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Modelling of heat affected zone in cylindrical steel elements surfaced by welding
Applied Mathematical Modelling 36 (2012) 1514–1528
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Modelling of heat affected zone in cylindrical steel elements surfaced by welding Jerzy Winczek ⇑ Institute of Mechanics and Machine Design Foundations, Technical University of Czestochowa, Dabrowskiego 73, 42-200 Czestochowa, Poland
a r t i c l e
i n f o
Article history: Received 14 March 2010 Received in revised form 29 August 2011 Accepted 1 September 2011 Available online 10 September 2011 Keywords: Temperature field Welding Surfacing Heat affected zone Modeling
a b s t r a c t Computational models of a temperature field in cylindrical steel elements surfaced by the following methods: controlled pitch, spiral welding sequence and spiral welding sequence with swinging motion of the welding head are presented in the paper. The lateral surface of regenerated cylindrical object, subjected to the welding heat source, has been treated as a plane rolled on cylinder and temperature field of repeatedly surfaced plain massive body was solved. Temperature rises, caused by overlaying consecutive welding sequences and self-cooling of areas previously heated, were taken into consideration in the solution. The computations of the temperature field for continuous casting steel machine roll made of 13CrMo4 steel were carried out. Ó 2011 Elsevier Inc. All rights reserved.
1. Introduction Surfacing is becoming more and more frequently used technological process for regeneration of machines parts, especially bulky ones and those undergoing intensive wear and tear or frequent damages. One of the method of surfacing rolled or cylindrical details is to associate rotational motion of an object with fed motion of the welding head along the lateral surface. Considering the way of this association three following methods described by Pilarczyk and Pilarczyk [1], McLoughlin [2] and Klimpel [3] can be distinguished: a controlled welding sequence pitch – with jumping travel after every full turn of the roll (Fig. 1(a)), with constant travel giving spiral path of welding sequences (Fig. 1(b)) and with constant travel of welding head along the axis of subject with simultaneous swinging motion of electrode across the welding sequence (Fig. 1(c)). In the modeling of temperature fields in welding processes two approaches prevail: numerical (mainly using finite element method FEM) and analytical (using integral transformation method and Green’s function). As an analytical solution to the heat conduction equation offers faster assessment of the temperature field, this method has found many supporters. Research into moving heat source was first presented by Rosenthal [4]. He obtained the temperature distribution equations for point, line and plane heat sources. Eagar and Tsai [5] modified Rosenthal’s model introducing 2-D Gaussian distributed heat source and formulated solution to the moving heat source on semi-infinite steel plate. Transient temperature distribution in gas tunsten arc welding (GTAW) was analyzed by applying three-dimensional model of the finite element by Na and Lee [6]. The solution to the temperature field developed by a ring heat source was presented, where intensity of heat flux at distance from arc center was determined with reference to the parameter defined as radius at which intensity of arc power falls to 5% of maximum intensity. Tekriwal and Mazumder [7] proposed finite element method (FEM) transient ⇑ Fax: +48 34 3605183. E-mail address: [email protected] 0307-904X/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.apm.2011.09.032
J. Winczek / Applied Mathematical Modelling 36 (2012) 1514–1528
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Nomenclature a A1, A3 c cq d h hlc l q q_ v t T t0 y0 z0 k
j q
thermal diffusivity (m2 s1) critical temperatures of the austenitisation specific heat (J/kg K) thermal capacity (J/K m3) diameter of roll (m) weld bead pitch (m) height of weld face (m) length of weld (m) power of hest source (W) individual power of voluminal heat source (W/m3) time (s) temperature (K, °C) time which has passed since initiating heat source (s) coordinate of the beginning of the weld (m) depth of heat source deposition (m) coefficient of heat conduction (W/mK) arc efficiency density (kg/m3)
temperature model in gas metal arc welding (GMAW). The model included thermal properties depending on temperature, effect of latent heat, convective and radiative boundary conditions. In this solution double ellipsoidal heat source model proposed by Goldak et al. [8] was used. Bo and Cho [9] derived an analytical solution to the transient temperature field in plate of finite thickness during single pass arc welding. The moving heat source with Gaussian distribution connected to an electric arc was assumed, likewise by Eagar and Tsai. Temperatures in a single-pass butt-welded pipe was studied by Karlsson and Josefson [10] using full 3D finite element model in pipe made of carbon-manganese steel. Thermal material properties, such as: heat capacity, thermal conductivity and convection, was used in this model. Parkitny et al. [11] calculated temperature field in the welded cylinders with rectangular cross sections. Algorithm was based on the method of elementary heat balances. Melting of electrode and parental material as well as the solidification of molten pool was considered. Mundra et al. [12] presented the computational analysis of the detailed prediction of heat transfer, phase change and fluid flow during welding with moving heat source. Basic equations were formulated in a reference frame connected to the heat source. Initially unknown velocity was the convective velocity of the fluid with motion of the heat source. This leads into additional terms in equations solved using control-volume-based computational method. The analytical solution to temperature field in fillet arc welds was derived by Jeong and Cho [13] for infinite plate of finite thickness bent under right angle. The analytical 3-D model of temperature field in semi-infinite body caused by 3-D moving heat source was formulated by Nguyen et al. [14] and experimentally validated by measurement of the transient temperature at various points in bead-on-plate specimens and in the middle of weld pool. Double ellipsoidal heat source proposed by Goldak et al. [8] was applied to the solution. Komanduri and Hou [15] presented FEM thermal analysis of arc welding process. The models of ring heat source and the models of disc heat source with pseudo-Gaussian distribution of heat intensity were applied. Data used in investigation were essentially the same as ones used by Tekriwal and Mazumder [7]. The numerical simulations of temperature field for Vgroove joint were carried out. Nguyen et al. [16] presented analytical approximate solution to single- and double-ellipsoidal model of the heat source for the modeling of temperature field in the finite thick plate. Authors also proposed solution in dimensional form for the ellipsoidal heat source. 3-D FEM analysis of influence prediction of the effect of shielded metal arc welding process parameters, such as: electrode diameter, speed of travel, current and voltage, on the temperature
Fig. 1. Surfacing methods of cylindrical elements: (a) controlled pitch, (b) with feed motion of welding head, (c) with feed motion of welding head and swinging motion of electrode.
