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Temperature Fields Produced by Traveling Distributed Heat Sources

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Temperature Fields Produced by Traveling Distributed Heat Sources Temperature Fields Produced by Traveling Distributed Heat Sources Use of a Gaussian heat distribution in dimensionless form indicates final weld pool shape can be predicted accurately for many welds and materials BY T. VV. EAGAR AND N.-S. TSAI ABSTRACT. The solution of a travelling cate that the Rosenthal solution gives Symbols used when presenting the distributed heat source on a semi-infinite good agreement with the actual weld solutions discussed below are defined in plate provides information about both bead size over several orders of magni­ Table 1. the size and the shape of arc weld pools. tude. However, the scatter can be as The results indicate that both welding much as a factor of three. In addition, the Formulation of the Solution process variables (current, arc length and point source solution does not provide travel speed) and material parameters any information concerning the shape of Rosenthal's solution for the tempera­ (thermal diffusivity) have significant the weld pool, since all transverse iso­ ture distribution produced by a steady effects on weld shape. The theoretical therms are assumed to be semicircular in state point heat source moving on the predictions are compared with experi­ shape. The question of weld pool shape surface of a semi-infinite plate using the mental results on carbon steels, stainless is of considerable interest of late due to coordinate system shown in Fig. 1 is given steel, titanium and aluminum with good wide variations in welding behavior of by: agreement. heat-to-heat lots (Ref. 8). One of the purposes of the present v(w + R) „ q 2a Introduction study is to determine what shape infor­ 27rkR (1) mation can be obtained from the solution It has been more than forty years since of a travelling distributed heat source. Christensen (Ref. 2) converted this to Rosenthal presented his solution of a Fortunately, several investigators have travelling point source of heat (Ref. 1) measured actual heat distributions in arcs the dimensionless form: which has been the basis for most subse­ on water-cooled copper anodes (Refs. 9, -<f + R*> quent studies of heat flow in welding. 10). Using these results, it is possible to e = n (2) Christensen put the results in dimension­ determine whether the presence of a R* less form in order to demonstrate that distributed rather than a point source of the solution applies to many materials heat can explain the range of weld shape where the operating parameter, n, over wide ranges of heat input (Ref. 2). variation measured by Christensen and includes the travel speed, the thermal Since that time, a number of refinements others. diffusivity and the net heat input to the have been offered (Refs. 3-6). Several In the following sections, a general workpiece. have attempted to use a more realistic solution of a travelling distributed heat The solution to a distributed heat distributed heat source (Refs. 3, 4, 6), but source is briefly presented. Next, a source as shown in Fig. 1 can be formu­ none have solved the entire temperature dimensionless solution of a travelling lated by the use of Green's functions. field for a travelling distributed source. Gaussian heat distribution is presented The steady-state heat conduction equa­ More recently, a general form of the with a number of results. These are then tion in a moving coordinate of travel travelling distributed heat source has compared with experimental weld pool speed v (Ref. 1) is: been offered, but calculations of the shapes. thermal field were limited and unex­ plained (Ref. 7). It should be emphasized at the outset d2T* d2J* 52T* that this solution retains all but one of the AJA -(A-Y T* = dwz + z + 2 Christensen's experimental results indi- simplifying assumptions used by Rosen­ dy dz \ 2a / thal; the assumptions include absence of convective or radiative heat flow, con­ stant average thermal properties and a 2a Q* (3) Paper presented at the 64th Annual AWS quasi-steady state semi-infinite medium. Convention in Philadelphia, Pennsylvania, dur­ The only change is the use of a distrib­ ing April 24-29, 1983. uted rather than a point source of heat. Despite these simplifications, it will be where T* = (T — T0) e , and Q* shown that the results not only agree T. W. EAGAR, Associate Professor —Materials (w,y,z,) is the heat source moving at a Engineering, and N. -S. TSAI are with the Mas­with the Rosenthal solution in the limit, speed of