equation in rectangular coordinates. In the case
of constant thermal conductivity, it reduces to

……………..2‐2 
Where the property α=k/ρC is again the thermal diffusivity of the material. Above Equation is
known as the Fourier-Biot equation,and it reduces to these forms under specified conditions:

2-1-2. Cylindrical Coordinates

The general heat conduction equation in cylindrical coordinates can be obtained from an
energy balance on a volume element in cylindrical coordinates, shown in Figure 2–3.
It can also be obtained directly from Eq. 2–1 by coordinate transformation using the
following relations between the coordinates of a point in rectangular and cylindrical
coordinate systems

Fig. 2-3

After lengthy manipulations, we obtain

………….2-3

2-1-3. Spherical Coordinates

The general heat conduction equations in spherical coordinates can be obtained from an
energy balance on a volume element in spherical coordinates, shown in Figure 2–4, by
following the steps outlined above. It can also be obtained directly from Eq. 2–1 by
coordinate transformation using the following relations between the coordinates of a point in
rectangular and spherical coordinate systems:

Fig. 2-4
Again after lengthy manipulations, we obtain

…………2-4

2-2. Steady state, 1-D conduction with no heatgeneration.
2-2-1. Plane slabs

The most simple conduction situation consists of one
dimension, steady heat transfer without sources or sinks
of heat. Consider, for example, the plane wall illustrated
in Fig. 2-5. In the Cartesian system, the conduction
equation reduces to the ordinary differential equation:

Assume that the boundary conditions have T = T1 and T
= T2 for x = 0 and L. The solution to this problem is the
Fig. 2-5
Familiar linear profile:

The numerical value of the rate of heat conduction through the wall is determined by
substituting the given values to be

Electrical Analogy –

By ohm’s law we know

V=IR

Here in heat transfer Temperature difference is analogous to potential difference, Rate of heat
transfer is analogous to flow of current. Then the thermal resistance can be given by,

(t1 – t2) = Q R th

Rth = L
kA

Heat Transfer through a composite wall
Let,

Then the heat conduction equations can be given by,

Wall with convection at end surfaces
Let,

Convective thermal resistance Q = hA ( Ts − T )
So,
(t1 – t2) = Q R th

1
.

Then the heat transfer equations can be given by,

Overall Heat transfer coefficient ( U )
Q = UA(thf − tcf )

2-2-2. Conduction through Cylinders

The generalized equation in cylindrical coordinate for 1-D heat transfer is given by

() 0 T2 Logarithmic temp. 
profile
1 . ∂ r. ∂t + qg = 0
r ∂r ∂r k

∫ ∫r21 T 2 2πkLdT T1 +r1
dr r2
= −
r Qr1 T 1

ln r 2 = 2πkL(T 1 − T 2)
r1 Q

Q = 2πkL(T 1 − T 2) T1 > T2
ln r2
r1

R th = ln r2
r1

2 π kL

Conduction through multilayered Cylinder with convection
Let,

hcf = Heat transfer coefficient for cold fluid h T
Tc = Temp. of cold fluid cf h
hhf = Heat transfer coefficient for hot fluid
Th = Temp. of hot fluid T h
c hf
Q = T1−T3 T T
∑ Rth 3
T1
2

Q = Th − Tc
ln r 2 r1 ln r 3 r 2
hhf 1 + 2πk 1L + 2πk 2 L + hcf 1
2πr1L 2πr 2L

2-2-3. Conduction in Spherical coordinate T
H2 yperbolic temp. profile
Assumptions-

i. Steady state Heat transfer
ii. 1 –D Heat transfer
iii. Homogeneous and isotropic material

1 ∂ r 2⎛⎜ ∂t ⎟⎞ + qg = 0 T
r2 ∂r ⎝ ∂r ⎠ k 1r

Q = 4πkr1r 2(T 1 − T 2) 1
r2 − r1
+

r
2

T > T
1  2

R th = r2 − r1
4π kr 1r 2

2-3. Critical thickness of Insulation
2-3-1. Plane slabs

By adding insulation there is no change in surface area so adding insulation would increase
conductive resistance but convective resistance will remain same. Hence heat transfer will
decrease by adding insulation. So here concept of critical thickness of insulation has no
significance.

2-3-2. Cylinders

r1= Radius of pipe

r = Radius of Insulation

T1 = Temperature of pipe

K ins = Thermal conductivity of Insulating material

hsur= convective heat transfer coefficient of surrounding fluid

T ∞ = Temperature of surrounding fluid

The conductive resistance for the insulator is given by,

R cond = ln r r1
2πkinsL

With increase in radius of insulation the conductive resistance will increase.

