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THE GATE ACADEMY's GATE Correspondence Materials consist of complete GATE syllabus in the form of booklets with theory, solved examples, model tests, formulae and questions in various levels of difficulty in all the topics of the syllabus. The material is designed in such a way that it has proven to be an ideal material in-terms of an accurate and efficient preparation for GATE. Quick Refresher Guide : is especially developed for the students, for their quick revision of concepts preparing for GATE examination. Also get 1 All India Mock Tests with results including Rank,Percentile,detailed performance analysis and with video solutions GATE QUESTION BANK : is a topic-wise and subject wise collection of previous year GATE questions ( 2001 – 2013). Also get 1 All India Mock Tests with results including Rank,Percentile,detailed performance analysis and with video solutions Bangalore Head Office: THE GATE ACADEMY Jayanagar 4th block E-Mail: info@thegateacademy.com Landline: 080-61766222
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Syllabus Heat Transfer
THE GATE ACADEMY PVT.LTD. H.O.: #74, Keshava Krupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750, info@thegateacamy.com © Copyright reserved. Web: www.thegateacademy.com
Syllabus for Heat Transfer
Modes of heat transfer, one dimensional heat conduction, resistance concept, electrical analogy,
unsteady heat conduction, fins; dimensionless parameters in free and forced convective heat
transfer, various correlations for heat transfer in flow over flat plates and through pipes;
thermal boundary layer; effect of turbulence; radiative heat transfer, black and grey surfaces,
shape factors, network analysis; heat exchanger performance, LMTD and NTU methods.
Analysis of GATE Papers
(Heat Transfer)
Year Percentage of marks Overall Percentage
2013 10.00
6.26%
2012 6.00
2011 4.00
2010 2.00
2009 9.00
2008 6.00
2007 8.00
2006 4.67
2005 6.67
Content Heat Transfer
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C O N T E N T S
Chapters Page No.
#1. Conduction 1 - 50
Introduction 1 – 2
One Dimensional Heat Conduction 2 – 8
Unsteady Heat Conduction 9 – 11
Critical Radius of Insulation 11 – 13
Heat Transfer Through Fins 13 – 17
Solved Examples 18 – 30
Assignment 1 31 – 34
Assignment 2 34 – 38
Answer Keys 39
Explanations 39 – 50
#2. Convection 51 - 97
Introduction 51 – 52
Convection Fundamentals 52 – 56
Forced Convection 56 – 66
Nusselt Numbers 67 – 68
Natural Convection 68 – 71
Solved Examples 72 – 85
Assignment 1 86 – 88
Assignment 2 88 – 90
Answer Keys 91
Explanations 91 – 97
#3. Radiation 98-137
Introduction 98
Blackbody Radiation 98 – 100
Radiative Properties 100 – 102
The View Factor 102 – 104
Radiation Heat Transfer 104 – 110
Solved Examples 111 – 123
Content Heat Transfer
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Assignment 1 124 – 128
Assignment 2 128 – 129
Answer Keys 130
Explanations 130 – 137
#4. Heat Exchanger 138-168
Introduction 138
Types of Heat Exchangers 138 – 139
The Overall Heat Transfer Coefficient 139 – 140
Analysis of Heat Exchanger 140 – 143
The Effectiveness – NUT Method 143 – 147
Solved Examples 148 – 158
Assignment 1 159 – 160
Assignment 2 160 – 161
Answer Keys 162
Explanations 162 – 168
Module Test 169 – 184
Test Questions 169– 177
Answer Keys 178
Explanations 178 – 184
Reference Books 185
Chapter 1 Heat Transfer
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CHAPTER 1
Conduction
Introduction
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. 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.
Consider steady heat conduction through a large plane wall of thickness x and area A, as
shown in Figure 1. The temperature difference across the wall is . 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,
W
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
Figure 1: Heat conduction through a large plane wall of thickness and area A
Chapter 1 Heat Transfer
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Which is called Fourier's law of heat conduction after J. Fourier, who expressed it first in his heat
transfer text in 1822. Here dT/dx 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.
Thermal Conductivity
The rate of conduction heat transfer under steady conditions can also be viewed as the defining
equation for thermal conductivity. Thus 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.
Thermal Diffusivity
The product , which is frequently encountered in heat transfer analysis, is called the heat
capacity of a material.
