Analytical Solutions for Transient Temperature in Semi-Infinite Body Subjected to 3-D Moving Egg Shape Heat

Source

Aniruddha Ghosh1, a, V.V.N.Sriram 2,b and Somnath Chattopadhyaya 3,c 1Assistant Professor in WBGS, ME Dept.,Govt College of Engg. & Textile Technology, Berhampore,

WB, India

2B.Tech Student,ISM,Dhanbad,India

3Associate Professor, ME&MME Dept, Indian School of Mines, Dhanbad, India a agmech74@gmail.com ,

c somuismu@gmail.com,

Keywords: Submerged Arc Welding Process, Gaussian Heat Distribution of Egg shape, Weld pool

geometry, Temperature Distribution.

Abstract. Determination of temperature distribution of submerged arc welded plates is essential while

designing submerged arc welding joint. The key role for the change of weld bead geometry dimension,

thermal stress, residual stress, tensile stress, hardness etc. is heat input. Heat input is the function of

temperature distribution of Submerged Arc Welding (SAW) process. An attempt is made in this paper

to find out the analytical solution for an egg shape heat density moving heat source in a semi-infinite

body with conduction and convection consideration. The solution has been obtained by integrating the

instant point heat source throughout the volume of the egg shape. Very good agreement between the

predicted transient temperatures and the measured ones at various points on submerged arc welded

plates has been obtained. The predicted geometry of the weld pool, HAZ width is also in good

agreement with the measured one. This may pave the way for the future applications of these solutions

in the problems such as microstructure modeling and residual stress/distortions and welding process

simulation.

Introduction

Lot of critical set of input variables i.e. current, voltage, electrode diameter, travel speed, wire feed

rate, stick out etc. are involved in submerged arc welding process[1].Temperature distribution patterns

depend on these welding process parameters because these are the functions of heat input[2].So, shape

of heat distribution is changed with the change of input parameters of SAW process[3].Study of

temperature distribution of welded plates is very essential for designing submerged arc welding

joint[4].An attempt to develop mathematical model of travelling heat source was made more than fifty

years ago[5].After that, lot of research works have been continuing on this area. Initially two

dimensional surface Gaussian heat source with effective arc radius has been adopted to find out the

temperature distribution on welded plates and weld pool geometry [6].This solution is an improved

version of estimation temperature distribution in the near heat source area in (x,y) plane but this

solution is failed to find out temperature distribution along Z direction. However this attempt has

indicated new direction for finding out temperature distribution on welded plates. Then an analytical

solution for transient temperature distribution for welded joint based on similar Gaussian heat

distribution is proposed. Different distribution parameters have also been introduced [7].These[6,7]

parameters are still limited to 2D heat source. But these solutions failed to describe the effect of

penetration. Recently this short coming has been overcome by considering 3D heat source.Goldak

et.al.[8] first described 3D double ellipsoidal mobile heat source and three dimensionally temperature

distribution has been calculated through Finite Element modeling. This could overcome the

shortcoming of the of the two dimensional Gaussian model to estimate the temperature distribution of

the welded plates with deeper weld bead penetration.Nguyen et. al. [10] described an analytical

solution of transient temperature distribution of a body (semi-infinite) subjected to three dimensional

heat density of semi ellipsoidal and double ellipsoidal mobile heat source. Very good agreement

between predicted and measured temperature distribution data achieved when double ellipsoidal heat

source has been assumed. But this analytical solution has some limitations. It is correct when both semi

ellipsoids are equal in size and this solution is valid only when cr = cf and fr =ff =1 where cr,cf are radii

of rear and front ellipsoid respectively and fr (resp. ff) is the portion of the heat deposited in the rare (

resp. front) ellipsoid. Victor et. al.[10] described a semi-analytical solution which has ability to

overcome the aforesaid limitations.Nguyen et. al.[11] again described an analytical approximate

solution for double ellipsoidal heat source in finite thick plates. In this work, they established that

approximate solution can be directly used for simulation of welding of finite thick plate without

applying the mirror method which is required in a semi-infinite body. This analytical approximate

solution is an effective tool for finding thermal stress, microstructure modeling etc.Many problems of

welding are involved with complicated geometries with complex boundary conditions or variable

properties, and cannot be solved analytically. In such cases, sufficiently accurate approximate solutions

can be obtained by computers using a numerical methods.Ravichandran et. al.[12] developed a model

of temperature distribution during circumferential arc welding of spherical and cylindrical components

using the finite element method and got very good results. Chandra [13] described the extension of

