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ABSTRACT. An innovative method forestimating the actual cooling rate in awelded section is presented. The methodis based on applying a weighting factor tothe Rosenthal analytical solutions forthick and thin plates. The factor is deter-mined from the heat-affected zone(HAZ) width, obtained from etched sec-tions, and reflects the actual response ofthe plate to the heat flow condition. Previ-ous formulations in the literature arebased on the assumption of either thin-plate or thick-plate conditions, while mostactual conditions lie somewhere betweenthe two extremes. Limited experimentalmeasurements of cooling rate, carried outby instrumented welding, showed goodagreement with predicted values. Themodel was further used to predict the peaktemperature profile across the HAZ.

Introduction

A typical arc weld thermal cycle con-sists of very rapid heating (several hun-dreds of degrees per second) to a peaktemperature, followed by relatively fastcooling (a few tens or hundreds of degreesper second) to ambient temperature. Themicrostructural changes in the weld zone,as well as the weld heat-affected zone(HAZ), are greatly dependent on theheating and cooling rates, which in turndepend on the weld heat input (a functionof arc energy, travel speed, and the ther-mal efficiency of the process), the platethickness/geometry, and the initial or in-terpass temperature. The microstructuralchanges will directly affect the propertychanges (whether mechanical or corro-

sion related) in the weld zone and theHAZ. Therefore, it is important to be ableto predict the actual thermal cycle charac-teristics such as peak temperature andcooling rate if microstructure is to be char-acterized and correlation with the proper-ties is sought. This becomes more signifi-cant if the effect of heat input on themicrostructural changes in the HAZ is tobe examined for a given material, as theheat input is only a rough, simplified pa-rameter specific to a welding process.Moreover, knowledge of the cooling rateis required for simulation approaches.

The most widely used and the bestknown analytical solutions to predict weldthermal history and cooling rate are thoseof Rosenthal (Refs. 1, 2). His approachwas based on the assumption of a movingpoint heat source on the plate surface, ne-glecting any heat transfer from the sur-face. It was also assumed that physical co-efficients were constant (i.e., independentof temperature), and all the energy fromthe welding equipment was transferred tothe arc. There have been many other rig-orous approaches with more realistic as-sumptions (e.g., Refs. 3–14) that take intoaccount surface heat transfer, an extendedor diffuse Gaussian heat source, depen-dence of thermal properties on tempera-ture, etc. All these modifications are

rather complicated. One of the findingsfrom these investigations (e.g., Ref. 8) wasthat the cooling rate (or cooling time) inthe HAZ did not depend on the locationof the point heat source and, therefore,could be found from Rosenthal’s solution.

Ashby and Easterling (Ref. 4) simpli-fied the two limiting solutions derived byRosenthal to obtain temperature/timeprofiles in the HAZ. One set of solutionswas derived for thick plates (assuming 3-D heat flow) and the other for thin plates(assuming 2-D heat flow). There is also anequation to determine a critical thicknessfor a given heat input (or a critical heatinput for a given plate thickness, if used inreverse) at which the 2-D conditionchanges to 3-D (Ref. 15). There have alsobeen some attempts to define dimension-less parameters that define the transitionpoint between 2-D and 3-D conditions(Refs. 16, 17). However, these criteria maybe too simplistic. Real welds are morelikely to lie between the two limiting solu-tions: a situation classified by some re-searchers as 2.5-D (Ref. 16),for whichthere is no simple solution (Ref. 1). Thequestion then is where a particular caselies with respect to the 2-D and 3-D con-ditions. The objective of this paper is to in-troduce a simple method to answer theabove question. The goal is to find the ac-tual values of parameters (as opposed tothe lower-bound or upper-bound values),such as cooling rate and peak temperatureprofile, for a given weld section. This canbe very useful as experimental measure-ment of weld thermal cycles, e.g., by em-bedding thermocouples in the HAZ, is insome cases impractical considering thesmall size of the HAZ. The approach, ex-plained in the next section, is based on de-riving a parameter (i.e., HAZ width) thatcan be measured by sectioning a sample,and using this parameter to calculate a

