[Bekir Sami Yilbas, Ahmet Z. Sahin (Auth.)] Fricti

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SpringerBriefs in Applied Sciences

and Technology

Manufacturing and Surface Engineering

Series editor

Joao Paulo Davim, Aveiro, Portugal

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Bekir Sami Yilbas Ahmet Z. Sahin

Friction Welding

Thermal and Metallurgical Characteristics

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Bekir Sami YilbasMechanical Engineering DepartmentKing Fahd University of Petroleum

and MineralsDhahran

Saudi Arabia

Ahmet Z. SahinKing Fahd University of Petroleum

and MineralsDhahranSaudi Arabia

ISSN 2191-530X ISSN 2191-5318 (electronic)ISBN 978-3-642-54606-8 ISBN 978-3-642-54607-5 (eBook)DOI 10.1007/978-3-642-54607-5Springer Heidelberg New York Dordrecht London

Library of Congress Control Number: 2014933787

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Preface

Friction welding can be used widely in industry because of its precision of

operation, high processing speed, and low cost. Friction welding has advantages of

incomplete molten state of the welded parts at the weld interface region. This isparticularly important for welding of dissimilar materials, in which case narrow

heat affected zone is favorable. To improve the end-product quality, care must be

taken to select proper welding parameters according to the sets of materials used in

the welding process. In addition, the development of high temperature gradients in

a short distance across the weld interface results in high stress levels in the welded

region. In some circumstances, this limits the practical applications of the welded

parts, in particular for weld sizes comparable to micro/nanoscales. Although

considerable research studies were carried out to minimize the welding defects,

further studies need to be carried out to explore the possible application of frictionwelding at micro/nanoscales. This is mainly because of the complicated nature of

the problem at micro/nanoscales. Since the process involves with multi-physics,

development of new model studies is required to capture the physical phenomena.

However, online experimentation of the welding process at micro/nano level is

extremely difficult and costly because of the limitations in sensing systems, which

operate at high temperatures during the friction welding process. On the other

hand, the model studies of the welding process provide useful insight into the

physical processes taking place during the welding and provide optimum operating

parameters for sound welds.Metallurgical and morphological changes in the welding region are important to

secure sound and quality welds for the practical applications. Since metallurgical

changes influence significantly mechanical properties of the weld sites, experi-

mental assessments of mechanical properties of resulting welds become essential.

Optimization of welding process, utilizing the statistical tools, improves

mechanical and metallurgical properties and assists to produce desirable welds for

the practical applications.

In this book, thermal analysis including thermal stress development during

friction welding is formulated at macro and micro levels. Equilibrium and non-

equilibrium heating situations, pertinent to friction welding, are classified and the

closed-form solutions of governing heat and momentum equations are presented.

Analytical solution is also extended to include two-dimensional heating situation

for non-equilibrium energy transfer in the welded region. Assessment of some

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metallurgical changes in the weld section and mechanical properties of welded

parts are included in the book. However, some cases related to modeling of friction

welding are not presented in this book due to space limitations and, therefore,

these cases are left for the future treatments.

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Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Reference. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Thermal Analysis of Friction Welding . . . . . . . . . . . . . . . . . . . . . 5

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Infinite Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2.1 Instantaneous Release of Heat. . . . . . . . . . . . . . . . . . . 6

2.2.2 Continuous Release of Heat . . . . . . . . . . . . . . . . . . . . 9

2.2.3 Moving Sources of Heat. . . . . . . . . . . . . . . . . . . . . . . 12

2.3 Semi-Infinite Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.3.1 Instantaneous Point Heat Release q (J)

on the Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.3.2 Instantaneous Line Heat Release q0 (J/m)

on the Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.3.3 Instantaneous Plane Heat Release q00 (J/m2

)

on the Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.3.4 Continuous Point Heat Release _q (W) on the Surface . . 21