Rosenthals method for the estimation of thermal field in a pipe with a mobile heat source by taking

many simplifying assumptions. Ahmet Akkus [14] described the effect of sheet thickness and current

density on cooling rate and distribution of temperature in resistance spot welding through his

experimental results. He found thicker sheets have higher temperatures in weld zone. Heat generation

in welding zone is affected by current density and sheet thickness.Veensra et. al.[15] measured

temperature between electrodes for RSW by applying thermal method and Bentley et.al.[16]

investigated temperature distribution in RSW by applying metallographic method.Kermanpur et.

al.[17] investigated experimentally distribution of temperature of GTAW process for butt weld

joint.Maheshwari et.al.[18] used thermocouples to measure temperature at different locations of GMA

welded plates. Temperature readings have been taken in every 10 seconds interval and temperature

profile has been generated for GMAW process which may be helpful to optimize the GMAW.Inspite of

more laborious and time consuming method, experimentally measured temperature distribution has

some advantages. One may guess the change of output parameters with the change of temperature

distribution accurately. Many researchers investigated analytical solution of thermal field [9, 10, 11]

only considering heat conduction, without considering heat lost through convection and radiation. But

practically three modes of heat transfer occur in welding process.Araya [19] has carried out the

numerical simulation of the temperature distribution generated by mobile heat source through control

volume approach. Here convection and radiation effect have been considered.Bianaco

et.al.[20,21]carried out two numerical methods for two and three dimensional models for evaluating

transient conductive fields due to moving heat source.Bianaco et.al.[22] also investigated transient

three dimensional temperature distribution numerically by COMSOL Multiphysics 3.2.Radiation and

convection modes of heat transfer from work piece surfaces as well as variable thermo physical

properties has been considered here. They found that surface heat transfer strongly affected the

temperature distributions in the worpiece.Ohring et.al.[23] considered radiative heat transfer,

evaporation, and viscous stress to find out temperature distribution but he assumed material properties

are constant.Mundra et.al.[24] considered specific heat and thermal conductivity values for solid and

liquid metal only for analysis of temperature distribution. Heat transfer mechanism in a molten pool is

extremely complex and its physics is not well understood till now[25].Biswas et.al.[25] considered a

numerical model based the finite element package ANSYS for single pass single sides submerged arc

welding of square butt joints. Using it, they determined the distortion of welded joint. In their study,

heat lost through natural heat convection has been considred.Many researchers tried to find out stress

distribution during welding process using different tools. One of them [26] carried out numerical

simulation of arc welding heat transfer and stress distribution for thin plates. For predicting temperature

and stress distribution of thin plates during arc welding, computational fluid dynamics and

computational solid mechanics have been utilized [26]. Postacioghu et.al.[27]investigated thermal

stress, strain during welding using linear theory of elasticity. They considered in their study that the

weld pool is approximated by an elliptic region which is constant in cross sectional shape and depth.

Few researchers [28] investigated thermal analysis of welding on Aluminum plates. For this study, they

considered three modes of heat transfer (i.e.conduction, convection, radiation).They solved the

equation of steady state temperature profiles of welded plates by finite difference method. They

compared calculated and measured temperature near the heat source and obtained good asgreement.In

the year 2005,Ali et.al.[29,30] described theory of relativistic heat conduction and relativistic moving

heat source which have given new direction to researchers for finding out temperature distribution ,

thermal stress etc. due to moving heat source. In this paper an improved version of shape of moving

heat source i.e. egg shape is considered and temperature distribution during submerged arc welding

HAZ width and weld pool dimensions are calculated and compared with experimental values. Finally

thermal stress developed during welding, has been estimated.

Experimental Procedure Results:

The experiments were conducted as per the design matrix randomly to avoid errors due to noise factors.