The convective thermal resistance is given by,

1
Rconv = hs × 2πrL

With increase in radius of insulation the convective resistance will decrease.

We can see that with increase in radius of insulation, conductive resistance increases and
convective resistance decreases.
The rate of heat transfer can be given by,

Q = T1 − T∞
Rcond + Rconv

So we can say by increasing radius of insulation rate of Heat transfer may increase or may
decrease.
To find the maximum heat transfer rate

dQ = 0
dr
We get the Critical radius of insulation for cylinder (rcr ) which is given by,

rcr = kins
hsur

The variation of rate of heat transfer with the radius is shown in Fig. 2-7

r − rCritical Thickness = cr 1

2-3-3. Spheres
The concept is similar to the cylinders.

Critical radius for sphere is given by,

(r )cr sphere = 2kins
hsur

2-4. Conduction with Heat generation

Many practical heat transfer applications involve the conversion of some form of energy into
thermal energy in the medium. Such mediums are said to involve internal heat generation,
which manifests itself as a rise in temperature throughout the medium. Some examples of
heat generation are resistance heating in wires, exothermic chemical reactions in a solid, and
nuclear reactions in nuclear fuel rods where electrical, chemical, and nuclear energies are
converted to heat, respectively.

Heat generation is usually expressed per unit volume of the medium, and is denoted by g ·,
whose unit is W/m3. For example, heat generation in an electrical wire of outer radius r0 and
length L can be expressed as

Where I is the electric current and Re is the electrical resistance of the wire. The temperature
of a medium rises during heat generation as a result of the absorption of the generated heat by
the medium during transient start-up period. As the temperature of the medium increases, so
does the heat transfer from the medium to its surroundings. This continues until steady
operating conditions are reached and the rate of heat generation equals the rate of heat
transfer to the surroundings. Once steady operation has been established, the temperature of
the medium at any point no longer changes.

2-4-1. Steady 1-D heat transfer through plane slab with heat generation

The generalized equation is,

In 1-D ∂ 2t + qg = 0 Max 
Integrate twice and get, ∂x 2 k
T
2 1 2 T 2
2. 1
X=0 L
Depending on the boundary condition this equation will give x X=L
Temperature profile.

Note- the temperature profile in case of Heat generation in plane slabs is parabolic.

2-4-2. Steady 1-D heat transfer through Cylinders with heat generation

The generalized equation is,

( )1 . ∂ r. ∂t + qg = 0 T
s
r ∂r ∂r k
Integrating twice we get, R
r = 0

T = − qgenr 2 + C1ln(r) + C 2 h, T∞
4k

1. For solid cylinder
Boundary conditions, at r = R, T = Ts

dT = 0
at r = 0, dr

We get,

( )T qg
= Ts + 4k R 2 −r 2

Energy balance,

qg ⎜⎛ πr 2 L ⎟⎞ = h ( 2πrL )(T s − T∞) = −k (2πrL)⎜⎛ dT ⎟⎞
⎝ ⎠ ⎝ dr ⎠

2. For hollow cylinder-
Boundary conditions, at r=R1, T=T1

at r=R2, T =T2

2-5. Unsteady state conduction

The temperature of a body, in general, varies with time as well as position. In rectangular
coordinates, this variation is expressed as T(x,y,z,t), where (x,y,z) indicates variation in the x,
y, and z directions, respectively, and t indicates variation with time. In the preceding chapter,
we considered heat conduction under steady conditions, for which the temperature of a body
at any point does not change with time. This certainly simplified the analysis, especially
when the temperature varied in one direction only, and we were able to obtain analytical
solutions. In this chapter, we consider the variation of temperature with time as well as
position in one- and multidimensional systems.

2-5-1. Lumped System analysis

In heat transfer analysis, some bodies are observed to behave like a “lump” whose interior
temperature remains essentially uniform at all times during a heat transfer process. The
temperature of such bodies can be taken to be a function of time only, T(t). Heat transfer
analysis that utilizes this idealization is known as lumped system analysis, which provides
great simplification in certain classes of heat transfer problems without much sacrifice from
accuracy.
Consider a small hot copper ball coming out of an oven (Fig. 2-8). Measurements indicate
that the temperature of the copper ball changes with time, but it does not change much with
position at any given time. Thus the temperature of the ball remains uniform at all times, and
we can talk about the temperature of the ball with no reference to a specific location.