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distribution were investigated by Mahapatra et al. [17]. A comparison of the results of numerical simulations and measured fusion zone and heat affected zone (HAZ) boundary for butt joint and one-sided fillet welds were presented. Kumar and Debroy [18] proposed three-dimensional model of numerical heat-transfer and fluid-flow to capture effects of tilt angle of fillet joint and welding positions on the temperature profiles, velocity fields, weld pool shape and weld pool free surface profile. Basic equations of conservation of mass, momentum and energy were solved using boundary fitted curvilinear coordinate system. Surface deformation of weld pool was calculated by minimizing the total surface energy. Wang et al. [19] used space–time FEM method to analyse transient thermal cycle during welding by solving transient convection–diffusion thermal equation. In the proposed method coordinate frame (Eulerian frame) is moving, while heat source is motionless. The model takes into account an addition of filler material during thermal analysis. Bibliography dealing with the multiple welding is rather meagre. Reed and Bhadeshia [20] proposed model describing thermal cycles occurring in multipass welds. In this method weld bead reinforcement was calculated and heat-flow equation for weld isotherms was used. The model was calibrated for multipass weld of steam-pipes. The method was used to theoretical investigation of the effect of interpass temperature, welding current and Ac3 temperature on the fraction of weld microstructure which becomes reaustenitised during fabrication of the weldment. Taljat et al. [21] presented numerical analysis (FEM) of own stresses caused by the field temperature during surfacing by spiral welding sequence. Fassani [22] compared thermal cycles during multipass gas metal arc welding (GMAW) calculated for models of point Rosenthal’s and one-dimensional Gausian distribution heat source. The analytical solution was presented to the one-dimensional problem – temperature distribution perpendicular to the welding direction. Comprehensive study of 3D temperature distribution in a thick tee made of stainless steel (316L) during multipass welding process was presented by Jiang et al. [23]. The computations were carried out in ABAQUS programe, where thermal physical properties (thermal conductivity, specific heat, density), dependent on temperature, were used in the thermal analysis. The 3D thermal model and ABAQUS were used by Deng and Murakawa [24] to calculate temperature field to predict welding residual stresses in multipass butt welded (metal inert gas (MIG) and metal active gas (MAG) welding processes) of pipes made of 9Cr-1Mo steel. The heat source was modeled as the surface heat source with a Gaussian distribution. Conductivity and specific heat were dependent on temperature. Influence of latent heat of phase transformations was taken into consideration. In computational procedure to analyse the temperature fields in multipass welds (tungsten inert gas (TIG) welding process) of pipe made of stainless steel (SUS304), Deng and Murakawa [25] applied volumetric heat source with a double ellipsoidal distribution proposed by Goldak as the heat of moving welding arc. Results of the numerical simulation were compared to experimental data from thermo-couples. More publications concern results of experimental research into the multipass welds. Murugan et al. [26] carried out an experimental analysis of the temperature field during multipass welding of 304 steel plates. The temperature curves at different distances from the weld path center line were plotted. The method of manual metal arc welding was chosen for experiment. Wojnowski et al. [27] measured weld thermal history (peaks of temperature), microhardness and carried out metalographic analysis of softened region in the samples made of CrMoV steel after multipass welding. Murugan et al. [28] measured thermal cycles (peaks of temperature attained during weld passes in low carbon steel) and residual stresses (used X-ray diffraction method) during each pass of welding. Kolhe and Datta [29] presented analysis results of microstructure and mechanical properties (Rockwell hardness, Vickers microhardness and impact energy) after multipass by submerged arc welding (SAW) method. The swinging motion of electrode was not taken into consideration in the quoted above works. Spiral welding with swinging motion of the welding head is becoming more and more popular welding method of cylindrical elements because of it high efficiency. A knowledge of changeable in time temperature field, similarly to welding with the use of controlled pitch and spiral welding technique, is fundamental for determining heat affected zone. It constitutes also a point of departure for calculations (on the basis of temperature changes velocity during cooling) of phase transformations as well as stress states. Thereby, it has not only cognitive significance, but also practical. 2. Analytical solution to the temperature field Performance of the concentrated moving heat source causing the temperature field changeable in time and space is the characteristic feature of surfacing by welding. Analytical description of this temperature field is obtained using equation of heat conduction in motionless Cartesian coordinate system (x, y, z):
rðkrTÞ þ q_ v ¼ cq
oT : ot
ð1Þ
Basically, to find description of the temperature field, one can use solution to basic Eq. (1) for infinite body with transient heat source Q applied in any point of the body with coordinates (x0 , y0 , z0 ) [30–32]:
Tðx; y; z; tÞ ¼
Q cqð4p atÞ3=2
ðx x0 Þ2 þ ðy y0 Þ2 þ ðz z0 Þ2 exp 4 at
! ð2Þ
The solution to temperature field, caused by the moving volumetric heat source with Gaussian distribution of the density power and traveling speed v [m/s] in fixed system of coordinates (x, y, z), was presented by Winczek [33]. The following
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assumptions were made: the material is homogenous and isotropic, physics-thermal properties do not depend on temperature, phase transformations are not taken into consideration, and during electrical welding process Joule heat are omitted. In the case of cylindrical elements with sufficiently big diameter (wall thickness > 25 mm – Mysliwiec [30] the surface subjected to welding heat source can be treated as a plane rolled on a cylinder. Then, the solution to the multiple welded plain elements described by Winczek [34,35] can be used to describe the temperature field changeable in time and space. In calculations of the temperature field during making n weld bead, temperature-rises connected with next transition of welding heat source DTH and self-cooling of already overlaid padding welds as well as already heated areas DTC were taken into consideration:
Tðx; y; z; z0 ; tÞ T 0 ¼ DT C þ DT H :
ð3Þ
2.1. Temperature field during welding using a conventional method of controlled weld pitch After unrolling the lateral surface of cylinder (Fig. 2), the temperature field in the point P with coordinates (x, y,z) caused by welding heat source with power q, depth z0 and traveling velocity v, at the time t 6 tk, where tk is time from the beginning to the end of kth weld welding, can be described as:
Tðx; y; z; z0 ; tÞ T 0 ¼
m¼2 X
j¼k1 X
m¼1
j¼1
AC
Z
tj
0
0
F C ðt Þdt þ Ak
Z
!
t 00
F H ðt Þdt
00
;
ð4Þ
t k1
t j1
while at time t > tk:
Tðx; y; z; z0 ; tÞ T 0 ¼
j¼k m¼2 XX m¼1 j¼1
AC
Z
tj
0
F C ðt 0 Þdt ;
t j1
where:
AC ¼
3 q ; 8 cqpaz0
Ak ¼
3 q v n v 2 t0 ; exp m 8 cqpaz0 2a 4a
Fig. 2. Schematic diagram for describing the temperature field during welding of cylindrical element using controlled weld pitch method.
ð5Þ
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nm ¼
pd pd þ
2
2
ð1Þm xð1Þm v ðt þ t0 Þ;
WH ðz0 ; t 00 Þ
00
F H ðt Þ ¼
t 00 þ t 0
p d
WC ðz0 ; t 0 Þ
F C ðt0 Þ ¼
WH ðz0 ; t Þ ¼
2
exp
t þ t0 t0
00
! n2m v 2 t00 ; exp 4aðt 00 þ t0 Þ 4a
z2 þ 2at00 1 z20
þ p2d ð1Þm xð1Þm v t 0 4aðt þ t0 t0 Þ
UðzÞerf UðzÞ
0;5
2ðat00 Þ
4at 00 z þ z0 ðz z0 Þ2 þ 2 exp 0;5 00 4at 00 z0 ð4pat Þ
WC ðz0 ; t0 Þ ¼ 1
z2 þ 2aðt t 0 Þ z20
erf
þ ðy y0i Þ2
!
z þ z0
erf
2
!
; !!
0;5
2ðat 00 Þ
0;5
ðz þ z0 Þ2 exp 4at 00
!
z þ z0 2ðaðt t 0 ÞÞ
z z0
z z0 ð4pat 00 Þ
!
UðzÞerf UðzÞ
0;5
; !!
z z0 2ðaðt t 0 ÞÞ
0;5
!! 4aðt t 0 Þ z þ z0 ðz z0 Þ2 z z0 ðz þ z0 Þ2 þ exp exp 0;5 0;5 4aðt t 0 Þ 4aðt t 0 Þ z20 ð4paðt t0 ÞÞ ð4paðt t0 ÞÞ
UðzÞ ¼
1 for z 2< 0; 1
z0 >
for z 2< z0 ;
1>
!
!!
;
t00 = t t0 , whereas t means current time of the total welding process. Quantity t0 characterizes the surface distribution of heat source, while r2B ¼ 4at 0 [36]. The total time from the beginning to the end of k-weld roll welding is described by relationship:
tk ¼ kðl=v Þ þ ðk 1Þtp ; and coordinate y0i of the beginning of i-weld can be calculated according to the formula:
y0i ¼ y01 þ ði 1Þh; where tp – idle time after making i-weld (time of idle motion of electrode with h pitch to the beginning of i + 1 weld), y01 – the coordinate of the beginning of the first weld roll, l – the length of weld defined by relationship:
l ¼ pðd þ 2hlc Þ: where d, diameter of the roll, hlc, height of face of weld. 2.2. Temperature field during a spiral welding After unrolling lateral surface of cylinder (Fig. 3) the temperature field in point P with coordinates (x, y,z) at the distance g from the source along the y axis and n1 and n2 along the x axis during kth roll weld can be described by following equations: – for time t 6 tk where t k ¼ tj¼k is the total time from the beginning to the end of kth weld welding:
T ðx; y; z; z0 ; t Þ T 0 ¼
j¼k1 X m¼2 X j¼1
m¼1
AC
Z
tj
0
F C ðt 0 Þdt þ
t j1
m¼2 X m¼1
Ak
Z
t
00
F H ðt00 Þdt ;
t k1
where
F H ðt00 Þ ¼
0
F C ðt Þ ¼
AC ¼
WH ðz0 ; t 00 Þ t 00 þ t 0
WC ðz0 ; t 0 Þ t þ t0 t0
3 q ; 8 cqpaz0
exp
! n2m þ g2 v 2 t00 ; 4aðt 00 þ t0 Þ 4a
p d exp
Ak ¼
2
! 2 2 þ p2d x þ v cos b t0 ð1Þm þ ðy v sin b t 0 y0i Þ ; 4aðt þ t 0 t 0 Þ
3 q v ðnm cos b þ g sin bÞ v 2 t0 exp ; 8 cqpaz0 2a 4a
ð6Þ
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Fig. 3. Schematic diagram for describing temperature field during spiral weld.
