Heat and Mass Transfer EBE-5214

Chapter 1
Introduction to Heat Transfer

Concept, modes of heat transfer, thermal conductivity of materials, measurement, Fourier’s

law, Newton’s law of cooling, heat transfer coefficient in convection, Stephan Boltzmann law

1.1. Introduction
Thermodynamically, heat is a form of energy that is transferred between two systems (or a
system and its surroundings) by virtue of temperature difference. However, the science of
thermodynamics deals with the amount of heat transfer as a system undergoes a process from one
equilibrium state to another, and makes no reference to how long the process will take. But in
engineering, we are often interested in the rate of heat transfer, which is the topic of the science of
heat transfer.
Classical thermodynamics treats the processes as through only the end states exist,
presuming that the system exists in a state of thermodynamic equilibrium. It provides no
information about the rates of irreversible flows. The heat flow with temperature difference is one
example of irreversible flow. In studies of processes that involve flow of heat or mass, the rate of
flow is an important parameter.
Heat: Heat is a form of energy and holds the same unit as of energy i.e., Joules. It is very
important to understand how heat is different from other forms of energy. Energy can flow from
one body to another only by two modes- heat or work. Heat is a form of energy which is transient
in nature and it flows from one point to another point only due to a difference in temperature,
while all other energy interactions that are not due to temperature difference can be signified as
work.
When two bodies at different temperatures come in contact with each other, the two
temperatures approach each other and after some time become equal. This equalization of
temperature of the bodies is on account of flow of energy in the form of heat from one body to
another. Therefore, heat may be defined as flow of energy from one body to another body by
virtue of temperature difference between them. The net flow of energy always occurs from high
temperature body towards low temperature body and this flow of heat stops the moment
temperature of both the bodies are equal. Thus, flow of heat is a transfer of energy occurring due
to a temperature difference between two bodies.
The driving potential or the force which causes the transfer of energy as heat is the
difference in temperature between systems. Other such transport processes are the transfer of
momentum, mass and electrical energy. In addition to the temperature difference, physical
parameters like geometry, material properties like conductivity, flow parameters like flow
velocity also influence the rate of heat transfer.
The purposes of study of heat transfer are:
 To estimate the rate of flow of energy as heat through the boundary of a system
under study (both under steady and transient conditions)
 To determine the temperature field under steady and transient conditions

The following points must be noted:

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 The heat flows from a hotter body to a colder body

 Heat lost by one body is equal to heat gained by the other
 In order to change the state of a body, heat is to be supplied at constant temperature
 Bodies expand on heating
 The weight of the body do not change on heating
 Heat flow is irreversible in nature
Heat can be transferred in three different ways: by conduction, convection, and radiation.
In most of the engineering applications, it is a combination of the two or three modes. Pure
conduction is found only in solids and convection is possible only in fluids. 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. Convection is the mode of heat 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. 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.
The study of heat transfer is directed to (i) the estimation of rate of flow of energy as heat
through the boundary of a system both under steady and transient conditions, and (ii) the
determination of temperature field under steady and transient conditions, which also will provide
the information about the gradient and time rate of change of temperature at various locations and
time. i.e. T (x, y, z, τ) and dT/dx, dT/dy, dT/dz, dT/dτetc. These two are interrelated, one being
dependent on the other. However explicit solutions may be generally required for one or the
other. The basic laws governing heat transfer and their application are as below:
a) First law of thermodynamics postulating the energy conservation principle: This law
provides the relation between the heat flow, energy stored and energy generated in a given
system. The relationship for a closed system is: The net heat flow across the system boundary
+ heat generated inside the system = change in the internal energy of the system. This will
also apply for an open system with slight modifications. The change in internal energy in a given
volume is equal to the product of volume density and specific heat cV and dT where the group cV
is called the heat capacity of the system.) The basic analysis in heat transfer always has to start
with one of these relations.
b) The second law of thermodynamics establishing the direction of energy transport as heat.
The law postulates that the flow of energy as heat through a system boundary will always be in
the direction of lower temperature or along the negative temperature gradient.
c) Newton’s laws of motion used in the determination of fluid flow parameters.
d) Law of conservation of mass, used in the determination of flow parameters.
e) The rate equations as applicable to the particular mode of heat transfer.

