ME535 1 Thermal Modelling

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Chapter 1 Thermal Modeling

Introduction

Analytical modeling

Instantaneous heat sources

Moving heat sources

Numerical simulations

1

Introduction

Thermal models are useful in calculating temperature fieldsduring welding, and predicting fusion zone widths, peaktemperatures and cooling rates in heat-affected zone.

Although the accuracy of simple analytical models is limited, thepredictions are helpful in understanding the effects of processparameters on metallurgical changes.

Consider an autogenous arc welding process (e.g. GTAW). If

2

,steady-state temperature field surrounding the heat source willbe established. The isotherms that define this field can bepredicted using analytical solutions for heat flow with a constantheat source.

2009/01/12 - Cont'd...



The governing equations, assuming the thermal properties

independent of temperature, are:

Governing Equations

for uniaxial,

for biaxial, and

for triaxial heat conduction.⎟⎟ ⎠

⎞⎜⎜⎝

⎛

∂∂

+∂∂

+∂∂

=∂∂

2

2

2

2

2

2

z

T

y

T

x

T a

t

T

2 xa

t ∂=

∂

⎟⎟ ⎠

⎞⎜⎜⎝

⎛

∂∂

+∂∂

=∂∂

2

2

2

2

y

T

x

T a

t

T

3

where T is the temperature, t the time, x, y and z the heat flow

direction, and a is the thermal diffusivity defined as:

where λ is the thermal conductivity, ρc the volume heat capacity.

ca ρ λ / =

The initial & boundary conditions for plane, line or point instantaneous

sources: T-T0=∞ for t=0 and x=0 (alternatively r=0 or R=0)

T-T =0 for t=0 and x≠0 (alternatively r>0 or R>0)

Instantaneous Heat Sources

T-T0=0 for 0<t< ∞ and x=±∞ (alternatively r=∞ or R=∞)

It is easy to verify that the following solutions satisfy both governing

equations, and initial/boundary conditions:

Plane source:

⎟⎟ ⎠

⎞⎜⎜⎝

⎛ −+=

at

x

at c

AQT T

4exp

)4(

/ 2

2 / 10π ρ

2

4

Point source:

where Q is the net heat input (energy) released at time t=0, T0 initial

temperature, A the cross section of the rod, d the plate thickness

⎟⎟ ⎠

⎜⎜⎝ −+=

at at cT T

4exp

)4(0

π ρ

⎟⎟ ⎠

⎞⎜⎜⎝

⎛ −+=

at

R

at c

QT T

4exp

)4(

2

2 / 30π ρ

22 y xr +=

222 z y x R ++=



Most of the heat supplied by the power source is transferred to the

workpiece, while a portion lost to the surroundings:

Arc Efficiency

==

Heat input per unit length of weld is:

Where Q net heat input

weld weld

V

EI H net

η =

5

tweld weld time

η arc efficiency

E arc voltage

I arc current

V welding speed

Typical Arc Efficiency Factors

6



Typical Thermal Properties

7

Stray Arc Strike

8

Arc strikes outside the area of permanent welds should be avoided on any

base metal. Cracks or blemishes caused by arc strikes shall be ground to a

smooth contour and checked to ensure soundness. [AWS D1.1, Section 3.10]



Cooling Time/Rate in Arc Strike

Arc strike can be modelled as a surface point source, in which the temperature rise

is twice as large as that for the point source in a heavy slab:

⎟⎟ ⎞

⎜⎜⎛ −+=RQ

T T exp2 2

2 / 30

To calculate the cooling time from 800 to 500oC, Δt8/5, use a group of typicalwelding parameters: Q=I (80A) x U (35V) x t (the duration of arc striking=0.1s) x η(0.75)=210, T0=20oC, for a low alloy steel with ρc=0.005Jmm-3 oC-1, a=5mm2s-1):

2 / 3

8

2 / 3

8

2

2 / 3

8

0800)54(005.0

210220

4

0exp

)4(

2800

t at at c

QT T

××

+=⎟⎟ ⎠

⎞⎜⎜⎝

⎛ −+==

π π ρ

9

2 / 3

5

2 / 3500)54(005.0

210220500

t T

××

+==π

st t t 14.036.05.0855 / 8 =−=−=Δ

[Grong, Metallurgical Modelil ing of Welding, 2nd edition, IOM, 1997]

CCT Diagram of AISI 1541

10

artensite FormedAD - hard, brittle,ydrogen cracking



T-2 Tanker USS Ponaganset

Only 3 years old, Ponagansetfractured into two sections while tiedat dockside in Boston in 1947.

