2151909 heat transfer e-note (thefreestudy.com) (1)

Department of Mechanical Engineering
Darshan Institute of Engineering &Technology
As Per New GTU Syllabus
Prepared by:
Mr. Dipak A. Solanki (M.E. – Thermal Science)
Mr. Bhavin J. Vegada (M.Tech – Thermal Engg.)

Sr. No. Chapters Page No.
1 Introduction 1.1 – 1.9
2 Steady State Heat Conduction 2.1 – 2.48
3 Heat Transfer from Extended Surfaces (Fin) 3.1 – 3.23
4 Transient (Unsteady State) Heat Conduction 4.1 – 4.13
5 Radiation: Processes and Properties 5.1 – 5.17
6 Radiation Heat Transfer 6.1 – 6.21
7 Convection 7.1 – 7.38
8 Boiling & Condensation 8.1 – 8.13
9 Heat Exchangers 9.1 – 9.35
BRIEF CONTENTS

Department of Mechanical Engineering Prepared By: Dipak A. Solanki
Darshan Institute of Engineering & Technology, Rajkot Page 1.1
1INTRODUCTION
Course Contents
1.1 Introduction
1.2 Thermodynamics and heat transfer
1.3 Application areas of heat transfer
1.4 Heat transfer mechanism
1.5 Conduction
1.6 Thermal conductivity
1.7 Convection
1.8 Radiation
1.9 References

1. Introduction Heat Transfer (2151909)
Prepared By: Dipak A. Solanki Department of Mechanical Engineering
Page 1.2 Darshan Institute of Engineering & Technology, Rajkot
1.1 Introduction
 Heat is fundamentally transported, or “moved,” by a temperature gradient; it flows or
is transferred from a high temperature region to a low temperature one. An
understanding of this process and its different mechanisms are required to connect
principles of thermodynamics and fluid flow with those of heat transfer.
1.2 Thermodynamics and Heat Transfer
 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 (Figure 1.1).
Fig. 1.1 Heat transfer from the thermos
 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 nonequilibrium phenomenon. Therefore, the
study of heat transfer cannot be based on the principles of thermodynamics alone.
 However, the laws of thermodynamics lay the framework for the science of heat
transfer. The first law requires that the rate of energy transfer into a system be equal

Heat Transfer (2151909) 1. Introduction
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 (Figure 1.2).
Fig. 1.2 Heat transfer from high temperature to low temperature
1.3 Application Areas of Heat Transfer
 Many ordinary household appliances are designed, in whole or in part, by using the
principles of heat transfer. Some examples:
 Design of the heating and air-conditioning system, the refrigerator and freezer, the
water heater, the iron, and even the computer, the TV, and the VCR
 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 (Figure 1.3)
Fig. 1.3 Application of heat transfer
 ENGINEERING HEAT TRANSFER
 The heat transfer problems encountered in practice can be considered in two groups:
i rating and
ii sizing problems.

 The rating problems deal with the determination of the heat transfer rate for an
existing system at a specified temperature difference.
 The sizing problems deal with the determination of the size of a system in order to
transfer heat at a specified rate for a specified temperature difference.
1.4 Heat Transfer Mechanisms
 Heat can be transferred in three different modes: conduction, convection, and
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.
1.5 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.
 The rate of heat conduction through a medium depends on the geometry of the
medium, its thickness, and the material of the medium, as well as the temperature
difference across the medium.
 We know that wrapping a hot water tank with glass wool (an insulating material)
reduces the rate of heat loss from the tank. The thicker the insulation, the smaller the
heat loss.
 We also know that a hot water tank will lose heat at a higher rate when the
temperature of the room housing the tank is lowered. Further, the larger the tank, the
larger the surface area and thus the rate of heat loss.
Fig. 1.4 Heat conduction through large plain wall
 Consider steady heat conduction through a large plane wall of thickness and
area A, as shown in figure 1.4. The temperature difference across the wall is
.

 Experiments have shown that the rate of heat transfer ̇ through the wall is doubled
when the temperature difference across the wall or the area A normal to the
direction of heat transfer is doubled, but is halved when the wall thickness L is
doubled.
 Thus we conclude that the rate of heat conduction through a plane layer is
proportional to the temperature difference across the layer and the heat transfer
area, but is inversely proportional to the thickness of the layer. That is,
( )( )
or
̇ ( ) ( )
 Where 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. In the limiting case of
, the equation above reduces to the differential form
̇ ( ) ( )
 Which is called Fourier’s law of heat conduction. Here ⁄ is the temperature
gradient, which is the slope of the temperature curve on a T-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 Eq. 1.2 ensures that heat transfer in the positive x direction is a
positive quantity.
 The heat transfer area A is always normal to the direction of heat transfer.
1.6 Thermal Conductivity
 The thermal conductivity of a material can be defined as the rate of heat transfer
through a unit thickness of the material per unit area per unit temperature difference.
 The thermal conductivity of a material is a measure of the ability of the material to
conduct heat.
 A high value for thermal conductivity indicates that the material is a good heat
conductor, and a low value indicates that the material is a poor heat conductor or
insulator.
 Note that materials such as copper and silver that are good electric conductors are
also good heat conductors, and have high values of thermal conductivity.
 Materials such as rubber, wood, and styrofoam are poor conductors of heat and have
low conductivity values.

1.7 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.
Fig. 1.5 Heat transfer by convection
 Consider the cooling of a hot block by blowing cool air over its top surface (Figure 1.5).
 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 Forced and Free (Natural) 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 (Figure 1.6).

 For example, in the absence of a fan, heat transfer from the surface of the hot block in
figure 1.5 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.
 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
̇ ( ) ( ) ( )
 Where h is the convection heat transfer coefficient in ⁄ , is the surface area
through which convection heat transfer takes place, is the surface temperature, and
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.
1.8 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
 Some people do not consider convection to be a fundamental mechanism of
heat transfer since it is essentially heat conduction in the presence of fluid
motion. But we still need to give this combined phenomenon a name, unless
we are willing to keep referring to it as “conduction with fluid motion.”
 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.

(at the speed of light) and it suffers no attenuation in a vacuum. This is how the energy
of the sun reaches the earth.
 The mechanism of the heat flow by radiation consists of three distinct phases:
i Conversion of thermal energy of the hot source into electromagnetic waves:
 All bodies above absolute zero temperature are capable of emitting radiant energy.
Energy released by a radiating surface is not continuous but is in the form of
successive and separate (discrete) packets or quanta of energy called photons. The
photons are propagated through the space as rays; the movement of swarm of
photons is described as the electromagnetic waves.
ii Passage of wave motion through intervening space:
 The photons, as carries of energy travel with unchanged frequency in straight paths
with speed equal to that of light.
iii Transformation of waves into heat:
 When the photons approach the cold receiving surface, there occurs reconversion of
wave motion into thermal energy which is partly absorbed, reflected or transmitted
through the receiving surface.
 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.
 The maximum rate of radiation that can be emitted from a surface at an absolute
temperature (in K) is given by the Stefan–Boltzmann law as
( ) ( )
 Where, is the energy radiated by black body, is the Stefan Boltzman constant.
⁄
 The radiation emitted by all real surfaces is less than the radiation emitted by a
blackbody at the same temperature, and is expressed as
( ) ( )
 Where, is a radiative property of the surface and is called emissivity; its value
depends upon surface characteristics and temperature. It indicates how effectively the
surface emits radiations compared to an ideal or black body radiator.
 Normally a body radiating heat is simultaneously receiving heat from other bodies as
radiation.
 Consider that surface 1 at temperature is completely enclosed by another black
surface 2 at temperature . The net radiant heat transfer is
( ) ( ) ( )
 Likewise, the net rate of heat transfer between the real surface (called gray surface) at
temperature to a surrounding black surface at temperature is

( ) ( ) ( )
 The net exchange of heat between the two radiating surfaces is due to the face that
one at the higher temperature radiates more and receives less energy for its
absorption.
 An isolated body which remains at constant temperature emits just as much energy by
radiation as it receives.
1.9 References
[1] Heat and Mass Transfer by D. S. Kumar, S K Kataria and Sons Publications.
[2] Heat Transfer – A Practical Approach by Yunus Cengel & Boles, McGraw-Hill
Publication.

2
STEADY STATE HEAT CONDUCTION
Course Contents
2.1 Introduction
2.2 Thermal resistance
2.3 Thermal conductivity of
material
2.4 General heat conduction
equation
2.5 Measurement of thermal
conductivity (Guarded hot
plate method)
2.6 Conduction through a plane
wall
2.7 Conduction through a
composite wall
2.8 Heat flow between surface and
surroundings: cooling and
heating of fluids
cylindrical wall
multilayer cylindrical wall
2.11 Conduction through a sphere
2.12 Critical thickness of insulation
2.13 Solved Numerical
2.14 References

2. Steady State Heat Conduction Heat Transfer (2151909)
2.1 Introduction
 The rate of heat conduction in a specified direction is proportional to the
temperature gradient, which is the rate of change in temperature with distance in
that direction. One dimensional steady state heat conduction through homogenous
material is given by Fourier Law of heat conduction:
Where,
= heat flux, heat conducted per unit time per unit area, ⁄
Q = rate of heat flow, W
A = area perpendicular to the direction of heat flow,
dt = temperature difference between the two surfaces across which heat is
passing, Kelvin K or degree centigrade
dx = thickness of material along the path of heat flow, m
 The ratio ⁄ represents the change in temperature per unit thickness, i.e. the
temperature gradient.
 The negative sign indicates that the heat flow is in the direction of negative
temperature gradient, so heat transfer becomes positive.
 The proportionality factor k is called the heat conductivity or thermal conductivity of
material through which heat is transfer.
 The Fourier law is essentially based on the following assumptions:
1. Steady state heat conduction, i.e. temperature at fixed point does not change
with respect to time.
2. One dimensional heat flow.
3. Material is homogenous and isotropic, i.e. thermal conductivity has a constant
value in all the directions.
4. Constant temperature gradient and a linear temperature profile.
5. No internal heat generation.
 The Fourier law helps to define thermal conductivity of the material.
 Assuming and , we obtain
 Hence thermal conductivity may be defined as the amount of heat conducted per
unit time across unit area and through unit thickness, when a temperature
difference of unit degree is maintained across the bounding surface.
 Unit of thermal conductivity is given by:

Heat Transfer (2151909) 2. Steady State Heat Conduction
[ ]
2.2 Thermal Resistance
 In systems, which involve flow of fluid, heat and electricity, the flow quantity is
directly proportional to the driving force and inversely proportional to the flow
resistance.
 In a hydraulic system, the pressure along the path is the driving potential and
roughness of the pipe is the flow resistance.
 The current flow in a conductor is governed by the voltage potential and electrical
resistance of the material.
 Likewise, temperature difference constitutes the driving force for heat conduction
through a medium.
Fig. 2.1 Concept of thermal resistance
 From Fourier’s law
⁄
 Thermal resistance, ⁄ , is expressed in the unit ⁄ .
 The reciprocal of thermal resistance is called thermal conductance and it represents
the amount of heat conducted through a solid wall of area A and thickness dx when
a temperature difference of unit degree is maintained across the bounding surfaces.
2.3 Thermal Conductivity of Materials
 Thermal conductivity is a property of the material and it depends upon the material
structure, moisture content and density of the material, and operating conditions of
pressure and temperature.
 Following remarks apply to the thermal conductivity and its variation for different
materials and under different conditions:
 In material thermal conductivity is due to two effects: the lattice vibrational waves
and flow of free electrons.

 In metals the molecules are closely packed; molecular activity is rather small and so
thermal conductivity is mainly due to flow of free electrons.
 In fluids, the free electron movement is negligibly small so conductivity mainly
depends upon the frequency of interactions between the lattice atoms.
 Thermal conductivity is highest in the purest form of a metal. Alloying of metals and
presence of other impurities reduce the conductivity of the metal.
 Mechanical forming (i.e. forging, drawing and bending) or heat treatment of metal
cause considerable variation in thermal conductivity. Conductivity of hardened steel
is lower than that of annealed steel.
 At elevated temperatures, thermal vibration of the lattice becomes higher and that
retards the motion of free electrons. So, thermal conductivity of metal decreases
with increases of temperature except the aluminium and uranium.
 Thermal conductivity of aluminium remains almost constant within the temperature
range of 130 to 370 .
 For uranium, heat conduction depends mainly upon the vibrational movement of
atoms. With increase of temperature vibrational movement increase so, conductivity
also increase.
 According to kinetic theory of, conductivity of gases is directly proportional to the
density of the gas, mean molecular speed and mean free path. With increase of
temperature molecular speed increases, so conductivity of gas increases.
Conductivity of gas is independent of pressure except in extreme cases as, for
example, when conditions approach that of a perfect vacuum.
 Molecular conditions associated with the liquid state are more difficult to describe,
and physical mechanisms for explaining the thermal conductivity are not well
understood. The thermal conductivity of nonmetallic liquids generally decreases with
increasing temperature. The water, glycerine and engine oil are notable exceptions.
The thermal conductivity of liquids is usually insensitive to pressure except near the
critical point.
 Thermal conductivity is only very weakly dependent on pressure for solids and for
liquids a well, and essentially dependent of pressure for gases at pressure near
standard atmospheric.
 For most materials, the dependence of thermal conductivity on temperature is
almost linear.
 Non-metallic solids do not conduct heat as efficiently as metals.
 Thermal conductivity of pure copper is 385 ⁄ and that of nickel
is 93 ⁄ .
 Monel metal, an alloy of 30% nickel and 70% copper, has thermal
conductivity of only 24 ⁄ .

 The ratio of the thermal and electrical conductivities is same for all metals at the
same temperature; and that the ratio is directly proportional to the absolute
temperature of the metal.
2.4 General Heat Conduction Equation
 The objective of conduction analysis is two fold:
i To determine the temperature distribution within the body
ii To make calculation of heat transfer.
 Fourier law of heat conduction is essentially valid for heat flow under uni-directional
and steady state conditions, but sometimes it is necessary to consider heat flow in
other direction as well.
 So for heat transfer in multi-dimensional, it is necessary to develop general heat
conduction equation in rectangular, cylindrical and spherical coordinate systems.
2.4.1 Cartesian (Rectangular) Co-ordinates:-
 Consider the flow of heat through an infinitesimal volume element oriented in a
three dimensional co-ordinate system as shown in figure 2.2. The sides dx, dy and dz
have been taken parallel to the x, y, and z axis respectively.
Fig. 2.2 Conduction analysis in cartesian co ordinates
 The general heat conduction equation can be set up by applying Fourier equation in
each Cartesian direction, and then applying the energy conservation requirement.
 If represents the thermal conductivity at the left face, then quantity of heat
flowing into the control volume through the face during time interval is given by:
 Heat influx
 During same time interval the heat flow out of the element will be,
 Heat efflux

 Heat accumulated within the control volume due to heat flow in the x-direction is
given by the difference between heat influx and heat efflux.
 Thus the heat accumulation due to heat flow in x-direction is
[ ]
[ ]
[ ]
 Likewise the heat accumulation in the control volume due to heat flow along the y-
and z-directions will be:
[ ]
[ ]
 Total heat accumulated due to heat transfer is given by
[ ( ) ( ) ( )]
 There may be heat source inside the control volume. If is the heat generated per
unit volume and per unit time, then the total heat generated in the control volume
equals to
 The total heat accumulated in the control volume due to heat flow along all the co-
ordinate axes and the heat generated within the control volume together increases
the internal energy of the control volume.
 Change in internal energy of the control volume is given by
 According to first law of thermodynamics heat accumulated within the control
volume due to heat flow along the co-ordinate axes

[ ( ) ( ) ( )]
 Dividing both sides by
( ) ( ) ( )
 This expression is known as general heat conduction equation for Cartesian co-
ordinate system.
 Note:- Homogeneous and isotropic material: A homogeneous material implies that
the properties, i.e., density, specific heat and thermal conductivity of the material
are same everywhere in the material system. Isotropic means that these properties
are not directional characteristics of the material, i.e., they are independent of the
orientation of the surface.
 Therefore for an isotropic and homogeneous material, thermal conductivity is same
at every point and in all directions. In that case and equation
becomes:
 The quantity ⁄ is called the thermal diffusivity, and it represents a physical
property of the material of which the solid element is composed. By using the
Laplacian operator , the equation may be written as:
 Equation governs the temperature distribution under unsteady heat flow through a
homogeneous and isotropic material.
 Different cases of particular interest are:
 For steady state heat conduction, heat flow equation reduces to:
or
 This equation is called Poisson’s equation.
 In the absence of internal heat generation, equation further reduces to:
or

 This equation is called Laplace equation.
 Unsteady state heat flow with no internal heat generation gives:
or
 This equation is called Fourier equation.
 For one-dimensional and steady state heat flow with no heat generation, the general
heat conduction equation is reduced to:
( )
 Thermal diffusivity:
 Thermal diffusivity of a material is the ratio of its thermal conductivity to the
thermal storage capacity . The storage capacity essentially represents thermal
capacitance or thermal inertia of the material.
 It signifies the rate at which heat diffuses in to the medium during change in
temperature with time. Thus, the higher value of the thermal diffusivity gives the
idea of how fast the heat is conducting into the medium, whereas the low value of
the thermal diffusivity shown that the heat is mostly absorbed by the material and
comparatively less amount is transferred for the conduction.
2.4.2 Cylindrical Co-ordinates:-
 When heat is transferred through system having cylindrical geometries like tube of
heat exchanger, then cylindrical co-ordinate system is used.
 Consider infinitesimal small element of volume
(a) (b)
Fig. 2.3 (a) Cylindrical co-ordinate system (b) An element of cylinder

Fig. 2.3 (c) Heat conduction through cylindrical element
 Assumptions:
1) Thermal conductivity , density and specific heat for the material do not vary
with position.
2) Uniform heat generation at the rate of per unit volume per unit time,
a) Heat transfer in radial direction,
 Heat influx
 Heat efflux
 Heat stored in the element due to flow of heat in the radial direction
[ ]
( )
( )
( )

( )
b) Heat transfer in tangential direction
 Heat influx
 Heat efflux
 Heat stored in the element due to heat flow in the tangential direction,
[ ]
( )
c) Heat transferred in axial direction
 Heat influx
 Heat efflux
 Heat stored in the element due to heat flow in axial direction,
[ ]

d) Heat generated within the control volume
e) Rate of change of energy within the control volume
 According to first law of thermodynamics, the rate of change of energy within the
control volume equals the total heat stored plus the heat generated. So,
* +
 Dividing both sides by
* +
or
* +
 which is the general heat conduction equation in cylindrical co-ordinates.
 For steady state unidirectional heat flow in the radial direction, and with no internal
heat generation, equation reduces to
( )
or
( )
( )

2.4.3 Spherical Co-ordinates:-
 When heat is transferred through system having spherical geometries like spherical
storage tank, ball of ball bearing, junction of thermocouple, then cylindrical co-
ordinate system is used.
 Consider infinitesimal small element of volume
 Assumptions:
1) Thermal conductivity , density and specific heat for the material do not vary
with position.
2) Uniform heat generation at the rate of per unit volume per unit time,
(a) (b)
(c)
Fig. 2.4 (a) Spherical co-ordinate system (b) An element of sphere
(c) Heat conducted through spherical element

a) Heat transferred through
 Heat influx
 Heat efflux
[ ]
b) Heat flow through
 Heat influx
 Heat efflux
[ ]
( )
( )
( )

c) Heat flow through
 Heat influx
 Heat efflux
 Heat stored in the element volume due to heat flow in the
[ ]
[ ]
[ ]
( )
d) Heat generated within the control volume
e) Rare of change of energy within the control volume
 According to first law of thermodynamics, the rate of change of energy within the
control volume equals the total heat stored plus the heat generated. So,
* ( ) ( )+
 Dividing sides by
* ( ) ( )+

 Which is the general heat conduction equation in spherical co-ordinates
 The heat conduction equation in spherical co-ordinates could also be obtained by
utilizing the following inter relation between the rectangular and spherical co-
ordinates.
 For steady state, uni-direction heat flow in the radial direction for a sphere with no
internal heat generation, equation can be written as
( )
 General one-dimensional conduction equation: The one-dimensional time
dependent heat conduction equation can be written as
( )
 Where n = 0, 1 and 2 for rectangular, cylindrical and spherical co-ordinates
respectively. Further, while using rectangular co-ordinates it is customary to replace
the r-variable by the x-variable.
2.5 Measurement of Thermal Conductivity (Guarded Hot Plate
Method)
 Construction
 The essential elements of the experimental set-up as shown in figure 2.5 are:
 Main heater placed at the centre of the unit. It is maintained at a fixed
temperature by electrical energy which can be metered.
 Guarded heater which surrounds the main heater on its ends. The guarded
heater is supplied electrical energy enough to keep its temperature same as that of
main heater.
Fig. 2.5 Elements of guarded hot plate method

 Function of the guarded heater is to ensure unidirectional heat flow and eliminates
the distortion caused by edge losses.
 Test specimens and which are placed on both sides of the heater.
 Cooling unit plates and are provided for circulation of cooling medium. Flow of
cooling medium is maintained to keep the constant surface temperature of
specimen.
 Thermocouples attached to the specimens at the hot and cold faces.
 Desired measurement
 From the Fourier’s law of heat conduction
 So to measure thermal conductivity k following measurements are required
 Heat flow Q from the main heart through a test specimen; it will be half of the total
electrical input to the main heater
 Thickness of the specimen X
 Temperature drop across the specimen ; subscripts h and c refer to the hot
and cold faces respectively
 Area A of heat flow; the area for heat flow is taken to be the area of main heater
plus the area of one-half of air gap between it and the guarded heater
 For the specimen of different thickness, the respective temperature at the hot and
cold faces would be different and then the thermal conductivity is worked out from
the following relation:
( )
 Where suffix 1 is for the upper specimen and 2 is for the lower specimen.
 Here Q is the total electrical input to the main heater.
2.6 Conduction Through a Plane Wall:-
 Consider one-dimensional heat conduction through a homogeneous, isotropic wall
of thickness with constant thermal conductivity and constant cross-sectional area
.
 The wall is insulated on its lateral faces, and constant but different temperatures
and are maintained at its boundary surfaces.
 Starting with general heat conduction equation in Cartesian co-ordinates
 For steady state, one dimensional with no heat generation equation is reduced to

or
 Integrate the equation with respect to x is given by
 The constants of integration are evaluated by using boundary conditions and here
boundary conditions are:
at and at
 When boundary conditions are applied
and
 So, integration constants are
 Accordingly the expression for temperature profile becomes
( )
 The temperature distribution is thus linear across the wall. Since equation does not
involve thermal conductivity so temperature distribution is independent of the
material; whether it is steel, wood or asbestos.
 Heat flow can be made by substitution the value of temperature gradient into
Fourier equation
[ ( ) ]
 Alternatively, The Fourier rate equation may be used directly to determine the heat
flow rate.
 Consider an elementary strip of thickness located at a distance from the
reference plane. Temperature difference across the strip is , and temperature
gradient is ⁄ .
 Heat transfer through the strip is given by

Fig. 2.6 Heat conduction through plane wall
 For steady state condition, heat transfer through the strip is equal to the heat
transfer through the wall. So integrate the equation between the limits, at
and at , thus
∫ ∫
 To determine the temperature at any distance from the wall surface, the Fourier
rate equation is integrated between the limit:
a) where the temperature is stated to be
b) where the temperature is to be worked out
 Thus,
∫ ∫
 Substituting the value of Q in above equation

( )
 The expression for the heat flow rate can be written as
⁄
 Where ⁄ is the thermal resistance to heat flow. Equivalent thermal circuit
for flow through a plane wall has been included in figure 2.6.
 Let us develop the condition when weight, not space, required for insulation of a
plane wall is the significant criterion.
 For one dimensional steady state heat condition
⁄
 Thermal resistance of the wall, ⁄
 Weight of the wall,
 Eliminating the wall thickness from expression
 From the equation when the product for a given resistance is smallest, the
weight of the wall would also be so. It means for the lightest insulation for a
specified thermal resistance, product of density times thermal conductivity should
be smallest.
2.7 Conduction Through a Composite Wall
 A composite wall refers to a wall of a several homogenous layers.
 Wall of furnace, boilers and other heat exchange devices consist of several layers; a
layer for mechanical strength or for high temperature characteristics (fire brick), a
layer of low thermal conductivity material to restrict the flow of heat (insulating
brick) and another layer for structural requirements for good appearance (ordinary
brick).
 Figure 2.7 shows one such composite wall having three layers of different materials
tightly fitted to one another.
 The layers have thickness , , and their thermal conductivities correspond to
the average temperature conditions.
 The surface temperatures of the wall are and and the temperatures at the
interfaces are and .

