Mathematical modeling of moving heat source shape for submerged arc welding process

ORIGINAL ARTICLE

Mathematical modeling of moving heat source shapefor submerged arc welding process

Aniruddha Ghosh & Himadri Chattopadhyay

Received: 2 March 2012 /Accepted: 20 June 2013 /Published online: 15 August 2013# Springer-Verlag London 2013

Abstract An attempt is made in this paper to find out theanalytical solution of the thermal field induced in a semi-infinite body by a moving heat source with Gaussian distri-bution by selecting appropriate inside volume for submergedarc welding process. Three different types of heat sourceshapes in the form of oval, double ellipsoidal, and conicalforms were considered and compared with the experimentalresult. The study shows that for heat input of submerged arcwelding process, the best suitable heat source shape is in theform of an oval. The study also shows two alternate ways ofpredicting the size of the heat-affected zone.

Keywords Submerged arc welding . Gaussian heatdistribution . Oval heat source shape

1 Introduction

Several critical input variables, e.g., current, voltage, elec-trode diameter, travel speed, wire feed rate, stick out, and thelike are involved in submerged arc welding. Temperaturedistribution during welding depends on these welding pro-cess parameters as heat input is a function of these parame-ters [1]. The study of temperature distribution of weldedplates is very essential for designing submerged arc weldingjoint [2]. An attempt at development of mathematical modelof traveling heat source was made more than 50 years ago.After that, lots of research work has been continuing in thisarea. Initially, two-dimension surface Gaussian heat source

with effective arc radius was adopted to find out temperaturedistribution on welded plates and weld pool geometry [3].This solution was an improved version of estimation temper-ature distribution near the heat source area in (x, y) plane, butthis solution is felt to find out temperature distribution alongthe Z direction. However, this attempt indicated a new direc-tion for finding out temperature distribution on welded plates.Then, an analytical solution for transient temperature distribu-tion for welded joint, based on similar Gaussian heat distribu-tion but different distribution parameters, was introduced [4].These researches [3, 4] are still limited to 2D heat source, sothese solutions are felt to describe the effect of penetration.Recently, this shortcoming has been overcome by considering3D heat source. To estimate the temperature of the weldedjoints with deeper weld bead penetration, Goldak et al. [5] firstdescribed 3D double ellipsoidal moving heat source and cal-culated three-dimensional temperature distribution throughfinite element modeling which could overcome the shortcom-ing of the two dimensional Gaussian model.

Nguyen et al. [6] presented an analytical solution oftransient temperature distribution of a semi-infinite bodysubjected to three-dimensional heat density of semi-ellipsoidal and double ellipsoidal mobile heat source. Verygood agreement between predicted and measured tempera-ture distribution data achieved assuming double ellipsoidalheat source. But there are still some limitations of this ana-lytical solution, i.e., this solution are valid only for identicalradii and heat dissipation of rear and front ellipsoid.Fachinotti and Cardona [7] proposed a semi-analytical solu-tion which was able to overcome the aforesaid limitations.Nguyen et al. [8] again described an approximate analyticalsolution for double ellipsoidal heat source in finite thickplates. Their approximate solution can be directly used forsimulation of welding of finite thick plate without the needfor applying the mirror method as required in a semi-infinitebody. It is an effective tool for finding thermal stress andmicrostructure modeling.

A. GhoshDepartment of Mechanical Engineering, Govt. College of Eng.& Textile Technology, Berhampore 742101, India

H. Chattopadhyay (*)Department of Mechanical Engineering,Jadavpur University, Kolkata 700032, Indiae-mail: [email protected]