nm ¼
pd 2
þ
pd 2
x þ v cos b ðt þ t 0 Þ ð1Þm ;
g ¼ y v sin bðt þ t0 Þ y0j ; tj ¼ jðl=v Þ; v, heat source velocity; y0i is the coordinate of the beginning of ith weld, l, length of the weld described by relationship:
l¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 c2w þ h ;
cw ¼ pðd þ 2hlc Þ;
and the angle between x axis and direction of heat source motion b = arcsin(h/l), – at time t > tk the temperature field is obtained in the following form:
Tðx; y; z; z0 ; tÞ T 0 ¼
j¼k X j¼1
AC
m ¼2 X m¼1
Z
tj
0
F C ðt 0 Þdt :
ð7Þ
t j1
2.3. Temperature field during spiral welding with swinging motion of welding head The velocity of heat source in relation to the object in the considered welding method is resultant of three velocities: the fed motion of welding head along the lateral surface of roll vg, swinging motion of electrode w and tangential velocity v of the electrode in relation to the rotating welded metal. Increases in temperature connected to already made welds (the cooling of earlier heated areas during welding) must be taken into account in the temperature field computations. After unrolling the lateral surface of cylinder (fig. 4) the temperature field in point P with coordinates (x,y, z), g meters away from the source along the y axis and n1 and n2 along the x axis during n, weld and k, swing of electrode can be described by the following equations: – at time t 6 tc, where tc total time for welding:
T ðx; y; z; z0 ; tÞ T 0 ¼
m¼2 X
j¼k1 X
m¼1
j¼1
AC
Z
jt p
ðj1Þtp
0
0
F C ðt Þdt þ AH
Z
!
t
jt p
00
F H ðt Þdt
00
:
ð8Þ
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Fig. 4. The calculation schematic diagram of temperature field during spiral bead surfacing with weaving-motion wire.
– at time t > tc:
T ðx; y; z; z0 ; t Þ T 0 ¼
m¼2 X
j¼kc X
m¼1
j¼1
AC
Z
jt p
ðj1Þt p
0
0
F C ðt Þdt þ AC
Z
!
tc 0
F C ðt Þdt
0
;
jtp
whereas:
AC ¼
3 q ; 8 cqpaz0
! k 3 q nm v gð1Þk w þ gv g ðv 2 þ w2 þ v 2g 2ð1Þ wv g Þt0 ; exp AH ¼ 8 cqpaz0 2a 4a 0
2 pdz þ ð1Þm p2dz x v t0 ðk 1Þt p ðk 1Þhe =2 ðn 1Þpdz 2 F ð z0 Þ F C ðt Þ ¼ exp @ 4aðt þ t 0 t 0 Þ t þ t0 t0 0
2 1 y y0 B ð1Þk B t 0 v g þ ð1Þk w t 0 ðk 1Þt p C A; 4aðt þ t0 t0 Þ
1 0 2 2 v 2 þ w2 þ v 2g 2ð1Þk wv g t00 1 n þ g m A F H ðt Þ ¼ exp @ t 0 þ t 00 4aðt 00 þ t 0 Þ 4a 00
("
! ! !# z2 þ 2at00 z þ z0 z z0 erf 1 UðzÞerf UðzÞ 0;5 0;5 z20 2ðat00 Þ 2ðat 00 Þ " ! !#) 4at 00 z þ z0 ðz z0 Þ2 z z0 ðz þ z0 Þ2 ; exp exp þ 2 0;5 4at 00 4at00 z0 ð4pat 00 Þ0;5 ð4pat 00 Þ
g ¼ y y0 B ð1Þk B þ ð1Þk wðt ðk 1Þtp þ t0 Þ ðt þ t0 Þv g ;
ð9Þ
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nm ¼
pdz 2
F ðz0 Þ ¼
þ ð1Þm
pdz 2
x v ðt ðk 1Þtp þ t 0 Þ ðk 1Þhe =2 þ ðn 1Þpdz ;
! !# " z2 þ 2aðt t 0 Þ z þ z0 z z0 erf 1 U ð z Þerf U ð z Þ 0;5 0;5 z20 2ðaðt t0 ÞÞ 2ðaðt t 0 ÞÞ " ! !# 4aðt t 0 Þ z þ z0 ðz z0 Þ2 z z0 ð z þ z0 Þ 2 þ exp exp ; 0;5 0;5 4aðt t0 Þ 4aðt t0 Þ z20 ð4paðt t 0 ÞÞ ð4paðt t 0 ÞÞ
Fig. 5. Filling up wear zone with welds.
Fig. 6. Maximum temperature in HAZ in roll conventional method of controlled pitch surfaced.
Fig. 7. HAZ during conventional surfacing: A1 = 760 °C, A3 = 795 °C.
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tp ¼ 2B=w;
dz ¼ d þ 2hlc ;
k ¼ truncðt=t p Þ þ 1;
n, number of current weld pad; S, weld bead pitch; m, the order of overlaying next padding welds coefficient, for m = 2 direction of the order of overlaying next padding welds is compatible with the direction of the y axis, while for m = 1 the direction is opposite, B, the amplitude of weld bead, distance between extreme position of the electrode in a zigzak motion in relation to weld axis is equal 2B, t0 characterized superficial distribution of the heat source.
Fig. 8. Isolines of maximum temperature in heat affected zone of spiral weld surfaced roll.
Fig. 9. HAZ during spiral welding: A1 = 760 °C, A3 = 795 °C.
Fig. 10. Isotherms in cross section at time t = 0.5 s after passage of electrode.
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3. Temperature field and heat-affected zone in regenerated CCS machine roll The rolls leading a steel sheet in continuous casting steel line undergo intense wear and tear as a result of major thermomechanical loads. Regeneration of the roll was made by welding its external surface. Numerical simulations of changes in temperature were made for a fragment of the roll with 0.23 m in diameter made of 13CrMo4 steel. Thermal properties of the welded object material were determined by a = 8E6 m2/s and cq = 5.2 MJ/m3 K. The critical temperatures of austenitization A1 = 760 °C and A3 = 795 °C were experimentally determined by 1405 Thermal Cycle Simulator Smitweld. Computations were conducted for the all three already mentioned above methods of welding. 3.1. Temperature field during welding using conventional method of controlled weld pitch The computations were carried out for the heat source of 7560 W power with Gaussian power density distribution determined by t0 = 1 s and depth z0 = 0.007 m moving with velocity v = 0.01 m/s. The power of heat source corresponds to surfacing at welding load voltage 27 V, amperage 280 A and arc efficiency j = 0.8. Numerical simulation was conducted for filling the wear zone up to 3 mm and weld pitch up to h = 6 mm (Fig 5). Isotherms of the maximum temperature in the heat affected zone (HAZ) of roll’s cross section welded using conventional method of the controlled weld pitch were illustrated in Fig. 6, while areas of total and partial transformation in HAZ were presented in Fig. 7.
Fig. 11. Isotherms in cross section at time t = 7.5 s after passage of electrode.
Fig. 12. Thermal cycles at point with coordinates (0.039, 0.113) from 6th weld area.