The C-Mn steel work piece (300x150x20 mm - 2 pieces) is cut and V groove of angle 60o

as per the

standards are prepared. The job was firmly fixed to a base plate by means of tack welding and then the

submerged arc welding was finally carried out. The welding parameters were recorded during actual

welding to determine their fluctuations, if any. The slag was removed and the job was allowed to cool

down. Welding is carried out for the square butt joint configuration. The job is cut at three sections by a

hacksaw cutter and the average values of the penetration, reinforcement height and width are recorded

using digital venire caliper of least count 0.02mm.

Temperatures are recorded at different points of the welded plates by FLUKE 574 infrared

thermometers and with the help of optical research microscope HAZ width(s) are measured.

Figure 1

Bead geometry, P-Penetration, H-Reinforcement height, W-Bead Width

Table 1

Observed Values for Bead Parameters for HAZ analysis

Sl.No. Job

No.

Voltage(V) Current(A) Travel

Speed(cm/

min)

Penetration

(mm)

Reinforcement

Height(mm)

Bead

Width

(mm)

HAZ

width(mm)

1 A1 25 350 17 6.70 2.38 17.96 1.20

2 A2 35 350 17 3.72 2.34 21.90 1.32

3 A3 25 450 17 6.69 3.16 21.00 1.40

4 A4 35 450 17 8.26 2.76 30.92 2.18

5 B1 25 350 30 5.28 1.00 13.94 1.05

6 B2 35 350 30 4.58 1.78 20.12 1.33

7 B3 25 450 30 6.60 2.25 15.90 1.20

8 B4 35 450 30 7.78 1.94 22.66 1.33

Table-2

Chemical composition of C-Mn steel work piece (in %)

C Sn Mn P S Cr Ni Mo Cu Al

0.18 0.36 1.58 0.023 0.027 0.06 0.03 0.01 0.04 0.05

Table-3

Temperature variation of different points on welded plates with respect to time

x

(cm)

y

(cm)

z

(cm)

Temperature(0C)

after 2minutes from the

starting of welding

process

Temperature(0C)

after 4minutes from

the starting of

welding process

Temperature(0C)

after 6 minutes from the

starting of welding

process

12 0 0 85 126 156

9 0 0 164 166 219

6 0 0 349 261 233

3 0 0 210 224 200

0 0 0 127 153 164

12 3.8 0 86 127 164

9 3.8 0 169 179 225

6 3.8 0 404 296 241

3 3.8 0 227 243 206

0 3.8 0 104 143 174

12 7.6 0 74 113 168

9 7.6 0 155 215 234

6 7.6 0 435 311 238

3 7.6 0 254 265 212

0 7.6 0 96 134 171

12 11.4 0 63 110 165

9 11.4 0 193 244 228

6 11.4 0 390 287 249

3 11.4 0 252 274 221

0 11.4 0 77 133 168

12 15.2 0 64 120 161

9 15.2 0 117 225 219

6 15.2 0 305 243 246

3 15.2 0 230 267 204

0 15.2 0 97 156 169

Figure 2

Surface and Contour plot: Temperature variation (0C) after 2minutes from the starting of welding

process of different points on welded plates with respect to time

Figure 3

Surface and Contour plot: Temperature variation (0C) after 2minutes from the starting of welding

process of different points on welded plates with respect to time(contour and surface plot)

Figure 4

Surface and Contour plot: Temperature variation (0C) after 2minutes from the starting of welding

process of different points on welded plates with respect to time

Mathematical Model of Transient Temperature Distribution - Part-A:

Development of Equation of 3D Egg shape:

Figure-5

Curves of 2-D egg shape and elliptical shape are indicated by blue and black colours respectively. 2-D

view of egg shape lies under the ellipse on the right side of the y-axis or negative x axis side.Eqation

of this oval shape curve (curve of blue colour) is assumed to be ax2+by

2 (f(x)) =1.

Figure-6

Gaussian heat density distribution of inside volume of an egg shape at z=0 plane.

Figure 7

Double Ellipsoidal Heat Source Configuration [31] i.e. it is a combination of two semi ellipsoids

(where, af = half of major axis of front portion of elliptical heat source and ar = half of major axis of

back portion of elliptical heat source).

For this study, assumptions are-

Work piece is a semi-infinite body.

Combinations of semi ellipsoids (as shown in fig-7) is assumed to be an oval shape whose 2D view

(curve of blue colour) is shown in fig-5.

af + ar (of fig-7) =a and y axis passes through mid of the 2-D egg shape curve.