Int J Adv Manuf Technol (2013) 69:2691–2701DOI 10.1007/s00170-013-5154-z

Many problems of welding encountered in practice in-volve complicated geometries with complex boundary con-ditions or variable properties, and cannot be solved analyti-cally. In such cases, sufficiently accurate approximate solu-tions can be obtained by computers using numericalmethods. Ravichandran et al. [9] developed a model oftemperature distribution during circumferential arc weldingof spherical and cylindrical components using the finiteelement method and got very good results. Employing manysimplifying assumptions in the mathematical model,Chandra [10] described the extension of Rosenthal’s methodfor the estimation of thermal field in a pipe with a mobileheat source. Many researchers gave importance on experi-mentations for finding out temperature distribution of thewelded plates. Akkus [11] investigated the effect of sheetthickness, current density on speed of cooling, and distribu-tion of temperature in resistance spot welding by experiment.In this work, it is found that thicker sheets have highertemperatures in the weld zone. Heat generation in thewelding zone was affected by current density and sheetthickness. Veenstra and Hults [12] measured the temperaturebetween electrodes for RSW by applying the thermal methodand Bentley et al. [13] investigated the temperature distribu-tion in RSW by applying the metallographic method.Kermanpur et al. [14] experimentally investigated gas tung-sten arc welding temperature distribution of the process forbutt weld. Maheshwari [15] used thermocouples to measurethe temperature at different locations of gas metal arc(GMA)-welded plates. Temperature readings are taken atevery 10-s interval and temperature profiles are generatedfor gas metal arc welding (GMAW) process which may behelpful to optimize the GMAW. In spite of more laboriousand time temperature, distribution obtained by experimenthas some advantages. Researchers [6–8] investigated analyt-ical solution of thermal field based on heat conduction,without considering heat lost through convection and radia-tion. But practically, the three modes of heat transfer occur inwelding process. Gutierrez and Araya [16] conducted thenumerical simulation of the temperature distribution generatedby mobile heat source by the approach of control volume.Here, convection and radiation effect have been considered.

Bianco et al. [17, 18] numerically solved the two- andthree-dimensional models for evaluating transient conduc-tive fields due to moving heat source. Bianco et al. [18]investigated transient three-dimensional temperature distri-bution numerically by COMSOL Multiphysics 3.2.Radiation and convection modes of heat transfer from work-piece surfaces as well as variable thermophysical propertiesare taken into account. They found that surface heat transferstrongly affected the temperature distributions in the work-piece. Assuming constant thermal properties of material,Ohring and Lugt [19] considered radiative heat transfer,evaporation, and viscous stress to find out temperature

distribution. Mundra et al. [20] considered specific heatand thermal conductivity values for solid and liquid metalonly.

From the above, it is clear that heat transfer mechanism ina molten pool is extremely complex and its physics is yet tobe fully understood. Biswas et al. [21] analyzed a numericalmodel with the aid of the finite element package ANSYS forsingle-pass single-sides submerged arc welding of squarebutt joints and the distortion of welded joint was determined.In their study, 2D heat source, heat lost through natural heatconvection is considered. However, actual heat source shapefor submerged arc welding process is of course threedimensional.

Sabapathy and Wahab [22, 23] and Klobčar et al. [24]attempted to find out the thermal field on welded plates forMMA welding process considering flatter and more evenlydistributed heat than Gaussian and found excellent compar-ison with the measured data. Ghosh et al. [25, 26] investi-gated the heat source shape for submerged arc weldingprocess. But in this work, near 10 % error in prediction oftransient temperature distribution was found.

In the preceding section, it has been shown that re-searchers have considered ellipsoidal and double ellipsoidalheat source shape compared with different welding process.In this analysis, an attempt is made to find out the analyticalsolution of the thermal field, induced in a semi-infinite body,by a moving heat source with Gaussian heat distribution withinside volume of oval shape for submerged arc weldingprocess. To our knowledge, for the first time, an oval heatsource shape has been considered here and validated withresults considering submerged arc welding process.

2 Experimental study

MEMCO semi-automatic welding machine with constantvoltage rectifier-type power source with a 1,200-A capacitywas used to join C-Mn steel plates (300×150×20 mm ESABSA1 (E8), 0.315 cm diameter, copper-coated electrode in coilform, and ESAB brand basic fluoride-type granular flux wasused). The experiments were conducted as per the designmatrix randomly to avoid errors due to noise factors. Twopieces of C-Mn steel plates were cut and V groove of 60°angle as per the standards were prepared. The chemical com-position of the work piece material is described in Table 1. Aroot opening of 0.1 cmwas selected to join the plates in the flatposition keeping it electrode positive and perpendicular to the

Table 1 Chemical composition of C-Mn steel work piece(in percent)

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

2692 Int J Adv Manuf Technol (2013) 69:2691–2701

plate. The job was firmly fixed to a base plate and then thesubmerged arc welding was finally carried out.

The welding parameters were recorded during actualwelding to determine their fluctuations, if any. The slagwas removed and the job was allowed to cool down.

Temperatures are recorded at different points of thewelded plates (as shown in Fig. 1) except welding line byinfrared thermometers (OMEGA SCOPE OS524E; temper-ature range, 2,482 °C; accuracy is ±1 % rdg or 2 °C which-ever is greater, resolution 1 °C, response time 10 ms).