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3.2. Temperature field in CSS machine roll during spiral welding The computations were carried out for the heat source of 7560 W power with Gaussian power density distribution determined by t0 = 1 s and z0 = 0.007 m, moving with velocity v = 0.01 m/s. The power of heat source corresponds to welding at arc voltage 27 V, amperage 280 A and j = 0.8. Isotherms of the maximum temperature in the heat affected zone (HAZ) of the roll’s cross section after spiral welding were presented in Fig. 8, while areas of total and partial transformation in HAZ – in Fig. 9. The Figs. 10 and 11 illustrate temperature distribution in the heat affected zone of spiral surfaced roll at particular times – 0.5 and 7.5 s after the passage of the electrode over the cross section during padding of the last weld. Finally, Figs. 12–14 illustrate the history of changes in temperature at the selected points of the cross section.
Fig. 13. Thermal cycles at point of contact with coordinates (0.012, 0.112) of the first and the second weld.
Fig. 14. Thermal cycles at point of contact with coordinates (0.048, 0.111) of 7th and 8th weld.
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3.3. Temperature field during spiral welding with swinging motion of the welding head The computations of temperature field were conducted in the heat affected zone of regenerated lateral surface of the roll, which wear zone of 2 mm in depth was welded with the swinging motion of electrode (Fig. 15). Numerical simulation of temperature field were made for the heat source of 4508 W power with Gaussian density power distribution determined by t0 = 1.53 s and z0 = 0.007 m characterizing volumetric model of the heat source. The power of heat source corresponds to welding at arc voltage 28 V, amperage 230 A and j = 0.7. Wear zone was filled up by four welds with 20 mm in the width and 1 mm in the face height. Technological parameters of welding head motion, defined by the velocity along the weld bead v = 0.001 m/s, frequency of crosswise swings f = 0, 2 1/s and amplitude A = 0.01 m, determine the welding head velocity along surfaced roll vg = 0.0275 mm/s, the velocity of electrode in swinging motion w = 0.008 m/ s and electrode pitch during one swing 0.005 m. Temperature distribution in the cross section of welded wear zone (parallel to roll axis) at time 1531.7 s since the beginning of welding process and 22 s after the passage of electrode over the section during padding the third weld is presented in Fig. 16.
Fig. 15. Surfaced roll wear zone filled by padding welds.
Fig. 16. Temperature field at time 1531.7 s since beginning welding process.
Fig. 17. HAZ of surfaced roll: A1 = 760 °C, A3 = 795 °C, solidus 1500 °C.
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Maximum temperatures in roll’s section enabled determination of the heat affected zone in longitudinal section of the roll using critical temperatures A1 and A3 (Fig. 17). The proposed model of temperature field makes it possible to analyse the temperature field at any point of regenerated detail. Thermal cycle at the point with coordinates (0.06, 0.105) from the heat affected zone is presented in Fig. 18 – cf. Figs. 16 and 17. Particular peaks correspond to consecutively surfaced welds. Swinging motion of the electrode caused multiple increases and decreases in temperature during padding the welds what was illustrated by thermal cycle fragments at the discussed point during making 2nd and 3rd weld – Fig. 19.
Fig. 18. Thermal cycle at point with coordinates (0.06, 0.105).
Fig. 19. Thermal cycle fragments at point with coordinates (0.06, 0.105) during making 2nd and 3rd weld.
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Fig. 20. Thermal cycle at point with coordinates (0.06, 0.100).
Fig. 21. Thermal cycle fragments at point with coordinates (0.06, 0.10) during padding 2nd and 3rd weld.
The influence of electrode swings on temperature ‘pulsation’ in the thermal cycle is diminishing with the distance from the heat source into the depth of material. This is noticeable on example of thermal cycles at the point (0.06, 0.100), what is 5 mm below the point analyzed previously (Figs. 20 and 21). 4. Conclusions The models, presented in this paper, made it possible to compute the temperature field in cylindrical steel elements welded using the controlled weld pitch method, spiral weld and spiral weld with swinging motion of the welding head.
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Analytical solutions to temperature field enabled observation of the thermal cycles in any points of the welded object as well as the analysis of temperature field in any roll’s longitudinal sections for the selected moment of the process. The proposed description gave the possibility to describe the characteristic heat affected zones: partial and total transformations as well as fusion. The solution to the temperature field formed the basis for further analysis of thermo-mechanical welding states (phase transformations, states of stress) of cylindrical steel elements, such as: cylinders, shafts, rolls of continuous casting steel machine, closures of an iron blast furnace and mill rolls. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]
[18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36]
J. Pilarczyk, J. Pilarczyk, Arc Welding and Surfacing of Metals, Slask, Katowice, 1996. p. 337. D. McLoughlin, Repair of continuous casting steel line rolls by submerged arc welding, Weld. Inst. Bull. 5 (1997) 85–88. A. Klimpel, Surfacing by Welding and Thermal Spraying, WNT, Warszawa, 2000. pp. 140–147. D. Rosenthal, Mathematical theory of heat distribution during welding and cutting, Weld. J. 5 (1941) 220s–234s. T.W. Eagar, N.S. Tsai, Temperature fields produced by traveling distributed heat sources, Weld. J. (1983) 346s–355s. S.J. Na, S.Y. Lee, A study on the tree-dimensional analysis of the transient temperature distribution in gas tungsten arc welding, Proc. Inst. Mech. Eng. (B3) (2001) 149–156. P. Tekriwal, J. Mazumder, Finite element analysis of three-dimensional transient heat transfer in GMA welding, Weld. J. (1988) 150s–156s. J. Goldak, A. Chakravarti, M. Bibby, A new finite element model for welding heat sources, Metall. Trans. B 15B (1984) 299–305. K.S. Bo, H.S. Cho, Transient temperature distribution in arc welding of finite thickness plates, Proc. Inst. Mech. Eng. B 204 (B3) (1990) 175–183. R.I. Karlsson, B.L. Josefson, Three dimensional finite element analysis of temperatures and stresses in a single – pass butt-welded pipe, Trans. ASME 112 (1990) 76–84. R. Parkitny, A. Pawlak, W. Piekarska, Temperature fields and stress states in welded tubes of rectangular cross section, in: Mechanical Effects of Welding, IUTAM Symposium Lulea, Sweden 1991, Springer-Verlag, Berlin, Heidelberg, 1991. K. Mundra, T. DebRoy, K.M. Kelkar, Numerical prediction of fluid flow and heat transfer in welding with a moving heat source, Numer. Heat Transfer A 29 (1996) 115–129. S.K. Jeong, H.S. Cho, An analytical solution to predict the transient temperature distribution in fillet arc welds, Weld. J. 6 (1997) 223s–232s. N.T. Nguyen, K. Matsuoka, N. Suzuki, Y. Maeda, Analytical solutions for transient temperature of semi-infinite body subjected to 3-D moving heat sources, Weld. J. 8 (1999) 265s–274s. R. Komanduri, Z.B. Hou, Thermal analysis of the arc welding process: Part I. General solutions, Metall. Mater. Trans. B 31B (2000) 1353–1370. N.T. Nguyen, Y.W. Mai, S. Simpson, A. Ohta, Analytical approximate solution for double ellipsoidal heat source in finite thick plate, Weld. J. (2004) 82s– 93s. M.M. Mahapatra, G.L. Datta, B. Pradhan, Three-dimensional finite element analysis to predict the effects of shielded metal arc welding process parameters on temperature distributions and weldment zones in butt and one-sided fillet welds, Proc. Inst. Mech. Eng. B. J. Eng. Manuf. 220 (2006) 837–845. A. Kumar, T. Debroy, Heat Transfer and fluid flow during gas–metal–arc fillet welding for various joint configurations and welding positions, Metall. Mater. Trans. A 38A (2007) 506–519. S. Wang, J. Goldak, J. Zhou, S. Tchernov, D. Downey, Simulation on the thermal cycle of a welding process by space-time convection-diffusion finite element analysis, Int. J. Thermal Stresses 48 (2009) 936–947. R.C. Reed, H.K.D. Bhadeshia, A simple model for multipass steel welds, Acta Metall. Mater. 42 (11) (1994) 3663–3678. B. Taljat, T. Zacharia, X-L. Wang, R. Keiser, R.W. Swindeman, Z. Feng, J. Jirinec, Numerical analysis of residual stress distribution in tubes with spiral weld cladding, Weld. J. (1998) 328s–335s. R.N.S. Fassani, Analytical modeling of multipass welding process with distributed heat source, J. Braz. Soc. Mech. Sci. Eng. XXV (3) (2003) 302–305. W. Jiang, K. Yahiaoui, F.R. Hall, Finite element predictions of temperature distributions in a multipass welded piping branch junction, J. Press. Vessel Technol. 