2.3.5 Continuous Line Heat Release _q0 (W/m)

on the Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.3.6 Uniform Heat Flux _q00 (W/m2) on the Surface . . . . . . . . 23

2.3.7 Continuous Strip (Along y-Axis) Heat Releaseon the Surface of Semi-Infinite Medium. . . . . . . . . . . . 24

2.3.8 Continuous Circular Disk Area (of Radius R) Heat

Release on the Surface of Semi-Infinite Medium. . . . . . 25

2.3.9 Moving Point-Source on the Surface . . . . . . . . . . . . . . 27

2.3.10 Moving Line-Source on the Surface. . . . . . . . . . . . . . . 28

2.3.11 Moving Infinite y-Strip Source (in x Direction)

on the Surface of a Semi-Infinite Solid . . . . . . . . . . . . 29

2.3.12 Moving Infinite y-Strip Source (in x Direction)

on the Surface of a Semi-Infinite Solid

with Convection Boundary. . . . . . . . . . . . . . . . . . . . . 30

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2.4 Slab (Plate) of Thickness L . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.4.1 Moving Point Source _q (W) on the Surface

of an Insulated Infinite Plate. . . . . . . . . . . . . . . . . . . . 31

2.4.2 Moving Line Source on the Surface

of an Insulated Infinite Plate . . . . . . . . . . . . . . . . . . . . 322.5 Thin Slab (Sheet) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.5.1 Spot Welding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.5.2 Moving Line Heat Release q Across the Thin

Sheets of Total Thickness d to Be Welded

Between the Two Electrodes Along x-Direction . . . . . . 34

2.6 Solid Rod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.6.1 Friction Welding of Long Rods . . . . . . . . . . . . . . . . . . 36

2.6.2 Time Variable Heat Source in Rod . . . . . . . . . . . . . . . 38

2.6.3 Finite Rod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412.7 Numerical Analysis of Friction Welding . . . . . . . . . . . . . . . . . 42

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3 Non-equilibrium Heating Situations . . . . . . . . . . . . . . . . . . . . . . . 47

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.2 Thermal Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.2.1 One-Dimensional Solid Case. . . . . . . . . . . . . . . . . . . . 48

3.2.2 Two-Dimensional Solid Case . . . . . . . . . . . . . . . . . . . 53

3.3 Findings and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583.3.1 Temperature and Stress Fields for Thermomechanically

Coupled One-Dimensional Semi-Infinite Solid . . . . . . . 58

3.3.2 Temperature Field in a Two-Dimensional

Solid Rod Case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.4 Metallurgical Changes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

3.5 Mechanical Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694.1 Thermal Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.2 Morphology, Metallurgical and Mechanical Properties . . . . . . . . 71

x Contents
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Chapter 1

Introduction

Abstract Friction welding is one of the effective joining techniques in industry.

Friction welding is the solid state welding and it offers an alternative weldingprocess for the joining the parts in particular electrical appliances, engine parts,

etc. The welding takes place when two surfaces, subjected to the joining, get in

mechanical contact and the surfaces are heated to the desired temperature through

frictional heat generation and, later, a forging pressure is introduced to weld the

parts. In the present chapter, introduction to friction welding is presented and the

welding mechanisms are described.

Keywords Friction welding Mechanism Process

Effective joining of the parts is one of the changes faced in industry. Although

many joining techniques including mechanical fastening, adhesive bonding, and

solid-phase welding are well established, friction welding offers an alternative

welding process for the joining the parts in particular electrical appliances, engine

parts, etc. Friction welding finds widespread industrial use as a mass-production

process for joining of materials. In the welding process, joining surfaces of the

parts are heated to the desired temperature through frictional heat and then a

forging pressure is introduced to weld the parts. Many ferrous and non-ferrous

alloys can be friction welded. Friction welding can be used to join materials ofdifferent thermal and mechanical properties. In some cases, the combinations of

materials cannot be joined by other welding techniques because of the formation

of brittle phases which make the joint poor in mechanical properties. The sub-

melting temperatures and short weld times of friction welding allow many com-

binations of materials to be joined.

