CALCULATION OF COUPLED PROBLEM BETWEEN TEMPERATURE AND PHASE TRANSFORMATION DURING GAS QUENCHING IN

Applied Mathematics and Mechanics (English Edition), 2006, 27(3):305- -311CEditorial Committee of Appl. Math. Mech, ISSN 0253- 4827CA LCULATION OF COUPLED PROBLEM BETWEENTEMPER ATURE AND PHASE TRANSFORMATIONDURING GAS QUENCHING IN HIGH PRESSURECHENG He-ming (程赫明)，WANG Hong-gang (王洪纲)，XIE Jian-bin (谢建斌)(Department of Engineering Mechanics, Kunming University of Science and Technology,Kumming 650093, P. R. China)(Contributed by WANG Hong gang, Original Member of Editorial Comitee, AMM)Abstract: The gas quenching is a modern, effective processing technology. On the basisof nonlinear surface heat-transfer coefficient obtained by Cheng during the gas quenching,the coupled problem between temperature and phase transformation during gas quenchingin high pressure was simulated by means of finite element method. In the numericalcalculation, the thermal physical properties were treated as the functions of temperatureand the volume fraction of phase constituents. In order to avoid effectual“oscillation"of the numerical solutions under smaller time step, the Norsette rational approximatemethod was used.Key words: gas quenching; surface heat- transfer coefficient; rational approximation;temperature; phase transformationChinese Library Classifcation: TG1563; O551.32000 Mathematics Subject Classification: 80M10; 80A22Digital Object Identifier(DOI): 10.1007/s 10483-006-0304-yIntroductionThe gas quenching is a modern, effective processing technology. Because of the smallerdifference of temperature between the surface and middle of specimen in gas quenching, theresidual stresses in specimen after gas quenching are smaller than that after water or oil quench-ing. This environment is polluted less with this technique, and it is prevailing in the industries.But the researches on the mechanism during gas quenching are behind its applications. In orderto obtain the distribution of residual stresses and perfect mechanical properties, it is necessaryto control the phase transformation and limit distortions. Because thermal strains and thermalstresses can not be measured, the numerical simulation technology is an effctive approach tounderstand the distribution and variety of thermal strains, thermal stresses and microstructure.In recent years, the numerical simulation of quenching processing in various quenching mediais prevailing in the world. Although there are now some special soft-wares, which can simulatequenching processing, namely DEFORM-2, DEFORM-3, the important problem in numericalsimulation is the boundary condition of stress and temperature. The calculating accuracy ofthermal stresses and strains is closely related with the calculating precision of temperature field.For gas quenching processing, the key parameter of the calculation of temperature is the surfaceheat-transfer coefficient.On the basis of nonlinear surface heat-transfer coefficient obtained bv Ref.[2] during thegas quenching, the coupled problem between temperat中国煤化工ion during gas* Received Aug.29, 2003; Revised Dec.02, 2005fYHCNMHGProject supported by the National Natural Science Foundation of China (No.10162002) and the KeyProject of Ministry of Education of China (No.204138)Corresponding author CHENG He-ming, Professor, Doctor, E-mail: chenghm@public.km.yn.cn306CHENG He-ming, WANG Hong-gang and XIE Jian-binquenching in high pressure was simulated by means of finite element method. In the numericalcalculation, the thermal physical properties were treated as the functions of temperature and thevolume fraction of phase constituents. In order to avoid effectual“oscillation" of the numericalsolutions under smaller time step, the Norsette rational approximate method was used.1 Surface Heat- Transfer CoefficientsDuring quenching， the surface heat- transfer coefficients have a great infuence upon themicrostructure and residual stresses in steel specimen. For this reason, many researches intothis property have been studied. The variation of this property with temperature has beenthe subject of investigations. The results obtained are very sensitive to small variation in theexperimental conditions, which may lead to considerable discrepancies in the value obtained.Therefore, it must be found necessary to determine the effect of temperature on the surfaceheat transfer coeficient while using the actual experimental conditions that were to be usedduring the subsequent determination of thermal stress and strain. Prince and Fletcher21 havean effectual method, which can determine the relationship between temperature and surfaceheat transfer cofficient during quenching of steel plate.n Ref.