The job is cut at three sections of welded plates when fluxhas been used. The samples are prepared by standard metal-lographic process and the average values of the penetration,reinforcement height are measured using digital venire cali-per of least count 0.002 cm. Table 1 depict the weld param-eters of SAW considered in the present work. The measuredvalues of weld dimensions and corresponding welding con-ditions are described in Table 2. In the present analysis, C-Mn mild steel has been used. Table 3 shows the temperature-dependent material properties [27] used for the transient heattransfer and heat-affected zone (HAZ) analysis.

3 Mathematical modeling

A three-dimensional thermal model through approximateanalytical solution is developed in the present work toanalyze the heat transfer and temperature distribution in

submerged arc welding. In the literature, models for bothconduction and convection are available. From the litera-ture survey, it is clear that the heat transfer mechanism in amolten pool is extremely complex and its physics is notwell understood. Although some progress has recently beenmade towards a proper modeling of convective heat trans-fer, these efforts are still directed towards a proper model-ing of convective heat transfer, these efforts are still direct-ed towards simple cases. The various material properties ofmetals in the molten state are also not authenticallyestablished. In arc welding, except for a small volume ofmetal, most portions of work specimen remain in the solidstate. Therefore, a three-dimensional thermal model wasconsidered to analysis the heat transfer and the resultingtemperature distribution.

In developing the thermal model, an attempt has beenmade to accommodate the actual welding conditions as faras possible. The stick-out and electrode diameter is howevernot considered. The heat source model, defining the distri-bution of the heat input due to the welding arc, has animportant effect on the heat distribution pattern in the vicin-ity of the weld zone. It is in this region where the fusion zoneand the heat-affected zone are formed. It is therefore impor-tant to study the effect of heat input distribution on the sizeand shape of these zones. For analyzing the heat flow patternon welded plates, the heat input can be treated as distributedheat source. In the literature, many heat source distributionsthrough Gaussian manner of inside volume ellipsoidal,

Fig. 1 Representation of axesand identification of few points(P1, P2, P3, P4) where readingof temperature was taken

Table 2 Observed values forbead parameters Sl. no. Job no. Voltage

(V)Current(A)

Travel speed(cm/min)

Penetration(mm)

Reinforcementheight (mm)

Bead width(mm)

1 A1 25 350 17 6.70 2.38 17.96

2 A2 35 350 17 3.72 2.34 21.90

3 A3 25 450 17 6.69 3.16 21.00

4 A4 35 450 17 8.26 2.76 30.92

5 B1 25 350 30 5.28 1.00 13.94

6 B2 35 350 30 4.58 1.78 20.12

7 B3 25 450 30 6.60 2.25 15.90

8 B4 35 450 30 7.78 1.94 22.66

Int J Adv Manuf Technol (2013) 69:2691–2701 2693

double ellipsoidal, double central conicoidal, and bell (2D),was described [25]. In the present work, oval heat source areconsidered and found as the most suitable heat source shape.

3.1 Gaussian heat distribution

It was found from the literature [6, 26] that the most suitableheat source is a combination of two semi-ellipsoidal shapes.Semi-major axis of one of these ellipsoids was af another was aras shown in Fig. 2. These combinations of two semi-ellipsoidalshapes was called double ellipsoidal heat source configuration.In the present study, it was found because of the experimentsthat the shape of weld pool geometry was oval. For job B4 inTable 2, the equation of oval weld pool that shape of weld poolgeometry was oval. For job B4 in Table 2, the equation of oval

weld pool geometry was x2

1:32þ y2

1:132� e0:3x ¼ 1 (as shown in

Fig. 3). So the heat source shape for this study was assumed asan oval shape (as shown in Fig. 3) whose equation is ax2+(by2+cz2)emx=1, where, m=0.3, for this study.

Where af+ar=a=semimajor axis, b=semiminor axis, c=-another semiprincipal axis of an ellipsoid whose equation isax2+by2+cz2=1. It can be realized in Fig. 3.Let us consider a fixed Cartesian reference frame x, y, and

z as shown in Fig. 1. Initially, an oval heat source (as shown

in Figs. 2 and 3) in which heat is distributed in a Gaussianmanner throughout the heat source’s volume was proposed.The heat density q(x, y, z) at a point (x, y, z) with in oval shapeis given by the following equation:

q x; y; zð Þ ¼ Ae− ax2þ by2þcz2ð Þ�emxð Þ ð1Þ

Where A is the Gaussian heat distribution parameter and a, b, c,and m are oval heat source parameters.