Fig. 2.7 Heat conduction through composite wall
 Under steady state conditions, heat flow does not vary across the wall. It is same for
every layer. Thus
 Rewriting the above expression in terms of temperature drop across each layer,
 Summation gives the overall temperature difference across the wall
( )
Then
 Where , is the total resistance.
 Analysis of the composite wall assumes that there is a perfect contact between
layers and no temperature drop occurs across the interface between materials.
2.8 Heat Flow Between Surface and Surroundings: Cooling and
Heating of Fluids
 When a moving fluid comes into contact with a stationary surface, a thin boundary
layer develops adjacent to the wall and in this layer there is no relative velocity with
respect to surface.

Fig. 2.8 Heat conduction through a wall separating two fluids
 In a heat exchange process, this layer is called stagnant film and heat flow in the
layer is covered both by conduction and convection processes. Since thermal
conductivity of fluids is low, the heat flow from the moving fluid of the wall is mainly
due to convection.
 The rate of convective heat transfer between a solid boundary and adjacent fluid is
given by the Newton-Rikhman law:
( )
 Where, is the temperature of moving fluid, is the temperature of the wall
surface, is the area exposed to heat transfer and h is the convective co-efficient.
The dimension of h is ⁄ .
 Heat transfer by convection may be written as
 Where h⁄ is the convection resistance.
 The heat transfer through a wall separating two moving fluids involves: (i) flow of
heat from the fluid of high temperature to the wall, (ii) heat conduction through the
wall and (iii) transport of heat from the wall to the cold fluid.
 Under steady state conditions, the heat flow can be expressed by the equations:
 Where h and h represent the convective film coefficients, is thermal conductivity
of the solid wall having thickness . These expressions can be presented in the form:

 Summation of these gives
( )
 The denominator h⁄ ⁄ h⁄ is the sum of thermal resistance of
difference sections through which heat has to flow.
 Heat flow through a composite section is written in the form
⁄
 Where, U is the overall heat transfer coefficient.
 It represents the intensity of heat transfer from one fluid to another through a wall
separating them.
 Numerically it equals the quantity of heat passing through unit area of wall surface in
unit time at a temperature difference of unit degree. The coefficient U has
dimensions of ⁄ .
 By comparing the equation
 So heat transfer coefficient is reciprocal of unit thermal resistance to heat flow.
 The overall heat transfer coefficient depends upon the geometry of the separating
wall, its thermal properties and the convective coefficient at the two surfaces.
 The overall heat transfer coefficient is particularly useful in the case of composite
walls, such as in the design of structural walls for boilers, refrigerators, air-
conditioned buildings, and in the design of heat exchangers.
2.9 Conduction Through a Cylindrical Wall
 Consider heat conduction through a cylindrical tube of inner radius , outer radius
and length .
 The inside and outside surfaces of the tube are at constant temperatures and
and thermal conductivity of the tube material is constant within the given
temperature range.
 If both ends are perfectly insulated, the heat flow is limited to radial direction only.
 Further if temperature at the inner surface is greater than temperature at the
outer surface, the heat flows radially outwords.

Fig. 2.9 Steady state heat conduction through a cylindrical wall
 The general heat conduction equation for cylindrical co-ordinate is given by
* +
 For steady state ⁄ unidirectional heat flow in the radial direction and with
no internal heat generation ( ) the above equation reduces to
( )
 Since,
( )
 Integration of above equation gives
 Using the following boundary conditions
and
 The constants and are
⁄ ⁄
 Using the values of and temperature profile becomes
⁄ ⁄
⁄
⁄

 Therefore in dimensionless form
⁄
⁄
 From the equation it is clear that temperature distribution with radial conduction
through a cylinder is logarithmic; not linear as for a plane wall.
 Further temperature at any point in the cylinder can be expressed as a function of
radius only.
 Isotherms or lines of constant temperature are then concentric circles lying between
the inner and outer cylinder boundaries.
 The conduction heat transfer rate is determined by utilizing the temperature
distribution in conjunction with the Fourier law:
[
⁄ ⁄
]
(
⁄
)
⁄
 In the alternative approach to estimate heat flow, consider an infinitesimally thin
cylindrical element at radius .
 Let thickness of this elementary ring be and the change of temperature across it
be .
 Then according to Fourier law of heat conduction
 Integrate the equation within the boundary condition
∫ ∫
⁄
 For conduction in hollow cylinder, the thermal resistance is given by:
⁄

 Special Notes
 Heat conduction through cylindrical tubes is found in power plant, oil refineries and
most process industries.
 The boilers have tubes in them, the condensers contain banks of tubes, the heat
exchangers are tubular and all these units are connected by tubes.
 Surface area of a cylindrical surface changes with radius. Therefore the rate of heat
conduction through a cylindrical surface is usually expressed per unit length rather
than per unit area as done for plane wall.
 Logarithmic Mean Area
 It is advantageous to write the heat flow equation through a cylinder in the same
form as that for heat flow through a plane wall.
Fig. 2.10 Logarithmic mean area concept
 Then thickness will be equal to and the area will be an equivalent area
. Thus
 Comparing equations 3.68 and 3.70
⁄
⁄ ⁄
 Where and are the inner and outer surface areas of the cylindrical tube.
 The equivalent area is called the logarithmic mean area of the tube. Further
⁄

 Obviously, logarithmic mean radius of the cylindrical tube is:
⁄
2.10 Conduction Through a Multilayer Cylindrical Wall
 Multi-layer cylindrical walls are frequently employed to reduce heat looses from
metallic pipes which handle hot fluids.
 The pipe is generally wrapped in one or more layers of heat insulation.
 For example, steam pipe used for conveying high pressure steam in a steam power
plant may have cylindrical metal wall, a layer of insulation material and then a layer
of protecting plaster.
 The arrangement is called lagging of the pipe system.
Fig. 2.11 Steady state heat conduction through a composite cylindrical wall
 Figure 2.11 shows conduction of heat through a composite cylindrical wall having
three layers of different materials.
 There is a perfect contact between the layers and so an equal interface temperature
for any two neighbouring layers.
 For steady state conduction, the heat flow through each layer is same and it can be
described by the following set of equations:
⁄
⁄

⁄
 These equations help to determine the temperature difference for each layer of the
composite cylinder,
 From summation of these equalities;
[ ]
 Thus the heat flow rate through a composite cylindrical wall is
 The quantity in the denominator is the sum of the thermal resistance of the different
layers comprising the composite cylinder.
 Where, is the total resistance
Fig. 2.12 Heat conduction through cylindrical wall with convection coefficient
 If the internal and external heat transfer coefficients for the composite cylinder as
shown in figure 2.12 are and respectively, then the total thermal resistance to
heat flow would be:

and heat transfer is given as
 Overall Heat Transfer Coefficient U
 The heat flow rate can be written as:
 Since the flow area varies for a cylindrical tube, it becomes necessary to specify the
area on which U is based.
 Thus depending upon whether the inner or outer area is specified, two different
values are defined for U.
 Equating equations of heat transfer
 Similarly
 Overall heat transfer coefficient may be calculated by simplified equation as follow
2.11 Conduction Through a Sphere
 Consider heat conduction through a hollow sphere of inner radius and outer
radius and made of a material of constant thermal conductivity.
Fig. 2.13 Steady state heat conduction through sphere

 The inner and outer surfaces are maintained at constant but different temperatures
and respectively. If the inner surface temperature is greater than outer
surface temperature , the heat flows radially outwards.
 General heat conduction equation in spherical coordinates is given as
* ( ) ( )+
 For steady state, uni-directional heat flow in the radial direction and with no internal
heat generation, the above equation is written as
( )
( )
 The relevant boundary conditions are
 Using the above boundary conditions values of constants are
 Substitute the values of constants in equation; the temperature distribution is given
as follow
( ) ( )
( )
[ ]
 In non dimensional form
( )
( )
 Evidently the temperature distribution associated with radial conduction through a
spherical is represented by a hyperbola.

 The conduction heat transfer rate is determined by utilizing the temperature
distribution in conjunction with the Fourier law:
⁄
 The denominator of the equation is the thermal resistance for heat conduction
through a spherical wall.
 In the alternative approach to determine heat flow, consider an infinitesimal thin
spherical element at radius and thickness .
 The change of temperature across it be . According to Fourier law of heat
conduction
 Separating the variables and integrating within the boundary conditions
∫ ∫
( )
⁄
 Heat conduction through composite sphere can be obtained similar to heat
conduction through composite cylinder. Heat conduction through composite sphere
will be:
⁄ ⁄ ⁄
 Further, if the convective heat transfer is considered, then
⁄ ⁄ ⁄ ⁄ ⁄

2.12 Critical Thickness of Insulation
 There is some misunderstanding about that addition of insulating material on a
surface always brings about a decrease in the heat transfer rate.
 But addition of insulating material to the outside surfaces of cylindrical or spherical
walls (geometries which have non-constant cross-sectional areas) may increase the
heat transfer rate rather than decrease under the certain circumstances.
 To establish this fact, consider a thin walled metallic cylinder of length l, radius and
transporting a fluid at temperature which is higher than the ambient temperature
.
 Insulation of thickness and conductivity k is provided on the surface of the
cylinder.
Fig. 2.14 Critical thickness of pipe insulation
 With assumption
a. Steady state heat conduction
b. One-dimensional heat flow only in radial direction
c. Negligible thermal resistance due to cylinder wall
d. Negligible radiation exchange between outer surface of insulation and
surrounding
 The heat transfer can be expressed as

 Where and are the convection coefficients at inner and outer surface
respectively.
 The denominator represents the sum of thermal resistance to heat flow.
 The value of and are constant; therefore the total thermal resistance will
depend upon thickness of insulation which depends upon the outer radius of the
arrangement.
 It is clear from the equation 2.85 that with increase of radius r (i.e. thickness of
insulation), the conduction resistance of insulation increases but the convection
resistance of the outer surface decreases.
 Therefore, addition of insulation can either increase or decrease the rate of heat
flow depending upon a change in total resistance with outer radius r.
 To determine the effect of insulation on total heat flow, differentiate the total
resistance with respect to r and equating to zero.
[ ]
 To determine whether the foregoing result maximizes or minimizes the total
resistance, the second derivative need to be calculated
[ ]
( ) ( )

 which is positive, so ⁄ represent the condition for minimum resistance and
consequently maximum heat flow rate.
 The insulation radius at which resistance to heat flow is minimum is called critical
radius.
 The critical radius, designated by is dependent only on thermal quantities and
.
 From the above equation it is clear that with increase of radius of insulation heat
transfer rate increases and reaches the maximum at and then it will decrease.
 Two cases of practical interest are:
 When
 It is clear from the equation 2.14a that with addition of insulation to bare pipe
increases the heat transfer rate until the outer radius of insulation becomes equal to
the critical radius.
 Because with addition of insulation decrease the convection resistance of surface of
insulation which is greater than increase in conduction resistance of insulation.
Fig. 2.14 Dependence of heat loss on thickness of insulation
 Any further increase in insulation thickness decreases the heat transfer from the
peak value but it is still greater than that of for the bare pipe until a certain amount
of insulation .
 So insulation greater than must be added to reduce the heat loss below the
bare pipe.
 This may happen when insulating material of poor quality is applied to pipes and
wires of small radius.
 This condition is used for electric wire to increase the heat dissipation from the wire
which helps to increase the current carrying capacity of the cable.

Fig. 2.15 Critical radius of insulation for electric wire
 When
 It is clear from the figure 2.14b that increase in insulation thickness always decrease
the heat loss from the pipe.
 This condition is used to decrease the heat loss from steam and refrigeration pipes.
 Critical radius of insulation for the sphere can be obtain in the similar way:
[ ]
[ [ ] ]
Ex 2.1.
A 30 cm thick wall of size is made of red brick ⁄ .
It is covered on both sides by layers of plaster, 2 cm thick ⁄ .
The wall has a window size of . The window door is made of 12 mm
thick glass ⁄ . If the inner and outer surface temperatures
are 15 and 40 , make calculation for the rate of heat flow through the wall.
Solution:
Given data:
Plaster: ⁄ ,
Red brick: ⁄ ,
Glass: ⁄ ,
, Total Area A = ,
Area of glass Window
 Total heat transfer from the given configuration is sum of the heat transfer
from composite wall and glass window. So,

 Heat transfer from the composite wall
Area of the wall,
Resistance of inner and outer plaster layers,
⁄
Resistance of brick work,
⁄
 Heat transfer from glass window
Resistance of glass,
⁄

So total heat transfer is given by
Ex 2.2.
A cold storage room has walls made of 200 mm of brick on the outside, 80 mm of
plastic foam, and finally 20 mm of wood on the inside. The outside and inside air
temperatures are and respectively. If the outside and inside
convective heat transfer coefficients are respectively 10 and 30 ⁄ , and the
thermal conductivities of brick, foam and wood are 1.0, 0.02 and 0.17 ⁄
respectively. Determine:
(i) Overall heat transfer coefficient
(ii) The rate of heat removed by refrigeration if the total wall area is 100
(iii) Outside and inside surface temperatures and mid-plane temperatures of
composite wall.
Solution:
Given data:
Brick: ⁄ ,
Plastic foam: ⁄ ,
Wood: ⁄ ,
, ⁄ , ⁄ ,

i. Over all heat transfer co-efficient U
 Convection resistance of outer surface
⁄
 Resistance of brick,
⁄
 Resistance of plastic foam,
⁄
 Resistance of wood,
⁄
 Convection resistance of inner surface
⁄
⁄
ii. The rate of heat removed by refrigeration if the total wall area is A = 100
( )
iii. Outside and inside surface temperatures and mid-plane temperatures of
composite wall
 Temperature of outer surface
 Temperature of middle plane
 Temperature of inner surface

Ex 2.3.
A furnace wall is made up of three layer of thickness 250 mm, 100 mm and 150 mm
with thermal conductivity of 1.65, k and 9.2 ⁄ respectively. The inside is
exposed to gases at with a convection coefficient of ⁄ and the
inside surface is at , the outside surface is exposed to air at with
convection coefficient of ⁄ . Determine:
(i) The unknown thermal conductivity k
(ii) The overall heat transfer coefficient
(iii) All surface temperatures
Solution:
Given data:
Layer 1: ⁄ ,
Layer 2: ⁄ ,
Layer 3: ⁄ ,
, ⁄ , ⁄ , Take A = 1 m2
i. Unknown thermal conductivity k
 Convection resistance of inner surface
⁄
 Resistance of layer 1,
⁄

⁄ ⁄
⁄
 Convection resistance of outer surface
⁄
Heat transfer by convection is given by
Heat transfer through composite wall is given by
⁄
⁄ ⁄
ii. Overall heat transfer co-efficient U
⁄
iii. All surface temperature
 Temperature of inner surface
 Temperature of outer surface

Ex 2.4.
A heater of 150 mm X 150 mm size and 800 W rating is placed between two slabs A
and B. Slab A is 18 mm thick with ⁄ . Slab B is 10 mm thick with
⁄ . Convective heat transfer coefficients on outside surface of slab A
and B are ⁄ and ⁄ respectively. If ambient temperature is
, calculate maximum temperature of the system and outside surface
temperature of both slabs.
Solution:
Given data:
,
Slab A: ⁄ , , ⁄
Slab B: ⁄ , , ⁄
i. Maximum temperature of the system
 Maximum temperature exist at the inner surfaces of both slab A and slab B
So, maximum temperature
 Under the steady state condition heat generated by the heater is equal to the
heat transfer through the slab A and slab B.

 Heat transfer through the slab A,
 Resistance of slab A,
⁄
 Convection resistance of outer surface of slab A
⁄
 Resistance of slab B,
⁄
 Convection resistance of outer surface of slab B
⁄
( ) { }
ii. Outside surface temperature of both slabs
 Heat transfer through slab A
 Outside surface temperature of slab A,
 Heat transfer through slab B
 Outside surface temperature of both slab B,
Ex 2.5.
A 240 mm dia. steam pipe, 200 m long is covered with 50 mm of high temperature
insulation of thermal conductivity ⁄ and 50 mm low temperature

insulation of thermal conductivity ⁄ . The inner and outer surface
temperatures are maintained at and respectively. Calculate:
(i) The total heat loss per hour
(ii) The heat loss per of pipe surface
(iii) The heat loss per of outer surface
(iv) The temperature between interfaces of two layers of insulation.
Neglect heat conduction through pipe material.
Solution:
Given data:
⁄ ⁄
i. Total heat loss per hour
 Resistance of high temperature insulation
⁄ ( ⁄ )
⁄
 Resistance of low temperature insulation
⁄ ( ⁄ )
⁄
⁄
⁄ ⁄⁄
ii. The heat loss per of pipe surface

⁄
iii. The heat loss per of outer surface
⁄
iv. The temperature between interfaces of two layers of insulation
Ex 2.6.
A hot fluid is being conveyed through a long pipe of 4 cm outer dia. And covered
with 2 cm thick insulation. It is proposed to reduce the conduction heat loss to the
surroundings to one-third of the present rate by further covering with some
insulation. Calculate the additional thickness of insulation.
Solution:
Given data:
i. Heat loss with existing insulation
 Resistance of existing insulation
⁄
ii. Heat loss with additional insulation
 Resistance of existing insulation
⁄

But, ⁄
⁄ ⁄
⁄ ⁄ ⁄
⁄
Ex 2.7.
A hot gas at with convection coefficient ⁄ is flowing through a
steel tube of outside diameter 8 cm and thickness 1.3 cm. It is covered with an
insulating material of thickness 2 cm, having conductivity of ⁄ . The outer
surface of insulation is exposed to ambient air at with convection coefficient of
⁄ .
Calculate: (1) Heat loss to air from 5 m long tube. (2) The temperature drop due to
thermal resistance of the hot gases, steel tube, the insulation layer and the outside
air. Take conductivity of steel ⁄ .
Solution:
Given data:
⁄ ⁄ , ⁄ , ⁄

i. Total heat loss to air from 5 m long tube, Q
 Convection resistance of hot gases
⁄
 Resistance of steel
⁄ ( ⁄ )
⁄
 Resistance of insulation
⁄ ( ⁄ )
⁄
 Convection resistance of outside air
⁄
ii. Temperature drop
 Temperature drop due to thermal resistance of hot gases
 Temperature drop due to thermal resistance of steel tube
 Temperature drop due to thermal resistance of insulation
 Temperature drop due to thermal resistance of outside air
Ex 2.8.
A pipe carrying the liquid at is 10 mm in outer diameter and is exposed to
ambient at with convective heat transfer coefficient of ⁄ . It is
proposed to apply the insulation of material having thermal conductivity of
⁄ . Determine the thickness of insulation beyond which the heat gain will be
reduced. Also calculate the heat loss for 2.5 mm, 7.5 mm and 15 mm thickness of
insulation over 1m length. Which one is more effective thickness of insulation?

Solution:
Given data:
⁄ , ⁄

i. Thickness of insulation beyond which heat gain will be reduced
 Critical radius of insulation
⁄ ⁄
ii. Heat loss for 2.5 mm thickness of insulation,
⁄ ( ⁄ )
⁄
⁄
iii. Heat loss for 7.5 mm thickness of insulation,
⁄ ( ⁄ )
⁄
⁄

iv. Heat loss for 15 mm thickness of insulation,
⁄ ( ⁄ )
⁄
⁄
Hence the insulation thickness of 15 mm is more effective
2.14 References:
Publication.
[3] Principles of Heat Transfer by Frank Kreith, Cengage Learining.