SUPPLEMENT TO THE WELDING JOURNAL, OCTOBER 2005Sponsored by the American Welding Society and the Welding Research Council

Estimation of Cooling Rate in the Welding ofPlates with Intermediate Thickness

Cooling rates estimated by Rosenthal’s thick- and thin-plate solutions can be modified by a weighting factor to account for

intermediate values of plate thickness

BY K. POORHAYDARI, B. M. PATCHETT, AND D. G. IVEY

KEYWORDS

Cooling RateHeat-Affected ZoneWeld Thermal CycleInstrumented WeldingTransformable SteelsRosenthal’s Analytical SolutionsGas Tungsten Arc WeldingHeat Input

K. POORHAYDARI (kioumars@ualberta.ca),D. G. IVEY, and B. M. PATCHETT are with theDepartment of Chemical and Materials Engineer-ing, University of Alberta, Alberta, Canada.

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weighting factor to apply to the thick- andthin-plate values of the parameters of in-terest. To compare the calculations withexperimental data, limited instrumentedautogenous GTAW (gas tungsten arcwelding with thermocouples embedded inhorizontal flat plates) was performed torecord the thermal cycles with a data a qui-sition system.

Theoretical Approach

The two limiting solutions to theRosenthal equations of a moving pointheat source for regions outside the fusionzone are laid out below (Ref. 15). The so-lutions give the temperature variationduring cooling as a function of time for agiven location, the peak temperature (TP)as a function of distance from the heatsource, and the weld time constant (Dt8–5),which is the cooling time from 800° to500°C (1472° to 932°F).

Thick plates (3-D):

Thin plates (2-D):The parameters in the above equations

are defined and their values are reportedin Table 1. Equation 9 calculates the criti-cal plate thickness, dC, over which thecrossover between the 2-D and 3-D condi-tions of heat flow takes place (Ref. 15).Note that the unique characteristic of

dq v

c T TC = ◊

-+

-

Ê

ËÁ

ˆ

¯˜

È

ÎÍÍ

˘

˚˙˙

/

2

1

500

1

800(9)

0 0

1/2

r

1 1

500

1

800

(8)2

02

02q

=

-( )-

-( )

Ê

Ë

ÁÁÁÁ

ˆ

¯

˜˜˜̃

T T

Dtq v

cd8 5

2

224

(7)- =( )/

plr q

T Te

q v

d c rP - =

Ê

ËÁ

ˆ

¯˜0

1 22

2(6)

p r

//

T Tq v

d ct

r

at- =

( )-

Ê

ËÁÁ

ˆ

¯˜̃0 1 2

2

44

(5)/

exp/

plr

1 1 1(4)

1 0 0q=

--

-

Ê

ËÁ

ˆ

¯˜

500 800T T

Dtq v

8 512

(3)- = /

plq

T Te

q v

crP - =

Ê

ËÁ

ˆ

¯˜0 2

2(2)

p r

/

T Tq v

t

r

at- = -

Ê

ËÁÁ

ˆ

¯˜̃0

2

2 4(1)

/exp

pl

Fig. 2 — Thermocouple/plate attachment on the bottom ofthe plate. The image was taken after welding.

Fig. 1 — Schematic for calibration (shifting and fitting) of the calculated peaktemperature profile.