1.2. Modes of Heat Transfer

1.2.1 Heat Transfer by Conduction
Thermal conduction is a process by which heat is transmitted by the direct contact between
particles of a body without any motion of the material as a whole. The phenomenon of
conduction heat transfer can be experienced by a simple experiment. Heat one end of a metal
rod. The other end of the rod will become hotter and hotter with the passage of time. Heat reaches
from the heated end of the rod to the other end by conduction through the material of the rod.
Conduction occurs in all media-solids, liquids, and gases when a temperature gradient exists. In

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opaque solids, it is the only mechanism by which the heat can flow. In the fluids, the molecules
have freedom of motion and energy is also transferred by movement of the fluid. However, if the
fluid is at rest, the heat is transferred by conduction. Heat transfer by conduction requires that the
temperature distribution in a medium is non-uniform, that is, a temperature gradient exists in the
body. Fourier noted in 1811 that the heat flow in a homogenous solid is directly proportional to
the temperature gradient. Consider a plane wall of thickness dx, whose area perpendicular to the
direction x is A as shown in Fig. 1.1. If the thickness of the wall is very small as compared to its
height and width, it is a case of one-dimensional (in direction x only) heat flow. Let one of the
faces of the wall is at a temperature T1 and the other at temperature T2. For the elemental
thickness dx, the temperature difference is dt. Then, the conduction heat flux q/A (the heat flux is
the heat flow rate per unit area of the surface), according to Fourier, is according to Fourier, is

Fig. 1.1 Conduction heat flow through a solid

q dt
 (1.1)
A dx
When the constant of the proportionality is inserted in above equation, we get
q dt
 k (1.2)
A dx
Fourier’s law of heat conduction which states that the heat flow by conduction in any
direction is proportional to the temperature gradient and area perpendicular to the flow
direction and is in the direction of the negative gradient.
The constant of proportionality ‘k’ is known as thermal conductivity of the material. It is a
physical property of a substance and characterizes the ability of a material to conduct heat. The
negative sign in the equation indicates that the heat flow is in the direction of falling temperature.
From Eq. (1.2), we get:
 q  dx
k     W mK (1.3)
 A  dt
Thus thermal conductivity determines the quantity of heat flowing per unit time through the
unit area with a temperature drop of 1 °C (K) per unit length.
The units used in the text for various parameters are: Q – W, (Watt), A – m2 , dT – °C or K
(as this is only temperature interval, °C and K can be used without any difficulty). x – m, k –
W/mK.
For simple shapes and one directional steady conditions with constant value of thermal
conductivity this law yields rate equations as below:

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Conduction, Plane Wall (Fig. 1.1), the integration of the equation 1.1 for a plane wall of
thickness, L between the two surfaces at T1 and T2 under steady condition leads to equation 1.4.
The equation can be considered as the mathematical model for such problems.
T1  T2
Q (1.4)
L
kA
From the above equation it must be understood that (T1 -T2 ) is driving potential and L/kA is
the resistance to the flow of heat.
Example 1.1: Determine the heat flow across a plane wall of 10 cm thickness with a constant
thermal conductivity of 8.5 W/mK when the surface temperatures are steady at 100°C and 30°C.
The wall area is 3m2 . Also find the temperature gradient in the flow direction.

This is also equal to – (100 – 30)/0.1 = – 700°C/m, as the gradient is constant all through the
thickness.