3HEAT TRANSFER FROM EXTENDED
SURFACES
Course Contents
3.1 Introduction
3.2 Steady flow of heat along a
rod (governing differential
equation)
3.3 Heat dissipation from an
infinitely long fin
3.4 Heat dissipation from a fin
insulated at the tip
3.5 Heat dissipation from a fin
losing heat at the tip
3.6 Fin performance
3.7 Thermometric well
3.9 References

3. Heat Transfer from Extended Surface Heat Transfer (2151909)
3.1 Introduction
 Heat transfer between a solid surface and a moving fluid is governed by the
Newton’s cooling law: ( ), where is the surface temperature
and is the fluid temperature.
 Therefore, to increase the convective heat transfer, one can
i Increase the temperature difference ( ) between the surface and the fluid.
ii Increase the convection coefficient . This can be accomplished by increasing the
fluid flow over the surface since h is a function of the flow velocity and the higher
the velocity, the higher the h.
iii Increase the contact surface area
 Many times, when the first option is not in our control and the second option (i.e.
increasing ) is already stretched to its limit, we are left with the only alternative of
increasing the effective surface area by using fins or extended surfaces.
 Fins are protrusions from the base surface into the cooling fluid, so that the extra
surface of the protrusions is also in contact with the fluid.
 Most of you have encountered cooling fins on air-cooled engines (motorcycles,
portable generators, etc.), electronic equipment (CPUs), automobile radiators, air
conditioning equipment (condensers) and elsewhere
3.2 Steady Flow of Heat Along A Rod (Governing Differential
Equation)
 Consider a straight rectangular or pin fin protruding from a wall surface (figure 3.1a
and figure 3.1b).
 The characteristic dimensions of the fin are its length L, constant cross-sectional area
and the circumferential parameter P.
Fig. 3.1a Schematic diagram of a rectangular fin protruding from a wall

Heat Transfer (2151909) 3. Heat Transfer from Extended Surface
Fig. 3.1b Schematic diagram of a pin fin protruding from a wall
 Thus for a rectangular fin
( )
and for a pin fin
 The temperature at the base of the fin is and the temperature of the ambient
fluid into which the rod extends is considered to be constant at temperature .
 The base temperature is highest and the temperature along the fin length goes on
diminishing.
 Analysis of heat flow from the finned surface is made with the following
assumptions:
i Thickness of the fin is small compared with the length and width; temperature
gradients over the cross-section are neglected and heat conduction treated one
dimensional
ii Homogeneous and isotropic fin material; the thermal conductivity k of the fin
material is constant
iii Uniform heat transfer coefficient h over the entire fin surface
iv No heat generation within the fin itself
v Joint between the fin and the heated wall offers no bond resistance;
temperature at root or base of the fin is uniform and equal to temperature of
the wall
vi Negligible radiation exchange with the surroundings; radiation effects, if any, are
considered as included in the convection coefficient h

vii Steady state heat dissipation
 Heat from the heated wall is conducted through the fin and convected from the
sides of the fin to the surroundings.
 Consider infinitesimal element of the fin of thickness dx at a distance x from base
wall as shown in figure 3.2.
Fig. 3.2 Heat transfer through a fin
 Heat conducted into the element at plane x
( ) ( )
 Heat conducted out of the element at plane ( )
( )
( ) ( )
 Heat convected out of the element between the planes x and ( )
( )( ) ( )
 Here temperature t of the fin has been assumed to be uniform and non-variant for
the infinitesimal element.
 According to first law of thermodynamic, for the steady state condition, heat
transfer into element is equal to heat transfer from the element

( ) ( )( )
( )( )
 Upon arrangement and simplification
( ) ( )
Let, ( ) ( )
 As the ambient temperature is constant, so differentiation of the equation is
Thus
( )
Where
√
 Equations 3.4 and 3.5 provide a general form of the energy equation for one
dimensional heat dissipation from an extended surface.
 The general solution of this linear homogeneous second order differential equation
is of the form
( )
 The constant and are to be determined with the aid of relevant boundary
conditions. We will treat the following four cases:
i Heat dissipation from an infinitely long fin
ii Heat dissipation from a fin insulated at the tip
iii Heat dissipation from a fin losing heat at the tip
3.3 Heat Dissipation From an Infinitely Long Fin
 Governing differential equation for the temperature distribution along the length of
the fin is given as,
( )
 The relevant boundary conditions are

Fig. 3.3 Temperature distribution along the infinite long fin
 Temperature at the base of fin equals the temperature of the surface to which the
fin is attached.
 In terms of excess temperature
or
 Substitution of this boundary condition in equation gives:
( )
 Temperature at the end of an infinitely long fin equals that of the surroundings.
 Substitution of this boundary condition in equation gives:
( )
 Since the term is zero, the equality is valid only if . Then from
equation 3.8 .
 Substituting these values of constant and in equation 3.7, following expression
is obtained for temperature distribution along the length of the fin.
( ) ( ) ( )
 Heat transfer from fin
 Heat transfer to the fin at base of the fin must equal to the heat transfer from the
surface of the fin by convection. Heat transfer to the fin at base is given as
( ) ( )
 From the expression for the temperature distribution (Equation 3.10)
( )

( ) [ ( ) ]
( )
 Substitute the value of ( ) in the equation 3.11
( )
But
√
√ ( ) ( )
 The temperature distribution (Equation 3.10) would suggest that the temperature
drops towards the tip of the fin.
 Hence area near the fin tip is not utilized to the extent as the lateral area near the
base. Obviously an increase in length beyond certain point has little effect on heat
transfer.
 So it is better to use tapered fin as it has more lateral area near the base where the
difference in temperature is high.
 Ingen-Hausz Experiment
Fig. 3.4 Setup of Ingen-Hausz’s Experiment
 Heat flow rates through solids can be compared by having an arrangement
consisting essentially of a box to which rods of different materials are attached
(Ingen-Hausz experiment).
 The rods are of same length and area of cross-section (same size and shape); their
outer surfaces are electroplated with the same material and are equally polished.
 This is to ensure that for each rod, the surface heat transfer will be same. Heat flow
from the box along the rod would melt the wax for a distance which would depend
upon the rod material. Let
= excess of temperature of the hot bath above the ambient temperature
= excess of temperature of melting point of wax above the ambient temperature
, , ……… = lengt s upto w ic wax melts.
 Then for different rods (treating each as fin of infinite length),

So
or
√ √ √
or
√ √ √
( )
or
 Thus, the thermal conductivity of the material of the rod is directly proportional to
the square of the length upto which the wax melts on the rod.
3.4 Heat Dissipation From a Fin Insulated At The Tip
 The fin is of any finite length with the end insulated and so no heat is transferred
from the tip.
 Therefore, the relevant boundary conditions are:
fin is attached.
Fig. 3.5 Heat dissipation from a fin insulated at the tip

or
 Substitution of this boundary condition in equation 3.6 gives:
( )
 As the tip of fin is insulated, temperature gradient is zero at end of the fin.
But
( )
( )
 Substitute the value of from equation 3.14 into equation 3.15
( )
( )
( )
[ ] ( )
 Substitute the value of in equation 3.14, we get
[ ] ( )
 Substitute the values of constant in equation 3.6, expression for temperature
distribution along the length of the fin is obtained
( ) ( )
 In terms of hyperbolic function, expression is given as
( )
( )
( )
 The rate of heat flow from the fin is equal to the heat conducted to the fin at the
base, so heat flow from the fin is given by
( ) ( )
( )
( )
( )

( )
( )
( )
( )
( ) ( ) ( ) ( )
 Substitute the value of equation 3.20 in equation 3.19, we get
( ) ( )
But
√
√ ( ) ( ) ( )
3.5 Heat Dissipation From a Fin Losing Heat At The Tip
 The fin tips, in practice, are exposed to the surroundings. So heat may be transferred
by convection from the fin tip.
Fig. 3.6 Heat dissipation from fin losing heat at the tip
 Therefore, relevant boundary conditions are
fin is attached.
or
 Substitution of this boundary condition in equation 3.6 gives:
( )

 As the fin is losing heat at the tip, i.e., the heat conducted to the fin at equals
the heat convected from the end to the surroundings
( ) ( )
 At the tip of fin, the cross sectional area for heat conduction equals the surface
area from which the convective heat transport occurs. Thus
( )
 Governing differential equation of fin is given as
( )
 Substitute above value in equation 3.23, we get
( )
 But,
 Substitute this value in equation 3.24
[ ]
 Substitue the value of from equation 3.22 in above equation
( ) [ ( ) ]
[ ]
[ ( )] [ ]
* +
( ) ( )
And

* +
( ) ( )
[
( )
( ) ( )
]
[
( ) ( ) ( )
( ) ( )
]
[
( )
( ) ( )
]
[
( ) ( )
]
( )
( ) ( )
( )
 Substituting these values of constants and in equation3.6, one obtains the
following expressiojn for temperature distribution along the length of the fin.
* +
( ) ( )
( )
( ) ( )
( )
( ) ( ) ( )
( ) ( )
( ) ( ) ( ) ( )
( ) ( )
( ) ( )
( ( ) ( )
)
( ) ( )
 Exporessing in terms of hyperbolic functions
( ) ( )
( ) ( )
( )
 The rate of heat flow from the fin is equal to the heat conducted to the fin at the
base, so heat flow from the fin is given by

( ) ( )
( ) [
( ) ( )
( ) ( )
]
( ) [
( ) , ( )-
( ) ( )
]
( ) ( ) [
( ) ( )
( ) ( )
]
 Substitute this value in equation 3.27
( ) [
( ) ( )
( ) ( )
]
But,
√
√ ( ) [
( ) ( )
( ) ( )
]
√ ( ) [
( )
( )
] ( )
3.6 Fin Performance
 It is necessary to evaluate the performance of fins to achieve minimum weight or
maximum heat flow etc.
 Fin effectiveness and fin efficiency are some methods used for performance
evaluation of fins
 Efficiency of fin:
 It relates the performance of an actual fin to that of an ideal or fully effective fin.
 In reality, temperature of fin drop along the length of fin, and thus the heat transfer
from the fin will be less because of the decreasing temperature difference towards
the tip of fin.
 A fin will be most effective, i.e., it would dissipate heat at maximum rate if the entire
fin surface area is maintained at the base temperature as shown in figure 3.7

Fig. 3.7 Ideal and actual temperature distribution in a fin
 Thus for a fin insulated at tip
√ ( ) ( )
( )( )
 The parameter represents the total surface area exposed for convective heat
flow. Upon simplification,
( )
√ ⁄
( )
( )
 Following poins are noted down from the above equation
i For a very long fin
( )
 Obviously the fin efficiency drops with an increase in its length.
 For small values of , the fin efficiency increases. When the length is reduced to
zero, then,
( )
 Thus the fin efficiency reaches its maximum vlaue of 100% for a tgrivial value of
, i.e., no fin at all.
 Actually efficiency of fin is used for the design of the fin but it is used for comparision
of the relative merits of fin of different geometries or material.
 Note that fins with triangular and parabolic profiles contain less material and are
more efficient than the ones with rectangular profiles, and thus are more suitable for
applications requiring minimum weight such as space applications.
 An important consideration in the design of finned surfaces is the selection of the
proper fin length L.
 Normally the longer the fin, the larger the heat transfer area and thus the higher the
rate of heat transfer from the fin.
 But also the larger the fin, the bigger the mass, the higher the price, and the larger
the fluid friction.

 Therefore, increasing the length of the fin beyond a certain value cannot be justified
unless the added benefits outweigh the added cost.
 Also, the fin efficiency decreases with increasing fin length because of the decrease
in fin temperature with length.
 Fin lengths that cause the fin efficiency to drop below 60 percent usually cannot be
justified economically and should be avoided.
 The efficiency of most fins used in practice is above 90 percent.
 Effectiveness of fin ( ):
 Fins are used to increase the heat transfer. And use of fin can not be recommended
unless the increase in heat transfer justifies the added cost of fin.
 In fact, use of fin may not ensure the increase in heat transfer. Effectiveness of fin
gives the increase in heat transfer with fin relative to no fin case.
 It represents the ratio of the fin heat transfer rate to the heat transfer rate that
would exist without a fin.
 Figure 3.8 shows the base heat transfer surface before and after the fin has been
attached.
 Heat transfer through the root area before the fin attached is:
( )
Fig. 3.8 Heat dissipation with and without fin
 After the attachment of an infinitely long fin, the heat transfer rate through the root
area becomes:
√ ( )
So, effectiveness of fin is given as
√ ( )
( )

√ ( )
 Following conclusions are given from the effectiveness of the fin
i If the fin is used to improve heat dissipation from the surface, then the fin
effectivenss must be greater than unity. That is,
√
But literature suggests that use of fins on surrface is justified only if the ratio
⁄ is greater than 5.
ii To improve effectiveness of fin, fin should be made from high conductive
manterial such as copper and aluminium alloys. Although copper is superior to
aluminium regarding to the thermal conductivity, yet fins are generally made of
aluminium because of their additional advantage related to lower cost and
weight.
iii Effectiveness of fin can also be increased by increasing the ratio of perimeter to
the cross sectional area. So it is better to use more thin fins of closer pitch than
fewer thicker fins at longer pitch.
iv A high value of film coefficient has an adverse effect on effectiveness. So fins are
used with the media with low film coefficient. Therefore, in liquid – gas heat
exchanger,such as car radiator, fins are placed on gas side.
 Relation between effeciency of fin and effectiveness of fin
√ ( ) ( )
( )( )
√ ( ) ( )
( )
( )( )
( )
( )
( )
 An increase in the fin effectiveness can be obtained by extending the length of fin
but that rapidly becomes a losing proposition in term of efficiency.
3.7 Thermometric Well
 Figure 3.9 shows an arrangement which is used to measure the temperature of gas
flowing through a pipeline.

 A small tube called thermometric well is welded radially into the pipeline. The well is
partially filled with some liquid and the thermometer is immersed into this liquid.
 When the temperature of gas flowing through the pipe is higher than the ambient
temperature, the heat flows from the hot gases towards the tube walls along the
well. This may cause temperature at the bottom of well to become colder than the
gas flowing around.
 So the temperature indicated by the thermometer will not be the true temperature
of the gas.
 The error in the temperature measurement is estimated with the help of the theory
of extended surfaces.
Fig. 3.9 Thermometric well
 The thermometric well can be considered as a hollow fin with insulated tip.
Temperature distribution is obtained as
( )
( )
 Where is the temperature of pipe wall, is the temperature of hot gas or air
flowing through the pipeline, and is the temperature at any distance x measured
from pipe wall along the thermometric well.
 If then
( )
( ) ( )
( )
 Where is the temperature recorded by the thermometer at the bottom of the
well.

Fig. 3.10 Use of thermometric well
 The perimeter of the protective well ( ) , and its cross-sectional
area . Therefore
Then
√ √ ( )
 From the equation 3.33 it is clear that diameter of the well does not have any effect
on temperature measurement by the thermometer.
 The error can be minimized by
i Lagging the tube so that conduction of heat along its length is arrested.
ii Increasing the value of parameter
 For a rectangle fin √ ⁄ .
 An increasing in can be affected by using a thinner tube of low thermal
conductivity or by increasing the convection co-efficient through finning the
manometric well
 The operative length is increased by inkling the pocket and setting its projection
beyond the pipe axis.
Ex. 3.1.
A cooper rod 0.5 cm diameter and 50 cm long protrudes from a wall maintained
at a temperature of 500 . The surrounding temperature is 30 . Convective heat
transfer coefficient is ⁄ and thermal conductivity of fin material is
⁄ . Show that this fin can be considered as infinitely long fin. Determine
total heat transfer rate from the rod.
Solution:
Given data:

, , , ,
⁄ , ⁄
⁄ ⁄ ( )
⁄
√ √ √
Fin can be considered as infinite long fin, if heat loss from the infinitely long rod is
equal to heat loss from insulated tip rod.
Heat loss from infinitely long rod is given by
( )
and heat loss from the insulated tip fin is given by
( ) ( )
These expressions provide equivalent results if ( )
Hence the rod can be considered infinite if
Since length of the rod (0.5 m) is greater than 0.256 m, rod can be considered as
infinitely long rod.
Heat loss from infinitely long rod is given by
( )
( )
Ex. 3.2.
Two rods A and B of equal diameter and equal length, but of different materials are
used as fins. The both rods are attached to a plain wall maintained at 160 , while
they are exposed to air at 30 . The end temperature of rod A is 100 while that of
the rod B is 80 . If thermal conductivity of rod A is 380 W/m-K, calculate the
thermal conductivity of rod B. These fins can be assumed as short with end
insulated.
Solution:
Given data:
Both rods are similar in their shape and size, connected to same wall and exposed to
same environment. So, for both the rods area and perimeters are equal and
following parameters are same.
, ,
For rod A: , ⁄
For rod B: ,
Temperature distribution for insulated tip fin is given by
( )
( )
And temperature at the free end,

( )
For rod A
( )
( )
For rod B
( )
( )
From above two calculation
√ ⁄
√ ⁄
√
( ) ⁄
Ex. 3.3.
A steel rod (k=30 W/m ), 12 mm in diameter and 60 mm long, with an insulated
end is to be used as spine. It is exposed to surrounding with a temperature of 60
and heat transfer coefficient of 55 W/m2 . The temperature at the base is 100 .
Determine : (i) The fin effectiveness (ii) The fin efficiency (iii) The temperature at
the edge of the spine (iv) The heat dissipation.
Solution:
Given data:
, , , ,
⁄ , ⁄
⁄
√ √ √
⁄ ⁄ ( )

i. Effectiveness of the fin
√ ( ) ( )
( )
√ ( )
√ ( )
ii. The fin efficiency
For a fin insulated at tip
√ ( ) ( )
( )( )
( )
√ ⁄
( ) ( )
iii. Temperature at edge of the spine
Temperature distribution for insulated tip fin is given by
( )
( )
And temperature at the free end,
( )
( )
( )
iv. The heat dissipation with insulated tip fin
( ) ( )
( ) ( )
Ex. 3.4.
A gas turbine blade made of stainless steel (k = 32 W/m-deg) is 70 mm long, 500
mm2 cross sectional area and 120 mm perimeter. The temperature of the root of
blade is 500 and it is exposed to the combustion product of the fuel passing from
turbine at 830 . If the film coefficient between the blade and the combustion gases
is 300 W/m2-deg, determine:
(i) The temperature at the middle of blade,
(ii) The rate of heat flow from the blade.

Solution:
Given data:
⁄ , ,
, , , ⁄ ,
√ √
( ) ( )
( ) ( )
( ) ( )
i. The temperature at the middle of blade
Temperature distribution for fin losing heat at the tip is given by
( ) ( )
( ) ( )
At the middle of the blade ⁄
( ) ( )
( ) ( )
( )
ii. Heat flow through the blade is given by
( ) [
( )
( )
]
( ) [ ]
The – ve sign indicates that the heat flows from the combustion gases to the blade.
Ex. 3.5.
An electronic semiconductor device generates 0.16 kj/hr of heat. To keep the surface
temperature at the upper safe limit of 75 , it is desired that the generated heat
should be dissipated to the surrounding environment which is at 30 . The task is
accomplished by attaching aluminum fins, 0.5 mm2 square and 10 mm to the surface.
Calculate the number of fins if thermal conductivity of fin material is 690 kj/m-hr-
deg and the heat transfer coefficient is 45 kj/m2-hr-deg. Neglect the heat loss from
the tip of the fin.
Solution:
Given data:

⁄ ⁄ ⁄
, , , ,
⁄ ⁄ ,
For square fin ,
√
Perimeter of the fin is given by
√ √
Heat loss from insulated tip fin is given by
( ) ( )
( )
Total number of fins required are given by
So, to dissipate the required heat 283 no. of fins are required.
3.9 References
Publication.

4TRANSIENT HEAT CONDUCTION
Course Contents
4.1 Introduction
4.2 Transient Conduction in
Solids with Infinite Thermal
Conductivity k
(Lumped Parameter Analysis)
4.3 Time Constant and Response
of a Thermocouple
4.4 Transient Heat Conduction in
Solid with Finite Conduction
and Convective Resistance
4.6 References

4. Transient Heat Conduction Heat Transfer (2151909)
4.1 Introduction
 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.
 But before steady-state conditions are reached, some time must elapse when a solid
body is suddenly subjected to a change in environment. During this transient period
the temperature changes, and the analysis must take into account changes in the
internal energy.
 This study is a little more complicated due to the introduction of another variable
namely time to the parameters affecting conduction. This means that temperature is
not only a function of location, as in the steady state heat conduction, but also a
function of time, i.e. ( ).
 Transient heat flow is of great practical importance in industrial heating and cooling,
some of the applications are given as follow
i Heating or cooling of metal billets;
ii Cooling of I.C. engine cylinder;
iii Cooling and freezing of food;
iv Brick burning and vulcanization of rubber;
v Starting and stopping of various heat exchanger unit in power plant.
 Change in temperature during unsteady state may follow a periodic or a non-
periodic variation.
 Periodic variation
 The temperature changes in repeated cycles and the conditions get repeated after
some fixed time interval. Some examples of periodic variation are given follow
i Variation of temperature of a building during a full day period of 24 hous
ii Temperature variation in surface of earth during a period of 24 hours
iii Heat processing of regenerators whose packings are alternately heated by flue
gases and cooled by air
 Non-periodic variation
 The temperature changes as some non-linear function of time. This variation is
neither according to any definite pattern nor is in repeated cycles. Examples are:
i Heating or cooling of an ingot in a furnace
ii Cooling of bars, blanks and metal billets in steel works
4.2 Transient Conduction in Solids with Infinite Thermal
Conductivity (Lumped Parameter Analysis)
 Even though no materials in nature have an infinite thermal conductivity, many
transient heat flow problems can be readily solved with acceptable accuracy by
assuming that the internal conductive resistance of the system is so small that the
temperature within the system is substantially uniform at any instant.

Heat Transfer (2151909) 4. Transient Heat Conduction
 This simplification is justified when the external thermal resistance (Convection
resistance) between the surface of the system and the surrounding medium is so
large compared to the internal thermal resistance (Conduction resistance) of the
system that it controls the heat transfer process.
 Consider a small hot copper ball coming out of an oven (Figure 4–1). Measurements
indicate that the temperature of the copper ball changes with time, but it does not
change much with position at any given time due to large thermal conductivity.
 Thus the temperature of the ball remains uniform at all times.
Fig. 4.1 Temperature distribution throughout the copper ball
 Consider a body of arbitrary shape of mass m, volume V, surface area , density ,
and specific heat initially at a uniform temperature (Figure 4–2).
Fig. 4.2 Lumped parameter analysis
 At time = 0, the body is placed into a medium at temperature , and heat transfer
takes place between the body and its environment, with a heat transfer coefficient
h. Let , but the analysis is equally valid for the opposite case.
 During a differential time interval , the temperature of the body falls by a
differential amount . An energy balance of the solid for the time interval can
be expressed as:
( ) ( )
( )
( )

 Negative sign indicates the decrease in internal energy. This expression can be
rearranged and integrated.
∫
( )
∫
( ) ( )
 The integration constant is evaluated from the initial conditions: .
Substitute the value of boundary condition in equation 4.1, we get
( )
 Substitute the value of in equation 4.1, we get
( ) ( )
( )
( )
( )
( )
( ) ( )
 Equation 4.2 is used to find the temperature at any instant .
 Following points can be made from the above equations:
i The body temperature falls or rises exponentially with time and the rate depends on
the parameter ( ⁄ ). Theoretically the body takes infinite time to approach
the temperature of surroundings and thus attain the steady state conditions.
However the difference between and becomes extremely small after a short
time and beyond that period the body temperature becomes practically equal to the
ambient temperature. The change in temperature of a body with respect to time is
shown in figure 4.3 for both cases (Heating and cooling)
Fig. 4.3 Change in temperature of body with respect to time
ii The quantity ( ⁄ ) has the dimensions of time and is called the thermal time
constant. Its value is indicative of the rate of response of a system to a sudden

change in the environmental temperature; how fast body will respond to a change in
the environmental temperature. It should be as small as possible for fast response of
the system to change in environmental temperature.
 Exponential term can be arranged in dimensionless term as follow:
( ) ( )
( ) ( )
 Where, ( ⁄ ) is the thermal diffusivity of the solid, and is a characteristic
length equal to the ratio of the volume of the solid to its surface area.
 The value of characteristic length of different geometry:
 The non-dimensional factor ( ⁄ ) is called the Fourier number, . It signifies the
degree of penetration of heating or cooling effect through a solid. For instance, a
large time would be required to obtain a significant temperature change for small
values of ( ⁄ ).
 The non-dimensional factor ( ⁄ ) is called the Biot number, . It gives the
indication of the ratio of internal (conduction) resistance to the surface (convection)
resistance.
 A small value of implies that the system has a small conduction resistance, i.e.
relatively small temperature gradient or nearly uniform temperature within the
system. In that case heat transfer is predominates by convective heat transfer
coefficient.
 Criteria for Lumped System Analysis
 Biot number is used to check the applicability of lumped parameter analysis. If Biot
number is less than 0.1, it has been proved that this model can be used without
appreciable error.
 The lumped parameter solution for transient conduction can be conveniently stated
as
( )
( )
( ) ( )
 Instantaneous and total heat flow rate

 The instantaneous heat flow rate may be obtained by using Newton’s law of
cooling. Heat transfer from the body at any instant is given as:
( ) ( )
 Where T is the temperature at any instant . Substitute the value of ( ) from
the equation no. 4.2. We get
( ) ( ) ( )
 Total heat flow rate
 Total heat flow rate can be obtained by integrating the equation 4.5 over the time
interval .
∫
∫ ( ) ( )
* ( )
[ ( ⁄ ) ]
⁄
+
( ) [ ( )]
( ) [ ( ) ] ( )
4.3 Time Constant and Response of a Thermocouple
 A Thermocouple is a sensor used to measure temperature. A thermocouple is
comprised of at least two metals joined together to form two junctions.
 One is connected to the body whose temperature is to be measured; this is the hot
or measuring junction. The other junction is connected to a body of known
temperature; this is the cold or reference junction.
 Therefore the thermocouple measures unknown temperature of the body with
reference to the known temperature of the other body.
 Measurement of temperature by a thermocouple is an important application of the
lumped parameter analysis.
 The response of a thermocouple is defined as the time required for the
thermocouple to reach the source temperature when it is exposed to it.
 Referring to the lumped-parameter solution for transient heat conduction;
( )
( )
( ) ( )

 It is evident that larger the parameter ⁄ , the faster the exponential term will
reach zero or more rapid will be the response of the thermocouple. A large value of
⁄ can be obtained either by increasing the value of convective coefficient, or
by decreasing the wire diameter, density and specific heat.
 The sensitivity of the thermocouple is defined as the time required by the
thermocouple to reach 63.2% of its steady state value. According to definition of
sensitivity
 Substitute the value in equation 4.7
( )
 The parameter ⁄ has units of time and is called time constant of the system
and is denoted by . Thus
( )
 Using time constant, the temperature distribution in the solids can be expressed as
( )
( )
( ) ( )
 The time constant represents the speed of response, i.e., how fast the thermocouple
tends to reach the steady state value. A large time constant corresponds to a slow
system response, and a small time constant represent a fast response. A low value of
time constant can be achieved for a thermocouple by
i Decreasing light metals the wire diameter
ii Using light metals of low density and low specific heat
iii Increasing the heat transfer coefficient
 Depending upon the type of fluid used, the response times for different sizes and
materials of thermocouple wires usually lie between 0.04 to 2.5 seconds.
 Note:- Once the time constant is measured, we have to wait for the that time to
measure the temperature within 63.2% of accuracy.