Another material property that appears in the transient heat conduction analysis is the thermal
diffusivity. Which represents how fast heat diffuses through a material and is defined as
⁄
Note that the thermal conductivity k represents how well a material conducts heat, and the heat
capacity represents how much energy a material stores per unit volume. Therefore, the
thermal diffusivity of a material can be viewed as the ratio of the heat conducted through the
material to the heat stored per unit volume. A material that has a high thermal conductivity or a
low heal capacity will obviously have a large thermal diffusivity. The larger the thermal
diffusivity. The faster the propagation of heat into the medium. A small value of thermal
diffusivity means that heat is mostly absorbed by the material and a small amount of heat will be
conducted further.
One Dimensional Heat Conduction
Heat transfer has direction as well as magnitude. The rate of heat conduction in a specified
direction is proportional to the temperature gradient, which is the change in temperature per
unit length in that direction. Heat conduction in a medium, in general, is three-dimensional and
time dependent. That is, and the temperature in a medium varies with position
as well as time. Heat conduction in a medium is said to be steady when the temperature does not
vary with time, and unsteady or transient when it does. Heat conduction in a medium is said to
Chapter 1 Heat Transfer
THE GATE ACADEMY PVT.LTD. H.O.: #74, Keshava Krupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750, info@thegateacamy.com © Copyright reserved. Web: www.thegateacademy.com Page 3
be one-dimensional when conduction is significant in one dimension only and negligible in the
other two dimensions, two-dimensional when conduction in the third dimension is negligible
and three-dimensional when conduction in all dimensions is significant. The governing
differential equation in such systems in rectangular, cylindrical and spherical coordinate
systems is derived in below section.
Rectangular Coordinates
Consider a small rectangular element of length , width and height , as shown in Figure 2.
Assume the density of the body is and the specific heat is C, an energy balance on this element
during a small time interval can be expressed as
(
x y z
+ (
x x y yz z
, (
, (
y
,
Noting that the volume of the element is the change in the energy content of
the element and the rate of heat generation within the element can be expressed as
x y z
x y z
Substituting into equation we get
x y z x y z
Dividing by x y z gives
y z
x
x z
y
x y
z
Figure 2: Three-dimensional heat conduction through a rectangular volume element
Chapter 1 Heat Transfer
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Nothing that the heat transfer area of the element for heat conduction in the directions are
respectively and taking the limit as and
yields
(
*
(
*
(
*
S v v F ’ w .
(
*
(
*
(
*
(
*
(
*
(
*
In the constant case of constant thermal conductivity
Where the property is again the thermal diffusivity of the materials and above
equation is known as the Fourier-Biot equation and it reduces to these forms under specified
conditions:
1. Steady-State: (called Poisson equation)
2. Transient, no heat generation: (called the Diffusion equation)
3. Steady state, no heat generation: (called the Laplace equation)
Note that in the special case of one dimensional heat transfer in the direction, the
derivatives with respect to y and z drop out.
Cylindrical Coordinates
The general heat conduction equation in cylindrical coordinates can be obtained from an energy
balance on a volume element in cylindrical coordinates, shown in Figure 3, by following the
Chapter 1 Heat Transfer
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steps just outlined. It can also be obtained directly by coordinate transformation using the
following relations between the coordinates of a point in rectangular and cylindrical coordinate
systems
After lengthy manipulations we obtain
(
*
(
*
(
*
Spherical Coordinates
The general heat conduction equations in spherical coordinates can be obtained from an energy
balance on a volume element in spherical coordinates, shown in Figure 4, by following the steps
outlined above. It can also be obtained directly by coordinate transformation using the following
relations between the coordinates of a point in rectangular and spherical coordinate systems
Again after lengthy manipulations, we obtain
(
*
(
*
(
*
Conduction through a cylindrical wall
For a cylinder at steady state, with no internal heat generation, the equation becomes
O b B. ’ w b equation as
Figure 3: A differential volume element in cylindrical coordinates.
Chapter 1 Heat Transfer
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Q = ( where
Comparing the above equation to that of heat transfer through a wall
Q = KA (T1 – T2 δ (T1 – T2) / (r2 – r1)
Where is the logarithmic mean area = (A2 – A1) / log (A2 / A1)
The above equations are applicable to any general heat conduction problem. The one
dimensional heat conduction is out particular area of interest as they result in ordinary
differential equations.
Conduction through sphere
Steady state, one dimensional with no heat generation equation in spherical co-ordinates is
(
*
O b B. ’ w b equation as
[
]
Figure 4: A differential volume element in spherical coordinates.
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