127 (2005) 7–12. D. Deng, H. Murakawa, Prediction of welding residual stress in multi-pass butt-welded modified 9Cr-1Mo steel pipe considering phase transformation effects, Comput. Mater. Sci. 37 (2006) 209–219. D. Deng, H. Murakawa, Numerical simulation of temperature field and residual stress in multi-pass welds in stainless steel pipe and comparison with experimental measurements, Comput. Mater. Sci. 37 (2006) 269–277. S. Murugan, P.V. Kumar, B. Raj, M.S.C. Bose, Temperature distribution during multipass welding of plates, Int. J. Press. Vessels Pipping 75 (1998) 891– 905. D. Wojnowski, Y.K. Oh, J.E. Indacochea, Metallurgical assessment of the softened HAZ region during multipass welding, Trans. ASME 122 (2000) 310– 315. S. Murugan, K. Sanjai, P.V. Rai Kumar, T. Jayakumar, B. Raj, M.S.C. Bose, Temperature distribution and residual stresses due to multipass welding in type 304 stainless steel and low carbon steel weld pads, Int. J. Pressure Vessels and Pipping 78 (2001) 307–317. K.P. Kolhe, C.K. Datta, Prediction of microstructure and mechanical of multipass SAW, J. Mater. Process. Technol. 197 (2008) 241–249. M. Mysliwiec, Thermo-Mechanical Basis of Welding, WNT, Warszawa, 1970. pp. 62–67. D. Radaj, Heat Effects of Welding, Temperature Field, Residual Stress, Distortion, Springer-Verlag, Berlin, Heidelberg, New York, London, Paris, Tokyo, 1992. pp. 38–39. K.E. Easterling, Modelling the weld thermal cycle and transformation behaviour in the hest affected zone, in: H.K.E. Cerjak, K.E. Easterling (Eds.), Mathematical Modelling of Weld Phenomena, The Institute of Materials, London, 1993, pp. 183–200. J. Winczek, Analysis of Elasto-Plastic States Along Welded Rods Changeable Loaded, Doctoral Dissertation, Technical University of Czestochowa. J. Winczek, Temperature fields during multipass surfaced cuboidal elements, in: Proceedings of Scn.-Tech. Conf. New materials – new technologies in ships building and machine-building industry, Swinoujscie, Poland, 1998. pp. 651–656. J. Winczek, Modelling of temperature field and phase transformations in elements regenerated by surfacing, Inform .Metal Technol. 2 (4) (2002) 121– 137. P.R. Vishnu, W.B. Li, K.E. Easterling, Heat-flow model for pulsed welding, Mater. Sci. Technol. 7 (1991) 649–659.
Contents lists available at SciVerse ScienceDirect
Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm
Modelling of heat affected zone in cylindrical steel elements surfaced by welding Jerzy Winczek ⇑ Institute of Mechanics and Machine Design Foundations, Technical University of Czestochowa, Dabrowskiego 73, 42-200 Czestochowa, Poland
a r t i c l e
i n f o
Article history: Received 14 March 2010 Received in revised form 29 August 2011 Accepted 1 September 2011 Available online 10 September 2011 Keywords: Temperature field Welding Surfacing Heat affected zone Modeling
a b s t r a c t Computational models of a temperature field in cylindrical steel elements surfaced by the following methods: controlled pitch, spiral welding sequence and spiral welding sequence with swinging motion of the welding head are presented in the paper. The lateral surface of regenerated cylindrical object, subjected to the welding heat source, has been treated as a plane rolled on cylinder and temperature field of repeatedly surfaced plain massive body was solved. Temperature rises, caused by overlaying consecutive welding sequences and self-cooling of areas previously heated, were taken into consideration in the solution. The computations of the temperature field for continuous casting steel machine roll made of 13CrMo4 steel were carried out. Ó 2011 Elsevier Inc. All rights reserved.
1. Introduction Surfacing is becoming more and more frequently used technological process for regeneration of machines parts, especially bulky ones and those undergoing intensive wear and tear or frequent damages. One of the method of surfacing rolled or cylindrical details is to associate rotational motion of an object with fed motion of the welding head along the lateral surface. Considering the way of this association three following methods described by Pilarczyk and Pilarczyk [1], McLoughlin [2] and Klimpel [3] can be distinguished: a controlled welding sequence pitch – with jumping travel after every full turn of the roll (Fig. 1(a)), with constant travel giving spiral path of welding sequences (Fig. 1(b)) and with constant travel of welding head along the axis of subject with simultaneous swinging motion of electrode across the welding sequence (Fig. 1(c)). In the modeling of temperature fields in welding processes two approaches prevail: numerical (mainly using finite element method FEM) and analytical (using integral transformation method and Green’s function). As an analytical solution to the heat conduction equation offers faster assessment of the temperature field, this method has found many supporters. Research into moving heat source was first presented by Rosenthal [4]. He obtained the temperature distribution equations for point, line and plane heat sources. Eagar and Tsai [5] modified Rosenthal’s model introducing 2-D Gaussian distributed heat source and formulated solution to the moving heat source on semi-infinite steel plate. Transient temperature distribution in gas tunsten arc welding (GTAW) was analyzed by applying three-dimensional model of the finite element by Na and Lee [6]. The solution to the temperature field developed by a ring heat source was presented, where intensity of heat flux at distance from arc center was determined with reference to the parameter defined as radius at which intensity of arc power falls to 5% of maximum intensity. Tekriwal and Mazumder [7] proposed finite element method (FEM) transient ⇑ Fax: +48 34 3605183. E-mail address: [email protected] 0307-904X/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.apm.2011.09.032
J. Winczek / Applied Mathematical Modelling 36 (2012) 1514–1528
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Nomenclature a A1, A3 c cq d h hlc l q q_ v t T t0 y0 z0 k
j q
thermal diffusivity (m2 s1) critical temperatures of the austenitisation specific heat (J/kg K) thermal capacity (J/K m3) diameter of roll (m) weld bead pitch (m) height of weld face (m) length of weld (m) power of hest source (W) individual power of voluminal heat source (W/m3) time (s) temperature (K, °C) time which has passed since initiating heat source (s) coordinate of the beginning of the weld (m) depth of heat source deposition (m) coefficient of heat conduction (W/mK) arc efficiency density (kg/m3)
temperature model in gas metal arc welding (GMAW). The model included thermal properties depending on temperature, effect of latent heat, convective and radiative boundary conditions. In this solution double ellipsoidal heat source model proposed by Goldak et al. [8] was used. Bo and Cho [9] derived an analytical solution to the transient temperature field in plate of finite thickness during single pass arc welding. The moving heat source with Gaussian distribution connected to an electric arc was assumed, likewise by Eagar and Tsai. Temperatures in a single-pass butt-welded pipe was studied by Karlsson and Josefson [10] using full 3D finite element model in pipe made of carbon-manganese steel. Thermal material properties, such as: heat capacity, thermal conductivity and convection, was used in this model. Parkitny et al. [11] calculated temperature field in the welded cylinders with rectangular cross sections. Algorithm was based on the method of elementary heat balances. Melting of electrode and parental material as well as the solidification of molten pool was considered. Mundra et al. [12] presented the computational analysis of the detailed prediction of heat transfer, phase change and fluid flow during welding with moving heat source. Basic equations were formulated in a reference frame connected to the heat source. Initially unknown velocity was the convective velocity of the fluid with motion of the heat source. This leads into additional terms in equations solved using control-volume-based computational method. The analytical solution to temperature field in fillet arc welds was derived by Jeong and Cho [13] for infinite plate of finite thickness bent under right angle. The analytical 3-D model of temperature field in semi-infinite body caused by 3-D moving heat source was formulated by Nguyen et al. [14] and experimentally validated by measurement of the transient temperature at various points in bead-on-plate specimens and in the middle of weld pool. Double ellipsoidal heat source proposed by Goldak et al. [8] was applied to the solution. Komanduri and Hou [15] presented FEM thermal analysis of arc welding process. The models of ring heat source and the models of disc heat source with pseudo-Gaussian distribution of heat intensity were applied. Data used in investigation were essentially the same as ones used by Tekriwal and Mazumder [7]. The numerical simulations of temperature field for Vgroove joint were carried out. Nguyen et al. [16] presented analytical approximate solution to single- and double-ellipsoidal model of the heat source for the modeling of temperature field in the finite thick plate. Authors also proposed solution in dimensional form for the ellipsoidal heat source. 3-D FEM analysis of influence prediction of the effect of shielded metal arc welding process parameters, such as: electrode diameter, speed of travel, current and voltage, on the temperature
Fig. 1. Surfacing methods of cylindrical elements: (a) controlled pitch, (b) with feed motion of welding head, (c) with feed motion of welding head and swinging motion of electrode.