4.4 Transient Heat Conduction In Solids With Finite Conduction
and Convective Resistance (0 < Bi < 100)
 In the lumped parameter analysis we assume that conductivity of the material is
infinite or variation of temperature within the body is negligible.
 But sometimes there may be variation of temperature with time and position.
 Consider a plane wall of thickness 2L, a long cylinder of radius ro, and a sphere of
radius ro initially at a uniform temperature Ti, as shown in figure 4.4.
 Note that all three cases possess geometric and thermal symmetry: the plane wall is
symmetry about its center plane (x = 0), the cylinder is symmetry about its centerline
(r = 0), and the sphere is symmetry about its center point (r = 0).
Fig. 4.4 Transient heat conduction in large wall, cylinder and sphere
 At a time , each geometry is placed in a large medium that is at a constant
temperature . Heat transfer takes place between these bodies and their
environments by convection with a uniform and constant heat transfer coefficient h.
 Temperature profile of plane wall
 The variation of temperature profile with respect to time in plane wall is shown in
figure 4.5.

Fig. 4.5 Transient heat conduction in large wall, cylinder and sphere
 When the wall is first exposed to the surrounding medium the entire wall is at its
initial temperature .
 But the wall temperature at the surface starts to drop as a result of heat transfer
from the wall to the surrounding medium. This creates a temperature gradient in the
wall.
 The temperature profile within the wall remains symmetric at all times about the
centre plane. The temperature profile gets flatter and flatter as times passes as a
result of heat transfer and finally becomes uniform at .
 The controlling differential equation for the transient heat conduction is:
 The appropriate boundary conditions are :
 at ; initially the wall is at uniform temperature
 ⁄ at ; symmetrical nature of the temperature profile within the plane
wall;
 ( ⁄ ) ( ) at . At the surface heat transfer by conduction is
equal to heat transfer by convection from the surface to medium.
 The solution of the controlling differential equation in conjunction with initial
boundary conditions would give an expression for temperature variation both with
time and position.
 The solution obtained after mathematical analysis indicate that
( ) ( )
 The temperature history becomes a function of Biot number ⁄ , Fourier number
⁄ and the dimensionless parameter ⁄ which indicates the location of point
within the plate where temperature is to be obtained. In case of cylinders and
spheres ⁄ is replaced by ⁄ .

 The Heisler charts give the temperature history of the solid at its mid plane, .
The temperatures at other locations are worked out by multiplying the mid-plane
temperature by correction factors read from correction charts.
 Following relation is used to measure temperature at any location
( ) ( ) ( )
 The Heisler charts are extensively used to determine the temperature distribution
and heat flow rate when both conduction and convection resistances are almost of
equal importance.
Ex. 4.1.
A spherical element of 40 mm diameter is initially at temperature of . It is
placed in boiling water for 4 minutes. After 4 minuts, at what temperature, the
spherical element will reach? If the same spherical element is initially at , find out
by lump theory that how much time will be taken by the element to reach at that
temperature? Take properties of the given spherical element as:
⁄ , ⁄ , ⁄ and heat transfer coefficient
⁄ .
Solution:
Given data:
, , ,
⁄
a. Find the temperature of spherical element after 4 min.
( )
( )
( )
( )
( )
( ( ) ) ( ( ) )
( )
( )
( ( ) )
( )
( )
( )

b. Find the time required to reach desired temperature of when initial
temperature is
( )
( )
( )
( )
( )
( ( ) ) ( ( ) )
( )
( )
( ( ) )
( )
( ) ( )
Ex. 4.2.
During a heat treatment process, spherical balls of 12 mm diameter are initially
heated to . Then they are cooled to by immersing them in an oil bath of
with convection coefficient ⁄ . Determine time required for cooling
process. What should be the convection coefficient if it is intended to complete the
cooling process in 10 minutes?
Thermo-physical properties of the balls are ⁄ , ⁄ ,
⁄ .
Solution:
Given data:
, , , , ⁄
⁄
a. Find the time required to obtain the required temperature.
( )
( )
( )
( )
( )
( ( ) ) ( ( ) )
( )
( )
( ( ) )
( )

( ) ( )
b. Find the convection co-efficient to complete the above process in 10 minutes.
( )
( )
( )
( )
( )
( ( ) ) ( ( ) )
( )
( )
( ( ) )
( )
( ) ( )
⁄
Ex. 4.3.
The temperature of an air stream flowing with a velocity of 3 m/s is measured by a
copper-constantan thermocouple which may be approximated as sphere of 3 mm in
diameter. Initially the junction and air are at a temperature of . The air
temperature suddenly changes to and is maintained at .
Take ⁄ , ⁄ , and ⁄ and ⁄ .
Determine: (i) Thermal time constant and temperature indicated by the
thermocouple at that instant (ii) Time required for the thermocouple to indicate a
temperature of (iii) Discuss the suitability of this thermocouple to measure
unsteady state temperature of fluid then the temperature variation in the fluid has a
time period of 30 seconds.
Solution:
Given data:
, , , ⁄ , ⁄
⁄
i. Thermal time constant and temperature indicated by it at that instant

( ) ( )
( )
Temperature at time .
( )
( )
( )
( )
( )
( )
( )
( )
ii. Time required for the thermocouple to indicate the temperature of
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
Since, the time constant ( ) is less than the time for the temperature change of
the fluid ( ), the thermometer will give a faithful record of the time varying
temperature of the fluid.
4.6 References:
Publication.

5RADIATION PROCESS AND
PROPERTIES
Course Contents
5.1 Introduction
5.2 Salient features and
characteristics of radiation
5.3 Wavelength distribution of
black body radiation:
Plank’s Law
5.4 Total Emissive Power:
Stefan-Boltzman law
5.5 Wien’s Displacement law
5.6 Relation between
Emissivity and absorptivity
of the body: Kirchoff’s Law
5.7 Plane and solid angle
5.8 Intensity of Radiation and
Lambert’s cosine law
5.9 Relation between the
normal intensity and
emissive power.
5.11 References

5. Radiation Process and Properties Heat Transfer (2151909)
5.1 Introduction
 Consider a hot object that is suspended in an evacuated chamber whose walls are at
room temperature (Figure 5.1). The hot object will eventually cool down and reach
thermal equilibrium with its surroundings.
 That is, it will lose heat until its temperature reaches the temperature of the walls of
the chamber.
 Heat transfer between the object and the chamber could not have taken place by
conduction or convection, because these two mechanisms cannot occur in a
vacuum.
 Therefore, heat transfer must have occurred through another mechanism that
involves the emission of the internal energy of the object. This mechanism is
radiation.
Fig. 5.1 Hot object in vacuum chamber
Fig. 5.2 Radiation heat transfer from hot to cold body
Unlike conduction and convection, heat
transfer by radiation can occur between
two bodies, even when they are separated
by a medium colder than both as shown in
figure 5.2.
Note:- Radiation differs from the other two heat transfer mechanisms in that it
does not require the presence of a material medium to take place. In fact, energy
transfer by radiation is fastest (at the speed of light) in a vacuum. Also, radiation
transfer occurs in solids as well as liquids and gases. In most practical
applications, all three modes of heat transfer occur concurrently at varying
degrees. But heat transfer through an evacuated space can occur only by
radiation. For example, the energy of the sun reaches the earth by radiation.

Heat Transfer (2151909) 5. Radiation Process and Properties
5.2 Salient Features and Characteristics of Radiation
 Radiation is the propagation and emission of energy in the form of electromagnetic
waves.
 The general phenomenon of radiation covers the propagation of electromagnetic
waves of all wavelengths, from short wavelength gamma rays to long wavelength
microwave.
Fig. 5.3 Electromagnetic wave spectrum
 Thermal radiation is that electromagnetic radiation emitted by a body as a result of
its temperature.
 Thermal radiation is limited to range of wavelength between 0.1 to 100 m, which
includes the entire visible and infrared and a part of the ultraviolet spectrum.
 Light is simply the visibleportion of the electromagneticspectrum that lies
between 0.40 and 0.76 m.
 A body that emits some radiation in the visible range is called a lightsource. The
sun is our primary light source.
 The radiation emitted by bodies at room temperature falls into the
infraredregion of the spectrum, which extends from 0.76 to 100 m.
( ) ( ) ( )
 The electromagnetic waves are emitted as a result of vibrational and
rotational movements of the molecular, atomic or sub atomic particles
comprising the matter. When body is excited by an oscillating electrical
signal, electronic or neutronic bombardment, chemical reaction etc, emission
of radiation occur.
 One form of radiation is differ from the other form of radiation by its
frequency and wavelength. The relation between frequency and wavelength
is given as
Wave length,

 Thermal radiation exhibit characteristics similar to those of light, and follow the
optical laws.
 Thermal radiation is continuously emitted by all matter whose temperature is above
absolute zero.
 Body at higher temperature emits energy at greater rate than bodies at low
temperature.
 Heat transfer by radiation depends upon the level of temperature unlike conduction
and convection.
5.2.1 Absorptivity, Reflectivity, and Transmissivity
 When thermal radiation ( )is incident on a surface, a part of the radiation may be
reflected by the surface ( ), a part may be absorbed by the surface ( ) and a part
may be transmitted through the surface ( ) as shown in figure 5.4.
 These fractions of reflected, absorbed, and transmitted energy are interpreted as
system properties called reflectivity, absorptivity, and transmissivity, respectively.
 Heat transfer by conduction and convection from the body at temperature of
1000 to surrounding at temperature of 800 is practically remains same for
the body at temperature of 900 to surrounding at temperature of 700 .
 Where as in the case of radiation heat transfer, heat transfer is not same even if
the temperature differences are same.
 Net heat transfer by radiation at elevated temperature is greater than heat
transfer at low temperature.
 Normally a body radiating heat is simultaneously receiving heat from other
bodies as incident radiation.
 Net heat exchange between two radiating surfaces is due to the fact that one at
high temperature radiates more and receives less energy for its absorption.
 An isolated body which remains at constant temperature emits just as much
energy radiation as it receives.
 The ultraviolet radiation includes the low-wavelength end of the
thermalradiation spectrum and lies between the wavelengths 0.01 and 0.40
m.Ultraviolet rays are to be avoided since they can kill microorganisms
andcause serious damage to humans and other living beings.
 About 12 percentof solar radiation is in the ultraviolet range.The ozone (O3)
layer inthe atmosphere acts as a protective blanket and absorbs most of this
ultravioletradiation.

Fig. 5.4: Reflection, absorption and transmitted energy
Thus using energy conservation,
Dividing these equation by
( )
Where
= absorptivity or fraction of total energy absorbed by the body
= reflectivity or fraction of total energy reflected from the body
= transmissivity or fraction of total energy transmitted through the body
The factors , and are dimensionless and vary from 0 to 1.
 A blackbody is defined as a perfect emitter and absorber of radiation. At a specified
temperature and wavelength, no surface can emit more energy than a blackbody.
 A blackbody absorbs all incident radiation, regardless of wavelength and direction.
Also, a blackbody emits radiation energy uniformly in all directions per unit area
normal to direction of emission. For black body = 1, and
 When a surface absorbs a certain fixed percentage of impinging radiations, the
surface is called gray body. A surface whose properties are independent of the
wavelength is known as a gray surface.
 A gray body is defined such that the monochromatic emissivity of the body is
independent of wavelength. For gray body
 The condition of constant absorptivity too is not satisfied by the real materials
and as such even a gray body remains a hypothetical concept like the black
body.
 In actual practice there does not exist a perfectly black body which will absorb
all incident radiations. Snow, with its absorptivity 0.985, is nearly black to the
thermal radiation.
 The absorptivity of a surface depends upon the direction of incident radiation,
temperature of the surface, composition and structure of the irradiated surface
and the spectral distribution of incident radiation.

 A body that reflects all the incident thermal radiations is called an absolutely white
body or specular body. For white body = 1, and
 A body that allows all the incident radiations to pass through it is calledtransparent
body or diathermanous. For such body , and
 Transmissivity varies with wave length of incident radiation. A material may be non-
transparent for a certain wavelength transparent for another. This type of material is
called selective transmitter.
 For opaque body, , and . It means that good absorbers are bad
reflector or vice-versa.
 The electrons, atoms, and molecules of all solids, liquids, and gases above
absolute zero temperature are constantly in motion, and thus radiation is
constantly emitted, as well as being absorbed or transmitted throughout the
entire volume of matter.
 That is, radiation is a volumetric phenomenon.
 A thin glass plate transmits most of the thermal radiations from the sun, but
absorbs in equally great measure the thermal radiations emitted from the low
temperature interior of a building.
 That’s the reason to use the glass in green house to trap the solar radiation in
low temperature space.
 Regular (specular) reflection implies that angle between the reflected
beam and the normal to the surface equals the angle made by the incident
radiation with the same normal.
 In a diffused radiation, incident beam is reflected in all directions.
(a) Specular radiation (b) Diffused radiation
Fig. 5.5 Specular and diffused radiations
 Most of the engineering materials have rough surfaces, and these rough
surfaces give diffused reflections.

5.2.2 Black Body Concept
 Consider a large cavity with small opening maintained at constant temperature as
shown in figure 5.7.
 The inner surface of the cavity is coated with the black lamp. A beam of thermal
radiation entering the hole strikes the inner surface. Since the inner surface has high
absoptivity, the major portion of the radiation is absorbed and only a small fraction
is reflected.
 The weak reflected beam does not find any way out and again strikes the inner
surface. Here it is again partly absorbed and partly reflected.
 Likewise the reflected radiation is successively absorbed and finally when is escapes
out, it has only a negligible amount of energy associated with it.
Fig.5.7 Black body concept
 Although a blackbody would appear black to the eye, a distinction should be
made between the idealized blackbody and an ordinary black surface.
 Any surface that absorbs light (the visible portion of radiation) would appear
black to the eye, and a surface that reflects it completely would appear white.
 Radiation in opaque solid is considered a surface phenomenon since the
radiation emitted only by the molecules at the surface can escape the solid
as shown in figure 5.6.
Fig. 5.6 Radiation in opaque solid

5.2.3 Spectral and Spatial Energy Distribution
 Spectral Energy Distribution: The radiation emitted by the body consists of
electromagnetic waves of various wavelengths. Distribution of radiation with wave
length is called spectral energy distribution as show in figure 5.8(a).
(a) Spectral energy distribution (b) Spatial energy distribution
Fig. 5.8 Spectral and spatial energy distribution
 Spatial (Directional) Energy Distribution: A surface emits the radiation in all
directions. The intensity of radiation is different in different direction. The
distribution of radiation along the direction is called spatial distribution.
5.3 Wavelength Distribution of Black Body Radiation: Plank’s Law
 The energy emitted by a black surface varies in accordance with wavelength,
temperature and surface characteristics of the body.
 Spectral blackbody emissive power (monochromatic emissive power) ( ) =
“amount of radiation energy emitted by a blackbody at an absolute temperature T
per unit time, per unit surface area, and per unit wavelength about the wavelength
.”
 Plank suggested the following law for the spectral distribution of emissive power:
( )
[ ⁄ ]
( )
Where,
h = plank constant,
 Considering that visible radiation occupies a very narrow band of the spectrum
from 0.4 to 0.76 _m, we cannot make any judgments about the blackness of a
surface on the basis of visual observations.
 For example, snow and white paint reflect light and thus appear white. But
they are essentially black for infrared radiation since they strongly absorb long-
wavelength radiation. Surfaces coated with lampblack paint approach idealized
blackbody behavior.

c = Velocity of light in vacuum,
k = Boltzman constant,
T = Absolute temperature of black body, K
The above expression is written as
( )
[ ⁄ ]
( )
Where,
⁄
⁄
 The variation of distribution of the monochromatic emissive power with wavelength
is called spectral energy distribution, and this has been shown in figure 5.9
 The following important features can be noted from this plot:
i The emitted radiation is a continuous function of wavelength. At any specified
temperature, it increases with wavelength, reaches a peak, and then decreases with
increasing wavelength.
Fig. 5.9 Radiation of black body as a function of wavelength and temperature
ii At any wavelength, the amount of emitted radiation increases with increasing
temperature.

iii As temperature increases, the pick of the curves shift to the left to the shorter
wavelength region. Consequently, a larger fraction of the radiation is emitted at
shorter wavelengths at higher temperatures.
iv The radiation emitted by the sun, which is considered to be a blackbody at 5780 K
(or roughly at 5800 K), reaches its peak in the visible region of the spectrum.
Therefore, the sun is in tune with our eyes.
v On theother hand, surfaces at T < 800 K emit almost entirely in the infraredregion and
thus are not visible to the eye unless they reflect light comingfrom other sources.
5.4 Total Emissive Power: Stefan-Boltzman law
 The total emissive power E of a surface is defined as the total radiant energy
emitted by the surface in all directions over the entire wavelength per unit surface
area per unit time.
 The basic rate equation for radiation transfer is based on Stefan-Boltzman law which
states that the amount of radiant energy emitted per unit area of black surface is
proportional to the fourth power of its absolute temperature.
( )
Where is the radiation coefficient of a black body.
 Total emissive power of black body can be obtained by integrating the
monochromatic emissive power over entire wavelength to
∫ ( ) ∫
[ ⁄ ]
( )
By simplifying the equation
( )
Where, is Stefan-Boltzmann constant, equal to ⁄ and T is the
absolute temperature in K.
 The Stefan-Boltzmann law helps us to determine the amount of radiations emitted in
all the directions and over the entire wavelength spectrum from a simple knowledge
of the temperature of the black body.
 Normally a body radiating heat is simultaneously receiving heat from other bodies as
radiation. Consider that surface 1 at temperature is completely enclosed by
another black surface at temperature . The net radiation heat flux is then given by
( ) ( )
5.5 Wien’s Displacement law
 Figure 5.9 shows that as the temperature increases the peaks of the curve also
increases and it shift towards the shorter wavelength.
 The wavelength, at which the monochromatic emissive power is a maximum, is
found by differentiating the Plank’s Equation with respect to and equating to zero.
( ) (
[ ⁄ ]
)

Solution of this equation is given as
( )
Where, is the wavelength at which emissive power is maximum.
 Wien’s displacement law may be stated as “The product of absolute temperature
and the wavelength at which the emissive power is maximum, is constant”
 It can be easily found out that the wavelength corresponding to the peak of the plot
( ) is inversely proportional to the temperature of the blackbody.
 It means that maximum spectral radiation intensity shifts towards the shorter
wavelength with rising temperature.
 The peak of the solar radiation, for example, occurs at
⁄ , which is near the middle of the visible
range.
 The peak of the radiation emitted by a surface at room temperature (T =
298 K) occurs at 9.72 m, which is well into the infrared region of the
spectrum.
 An electrical resistance heater starts radiating heat soon after it is plugged
in, and we can feel the emitted radiation energy by holding our hands
facing the heater. But this radiation is entirely in the infrared region and
thus cannot be sensed by our eyes. The heater would appear dull red
when its temperature reaches about 1000 K, since it will start emitting a
detectable amount (about 1 W/m2
· m) of visible red radiation at that
temperature.
 As the temperature rises even more, the heater appears bright red and is
said to be red hot. When the temperature reaches about 1500 K, the
heater emits enough radiation in the entire visible range of the spectrum
to appear almost white to the eye, and it is called white hot.
 Although it cannot be sensed directly by the human eye, infrared radiation
can be detected by infrared cameras, which transmit the information to
microprocessors to display visual images of objects at night.
 Rattlesnakes can sense the infrared radiation or the “body heat” coming
off warm-blooded animals, and thus they can see at night without using
any instruments.
 A surface that reflects all of the light appears white, while a surface that
absorbs all of the light incident on it appears black. (Then how do we see a
black surface?)
 It should be clear from this discussion that the color of an object is not due
to emission, which is primarily in the infrared region, unless the surface
temperature of the object exceeds about 1000 K.

5.6 Relation Between Emissivity and Absorptivity of the Body:
Kirchoff’s Law
 Consider two surfaces, one absolutely black at temperature and the other non-black
at temperature T. The surfaces are arranged parallel to each other and so close that
radiation of one falls totally on the other.
Fig. 5.11 Heat transfer between black and non black surface
 The radiant energy E emitted by the non-black surface impinges on the black surface and
gets fully absorbed. Likewise the radiant energy emitted by the black surface strikes
the non-black surface. If the non-black surface has absorptivity , it will absorb
 Instead, the color of a surface depends on the absorption and reflection
characteristics of the surface and is due to selective absorption and
reflection of the incident visible radiation coming from a light source such as
the sun or an incandescent light bulb.
 A piece of clothing containing a pigment that reflects red while absorbing
the remaining parts of the incident light appears “red” to the eye (Fig. 5.10).
Leaves appear “green” because their cells contain the pigment chlorophyll,
which strongly reflects green while absorbing other colors.
Fig. 5.10 Reflection of incident light from the surface

radiations and the remainder ( ) will be reflected back to black body where
it will be fully absorbed. If the both surfaces are at same temperature then the net heat
transfer is equal to zero. Net heat transfer for the non-black body is given as
 “The ratio of the emissive power of a certain non-black body E to the emissive power of
black body , both bodies being at the same temperature, is called the emissivity of the
body”.
( )
 Kirchoff’s law can be stated as: “The emissivity and absorptivity of a real surface are
equal for radiation with identical temperature and wavelength.” It means that perfect
absorber is also a perfect radiator.
5.7 Plane and Solid Angle
 Plane angle is defined by a region by the rays of a circle, and is measured as the ratio of
the element of arc of length l on the circle to the radius r of the circle. Mathematically
⁄
Fig. 5.12 Plane and solid angle
 The solid angle is defined by a region by the rays of a sphere, and is measured as:
( )
Where
= projection of the incident surface normal to the line of propagation
= area of incident surface
= angle between the normal to the incident surface and the line of propagation
= length of the line of propagation between the radiating and the incident surfaces
 Emissivity is used to find out the emissive power of the gray surface.

Fig. 5.13 Relationship between A, Anand
5.8 Intensity of Radiation and Lambert’s Cosine Law
 “Intensity of radiation I is the energy emitted (of all wave lengths) in a particular
direction per unit surface area and through a unit solid angle”.
 The area is projected area of the surface on a plane perpendicular to the direction of
radiation.
 Intensity of radiation varies with the angle normal to the surface and is given by
Lambert’s cosine law.
 Let us try to quantify the size of a slice of pizza. One way of doing that is
to specify the arc length of the outer edge of the slice, and to form the
slice by connecting the endpoints of the arc to the center.
 A more general approach is to specify the angle of the slice at the center,
as shown in Figure 5.14, this angle is called plain angle
Fig. 5.14 (a)Slice of pizza of plain angle (b)Slice of watermelon of solid angle
 Now consider a watermelon, and let us attempt to quantify the size of a
slice. Again we can do it by specifying the outer surface area of the slice
(the green part), or by working with angles for generality.
 Connecting all points at the edges of the slice to the center in this case
will form a three-dimensional body (like a cone whose tip is at the
center), and thus the angle at the center in this case is properly called the
solid angle.

 lambert’s cosine law “the intensity of radiation in a direction from the normal to a
black emitter is proportional to cosine of the angle ”.
Fig. 5.15 Lambert cosine law
 If denotes the normal intensity and represents the intensity at angle from the
normal, then
( )
 Apparently the energy radiated out decreases with increase in and becomes zero at
.
Fig. 5.16 Radiation emitted at angle
 When the collector is oriented at an angle from the normal to the emitter, then
the radiations striking and being absorbed by the collector can be expressed as:
( )
( )
 Where, is the solid angle subtended by the collector at the surface of the emitter
.