Table 1 — Definition, Units, and Values (for Carbon Steel) of the Parameters Used in Equations1–9 (Ref. 4)

Symbol Definition and unit Value

T Temperature, °C (°F) —TP Peak temperature, °C (°F) —t Time, s —Dt8–5 Cooling time from 800° to 500°C, s —r Radial/lateral distance from weld, m (in.) —T0 Initial temperature, °C (°F) 25 (77)l Thermal conductivity, Js–1m–1°C–1 (Js–1in.–1°F–1) 41 (1.87)a Thermal diffusivity, m2/s (in.2/s) 9.1 ¥ 10–6 (14.1 x 10–3)rc Specific heat per unit volume, Jm–3°C–1 (Jin.–3°F–1) 4.5 ¥ 106 (0.13 x 103)d Plate thickness, m (in.) 8 ¥ 10–3 (0.31)q Arc power, J/s —v Travel speed, m/s (in./s) —q/v Heat input, J/m (J/in.) —

Fig. 3 — Transverse section of a 0.5 kJ/mm (12.7kJ/in.) weld sample showing the location of a ther-mocouple tip in the HAZ. The dark background areais Bakelite™ used for metallographic mounting. Theother regions are base metal (BM) and weld metal(WM).

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cooling rate is that it is independent of thedistance from the heat source, at least inthe HAZ. This fact has also been con-firmed by numerical analysis (e.g., Ref.18) as well as experiments (e.g., Ref. 19),where researchers have found that coolingrate is only dependent on heat input, plategeometry/thickness, and plate initial tem-perature (i.e., preheating). This assump-tion is widely accepted in the welding in-dustry (e.g., Refs. 12, 15, 20). One abstractparameter appearing in these equations isheat input, q/v, which in electric arcprocesses is defined as

E is the arc voltage in V, I is arc currentin A, and h is the thermal efficiency. Ther-mal efficiency, which is in fact the ratio be-tween the actual heat input to the work-piece and the energy output of the weldingmachine, is lower than 1, because somepart of the arc energy is dissipated to thesurroundings by radiation, convection, orconduction, and is therefore lost. In thispaper, the term actual heat input (a.k.a. ef-fective heat input or net heat input) refersto q/v with h < 1, and the term nominalheat input refers to q/v with h = 1.

The main idea here is to evaluate theactual state of heat flow, whether 2-D, 3-D, or something in between, for which theactual response of the joint to the weldthermal cycle can be used. It is well knownthat the visible HAZ boundaries in trans-formable steels correspond to the TS(solidus temperature or peritectic temper-ature) on the side adjacent to the weldzone and to the Ac1 (the lower critical tem-

perature upon heating), or simply A1 (Ref.15), on the side adjacent to the unaffectedbase metal. Therefore, the HAZ width, W,can be simply formulated as

W = rA1 – rTs (11)

Finding rA1 and rTs from Equations 2 and6 and substituting them into Equation 11will yield the HAZ width for thick- andthin-plate assumptions, respectively:

WThin-plate = C2 · (q/v) (14)

Based on the assumption that the actual situation lies between the two lim-iting solutions, the actual HAZ width canbe related to the above values (Equations12 and 14) as

WActual = WThick-plate+ F · (WThin-plate – WThick-plate) (16)

F, varying between 0 and 1, is theweighting factor that calculates the devia-tion from the thick-plate solution with re-spect to the thin-plate solution. It can beshown that Equation 16 can be easily de-rived, if the same kind of relationship forestimation of rA1 and rTs was consideredinitially (i.e., the following equations), and

Ce

T A

d c T T A T

S

S

2

1

2

1

0 1 0

2

2(15)

=È

ÎÍÍ

˘

˚˙˙

◊-( )

-( ) -( )È

Î

ÍÍÍ

˘

˚

˙˙˙

p

r

Ce c

A T T TS

1

1

2

1 0

1

2 0

1

2

2

1 1(13)

=È

ÎÍÍ

˘

˚˙˙

◊

-( )-

-( )

È

Î

ÍÍÍÍÍ

˘

˚

˙˙˙˙˙

p r

W C q vThick plate- = ◊( )1

1

2 (12)/

q vE I

v/

. .= h(10)

Fig. 4 — The actual weld thermal cycles measured in the HAZ close to the weldinterface for different weld samples with heat inputs of 0.5, 1.5, and 2.5 kJ/mm(12.7, 38.1, and 63.5 kJ/in.). The cooling time from 800° to 500°C (1472° to932°F) increases dramatically as the heat input increases.