v. sachusetts Institute of Technology, Cambridge, but that they are capable of explaining A derivation of the Green's function Massachusetts. most of the experimental scatter. which satisfies equation (3) and suitable 346-s | DECEMBER 1983 where <5Q is the amount of heat located Table 1—List of symbols at position (x',y',z') at time t'. The solution of an instantaneous a - thermal diffusivity Gaussian heat source is the superposition c — specific heat of a series of point heat source solutions G — Green's function k — thermal conductivity over the distributed region. By substitut­ n — operating parameter (n = qv/ e ing the Gaussian distributed heat source 2 <U ' 47ra pc[Tc - T0] for the point heat source Q, this super­ q — net heat input per unit time (power) position is performed by the integration Q — power distribution as shown below and derived in Appen­ o* - heat source moving at a speed of v dix B. R — distance to the center of arc 2 2 2 /2 (R = (w + y + z )' ) CO /»co R* — dimensionless distance from the cen­ 2 2 2 dT, (8) q(r)=qm-e-' ' °- ter of the arc (R* = \A + -A + f2]'/2 / T — temperature dx' dy' -co J -oo lo — initial temperature , (x'2 + y'2)/2„2 — critical temperature qe_ dt' Tc 2 3/2 u — dimensionless distribution parameter 2w pc[4ira(t- t')] (u = v<r/2a) V — travel speed of arc 2 2 2 w — distance in x direction in a moving (x-x') + (y-y') + (z-z') coordinate of speed v (w = x — vt) 4a(t-t') V — distance in y direction z — distance in z direction x24-y2 ; a — distribution parameter [qdt'jexp 4a(t-t')4-2<T2 4a(t-t') — density P /2 2 <5Q — incremental amount of heat 2irp c[47ra(t - t')]' [2a(t - t') 4- cr ] T — dimensionless time ft — dimensionless temperature This corresponds to the rise of temper­ (0 = P- To]/[Tc - T0]) i. — dimensionless distance in the moving ature during a very short time interval coordinate (£ = vw/2a) from time, t', to t' + dt' due to an i — dimensionless distance y amount of heat qdt' released on the if — dimensionless distance z surface. OO — infinity When considering the Gaussian heat Fig. 1 — A Gaussian distributed heat source moving on the surface of a semi-infinite plate, source travelling with a constant speed v, where the distribution parameter, rr, is the the total increase of temperature is the boundary conditions is given in Appendix standard deviation of the Gaussian function, sum of all such contributions in the time A. The resulting Green's function for a and ((, \l, J) are the dimensionless distances interval from t' = 0 to t' = t. A simpler distributed surface heat source Q* is: from the center of the arc in a coordinate expression of the solution is expected if moving at speed, v, in the x direction the solution is presented in a moving coordinate system with travel speed v. G(w|w',y|y') = This summation can be carried out again by integration: __^([w-w']2+[y-y']2 + z2)'/2 2 e 2a Q, with a distribution parameter tr. The (4) T-T = dT ([w-w']24-[y-y']24-z2)'/2 parameter, cr, has dimensions of length 0 , I <' and can be considered as the half-width U t'-o of the arc. Q is then given by: tdt' (t-f) -Vi where (w',y',z') is the location of the 2 heat source and (w,y,z) is the point of 7rpc(47ra)' I 02a(t-t')-F<r -(x2 + y2)/2 2 interest. The temperature distribution is Q(x,y) = ^e ff (6) expressed by an integration of the prod­ (x-vt')2 + y2 z2 uct of the Green's function and the heat 2 (9) distribution over the surface of interest With the Gaussian heat source, a solu­ 4a(t - t') + 2<r - 4a(t — t') (Ref. 11): tion other than the Green's function can be derived that will reduce the double This is similar to the solution obtained integration to a single integration and by Cline and Anthony (Ref. 13). For permit expression in dimensionless form. simplicity, let t"=t —t', then T - T0 = J j G(w|w',y|y') The heat conduction equation is solved dt" = dt' and x — vt' = w 4-vt", first in a fixed coordinate using an instan­ where w = x — vt. Equation (9) can be taneous Gaussian heat source. It is then written as: -— (w-w')Q* (! w e 2a —— — ^}ds' (5) integrated with respect to time in a mov­ ing coordinate to provide a quasi-steady { t"-'/2 state travelling solution. The solution, -T0= rv /2 2 which satisfies the differential equation of J o 7rpc(4*-a) 2at" 4- <r This is the general solution to a travel­ heat conduction in fixed coordinates, is 2 2 ling heat source of arbitrary distribution (Ref. 12): w2 + y + 2wvt" + v t"2 •4at" Q*. However, the solution is complicated -tat" + 2(7 2 (10) due to the double integral, and the results <5Q dt' dT - are not easily expressed in dimensionless t pc(47ra[t-t'])3 form.
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