5.9 Relation Between the Normal Intensity and Emissive Power
 Consider the emission of radiation by a differential area element dA of a surface, as
shown in Figure 5.17. Radiation is emitted in all directions into the hemispherical
space.
Fig. 5.17 Emission of radiation from differential element dA into hemispherical shape
( ) ( )( )
( )
 Then the radiations leaving the emitter and striking the collector is:
 Substitute the value of and in the above equation
( )
 The total energy radiated by the emitter and passing through a hemispherical region
can be worked out by integrating the above equation over the limits
Thus,
∫ ∫
⁄
∫
( ) ( )
 But the total emissive power of the emitter with area and the temperature T is also
given by:
 Combining the above equations, we get
( )
 Thus for a unit surface, the intensity of normal radiation is the ⁄ times the emissive
power .

Ex. 5.1.
A furnace emits radiation at 2000 K. treating it as a black body radiation calculate:
(i) Monochromatic radiant flux density at 1μm wave length.
(ii) Wave length at which emission is maximum and corresponding radiant flux
density.
(iii) Total emissive power,
Solution:
Given data:
,
i. Monochromatic emissive power at wave length
From plank’s law of distribution
( )
[ ⁄ ]
( )
( )
[ ⁄ ]
⁄
ii. Wave length at which emission is maximum and radiant flux density
From Wien’s displacement law:
Maximum radiant flux density,
( )
( ) ⁄
iii. Total emissive power. From Stefan – Boltzman law:
⁄
5.11 References
[2] Heat Transfer – A Practical Approach by Yunus Cengel& Boles, McGraw-Hill
Publication.

6RADIATION HEAT TRANSFER
Course Contents
6.1 Introduction
6.2 Heat exchange between two
black surfaces: Shape Factor
6.3 Shape factor algebra and
salient features of the shape
factor
6.4 Shape factor relations
6.5 Electrical network approach
for radiation heat exchange
6.6 Radiation heat exchange
between non-black bodies
6.7 Radiation shields
6.8 Radiation Heat Transfer in
Three Surface Enclosure
6.10 References

6. Radiation Heat Transfer Heat Transfer (2151909)
6.1 Introduction
 Till now we have discussed fundamental aspects of various definitions and laws. Now
we will study the heat exchange between two or more surfaces which is of practical
importance.
 The two surfaces which are not in direct contact, exchanges the heat due to
radiation phenomena. The factors those determine the rate of heat exchange
between two bodies are the temperature of the individual surfaces, their
emissivities, as well as how well one surface can see the other surface. The last
factor is known as view factor, shape factor, angle factor or configuration factor.
6.2 Heat Exchange Between Two Black Surfaces: Shape Factor
 Consider heat exchange between elementary areas and of two black
radiating bodies having areas and respectively.
 The elementary areas are at a distance r apart and the normals to the areas make
angles and with the line joining them. The surface is at temperature and
the surface is at temperature .
Fig. 6.1 Radiant heat exchange between two black surfaces
 If the surface subtends a solid angle at the centre of the surface , then
radiant energy emitted by and impinging on (and absorbed by) the surface
is:
Where,
Intensity of radiation at an angle with normal to the surface and is given
by
Intensity of radiation normal to the surface
Projected area of normal to the line joining and

Heat Transfer (2151909) 6. Radiation Heat Transfer
But
 Integration of equation 6.1 over finite areas and gives:
∫ ∫
 The solution of this equation is simplified by introducing a term called radiation
shape factor, geometrical factor, configuration factor or view factor. The shape
factor depends only on the specific geometry of the emitter and collection surfaces,
and is defined as:
 “The fraction of the radiative energy that is diffused from one surface element and
trike the ther urfa e ire tly with i terve i g rele ti .”
 The radiation shape factor is represented by the symbol i which means the shape
factor from a surface i to another surface . Thus the radiation shape factor of
surface to surface is
∫ ∫
∫ ∫
 From the equation no. 6.2 and 6.3, the radiation leaving and striking is given by
 Similarly the energy leaving and striking is
 and the net energy exchange from to is :
 When the surfaces are maintained at the same temperatures, and , there can
be no heat exchange,
{ }

 Since and are non-zero quantities,
 The above result is known as a reciprocity theorem. It indicates that the net radiant
interchange may be evaluated by computing one way configuration factor from
either surface to the other. Thus net heat exchange between surfaces and is
6.3 Shape factor algebra and salient features of the shape factor
 The salient features for complex geometries can be derived in terms of known shape
factors for other geometries. For that the complex shape is divided into sections for
which the shape factors is either known or can be readily evaluated.
 The known configuration factor is worked out by adding and subtracting known
factors of related geometries. The method is based on the definition of shape factor,
the reciprocity principal and the energy conservation law.
 The inter-relation between various shape factors is called factor algebra.
 Salient features of shape factor:
 The value of shape factor depends only on the geometry and orientation of surfaces
with respect to each other. Once the shape factor between two surfaces is known, it
can be used for calculating the radiant heat exchange between the surfaces at any
temperature.
 The net heat exchange between surfaces is
 When the surfaces are thought to be black and are maintained at the
same temperature , there is no heat exchange and as such
Since and are non-zero quantities,
 This reciprocal relation is particular useful when one on the shape factor is unity.
 All the radiation streaming out from an inner sphere (surface 1) is intercepted by the
enclosing outer sphere (surface 2). As such the shape factor of inner sphere (surface
1) with respect to the enclosure is unity and the shape factor of outer sphere
(surface 2) can be obtained by using reciprocal relation.
 Equation 6.7 applies only to black surfaces and must not be used for surfaces
having emissivities very different from unity.

Fig. 6.2 Two concentric spheres
 The radiant energy emitted by one part of concave surface is intercepted by another
part of the same surface. Accordingly a concave surface has a shape factor with
respect to itself. The shape factor with respect to itself is denoted by .
(a) Flat surface (b) Convex surface (c) Concave surface
Fig. 6.3 Shape factor of surface with respect to itself
 For a flat or convex surface, the shape factor with respect to itself is zero.
6.4 Shape Factor Relations
6.4.1 The Reciprocity Relation
 The view factor and are not equal to each other unless the area of the two
surfaces are. That is,
 We have already discussed that the view factors are related to each other is given by
 This relation is know as reciprocity relation or the reciprocity rule.
6.4.2 Summation Rule
 Any radiating surface will have finite area and therefore will be enclosed by many
surfaces.
 For radiation heat transfer analysis, radiating surface is considered as a part of the
enclosure.

 Even openings are treated as imaginary surfaces with radiation properties equivalent
to those of the opening.
 The conservation of energy principle requires that the entire radiation leaving any
surface i of an enclosure be intercepted by the surfaces of the enclosure.
 Therefore, the sum of the view factors from surface i of an enclosure to all surfaces
of the enclosure, including to itself, must equal unity. This is known as the
summation rule for an enclosure and is expressed as (Figure 6.4)
Fig. 6.4 Radiaton leaving the surface i of an enclosure intercepted by completely by
the surface of enclosure
∑
 Where, N is the number of surfaces of the enclosure.
6.4.3 The Superposition Rule
 If one of the two surfaces (say ) is divided into sub areas , , ….. , then
∑
 With respect to figure 6.5, when the radiating surface has been split up into areas
and ,
 Obviously
 If the receiving surface is divide into subareas and ,
∑
 Applying the summation rule to surface 1 of a three-surface enclosure,

Fig. 6.5 Superposition rule
6.4.4 The Symmetry Rule
 Identical surfaces that are oriented in an identical manner with respect to another
surface will intercept identical amounts of radiation leaving that surface.
Fig. 6.6 Symmetry rule
 So, the symmetry rule can be expressed as two or more surfaces that posse
symmetry about a third surface will have identical view factors from that surface.
From the figure 6.6
6.5 Electrical Network Approach For Radiation Heat Exchange
 Solution of the radiation heat transfer problem can be obtained by reducing the
actual system to an equivalent electrical network and then solving that network. To
understand the concept, first some terminology should be defined.
 Radiosity : It indicates the total radiant energy leaving a surface per unit time per
unit surface area. It is the sum of the radiation emitted from the surface and the
reflected portion of any radiation incident upon it.
 Irradiation : it indicates the total radiant energy incident upon a surface per unit
time per unit area; some of it may be reflected to become a part of the radiosity of
the surface.
 Thus if the transmitting surface is sub divided, the shape factor for that surface
with respect to the receiving surface is not equal to the sum of the individual shape
factors.
 Apparently the shape factor from a radiating surface to a subdivided receiving
surface is simply the sum of the individual shape factors.

Fig. 6.7 Surface radiosity and irradation
 According to the definition of the radiosity, total energy leaving the surface is given
by
 Where is the emissive power of a perfect black body at the same temperature. As
no energy is transmitted through the opaque body, and so
 r i g t Kir h ff’ law, the a rptivity of the surface is equal to emissivity .
Therefore,
 The rate at which the radiation leaves the surface is given by the difference between
its radiosity and irradiation.
⁄
 This equation can be represented in the form of an electrical network as shown in
figure 6.8. The factor ⁄ is related to the surface properties of the radiating
body and is called the surface resistance to radiation heat transfer.
Fig. 6.8 Electrical analogy of surface resistance to radiation
 Equation 6.21 can be written in the form of electrical network as
⁄

 Where, is the surface resistance and given by
 Now consider the radiant heat exchange between two non-black surfaces. Out of
total radiation leaving the surface 1, only a fraction is received by the
other surface 2. Similarly the heat radiated by surface 2 and received by surface 1 is
. So net heat transfer between two surfaces is given by
 From the recirpicity theorem :
⁄
 Equation 6.25 can be represented by an electrical circuit as shown in figure 6.9. The
factor ⁄ is related to distance between two bodies and its geometry, and is
called space resistance to radiation heat transfer.
Fig. 6.9 Electrical analogy of space resistance to radiation
 Equation 6.9 can be written in the form of electrical network as
⁄
 Where, is the space resistance and given by
 Radiation heat transfer can be represented by electrical network, consisting of two
surface resistances of two radiating bodies and the space resistance between them
as shown in figure 6.10.
Fig. 6.10 Electrical analogy of radiation heat transfer between two surfaces
 For black body , so surface resistance of the black body is equal to zero.
 So, from equation 6.22 radiosity is equal to emissive power of the black body

 The net heat exchange between two gray surfaces is given by
 Where and are surface resistances and is space resistance, equation 6.28
can be written as,
⁄ ⁄ ⁄
⁄ ⁄ ⁄
⁄ ⁄ ⁄ ⁄
( )
 Where, ( ) is called gray body factor and is given by
( )
⁄ ⁄ ⁄ ⁄
6.6 Radiation Heat Exchange Between Non-Black Bodies
6.6.1 Small Object in a Large Cavity
Fig. 6.11 Small object in a large cavity (enclosure)
 All the radiations emitted by object 1 reach and are absorbed by object 2, and area
of object 1 is very small compare to area of object 2. So,
 Substitute the above value in equation 6.29
( )
⁄
 When the heat exchange is between two black surfaces, the surface resistance
becomes zero as . The gray body factor ( ) becomes equal to space
factor in the equation 6.29.

6.6.2 Infinite Large Parallel Plates
 All the radiations emitted by plane one reach and are absorbed by other plane, and
areas of the two planes are infinite. So,
Fig. 6.12 Infinite large parallel plates
( )
⁄ ⁄
⁄ ⁄
6.6.3 Infinite Long Concentric Cylinders or Sphere
(a) Concentric cylinder (b) Concentric sphere
Fig. 6.13 Infinite long concentric cylinder and sphere
 The inner cylinder or sphere of area sees only the outer surface and not itself. So,
( )
⁄ ⁄ ⁄
⁄ ⁄ ⁄

6.7 Radiation Shields
 Radiation heat transfer between two surfaces can be reduced greatly by inserting a
thin, highly reflectivity (low-emissivity) sheet of material between the two surfaces.
Such highly reflective thin plates or shells are called radiation shields.
 Consider two infinite parallel plates as shown in figure 6.14. Radiation network for
the radiation heat transfer consists of two surface resistances and one space
resistance as shown in figure 6.14.
Fig. 6.14 Heat exchange between two infinite parallel planes without radiation shields
 With no radiation shields, the net heat exchange between the infinite parallel plates
is given by
⁄ ⁄ ⁄
 For parallel plates configuration,
⁄ ⁄
 When , the above equation becomes
⁄
 Now consider a radiation shield placed between these two plates as shown in figure
6.15.
 The radiation network of this geometry is constructed by drawing a surface
resistance associated with each surface and connecting these surface resistances
with space resistances, as shown in figure 6.15.

Fig. 6.15 Radiation heat exchange between two infinite parallel plates with radiation
shield
 The resistances are connected in series, and thus the rate of radiation heat transfer
is given as
⁄ ⁄ ( )⁄ ( )⁄ ⁄ ⁄
⁄ ⁄ ( ⁄ ⁄ )
 When , the above equation becomes
⁄
 Comparison of expressions 6.38 and 6.40 shows that ratio of heat flow with a
radiation shield becomes just half of what it would have been without the radiation
shield.
 If n-radiation shields are inserted between the two planes, then
I. There will be two surface resistances for each radiation shield, and one for each
radiating surface. When emissivity of all the surfaces are equal, then all the
surface resistances will have same value ⁄ .
II. There would be space resistance and configuration factor for each will be
unity.
 So, the total resistance for n number of radiation shield is given by
( )

 And therefore the heat exchange with n-shields is given by
⁄
 A comparison expressions 6.38 and 6.41 does indicate that the presence of n-shields
reduces the radiant heat transfer by a factor of .
 Under steady state conditions, the shield attain a uniform temperature of .
Temperature of radiation shield can be obtained by comparing the heat transfer
between surface 1 and shield with heat transfer between surface 1 and surface 2.
6.8 Radiation Heat Transfer in Three-Surface Enclosure:
 Consider an enclosure consisting of three opaque, diffuse, and gray surfaces as
shown in figure 6.16.
 The radiation network of this geometry is obtained by drawing a surface resistance
associated with each of the three surfaces and connect these surface resistances
with space resistances as shown in figure 6.16.
 The three equations for the determination of the radiosity , and are obtained
from the requirement that the algebraic sum of the currents at each node must
equal zero. Hence,
{ }
 Once the radiosities are available, the net rate of radiation heat transfers at each
surface can be determined from the following equation:
∑ ∑ ( ) ∑

 Above equation is used to find the net radiation heat transfer from surface i which is
enclosed by N no. of surfaces.
 Set of equations can be obtained from the equation 6.42 for the different
configuration.
 Net rate of heat transfer from the reradiating surface is equal to zero.
Ex. 6.1.
Determine the view factors from the base of the pyramid shown in figure 1 to each
of its four side surfaces. The base of the pyramid is a square, and its side surfaces are
isosceles triangles.
Figure 1 Square pyramid
Solution:
According to reciprocity principal
And
So, from above equations
Ex. 6.2.
Consider a cylindrical furnace with radius = 1m and height = 1m as shown in figure
3. Take σ = 5.67 X 0-8 W/m2K4
Figure 3 Cylindrical furnace

Determine the net rate of radiation heat transfer at each surface during the steady
operation and explain how these surfaces can be maintained at specified
temperatures.
Solution:
Given data:
,
, ,
, ,
Determine: , and
The view factor from the base to side surface is determined by applying the
summation rule.
Since base surface is flat so, .
Top and bottom surfaces are symmetric about the side surface so,
and . The view factor is determine from the reciprocity relation,
( ) ( )
Figure 4 radiation network associated with three surface enclosure
 Radiosities at each surface can be determined from the following equations (Figure
4)

Substitute the value in above equations
⁄ ⁄ ⁄
⁄ ⁄ ⁄
As the surface 3 is a black body so,
⁄ ⁄ ⁄
⁄ ⁄ ⁄
Solving these equations for and gives
⁄ ⁄ ⁄
Then the net rates of radiation heat transfer at the three surfaces are determined
from following equations
[
⁄ ⁄
]
[
⁄ ⁄
]
[
⁄ ⁄
]
[
⁄ ⁄
]
[
⁄ ⁄
]
[
⁄ ⁄
]
To maintain the surfaces at the specified temperatures, we must supply heat to the
top surface continuously at a rate of 27.6 kW while removing 2.13 kW from the base
and 25.5 kW from the side surfaces.

Ex. 6.3.
The flat floor of a hemispherical furnace is at 800 K and has emissivity of 0.5. The
corresponding values for the hemispherical roof are 1200 K and 0.25. Determine the
net heat transfer from roof to floor. Take .
Solution:
Given data:
, , , , Take
Figure 6 Schematic and network diagram of hemispherical furnace
⁄ ⁄ ⁄
All the radiations from the floor reach the floor and hence
⁄ ⁄ ⁄
For the given configuration
⁄
⁄ ⁄
⁄
The negative sign indicates that heat flow is from roof to floor.
Ex. 6.4.
Determine net radiation heat transfer per m2 for two infinite parallel plates held at
temperature of 800 K and 500 K respectively. Emissivities of hot and cold plates are
0.6 and 0.4 respectively. Now it is intended to reduce the heat transfer to 40% of
original value by placing a radiation shied between the plates. Calculate the
emissivity of the shield and its equilibrium temperature.
Solution:
Given Data:
,

, ,
Figure 7 Two parallel plates and network diagram
 With no radiation shields, the net heat exchange between the infinite parallel plates
is given by
⁄ ⁄ ⁄
⁄ ⁄
⁄ ⁄
⁄
⁄
Figure 8 Radiation shield placed between two parallel plates and network diagram

 With radiation shields, the net heat exchange between the infinite parallel plates is
given by
⁄ ⁄ ( )⁄ ( )⁄ ⁄ ⁄
⁄ ⁄ ⁄
⁄ ⁄ ⁄
⁄ ⁄ ⁄
⁄
⁄
⁄
So, emissivity of the radiation shield is 0.347.
Temperature of radiation shield:-
Heat transfer from plate 1 to plate 2 is equal to the heat transfer from plate 1 to
radiation shield 3.
⁄ ⁄ ( )⁄
⁄ ⁄
⁄ ⁄
⁄ ⁄
⁄
6.10 References:

Publication.

Department of Mechanical Engineering Prepared By: Bhavin J. Vegada
7
CONVECTION
Course Contents
7.1 Introduction to Convection
7.2 Newton-Rikhman Law
7.3 Free and Forced Convection
7.4 Dimensional Analysis
7.5 Dimensionless Numbers &
Their Physical Significance
7.6 Dimensional Analysis Applied
to Forced Convection
7.7 Dimensional Analysis Applied
to Free Convection
7.8 Empirical Co-relations for Free
& Forced Convection
7.9 Thermal and Hydrodynamic
Boundary Layer
7.10 Derivation of Differential
Convection Equations
A. Continuity Equation
B. Momentum Equation
C. Energy Equation
7.11 Von-Karman Integral
Momentum Equation
7.12 Solution for Velocity Boundary
Layer
7.14 References

7. Convection Heat Transfer (2151909)
Prepared By: Bhavin J. Vegada Department of Mechanical Engineering
7.1 Introduction to Convection
 Thermal convection occurs when a temperature difference exists between a solid
surface and a fluid flowing past it.
Fig. 7.1 Convection Phenomena
 It is well known that a hot plate of metal will cool faster when placed in front of a fan
than when exposed to still air.
 For example,
We know that the velocity at which the air blows over the hot plate obviously
influences the heat transfer rate. But does it influence the cooling in a linear way?
i.e. if the velocity is doubled, will the heat transfer rate doubled?
Relation with conduction:
 As shown in Fig. 7.1 the velocity of fluid layer at the wall will be zero, the heat must
be transferred by conduction at that point.
 Thus we might compute the heat transfer using Fourier’s equation of conduction i.e.
with the thermal conductivity of fluid and the fluid temperature
gradient at wall.
 Why then, if the heat flows by conduction in this layer, do we speak of “Convection”
heat transfer and need to consider the velocity of the fluid?
 The answer is that the temperature gradient is dependent on the rate at which the
fluid carries the heat away; a high velocity produces a large temperature gradient,
and so on.
 It must be remembered that the physical mechanism of heat transfer at the wall is a
conduction process.
Heated Wall
Free Stream

Heat Transfer (2151909) 7. Convection
7.2 Newton-Rikhman Law OR Newton’s Law of Cooling OR
Convection Rate Equation
 The appropriate convection rate equation for the convective heat transfer between
a surface and an adjacent fluid is given by Newton’s law of cooling:
( ) ( )
Where,
Convective heat flow rate
Surface area exposed to heat transfer
Surface temperature of solid and
Temperature of the fluid (Stagnant or Undisturbed)
The Convective heat transfer co-efficient or The film co-efficient or The surface
conductance
 The heat transfer co-efficient is sometimes called the film conductance or surface
conductance because of its relation to the conduction process in the thin stationary
layer of fluid at the wall surface.
 Unit of Convective heat transfer co-efficient: ⁄ or ⁄ or ⁄
 The value of film co-efficient is dependent upon:
1. Surface conditions: Roughness & Cleanliness
2. Geometry and orientation of surface: Plate, Tube and Cylinder placed
horizontally or vertically.
3. Thermo-physical properties of the fluid: Density, Viscosity, Specific heat, Co-
efficient of expansion and thermal conductivity.
4. Nature of fluid flow: Laminar or Turbulent
5. Boundary layer configuration
6. Existing thermal conditions.
The film co-efficient ( ) depends on viscosity of fluid because………
The viscosity influences the velocity profile and correspondingly the energy
transfer rate in the region near the wall.