“The presence of the sharp cracks inthe arc strike located in a region of high residual stress resulting fromthe fillet weld provided the necessaryconditions for fracture initiation.”

Also, “the failure temperature was° "

11

Transition“ temperature (50°F) of thesource plate”; “the steel was typical

of the World War II productionmaterial which features relativelyhigh carbon and low manganesecontents”.

[Unclassified defense document: Pellini and Puzak, NRL

Report 5920, March 12, 1963, U.S. Naval Research

Laboratory, Washington, D.C.]

Cooling Time/Rate in Resistance Spot Welding

Assume current I = 8kA, voltage drop between two electrodes E = 1.6V,duration t = 0.3s, efficiency η = 0.5, room temperature T0 = 20oC, platethickness d = 2 mm thick, low alloy steel: ρc = 0.005Jmm-3 oC-1, a = 5mm2s-1

Estimate the cooling time from 800 to 500oC:

12

ctile to Brittleansition Temp



Moving Heat Sources

In most welding processes, the heat source moves at a constant speed

along a straight line, and the heat supplied by the power source is constant. A pseudo-steady state temperature field around the heat source will be

developed, which means this temperature field moves in the plate alongwith the heat source. To determine T at any given point, P, we need thefollowing simplifying assumptions:

Steady-state heat flow

Negligible heat of fusion

Constant thermal

13

properties

No heat losses from

the workpiece surface No convection in

the weld pool2-D Model

Assume a thin plate, negligible temperature variations through the thickness, andtwo-dimensional heat flow:

Two-Dimensional Rosenthal’s Solution

d

14

Where, r = (x2+y2)1/2 and K modified Bessel function of the second kind and zero order:

⎟ ⎠

⎜⎝

⎟ ⎠

⎜⎝ −+=

α α π 22exp

2),( 00

rV K

V

kd

QT t r T

( ) ⎥⎦

⎤⎢⎣

⎡ ⋅⋅−

⋅+−−≅ ...

8!3

)531(

8!2

)31(

8!1

11)exp(

2 33

2

22

22

0 Z Z Z

Z Z

z K π

[S. Kou, Welding Metallurgy,

2nd edition, Wiley, 2003]



Assume a semi-infinite workpiece and three dimensional heat flow:

Three-Dimensional Rosenthal’s Solution

15

The temperature solution is:

Where, R = (x2+y2+z2)1/2.

⎟ ⎠ ⎞⎜

⎝ ⎛ +−+=

α π 2)(exp

2),( 0

V R xkRQT t r T

[S. Kou, Welding Metallurgy, 2nd edition, Wiley, 2003]

In two solutions for the temperature field around a moving heat source, aneffective plate thickness compares the thickness of the plate with thedimensions of the weld and indicates which solution is suitable:

Effective Plate Thickness

where Tc is the temperature of interest, usually the melting temperature.

If τ ≤ 0.6, use the thin plate (2D) solution,

0.6 < τ < 0.9, use both and choose the most conservative.

⎟⎟ ⎠

⎜⎜⎝

⎛ −=

net

c

H

T T cd

)( 0 ρ τ

16

τ ≥ 0.9, use the thick plate (3D) solution.



The Rosenthal Equations are based on a number of assumptions that simplify the

mathematical treatment of heat flow in welding. This makes the model relativelyeasy to use; however, there are a number of factors that limit the accuracy of

Rosenthal Model’s Limitations

1. Arc efficiency may not be accurately known.

2. Constant thermal properties are assumed, but really are temperaturedependant.

3. Radiation and convective losses from surface are ignored.

4. Fluid flow is neglected.

5. Complex geometry is ignored.

17

6. Heat is actually put in over an area, affecting predicted weld sizes, heating andcooling rates.

7. In arc processes, voltage and current each have their own roles, voltage affectswidth more and current affects depth more.