Table 2 — HAZ Widths and Weighting Factors for Three Weld Samples

Nominal heat HAZ width, mm (in.) Weightinginput, kJ/mm Thick-plate Thin-plate Measured factor, F(kJ/in.) (Eq. 12) (Eq. 14) (Eq. 19)

0.5 1.76 2.05 1.86 0.34(12.7) (0.069) (0.081) (0.073)1.5 3.05 6.16 4.80 0.56(38.1) (0.120) (0.243) (0.189)2.5 3.93 10.27 7.40(a) 0.55(63.5) (0.155) (0.404) (0.291)

(a) An estimated value based on the extension of the HAZ boundary curves, as the HAZ/BM boundary intersectsthe bottom of the plate for the heat input of 2.5 kJ/mm.

Fig. 5 — Variation of HAZ width with heat input. The experimental values liebetween the upper- and lower-bound lines (thin- and thick-plate solutions) pre-dicted from the Rosenthal theory.

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then substituted into Equation 11.rA1-Actual = rA1–Thick-plate+ F · (rA1–Thin-plate – rA1–Thick-plate) (17)

rTs-Actual = rTs–Thick-plate+ F · (rTs–Thin-plate – rTs–Thick-plate) (18)

Rearranging Equation 16 gives thevalue of F for any given weld sample:

F = (WActual – WThick-plate)/(WThin-plate – WThick-plate) (19)

It is proposed that the thermal para-meters of importance can be estimated byweighting the thick- and thin-plate solu-tions, using the weighting factor F:

DtActual = DtThick-plate+ F · (DtThin-plate – DtThick-plate) (20)

TP,Actual = TP,Thick-plate+ F · (TP,Thin-plate – TP,Thick-plate) (21)

Equation 20 is similar to one proposed

by Hess et al. (Ref. 19), who proposed thata “correction factor,” which would repre-sent “the degree of infiniteness,” shouldbe calculated by measuring the instanta-neous cooling rate at a particular timefrom an experimentally obtained thermalcycle. This value would then be comparedwith the corresponding values obtainedfrom Rosenthal’s equations for thick- andthin-plate conditions through Equation20. This correction factor would be used toobtain corrected values of cooling rate atother times (or temperatures). The ap-proach adopted here, and shown to be re-liable in the next section, demonstratesthat the correction factor can be obtainedfrom the measurement of the HAZ widthindependently, and therefore the instru-mented welding test can be eliminated.

Note, as mentioned earlier, the locationof the heat source does not affect the cool-ing rate but shifts the peak temperatureprofile relative to the plate geometry. Inreality, the heat source may not coincidewith the top surface of the plate. Going

back to the fundamental assumption thatthe location and peak temperatures of theHAZ boundaries are known, the exactpeak temperature profile can be plottedwith respect to the distance from the topsurface of the plate, which is a tangible ref-erence point. Figure 1 shows schematicallyhow a calculated profile (either based onthick-plate/thin-plate assumptions orweighted as proposed by Equation 21) isshifted and fitted into the known locationsdesignated by points A (representing thesolidus temperature) and B (representingthe A1 temperature). Equation 22 formu-lates this mathematically.

(22)

ST=(WC–WM).TS–TPTS–A1 (23)

The symbol * on r values in Equation 22designates the distance of a given point inthe plate (specified by a peak tempera-ture) from the top surface. Otherwise, rdesignates the distance with reference tothe heat source. Therefore, rTp values area series of points for which TP is calculatedfrom Equation 21; rTs is one of thesepoints where TP = TS, and r*

Ts is measuredfrom the weld section. WC and WM are thecalculated and measured values, respec-tively, of the HAZ width. As shifting theentire curve from the point A' to A in Fig.1 (represented by the initial terms inEquation 22) may move an arbitrary pointC' to C'' and not necessarily to C, a furtheradjustment is needed. This adjustment isrepresented by the term STp in Equation22 and calculated from Equation 23,which provides an additional shift (e.g.,

r r r r STp Tp Ts Ts Tp* *= - -Ê

ˈ¯ -

Fig. 6 — Calibrated peak temperature profiles for a nominal heat input of0.5 kJ/mm (12.7 kJ/in.). The weighted profile is plotted along with the thick-and thin-plate solutions.