Equation of this 3D egg shape will be-

ax2+ (by

2 + cz

2) (f(x)) =1.Where f(x) =

Where a, b, c, m are the oval shape heat source parameters.

Gaussian Heat Distribution:

Figure 8

Sketch of the work piece (semi-infinite body) for submerged Arc Welding process.

Figure 9

Heat density distribution pattern on welded plates, when electrode of submerged arc welding is just

starting to move along welding line

Figure 10

Heat density distribution (in side volume of an egg shape) pattern on welded plates, when electrode of

submerged arc welding is in the middle position of welding line during welding.

Let us consider a fixed Cartesian reference frame x, y, z as shown in fig-8. Initially proposed an egg

shape heat source in which heat is distributed in a Gaussian manner throughout the heat sources

volume. The heat density q(x, y, z) at a point(x, y, z) with in oval shape is given by the following

equation:

q(x, y, z) = q (0) ( )( ))) (1)

[Where q(0) is Gaussian heat distribution parameter and a, b, c,m are egg shape heat source

parameters]

If Q0 is the total heat input, then

2Q0 = )

dxdydz

Or q (0) =

Q0

Oval shape heat distribution equation is

q(x, y, z) =

Q0 )( ))) (2)

Here,Q0=IV ;

V, I, =welding voltage, current and arc efficiency respectively.

Arc efficiency is taken 1 for submerged arc welding process.

Analytical solution: Transient temperature field of egg shape heat source in a semi-infinite body is

based on solution for the instant point source that satisfied the following differential equation of heat

conduction of fixed coordinates [9].

dTt =

)

exp (- ) ) )

)) (3)

Where =thermal diffusivity, cp=specific heat, =mass density; t, t =time; dTt=transient temperature

change due to the point heat source dQ at time t ;( x, y, z) = location of instant point heat source dQ

at time t.

Let us consider the solution of an instant egg shape heat source as a result of superposition of a series

of instant point heat source over the volume of the distributed Gaussian heat source. Substitute eqn.No.

(2) into eqn.No. (3) and integration over the volume of theuheatusourceuofuoval shapes gives

dTt=

)

) ) )

))

(

) )) dxdydz

Let, for the sake of simplicity of above integration procedure

dTt=

)

) ) )

))

(

) )) dxdydz

=

) (4)

Where,

I =

)

))

( ) dx

Iz=

)

))

( ) ))

dz [f (x) = ]

= )

) )

)

) ( )

Iy =

)

))

( ) ))

dy

= )

) )

)

) ( )

So,

I=[ ) ) ) ) )

( ( ) )

( ( ) ))

] (x)

+

[ ) ) ) ) )

( ( ) )

( ( ) ))

)

] (x) When heat source is moving with constant speed v from time t=0 to t=t, the increase of temperature

during this time is equivalent to the sum of all the contributions of the moving heat source during the

travelling time as

T - T0=

)

dt (where =f (x-vt), =f(x)) (5)

Where, T=temperature at any point (x, y, z) on welded plates at any time t during welding and T0

=temperature of plates just before the starting of welding i.e. atmospheric temperature.

Boundary condition:

In this thermal model natural heat convection (i.e. Newtons law of cooling) is applied by considering

hear loss due to convection over the surface.

Figure 11

A very small brick of welded plates whose surface area A, Volume V,coordinate of coordinate of the

centroid of the upper surface (x,y,z)(welded plated are assumed to be simple brick type solids)

Let, during welding process, due to convective heat lost from a small brick of welded plates (fig-8)

temperature is decreased by an differential amount dT in the differential time interval dt .An energy

balance of solid for the time interval dt can be expressed as

(Convective heat transfer from the body during dt)= (change in the energy of the body during dt)

Or, hA (T-T0)dt = VcpdT (h= convective heat transfer coefficient)

Integrating we get

T(x, y, z, t) T0 = (T-T0)e-nt

(6)

Where, T(x, y, z, t) is the temperature at time t at the point(x, y, z) on welded plates, (T-T0) is as

described in equation No-5 and n=

=

as V=Al(l=thickness of welded plates).

It has been found from literature [25] that thermal conductivity (K), specific heat (Cp), young modulus

(E), Thermal Expansion Coefficient ( ) are changed with the change of temperature.