7.3 Free and Forced Convection
 With respect to the cause of fluid flow, two types of convection are distinguished:
1. Free Convection or Natural Convection and
2. Forced Convection.
1. Free Convection or Natural Convection
 When a surface is maintained in still fluid at a temperature higher or lower than that
of the fluid, a layer of fluid adjacent to the hot or cold surface gets heated or cooled
by conduction.
 A density difference is created between this adjacent layer and the still fluid
surrounding it.
 The density difference introduces a buoyant force causing flow of fluid near the
surface.
 Heat transfer under such conditions is known as Free or Natural Convection.
 Thus, “Free or Natural convection is the process of heat transfer which occurs due
to movement of the fluid particles by density changes associated with temperature
differential in a fluid.”
 This mode of heat transfer occurs very commonly, some of the examples are:
I. House heating system
II. The cooling of transmission lines, electric transformers and rectifiers.
2. Forced Convection
 Flow of fluid is caused by a pump, a fan or by the atmospheric winds.
 These mechanical devices speeds up the heat transfer rate.
 In free convection flow velocities encountered are lower compared to flow velocities
in forced convection, consequently the value of convection co-efficient is lower, and
for a given rate of heat transfer larger area could be required.
 Examples of forced convection are: cooling of I.C. Engines, Air conditioner, Heat
exchangers, etc.
 The rate of heat transfer is calculated using the equation 7.1.
Table 7.1 Typical values of convective co-efficient
Sr. No. Free Convection Forced convection
1 Air – 3 to 7 W/m2
K
Air & Super heated steam – 30 to
300 W/m2
K
2 Gases – 2 to 20 W/m2
K Oil – 60 to 3000 W/m2
K
3 Liquids – 30 to 300 W/m2
K Water – 3000 to 10000 W/m2
K

7.4 Dimensional Analysis
 “Dimensional analysis is a mathematical technique which makes use of the study of
the dimensions for solving several engineering problems.”
 Dimensional analysis has become an important tool for analyzing fluid flow
problems. It is specially useful in presenting experimental results in a concise form.
 There are two methods are used in dimensional analysis: 1) Rayleigh’s Method and
2) Buckingham’s -Theorem.
Buckingham’s -Theorem
“If there are variables (independent and dependent variables) in a physical
phenomenon and if these variables contain fundamental dimensions, then the
variables are arranged into ( ) dimensionless terms; each terms are called -
terms.”
System of Dimensions:
 In the area of heat transfer, two more dimensions namely the temperature
difference ( ) and the heat ( ) are also taken as fundamental quantities.
 Here heat ( ) can be expressed in terms of MLT. So the fundamental quantities are
mass, length, time and temperature; designated by the M,L,T,θ respectively.
 Temperature is specially used in compressible flow and heat transfer phenomena.
Table 7.2 Quantities used in fluid mechanics and heat transfer & their dimensions
Sr.
No.
Quantity Symbol Units (SI)
Dimensions
(MLTθ System)
Dimensions
(MLTθH System)
A Fundamental
1 Mass M Kg M1
L0
T0
θ0
M1
L0
T0
θ0
H0
2 Length L m M0
L1
T0
θ0
M0
L1
T0
θ0
H0
3 Time T Sec M0
L0
T1
θ0
M0
L0
T1
θ0
H0
4 Temperature θ K M0
L0
T0
θ1
M0
L0
T0
θ1
H0
5 Heat Q, H Joule M1
L2
T-2
M0
L0
T0
θ0
H1
B Geometric
1 Area A m2
L2
L2
2 Volume V m3
L3
L3
C Kinematic
1 Linear Velocity u, v m/s L1
T-1
L1
T-1

2 Angular Velocity ω rad/s T-1
T-1
3 Acceleration a m/s2
L1
T-2
L1
T-2
4 Angular
Acceleration
α rad/s2
T-2
T-2
5 Discharge Q m3
/sec L3
T-1
L3
T-1
6 Kinematic Viscosity ν m2
/sec L2
T-1
L2
T-1
D Dynamic
1 Force / Resistance F/R N (kg-m/s2
) M1
L1
T-2
M1
L1
T-2
2 Density ρ Kg/ m3
M1
L-3
M1
L-3
3 Specific Weight w N/ m3
M1
L-2
T-2
M1
L-2
T-2
4 Dynamic Viscosity μ Kg/m-sec M1
L-1
T-1
M1
L-1
T-1
5 Work, Energy W, E N-m (Joule) M1
L2
T-2
H1
6 Power P Watt (J/sec) M1
L2
T-3
T-1
H1
E Thermodynamic
1 Thermal
Conductivity
K W/m-K M1
L1
T-3
θ-1
L-1
T-1
θ-1
H1
2 Specific Heat Cp, Cv kJ/kg-K L2
T-2
θ-1
M-1
θ-1
H1
3 Heat Transfer Co-
efficient
h W/m2
-K M1
T-3
θ-1
L-2
T-1
θ-1
H1
4 Gas Constant R J/kg-K L2
T-2
θ-1
M-1
θ-1
H1
5 Thermal Diffusivity α m2
/sec L2
T-1
L2
T-1
7.5 Dimensionless Numbers & Their Physical Significance
1. Reynolds Number (Re)
 It is defined as a ratio of inertia force to viscous force.
( )
 It indicates the relative importance of the inertial and viscous effects in a fluid
motion.

 At low Reynolds number, the viscous effect dominates and the fluid motion is
laminar.
 At high Reynolds number, the inertial effects lead to turbulent flow.
 Reynolds number constitutes an important criterion of kinematic and dynamic
similarity in forced convection heat transfer.
2. Prandtl Number (Pr)
“It is the ratio of kinematic viscosity to thermal diffusivity of the fluid”.
( ⁄ )
( )
 The kinematic viscosity represents the momentum transport by molecular friction
and thermal diffusivity represents the heat energy transport through conduction.
 Pr provides a measure of the relative effectiveness of momentum and energy
transport by diffusion.
 For highly viscous oils, Pr is quite large (100 to 10000) and that indicates rapid
diffusion of momentum by viscous action compared to the diffusion of energy.
 For gases, Pr is about 1, which indicates that both momentum and heat dissipate
through the field at about the same rate.
 The liquid (liquid sodium or liquid potassium) metals have Pr = 0.003 to 0.01 and that
indicates more rapid diffusion of energy compared to the momentum diffusion rate.
 The Prandtl number is connecting link between the velocity field and the
temperature field, and its value strongly influences relative growth of velocity and
thermal boundary layers.
 Mathematically,
( ) ( )
Where,
Thickness of velocity boundary layer
Thickness of thermal boundary layer
For,
Oil Gases Liquid Metals

3. Nusselt Number (Nu)
 Nu established the relation between convective film co-efficient ( ), thermal
conductivity of the fluid ( ) and a significant length parameter ( ) of the physical
system.
( )
 To understand the physical significance of the Nu, consider a fluid layer of thickness
and temperature difference as shown in Fig. 7.2.
Fig. 7.2 Heat transfer through the fluid layer
 Heat transfer through the fluid layer is by convection when the fluid involves some
motion and by conduction when the fluid layer is motionless.
 Heat flux (The rate of heat transfer per unit surface area) in either case is,
̇ ̇
Taking their ratios,
̇
̇
 The Nusselt number is a convenient measure of the convective heat transfer co-
efficient.
 The larger the Nusselt number, the more effective the convection.
 The for a fluid layer represents heat transfer across the layer by pure
conduction.
 For a given Nu, h is directly proportional to thermal conductivity of the fluid and
inversely proportional to the significant length parameter.
4. Grashoff Number (Gr)
 It indicates the relative strength of the buoyant to viscous forces.
( )
Fluid
Layer
̇

( )
( )
( )
( )
 Obviously the Grashoff number represents the ratio of Buoyant force and Inertia
force to the square of the Viscous force.
 Grashoff number has a role in free convection.
 Free convection is usually suppressed at sufficiently small Gr, begins at some
critical value of Gr and then becomes more and more effective as Gr increases.
5. Stanton Number (St)
 “It is the ratio of heat transfer co-efficient to the flow of heat per unit temperature
rise due to the velocity of fluid”.
( )
( ) ( )
( )
 It should be noted that Stanton number can be used only in co-relating forced
convection data (since the expression contains velocity, ).
6. Peclet Number (Pe)
 “It is the ratio of mass heat flow rate by convection to the flow rate by conduction
under an unit temperature gradient and through a thickness ”.
( )
⁄
( )
( )
 The Peclet number is a function of Reynolds number and Prandtl number.

7. Graetz Number (G)
 “It is the ratio of heat capacity of fluid flowing through the pipe per unit length to the
conductivity of pipe material.”
⁄
( )
( )
( ) ( )
Where,
and are the diameter and length of pipe respectively.
( )
( )
( ) ( )
 Graetz number is merely a product of a constant and the Peclet number.
7.6 Dimensional Analysis Applied to Forced Convection
 Let us now consider the case of a fluid flowing across a heated tube.
 The heat transfer co-efficient is a function of the following variables:
( ) ( )
( ) ( )
Heat transfer co-efficient
Fluid density
Tube diameter
Fluid velocity

Fluid viscosity
Specific heat
Thermal conductivity
 Total number of variables,
Number of fundamental dimensions, (i.e. M, L, T, θ)
Total number of
 Hence equation 7.13 may be written as,
( ) ( )
 Selecting as a repeating variables.
( ) ( ) ( ) ( ) ( )
By solving above equations, we get,
( ) ( )
( ) ( ) ( ) ( ) ( )

( ) ( )
( ) ( ) ( ) ( ) ( )
( ) ( )
Put the values of in equation 7.14, we get,
( )
( )
( )
( ) ( )
 Hence Nusselt number is a function of Reynolds number and Prandtl number for
forced convection.
7.7 Dimensional Analysis Applied to Free Convection
 Let us now consider the case of natural convection from a vertical plane wall to an
adjacent fluid.
 The free convection heat transfer co-efficient ( ) depends upon the variables;

 Since the fluid circulation in free convection is due to the difference in density
between the various fluid layers due to temperature gradient and not by external
agency. Therefore, velocity ( ) is no longer an independent variable but depends
upon the following factors:
(i) (The co-efficient of thermal expansion of the fluid)
(ii) (Acceleration due to gravity)
(iii) (The difference of temperature between the heated surface and the
undisturbed fluid)
 Thus, heat transfer co-efficient ( ) can be expressed as follows:
( ) ( )
( ) ( )
Heat transfer co-efficient
Fluid density
Characteristic length
Fluid viscosity
Specific heat
Thermal conductivity
Buoyant force
 Total number of variables,
Number of fundamental dimensions, (i.e. M, L, T, θ)
Total number of
 Hence equation 7.20 may be written as,
( ) ( )
 Selecting as a repeating variables.
( ) ( ) ( ) ( ) ( )

( )
( ) ( )
( ) ( ) ( ) ( ) ( )
( ) ( )
( ) ( ) ( ) ( ) ( )

( ) ( )
Put the values of in equation 7.21, we get,
( )
.
( )
/
.
( )
/
( ) ( )
 Hence Nusselt number is a function of Grashoff number and Prandtl number for
natural or free convection.
7.8 Empirical Co-relations for Free & Forced Convection
 Mathematical analysis of convective heat problems is complicated due to the large
number of variables involved.
 Majority of the convective problems are, therefore, analysed through the technique
of dimensional analysis supported by experimental investigations. The dimensional
analysis helps to develop certain correlations for the convective coefficient.
 The constants and exponents appearing in these correlations for a particular
situation are worked out through experiments.
 Use “Heat & Mass Transfer by Dr. D. S. Kumar” to see different empirical co-relations
for free and forced convection for different cases. (Equations should be given in
examination so no need to remember)
 Some of the important terminology associated with this topic is explained below:
Bulk Temperature & Mean Film Temperature
 The physical properties (µ, ρ, Cp, k) of a fluid are temperature dependent.
Key Notes:
 In natural or free convection, the flow is produced by buoyant effects
resulting from temperature difference. These effects are included in the
Grashoff number.
 Reynolds number is important in the case of forced convection and
similarly the Grashoff number is important in the case of free
convection.

 The accuracy of the results obtained by using theoretical relations and the
dimensionless empirical co-relations would depend upon the temperature chosen
for the evaluation of these properties.
 No uniform procedure has been attained in the selection of this reference
temperature.
 However, it is customary to evaluate the fluid properties either on the basis of bulk
temperature or the mean film temperature.
Mean Bulk Temperature:
 The mean bulk temperature (tb) denotes the equilibrium temperature that would
result if the fluid at a cross section was thoroughly mixed in an adiabatic container.
 For internal flow (Heat exchangers), the fluid flowing through the tubes may be
heated or cooled during its flow passage. The bulk temperature is then taken to be
the arithmetic mean of the temperatures at inlet to and at exit from the heat
exchanger tube; i.e.
( )
Mean Film Temperature:
 It is the arithmetic mean of the surface temperature ( ) of a solid and the
undisturbed temperature ( ) of the fluid which flows over the surface. i.e.
( )
Characteristic Length OR Equivalent Diameter
 Characteristic length ( ) or Diameter ( ) has appeared in the dimensionless
numbers discussed in the Art. 7.5.
 The pipe and the flat plate are the simplest geometries for the occurrence of a flow.
However in many instances some complicated geometries are also used and hence
all the calculations of convective heat transfer become much more complicated and
difficult.
 In order to avoid such difficulties, the concept of an equivalent circular tube is used.
This is a tube which would present the same resistance against the flow or would
secure the same heat transfer as the duct usually used under comparable conditions.
 The diameter of an equivalent tube is known as equivalent diameter ( ) or
characteristic length ( ). The equivalent diameter is usually defined as;
( )
Where,

Cross-sectional area and
Perimeter
 The equivalent diameter or characteristic length of few geometries are given below:
For Rectangular Duct:
( )
For Rectangular Annulus:
( )
[( ) ( )]
( )
[( ) ( )]
When, and
( )
For Annulus: (Refer Fig.)
( )
[ ( ) ]
For Annulus: (Refer Fig.)
( )
( )

7.9 Thermal and Hydrodynamic Boundary Layer
 The concept of boundary layer was first introduced by L. Prandtl in 1904 and since
then it has been applied to several fluid flow problems.
A. Hydrodynamic Boundary Layer: Flat Plate
 “When a fluid flows around an object, their exist a thin layer of fluid close to the
solid surface within which shear stresses significantly influence the velocity
distribution. The fluid velocity varies from zero at the solid surface to the velocity of
free stream flow at a certain distance away from the solid surface. This thin layer of
changing velocity has been called the hydrodynamic boundary layer.”
 Consider the parallel flow of a fluid over a flat plate as shown in Fig. 7.3.
Fig. 7.3 Development of a boundary layer on a flat plate
 The edge facing the direction of flow is called leading edge. The rear edge is called
the trailing edge.
 The coordinate is measured along the plate surface from the leading edge of the
plate in the direction of flow, and is measured from the surface in the normal
direction.
 The fluid approaches the plate in the direction with a uniform velocity , which
is practically identical to the free stream velocity of the fluid.
 The velocity of the fluid particles in the first fluid layer adjacent to the plate becomes
zero because of the no – slip condition.
 This motionless layer slows down the particles of the neighboring fluid layer as a
result of friction between the particles of these two adjoining fluid layers at different
velocities.
 This fluid layer then slows down the molecules of the next layer and so on.

 Thus the presence of the plate is felt up to some normal distance (thickness of
velocity boundary layer) from the plate beyond which the free stream velocity
remains unchanged.
 As a result, the component of the fluid velocity varies from 0 at to
nearly at
 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 or
hydrodynamic boundary layer.
 The thickness of boundary layer ( ) increases with distance from the leading edge;
as more and more fluid is slowed down by the viscous effects, becomes unstable and
breaks into turbulent boundary layer.
 In turbulent boundary layer, a very thin layer near the smooth surface remains
laminar, called laminar sub-layer.
 For the flow over a flat surface, if Reynolds No. is less than 5 X 105
, the flow is
laminar and velocity distribution is parabolic.
 The boundary layer thickness ( ):
“It is arbitrarily defined as that distance from the plate surface in which the velocity
reaches 99% of the velocity of the free stream ( )”
The hypothetical line of divides the flow over a plate into two regions:
(a) The boundary layer region, in which the viscous effects and the velocity changes
are significant and (b) The irrotational flow region, in which the frictional effects are
negligible and the velocity remains essentially constant.
B. Thermal Boundary Layer
 Whenever a flow of fluid takes place over a heated or cold surface, a temperature
field is set-up in the field next to the surface. The zone or thin layer wherein the
temperature field exists is called the thermal boundary layer.
 The temperature gradient results due to heat exchange between the plate and the
fluid.
 Consider the flow of a fluid at a uniform temperature of over a hot flat plate at
temperature as shown in Fig. 7.4.
 The fluid particles in the layer adjacent to the surface will reach thermal equilibrium
with the plate and assume the surface temperature . 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 at
the surface to sufficiently far from the surface.

Fig. 7.4 Thermal boundary layer during flow of cold fluid over a warm plate
 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 at any location along the surface is
defined as the distance from the surface at which the temperature difference
( ) equals ( ).
 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.
 If the approaching free stream temperature is above the plate surface
temperature , the thermal boundary layer will have the shape as depicted in Fig.
7.5.
Fig. 7.5 Temperature profile in T.B.L. when warm fluid flows over a cold plate
 The temperature of the fluid changes from a minimum at the plate surface to the
temperature of the main stream at a certain distance from the surface.
 At point A, the temperature of the fluid is the same as the surface temperature .
 The fluid temperature increases gradually until it acquires the free stream
temperature .

 The distance measured perpendicularly to the plate surface, denotes the
thickness of thermal boundary layer at a distance from the leading edge of the
plate.
Relation between Thermal & Velocity Boundary Layer
 The velocity profile of the velocity boundary layer is
dependent primarily upon the viscosity of the fluid.
 The temperature profile of the thermal boundary layer is
depends upon the flow velocity, specific heat, viscosity and
thermal conductivity of the fluid.
 The thermo-physical properties of the fluid affect the relative
magnitude of and , and the non-dimensional Prandtl
number ( ) constitutes the governing parameter:
(i) When
(ii) When
(iii) When
Fig. 7.6 Relation between thermal and hydrodynamic boundary
layer for different Prandtl number

7.10 Derivation of Differential Convection Equations
 Consider an infinitesimal two dimensional control volume ( ).
 Assume that:
1. Flow is steady and fluid is incompressible.
2. Fluid viscosity is constant.
3. Shear in y-direction is negligible.
4. No pressure variations in the flow field.
5. Fluid is continuous both in space (i.e. no voids occur in the fluid) and time
(i.e. mass is neither created nor destroyed).
A. Conservation of Mass – The Continuity Equation
Fig. 7.7 Differential control volume for mass balance – Continuity equation
 Let represents the velocity of fluid flow at the face AD and hence velocity of fluid
motion at surface BC will be , -.
Velocity
Boundary
Layer

 Similarly the fluid velocity at the bottom face AB and at the top face CD are and
, - respectively.
 According to conservation of mass principle,
( )
 The mass flow entering the face AD of the control volume during time interval ,
( )
( ) ( )
 During the same time interval, mass of fluid flowing out from face BC,
( ) ( )
 Similarly the mass flow entering the bottom face AB is and the mass leaving
the top face Dc is ( ) .
 From equation 7.30
( ) ( )
Simplification gives,
( )
 Equation 7.33 is the mass continuity equation for 2-D, Steady flow of an
incompressible fluid.
B. Force or Momentum Equation
‒ For a 2-D infinitesimal control volume ( ) within the
boundary layer region, the viscous forces acting along with the momentum of fluid
entering and leaving the elementary volume have been indicated in Fig. 7.8.
‒ Newton’s second law of motion is applied to the control volume. The statement
resulting from the application is,
Sum of applied forces in - direction = rate of change of - directional momentum
 In boundary layer analysis we are interested in the - directional forces. The
resulting equation is known as momentum equation (for - direction).

Fig. 7.8 Force and momentum balance for control volume
‒ The momentum flux in the x direction is product of mass flow rate through a
particular side of control volume and - directional velocity component at that
point.
‒ The rate of momentum entering the face AD of control volume,
( )
‒ The rate of momentum leaving the face BC of control volume,
( )
[ ]
‒ The rate of momentum in - direction associated with mass enters the bottom face
AB of control volume,
( )
‒ The rate of momentum in - direction leaves the top face CD of control volume,
( )
[ ]
‒ The net or resultant momentum transfer in - direction,
̇ ( )
̇
( )
( )
( )
Control
Volume

= (Rate of momentum leaving the face BC & face CD) – (Rate of momentum entering
the face AD & face AB)
[ ] [ ]
[ ]
[ { } ]
( ) ( ) ( )
‒ The forces acting in x direction are viscous and pressure forces.
Pressure Forces:
‒ The pressure force on the face AD,
‒ The pressure force on the face BC (in opposite direction),
[ ]
Viscous Forces:
‒ The viscous force at the face AB (in negative - direction),
( )
‒ The viscous force at the face CD,
( )
(
, -
)
, -
‒ Net forces in x direction,
( )

From equations 7.34 and 7.35 get,
( )
( ) ( )
 The above equation is called momentum equation for the laminar boundary layer
with constant properties.
 If the pressure changes on two side of control volume is negligible then above
equation reduces to,
( )
C. Energy Equation for Thermal Boundary Layer
 Consider an element of dimensions ( ) in the boundary layer.
 The rate of temperature change in the direction is being presumed small and as
such conduction is to be considered only in the direction.
 Further, the convective terms in the and directions have been written in terms of
mass, temperature and specific heat, which is assumed constant.
Fig. 7.9 Differential control volume for conservation of energy
( ) ( )
̇
( )
̇ ( )
( ) ( )
( ( ) )
Control
Volume

 According to principle of conservation of energy for the steady state condition, the
algebraic sum of total heat due to convection, conduction and viscous effects equals
to zero. Thus,
( )
 The energy convected in direction,
Energy influx,
( )
Energy efflux,
( ) ( )
By neglecting the product of small quantities, we get,
[ ]
Net energy convected in direction,
[ ] ( )
 Similarly the net energy convected in direction,
( ) [ ]
[ ] ( )
 The heat conduction in direction,
[ { ( ) }]
( )
Viscous Heat Generation:
 Due to relative motion of fluid in the boundary layer (fluid on the top face of the
control volume moves faster than fluid on the bottom face), there will be viscous
effects which will cause heat generation.

( )
 This force will act through a distance S which can be determined by the relative
velocity of fluid flow at the upper and lower faces of the element;
( )
( ) ( )
 From equation 7.38, we get,
[ ] [ ] ( )
( ) ( )
[ ( )] ( ) ( )
 From the continuity equation for 2-D flow, we have,
( ) ( )
 Equation 7.44 is the differential energy equation for flow past a flat plate.
 If viscous heat generation is neglected, the energy equation takes the form,
( )
( )
(Note: It may be noted that the energy equation (7.46) is similar to be momentum
equation (7.37) further the kinematic viscosity and the thermal diffusivity have
the same dimensions.)