Table 3 — Cooling Time and Cooling Rate from 800° to 500°C (1472° to 932°F) for Three WeldSamples

Nominal Cooling time (Dt8–5), s Mean cooling rate, °C/sheat input, (°F/s)(kJ/mm) (Eq. 26)(kJ/in.) Thick-plate Thin-plate Estimated Measured Estimated Measured

(Eq. 3) (Eq. 7) (Eq. 20)

0.5 1.2 2.6 1.7 1.8±0.1 178.8 171.0±6.2(12.7) (321.8)(307.8±11.2)1.5 3.6 23.6 14.8 17.2±1.5 20.2 16.8±1.2(38.1) (36.4) (30.2±2.2)2.5 5.9 65.6 38.6 37.7±10.1 7.8 8.0±2.9(63.5) (14.0) (14.4±5.2)

Fig. 7 — Peak temperature profiles weighted and fitted for three different weldsamples; heat input values of 0.5, 1.5, and 2.5 kJ/mm (12.7, 38.1, and 63.5kJ/in.).

P

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point C'' to C or point B'' to B) propor-tional to the temperature difference in TPfrom the reference point used here (i.e.,TS). Note that this adjustment is also pro-portional to the difference between thecalculated and measured values of theHAZ width, which is considerable whenthe calculations are based on thick- orthin-plate assumptions. For the casewhere calculation of peak temperature isbased on weighted values (i.e., Equation21), the adjustment is minimal (even nulltheoretically as the calculated widthshould represent the actual width in thefirst place, but minimal due to accumu-lated errors from rounding throughout thecalculations).

The following equations can be used tofind the critical temperatures A1 and Tm(melting point, which is an upper-boundvalue for Ts), in case this information isnot readily available. The alloying addi-tions are in wt-% and the temperaturesare calculated in K (Ref. 21).

A1 = 996 – 30 Ni – 25 Mn – 5 Co + 25 Si + 30 Al + 25 Mo + 50 V (24)

Tm = 1810 – 90 C (25)

Experimental Procedure

A plate of Grade 690 microalloyedsteel (containing 0.08% C and microal-loyed mainly with Ti, Nb, and V), 8 mm(0.31 in.) in thickness, was used in thisstudy. The designation 690 refers to thespecified minimum yield strength inmegapascal (Canadian Standards Associ-ation designation (Ref. 22)), equivalent to100 in ksi. The plate was cut into pieceswith approximate dimensions of 285 mm(11.22 in.) by 165 mm (6.50 in.) to be usedfor instrumented welding. The oxidescales were removed from both surfacesby milling to allow for good arc conductiv-ity and stability, as well as to remove smallbumps on the surface and make them veryflat. Holes were drilled about 2 in. (50.8mm) apart on the bottom side of the platesalong the central line with variable depths.Four thermocouples (three of type K andone of type S) were used for each experi-ment. The type S thermocouple was usedfor the hole close to the weld interface,which experienced the highest tempera-tures. Thermocouple wires, 0.25 mm(0.010 in.) in diameter were passedthrough a ceramic insulator, 3.18 or 1.59 (1⁄8 or 1⁄16 in.) OD, and spot welded at oneend to make the junction. The thermo-couples were secured to the plate withsprings as shown in Fig. 2, which providedgentle pressure on the back of the ceramicinsulator to maintain contact between thethermocouple junction and the plate

throughout welding. Welding was performed using DCEN

autogenous GTAW with DC current and atungsten electrode used in negative polarity. Nominal heat inputs of 0.5, 1.5,and 2.5 kJ/mm (12.7, 38.1, and 63.5 kJ/in.)were applied by selecting an arc voltage of12.2 V, an arc current of 150 A, and travelspeeds of ~ 3.6, 1.2, and 0.7 mm/s, respectively. After welding, the plateswere precision cut transversely from thecenter of the thermocouple holes, andthen mounted, ground, polished, andetched, to reveal the weld interface andthe HAZ. Figure 3 shows a cross section-for one of the samples.