The denominator in equation 1.4, namely L/kA can be considered as thermal resistance
for conduction. An electrical analogy is useful as a concept in solving conduction problems and
in general heat transfer problems. We shall discuss it later in this chapter.
1.2.1.1. Thermal Conductivity: It is the constant of proportionality in Fourier’s equation and
plays an important role in heat transfer. The unit in SI system for conductivity is W/mK. It is a
material property. Its value is higher for good electrical conductors and single crystals like
diamond. Next in order are alloys of metals and non metals. Liquids have conductivity less than
these materials. Gases have the least value for thermal conductivity.
In solids heat is conducted in two modes. 1. The flow of thermally activated electrons and 2.
Lattice waves generated by thermally induced atomic activity. In conductors the predominant

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mode is by electron flow. In alloys it is equal between the two modes. In insulators, the lattice
wave mode is the main one. In liquids , conduction is by atomic or molecular diffusion.
In gases conduction is by diffusion of molecules from higher energy level to the lower level.
Thermal conductivity is formed to vary with temperature. In good conductors, thermal
conductivity decreases with temperature due to impedance to electron flow of higher electron
densities. In insulators, as temperature increases, thermal atomic activity also increases and
hence thermal conductivity increases with temperature. In the case of gases, thermal
conductivity increases with temperature due to increased random activity.

1.2.1.2. Thermal Insulation: In many situations to conserve heat energy, equipments have to
be insulated. Thermal insulation materials should have a low thermal conductivity. This is
achieved in solids by trapping air or a gas in small cavities inside the material. It may also be
achieved by loose filling of solid particles. The insulating property depends on the material as
well as transport property of the gases filling the void spaces. There are essentially three types of
insulating materials:
 Fibrous: Small diameter particles or filaments are loosely filled in the gap between
surfaces to be insulated. Mineral wool is one such material, for temperatures below
700°C. Fiber glass insulation is used below 200°C. For higher temperatures refractory
fibers like Alumina (Al2 O3 ) or silica (S1 O2 ) are useful.
 Cellular: These are available in the form of boards or formed parts. These contain voids
with air trapped in them. Examples are polyurethane and expanded polystyrene foams.
 Granular: These are of small grains or flakes of inorganic materials and used in
preformed shapes or as powders.
The effective thermal conductivity of these materials is in the range of 0.02 to 0.04 W/mK.
1.2.2 Heat Transfer by Convection
The term is applied to transport of heat as a volume of liquid or gas moves from a region of
one temperature to that of another temperature. Thus, the transport of heat is linked with the
movement of the medium itself. The convection can be observed in liquids if we carry out a
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simple experiment. Consider a pot with water which is placed over a burner. The water at the
bottom of the pot is heated and becomes less dense than before due to its thermal expansion.
Thus, the water at the bottom, which is less dense than the cold water in the upper portion, rises
upwards. It transfers its heat by mixing as it rises. The movement of the water, referred as the
convection currents, can be observed by putting a few crystals of potassium permanganate in the
bottom of the pot. Density differences and the gravitational force of the earth act to produce a
force known as buoyancy force, which drives the flow. When the flow is due to the density
differences only, it is called natural or free convection. Density differences may also be caused by
the composition gradients. For example, mixture of water vapour and air rises mainly due to the
lower density of water vapour present in the moist air.

Figure 1.3 Convection process in liquid

Convection is termed as forced if the fluid is forced to flow over a surface or in a duct by
external means such as a fan, pump, or blower, that is , the forced convection implies
mechanically induced flow. Heat transfer processes involving change of phase of a fluid (boiling
of liquids or condensation of vapour) are also considered to be convection because of the motion
of fluid that is set up due to the rising vapour bubbles during boiling or the falling liquid droplets
during the condensation. The heat transfer by convection is always accompanied by conduction.
The combined process of heat transfer by convection and conduction is referred to as convective
heat transfer. The heat transfer, between a solid surface and a fluid, is expressed by Newton’s law
of cooling as:
q  hAT (1.5)
where ΔT is the temperature difference between the bulk fluid and the surface, ‘A’ is the
area of the surface transferring heat, and ‘h’ is known as heat transfer coefficient or film
coefficient. In general, the value of the heat transfer coefficient ‘h’ depends on the fluid-flow
conditions, the thermo-physical properties of the fluid and the type of flow passage.
If the temperature difference between two finite points within the fluid is (T1-T2), then the
rate of heat transfer by convection can be expressed as:
(T1  T2 )
q  hA(T1  T2 )  (1.6)
1
hA
where, Q W. Am2 , T1 , T2 °C or K, h W/m K.
2

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The quantity 1/hA is called convection resistance to heat flow. The equivalent circuit is
given in Fig. 1.4(b).