If Q0 is the total heat input, then

2Q0 ¼Z ∞

−∞

Z ∞

−∞

Z ∞

−∞q x; y; zð Þdxdydz

or A ¼ 2ffiffiffiffiffiffiffiabc

p

π3=2� 1

em24a

� Q0

ð2Þ

The oval-shaped heat distribution equation is:

q x; y; zð Þ ¼ 2ffiffiffiffiffiffiffiabc

p

π3=2� 1

em24a

� Q0e− ax2þ by2þcz2ð Þ�emxð Þ ð3Þ

Table 3 Variation of thermomechanical properties with re-spect to temperature of C-Mnsteel [27]

Temperature(°C)

Thermal conductivity(W/mK)

Specific heat(J/kgK)

Thermal expansioncoefficient (10−6/°C)

Young modulus(GPa)

Poissonratio

0 52 450 10 200 0.28

100 51 500 11 200 0.31

300 46 565 12 200 0.33

450 41 630.5 13 150 0.34

550 37.5 705.5 14 110 0.36

600 36 773.3 14 88 0.37

720 31 1,080 14 20 0.37

800 26 931 14 20 0.42

1,450 29.5 438 15 2 0.47

1,510 29.7 400 15 0.2 0.5

1,580 29.7 735 15 0.00002 0.5

5,000 42 400 15.5 0.00002 0.5

Fig. 2 Double ellipsoidal heatsource described. Doubleellipsoidal heat sourceconfiguration, i.e., it is acombination of two semi-ellipsoidal (one has major axis afand other ar). For this study,combination of semi-ellipsoids isassumed to be an oval shape


Here,

Q0 ¼ I � V � ŋ; ð4ÞV, I, ŋ=welding voltage, current, and arc efficiency, respec-tively. Arc efficiency is taken 0.9 for submerged arc weldingprocess.

The transient temperature field of the oval-shaped heatsource in a semi-infinite body is based on the solution for theinstant point source that satisfied the following differentialequation of heat conduction of fixed coordinates [6].

3.2 Moving double oval heat source problem

Let us consider a heat source located at x=0 at time t=0moves with constant velocity v along the x-axis and heatemitted at a points (x, y, z) at an instant t by the oval heatsource. Mathematical expression of the oval heat source istaken as:

q x; y; zð Þ ¼ 2ffiffiffiffiffiffiffiabc

p

π3=2� 1

em24a

� Q0e− ax2þ by2þcz2ð Þ�emxð Þ ð5Þ

3.3 Induced temperature field

Heat conduction in a homogeneous solid is governed by thelinear partial differential equation

k∇2T þ q ¼ ρcp∂T∂t

ð6Þ

Where T=T(x, y, z, t) is the temperature at point (x, y, z) attime t, q is the heat source, ρ is the density, c is the heatcapacity, and k is the thermal conductivity of the plates ofwelded plates.

The fundamental solution of Eq. (6) is the Green function,i.e.,

G x−x0; y−y0; z−z0; t−t0ð Þ

¼ 1

ρcp 4απ t−t0ð Þ½ �3=2e

−x−x0ð Þ2þ y−y0ð Þ2þ z−z0ð Þ2

4α t−t0ð Þ

�h ið7Þ

Where α=k/(ρc) is the thermal diffusivity.Equation (7) gives the temperature increment at point (x,

y, z) and at instant t due to an instantaneous unit heat sourceapplied at point (x′, y′, z′) at instant t′, assuming the body tobe infinite with an initial homogeneous temperature. Then,due to the linearity of Eq. (7), the temperature variationinduced at point (x, y, z) at time t by instantaneous heatsource of magnitude

q x0; y0; z0; t0ð Þapplied at x0; y0; z0ð Þat time t0 is

q x0; y0; z0; t0ð ÞG x−x0; y−y0; z−z0; t−t0ð Þð8Þ

Assuming that heat has been continuously generated atpoint (x′, y′, z′) from t′=0, the temperature increment at point(x, y, z) at time t is

Z t

0

q x0; y0; z0; t0ð Þ G x−x0; y−y0; z−z0; t−t0ð Þdt0 ð9Þ

If we assume that the heat has been continuouslygenerated from t′=0 throughout an infinite medium, thetemperature increment at any point (x, y, z) and at anyinstant t takes the form

ΔT x; y; z; tð Þ ¼Z t

0q x0; y0; z0; t0ð Þ G x−x0; y−y0; z−z0; t−t0ð Þdx0dy0dz0dt0

ð10Þ

Fig. 3 Comparison of ellipse

( x2

1:32þ y2

1:132¼ 1 ) and oval

( x2

1:32þ y2

1:132� e0:3x ¼ 1 ) shapes.