Friction welding is achieved by the frictional heat generated between the

components that are pressed together as a result of friction and pressure. As a result

of the heat generated the component surfaces that are in contact soften, become

plasticized and mix together. After the frictional operation and the relative motion

are terminated the interface region cools down and a strong bond is achieved after

hardening. Depending on the type of the processes involved friction welding can

be classified into a number of different types.

B. S. Yilbas and A. Z. Sahin, Friction Welding, SpringerBriefs in Manufacturing

and Surface Engineering, DOI: 10.1007/978-3-642-54607-5_1, The Author(s) 2014

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Linear Friction Welding: Friction welding achieved by the linear relative

motion across the interface is named linear friction welding. This type of welding

is suitable for components that are difficult to join by other welding techniques. It

is successfully used in aerospace industry to join complex parts such as turbine

blades in gas turbines.Rotary Friction Welding: This is most commonly used method in friction

welding. In rotary friction welding a rotating component is pressed against a

stationary component to achieve bonding. This type of friction welding is suitable

for carbon steel and other metal applications. Dissimilar materials can also be

joined by this technique.

Spin Welding: This type of friction welding is generally used for thermoplastic

materials where the material in the vicinity of the interface softens and moves out-

wards as a result of spinning and pressure. Once a homogeneous layer of soft material

is available at the interface sufficient force is applied to join the parts together.Friction Stir Welding: Friction stir welding is achieved by a non-consumable

tool that does not soften during the operation. The tool is pressed on the interface

of the components to be welded together. The tool softens both of the components

around the interface and mixes the softened material from both of the component

around the interface to provide bonding.

Inertia Friction Welding:In inertia friction welding the required energy for the

joining the components is obtained from the stored kinetic energy in a flywheel or

the welding machine. One of the components is held stationary while the other

component is attached to the rotating flywheel. As soon as the components arebrought into contact the kinetic energy of the flywheel is converted into frictional

heat that is used for the welding of the components.

Friction Stud Welding: In this friction welding process a high speed rotating

stud is pressed against a stationary substrate. Thus the frictional heat softens the

region of contact and provides the joint. Friction stud welding is suitable for

special applications where other conventional welding techniques may not be

applicable such as underwater welding. However, the cost of this kind of welding

is high and therefore the applicability may be limited.

Friction Surfacing: A coating material is used in the interface of the compo-nents to be joined for this type of welding. The frictional heat generated turns the

coating material into a plastic layer which consequently joins the components

together when the joint is cooled. Since the type of material used as the coating

layer the metallurgical and physical properties could be very different from those

of the base component.

In friction welding, thermal energy in terms of heat generation, which is nec-

essary for welding, is produced by direct conversion of mechanical energy into

thermal energy at the interface of the workpieces. Friction welds can be produced

by holding a non-rotating solid part in contact with a rotating part under gradually

increasing pressure until welding temperature is reached at the interface between

the stationary and rotating parts. Stopping the rotating part rapidly and applying

the forging load completes the welding process. The proper alignments of the

2 1 Introduction


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stationary and rotating parts are critical to achieve the sound weld. On the other

hand, frictional wear removes irregularities and asperities from the surfaces of the

parts during the welding process. Therefore, clear and smooth interfacial surfaces

of the welded parts are resulted after completing the welding.

Friction welding refers to a group of non-fusion joining processes, in which ajoint is produced by rotating one part against another while applying an axial

force. Two types of friction welding use rotational motion. In the continuous-

drive friction welding method, energy is supplied to the interface at constant

rotational speed by an electric motor. In the inertia welding method, energy is

derived from a flywheel of predetermined size, running at a predetermined initial

speed. In the continuous drive method, one of the components to be welded is

held stationary while the other component is rotated at a specified speed. The two

components are then brought together under axial pressure for a certain time

period or until a predetermined burn-off is produced. The drive is then declut-ched, and the rotating component is quickly brought to a halt while the axial

pressure is maintained or increased to a higher forging pressure. In the inertia

welding method, one of the components to be welded is held stationary while the

second component is clamped in a spindle chuck, usually with attached flywheels.