Relation of these material properties with temperature [25]are given below -

K=2.210-5

T2 0.014T +52 (7)

Cp = 0.00081T2 + 1.3T+0.036 (8)

= - 3.110-6 T2 + 0.0079T +10 (9)

E= 39.610-5

T2 -0.3T +230 (10)

From the equation No. 6,7,8,9 & measured temperature data of table No. 3 and with the help of

MATLAB 7.0, values of m, a, b, c have been calculated. These are a=36984.35, b=24758.12,

c=29957.32, m=0.2.

Table 3

Comparison of Predicted and Experimental data of temperature distribution

Time(minute) Measured temperature data(0C) Predicted temperature data(

0C)

0 0 0

2 182 190

4 215 226.5

6 248 261.5 Comparison of Predicted(from eqation No.-6,7,8,9 )and measured Temperature distribution at the point

x=0,y=0,z=0 on the plate for square butt welding of 20 mm thick plate (for B4 job; heat input-

2.84kJ/mm, , =7833kg/m3, h=15W/m2-0C, a=36984.35, b=24758.12, c=29957.32, m=0.2) has been

made.

Figure 12

Comparison of Predicted and measured temperature distribution ( with the help of data(s) of table No-

3)

Prediction of Weld Bead Penetration: Weld Bead Penetration of a structural steel is the region heated

from atmospheric temperature (i.e., 300C) to the temperature the melting point temperature of welded

materials (i.e.14510C). Putting these values in the equation No (6) Penetration(s) have been calculated

at y=0, x=vt, t=t, which are tabulated below.

0

50

100

150

200

250

300

0 2 4 6

Tem

pe

ratu

re(0

C)

Time (in minute)

Measured data

Predicted data

Table 4

Comparison of predicted and experimental values of penetration

Penetration

(mm)

(Experimental Values)

Penetration

(mm)

(Predicted Values)

% error

6.70 7.36 10

3.72 4.1 9

6.69 7.34 9.5

8.26 9.08 10

5.28 5.78 9.8

4.58 5.03 9.9

6.60 7.26 9.8

7.78 8.55 9.9

Figure 13

Comparison of predicted and experimental results for weld bead penetration ( with the help of data(s)

of table No-4)

Prediction of Weal bead Width: Weld Bead width of a structural steel is the region heated from

atmospheric temperature (i.e., 303K) to the temperature the melting point temperature of welded

materials (i.e.1684K). Putting these values in the equation No (5) half of weld bead(s) have been

calculated at z=0, x=vt, t=t, after that multiplying by two bead width(s) have been calculated, which

are tabulated below.

1 2 3 4 5 6 7 8

Series1 6.7 3.72 6.69 8.26 5.28 4.58 6.6 7.78

Series2 7.36 4.1 7.34 9.08 5.78 5.03 7.26 8.55

0123456789

10

We

ld B

ead

Pe

ne

trat

ion

(mm

)

Comparison of Predicted and Experimental results

Table 5

Comparison of predicted and experimental values of Weld Bead Width

Weld Bead Width

(mm)

(Experimental Values)

Weld Bead Width

(mm)

(Predicted Values)

% error

17.96 19.73 9.9

21.90 23.91 9.2

21.00 23.00 9.6

30.92 34.01 10

13.94 15.29 9.7

20.12 22.09 9.8

15.90 17.47 9.9

22.66 24.90 9.9

Figure 14

Comparison of predicted and experimental values of Weld Bead Width ( with the help of data(s) of

table No-5)

Prediction of HAZ width with the help of equation No-6:

The heat-affected zone (HAZ) is the area of base material, either a metal or a thermoplastic, which has

had its microstructure and properties altered by welding or heat intensive cutting operations. The heat

input in the welding process and subsequent re-cooling causes these changes of microstructure in the

neighboring area of the weld. The extent and magnitude of property change depends primarily on the

characteristics of the base material, the weld, filler metal, and the volume and concentration of heat

input by the welding process. The thermal diffusivity of the base material plays a significant roleif the diffusivity is high, the material cooling rate is high and the HAZ is relatively small. Alternatively, a

low diffusivity leads to slower cooling and a larger HAZ. The amount of heat inputted by the welding

process plays an important role as well, as processes like oxyfuel welding use high heat input and

increase in the size of the HAZ. Processes like laser beam welding and electron beam welding give a

highly concentrated, limited amount of heat, resulting in a small HAZ. Arc welding falls between these