Results and Discussion

Figure 4 shows representative thermalcycles (for three different heat inputs), ex-perimentally obtained at locations close tothe weld interface. The thermal cycle isvery fast for a nominal heat input of 0.5kJ/mm (12.7 kJ/in.) and gets considerablyslower as the heat input increases to nom-inal values of 1.5 and 2.5 kJ/mm (38.1 to63.5 kJ/in.). The curves (not all shownhere) were analyzed to obtain coolingtimes and peak temperature profiles, asdescribed below.

Weighting Factors

Figure 5 compares the values of theHAZ width measured directly below thecenterline (on sections without the ther-mocouple holes) for several weld samples(nominal heat inputs 0.5–2.5 kJ/mm), withthe theoretical thick- and thin-plate solu-tions found from Equations 12 to 15. Ad-ditional weld samples with nominal heatinputs of 0.7 and 1.0 kJ/mm, producedwithout thermocouple instrumentation,were sectioned and their HAZ widths areincluded in Fig. 5. Again the welding pa-rameter that varied to produce the re-quired heat input was the travel speed;other parameters were as stated earlier.These additional samples provided data tosee if the HAZ width was linear with re-spect to the heat input. As shown in Fig. 5,the dependence deviates from linearity.The calculated weighting factors for thethree weld samples, for which instru-mented welding was performed, are re-ported in Table 2.

A value of Ae1 = 695°C (1283°F) wasused here, which was obtained from theCCT (continuous cooling transformation)diagram for the specific Grade 690 mi-croalloyed steel (Ref. 23). This value issomewhat higher than the equilibriumvalue of A1 = 684°C (1263°F) obtainedfrom Equation 24. As Equation 25 calcu-lates Tm rather than the solidus tempera-

ture, a value of TS = 1500°C (2732°F) wasused here, which is slightly lower than theTm = 1530°C (2786°F) obtained fromEquation 25. This value also lies betweenthe peritectic temperatures for the Fe-Cand Fe-Mn binary systems (i.e., 1493° and1504°C, or 2719° and 2739°F, respectively(Ref. 24)). A value of h = 0.75 was used inEquation 10 to obtain the actual values ofheat input. Thermal efficiency is not easyto determine, as it depends on many fac-tors such as the weld process, weldingequipment and setup, travel speed, thematerial to be welded (anode work func-tion), arc voltage, and current. In GTAW,thermal efficiency is highest for DCENand increases as the travel speed in-creases. Fuerschbach and Knorovsky(Ref. 25) reported arc efficiencies ofaround 0.7–0.8 for GTAW-DCEN afterconducting calorimetric tests on severalsteels and a Ni-based alloy. Arc efficiencyreached a plateau of 0.8 as the travel speedincreased. Values within the same rangewere also confirmed by Kou (Ref. 26) andused by others (e.g., Ref. 12). Althoughmuch lower values for the thermal effi-ciency in GTAW have been reported in theliterature elsewhere (e.g., Refs. 1, 5, 15), avalue of h = 0.75 obtained through morerecent calorimetric experiments wasdeemed to be more reasonable for thesetup and welding characteristics usedhere, which is in agreement with the valuereported in the ASM Handbook (Ref. 27).