Peak temperature in HAZ at a distance (y) from the weld centreline

2-D

Adam’s Equations

00

11

T T H

πρcyd

T T mnet p −+=

−

3-D

Weld width

2-D

0

2

0

1

22

44.51

T T a

Vy

EIV

π ka

T T m p −+

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟ ⎠

⎞⎜⎝

⎛ +=− η

V

a

d T T cV

QW

5

4

2

−

−

=

18

3-D

m

)(5

16

5

8

0

2

T T cV

Q

V

W W

m −=+

πρ

α

[Adams, C.M., Jr., Welding Journal, 37(5): 210s-215s]

[Jhaveri, Moffat, Adams, Welding Journal 41(1): 12s-16s]



Centreline cooling rate

2-D

Adam’s Equations (Continued)

3

0

2

)(2 T T d

ck T

c −⎟⎟ ⎞

⎜⎜⎛

=∂

ρ π

[Adams, C.M., Jr., Welding Journal, 37(5): 210s-215s]

[Jhaveri, Moffat, Adams, Welding Journal 41(1): 12s-16s]

3-D

Centreline cooling time

2-D

net

net

c

H

T T k

t

T 2

0 )(2 −=

∂∂ π

⎥⎦

⎤⎢⎣

⎡

−−

−=Δ

2

0

2

0

2

5 / 8)800(

1

)500(

1

4

) / (

T T ck

d H t net

ρ π

19

3-D

Where Tc is the temperature at which the cooling rate is calculated. For steels, this isusually the pearlite nose temperature on the CCT diagram, when consideringwhether pearlite will form. Note: this equation is only applicable at the weldcenterline in the direction parallel to travel.

⎥⎦

⎤⎢⎣

⎡

−−

−=Δ

)800(

1

)500(

1

2 00

5 / 8T T k

H t net

π

Temperature Fields in Welding 1018 Steel The effect of increasing the weld travel speed can be shown using the 3D

Rosenthal solution.

Increasing the travel speed results in an elongation of the isotherms and weldpool.

Note that the heat input must be increased in order to maintain a constant weldwidth when the travel speed is increased.

[S. Kou, Welding Metallurgy, 2nd edition, Wiley, 2003]

20

Welding speed 2.4 mm/s 6.2 mm/s

Heat input 3200W 5000W



Thermal Cycles in Welding 1018 Steel

The temperature vs. time plot is shown for points at a variety of distances from

the weld centerline. The infinite peak temperature at the origin of the coordinate system is the result

’o e s ngu a r y pro em n osen a s so u ons. [S. Kou, Welding Metallurgy,

2nd edition, Wiley, 2003]

21

Welding speed 2.4 mm/s 6.2 mm/s

Heat input 3200W 5000W

Numerical Simulations Modern computer technology allows numerical solutions (such as finite

element models) for 2-D or 3-D heat flow during welding without mostassumptions of Rosenthal’s analytical solutions required.

This level of complexity needed for most practical problems.

Essential to capture complexity of multi-material problems,temperature-dependent properties, complex geometry, etc.

Limitations:

Too easy to use without real insight into underlying behaviour –“garbage in = garbage out”

22

Each calculation covers one set of material/process conditions –cumbersome and slow to cover large parameter space

Numerical models may give better agreement with experimental results;however, the Rosenthal equations are very simple in application andprovide a good qualitative estimate of what can be expected.



Effect of Power Density Profiles

GTAW:

Heat input = 860 W

Welding speed =5.5 mm/s

23[S. Kou and Y. Le, Metall. Trans., 14A, 1983, p. 2245]

Effects of Electrode Tip Geometry in GTAW

24

Heat input = 860 W

Welding speed =5.5 mm/s

[S. Kou and Y. Le, Metall. Trans., 14A, 1983, p. 2245] GTAW



Power Densities of Heating Sources

Air/fuel

gas flame

Electroslag,

Oxyaetylene

flame, Thermit

Friction

Arc

Resistance Laser beam

Electron beam

25

102 107106105104103

W/cm2

[ASW Handbook, Vol. 8, 1993]

Power Density for Various Welding Processes

26[IIW, The Physics of Welding, 2nd Edition (edited by J.F. Lancaster), 1986]]



HAZ Width as Function of Power Density

27[ASM Handbook, Vol. 6, Welding, Brazing and Soldering, 1993]

GTAW of 3.2-mm-thick AA6061

Thermal cycles[S. Kou, Welding

Metallurgy,

2nd edition,

Wiley, 2003]

28

Fusion boundaries

and isotherms

ME535 1 Thermal Modelling

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