Fig. 2-8
Now let us go to the other extreme and consider a large roast in an oven. If you have done
any roasting, you must have noticed that the temperature distribution within the roast is not
even close to being uniform. You can easily verify this by taking the roast out before it is
completely done and cutting it in half. You will see that the outer parts of the roast are well
done while the center part is barely warm. Thus, lumped system analysis is not applicable in
this case. Before presenting a criterion about applicability of lumped system analysis, we
develop the formulation associated with it.

Criteria for lumped system-

The lumped system analysis certainly provides great convenience in heat transfer analysis,
and naturally we would like to know when it is appropriate to use it. The first step in

establishing a criterion for the applicability of the lumped system analysis is to define a
characteristic length as

L= V
As

Biot number (Bi) – Ratio of Internal Conductive resistance and external convective
resistance.

L = hL = Bi
kA ksolid

1
hA

When a solid body is being heated by the hotter fluid surrounding it (such as a potato being
baked in an oven), heat is first convected to the body and subsequently conducted within the
body. The Biot number is the ratio of the internal resistance of a body to heat conduction to
its external resistance to heat convection. Therefore, a small Biot number represents small
resistance to heat conduction, and thus small temperature gradients within the body.

It is generally accepted that lumped system analysis is applicable if

Bi < 0.1
Equation for time taken by a substance to reach a particular temperature

T − T∞ − hA s ( t ) T  > T
∞i
= e ρVc p
Ti − T∞

The variation of temperature with time is exponential
Temperature

T

Time

In an unsteady heat transfer (lumped system) analysis the temperature changes exponentially
with time i.e. if the body absorbs heat the temperature will increase exponentially and if the
body loses heat the temperature will decrease exponentially.

Initially to reach a particular temperature difference the time taken by the body will be less
but after some time to reach the same temperature difference the time taken by the body will
be more.

2-6. Heat transfer through extended surfaces (fins)

The rate of heat transfer from a surface at a temperature Ts to the surrounding medium at Tis
given by Newton’s law of cooling as

Where As is the heat transfer surface area and h is the convection heat transfer coefficient.
When the temperatures Ts and T∞ are fixed by design considerations, as is often the case,
there are two ways to increase the rate of heat transfer: to increase the convection heat
transfer coefficient h or to increase the surface area As. Increasing h may require the
installation of a pump or fan, or replacing the existing one with a larger one, but this
approach may or may not be practical. Besides, it may not be adequate. The alternative is to
increase the surface area by attaching to the surface extended surfaces called fins made of
highly conductive materials such as aluminum. Finned surfaces are manufactured by
extruding, welding, or wrapping a thin metal sheet on a surface. Fins enhance heat transfer
from a surface by exposing a larger surface area to convection and radiation.

Finned surfaces are commonly used in practice to enhance heat transfer, and they often
increase the rate of heat transfer from a surface severalfold. The car radiator shown in Fig. 2-
9 is an example of a finned surface. The closely packed thin metal sheets attached to the hot
water tubes increase thesurface area for convection and thus the rate of convection heat
transfer from the tubes to the air many times. There are a variety of innovative fin designs
available in the market, and they seem to be limited only by imagination (Fig.2-10).

Fig. 2-9 Fig. 2-10

In the analysis of fins, we consider steady operation with no heat generation in the fin, and
we assume the thermal conductivity k of the material to remain constant. We also assume the
convection heat transfer coefficient h to be constant and uniform over the entire surface of the
fin for convenience in the analysis. We recognize that the convection heat transfer coefficient

h, in general, varies along the fin as well as its circumference, and its value at a point is a
strong function of the fluid motion at that point. The value of h is usually much lower at the
fin base than it is at the fin tip because the fluid is surrounded by solid surfaces near the base,
which seriously disrupt its motion to the point of “suffocating” it, while the fluid near the fin
tip has little contact with a solid surface and thus encounters little resistance to flow.
Therefore, adding too many fins on a surface may actually decrease the overall heat transfer
when the decrease in h offsets any gain resulting from the increase in the surface area.

Generalized differential equation for a Fin

Let, P = Perimeter θ = T − T∞
Ac = Area of cross section of fin
θ 0 = T 0 − T∞
Where,
∂ 2θ − m2θ = 0
∂x 2

m = hP
kA c

Infinitely long fin
For sufficiently long fin Temperature at fin tip will approach the environment temperature,
i.e.

Ttip = T∞

Heat transfer rate is given by,
Q = hPkAc ×θ 0

2-6-1. Fin performance

Fin Efficiency
• Along the length the temperature of fin decreases.
• If kept at same temperature efficiency will be maximum.