1 2 3 4 5 6 7 8

Series1 17.96 21.9 21 30.92 13.94 20.12 15.9 22.66

Series2 19.73 23.91 23 34.01 15.29 22.09 17.47 24.9

05

10152025303540

We

ld B

ead

Wid

th(m

m)

Comparison of predicted and experimental values of

Weld Bead Width

two extremities, with the individual processes varying somewhat in heat input]. HAZ width of a

structural steel is the region heated from recrystalization temperature (i.e., 7000C) to the temperature

just below the melting point temperature of welded materials (i.e.14510C). Putting these values in the

equation No (6) HAZ width(s) have been calculated at z=0, x=vt, t=t, which are tabulated below.

Table-6

Predicted HAZ width(s)

Sl.No. Job

No.

Predicted

HAZ

width(mm)

1 A1 1.50

2 A2 1.90

3 A3 1.89

4 A4 2.20

5 B1 1.12

6 B2 1.60

7 B3 1.30

8 B4 1.90

Table 7

Comparison predicted and measured HAZ width

Sl No. measured HAZ width predicted HAZ width % of error

1 1.20 1.50 0.97

2 1.32 1.90 43.94

3 1.40 1.89 35.00

4 2.18 2.20 0.92

5 1.05 1.12 6.67

6 1.33 1.60 20.30

7 1.20 1.30 8.33

8 1.33 1.90 42.86

Prediction of HAZ width from the concept of variation of heat density on welded plates:

Figure 15

Cross sectional view of oval shape bead geometry (curve-1) and oval shape HAZ boundary (curve-2)

for submerged arc welding process

Let equation of egg Shape Bead Geometry (curve-1) Ax2+ (By

2+Cz

2) f(x) =1, where A, B, C are the

egg Shape Bead Geometry parameters.

It has been found from literature [4] that, q (A, 0, 0) =q (0) =0.05 q (0).

a= )

. Similarly, b=

)

, c=

)

Let equation of egg Shape heat affected zone boundary (curve-2) Ax2+ (By2+Cz2) f(x) =1, where A,

B, C are the egg Shape heat affected zone boundary parameters.

Heat density at the boundary of egg Shape Bead Geometry= cp(Tm T0), where density of welded

material, cp is the specific heat of this material, Tm melting point temperature of welded material, T0 is

the atmospheric temperature.

Heat density at the boundary of heat affected zone = C (Thaz T0), where Thaz is the recrystilization

temperature of welded material.

So, )

) =

)

) ,

Or, )

) =

)

) =

) ( )( )))

) ( )( )))

Or, )

) =

)

)

It has been found from literature that Tm =14640C,Thaz = 690

0C, )

=0.05q (0) and assuming

atmospheric temperature is equal to 300C.

=1.1224 (11)

Table -8: Observed Values for Bead Parameters and calculated HAZ width from equation no-6.

Sl.No. Job

No.

Bead

Width

(mm)

Value of

B

Value of

B HAZ

width(B-B)

1 A1 17.96 8.98 10.08 1.10

2 A2 21.90 10.95 12.29 1.34

3 A3 21.00 10.50 11.79 1.29

4 A4 30.92 15.46 17.35 1.89

5 B1 13.94 6.97 7.82 0.85

6 B2 20.12 10.06 11.29 1.23

7 B3 15.90 7.95 8.92 0.97

8 B4 22.66 11.33 12.72 1.39

Table-9

Comparison predicted and measured HAZ width

Sl No. Measured HAZ width Predicted HAZ width % of error

1 1.20 1.10 8.33 2 1.32 1.34 1.52 3 1.40 1.29 7.86 4 2.18 1.89 13.30 5 1.05 0.85 19.05 6 1.33 1.23 7.52 7 1.20 0.97 19.17 8 1.33 1.39 4.51

Calculation of Thermal Stress along y axis:

Figure 16

Let welded plates are divided many very small parts. Dimension of one part is dx, dy, dz. Due to

change of temperature, force (dF) will be generated on this part as welded plates are clamed during

welding.