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J. Winczek / Applied Mathematical Modelling 36 (2012) 1514–1528
distribution were investigated by Mahapatra et al. [17]. A comparison of the results of numerical simulations and measured fusion zone and heat affected zone (HAZ) boundary for butt joint and one-sided fillet welds were presented. Kumar and Debroy [18] proposed three-dimensional model of numerical heat-transfer and fluid-flow to capture effects of tilt angle of fillet joint and welding positions on the temperature profiles, velocity fields, weld pool shape and weld pool free surface profile. Basic equations of conservation of mass, momentum and energy were solved using boundary fitted curvilinear coordinate system. Surface deformation of weld pool was calculated by minimizing the total surface energy. Wang et al. [19] used space–time FEM method to analyse transient thermal cycle during welding by solving transient convection–diffusion thermal equation. In the proposed method coordinate frame (Eulerian frame) is moving, while heat source is motionless. The model takes into account an addition of filler material during thermal analysis. Bibliography dealing with the multiple welding is rather meagre. Reed and Bhadeshia [20] proposed model describing thermal cycles occurring in multipass welds. In this method weld bead reinforcement was calculated and heat-flow equation for weld isotherms was used. The model was calibrated for multipass weld of steam-pipes. The method was used to theoretical investigation of the effect of interpass temperature, welding current and Ac3 temperature on the fraction of weld microstructure which becomes reaustenitised during fabrication of the weldment. Taljat et al. [21] presented numerical analysis (FEM) of own stresses caused by the field temperature during surfacing by spiral welding sequence. Fassani [22] compared thermal cycles during multipass gas metal arc welding (GMAW) calculated for models of point Rosenthal’s and one-dimensional Gausian distribution heat source. The analytical solution was presented to the one-dimensional problem – temperature distribution perpendicular to the welding direction. Comprehensive study of 3D temperature distribution in a thick tee made of stainless steel (316L) during multipass welding process was presented by Jiang et al. [23]. The computations were carried out in ABAQUS programe, where thermal physical properties (thermal conductivity, specific heat, density), dependent on temperature, were used in the thermal analysis. The 3D thermal model and ABAQUS were used by Deng and Murakawa [24] to calculate temperature field to predict welding residual stresses in multipass butt welded (metal inert gas (MIG) and metal active gas (MAG) welding processes) of pipes made of 9Cr-1Mo steel. The heat source was modeled as the surface heat source with a Gaussian distribution. Conductivity and specific heat were dependent on temperature. Influence of latent heat of phase transformations was taken into consideration. In computational procedure to analyse the temperature fields in multipass welds (tungsten inert gas (TIG) welding process) of pipe made of stainless steel (SUS304), Deng and Murakawa [25] applied volumetric heat source with a double ellipsoidal distribution proposed by Goldak as the heat of moving welding arc. Results of the numerical simulation were compared to experimental data from thermo-couples. More publications concern results of experimental research into the multipass welds. Murugan et al. [26] carried out an experimental analysis of the temperature field during multipass welding of 304 steel plates. The temperature curves at different distances from the weld path center line were plotted. The method of manual metal arc welding was chosen for experiment. Wojnowski et al. [27] measured weld thermal history (peaks of temperature), microhardness and carried out metalographic analysis of softened region in the samples made of CrMoV steel after multipass welding. Murugan et al. [28] measured thermal cycles (peaks of temperature attained during weld passes in low carbon steel) and residual stresses (used X-ray diffraction method) during each pass of welding. Kolhe and Datta [29] presented analysis results of microstructure and mechanical properties (Rockwell hardness, Vickers microhardness and impact energy) after multipass by submerged arc welding (SAW) method. The swinging motion of electrode was not taken into consideration in the quoted above works. Spiral welding with swinging motion of the welding head is becoming more and more popular welding method of cylindrical elements because of it high efficiency. A knowledge of changeable in time temperature field, similarly to welding with the use of controlled pitch and spiral welding technique, is fundamental for determining heat affected zone. It constitutes also a point of departure for calculations (on the basis of temperature changes velocity during cooling) of phase transformations as well as stress states. Thereby, it has not only cognitive significance, but also practical. 2. Analytical solution to the temperature field Performance of the concentrated moving heat source causing the temperature field changeable in time and space is the characteristic feature of surfacing by welding. Analytical description of this temperature field is obtained using equation of heat conduction in motionless Cartesian coordinate system (x, y, z):
rðkrTÞ þ q_ v ¼ cq
oT : ot
ð1Þ
Basically, to find description of the temperature field, one can use solution to basic Eq. (1) for infinite body with transient heat source Q applied in any point of the body with coordinates (x0 , y0 , z0 ) [30–32]:
Tðx; y; z; tÞ ¼
Q cqð4p atÞ3=2
ðx x0 Þ2 þ ðy y0 Þ2 þ ðz z0 Þ2 exp 4 at
! ð2Þ
The solution to temperature field, caused by the moving volumetric heat source with Gaussian distribution of the density power and traveling speed v [m/s] in fixed system of coordinates (x, y, z), was presented by Winczek [33]. The following
J. Winczek / Applied Mathematical Modelling 36 (2012) 1514–1528
1517
assumptions were made: the material is homogenous and isotropic, physics-thermal properties do not depend on temperature, phase transformations are not taken into consideration, and during electrical welding process Joule heat are omitted. In the case of cylindrical elements with sufficiently big diameter (wall thickness > 25 mm – Mysliwiec [30] the surface subjected to welding heat source can be treated as a plane rolled on a cylinder. Then, the solution to the multiple welded plain elements described by Winczek [34,35] can be used to describe the temperature field changeable in time and space. In calculations of the temperature field during making n weld bead, temperature-rises connected with next transition of welding heat source DTH and self-cooling of already overlaid padding welds as well as already heated areas DTC were taken into consideration:
Tðx; y; z; z0 ; tÞ T 0 ¼ DT C þ DT H :
ð3Þ
2.1. Temperature field during welding using a conventional method of controlled weld pitch After unrolling the lateral surface of cylinder (Fig. 2), the temperature field in the point P with coordinates (x, y,z) caused by welding heat source with power q, depth z0 and traveling velocity v, at the time t 6 tk, where tk is time from the beginning to the end of kth weld welding, can be described as:
Tðx; y; z; z0 ; tÞ T 0 ¼
m¼2 X
j¼k1 X
m¼1
j¼1
AC
Z
tj
0
0
F C ðt Þdt þ Ak
Z
!
t 00
F H ðt Þdt
00
;
ð4Þ
t k1
t j1
while at time t > tk:
Tðx; y; z; z0 ; tÞ T 0 ¼
j¼k m¼2 XX m¼1 j¼1
AC
Z
tj
0
F C ðt 0 Þdt ;
t j1
where:
AC ¼
3 q ; 8 cqpaz0
Ak ¼
3 q v n v 2 t0 ; exp m 8 cqpaz0 2a 4a
Fig. 2. Schematic diagram for describing the temperature field during welding of cylindrical element using controlled weld pitch method.
