Heat and Mass Transfer during welding

Energy -> Conduction of heat

Material -> Diffusion of atoms, vacancy, ect…

Diffusion equation

Conduction of heat (1)

Fourier's Law

Heat flux (J/m2s)

Thermal conductivity (J/sKm, W/Km)

Temperature (K)

Diffusion equation

Specific heat (J/kgK)

Density (kg/m3)

Diffusion coefficient (m2/s)

Diffusion equation for Cartesian coordinate system

2

2

2

2

2

2

z

T

y

T

x

TD

t

T

T grad q

TDTcct

T grad div grad div div

1

q

cD

Conduction of heat (2)

Solutions for special boundary conditions

• Steady state (t=), one dimensional

• Steady state (t=), axially symmetric

• Steady state (t=), center symmetric

02

2

x

TD

t

T

01

2

2

r

T

rr

TD

t

T

02

2

2

r

T

rr

TD

t

T

21)( CxCxT

21 ln)( CrCrT

21

1)( C

rCrT

Conduction of heat (3)

D (m2/s):thermal diffusion coefficient

(kg/m3):density, and

c (J/(K kg)):specific.

Basic solutions of diffusion equation

• one dimensional area heating; Q* (J/m2)

• two dimensional linear heating; Q* (J/m)

• three dimensional point heating; Q* (J)

as a function of time t and position (x,y,z),

for the heat input at the position (x h z).

Dt

x

Dtc

QT

4exp

4

12

x

Dt

yx

Dtc

QT

4exp

4

122

hx

Dt

zyx

Dtc

QT

4exp

4

1222

2/3

zhx

2

2

2

2

2

2

z

T

y

T

x

TD

t

T

(x,h,z )

xy

z

(x,h,_)

xy

z

(x,_,_)x

z

y

Conduction of heat (4)

D (m2/s):thermal diffusion coefficient

(kg/m3):density, and

c (J/(K kg)):specific.

Basic solutions of diffusion equation

• one dimensional area heating

for heat inputs

Q* (J/m2)

• two dimensional linear heating

Q* (J/m)

• three dimensional point heating

Q* (J)

as a function of time t and position (x,y,z),

for the heat input at the position (x h z).

Dt

x

Dtc

QT

4exp

4

12

x

Dt

yx

Dtc

QT

4exp

4

122

hx

Dt

zyx

Dtc

QT

4exp

4

1222

2/3

zhx

-20 -10 0 10 20

X-AXIS

0

0.2

0.4

Y-A

XIS

O Ar

Laser beam

2

2

2

2

2

2

z

T

y

T

x

TD

t

T

Dt

zyx

Dtc

QT

4exp

4

1222

2/3

zhx

Exercise 1 Instantly energy is inputted onto a point O on the

surface of a semi-infinite body. The temperature is measured at another point A on the surface of the body. The distance OA is r (m) and the energy input is Q (J). Thermal diffusion coefficient, density, and specific heat of the body are D (m2/s), (kg/m3), and c (J/(K kg)). Write a function of the temperature change at the point A as a function of time t (s), under the assumption that the material never melt nor evaporate,

and 100% of the energy input is absorbed in the body.

Exercise 1’

The heat source moves from t=tstart (s) to tend (s). The distance OA is r (m), the energy input is q (J/s), and the velocity is v (m/s). Thermal diffusion coefficient, density, and specific heat of the body are D (m2/s), (kg/m3), and c (J/(K kg)). Write a function of the temperature change at the point A as a function of time t (s) in a style of integral.

O

A

vLaser beam

r

q

vO

A

v

r

q

v

t=0t=t t=-

Exercise 2 Instantly energy is inputted onto a point O on the

surface of a thin plate. The temperature is measured at another point A on the plate, whose thickness is h (m). The distance OA is r (m) and the energy input is Q (J). Thermal diffusion coefficient, density, and specific heat of the body are D (m2/s), (kg/m3), and c (J/(K kg)). Write a function of the temperature change at the point A as a function of time t (s), under the assumption that the material never melt nor evaporate, and temperature distribution in depth direction can be negligible because of small thickness and no radiant energy loss.