The flywheel and chuck assembly is then rotated to a certain speed to store a

predetermined amount of energy. The drive to the flywheel is declutched, and the

two components to be welded are brought together under axial pressure. Friction

between the parts decelerates the flywheel converting stored energy to frictional

heat. In general, a small projection at the center of one of the weld members isused to ensure a proper heating and forging action for welding large-diameter

bars. Friction welding is very tolerant of the pre-weld interface conditions:

consequently, roughly-formed ends even with a degree of oxidized surfaces can

be used without affecting the weld strength. This is because of the fact that

rotational phase of friction welding initially scours the weld interface and

removes impurities. As the frictional heat generation at the interface increases,

the materials soften and a condition of full-face intimate contact is achieved.

Friction welding can be achieved at high production rates, and therefore is eco-

nomical in operation.Friction welding process is involved with equipment, which is easy to construct.

A schematic view of a typical friction welding apparatus (friction welder) is shown

in Fig.1.1. A friction welder can operate at different applied load conditions,

which depend on the parts size and parts material. The maximum typical load for

metallic parts is in the order of 120 KN. In general, the welder motor has variable

speed, which could be controlled by a computer. A typical welder speed for

metallic parts is in the order of 3500 rpm. The friction welder operating param-

eters include rotational speed, friction pressure, friction time, forging pressure,

feed rate, brake delay time, upset delay time, and upsetting time. The operating

parameters can be controlled by a computer for a desired end product quality.

A typical forcetime curve is shown in Fig. 1.2for welding of metallic parts [1].

1 Introduction 3


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Reference

1. Yilbas BS, Sahin AZ, Coban A, Abdul Aleem BJ (1995) Investigation into the properties offriction welded aluminum bars. J Mater Process Tech 54:7681

Fig. 1.2 A typical time-load curve used during the friction welding process [1]

Fig. 1.1 Shows a schematic view of friction welding equipment (friction welder) [1]

4 1 Introduction


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Chapter 2

Thermal Analysis of Friction Welding

Abstract Thermal energy is generated during the friction welding process. In this

case, the solid surfaces rub against each other and heat is generated as a result offriction. Since the process is trasient and involves with axis-symmetric heating

situation, formulation of the thermal analysis becomes essential. In this chapter,

thermal analysis based on Fourier heat conduction is introduced and the solution of

conduction equation is obtained for appropriate boundary conditions.

Keywords Friction welding Heating analysis Temperature

2.1 Introduction

Welding of solid materials is achieved by providing thermal energy in the form of

heat for melting or softening the interface between the two materials and bringing or

pressing them together. Friction is one of the methods of generating the required

thermal energy for welding process. As the solid surfaces rub against each other heat

is generated as a result of friction. The heat generated due to friction subsequently

diffuses through the bulk of the contacting solid materials. As the heat is necessary

for obtaining sound welds, it also affects the mechanical as well as the micro-

structural properties of the welded materials in the vicinity of the welding interface.

Thermal analysis of friction welding is carried out to determine the resulting

temperature distribution around the welding interface and thus allows determi-

nation of the high temperature effects on the micro-structure of the materials as

well as the quality of the weld. The thermal analysis related to the friction welding

is carried out in line with the previous studies [112].

As an elementary example, consider two solid objects with flat surfaces pressed

together with a force F and sliding against each other with a relative velocity of V.