So,

dF= )

dxdz [where E = modulus of elasticity]

F= )

Thermal Stress along y axis=

So, thermal stress will be developed along y axis at any time t

=

(12)

(Where L1,L2,th are the length of the welded plates along x direction, length of the welded plates along

y direction, thickness of the welded plates along z direction respectively)

Similarly, Thermal stress along x, z direction can be calculated.

Figure 17

Comparison between Predicted(with the help of eqation No-9,10,12 )and measured Thermal Stress

along y axis.Here temperature value represents the average temperature of four corner points.

Coordinates of these points are (0,0,0),(0.15,0,0),(0.15,0.15,0),(0,0.15,0) at different times.

Mathematical Model of Transient Temperature Distribution Part-B:

Let T(x,y,t)= d0 + d1 x+ d2 x2+ d3 y+ d4 y

2+ d5 t+ d6 t

2 (13)

With the help of measured temperature data of table No-3 ,SYSTAT software and matlab 7.0 , values

of d0, d1 ,d2 ----------- of equation No-13, have been calculated which are tabulated below.

Table 10

Values of coefficients of equation No-13

d0 168.5 d1 59.09 d2 22.99 d3 -5.238 d4 -1.295

d5 -72

d6 0.01

Table 11

Comparison of predicted and mesured temperature values

x

(cm)

y

(cm)

z

(cm)

Measured

temperature(0C)

after 2minutes from the

starting of welding

process

Predicted

temperature(0C)

after 2minutes from

the starting of

welding process

from equation No-

Error (in %)

12 0 0 85 87.55 3

9 0 0 164 170.56 4

6 0 0 349 355.98 2

3 0 0 210 216.30 3

0 0 0 127 135.89 7

12 3.8 0 86 93.74 9

9 3.8 0 169 174.07 3

6 3.8 0 404 428.24 6

3 3.8 0 227 233.81 3

0 3.8 0 104 108.16 4

12 7.6 0 74 76.22 3

9 7.6 0 155 156.55 1

6 7.6 0 435 474.15 9

3 7.6 0 254 259.08 2

0 7.6 0 96 100.80 5

12 11.4 0 63 66.15 5

9 11.4 0 193 206.51 7

6 11.4 0 390 393.90 1

3 11.4 0 252 259.56 3

0 11.4 0 77 80.85 5

12 15.2 0 64 65.28 2

9 15.2 0 117 118.17 1

6 15.2 0 305 326.35 7

3 15.2 0 230 232.30 1

0 15.2 0 97 99.91 3

Goodness of fit for equation No. 13

The sum of squares due to error (SSE) =5.481104

R-square = 0.7849

Adjusted R-square = 0.7283

Root mean squared error (RMSE) = 53.71

So mathematical model of Transient Temperature Distribution Part-B is adequate.

Conclusion

1. Shape of heat distribution on welded plate is egg shape for Submerged Arc Welding process and heat

source parameters of this heat source can be measured from the dimension of bead geometry.

2. Transient temperature distribution on welded plate can be calculated with the help of Gaussian Oval

shape Heat distribution technique.

3. In this study, analytical solutions for the transient temperature field of a semi infinite body

subjected to 3-D power density moving heat source (such as oval shape heat source, which is

first time attempted in this work) were found and experimentally validated. The analytical

solution for oval shape heat source was used to calculate transient temperatures at selected

points on welded plates which are welded by taking x- axis along welding line, origin is starting

point of welding, y-axis is perpendicular to welding line and z-axis towards plate thickness.

Both numerical and experimental results from this study have showed that the present analytical

solution could offer a very good prediction for transient temperatures near the weld pool, as

well as simulate the complicated welding path. Furthermore, very good agreement between the

calculated and measured temperature data indeed shows the creditability of the newly found

solution and potential application for various simulation purposes, such as thermal stress,

residual stress calculations and microstructure modeling.

4. Bead geometry dimensions have been calculated with the help of analytical solution (described

in this paper). Very good agreement between the calculated and experimental values has been

achieved.

5. Prediction of HAZ width has been made with the help of three dimension transient temperature

distribution equation .It is also new technique which is not previously applied.

6. Good agreements between calculated measured HAZ width(s) have also been found for this case.

7. Good agreements between calculated (from equation No-13) and measured temperatures have also been found.

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