Assumptions made to derive energy equation:
1. Steady incompressible flow
2. Negligible body forces, viscous heating and conduction in flow direction.
3. Constant fluid properties evaluated at the film temperature,
( )
.
7.11 Von-Karman Integral Momentum Equation
 Approximate solution of momentum equation.
 Used to find out the frictional drag on smooth flat plate for both laminar and
turbulent boundary layer.
 Neglecting pressure and gravity forces.
 Fig. 7.10(a) shows a fluid flowing over a thin plate with a free stream velocity
 Consider a small length of the plate at a distance from the leading edge as
shown in Fig. 7.10(a).
 The enlarged view of the small length of the plate is shown in Fig. 7.10(b).
 Consider unit width of plate perpendicular to the direction of flow.
Fig. 7.10 Momentum equation for boundary layer by Von Karman
 Let ABCD be a small element of a boundary layer where the edge DC represents the
outer edge of the boundary layer.
 Mass rate of fluid entering through face AD,
̇ ∫ ( ) ∫
 Mass rate of fluid leaving through face BC,
̇ ̇
( ̇ )
y
A B
C
D
x dx
Boundary
Layer
Thin Smooth
Flat Plate
( ) ( )

̇ ∫ *∫ +
 No mass can enter the control volume ABCD through its solid wall AB.
 Therefore the continuity requirement then stipulates that the mass increment
*∫ + must represent the mass flow rate that enters the control volume
ABCD through face CD with free stream velocity
 The corresponding momentum fluxes are:
 Momentum rate of fluid entering the control volume in direction through AD,
∫
∫ ( )
 Momentum rate of fluid leaving the control volume in direction through BC,
∫ *∫ + ( )
 Momentum rate of fluid entering the control volume in direction through DC,
*∫ + ( )
 In the absence of any pressure and gravity forces, the drag or shear force ( )
at the plate surface must be balanced by the net momentum change for the control
volume.
 Therefore, as per momentum principle the rate of change of momentum on the
control volume ABCD must be equal to the total force on the control volume in the
same direction.
( )
( ) *∫ + *∫ +

*∫ ( ) +
*∫ ( ) + ( )
 Equation 7.50 is the Von Karman momentum integral equation for the hydrodynamic
boundary layer.
 The integral equation expresses the wall shear stress as a function of the non
dimensional velocity distribution
7.12 Solution for Velocity Boundary Layer
Method of solution for velocity boundary layer
1. Exact solution (Blasius solution)
2. Approximate solution (Von Karman solution)
1. Blasius Solution:-
 Thickness of velocity boundary layer,
√
Where,
δ = Thickness of velocity boundary layer
 The local skin friction co-efficient,
√
 Average skin friction co-efficient,
̅̅̅
√

Where,
2. Von Karman Integral Momentum Equation Solution:-
 Thickness of velocity boundary layer,
√
Where,
δ = Thickness of velocity boundary layer
 The local skin friction co-efficient,
√
 Average skin friction co-efficient,
̅̅̅
√
Where,
Important Notes:
[1] The average skin friction co-efficient is quite often referred to as the drag co-
efficient.
[2] For the flow over a flat surface, if Reynolds No. is less than 5 X 105
, the flow is
laminar.
[3] When the plate is heated over the entire length, the hydrodynamic and thermal
boundary layer thicknesses are related to each other by the expression,
( ) ⁄
[4] Pohlhausen has suggested the following relation for general case,
( ) ⁄
[5] The local Nusselt no. for laminar flow is given by,
( ) ( )

[6] The Average Nusselt no. for laminar flow is given by,
̅̅̅̅
̅
( ) ( )
[7] The mass flow rate at any position in the boundary layer is given by,
∫ ∫ [ { ( ) ( ) }]
( ) ( )
Therefore, mass entrainment through the boundary layer is given by,
( )
Ex 7.1. [GTU; Jan-2013; 7 Marks]
A hot plate of 400mm x 400mm at 100°C is exposed to air at 20°C. Calculate heat loss
from both the surfaces of the plate if (a) the plate is kept vertical (b) plate is kept
horizontal. Air properties at mean temperature are ρ = 1.06 kg/m3
, k = 0.028 W/m-k,
Cp = 1.008 KJ/kg-k, and ν = 18.97 x 10-6
m2
/s.
Use following correlations:
( )
( )
( )
Solution: Given Data:
Properties of air @
ρ = 1.06 kg/m3
k = 0.028 W/m-k
Cp = 1.008 KJ/kg-k
ν = 18.97 x 10-6
m2
/s
To be Calculated:
a) for vertical plate
b) for horizontal plate
 Coefficient of expansion,
( )
 Grashoff Number,
( )

( )
( )
 Prandtl Number,
 For Vertical Plate:
Nusselt Number,
( )
( )
Convective Heat Transfer Coefficient,
⁄
Heat Transfer,
( )
( ) ( )
 For Horizontal Plate:
For Upper Surface
Nusselt Number,
( )
( )
⁄
Heat Transfer,
( )
( ) ( )
For Lower Surface
Nusselt Number,
( )

( )
⁄
Heat Transfer,
( )
( ) ( )
 Heat Transfer from Both Surfaces,
Ex 7.2. [GTU; Dec-2011; 7 Marks]
A steam pipe 8 cm in diameter is covered with 3 cm thick layer of insulation which
has a surface emissivity of 0.9. The surface temperature of the insulation is 80 °C and
the pipe is placed in atmospheric air at 24 °C. Considering heat loss by both radiation
and natural convection calculate:
(a) The heat loss from the 7 m length of pipe.
(b) The overall heat transfer coefficient and the heat transfer coefficient due to
radiation alone.
The thermo physical properties of air at mean film temperature of 52°C are as
following:
ρ = 1.092 kg/m3
, Cp = 1.007 KJ/kg-°C, μ = 19.57×10-6
kg/ms, k = 27.81×10-3
W/m-°C
(where the notations have their usual meaning.)
Use empirical correlation for horizontal cylinders as,
( )
Solution: Given Data: To be Calculated:
a)
b)
 Characteristic length for horizontal cylinder,
( )
 Coefficient of expansion,

( )
 Grashoff Number,
( )
( )
 Prandtl Number,
 Nusselt Number,
( )
( )
 Convective Heat Transfer Coefficient,
⁄
 Heat Transfer by Convection,
( )
( ) ( )
( ) ( )
 Heat Transfer by Radiation,
( )
( ) [( ) ( ) ]
 Total Heat Transfer Rate:
 Overall Heat Transfer Coefficient:
( )
( ) ( )
⁄
 Heat Transfer Coefficient by Radiation:

( )
( ) ( )
⁄
Ex 7.3. [GTU; May-2012; 7 Marks]
The air at atmospheric pressure and temperature of 30°C flows over one side of plate
of a velocity of 90 m/min. This plate is heated and maintained at 100°C over its entire
length. Find out the following at 0.3 and 0.6 m from its leading edge. (a) Thickness of
velocity boundary layer and thermal boundary layer. (b) Mass flow rate which enters
the boundary layer between 0.3 m and 0.6 m per metre depth of plate. Assume unit
width of plate. Properties of air at 30°C: ρ = 1.165 kg/m3
, v = 16 × 10-6
m2
/s, Pr =
0.701, Cp = 1.005 kJ/kg-K, k = 0.02675 W/m-K.
⁄
To be Calculated:
a)
b) ̇
 Prandtl Number,
 Reynolds Number,
⁄
⁄
By Using Von-Karman Solution:-
 Thickness of Velocity Boundary Layer:
At distance 0.3 m,
√ √
At distance 0.6 m,
√ √
 Thickness of Thermal Boundary Layer:
At distance 0.3 m,
⁄

( ) ( )
At distance 0.6 m,
( ) ( )
 Mass Flow Rate:
̇ ( ) ( )
̇ ⁄
7.14 References
[2] Heat and Mass Transfer by R. K. Rajput, S. Chand Publications.
[3] Heat and Mass Transfer by P.K. Nag, McGraw-Hill Publication.
[4] Heat and Mass Transfer by Mahesh M Rathore, McGraw-Hill Publication.
Publication.
[6] National Programme on Technology Enhanced Learning (NPTEL), A Joint Initiate by
IIT’s and IISc. (Web: http://nptel.ac.in/)

8BOILING & CONDENSATION
Course Contents
8.1 Introduction
8.2 Boiling
8.3 Types of Boiling
8.4 Boiling Regimes
8.5 Bubble Growth
8.6 Condensation
8.7 Dropwise and Filmwise
Condensation
8.8 References
8.9 GTU Paper Analysis

8. Boiling and Condensation Heat Transfer (2151909)
8.1 Introduction
 When the temperature of a liquid at a specified pressure is raised to the saturation
temperature (Tsat), at that pressure Boiling occurs.
 Likewise, when the temperature of a vapor is lowered to saturation temperature
(Tsat), Condensation occurs.
 Boiling and Condensation are considered to be forms of convection heat transfer
since they involve fluid motion, such as the rise of the bubbles to the top and the
flow of condensate to the bottom.
 Boiling and Condensation differ from other forms of convection, in that they depend
on the latent heat of vaporization (hfg) of the fluid and the surface tension (σ) at the
liquid vapor interface, in addition to the properties of the fluid in each phase.
 During a phase change, large amount of heat (due to large latent heat of
vaporization released or absorbed) can be transferred essentially at constant
temperature.
 The phenomenon’s are quite difficult to describe due to change in fluid properties
(density, specific heat, thermal conductivity, viscosity, etc.) and due to
considerations of surface tension, latent heat of vaporization, surface characteristics
and other features of two phase flow.
 Heat transfer co-efficient h associated with boiling and condensation are typically
much higher than those encountered in other forms of convection processes that
involve a single phase.
8.2 Boiling
 Boiling is the convective heat transfer process that involves a phase change from
liquid to vapor state.
 Boiling is a liquid to vapor phase change process just like evaporation, but there are
significant differences between the two. Evaporation occurs at the liquid–vapor
interface when the vapor pressure is less than the saturation pressure of the liquid at
a given temperature. Examples of evaporation are: drying of clothes, the
evaporation of sweat to cool human body and the rejection of waste heat in wet
cooling towers. Note that evaporation involves no bubble formation or bubble
motion.
 Boiling, on the other hand, occurs at the solid–liquid interface when a liquid is
brought into contact with a surface maintained at a temperature Ts sufficiently
above the saturation temperature Tsat of the liquid. At 1 atm, for example, liquid
water in contact with a solid surface at 110°C will boil since the saturation
temperature of water at 1 atm is 100°C.

Heat Transfer (2151909) 8. Boiling and Condensation
 Heat is transferred from the solid surface to the liquid, and the appropriate form of
Newton’s law of cooling is, ( )
Where, ( ) is termed the excess temperature.
 The boiling process is characterized by the rapid formation of vapor bubbles at the
solid–liquid interface that detach from the surface when they reach a certain size
and attempt to rise to the free surface of the liquid.
Applications of Boiling
 Steam production.
 Absorption of heat in refrigeration and Air-conditioning systems.
 Greater importance has recently been given to the boiling heat transfer
because of developments of nuclear reactors, space-crafts and rockets,
where large quantities of heat are produced in a limited space and are to be
dissipated at very high rates.
8.3 Types of Boiling
A. Classification of boiling on the basis of the presence of bulk fluid
motion
1. Pool Boiling
 The liquid above the hot surface is stationary.
 The only motion near the surface is because of free convection and the motion of
the bubbles under the influence of buoyancy.
 The pool boiling occurs in steam boilers. Pool boiling of a fluid can also be
achieved by placing a heating coil in the fluid.
2. Forced Convection Boiling / Flow Boiling
 The fluid motion is induced by external means such as pump.
 The liquid is pumped and forced to move in a heated pipe or over a surface in a
controlled manner.
 The free convection and the bubble induced mixing also contribute towards the
fluid motion.

(a) Pool Boiling (b) Flow Boiling (c) Sub-cooled Boiling (d) Saturated Boiling
Fig. 8.1 Classification of boiling
B. Classification of boiling on the basis of the presence of bulk liquid
temperature
1. Sub-cooled or Local Boiling
 The temperature of liquid is below the saturation temperature and boiling takes
place only in vicinity of the heated surface.
 The vapor bubbles travel a short path and then vanish; apparently they condense
in the bulk of the liquid which is at a temperature less than a boiling point or
saturation temperature.
2. Saturated Boiling
 The temperature of the liquid exceeds the saturation temperature.
 The vapor bubbles generated at the solid surface(solid-liquid interface) are
transported through the liquid by buoyancy effects and eventually escape from
the surface (liquid-vapor interface).
 The actual evaporation process then sets in.
8.4 Boiling Regimes
 Whether the boiling phenomenon corresponds to pool boiling or forced circulation
boiling, there are some definite regimes of boiling associated with progressively
increasing heat flux.
 Nukiyama (1934) was the first to identify different regimes of pool boiling using the
apparatus of Fig. 8.2. These different regimes can be illustrated by considering an
electrically heated horizontal nichrome/Platinum wire submerged in a pool of liquid
at saturation temperature.
 Fig. 8.3 shows the relationship between heat flux and the temperature excess (Ts –
Tsat); Where,

Ts = Temperature of the hot surface
Tsat = Saturation temperature corresponding to the pressure at which the
liquid is being evaporated.
 The heat flux is easily controlled by voltage drop across a wire of fixed resistance.
 Although the boiling curve given in Fig. 8.3 is for water, the general shape of the
boiling curve remains the same for different fluids.
 Different boiling regimes are:
A. Natural Convection Boiling
B. Nucleate Boiling
C. Film Boiling
Fig. 8.2 Nukiyama’s power controlled heating apparatus for demonstrating the boiling curve
A. Natural / Free Convection Boiling (up to point A on Boiling curve)
 The boiling takes place in a thin layer of liquid which adjoins the heated surface.
 The liquid in the immediate vicinity of the wall becomes superheated, i.e.
temperature of the liquid exceeds the saturation temperature at the given
pressure.
 The superheated liquid rises to the liquid-vapor interface where evaporation
takes place.
 The fluid motion is by free convection effects.
 The heat transfer rate increases, but gradually, with growth in a temperature
excess.

Fig. 8.3 Boiling curve for saturated water at atmospheric pressure
B. Nucleate Boiling (between point A & C on Boiling curve)
 When the liquid is overheated in relation to saturation temperature, vapor
bubbles are formed at certain favorable spots called the Nucleation or Active
sites. Point A is referred as the onset of nucleate boiling, ONB.
 The nucleate boiling regimes can be separated into two distinct regions:
A – B:-
o Isolated bubbles are formed at various nucleation sites, on the heated
surface but these bubbles get condensed in the liquid after detaching
from the surface.
B – C:-
o Heater temperature is further increased. Bubbles forms at very high rates
and they form continuous columns of vapor in the liquid.
o The liquid is quite hot and the bubbles do not condense in it.
o These bubbles rise to the free surface, where they break-up and release
its vapor content and that helps in rapid evaporation.

o The space vacated by the rising bubbles is filled by the liquid in the
vicinity of the heated surface, and the process is repeated.
o The agitation or stirring caused by the entrainment of the liquid to the
heated surface and rapid evaporation is responsible for the increased
heat transfer co-efficient and heat flux in the nucleate boiling region.
o The heat flux hence reaches maximum at point C, which is called the
critical /maximum heat flux, qmax.
o Nucleate boiling is the most desirable boiling regime in practice because
high heat transfer rates can be achieved in this regime with relatively
small values of ΔTexcess.
C. Film Boiling (beyond point C on Boiling curve)
Transition Boiling (between point C & D)
 As the heater temperature and thus ΔTexcess is increased past point C, the heat
flux decreases as shown in Fig. 8.3.
 This is because a bubble formation is very rapid; the bubbles blanket the heating
surface and prevent the incoming fresh liquid from taking their place.
 A large fraction of the heating surface is covered by a vapor film, which acts as an
insulation due to the low thermal conductivity of the vapor.
 In the transition boiling regime, both nucleate and film boiling partially occurs.
 Nucleate boiling at point C is completely replaced by film boiling at point D.
 Operation in the transition boiling regime, which is also called the unstable film
boiling regime, is avoided in practice.
Beyond point D
 In this region the heated surface is completely covered by a continuous stable
vapor film.
 The temperature differences are so large that radiant heat flux becomes
significant, and the heat flux curve begins to rise upward with increasing ΔTexcess.
That marks the region of stable film boiling.
 The phenomenon of stable film boiling is referred as “Leidenfrost effect” and
point D, where the heat flux reaches a minimum, is called the Leidenfrost point.
Burn out point (Point F)
 In order to move beyond point C, where qmax occurs, we must increase the
heated surface temperature (Ts).

 To increase Ts, however we must increase the heat flux. But the fluid can not
receive this increased energy beyond point C, and the heated surface
temperature (Ts) to rise even further.
 If the surface temperature exceeds the temperature limit of the wall material,
burn out (structural damage & failure) of the wall occurs.
8.5 Bubble Growth
 The bubble formation in nucleate boiling is greatly influenced by the nature and
condition of the heating surface and surface tension at the solid-liquid interface
(Shape, size and inclination of bubbles, however do not have much effect on the
heat transfer rate).
 The surface tension signifies wetting capability of the surface with the liquid (i.e. low
surface tension → Highly wetted surface) and that influences the angle of contact
between the bubble and solid surface.
 Any contamination of the surface would affect its wetting characteristics and
influence the size and shape of the vapor bubbles.
 If the surface tension of the liquid is low, it tends to wet the surface (fully wetted
surface), so that the bubble is readily pushed by the liquid and rises. The vapor
bubbles tend to become globular or oval in shape as shown in Fig. 8.4(a) (iii) and
they are disengaged from the surface.
Fig. 8.4(a) Wetting characteristics for typical vapor bubbles
 In case of liquid having intermediate surface tension (partially wetted surface) a
momentary balance may exist between the bubbles and solid surface so that it is
necessary to form larger bubbles before the buoyant force can free them from the
surface; the shape of the bubble is shown in Fig. 8.4(a) (ii).
 On the unwetted surface, the bubbles spread out as shown in Fig. 8.4(a) (i); forming
a wedge between the water and heating surface, thereby allowing hydrostatic forces
to resist the action of buoyancy.

 The formation of bubble with fully wetted surface as shown in Fig. 8.4(a) (iii) gives
high heat transfer rate compared with the bubble shapes shown in Fig. 8.4(a) (i) and
(ii); because the area covered by the insulating vapor film is the smallest.
 Experimental evidence does indicate that the vapor bubbles are not always in
thermodynamic equilibrium with the surrounding liquid.
Fig. 8.4(b) Force balance for a spherical bubble
 The vapor inside the bubble is not necessarily at the same temperature as the liquid
and the vapor pressure Pv inside the bubble exceeds the liquid pressure Pl acting
from outside of the bubble. Fig. 8.4(b) indicates one such spherical bubble with
various forces acting on it.
i. The resultant pressure (Pv– Pl) acts on area πr2
and the pressure force equals
πr2
(Pv– Pl).
ii. The surface tension σ of the vapor-liquid interface acts on the interface
length 2πr and the surface tension force equals 2πrσ.
 Under equilibrium conditions, the pressure force is balanced by the surface
tension force. Thus,
( )
( ) ( )
 The vapor may be considered as a perfect gas for which the Clayperon equation
may be used, which is given below:
( )
 From equation (8.1) and (8.2) we can derive,
[ ] ( )
 Equation (8.3) is the equilibrium relationship between the bubble radius and the
amount of superheat.

 A bubble of radius r will grow if ( ) ( ) otherwise it will
collapse. Here Tlis the temperature of the liquid surrounding the bubble.
 The bubble diameter Db at the time of detachment from the surface can be
worked out from the relation proposed by Fritz:
√
( )
( )
Where, β is the angle of contact and the empirical constant Cd has the value
0.0148 for water bubbles.
Factors affecting the nucleate pool boiling
1) Material, shape and condition of the surface:
Under identical conditions of pressure and temperature difference, the
boiling heat transfer coefficient is different for different metals; copper has
a high value compared to steel. Further a rough surface gives a better heat
transmission then when the surface is either smooth or has been coated to
weaken its tendency to get wetted.
2) Pressure:
The temperature difference between the heating surface and the bulk and
hence the rate of bubble growth is affected by pressure. The maximum
allowable heat flux for a boiling liquid increases with pressure until critical
pressure is reached and thereafter it declines.
3) Liquid properties:
Experiments have shown that the bubble size increases with the dynamic
viscosity of the liquid. With increase in bubble size, the frequency of
bubble formation decreases and that result in reduced rate of heat
transfer.

8.6 Condensation
 “Condensation occurs when the temperature of a vapor is reduced below its
saturation temperature corresponding to the vapor pressure.”
 This is usually done by bringing the vapor into contact with a solid surface whose
temperature, Ts is below the saturation temperature Tsat of the vapor.
 The latent energy of the vapor is released, heat is transferred to the surface, and the
condensate is formed.
 The condensation can also occur on the free surface of a liquid or even in a gas when
the temperature of the liquid or the gas to which the vapor is exposed is below Tsat.
 In this chapter we will consider surface condensation only.
 Depending upon the behavior of condensate upon the cooled surface, the
condensation process has been categorized into two distinct modes: (A) Film wise
condensation and (B) Drop wise condensation.
8.7 Drop wise and Film wise Condensation
Fig. 8.5 Film wise and Drop wise Condensation
A. Film wise condensation
 The liquid condensate wets the solid surface, spread out and forms a continuous
film over the entire surface.
 The liquid flows down the cooling surface under the action of gravity and the
layer continuously grows in thickness because of newly condensing vapors.
 The continuous film offers resistance and restricts further transfer of heat
between the vapor and the surface.

 Film condensation only occurs when a vapor relatively free from impurities, is
allowed to condense on a clean surface.
 Film condensation is generally a characteristic of clean, uncontaminated
surfaces.
B. Drop wise condensation
 The liquid condensate collects in droplets and does not wet the solid cooling
surface.
 The droplets develop in cracks, pits and cavities on the surface, grow in size,
break away from the surface, knock-off other droplets and eventually run-off the
surface without forming a film.
 A part of the condensation surface is directly exposed to the vapor without an
insulating film of condensate liquid.
 Evidently there is no film barrier to heat flow and higher heat transfer rates are
experienced.
 Drop wise condensation has been observed to occur either on highly polished
surfaces, or on surfaces contaminated with impurities like fatty acids and organic
compounds.
 Drop wise condensation gives co-efficient of heat transfer generally 5 to 10 times
larger than with film condensation.
 It is therefore common practice to use surface coatings that inhibit wetting, and
hence simulate drop wise condensation.
 Silicon, Teflon and an assortment of waxes and fatty acids are often used for this
purpose.
 However such coatings gradually lose their effectiveness due to oxidation,
fouling or outright removal and film condensation eventually occurs.
 Although it is desirable to achieve drop wise condensation in industrial
applications, it is often difficult to maintain this condition.
 Condenser design calculations are often based on the assumption of film
condensation.

8.8 References
[3] Fundamentals of Heat and Mass Transfer by Frank P. Incropera, John Wiley & Sons
Publication.
Publication.
 Influence of the presence of non-condensable gases
The presence of non-condensable gas such as air in a condensing vapor produces
a detrimental (negative) effect on the heat transfer coefficient.
It has been observed that even with a few percent by volume of air in steam the
condensation heat transfer coefficient is reduced by more than 50%.
This is owing to the fact that when a vapor (containing non-condensable gas)
condenses, the non-condensable gas is left at the surface.
Any further condensation at the surface will occur only after incoming vapor has
diffused through this non-condensable gas collected in the vicinity of the
surface.
The non-condensable gas adjacent to the surface acts as a thermal resistance to
the condensation process. The rate of condensation decreases greatly when the
condensable vapor is contaminated with even very small amounts of non-
condensable gases.
As the presence of non-condensable gas in a condensing vapor is undesirable,
the general practice in the design of a condenser should be to vent the non-
condensable gas to the maximum extent possible.

9
HEAT EXCHANGERS
Course Contents
9.1 Introduction
9.2 Types of Heat Exchangers
9.3 Heat Exchanger Analysis
9.4 Overall Heat Transfer Co-
efficient
9.5 Fouling Factor
9.6 Logarithmic Mean
Temperature Difference
(LMTD)
9.7 Correction Factors for
Multi-pass Arrangement
9.8 Effectiveness and NTU for
Parallel & Counter Flow
Heat Exchanger
9.10 References

9. Heat Exchangers Heat Transfer (2151909)
9.1. Introduction
“Heat exchanger is process equipment designed for the effective transfer of heat
energy between two fluids; a hot fluid and a coolant”. The purpose may be either to
remove heat from a fluid or to add heat to a fluid.
Examples of heat exchangers:
 Intercoolers and pre-heaters
 Condensers and boilers in steam plant
 Condensers and evaporators in refrigeration unit
 Regenerators
 Automobile radiators
 Oil coolers of heat engine
 Evaporator of an ice plant and milk-chiller of a pasteurizing plant
The heat transferred in the heat exchanger may be in the form of latent heat (i.e. in
boilers & condensers) or sensible heat (i.e. in heaters & coolers).
9.2. Types of Heat Exchangers
Many types of heat exchangers have been developed to meet the widely varying
applications. Heat exchangers are typically classified according to:
A. Nature of heat exchange process:
I. Direct contact or open heat exchanger
 Complete physical mixing of hot and cold fluid and reach a common
temperature.
 Simultaneous heat and mass transfer.
 Use is restricted, where mixing between two fluids is harmful.
 Examples: (i) Water cooling towers - in which a spray of water falling from the
top of the tower is directly contacted and cooled by a stream of air flowing
upward and (ii) Jet condensers.
II. Regenerators
 In a regenerator the hot fluid is passed through a certain medium called “matrix”,
serves as a heat storage device.
 The heat is transferred and stored in solid matrix and subsequently transferred
to the cold fluid.