You can use below eq..

O A

r

Laser beam

h

lengthunit per input heat theis (J/m) where

4exp

4

122

Q

Dt

yx

Dtc

QT

hx

O

A

v

r

q

v

t=0t=t t=-

h

Exercise 2’

The heat source moves from t=tstart (s) to tend (s). The distance OA is r (m), the energy input is q (J/s), and the velocity is v (m/s). Thermal diffusion coefficient, density, and specific heat of the body are D (m2/s), (kg/m3), and c (J/(K kg)). Write a function of the temperature change at the point A as a function of time t (s) in a style of integral.

Exercise 3 Instantly energy Q (J) is inputted at a cross section

(x=x) of the rod. The are of the cross section is S (m2). Thermal diffusion coefficient, density, and specific heat of the rod are D (m2/s), (kg/m3), and c (J/(K kg)). Write a function of the temperature distribution as a function of time t (s) and position x (m), under the assumption that the material never melt nor evaporate, and temperature distribution in the cross section can be negligible because of its small area and no radiant energy loss.

You can use below eq..

areaunit per input heat theis )(J/m where

4exp

4

1

2

2

Q

Dt

x

Dtc

QT

x

S

0 xx

Exercise 3’

Instantly energy is distributed at the half part (x>0) of the rod. The distributed energy density is q (J/m). Thermal diffusion coefficient, density, and specific heat of the rod are D (m2/s), (kg/m3), and c (J/(K kg)). Write a function of the temperature distribution as a function of time t (s) and position x (m).

Dt

x

Dtc

SQT

4exp

4

12

x

S

0 xx

S

0 xx

0 x

x

Heat input

q (J/m)

x

qx (J)

Dt

x

c

Sq

Dt

x

c

SqT

4erfc

2

1

4erf1

2

1

erferf

erf1erf1erfc

Temperature (K) of the part (x>0) at t=0 ! q/S (J /m3) :energy input per unit volume <- q (J/m), and S (m2) c (J/K /m3) :specific heat per unit volume <- (kg/m3), and c (J/(K kg)). :normalized distance <- D (m2/s), and t (s)

-4 -2 0 2 4

Distance x (m)

0.0

0.2

0.4

0.6

0.8

1.0

Fun

cti

on

=01 2

3

10

100

xerfc

2

1 Dt

x

4

Exercise 4 Please see below fig.. Instantly energy Q (J) is

inputted onto the surface O of a terminal of the rod, whose cross section is S (m2). The temperature is measured at a plain A. The distance OA is r (m). Thermal diffusion coefficient, density, and specific heat of the rod are D (m2/s), (kg/m3), and c (J/(K kg)). Write a function of the temperature change at the plain A as a function of time t (s), under the assumption that the material never melt nor evaporate, and temperature distribution in the cross section can be negligible because of the small cross section and no radiant energy loss.

You can use below eq..

r

AO

S

areaunit per input heat theis )(J/m where

4exp

4

1

2

2

Q

Dt

x

Dtc

QT

x

Exercise 4’

Continuously, the energy is inputted onto the surface O from t=0 (s). The energy input is q (J/s). The temperature is measured at a plain A. The distance OA is r (m). Thermal diffusion coefficient, density, and specific heat of the rod are D (m2/s), (kg/m3), and c (J/(K kg)). Write a function of the temperature change at the plain A as a function of time t (s).

r

AO

S

Dt

r

Dtc

SQT

4exp

4

12 2

t0

Heat input

q (J/s)

t-

q (J)

A g - C u p h ase d i ag r am .

A ssessed C u -N i p h ase d iag r am . L iqu idus and solidus ar e obtain ed f rom th e ex per im en tal data of

[71B as], [71F ee], an d [71Sch ]. T h e low -tem p er atu r e p or t ion of th e d iagr am is ob tain ed f r om

th er m ody n am ic m odel in g.

Cu/Ni Cu/Ag

700 �

Cu (Ni or Ag)

Cu Ag

Cu Ni