The power consumed against the frictional force Ff= lF is converted to thermal

energy generation at the interface. The rate of thermal energy generation is given by

_Q FfV lFV 2:1

B. S. Yilbas and A. Z. Sahin, Friction Welding, SpringerBriefs in Manufacturing

and Surface Engineering, DOI: 10.1007/978-3-642-54607-5_2, The Author(s) 2014

5


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where l is the coefficient of friction. If the contact area is A then the rate of

thermal energy generation per unit area becomes

q _Q

AlFV

A : 2:2

Example 1 A solid block of 100 kg mass slides along a horizontal concrete

pavement with a speed of 2 m/s. The coefficient of friction between the block and

the pavement is estimated to be 0.4. Determine the rate of thermal energy gen-

eration in W.

Solution:

The force F acting on the pavement due to the weight of the block is

F m g 100 9:81 981 NTherefore, the thermal energy generation as a result of friction between the

solid block and the pavement is obtained by using Eq. (2.1)

_Q lFV 0:4 981 2 784:8 WThe thermal energy generated at the interface diffuses through the solid objects.

The amount of thermal energy diffusion in each of the solids depends on the

thermal conductivity of the solids.

The temperature distribution in solids can be determined by solving the heatconduction equation. If the thermo-physical properties are assumed constant and

no phase change (i.e. no melting of solids) is considered the heat conduction

equation is given by

oT

ot ar2T 2:3

wherea k=qCP is the thermal diffusivity, k is the thermal conductivity, q is thedensity and CPis the specific heat. Solution of Eq. (2.3) with appropriate initial and

boundary conditions yields the transient temperature distribution inside the solids.

2.2 Infinite Medium

2.2.1 Instantaneous Release of Heat

One of the fundamental solutions of the heat conduction equation in relation to the

welding process is for the instantaneous point source in an infinite medium(Fig.2.1). In this case the heat liberated at a point diffuses in all directions in the

medium. The solution for the transient temperature distribution in this case is

given by

6 2 Thermal Analysis of Friction Welding


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Figure2.3 shows the temperature profiles in a large size steel material after an

instantaneous release of 100 kJ/m of heat along the z-axis at time t = 0. Each ofthe curves in Fig. 2.3represents the temperature profile at different times after the

release of the heat, namely, at t = 0.25, 0.5, 0.75, 1.0, and 1.25 s. The temperature

on the z axis at t = 0.25 s is more than 1000 K and it decreases sharply as a result

of heat diffusion.

Similarly, if the instantaneous heat release occurs on the entire y-z plane, then

the heat diffuses along the x direction. Thus the solution of the heat conduction

equation yields a transient one-dimensional temperature distribution in the form

Tx; t Ti q00

qCP 4pat 1=2exp x2

4at 2:6

where q00 (J/m2) is the amount of heat released instantaneously over the y-z planeper unit area at time t = 0. Figure2.4shows the temperature profiles in this case

Fig. 2.2 The temperature

distribution in a large size

steel material after an

instantaneous thermal energy

release of 1 kJ at the origin.

(t = 0.25, 0.5, 0.75, 1.0 and

1.25 s)



steel material after a thermal

energy release of 100 kJ/m

along the z-axis at time = 0.

(t = 0.25, 0.5, 0.75, 1.0 and

1.25 s)



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for the instantaneous heat release of 10 MJ/m2 on the y-z plane at time t = 0. The

material considered in this case is also steel with the same thermophysical char-

acteristics mentioned above. The temperature profiles show similar behavior to the

cases mentioned above, however, the peak values of temperature are different.

This is because the instantaneous heat release takes place in a larger area and the

diffusion of heat occurs in a larger volume. In this case the change (or decrease) of

peak temperature during the time period t = 0.25 to 1.25 s is around 120 K.

The fundamental solutions given by the above three equations provide infor-mation on the diffusion of heat in the solid medium with no boundaries. However,

they do not adequately describe the thermal energy diffusing during the welding

process. This is because the thermal energy generation in a typical welding process

is not spontaneous and the domain is normally finite.