In most of the heat exchangers, heat is transferred between hot and cold fluid streams across
a solid wall. In such cases, it is convenient to combine the two film coefficients (i.e., of the hot
and cold fluid streams) to give a single coefficient known as overall heat transfer coefficient U,
which is defined as:
q
U
AT
Here ΔT is the temperature difference between the two fluids.
Example 1.2: Determine the heat transfer by convection over a surface of 0.5 m area if the
2

surface is at 160°C and fluid is at 40°C. The value of convective heat transfer coefficient is 25
W/m K. Also estimate the temperature gradient at the surface given k = 1 W/mK.
2

Solution: Refer to Fig. 1.4a and equation 1.6

Q = hA (T1 – T2 ) = 25 × 0.5 × (160 – 40) W = 1500 W or 1.5 kW
The resistance = 1/hA = 1/(25 × 0.5) = 0.08°C/W.
The fluid has a conductivity of 1 W/mK, then the temperature gradient at the surface is
Q = – kA dT/dy
Therefore, dT/dy = – Q/kA
= – 1500/1.0 × 0.5 = – 3000°C/m.
The fluid temperature is often referred as Tfor indicating that it is the fluid temperature
well removed from the surface. The convective heat transfer coefficient is dependent on several
parameters and the determination of the value of this quantity is rather complex, and is discussed
in later chapters.
1.2.3 Heat Transfer by Radiation
The thermal radiation is the process of heat propagation by means of electromagnetic waves
produced by virtue of the temperature of the body. It depends both on the temperature and an
optical property known as emiss ivity ‘' of the body. In contrast to the conduction and convection
heat transfers, radiation can take place through a perfect vacuum. Solids, liquids and gases may
radiate energy. For example, water vapour and carbon dioxide are the principal sources of the
gaseous radiation in furnaces. Boltzmann established that the rate at which a body gives out the
heat by radiation is proportional to the fourth power of the absolute temperature of the body, that
is,

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q AT 4 (1.7)
When the constant of the proportionality is inserted,
q   AT 4 (1.8)
where the constant of proportionality ‘σ’ is known as Stefan–Boltzmann constant, and ‘A’ is
the surface area of the body. A body may absorb, transmit, or reflect radiant energy. A blackbody
absorbs the entire radiation incident on it. Thus, it is a perfect absorber. Technically, the
blackbody is a hypothetical body. It does not necessarily refer to the color of the body, though
bodies black in color usually absorb most. A blackbody is also a perfect or ideal radiator, for
which the emissivity  = 1. For real bodies,  is less than 1 and they do not emit as much energy
as a blackbody.
The net heat radiated between two black bodies 1 and 2 at temperatures T1 and T2 that see
each other completely (i.e., they exchange heat by radiation between themselves only), the net
energy exchange is proportional to the difference in T1 4 and T2 4 . Thus the radiation exchange
between bodies which are not black is quite complex and will be dealt with in detail later.
q12   A T14  T2 4  (1.9)
Thermal radiation is part of the electromagnetic spectrum in the limited wave length range
of 0.1 to 10 m and is emitted at all surfaces, irrespective of the temperature. Such radiation
incident on surfaces is absorbed and thus radiation heat transfer takes place between surfaces at
different temperatures. No medium is required for radiative transfer but the surfaces should be in
visual contact for direct radiation transfer. The rate equation is due to Stefan-Boltzmann law
which states that heat radiated is proportional to the fourth power of the absolute temperature of
the surface and heat transfer rate between surfaces is given in equation 1.4 is slightly modified in
case the two objects are vividly oriented. The situation is represented in Fig. 1.5 (a).
q12  F A T14  T2 4 
where, F—a factor depending on geometry and surface properties,
σ—Stefan Boltzmann constant 5.67 × 10–8 W/m2 K4 (SI units)
Am2 , T1 , T2 K (only absolute unit of temperature to be used).
This equation can also be rewritten as.