•

η = Qactual
Qideal

Fin Effectiveness

Effectiveness must be greater than 1

3. Convection

So far, we have considered conduction, which is the mechanism of heat transfer through a
solid or a quiescent fluid. We now consider convection, which is the mechanism of heat
transfer through a fluid in the presence of bulk fluid motion.

Convection is classified as natural (or free) and forced convection, depending on how the
fluid motion is initiated. In forced convection, the fluid is forced to flow over a surface or in a
pipe by external means such as a pump or a fan. In natural convection, any fluid motion is
caused by natural means such as the buoyancy effect, which manifests itself as the rise of
warmer fluid and the fall of the cooler fluid. Convection is also classified as external and
internal, depending on whether the fluid is forced to flow over a surface or in a channel

Physical mechanism of convection
Consider steady heat transfer through a fluid contained between two parallel plates
maintained at different temperatures, as shown in Figure 3-1. The temperatures of the fluid
and the plate will be the same at the points of contact because of the continuity of
temperature. Assuming no fluid motion, the energy of the hotter fluid molecules near the hot
plate will be transferred to the adjacent cooler fluid molecules. This energy will then be
transferred to the next layer of the cooler fluid molecules. This energy will then be transferred
to the next layer of the cooler fluid, and so on, until it is finally transferred to the other plate.
This is what happens during conduction through a fluid. Now let us use a syringe to draw
some fluid near the hot plate and inject it near the cold plate repeatedly. You can imagine that
this will speed up the heat transfer process considerably, since some energy is carried to the
other side as a result of fluid motion.

Fig. 3-1

Despite the complexity of convection, the rate of convection heat transfer is observed to be
proportional to the temperature difference and is conveniently expressed by Newton’s law of
cooling as

Judging from its units, the convection heat transfer coefficient h can be defined as the rate of
heat transfer between a solid surface and a fluid per unit surface area per unit temperature
difference.
You should not be deceived by the simple appearance of this relation, because the convection
heat transfer coefficient h depends on the several of the mentioned variables, and thus is
difficult to determine. When a fluid is forced to flow over a solid surface that is nonporous
(i.e., impermeable to the fluid), it is observed that the fluid in motion comes to a complete
stop at the surface and assumes a zero velocity relative to the surface. That is, the fluid layer
in direct contact with a solid surface “sticks” to the surface and there is no slip. In fluid flow,
this phenomenon is known as the no-slip condition, and it is due to the viscosity of the fluid
(Fig. 3-2).

Fig. 3-2
An implication of the no-slip and the no-temperature jump conditions is that heat transfer
from the solid surface to the fluid layer adjacent to the surface is by pure conduction, since
the fluid layer is motionless, and can be expressed as

Where T represents the temperature distribution in the fluid and ∂T/∂y is the temperature
gradient at the surface. This heat is then convected away from the surface as a result of fluid
motion. Note that convection heat transfer from a solid surface to a fluid is merely the
conduction heat transfer from the solid surface to the fluid layer adjacent to the surface.
Therefore, we can equate both the heat flux to obtain

For the determination of the convection heat transfer coefficient when the temperature
distribution within the fluid is known.
Velocity Boundary layer
Consider the parallel flow of a fluid over a flat plate, as shown in Fig. 3-3. Surfaces that are
slightly contoured such as turbine blades can also be approximated as flat plates with
reasonable accuracy. The x-coordinate is measured along the plate surface from the leading
edge of the plate in the direction of the flow, and y is measured from the surface in the
normal direction. The fluid approaches the plate in the x-direction with a uniform upstream
velocity of V, which is practically identical to the free-stream velocity u∞ over the plate away
from the surface (this would not be the case for cross flow over blunt bodies such as a
cylinder).

Fig. 3-3

The region of the flow above the plate bounded by in which the effects of the viscous
shearing forces caused by fluid viscosity are felt is called the velocity boundary layer. The
boundary layer thickness, is typically defined as the distance y from the surface at which u
=0.99u∞.
The hypothetical line of u = 0.99u∞ divides the flow over a plate into two regions: the
boundary layer region, in which the viscous effects and the velocity changes are significant,
and the in viscid flow region, in which the frictional effects are negligible and the velocity
remains essentially constant.

Thermal Boundary layer

We have seen that a velocity boundary layer develops when a fluid flows over a surface as a
result of the fluid layer adjacent to the surface assuming the surface velocity (i.e., zero
velocity relative to the surface). Also, we defined the velocity boundary layer as the region in
which the fluid velocity varies from zero to 0.99u∞. Likewise, a thermal boundary layer
develops when a fluid at a specified temperature flows over a surface that is at a different
temperature, as shown in Fig. 3-4.