ð5Þ
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J. Winczek / Applied Mathematical Modelling 36 (2012) 1514–1528
nm ¼
pd pd þ
2
2
ð1Þm xð1Þm v ðt þ t0 Þ;
WH ðz0 ; t 00 Þ
00
F H ðt Þ ¼
t 00 þ t 0
p d
WC ðz0 ; t 0 Þ
F C ðt0 Þ ¼
WH ðz0 ; t Þ ¼
2
exp
t þ t0 t0
00
! n2m v 2 t00 ; exp 4aðt 00 þ t0 Þ 4a
z2 þ 2at00 1 z20
þ p2d ð1Þm xð1Þm v t 0 4aðt þ t0 t0 Þ
UðzÞerf UðzÞ
0;5
2ðat00 Þ
4at 00 z þ z0 ðz z0 Þ2 þ 2 exp 0;5 00 4at 00 z0 ð4pat Þ
WC ðz0 ; t0 Þ ¼ 1
z2 þ 2aðt t 0 Þ z20
erf
þ ðy y0i Þ2
!
z þ z0
erf
2
!
; !!
0;5
2ðat 00 Þ
0;5
ðz þ z0 Þ2 exp 4at 00
!
z þ z0 2ðaðt t 0 ÞÞ
z z0
z z0 ð4pat 00 Þ
!
UðzÞerf UðzÞ
0;5
; !!
z z0 2ðaðt t 0 ÞÞ
0;5
!! 4aðt t 0 Þ z þ z0 ðz z0 Þ2 z z0 ðz þ z0 Þ2 þ exp exp 0;5 0;5 4aðt t 0 Þ 4aðt t 0 Þ z20 ð4paðt t0 ÞÞ ð4paðt t0 ÞÞ
UðzÞ ¼
1 for z 2< 0; 1
z0 >
for z 2< z0 ;
1>
!
!!
;
t00 = t t0 , whereas t means current time of the total welding process. Quantity t0 characterizes the surface distribution of heat source, while r2B ¼ 4at 0 [36]. The total time from the beginning to the end of k-weld roll welding is described by relationship:
tk ¼ kðl=v Þ þ ðk 1Þtp ; and coordinate y0i of the beginning of i-weld can be calculated according to the formula:
y0i ¼ y01 þ ði 1Þh; where tp – idle time after making i-weld (time of idle motion of electrode with h pitch to the beginning of i + 1 weld), y01 – the coordinate of the beginning of the first weld roll, l – the length of weld defined by relationship:
l ¼ pðd þ 2hlc Þ: where d, diameter of the roll, hlc, height of face of weld. 2.2. Temperature field during a spiral welding After unrolling lateral surface of cylinder (Fig. 3) the temperature field in point P with coordinates (x, y,z) at the distance g from the source along the y axis and n1 and n2 along the x axis during kth roll weld can be described by following equations: – for time t 6 tk where t k ¼ tj¼k is the total time from the beginning to the end of kth weld welding:
T ðx; y; z; z0 ; t Þ T 0 ¼
j¼k1 X m¼2 X j¼1
m¼1
AC
Z
tj
0
F C ðt 0 Þdt þ
t j1
m¼2 X m¼1
Ak
Z
t
00
F H ðt00 Þdt ;
t k1
where
F H ðt00 Þ ¼
0
F C ðt Þ ¼
AC ¼
WH ðz0 ; t 00 Þ t 00 þ t 0
WC ðz0 ; t 0 Þ t þ t0 t0
3 q ; 8 cqpaz0
exp
! n2m þ g2 v 2 t00 ; 4aðt 00 þ t0 Þ 4a
p d exp
Ak ¼
2
! 2 2 þ p2d x þ v cos b t0 ð1Þm þ ðy v sin b t 0 y0i Þ ; 4aðt þ t 0 t 0 Þ
3 q v ðnm cos b þ g sin bÞ v 2 t0 exp ; 8 cqpaz0 2a 4a
ð6Þ
J. Winczek / Applied Mathematical Modelling 36 (2012) 1514–1528
1519
Fig. 3. Schematic diagram for describing temperature field during spiral weld.
nm ¼
pd 2
þ
pd 2
x þ v cos b ðt þ t 0 Þ ð1Þm ;
g ¼ y v sin bðt þ t0 Þ y0j ; tj ¼ jðl=v Þ; v, heat source velocity; y0i is the coordinate of the beginning of ith weld, l, length of the weld described by relationship:
l¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 c2w þ h ;
cw ¼ pðd þ 2hlc Þ;
and the angle between x axis and direction of heat source motion b = arcsin(h/l), – at time t > tk the temperature field is obtained in the following form:
Tðx; y; z; z0 ; tÞ T 0 ¼
j¼k X j¼1
AC
m ¼2 X m¼1
Z
tj
0
F C ðt 0 Þdt :
ð7Þ
t j1
2.3. Temperature field during spiral welding with swinging motion of welding head The velocity of heat source in relation to the object in the considered welding method is resultant of three velocities: the fed motion of welding head along the lateral surface of roll vg, swinging motion of electrode w and tangential velocity v of the electrode in relation to the rotating welded metal. Increases in temperature connected to already made welds (the cooling of earlier heated areas during welding) must be taken into account in the temperature field computations. After unrolling the lateral surface of cylinder (fig. 4) the temperature field in point P with coordinates (x,y, z), g meters away from the source along the y axis and n1 and n2 along the x axis during n, weld and k, swing of electrode can be described by the following equations: – at time t 6 tc, where tc total time for welding:
T ðx; y; z; z0 ; tÞ T 0 ¼
m¼2 X
j¼k1 X
m¼1
j¼1
AC
Z
jt p
ðj1Þtp
0
0
F C ðt Þdt þ AH
Z
!
t
jt p
00
F H ðt Þdt
00
:
ð8Þ
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J. Winczek / Applied Mathematical Modelling 36 (2012) 1514–1528
Fig. 4. The calculation schematic diagram of temperature field during spiral bead surfacing with weaving-motion wire.
– at time t > tc:
T ðx; y; z; z0 ; t Þ T 0 ¼
m¼2 X
j¼kc X
m¼1
j¼1
AC
Z
jt p
ðj1Þt p
0
0
F C ðt Þdt þ AC
Z
!
tc 0
F C ðt Þdt
0
;
jtp
whereas:
AC ¼
3 q ; 8 cqpaz0
! k 3 q nm v gð1Þk w þ gv g ðv 2 þ w2 þ v 2g 2ð1Þ wv g Þt0 ; exp AH ¼ 8 cqpaz0 2a 4a 0
2 pdz þ ð1Þm p2dz x v t0 ðk 1Þt p ðk 1Þhe =2 ðn 1Þpdz 2 F ð z0 Þ F C ðt Þ ¼ exp @ 4aðt þ t 0 t 0 Þ t þ t0 t0 0
2 1 y y0 B ð1Þk B t 0 v g þ ð1Þk w t 0 ðk 1Þt p C A; 4aðt þ t0 t0 Þ
1 0 2 2 v 2 þ w2 þ v 2g 2ð1Þk wv g t00 1 n þ g m A F H ðt Þ ¼ exp @ t 0 þ t 00 4aðt 00 þ t 0 Þ 4a 00
("
! ! !# z2 þ 2at00 z þ z0 z z0 erf 1 UðzÞerf UðzÞ 0;5 0;5 z20 2ðat00 Þ 2ðat 00 Þ " ! !#) 4at 00 z þ z0 ðz z0 Þ2 z z0 ðz þ z0 Þ2 ; exp exp þ 2 0;5 4at 00 4at00 z0 ð4pat 00 Þ0;5 ð4pat 00 Þ
g ¼ y y0 B ð1Þk B þ ð1Þk wðt ðk 1Þtp þ t0 Þ ðt þ t0 Þv g ;
ð9Þ
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nm ¼
pdz 2
F ðz0 Þ ¼
þ ð1Þm
pdz 2
x v ðt ðk 1Þtp þ t 0 Þ ðk 1Þhe =2 þ ðn 1Þpdz ;
! !# " z2 þ 2aðt t 0 Þ z þ z0 z z0 erf 1 U ð z Þerf U ð z Þ 0;5 0;5 z20 2ðaðt t0 ÞÞ 2ðaðt t 0 ÞÞ " ! !# 4aðt t 0 Þ z þ z0 ðz z0 Þ2 z z0 ð z þ z0 Þ 2 þ exp exp ; 0;5 0;5 4aðt t0 Þ 4aðt t0 Þ z20 ð4paðt t 0 ÞÞ ð4paðt t 0 ÞÞ
Fig. 5. Filling up wear zone with welds.