2.2.2 Continuous Release of Heat

Now, consider a steady (continuous) case of thermal energy generation in an

infinite medium. In this case there is no transient term in the governing heat

conduction equation and therefore Eq. (2.3) simplifies to

r2T 0 2:7If the thermal energy generation occurs at a rate _q (W) at the origin (r = 0) the

temperature distribution in the medium is obtained to be

Tr; t Ti _q

qCP 4par erfc r

2ffiffiffiffi

atp 2:8where r

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 y2 z2

p . It should be noted that as t! 1 the term erfc (0) = 1and

therefore the steady-state temperature distribution in the medium is obtained to be



steel material after a thermal

energy release of 10 MJ/m2

on the y-z plane at time = 0.

(t = 0.25, 0.5, 0.75, 1.0 and

1.25 s)

2.2 Infinite Medium 9


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Tr Ti _qqCP 4par : 2:9

Figure2.5shows the temperature distribution in a steel material for the case of

a continuous thermal energy release of 100 W at the origin. The steady temper-

ature distribution in the material varies inversely with the radial distance as given

in Eq. (2.9).

Example3 Consider the point energy released in the large steel material as given

in Example 1 to be continuous, i.e. 1 kW. Determine the temperature at a location

5 mm away from the location of the heat release after 2 s.

Solution:

In this case the solution is given by Eq. (2.8)

Tr; t Ti _qqCP 4par erfc

r

2

ffiffiffiffiat

p

T0:005; 2 300 10007800 473 4p 1:172 105 2 erfc

0:005

2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1:172 105 2p

317 K

The temperature in this case is lower when compared with the solution given in

Example 2. The reason for this is because the heat release in the current case is not

instantaneous and therefore it is slower that the case in Example 2. Therefore, the

temperature of the material around the spot where the heat is released continues to

increase and approaches a steady value. This value for t ? infinity can be shown

to be 336.8 K. For the case of instantaneous heat release, however, the temperatureat the given location increases initially and then decreases as the wave of heat

passes through that point.

Fig. 2.5 The radial

temperature distribution at

different times in a steel

material for a continuous

release of 100 W rate of heat

at the origin (t = 10-4, 10-3,

10-2, 10-1 and 1 s)



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In case of continuous heat release along the z-axis at a rate of _q0 (W/m) thetemperature distribution in the infinite medium becomes

Tr; t Ti _q0

4pqCPaEi

r2

4at

2:10

where r ffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 y2p . Since for t! 1 the term Ei0 ! 1 there is no steadytemperature distribution in this case. Figure2.6shows the temperature profiles atvarious times (t = 0.01, 0.1, 1, 10 and 100 s) in a steel material when a continuous

release of 10 kW/m rate of heat takes place along the z-axis. The heat diffuses

through the bulk of the material and therefore the temperature in the material

continues to increase.

Example 4 A high resistance electric wire is located in a large steel block

releasing 10 kW/m heat energy. The initial temperature of the block is 25 C.

Estimate the temperature 5 mm away from the wire after 1 min of heating.

Solution:According to Eq. (2.10) the temperature is obtained after substituting the given

information:

T0:005; 60 298 100004p 7800 473 1:172 105Ei

0:0052

4 1:172 105 60

374:5 K

For the case of thermal energy generation over the y-z plane at a rate of _q00 (W/m2) the temperature distribution in the infinite medium is given by

Tx; t Ti _q00

2k

ffiffiffiffiffiffiffi4at

p

r exp x

2

4at

xerfc x

2ffiffiffiffi

atp

" # 2:11

Fig. 2.6 The radial

temperature distribution at

different times in a steel

material for a continuous

release of 10 kW/m rate of

heat along the z-axis

(t = 0.01, 0.1, 1, 10 and

100 s)