The finding that the experimental val-ues for lower heat-input samples arecloser to the thick-plate solution in Fig. 5is reasonable, as the weld zone sizes forthese samples are relatively small with re-spect to the thickness of the plate. UsingEquation 9 in reverse gives a critical heatinput of 0.17 kJ/mm (4.3 kJ/in.), corre-sponding to a nominal heat input of 0.23kJ/mm (5.7kJ/in.) assuming h = 0.75, fora fixed plate thickness of 8 mm (0.31 in.).However, as can be seen, assuming heattransfer behavior is 2-D above the criticalvalue is not reasonable, as none of the actual data lie on the thin-plate solutionline. For the 2-D situation to prevail, theweld metal zone should span over thewhole width of a thin plate, so that theheat transfer occurs only within the planeof the plate, which is not the case even forthe highest heat input used here.

Cooling Time/Rate

Table 3 compares the predicted values ofcooling time and cooling rate for nominalheat inputs of 0.5, 1.5, and 2.5 kJ/mm(12.7, 38.1, and 63.5 kJ/in.), with those ex-perimentally obtained. Note, that meancooling rate for the temperature range800° to 500°C (1472° to 932°F) is calcu-lated in °C/s from a simple equation:

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The experimental values reported inTable 3 are the arithmetic means amongthe thermocouple readings, as long as thepeak temperature is above 800°C(1472°F). The predicted range of coolingtimes for a 0.5 kJ/mm (12.7 kJ/in.) weldsample was found to be between 1.2 and2.6 s. This translates to mean cooling ratesapproximately between 110° and 250°C/s(~200° to 450°F/s). The predicted rangesof mean cooling rates for the other weldsamples were found to be approximately13°–85°C/s (23°–153°F/s) for a heat inputof 1.5 kJ/mm (38.1 kJ/in.) and 5°–50°C/s(9°–90 °F/s) for a heat input of 2.5 kJ/mm(63.5 kJ/in.). The wide range of coolingvalues suggests a great deal of uncertainty.Having more specific numbers, using themethod proposed here, is essential for anysimulation or microstructure/propertycorrelation studies.

The standard deviation of measuredcooling time values increased relative tothe average value as the heat input in-creased due to heat accumulation in theplate during welding. Heat accumulationincreased the effective initial temperaturein the plate as welding continued. For in-stance, the first thermocouple for 2.5kJ/mm (63.5 kJ/in.) welding recorded acooling time of 23.5 s while the next threethermocouples recorded cooling times of38.0, 42.6, and 46.8 s in sequence. This phe-nomenon would not happen for lower heatinputs (low relative to the thickness andarea of the plate), as the travel speed wouldbe high enough to minimize the tempera-ture rise during welding. Using a largerplate would also result in a smaller stan-dard deviation. Nevertheless, the range isstill much smaller than the range betweenthe thick- and thin-plate values.

Peak Temperature Profiles

A weighted peak temperature profile,obtained from Equation 21 and calibratedby Equation 22, is shown in Fig. 6 (solidline) for a weld sample with a nominalheat input of 0.5 kJ/mm (12.7 kJ/in.). Thetheoretical results with thin- and thick-plate assumptions are also plotted(dashed lines). All profiles are calibratedwith respect to the actual plate geometryand location of the HAZ boundaries ac-cording to Equation 22, to show the dif-ferences in the shapes (curvatures) only.As can be seen, the different assumptionsdo not markedly change the shape of thepeak temperature profiles. For this heatinput (0.5 kJ/mm), the correspondingweighting factor (Table 2) predicts a pro-

file with a shape close to that predicted bythe thick-plate solution.

The weighted and calibrated peak tem-perature profiles for the three values ofheat input were determined and are plot-ted in Fig. 7. Some of the values experi-mentally obtained are indicated too. Con-tact between the thermocouple tip and theplate was not well maintained for all ther-mocouples throughout welding. There-fore, only those that maintained good con-tact are reported here. Spot welding of thethermocouple junctions to the plate (e.g.,Ion et al. (Ref. 21)) would be ideal, as thebest response time is expected from smallbead volumes with maximized surface con-tact. Spot welding at the bottom of drilledholes was impractical for the present setup,consisting of small-diameter holes andsmall thermocouple junction sizes.