Fig. 3-4

Consider the flow of a fluid at a uniform temperature of T∞ over an isothermal flat plate at
temperature Ts. The fluid particles in the layer adjacent to the surface will reach thermal
equilibrium with the plate and assume the surface temperature Ts. These fluid particles will
then exchange energy with the particles in the adjoining-fluid layer, and so on. As a result, a
temperature profile will develop in the flow field that ranges from Ts at the surface to T∞
sufficiently far from the surface. The flow region over the surface in which the temperature
variation in the direction normal to the surface is significant is the thermal boundary layer.
The thickness of the thermal boundary layer t at any location along the surface is defined as
the distance from the surface at which the temperature difference T - Ts equals 0.99(T∞ -Ts).
Note that for the special case of Ts = 0, we have T = 0.99T∞ at the outer edge of the thermal
boundary layer, which is analogous to u = 0.99u∞ for the velocity boundary layer.
The thickness of the thermal boundary layer increases in the flow direction, since the effects
of heat transfer are felt at greater distances from the surface further downstream.

The convection heat transfer rate anywhere along the surface is directly related to the
temperature gradient at that location. Therefore, the shape of the temperature profile in the
thermal boundary layer dictates the convection heat transfer between a solid surface and the
fluid flowing over it. In flow over a heated (or cooled) surface, both velocity and thermal
boundary layers will develop simultaneously. Noting that the fluid velocity will have a strong
influence on the temperature profile, the development of the velocity boundary layer relative
to the thermal boundary layer will have a strong effect on the convection heat transfer.

Nusselt number

In convection studies, it is common practice to nondimensionalize the governing equations
and combine the variables, which group together into dimensionless numbers in order to
reduce the number of total variables. It is also common practice to nondimensionalize the
heat transfer coefficient h with the Nusselt number, defined asRatio of Heat Convected to
Heat Conducted.

NU = hL
kfluid

Where k is the thermal conductivity of the fluid and L is the characteristic length.

Characteristic Dimension for Nusselt no.

1. For Pipe Flow – L = Diameter of Pipe (D)

2. For Flow over Plate – L = Distance from leading edge (x)

3. For any Conduit – L = Equivalent Diameter

L = Deq

Deq = 4 A
P

Where, A = Area of cross section

P = Perimeter

Reynolds Number

The transition from laminar to turbulent flow depends on the surface geometry, surface
roughness, free-stream velocity, surface temperature, and type of fluid, among other things.
After exhaustive experiments in the 1880s, Osborn Reynolds discovered that the flow regime
depends mainly on the ratio of the inertia forces to viscous forces in the fluid. This ratio is
called the Reynolds number, which is a dimensionless quantity, and is expressed for external
flow as Ratio of Inertia force and viscous force

Re = ρVL
µ

Where, V = Velocity of fluid

µ = Dynamic viscosity

L = Characteristic Dimension

ρ = Mass density

At large Reynolds numbers, the inertia forces, which are proportional to the density and the
velocity of the fluid, are large relative to the viscous forces, and thus the viscous forces
cannot prevent the random and rapid fluctuations of the fluid. At small Reynolds numbers,
however, the viscous forces are large enough to overcome the inertia forces and to keep the
fluid “in line.” Thus the flow is turbulent in the first case and laminar in the second.

Reynolds's no. (Re)

(a) For Pipe Flow – L = Diameter of Pipe (D)

if Re 2000 then Laminar Flow

Re 4000 then Turbulent Flow

2000 Re 4000 then Transitional Flow

(b) For any Conduit - L = Equivalent Diameter

L = Deq
Deq = 4 A

P

(c). Flow Over Plane Surface L = Distance from leading edge (x)

(Re)x = ρU∞x
µ

(Re)x = Local Reynolds's no.

(Re) L = ρU∞L
µ

If (Re)L 2*105 then Flow is Laminar

If (Re)L 5*105 then Flow is Turbulent

Prandtl Number

The relative thickness of the velocity and the thermal boundary layers is best described by the
dimensionless parameter Prandtl number, defined as

Prandtl no. depends on properties of fluid
Pr can be = 1 i.e. = α, δt = δ

1 i.e. α, δt δ

1 i.e. α, δt δ

The Prandtl numbers of gases are about 1, which indicates that both momentum and heat
dissipate through the fluid at about the same rate. Heat diffuses very quickly in liquid metals
(Pr< 1) and very slowly in oils (Pr> 1) relative to momentum. Consequently the thermal
boundary layer is much thicker for liquid metals and much thinner for oils relative to the
velocity boundary layer.