Fig. 6. Maximum temperature in HAZ in roll conventional method of controlled pitch surfaced.
Fig. 7. HAZ during conventional surfacing: A1 = 760 °C, A3 = 795 °C.
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tp ¼ 2B=w;
dz ¼ d þ 2hlc ;
k ¼ truncðt=t p Þ þ 1;
n, number of current weld pad; S, weld bead pitch; m, the order of overlaying next padding welds coefficient, for m = 2 direction of the order of overlaying next padding welds is compatible with the direction of the y axis, while for m = 1 the direction is opposite, B, the amplitude of weld bead, distance between extreme position of the electrode in a zigzak motion in relation to weld axis is equal 2B, t0 characterized superficial distribution of the heat source.
Fig. 8. Isolines of maximum temperature in heat affected zone of spiral weld surfaced roll.
Fig. 9. HAZ during spiral welding: A1 = 760 °C, A3 = 795 °C.
Fig. 10. Isotherms in cross section at time t = 0.5 s after passage of electrode.
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3. Temperature field and heat-affected zone in regenerated CCS machine roll The rolls leading a steel sheet in continuous casting steel line undergo intense wear and tear as a result of major thermomechanical loads. Regeneration of the roll was made by welding its external surface. Numerical simulations of changes in temperature were made for a fragment of the roll with 0.23 m in diameter made of 13CrMo4 steel. Thermal properties of the welded object material were determined by a = 8E6 m2/s and cq = 5.2 MJ/m3 K. The critical temperatures of austenitization A1 = 760 °C and A3 = 795 °C were experimentally determined by 1405 Thermal Cycle Simulator Smitweld. Computations were conducted for the all three already mentioned above methods of welding. 3.1. Temperature field during welding using conventional method of controlled weld pitch The computations were carried out for the heat source of 7560 W power with Gaussian power density distribution determined by t0 = 1 s and depth z0 = 0.007 m moving with velocity v = 0.01 m/s. The power of heat source corresponds to surfacing at welding load voltage 27 V, amperage 280 A and arc efficiency j = 0.8. Numerical simulation was conducted for filling the wear zone up to 3 mm and weld pitch up to h = 6 mm (Fig 5). Isotherms of the maximum temperature in the heat affected zone (HAZ) of roll’s cross section welded using conventional method of the controlled weld pitch were illustrated in Fig. 6, while areas of total and partial transformation in HAZ were presented in Fig. 7.
Fig. 11. Isotherms in cross section at time t = 7.5 s after passage of electrode.
Fig. 12. Thermal cycles at point with coordinates (0.039, 0.113) from 6th weld area.
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3.2. Temperature field in CSS machine roll during spiral welding The computations were carried out for the heat source of 7560 W power with Gaussian power density distribution determined by t0 = 1 s and z0 = 0.007 m, moving with velocity v = 0.01 m/s. The power of heat source corresponds to welding at arc voltage 27 V, amperage 280 A and j = 0.8. Isotherms of the maximum temperature in the heat affected zone (HAZ) of the roll’s cross section after spiral welding were presented in Fig. 8, while areas of total and partial transformation in HAZ – in Fig. 9. The Figs. 10 and 11 illustrate temperature distribution in the heat affected zone of spiral surfaced roll at particular times – 0.5 and 7.5 s after the passage of the electrode over the cross section during padding of the last weld. Finally, Figs. 12–14 illustrate the history of changes in temperature at the selected points of the cross section.
Fig. 13. Thermal cycles at point of contact with coordinates (0.012, 0.112) of the first and the second weld.
Fig. 14. Thermal cycles at point of contact with coordinates (0.048, 0.111) of 7th and 8th weld.
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3.3. Temperature field during spiral welding with swinging motion of the welding head The computations of temperature field were conducted in the heat affected zone of regenerated lateral surface of the roll, which wear zone of 2 mm in depth was welded with the swinging motion of electrode (Fig. 15). Numerical simulation of temperature field were made for the heat source of 4508 W power with Gaussian density power distribution determined by t0 = 1.53 s and z0 = 0.007 m characterizing volumetric model of the heat source. The power of heat source corresponds to welding at arc voltage 28 V, amperage 230 A and j = 0.7. Wear zone was filled up by four welds with 20 mm in the width and 1 mm in the face height. Technological parameters of welding head motion, defined by the velocity along the weld bead v = 0.001 m/s, frequency of crosswise swings f = 0, 2 1/s and amplitude A = 0.01 m, determine the welding head velocity along surfaced roll vg = 0.0275 mm/s, the velocity of electrode in swinging motion w = 0.008 m/ s and electrode pitch during one swing 0.005 m. Temperature distribution in the cross section of welded wear zone (parallel to roll axis) at time 1531.7 s since the beginning of welding process and 22 s after the passage of electrode over the section during padding the third weld is presented in Fig. 16.
Fig. 15. Surfaced roll wear zone filled by padding welds.
Fig. 16. Temperature field at time 1531.7 s since beginning welding process.
Fig. 17. HAZ of surfaced roll: A1 = 760 °C, A3 = 795 °C, solidus 1500 °C.
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Maximum temperatures in roll’s section enabled determination of the heat affected zone in longitudinal section of the roll using critical temperatures A1 and A3 (Fig. 17). The proposed model of temperature field makes it possible to analyse the temperature field at any point of regenerated detail. Thermal cycle at the point with coordinates (0.06, 0.105) from the heat affected zone is presented in Fig. 18 – cf. Figs. 16 and 17. Particular peaks correspond to consecutively surfaced welds. Swinging motion of the electrode caused multiple increases and decreases in temperature during padding the welds what was illustrated by thermal cycle fragments at the discussed point during making 2nd and 3rd weld – Fig. 19.
Fig. 18. Thermal cycle at point with coordinates (0.06, 0.105).
Fig. 19. Thermal cycle fragments at point with coordinates (0.06, 0.105) during making 2nd and 3rd weld.
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Fig. 20. Thermal cycle at point with coordinates (0.06, 0.100).
Fig. 21. Thermal cycle fragments at point with coordinates (0.06, 0.10) during padding 2nd and 3rd weld.
The influence of electrode swings on temperature ‘pulsation’ in the thermal cycle is diminishing with the distance from the heat source into the depth of material. This is noticeable on example of thermal cycles at the point (0.06, 0.100), what is 5 mm below the point analyzed previously (Figs. 20 and 21). 4. Conclusions The models, presented in this paper, made it possible to compute the temperature field in cylindrical steel elements welded using the controlled weld pitch method, spiral weld and spiral weld with swinging motion of the welding head.
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Analytical solutions to temperature field enabled observation of the thermal cycles in any points of the welded object as well as the analysis of temperature field in any roll’s longitudinal sections for the selected moment of the process. The proposed description gave the possibility to describe the characteristic heat affected zones: partial and total transformations as well as fusion. The solution to the temperature field formed the basis for further analysis of thermo-mechanical welding states (phase transformations, states of stress) of cylindrical steel elements, such as: cylinders, shafts, rolls of continuous casting steel machine, closures of an iron blast furnace and mill rolls. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]
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