Applications and Limitations

The thermal cycles and the calculatedpeak temperature profiles have been usedfor kinetic analysis of carbide particle dis-solution and coarsening in the HAZ, whilethe cooling rate estimations have beenused for interpretation of phase transfor-mations and carbide reprecipitation (Ref.28). Although this approach was geared totransformable steels in this paper, it is ex-pected that the concept can be applied toother materials, as long as the HAZ hasvisible boundaries in the etched macrosection and the peak temperatures for theboundaries are known.

One important practical point in exam-ining the weld weld zone, or in selecting thewelding parameters to reach a certaincooling rate in the joint, is that cooling rateis not the same in all radial directions on atransverse section. It has been observed ex-perimentally in very early measurements(Ref. 19) that on plates of intermediatethickness (2.5-D heat flow condition), thecooling rate is higher at the top of the plateadjacent to either side of the bead than atthe bottom (directly under the depositedmetal) and the HAZ width is smaller. Thisis due to the difference in the availability ofquenching material at different radial di-rections. It is expected that the method ofcorrection, based on HAZ width pre-sented here, will be capable of determiningthe actual cooling rate along the directionof interest, by measuring the HAZ widthalong that direction. This requires moreinvestigation. Note that some uncertaintyabout the material thermal propertiesand/or welding parameters may be less im-portant for the proposed method. The pro-posed method compares theoretical solu-tions with, and fits them to, the actualbehavior of the welded section, whose be-havior is based on effective values of thethermal parameters, probably unknown tothe investigator.

The method introduced in this paperwas used for bead-on-plate or autogenouswelding on flat plates, which is used widelyfor mechanical and microstructural exam-inations of the weld HAZ. One shouldnote that the original Rosenthal thick- andthin-plate solutions were derived for suchconditions. The general concept behindthe method, however, may be applied toweld sections with other geometries, aslong as the initial upper-bound and lower-bound equations applicable to thosegeometries are available, or the Rosenthalsolutions are modified so that they can beapplied to other geometries. This alsoneeds further investigation. Examinationof a wider range of welding variablesneeds to be considered as future work.

Summary

An innovative method for estimationof cooling rate in a welded section was pre-sented in this paper. The method is basedon applying a weighting factor to theRosenthal analytical solutions for thickand thin plates. The factor is determinedfrom the HAZ width, obtained frometched sections, which reflects the actualresponse of the plate to the heat flow con-ditions. While the lower- and upper-bound solutions to Rosenthal’s equation,derived by assuming thick- and thin-plateconditions, provide a wide range of valuesfor cooling rate during the weld thermalcycle, the method and formulations pre-sented here yielded specific numbers veryclose to those obtained experimentally.The advantage of this method is the sim-plicity of the calculations and the experi-mental procedure; only a macro-etchedweld section is needed. Instrumentedwelding tests (by embedding thermocou-ples) can be eliminated. The method wasalso used to obtain a weighted peak tem-perature profile and to calibrate the shapeand size of the profile.

The method was applied only to hori-zontal flat plates, where the Rosenthalequations directly apply. More work isneeded for application of this method (orthe general idea) to plates with othergeometries/positions and examination of awider range of welding variables. Never-theless, the limited experimental work hasdemonstrated the promise of this simplemethod, as a first step in the extraction ofuseful information from a weld HAZ sec-tion that contains thermal history infor-mation. This information can be appliedto correlation studies involving mi-crostructure, simulation, and modeling.

Acknowledgments

The authors wish to thank the NationalSciences and Engineering ResearchCouncil (NSERC) of Canada and IPSCO

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Inc. for their financial support of this pro-ject. The authors are also grateful to ClarkBicknell for welding, Les Dean for writingthe LabVIEW 5.0 code for thermal dataacquisition, and Walter Boddez for discus-sions on thermocouple fabrication andsupply purchases.

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