Mathematical relation between Hydrodynamic B.L.(δ) and Thermal B.L. (δt)
Relation between Hydrodynamic B.L. (δ) and Thermal B.L. (δt) depends on Prandtl number
and is given by

δ t = δ (Pr) −0.333

Blausius result for hydrodynamic boundary layer thickness,
5

√

3-1. Forced Convection

Since the heat transfer coefficient is a direct function of the temperature gradient next to the
wall, the physical variables on which it depends can be expressed as follows: h=f(fluid
properties, velocity field ,geometry, temperature etc.) As the function is dependent on several
parameters, the heat transfer coefficient is usually expressed in terms of correlations
involving pertinent non-dimensional numbers. Forced convection: Non-dimensional
groupings

In forced Convection Dimensionless analysis is done to find heat transfer coefficient.

3-1-1. Flow over plane surface.

Consider the parallel flow of a fluid over a flat plate of length L in the flow direction, as
shown in Figure 3-5. The x-coordinate is measured along the plate surface from the leading
edge in the direction of the flow. The fluid approaches the plate in the x-direction with
uniform upstream velocity V and temperature T∞. The flow in the velocity boundary layer
starts out as laminar, but if the plate is sufficiently long, the flow will become turbulent at a
distance xcr from the leading edge where the Reynolds number reaches its critical value for
transition.

Fig. 3-5
The transition from laminar to turbulent flow depends on the surface geometry, surface

roughness, upstream velocity, surface temperature, and the type of fluid, among other things,
and is best characterized by the Reynolds number. The Reynolds number at a distance x from
the leading edge of a flat plate is expressed as

Re = ρVL
µ

Laminar flow over Flat Plate Nux = 0.332 Rex0.5 Pr0.33
When (Re)L< 2*105 NuL = 0.664 ReL0.5 Pr0.33
Where,

Nux = Local nusselt no.

Nux = hx × x
k

NuL = Average nusselt no.

NuL = hL
k

Turbulent flow over Flat plate
When

ReL> 5*105

Nu = 0.036 Re 0.8 Pr0.33

3-1-2. Flow through Pipe or any conduit ……DittusBoelter Equation
Turbulent flow
When, Re> 4000

Nu = 0.023 Re0.8 Prn

n = 0.3 when the fluid is getting cooled

n= 0.4 when the fluid is getting heated

Laminar flow
When, Re< 2000

There can be two cases in laminar flow through pipes

1. Uniform heat flux - Nu = 4.36
2. Constant wall temperature – i. Slug flow - Nu = 5.78

ii. Fully developed flow - Nu = 3.66

4. Radiation

4-1. Basics of radiation

All bodies constantly emit energy by a process of electromagnetic radiation. The intensity of
such energy flux depends upon the temperature of the body and the nature of its surface. Most
of the heat that reaches you when you sit in front of a fire is radiant energy. Radiant energy
browns your toast in an electric toaster and it warms you when you walk in the sun. Objects
that are cooler than the fire, the toaster, or the sun emit much less energy because the energy
emission varies as the fourth power of absolute temperature. Very often, the emission of
energy, or radiant heat transfer, from cooler bodies can be neglected in comparison with
convection and conduction. But heat transfer processes that occur at high temperature, or
with conduction or convection suppressed by evacuated insulations, usually involve a
significant fraction of radiation.

Provost Law

Every substance above absolute zero (0 k) emits radiation. These radiations are in the form of
electromagnetic waves.

• These waves travel with the speed of light (3×108 m/s).
• Radiation emitted over wavelength of 0.1 µm to 100 µm is known as thermal

radiation.
• Unlike Conduction and Convection, no material medium is required for radiation.

Although all electromagnetic waves have the same general features, waves of different
wavelength differ significantly in their behavior. The electromagnetic radiation encountered
in practice covers a wide range of wavelengths, varying from less than 10-10µmfor cosmic
rays to more than 1010µm for electrical power waves. The electromagnetic spectrum also
includes gamma rays, X-rays, ultraviolet radiation, visible light, infrared radiation, thermal
radiation, microwaves, and radio waves.

Black bodies

The model for the perfect thermal radiator is a so-called black body. This is a body which
absorbs all energy that reaches it and reflects nothing. The term can be a little confusing,
since such bodies emit energy. Thus, if we possessed infrared vision, a black body would
glow with “color” appropriate to its temperature. Of course, perfect radiators are “black” in
the sense that they absorb all visible light (and all other radiation) that reaches them.