It was found from the experimentthat the shape of weld poolgeometry is oval and theequation of weld pool geometryfor the B4 job (of Table 2) isx2

1:32þ y2

1:132� e0:3x ¼ 1


Then, the temperature induced by the oval heat sourcedefined by equation is

ΔT x; y; z; tð Þ ¼Z t

0

1

2� Q0

ρcpπ32 4πα t−t0ð½ �3=2

� 2ffiffiffiffiffiffiffiabc

p

π3=2� 1

em24a

� I x � Iy � I z dtf ð11Þ

Finally, by assuming the body was initially at the homo-geneous temperature T0, the temperature field is defined by

T x; y; z; tð Þ−T0 ¼Z t

0

dt0

8ρcpπ32 πα t−t0ð½ �3=2

�ffiffiffiffiffiffiffiabc

p

π3=2� 1

em24a

� Q0 � Ix � Iy � I zdt0 ð12Þ

Where α=thermal diffusivity; cp=specific heat ρ=massdensity; t, t′=time; dTt′=transient temperature change due tothe point heat source dQ at time t′; (x′, y′, z′)=location ofinstant point heat source dQ at time t′.

Here:

I z ¼Z ∞

−∞e

−z−z0ð Þ24α t−t0ð Þ

� �� e− cz02ð Þemx0 Þh i

dz0

¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4πα t−t0ð Þp

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4cαf x0ð Þ t−t0ð Þ þ 1

p � e− cemx0 z24cf x0ð Þα t−t0ð Þþ1

ð13Þ

Iy ¼Z ∞

−∞e

−y−y0ð Þ24α t−t0ð Þ

� �� e− cy02ð Þemx0 Þh i

dy0

¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4πα t−t0ð Þp

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4bαemx0 t−t0ð Þ þ 1

p � e− bemx0 y24bemx0 α t−t0ð Þþ1

ð14Þ

Ix ¼Z ∞

−∞e

−x−x0ð Þ2

4α t−t0ð Þ

� �� e− ax02ð Þh i

� Iy � Izd x0

¼ 4πα t−t0ð Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4bα t−t0ð Þemx þ 1ð Þ 4cα t−t0ð Þemx þ 1ð Þp e

− by2emx

4bα t−t0ð Þemxþ1ð Þ −bz2emx

4cα t−t0ð Þemxþ1ð Þ

� �� e−ax

2

" #� xð Þ

þ4πα t−t0ð Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

4bα t−t0ð Þe−mx þ 1ð Þ 4cα t−t0ð Þe−mx þ 1ð Þp e− by2e−mx

4bα t−t0ð Þemxþ1ð Þ −bz2e−mx

4cα t−t0ð Þemxþ1ð Þ

� �� e

−ax2− 4x2

4α t−t0ð Þ" #

� xð Þ

ð15Þ

Ix has been calculated by applying numerical methodtaking appropriate values of integration upper and lowerlimit. When heat source is moving with constant speed vfrom time t′=0 to t′=t, the increase of temperature during thistime is equivalent to the sum of all the contributions of themoving heat source during the traveling time as:

T x; y; z; tð Þ−T0 ¼Z t

0

1

2� 1

8ρcpπ32 πα t−t0ð Þ½ �3=2


p

π3=2� 1

em24a

� Q0 � I x1 � Iy

� I z d t0 ð16Þ

Where; Ix1 ¼ f x−vt0ð Þ; when Ix ¼ f xð Þð Þ

3.4 Calculation of oval-shaped bead geometry parameters

Let A, B, and C the oval-shaped bead geometry parameters. Ithas been in the literature [2], q (A, 0, 0)=q (0)eαA2=0.05 q (0)

a ¼ ln 20ð ÞA2 ð17Þ

Similarly,

b ¼ ln 20ð ÞB2 ð18Þ

c ¼ ln 20ð ÞC2 ð19Þ

The values A, B, and C can be measured from weld beadgeometry, B=half of the bead width, C=penetration, and


A=half of the major axis of oval shape=1.15×B (experimen-tally found, i.e., through weld pool measurement for sub-merged arc welding process). Experimentally measuredvalues A, B, C are applied to find out the values temperaturedistribution of Eq. 16.

