heat transfer : mechanical engineering, the gate academy

HEAT TRANSFER

For

Mechanical Engineering

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Syllabus Heat Transfer

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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


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




Answer Keys 91


#3. Radiation 98-137

Introduction 98

Blackbody Radiation 98 – 100

Radiative Properties 100 – 102

The View Factor 102 – 104

Radiation Heat Transfer 104 – 110


Content Heat Transfer

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Answer Keys 130


#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




Answer Keys 162


Module Test 169 – 184

Test Questions 169– 177

Answer Keys 178


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



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



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



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



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.



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.

heat transfer : mechanical engineering, the gate academy

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