It is necessary to have an experimental method for making a perfectly black body. The
conventional device for approaching this ideal is called by the German term hohlraum, which
literally means “hollow space”. Figure 4-1 shows how a hohlraum is arranged. It is simply a
device that traps all the energy that reaches the aperture.

Fig. 4-1
The distribution of energy incident on a translucent slab.
What are the important features of a thermally black body? First consider a distinction
between heat and infrared radiation. Infrared radiation refers to a particular range of
wavelengths, while heat refers to the whole range of radiant energy flowing from one body to
another. Suppose that a radiant heat flux, q, falls upon a translucent plate that is not black, as
shown in Fig. 4-2 A fraction, α, of the total incident energy, called the absorptance, is
absorbed in the body; a fraction, ρ, called the reflectance, is reflected from it; and a fraction,
τ, called the transmittance, passes through. Thus

1=α+ρ+τ
This relation can also be written for the energy carried by each wavelength in the distribution
of wavelengths that makes up heat from a source at any temperature:

1=αλ+ρλ+τλ
All radiant energy incident on a black body is absorbed, so that αb or αλb = 1 and ρb = τb = 0.
Furthermore, the energy emitted from a black body reaches a theoretical maximum, which is
given by the StefanBoltzmann law. We look at this next.

Fig. 4-2

White body

A body is said to be a white body if it reflects all the radiation incident on it.
So, for a white body

ρ = 1, τ = 0 and α = 0
Note- color of a body has nothing to do with black or white body. It is the surface condition
that defines whether the body is black or non-black.
Snow which appears white in color is a black body (α = 0.96) Because it absorbs maximum
radiation falling on it.

4-2. Laws of radiation

Plank’s law

Plank’s law gives the monochromatic emissive power of a black body. Here monochromatic
means at a particular wavelength.

1
Where, T= temperature in kelvin

Stefan-Boltzmann law

The flux of energy radiating from a body is commonly designated e(T) W/m2. The symbol
eλ(λ,T) designates the distribution function of radiative flux in λ, or the monochromatic
emissive power.
This law gives the total emissive power of a black body at a particular temperature. (here
total means summed over all wavelengths)

The Stefan-Boltzmann law is

where the Stefan-Boltzmann constant, σ, is 5.67×10−8 W/m2·K4, and T is the absolute
temperature.

Wien’s law

The very accurate measurements of the black-body energy spectrum by Lummer and
Pringsheim (1899) are shown in Fig. 4-3. The locus of maxima of the curves is also plotted. It
obeys a relation called Wien’s law:

Fig. 4-3 Monochromatic emissive power of a black body at several temperatures—predicted
and observed.

About three-fourths of the radiant energy of a black body lies to the right of this line in Fig.
4-3 Notice that, while the locus of maxima leans toward the visible range at higher
temperatures, only a small fraction of the radiation is visible even at the highest temperature.

Observations-

I. The emission is continuous over all wavelengths
II. As the temperature increases the wavelength at which the maximum emissive power

occurs decreases
III. .

4-3. Irradiation (G) surface is known as irradiation.
G
The total radiation incident on the

4-4. Emissivity ( )

It is defined as ratio of emissive power of a body to the emissive power of a black body. (At a
temp. T )

Note- For a black body emissivity is equal to one.

4-5. Types of reflection –

There are 3 types of reflection.
a) Specular
b) Directional
c) Diffuse

4-6. Grey surface

A surface is said to be grey surface if radiation properties like emissivity, absorptivity are
independent of wavelength.
Note- No surface is truly grey or diffuse and this assumption is made to simplify the analysis.

4-7. Kirchhoff’s law

According to Kirchhoff’s law under thermal equilibrium the ratio of emissive power of a
body (e) to its absorptivity is constant.
i.e.
so,
andαb = 1
hence

Under thermal equilibrium emissivity is equal to absorptivity.

Radiant heat exchange

Suppose that a heated object (1 in Fig. 4-4a) radiates only to some other object (2) and that
both objects are thermally black. All heat leaving object 1 arrives at object 2, and all heat
arriving at object 1 comes from object 2. Thus, the net heat transferred from object 1 to object
2, Qnet, is the difference between Q1 to 2= A1eb(T1)

Fig. 4-4The net radiant heat transfer from one object to another.
and Q2 to 1=A1eb(T2)
If the first object “sees” other objects in addition to object 2, as indicated in Fig. 4-4b, then a
view factor (sometimes called a configuration factor or a shape factor), F1–2, must be
included in eqn

We may regard F1–2 as the fraction of energy leaving object 1 that is intercepted by object 2.

Kirchhoff’s law states that a body in thermodynamic equilibrium emits as much energy as it
absorbs in each direction and at each wavelength.