(1.10)
where the denominator is referred to as radiation resistance (Fig. 1.5)

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Example 1.3: A surface is at 200°C and has an area of 2m2 . It exchanges heat with another
surface B at 30°C by radiation. The value of factor due to the geometric location and emissivity is
0.46. Determine the heat exchange. Also find the value of thermal resistance and equivalent
convection coefficient.
Solution: Refer to equation 1.9 and 1.10 and Fig. 1.5.
T1 = 200°C = 200 + 273 = 473K, T2 = 30°C = 30 + 273 = 303K.
(This conversion of temperature unit is very important)
= 5.67 × 10–8 , A = 2m2 , F = 0.46.
Therefore, Q = 0.46 × 5.67 × 10–8 × 2[4734 – 3034 ]
= 0.46 × 5.67 × 2 [(473/100)4 – (303/100)4 ]
(This step is also useful for calculation and will be followed in all radiation problems taking 10–8
inside the bracket).
Therefore, Q = 2171.4 W
Resistance can be found as
Q = ΔT/R, R = ΔT/Q = (200–30)/2171.4
Therefore, R = 0.07829°C/W or K/W
Resistance is also given by 1/hrA.
Therefore, hr = 6.3865 W/m2 K
Check: Q = hrAΔT = 6.3865 × 2 × (200–30) = 2171.4 W
The denominator in the resistance terms is also denoted as hrA. where
hr = F(T1 + T2 )(T1 2 + T2 2 )
and is often used due to convenience approximately
3
T T 
hr  F  1 2 
 2 
The determination of F is rather involved and values are available for simple configurations
in the form of charts and tables. For simple cases of black surface enclosed by the other surface F
= 1 and for non black enclosed surfaces F = emissivity (defined as ratio of heat radiated by a
surface to that of an ideal surface).

1.3. Combined modes of heat transfer

Previous sections treated each mode of heat transfer separately. But in practice all the three
modes of heat transfer can occur simultaneously. Additionally heat generation within the solid
may also be involved. Most of the time conduction and convection modes occur simultaneously
when heat from a hot fluid is transferred to a cold fluid through an intervening barrier. Consider
the following example. A wall receives heat by convection and radiation on one side. After
conduction to the next surface, heat is transferred to the surroundings by convection and
radiation. This situation is shown in Fig. 1.7.
The heat flow is given by equation 1.11.

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(1.11)
where hr1 and h r2 are radiation coefficients and h 1 and h 2 are convection coefficients

Example 1.4: A slab 0.2 m thick with thermal conductivity of 45 W/mK receives heat from a
furnace at 500 K both by convection and radiation. The convection coefficient has a value of 50
W/m2 K. The surface temperature is 400 K on this side. The heat is transferred to surroundings at
T∞2 both by convection and radiation. The convection coefficient on this side being 60 W/m 2 K.
Determine the surrounding temperature. Assume F = 1 for radiation.

1.3.1. Overall Heat Transfer Coefficient: Often when several resistances for heat flow is
involved, it is found convenient to express the heat flow

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Q  UAT (1.12)
where ‘U’ is termed as overall heat transfer coefficient having the same unit as
convective heat transfer coefficient, h. The value of U can be obtained for a given area A by
equation 1.13.

(1.13)
where R1 , R2 , R3 , ...... are the resistances in series calculated based on the areas A1 , A2 , A3
etc.

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