Actually thermal diffusivity, thermal conductivity, and thespecific heat of the material changed with temperaturechange. So, to get better results again, approximate solutionof transient temperature distribution has been developedconsidering variation of thermomechanical properties w.r.t.temperature, which is described below. From Eqs. (1) to (10), it can be written that

Z T

T0

pcpα3=2dT ¼

Z t

0

1

2� 1

π32 4π t−t0ð Þ½ �3=2


p

π3=2� 1

em24a

� Q0 � Ix1 � Iy � I z dt0

Or

Z T

T0

kα1=2dT ¼Z t

0

1

2� 1

π32 4π t−t0ð Þ½ �3=2


p

π3=2� 1

em24a

� Q0 � Ix1 � I y � I z dt0 asα ¼ k

ρcp

� � ð20Þ

or from the data of thermomechanical properties in Table 3and with the help of MATLAB, relations between thermomechanical properties and temperature have been developed

which are described in Table 4. Putting the mathematicalexpression in Eq. (20), we get

Z T

T0

2:7� 10−10 � T 2−1:210−6 � T þ 0:0035� �� −0:01T þ 52ð Þ dT

¼Z t

0

1

2� 1

π32 4π t−t0ð Þ½ �3=2


p

π3=2� 1

em24a

� Q0 � Ix1 � Iy � I z dt 0

Or

−0:0675� 10−12 � T4−T40

�� þ 0:86� 10−8 � T3−T30

� �−31:2� 10−6 � T2−T 2

0

� �þ 0:182 T−T0ð Þ¼

Z t

0

1

8� π92 t−t0ð Þ½ �3=2


p� 1

em24a

� Q0 � Ix1 � I y � I z dt 0

ð21Þ

Calculation of a, b, and c have been done with the helpof Eqs. 17, 18, and 19. Solution of this equation (Eq. 21)has been completed with the help of Horner’s method.The value of m is equal to 0.2 and has been assumed forthis study.

Now the heat input (Q0)is the sum of conductive heat flowand convective heat flow by neglecting radiation. For a blockof dimension dx, dy, dz, and convective heat transfer coeffi-cient h, it can be written that

Heat input Q0ð Þ ¼ ρ� dx� dy� dz� cp � T−T0ð ÞConvectiveheat lost Qconvð Þ ¼ h� dx� dy� T−T0ð ÞSo;

Qconv

Q0¼ h

cpLρas dz ¼ plate thickness ¼ Lð Þ

By putting the material properties, this becomes

Qcond ¼ 0:9Q0

So, Eq. (21) can be modified as:

−0:0675� 10−12 � T4−T40

� �þ 0:86� 10−8 � T 3−T30

� �−31:2� 10−6 � T2−T 2

0

� �þ 0:182 T−T0ð Þ

¼Z t

0

0:9

8� π92 t−t0ð Þ½ �3=2


p� 1

em24a

� Q0 � I x1 � Iy � I z dt 0 ð22Þ

Table 4 Mathematical relation of thermomechanical properties andtemperature

Sl. no Relation with temperaturethermo mechanicalproperty

Mathematical model RegressioncoefficientR2 (%)

1 K (−0.01T+52) 96

2 α1/2 (2.7×10−10×T2

−1.210−6×+0.0035)99


Fig. 4 Comparison of measuredand estimated temperaturedistribution through analyticalsolution at different points onwelded plate; a (25, 10, 0),b (25, 15, 0), c (25, 20, 0),and d (0, 0, 20). In y direction


(T(x, y, z, t) -T0) is considered

Calculation of a, b, and c have been done with the help ofEqs. (17), (18), and (19). The solution of this equation(Eq. (22)) has been completed with the help of Horner’smethod.

4 Results and discussion

With the help of Eq. (22), transient temperature was calcu-lated and compared with measured data which are shown inFig. 4. Comparisons are made among results from doubleellipsoidal, double conical, and oval-shaped heat source.While the trends of temperature variation are similar, it wasfound that the best match with the experimental data was forthe oval source. Such comparisons were done at differentlocations and the oval-shaped source was found to be mostsuitable.

4.1 Prediction of penetration

Weld bead penetration of a structural steel is the regionheated from atmospheric temperature (i.e., 723 °C) to themelting point temperature of welded materials (i.e.,1,464 °C). Putting these values in Eq. (22), penetration(s)can be calculated as y=0, x=vt′, t=t′=travel time of electrode,which are tabulated in Table 5.

4.2 Prediction of weld bead width

Weld bead width of a structural steel is the region heatedfrom atmospheric temperature (i.e., 723 °C) to the tempera-ture the melting point temperature of welded materials (i.e.,1,464 °C). Putting these values in Eq. (22), half of weldbead(s) can be calculated as z=0, x=vt′, t= t′=travel time ofelectrode and after that by multiplying by two bead width(s);the results are tabulated in Table 6.