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It is clear that there is no steady-state solution for this case either since the first

term in the equation is proportional toffiffi

tp

. The temperature anywhere in the

medium including that on the y-z plane increases continuously with time pro-

portional toffiffi

tp

. For x = 0 the temperature on the y-z plane is obtained to be

T

0; t

Ti

_q00

kffiffiffiffiat

pr 2:12

Figure2.7shows the temperature distribution in the direction of x-axis for the

case of continuous rate of heat release of 10 MW/m2 on the y-z plane. Each of the

curves in Fig. 2.7represents the temperature distribution at time t = 1, 2, 3, 4, and

5 s, respectively. The temperature in the bulk of the material increases in a con-

tinuous manner as a result of diffusion of heat away from the y-z plane. The

temperature increase at the y-z plane is proportional toffiffi

tp

as shown in Fig. 2.8.

2.2.3 Moving Sources of Heat

In most practical situations the heat source moves along the medium. Therefore,

for accurate determination of the temperature distribution in the medium, moving

heat sources must be considered. When the heat source or the medium moves

along the x-direction the governing equation for conduction heat transfer becomes

VoT

ox

a

r2T

2:13


distribution at different times

in a steel material for a

continuous release of

10 MW/m2 rate of over the y-

z plane (t = 1, 2, 3, 4 and

5 s)



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2.2.3.1 Moving Point-Source

Now, consider a steadily moving point heat source in the x-direction with a speedof V in an infinite medium. The temperature distribution on the coordinates

moving with the point source is named the quasi-steady state solution. This

solution can be expressed as

Tx;y;z Ti _qqCP 4par exp

Vrx2a

2:14

where rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

x2 y2 z2p

. It should be noted that when the velocity goes to zero

the solution approaches the steady-state solution for the case of continuous point

source given in Eq. (2.9). Figure2.9shows the temperature distribution along the

x-axis in a steel material in which a moving point release of heat at a rate of 100 W

occurs along the x-axis. The speed of moving point heat source along the x-

direction is 0.01 m/s.


variation with time on the y-z

plane in the steel material for

a continuous release of

100 kW/m2 rate of over the

y-z plane

Fig. 2.9 Temperature

distribution in a steel material

along the x-axis subjected to

the moving point source of

100 W along the x-axis with

a speed of 0.01 m/s



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Example5 A point source of 100 W moves in a large steel material with a speed

of 10 mm/s. The temperature of the material far away from the source of heat

release is 300 K. Determine the temperature 2 mm behind and 2 mm in front of

the moving front.

Solution:

Equation (2.14) can be written in the one-dimensional form as:

Tx Ti _qqCP 4pa xj j exp

V xj j x2a

2 mm in front:

T0:002 300 1007800

473

4p

1:172

105

0:002

exp

0:01 0:002 0:002 2

1:172

105

316:8 K

2 mm behind:

T0:002 300 1007800 473 4p 1:172 105 0:002 exp

0:01 0:002 0:002 2 1:172 105

392:5 K

2.2.3.2 Moving Line-Source

If the moving source is a line heat source along the z direction and moving along

the x-axis the quasi-steady state solution for the temperature on the moving

coordinate system is obtained to be

Tx;y Ti _q0

2pkexp Vx

2a

K0

Vr

2a

2:15

Figure2.10shows the temperature distribution in the steel material along the x-

axis when a line source of heat of 100 kW/m (along the z-axis) moves in thedirection of x-axis. The speed of the line heat source is taken as 0.01 m/s.

2.2.3.3 Moving Plane-Source

Finally when the heat source is uniformly distributed on the y-z plane that is

moving along the x-axis with a velocity V the quasi steady state solution of the

temperature on the moving coordinate axis becomes

Tx Ti _q00

qCPVexp Vx

a

1 sgnx2

2:16



24/79

where sgn(x) is the Signum function which is defined as

sgnx xxj j

1 if x\00 if x 01 if x[0

8

[Bekir Sami Yilbas, Ahmet Z. Sahin (Auth.)] Fricti

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