Heat Transfer (2151909) 9. Heat Exchangers
 The effectiveness of regenerator is depends upon the heat capacity of the
regenerating material and the rate of absorption and release of heat.
 In a fixed matrix configuration, the hot and cold fluids pass alternately through a
stationary matrix, and for continuous operation two or more matrices are
necessary, as shown in Fig. 9.1(a). One commonly used arrangement for the
matrix is the “packed bed”. Another approach is the rotary regenerator in which
a circular matrix rotates and alternately exposes a portion of its surface to the
hot and then to the cold fluid, as shown in Fig. 9.1(b).
Fig.9.1 (a) Fixed dual-bed regenerator (b) Rotary regenerator
III. Recuperators
 In this type of heat exchanger the hot and cold fluids are separated by a wall and
heat is transferred by a combination of convection to and from the wall and
conduction through the wall. The wall can include extended surfaces, such as
fins.
 Majority of the industrial applications have recuperator type heat exchangers.
B. Relative direction of motion of fluids
I. Parallel flow
 Hot and cold both the fluids flow in the same direction
II. Counter flow
 Flow of fluids is opposite in direction to each other
 Gives maximum heat transfer rate

Fig.9.2 Different flow regimes and temperature profiles in a double-pipe heat exchanger
III. Cross flow arrangement
 Two fluids are directed perpendicular to each other.
 Examples: Automobile radiator and cooling unit of air-conditioning duct.
 The flow of the exterior fluid may be by forced or by natural convection.
 Fig.9.3 shows different configurations used in cross-flow heat exchangers.
Fig.9.3 Different flow configurations in cross-flow heat exchangers

C. Mechanical design of heat exchange surface
I. Concentric tube heat exchanger
 Two concentric pipes.
 Each carrying one of the fluids.
 The direction of flow may correspond to parallel or counter flow arrangement as
shown in Fig.9.2.
II. Shell & tube heat exchanger
 One of the fluids is carried through a bundle of tubes enclosed by a shell and
other fluid is forced through shell and flows over the outside surface of tubes.
 The direction of flow for either or both fluids may change during its passage
through the heat exchanger.
Fig.9.4 Shell & tube heat exchanger with one shell pass and one tube pass (1-1 exchanger)
III. Multiple shell & tube passes
 Single-pass: Two fluids may flow through the exchanger only once as shown in
Fig.9.4.
 Multi-pass: One or both fluids may traverse the exchanger more than once as
shown in Fig.9.5.
 Baffles are provided within a shell which cause the fluid surrounding the tubes
(shell side fluid) to travel the length of shell a no. of times.
 An exchanger having n – shell passes and m – tubes passes is designed as n-m
exchanger.
 A multiple shell & tube exchanger is preferred to ordinary counter flow design
due to its low cost of manufacture, easy dismantling for cleaning and repair and
reduced thermal stresses due to expansion.

Fig. 9.5 Shell & tube heat exchangers. (a) One shell pass and two tube passes. (b) Two
shell passes and four tube passes.
D. Physical state of heat exchanging fluids
The direction of flow is immaterial in these cases and the LMTD will be the same for
both parallel flow, counter flow and other flow types. Refer Fig. 9.6.
I. Condenser
 The temperature of hot fluid will remain constant throughout the heat
exchanger. (only latent heat is transferred)
II. Evaporator
 The temperature of cold fluid will remain constant throughout the heat
exchanger. (only latent heat is transferred)
Fig. 9.6 (a) Condensing (b) Evaporating

9.3. Heat Exchanger Analysis
 Fig. 9.7 represents the block diagram of a heat exchanger.
 The governing parameters are:
I. Overall heat transfer co-efficient (U) due to various modes of heat
transfer
II. Heat transfer surface area
III. Inlet and outlet fluid temperatures
Fig. 9.7 Overall energy balance in heat exchanger
 Assuming there is no loss of heat to the surroundings and potential and kinetic
energy changes are negligible.
 From the energy balance in the heat exchanger,
Heat given up by the hot fluid,
̇ ( )
Heat picked up by the cold fluid,
̇ ( )
Total heat transfer rate in the heat exchanger is given by,
( )
Where,
U = Overall heat transfer co-efficient between the two fluids

A = Effective heat transfer area
θm = Appropriate mean value of temp. difference or logarithmic mean temp.
difference
9.4. Overall Heat Transfer Co-efficient
 A heat exchanger is essentially a device in which energy is transferred from one fluid
to another across a good conducting solid wall.
 The rate of heat transfer between two fluids is given by,
∑
∑
( )
(a) Plane Wall (b) Cylindrical Wall
Fig. 9.8 Thermal resistance network for (a) plane and (b) cylindrical separating wall
 When the two fluids of the heat exchanger are separated by a plane wall as shown in
Fig. 9.8 (a), the thermal resistance comprises:
(i) Convection resistance due to the fluid film at the inner surface
(ii) Wall conduction resistance
(iii) Convection resistance due to fluid film at the outer surface
Wall
Wall

( )
 A plane wall has a constant cross-sectional area normal to the heat flow i.e.
( )
 For a cylindrical separating wall as shown in Fig. 9.8 (b), the cross-sectional area of
the heat flow path is not constant but varies with radius.
 It then becomes necessary to specify the area upon which the overall heat transfer
co-efficient is based. Thus depending upon whether the inner or outer area is
specified, two different values are defined for overall heat transfer co-efficient U.
( )
Since,
( )
 If resistance due to material is neglected then,
( )
 Further if the wall thickness is small i.e.
( )
 Similarly for outer surface,
( )
 If resistance due to material is neglected and wall thickness is assumed to be very
small then we get,
( )
 Overall heat transfer co-efficient for different applications are given in Table 9.1.

Table 9.2 Typical fouling
factors
Type of fluid
Fouling Factor
Rf, m2
K/W
Sea water
Below 325K
Above 325K
0.00009
0.0002
Treated boiler
feedwater
above 325 K
0.0002
Fuel oil 0.0009
Industrial air 0.0004
Refrigerating
liquid
0.0002
Steam 0.00009
9.5. Fouling Factor
 Equations 9.3 to 9.10 are essentially valid only for clean and un-corroded surface.
 However during normal operation the tube surfaces get covered by deposits of ash,
soot (smoke), dirt and scale etc. This phenomenon of rust formation and deposition
of fluid impurities is called Fouling.
 The surface deposits increase thermal resistance with a corresponding drop in the
performance of the heat exchange equipment.
 Since the thickness and thermal conductivity of the scale deposits are difficult to
determine, the effect of scale on heat flow is considered by specifying an
“Equivalent Scale Heat Transfer Co-efficient”, ( ).
 If and denote the heat transfer co-efficient for the scale formed on the inside
and outside surfaces respectively, then the thermal resistance due to scale
formation on the inside surface is,
And thermal resistance due to scale formation on the outer surface is,
 With the inclusion of these resistances at the inner and outer surfaces,
Table 9.1 Representative values of the
overall heat transfer co-efficient in heat
exchangers
Type of heat exchanger U, W/m2
°C
Water-to-water 850–1700
Water-to-oil 100–350
Water-to-gasoline 300–1000
Feedwater heaters 1000–8500
Steam-to-light fuel oil 200–400
Steam-to-heavy fuel oil 50–200
Steam condenser 1000–6000
Gas-to-gas 10–40

( )
 For the inner surface,
 For the outer surface,
 Fouling Factor ( ):
The reciprocal of scale heat transfer co-efficient is called the fouling factor(
). It can be determined experimentally by testing the heat exchanger in both the
clean and dirty conditions.
 Values of typical fouling factors for different condition are given in Table 9.2.
Important Points
 The overall heat transfer co-efficient (U) depends upon the flow rate
and properties of the fluid, the material thickness and surface
condition of tubes and the geometrical configuration of the heat
exchanger.
 High conducting liquids such as water and liquid metals give higher
values of heat transfer co-efficient (h) and overall heat transfer co-
efficient (U).
 For an efficient and effective design, there should be no high thermal
resistance in the heat flow path; all the resistance in the heat
exchanger must be low.

9.6. Logarithmic Mean Temperature Difference (LMTD)
 During heat exchange between two fluids, the temperature of the fluids, change in
the direction of flow and consequently there occurs a change in the thermal head
causing the flow of heat.
 In a parallel flow system, the thermal head (temperature potential) causing the flow
of heat is maximum at inlet and it goes on diminishing along the flow path and
becomes minimum at the outlet.
 In a counter flow system, both the fluids are in their coldest state at the exit.
 To calculate the rate of heat transfer by the expression, an average
value of the temperature difference (i.e. LMTD) between the fluids has to be
determined.
Assumptions made to derive expression for LMTD:
1. The overall heat transfer co-efficient, U is constant.
2. The flow conditions are steady.
3. The specific heats and mass flow rate of both fluids are constant.
4. There is no loss of heat to surrounding i.e. the heat exchanger is perfectly insulated.
5. There is no change of phase either of the fluid during the heat transfer.
6. The changes in potential and kinetic energies are negligible.
7. Axial conduction along the tubes of the heat exchanger is negligible.
Fig. 9.9(a) Temperature changes of fluids during counter flow arrangement

 LMTD for Counter Flow Heat Exchanger
 Consider heat transfer across an element of length at a distance from the
entrance side of the heat exchanger as shown in Fig. 9.9(a).
 Let at this section, the temperature of the hot fluid be and that of cold fluid be .
 Heat flow ( ) through this elementary length is given by,
( ) ( )
Where, ( ) is the temperature difference between the fluids and hence
.
 Due to heat exchange, the temperature of hot and cold fluid decreases by and
respectively in the direction of heat exchanger length (Refer Fig. 9.9(a)).
 Then, heat exchange between the fluids for a given elementary length is given as,
( )
Where,
Heat capacity of hot fluid
Heat capacity of cold fluid
Mass flow rate of hot fluid
Mass flow rate of cold fluid
Specific heat of hot fluid
Specific heat of cold fluid
 From equation 9.13,
[ ]
[ ] ( )
Put value of from equation 9.12,
[ ]
[ ]
By integrating,

∫ ∫ [ ]
[ ] ( )
 Now total heat transfer rate between the two fluids is given by,
( )
( )
From equation 9.15,
[ ]
[( ) ( )]
For counter flow heat exchanger,
We get,
[ ]
[ ]
( )
Where,
[ ] [ ]
is called Logarithmic Mean Temperature Difference (LMTD).
 LMTD for Parallel Flow Heat Exchanger
entrance side of the heat exchanger as shown in Fig. 9.9(b).
 Let at this section, the temperature of the hot fluid be and that of cold fluid be .
( ) ( )
.

Fig. 9.9(b) Temperature changes of fluids during parallel flow arrangement
 In parallel flow, due to heat exchange, the temperature of the hot fluid decreases by
and the temperature of cold fluid increases by in the direction of heat
exchanger length (Refer Fig. 9.9(b)).
( )
Where,
Heat capacity of hot fluid
Heat capacity of cold fluid
Mass flow rate of hot fluid
Mass flow rate of cold fluid
Specific heat of hot fluid
Specific heat of cold fluid
 From equation9.18,
[ ]
[ ] ( )
Length
Temp.

[ ]
[ ]
By integrating,
∫ ∫ [ ]
[ ] ( )
 Now total heat transfer rate between the two fluids is given by,
( )
( )
From equation 9.20,
[ ]
[( ) ( )]
For parallel flow heat exchanger,
We get,
[ ]
[ ]
( )
Where,
[ ] [ ]
is called Logarithmic Mean Temperature Difference (LMTD).

 Arithmetic Mean Temperature Difference (AMTD)
 When the temperature variation of the fluids is relatively small, then temperature
variation curves are approximately straight lines (as in condenser and evaporator)
and sufficiently accurate results are obtained by taking the arithmetic mean
temperature difference (AMTD).
( ) ( )
( )
 Temperature changes of mediums during condensation and evaporation is shown in
Fig. 9.6 (Page no. 9.6).
9.7. Correction Factors for Multi-pass Arrangements
 The relation
[ ]
for LMTD is essentially applicable for the single pass heat
exchangers.
 The effect of multi-tubes, several shell passes or cross flow in an actual flow
arrangement is considered by identifying a correction factor F such that,
( )
 F depends on geometry of the heat exchanger and the inlet and outlet temperatures
of hot and cold fluid streams.
[ ]
Special Case:-
If then,
By applying L’Hospital’s rule,
We get,

 Correction factors for several common arrangements have been given in Figs. 9.10 to
9.13.
 The data is presented as a function of two non-dimensional temperature ratios P and
R. the parameter P is the ratio of the rise in temperature of the cold fluid to the
difference in the inlet temperatures of the two fluids and the parameter R defines
the ratio of the temperature drop of the hot fluid to temperature rise in the cold
fluid.
( )
 Since no arrangement can be more effective than the conventional counter flow, the
correction factor F is always less than unity for shell and tube heat exchanger.
 Its value is an indication of the performance level of a given arrangement for the
given terminal fluid temperatures.
 When a phase change is involved, as in condensation or boiling, the fluid normally
remains at essentially constant temperature. For these conditions, P or R becomes
zero and we obtain
Fig. 9.10 Correction-factor plot for exchanger with one shell pass and two, four, or any
multiple of tube passes

Fig. 9.11 Correction-factor plot for exchanger with two shell passes and four eight or any
multiple of tube passes
Fig. 9.12 Correction factor plot for single pass cross-flow heat exchanger with both fluids
unmixed

Fig. 9.13 Correction factor plot for single-pass flow heat exchanger, one fluid mixed and the
other unmixed
9.8. Effectiveness and Number of Transfer Units (NTU)
 The concept of LMTD for estimating/analyzing the performance of a heat exchanger
unit is quite useful only when the inlet and outlet temperature of the fluids are
either known or can be determined easily from the relevant data.
 In normal practice the useful design is however based on known fluid inlet
temperatures and estimated heat transfer co-efficients. The unknown parameters
may be the outlet conditions and heat transfer or the surface area required for a
specified heat transfer.
 An analysis/estimate of the heat exchanger can be made more conveniently by the
NTU approach, which is based on the capacity ratio, effectiveness and number of
transfer units.
Capacity Ratio (C):
 The product of mass and specific heat ( ) of a fluid flowing in a heat exchanger
is termed as the Capacity rate. It indicates the capacity of the fluid to store energy at
a given rate.
 “The ratio of minimum to maximum capacity rate is defined as Capacity ratio ( ) ”
 Let,

Capacity rate of the hot fluid,
Capacity rate of the cold fluid,
 In parallel or counter flow, hot or cold fluid may have the minimum value of capacity
rate.
If
If
For counter flow heat exchanger,
Table 9.3 Table 9.4
Effectiveness of Heat Exchanger ( ):
 “The effectiveness of a heat exchanger is defined as the ratio of energy actually
transferred to the maximum possible theoretical energy transfer.”
( )
 Actual heat transfer,
( ) ( ) ( )
( ) ( )
( )
If then,
( ) ( )
( )
If then,

 A maximum possible heat transfer rate is achieved if a fluid undergoes temperature
change equal to the maximum temperature difference available.
 As described in Table 9.3 and Table 9.4, we may write the general expression,
( ) ( )
 The effectiveness of heat exchanger is then,
( )
( )
( )
( )
( )
( )
( )
( )
 If
( )
( )
 If
( )
( )
The subscript on designates the fluid which has the minimum heat capacity rate.
Number of Transfer Units (NTU):
 The group is called the number of transfer units (NTU).
( )
( )
( )
 NTU is a dimensionless parameter.
 It is a measure of the (heat transfer) size of the heat exchanger.
 The larger the value of NTU, the closer the heat exchanger reaches its
thermodynamic limit of operation.
Effectiveness for the parallel flow heat exchanger:
entrance side of the heat exchanger as shown in Fig. 9.9(b).
( ) ( )

.
 In parallel flow, due to heat exchange, the temperature of the hot fluid decreases by
and the temperature of cold fluid increases by in the direction of heat
exchanger length (Refer Fig. 9.9(b)).
( )
[ ]
[ ] ( )
[ ]
[ ]
By integrating,
∫ ∫ [ ]
[ ] ( )
( ) [ ] ( )
 From the definition of effectiveness,
( )
( )
( )
( )
And
( )
( )
( )
( )

Values of outlet temperatures,
( )
And
( )
( ) ( ) [ ]
( ) [ [ ]]
Substituting this value in equation 9.33, we get,
(
( ) [ * +]
)
[ ]
[ [ ]] [ ]
[ [ ]] [ [ ]]
[ * +]
* +
[ * +]
* +
Now, if
Therefore and
Then we get,
[ * +]
* +
But,

and Capacity ratio,
[ [ ]]
[ ]
( )
 Equation 9.34 is the effectiveness of the parallel flow heat exchanger with hot fluid
having the minimum capacity rate.
 The same relationship would result when the analysis is made with the cold fluid
having minimum capacity rate.
Effectiveness of a parallel flow heat exchanger is,
[ [ ]]
[ ]
( )
Effectiveness for the counter flow heat exchanger:
entrance side of the heat exchanger as shown in Fig. 9.9(a).
( ) ( )
.
 Due to heat exchange, the temperature of hot and cold fluid decreases by and
respectively in the direction of heat exchanger length (Refer Fig. 9.9(a)).
( )
[ ]
[ ] ( )
[ ]

[ ]
By integrating,
∫ ∫ [ ]
[ ] ( )
( ) [ ] ( )
 From the definition of effectiveness,
( )
( )
( )
( )
And
( )
( )
( )
( )
Values of outlet temperatures,
( )
And
( )
Substituting this value in equation 9.40, we get,
( )
( )
[ { }]
( )
( )
[ { }]
( ) * +
( ) * +
[ { }]
* +
* +
[ { }]

[ { }] [ ]
[ { }] [ { }]
[ { }] [ [ { }]]
* , -+
[ * , -+]
Now, if
Therefore and
Then we get,
* , -+
[ * , -+]
But,
and Capacity ratio,
[ ( )]
[ [ ( )]]
( )
 Equation 9.41 is the effectiveness of the counter flow heat exchanger with cold fluid
having the minimum capacity rate.
 The same relationship would result when the analysis is made with the hot fluid
having minimum capacity rate.
Effectiveness of a counter flow heat exchanger is,
[ ( )]
[ [ ( )]]
( )
Limiting values of capacity ratio, C:
 Two limiting cases of practical interest are:
1) During the process of boiling and condensation, only a phase change takes place
and one fluid remains at constant temperature throughout the exchanger.
By definition, the specific heat represents the change of enthalpy with respect to
temperature, i.e., ⁄ . With temperature difference being zero, the

effective specific heat and consequently the heat capacity tends to infinity. In
that case and . The expression for effectiveness (both for
parallel and counter flow) then reduces to,
( ) ( )
2) The effectiveness is the lowest in the other limiting case of , which
is realized when the heat capacity rates of the two fluids are equal.
In a counter flow double pipe heat exchanger ,water is heated from 25°C to 65°C by
oil with specific heat of 1.45 kJ/kg K and mass flow rate of 0.9 kg/s. The oil is cooled
from 230°C to 160°C. If overall Heat transfer coefficient is 420 W/m2
°C. calculate
following:
a) The rate of heat transfer
b) The mass flow rate of water , and
c) The surface area of heat exchanger
⁄
̇
To be Calculated:
a)
b) ̇
c)
 From Energy balance equation,
̇ ( ) ̇ ( )
 Rate of Heat Transfer:
̇ ( )
( )
 Mass of cooling water:
̇ ( )
̇ ( )
̇
 For Counter flow heat exchanger,

Log Mean Temperature Difference (LMTD),
 Surface area:
A heat exchanger is to be designed to condense 8 kg/sec of an organic liquid
(tsat=80°C, hfg=600 KJ/kg) with cooling water available at 15°C and at a flow rate of
60 kg/sec. The overall heat transfer coefficient is 480 W/m2
°C calculate:
a) The number of tube required. The tubes are to be of 25 mm outer diameter, 2 mm
thickness and 4.85 m length
b) The number of tube passes. The velocity of the cooling water is not to exceed 2
m/sec.
̇
̇
To be Calculated:
a)
b)
 From Energy Balance Equation,
̇ ̇ ( )
 Rate of Heat Transfer,
̇
 Cooling Water Outlet Temperature,
̇ ( )
( )

 For Condenser (By Considering Parallel Flow),
 Total Surface Area,
 Total Number of Tubes:
 Number of Tubes Per Pass,
̇
̇
 Number of Passes:
A parallel flow heat exchanger has its tubes of 5 cm internal and 6 cm external
diameter. The air flows inside the tubes and receives heat from hot gases circulated in
the annular space of the tube at the rate of 100 kW. Inside and outside heat transfer
coefficients are 250 W/m2
K and 400 W/m2
K respectively. Inlet temperature of hot
gases is 500 °C, outlet temperature of hot gases is 300 °C, inlet temperature of air
50°C, Exit temperature of air 140 °C. Calculate :
a) Overall heat transfer coefficient based on outer surface area
b) Length of the tube required to affect the heat transfer rates. Neglect the thermal
resistance of the tube.
c) If each tube is 3 m length find the number of tubes required.

a)
b)
c)
̇ ( ) ̇ ( )
 Overall Heat Transfer Co-efficient:
(Note: Here tube wall thermal resistance & effect of fouling is neglected so overall
heat transfer co-efficient will remain same for outer & inner surface area.)
 For Parallel Flow Heat Exchanger,
 Total Surface Area,
 Length of Tubes:
 Total Number of Tubes Required if :

Ex 9.4. [GTU; Jan-2013; 7 Marks]
A heat exchanger is used to cool hot water from 80°C other to 60°C by transferring
heat to other stream of cold water enters the heat exchanger at 20°C and leave at
40°C. Should this heat exchanger operate under parallel flow or counter flow
conditions? Also determine the exit temperatures if the flow rates of the fluids are
doubled.
a) ̇
b) ̇
 The outlet temperature of cold fluid is less than the outlet temperature of hot
fluid. Such a temperature profile is possible in parallel flow arrangement, and
hence the exchanger should operate in a parallel flow mode.
̇ ( ) ̇ ( )
Since both fluids have equal temperature difference,
̇ ̇
 Heat Capacity Ratio,
 Effectiveness,
( )
( )
( )
( )
 Effectiveness for Parallel Flow Heat Exchanger,
[ ( )]
[ ( )]

[ ]
 When flow rates of the fluids are doubled, the thermal capacity rates of the hot
and cold fluids will still be equal and accordingly the heat capacity ration C will be
unity. Also,
( ) ( )
Hence,
( ) ( )
( )
 New Effectiveness,
[ ( )]
[ ( )]
 New Effectiveness in terms of Temperatures,
( )
( )
( )
( )
( )
( )
And,
( )
( )
Hot oil enters into a counter flow heat exchanger at 150°C and leaves at 40°C. The
mass flow rate of oil is 4500 kg/hr and its specific heat is 2 kJ/kg-K. The oil is cooled
by water which enters the heat exchanger at 20°C. The overall heat transfer co-
efficient is 1400 W/m2
K. The exit temperature is not to exceed 80°C. Using
effectiveness-NTU method, find
a) Mass flow rate of water
b) Effectiveness of heat exchanger
c) Surface area required.

̇
⁄
Take,
⁄
To be Calculated:
a) ̇
b)
c)
 Mass Flow Rate:
From Energy Balance Equation,
̇ ( ) ̇ ( )
( ) ̇ ( )
̇
 Heat Capacity Rate of the Hot Fluid, ̇
 Heat Capacity Rate of the Cold Fluid, ̇
Here,
and
 Heat Capacity Ratio,
 Effectiveness:
( )
( )
( )
( )
( )
 Effectiveness for Counter Flow Heat Exchanger,
[ ( )]
[ ( )]
[ ( )] [ ( )]
[ ( )] [ ( )]
[ ( )] ( )
[ ( )]
[ ( )] ( )
( )
( )
 Surface Area:

9.10 References
[3] Heat and Mass Transfer by P.K. Nag, McGraw-Hill Publication.
[4] Heat and Mass Transfer by Mahesh M Rathore, McGraw-Hill Publication.
Publication.
[6] National Programme on Technology Enhanced Learning (NPTEL), A Joint Initiate by
IIT’s and IISc. (Web: http://nptel.ac.in/)

2151909 heat transfer e-note (thefreestudy.com) (1)

More Related Content

What's hot (20)

Similar to 2151909 heat transfer e-note (thefreestudy.com) (1) (20)

Recently uploaded (20)

2151909 heat transfer e-note (thefreestudy.com) (1)