4.3 Prediction of HAZ width

HAZ width of a C-Mn steel is the region heated from lowercritical temperature (i.e., 723 °C) to the temperature justbelow the melting point temperature of welded materials(i.e., 1,464 °C). Putting these values in Eq. (22) HAZwidth(s) can be calculated as z=0, x=vt′, t= t′=travel timeof electrode which are described in Table 7.

The HAZ can be predicted in an alternative manner as-suming the equation of the oval-shaped heat-affected zoneboundary as A′x2+(B′y2+C′z2) f(x)=1, where A′, B′, and C′are the oval-shaped heat-affected zone boundary parameters.

Heat density at the boundary of oval-shaped beadgeometry=ρ×cp×(Tmelt−T0), where ρ is density of weldedmaterial, cp is the specific heat of this material, Tm is meltingpoint temperature of welded material, and T0 is the atmo-spheric temperature.

Heat density at the boundary of heat-affectedzone=ρ×cp×(Thaz−T0), where Thaz is the lower critical tem-perature (i.e., 723 °C) of the welded material.

So, �cp� Tmelt−T0ð Þρ�cp� Tmelt−T0ð Þ ¼ q 0;B;0ð Þ

q 0;B0;0ð Þ ; leading to, Tmelt–T0ð ÞThaz–T0ð Þ ¼ q 0;B;0ð Þ

q 0;B0;0ð Þ

¼ q 0ð Þ e− ax2þ by2þcz2ð Þ f xð Þð Þð Þ at x¼0;y¼B;z¼0

q 0ð Þ e− ax2þ by2þcz2ð Þ f xð Þð Þð Þ at x¼0;y¼B0;z¼0Table 5 Comparison of predicted and experimental values ofpenetration

Penetration (mm) % error

Experimental values Predicted values

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

Table 6 Comparison of predicted and experimental values of weldbead width

Weld bead width (mm) % error

Experimental values Predicted values

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

Table 7 Predicted HAZ width in millimeter

Slno.

Value ofB

Calculated fromEq. (23)

Calculated fromEq. (22)

Measuredvalue

1 1.1 1.2 1.2 1.12

2 1.3 1.5 1.4 1.33

3 1.3 1.4 1.4 1.32

4 1.9 2.1 2.0 1.92

5 0.9 0.9 0.9 0.91

6 1.2 1.4 1.3 1.22

7 1.0 1.1 1.1 1.01

8 1.4 1.5 1.5 1.56


Finally, Tmelt–T0ð ÞThaz–T0ð Þ ¼ q 0ð Þe−bB2

q 0ð Þe−bB02

As,

Tmelt ¼ 1; 464 �C; Thaz ¼ 723 �C; q 0ð Þe−bB2 ¼ 0:05q 0ð Þand assuming atmospheric temperature is equal to 30 °C, one

can write B0B ¼ 1:1224 leading to (B ′−B)=0.1224×B

In other words,

HAZ width ¼ 0:1224� B ð23ÞWith the help of Table 2 and Eq. (23), HAZ width(s) havebeen calculated and shown in Table 7.

4.4 Validation of predicted data of HAZ width

To validate predicted HAZ width, measurement of hardnesson welded zone and nonwelded zone (as shown in Fig. 5)was carried out. It was found by measuring hardness (as

shown in Fig. 5b,c) that at just below the fusion zone, thehardness values are low. Around 1.56 mm in both sides at justbelow the fusion zone, the hardness values (as shown inFig. 5b) are comparatively low. This low hardness and prom-inent grain growth portion is the HAZ. The experimental valueagrees with the theoretical with maximum error of around 4 %.

5 Conclusions

The study shows that the transient temperature filed forsubmerged arc welding can be solved using analytical meth-od. Among different forms of heat source shapes, it is ob-served that the oval shape predicts a better transient temper-ature history. This work also shows that the heat-affectedzone size could be predicted from the analytical solution withreasonable accuracy.

Fig. 5 Hardness variation ofwelded plate, a hardness valuesare recorded at different gridjunctions of the welded specimen(distance between two gridpoints is equal to 2 mm),b Fringe plot of hard variation ofsubmerged arc welded plates.Here, HAZ width is equal to0.156 or 1.56 mm (for the job-A4, heat input for the process is2.84 kJ/mm), c identification ofdifferent portion of submergedarc-welded plates


References

1. Gunarajan V, Murugan N (2002) Prediction of heat affected zonecharacteristics in submerged arc welding of structural steel pipes.Weld Res 94–98

2. Goldak JA, Akhlaghi M (2005) Computational welding mechanics.Springer, New York, pp 71–115

3. Eager TW, Tsai NS (1983) Temperature fields produced by travel-ing distributed heat sources. Weld J 62(12):346–355

4. Jeong SK, Cho HS (1997) An analytical solution to predict the tran-sient temperature distribution in fillet arc welds. Weld J 76(6):223–232