Shape factor

Important points related to shape factor.
I. Reciprocity theorem –
A 1 F 12 = A 2 F 21
II. Summation rule –
F 11 + F 12 + F 13 + ……… F 1n = 1

Radiative heat exchange between two enclosures

Radiative heat exchange between two black surfaces :

→ Radiation emitted by (1) = σ A1T14
→ Radiation emitted by (1) and talling on (2) = σ A1T14 F12
→ As (2) is black body it i.e. absorbs all = σ A1T14 F12
Similarly,
Radiation emitted by (2) = σ A2T24
emitted by (2) absorbed by (1) = σ A2T24 F21

Net radiative heat exchange (Q)
Net radiative heat exchange (Q) = σ A1T14 F12 − σ A2T24 F21
A1F12 = A2 F21

( )Q. = σ A1F12 T14 − T24

σ (T14 -T24 ) = Q
A1 F12

(J1 -J2 ) = Q
A1F12 / A2 F21

Radiosity (J) : The total radiation leading the surface is known as Radiosity .

Ioque τ = 0

J = ρG + ε eb 2 emitted
↓

Reflected

Assumption

(i) The surfaces are assumed to be
(a) gray
(b) Diffuse
(c) Opaque (τ= 0)
G = J − ε eb

ρ

α + ρ +τ =1

ρ =1−α

but α = ε fom kir chotl's law

G = J − ε eb
1−ε

Electrical analogy

Net radiation leaving the surface = J – G

J = J − ε eb
1−ε

J − ε J− J + ε eb = ε (eb − J )
1− ε 1−ε

Q = ε (eb − J )
A 1−ε

Q = ε A(eb − J )
1−ε

eb − J = Q ⎛ 1−ε ⎞
⎜⎝ εA ⎠⎟

1− ε → surface resist an ce
EA

For a black body as ε is 1 the surface resistance is zero for a black body.

The radiation leaving 1, is J1A1
The fraction of radiation falling on 2 is J1A1f12
Similarly radiation leaving 2 and falling on 1 is J2A2f21
Net radiative heat exchange
Q = J1A1f12 - J2A2f21
Q = (J1 – J2) A1F12

J1 − J2 = Q ⎡1⎤
⎢ ⎥
⎣ A1 F12 / A2 F21 ⎦

↓

space / shape registance

eb1 − eb2 = Q ⎣⎢⎡1ε−1 Aε11 + 1 + 1− ε2 ⎤
A1F12 ε 2 A2 ⎥
⎦

Q = 1− ε1 eb1 − eb2 1−ε2
+ 1 +

ε1 A1 A1F12 ε 2 A2

Q12 = σ (T14 − T22 )
1− ε1 + 1 + 1− ε2
ε1 A1 A1F12 ε 2 A2

* Radiative heat transfer between two enclosure (Two body enclosure) :
Tow body enclosure means heat transfer (radiative) is occurring only between two bodies,
so F11 + F12 = 1
case (i) - A small body kept in a large room –
F11 + F12 = 1

F12 = 1

A1 0
A2

Q = 1− ε1 (eb1 − eb2 ) ε2
+ 1 + 1−

ε1 A1 A1F12 ε 2 A2

Q = A1 (eb1 − eb2 )
1 ⎛ 1 − ε ⎞
ε1 −1+1+ 0 ⎜ ε 2 ⎟
⎝ 2 ⎠

Q= ε1 A1σ (T14 − T24 )
Case (ii) Two large parallel plates

F11 + F12 = 1

F12 = 1

A1 = A2 = A

Q = 1− ε1 eb1 − eb2 1− ε2
+ 1 +

ε1 A1 A1F12 ε 2 A2

Q = A(eb1 − eb2 )
1+ 1 −1
ε1 ε2

Case (ii) Radiative heat exchange between two concentric cyinderse :

F11 + F12 = 1

F12 = 1

Q = 1− ε1 eb1 − eb2 1−ε2
+ 1 +

ε1 A1 A1F12 ε 2 A2

Q = A1σ (T14 − T24 )
1− ε1 A1 ⎡1 −1⎥⎤
ε1 + A2 ⎢ ⎦ +1
⎣ ε
2

Q = A1σ (T14 − T24 )

1 ⎛ D1 ⎞2 ⎛ 1 − ε 2 ⎞
ε1 ⎜ D2 ⎟ ⎜ ε ⎟
⎝ ⎠ ⎝ 2 ⎠

* Radiation shield : are used to reduce heat transfer therefore radiation shield must have high
reflective power.