5. Goldak J, Chakravarti A, Bibby M (1985) A double ellipsoidal finiteelement model for welding heat source, IIW Doc. No212-603-85

6. Nguyen NT et al (1999) Analytical solution for transient tempera-ture of semi-infinite body subjected to 3-D moving heat sources.Weld Res 265–274

7. Fachinotti VD, Cardona A (2008) Semi-analytical solution of thethermal field induced by a moving double-ellipsoidal welding heatsource in a semi-infinite body. Asoc Argent de Mecanica Comput10–13:1519–1530

8. Nguyen NT, Mai YW, Simpson S, Ohta A (2004) Analyticalapproximate solution for double ellipsoidal heat source in finitethick plate. Welding Res 82–93

9. Ravichandran G, Raghupathy VP, Ganesan N (1996) Analysis oftemperature distribution during circumferential welding of cylin-drical and spherical components using the finite element method.Comput Struct 59:225–255

10. Chandra U (1985) Determination of residual stresses due to girth buttwelds in pipes. Trans ASME J Press Vessel Technol 107:178–184

11. Akkus A (2009) Temperature distribution study in resistance spotwelding. J Sci Ind Res 68:199–202

12. Veenstra PC, Hults A (1969) A thermal model of spot weldingprocess. Greve, Eindhoven

13. Bentley KP, Greenwod JA, Knowlson P, Baker RG (1963) Tem-perature distribution in spot welds. Br Weld J 10:613–619

14. Kermanpur A, Shamanian M, Esfahani V, Yeganesh V (2009)Three-dimensional thermal simulation and experimental investiga-tion of GTAW circumferentially butt-welded Incoloy 800 pipes. JMater Proc Technol 199:110–121

15. Maheshwari A, Harish KA, Surendra K and Coolant S (2010)Experimental determination of weld pool temperature and to generate

temperature profile for GMAW, Proc. 2nd International Conferenceon ‘Production and industrial engineering’, NIT, Jalandhar, India,pp 45

16. Gutierrez G, Araya JG (2003) Temperature distribution in a finitesolid due to a moving laser beam. Proc. IMECE2003-43545, 2003ASME Int. Mech. Eng. Congr., Washington, D.C. November 15–21

17. Bianco N, Manca O, Naso V (2004) Numerical analysis of transienttemperature fields in solids by a moving heat source, HEFAT2004,3rd Int. Conf. on Heat Transfer, Fluid Mechanics and Thermody-namics, paper no BN2, 21–24

18. Bianco N, Manca O, Nardini S (2006) Transient heat conduction insolids irradiated by a moving heat source. Proc. COMSOL UserConference 2006 Milano

19. Ohring S, Lugt HJ (1999) Numerical simulation of a time dependent3D GMAweld pool due to a moving Arc. Weld J 79(120):416–171

20. Mundra K et al (1997) Weld metal microstructure calculation fromfundamentals of transport phenomena in the arc welding of low-alloy steels. Weld J 76(4):163–171

21. Biswas P, Mandal NR (2009) Thermomechanical finite elementanalysis and experimental investigation of single-pass single-sided submerged arc welding of C-Mn steel plates. J Eng Manuf224:627–639

22. Sabapathy PN, Wahab MA (2000) The prediction of burn-troughduring in service welding of gas pipelines. Int J Press Vessel Pip77:669–677

23. Sabapathy PN, Wahab MA, Painter MJ (2001) Numerical modelsof in-service welding of gas pipelines. J Mater Process Technol118:14–21

24. Klobčar D, Tušek J, Taljat B (2004) Finite element modeling ofGTAweld surfacing applied to hot-work tooling. Comput Mater Sci31:368–378

25. Ghosh A et al (2011) Prediction of temperature distribution onsubmerged arc welded plates through Gaussian heat distributiontechnique. Adv Mater Res 284–286:2477–2480

26. Ghosh A et al (2011) Prediction of weld bead penetration,transient temperature distribution & HAZ width of submergedarc welded structural steel plates. Defect Diff Forum 316–317:135–152

27. Mahapatra MM, Datta GL, Pradhan B, Mandal NR (2006) Three-dimensional finite element analysis to predict the effects of SAWprocess parameters on temperature distribution and angular distor-tions in single-pass butt joints with top and bottom reinforcements.Int J Pres Vess Piping 83:721–729


Documents

Mathematical modeling of moving heat source shape for submerged arc welding process