[1], the explicit finite diference method, nonlinear estimate method and the experi-mental relation between temperature and time during quenching have been used to solve theinverse problem of heat conduction. The relationships between surface temperature and surfaceheat transfer cofficient of cylinder have been given (as shown in Fig.1). Figure 2 denotes thecomparison of surface heat-transfer cofficient between with and without phase transformation.0.0025,1 with phase transformnation0.0015-/ 2 without phase transformation豆≥0.00050.0005-0200400600800 1000弓06200600800Surface temperature T/'CSurface temperature T/CFig.1 Surface heat-transfer coefficient during Fig.2 Comparison of surface heat-transfer coef-gas quenchingficient between with and without phasetransformationFrom Fig.1 and Fig.2, the some conclusions can be obtained. (1) The surface heat-transfercofficients appear the stronger nonlinear property for surface temperature. Therefore, it isinaccurate to simulate the processing of gas quenching by means of linear surface heat-transfercoefficients. (2) In the initial stage of gas quenching, there are stronger heat exchanges be-tween specimen and gas, unlike in the initial stage of liquid quenching. Subsequently, it keepsapproximately constant. (3) In the stage of martensite phase transformation, the surface heat-transfer cofficients quickly rose. It is shown that martensite phase transformations have a greatinfuence upon the surface heat-transfer cofficients. In numerical simulation of temperature,thermal stresses and strains during gas quenching, the coupled effect between temperatures andphase transformations must be taken into account.中国煤化工2 Mathematical Model of Martensite in.JH| CNMH GIn the past a number of studies have dealt with prediction of microstructure evolution ofTemperature and Phase Transformation during Gas Quenching307steel during cooling. Up to the present, it has not been possible to calculate volume the frac-tion of constituent of phase transformation with rein theoretical expression3-5. Usually, thetemperature-time curve is discretized in series of isothermal steps. On each step the volumefraction of new phase formed is calculated by using isothermal kinetics. The isothermal trans-formation kinetics of ferrite/ pearlite and bainite is modeled according to the law developed byJohnson- Mehl3-5].φk= 1- exp(-bgt"k),(1)in which, bh and nk can be determined by the experimental data of isothermal phase transforma-tion4, k=1 ferrite/pearlite, k=2 bainite. bk and nk were approximated by polynomialsl3- -5],i.e.,lgbk = Cok + CixT + C2kT2 + C3kTr,(2)nk= Dok+ D1rT + D2kT2 + D3kT8.(3)The constants Cjk and Djn(j = 0, 1, 2, 3) were determined by the TTT diagram of 45 steel andminimum quadrate method, and were listed in Table 1.Table 1 Values of constants Cjk and Djxl3]Cjk/Djk CokCrkC2C3kDokD1D2kD3kk= 1328.315.200.0023480.0000012147.262.560.00092450000004691k:=263.58 0.001854- 0.0000242 :0.0- 21.93 0.1102-0.0001216.0In order to calculate volume fraction of constituent during continuous cooling by means ofEq.(1), we adopted the technique developed by Ivan Tzitellkov[5], i.e., the temperature- timecurve is discretized in series of isothermal steps. On each step the volume fraction of new phaseformed is calculated by using isothermal kinetics. The time transformation τ* of each stepshould be「1g(1 - φm)]"keTm+1(4)- bxTm+1Therefore, Eq.(1) becomesφ=1-exp[-bx(Ot++)"].(5)For the volume fraction of the constituent in martensite, we calculate it by means of followingformula[1:φM=(1-φ1 - φ2)(T-Mp2.57(6)M-T)where Ms and Mf denote initial temperature and final temperature of martensite transforma-tion, respectively.3 Functional and Finite Element Formula Temperature FieldA φ20 mm x 60 mm cylinder of 45 steel was quenched by 10 bar nitrogen gas from a tem-perature of 860°C. The nonlinear heat conduct equation is02T .02T.、10ToT(7)8z28r2'“r 8r0twhere C, and入denote the specific heat capacity a中国煤化工y, respectively,which are the function of temperature and volume:MYHCNMHGof the phasetransformation. The boundary condition of heat transr、oT。=h(T)(Ta- T);(8)308CHENG He-ming, WANG Hong-gang and XIE Jian-binthe initial condition isTlt=0= To(xk)，k= 1,2,3,(9)here h(T) is the surface heat-transfer coefficient, which is the function of temperature and thvolume fraction of phase constituents and is determined by finite diference method, nonlinearestimate method and the experimental relationships between time and temperaturel1l. Ta, Toand To denote the surface temperature of a cylinder, the temperature of quenching media andinitial temperature, respectively. The functional6l above problem is( 8T8TKn=-+ pCrTdSdtOt (1..{龄[()+(\)](10)r)sdt,where入=φnλk, Cvρ=φ:CuPk;(11)k=1h(Tn-1) is the surface heat-transfer cofficient at tn-1. T is fix in variational operationl6l.φk，λk: and Cupr(k=1 austenite, k=2 ferrite/ pearlite, k:=3 bainite, k=4 martensite) denote thevolume fraction of constituent, thermal conductivity and specific heat capacity, respectively. If8 nodes iso-parameter element is adopted, the finite element formula isKT+MT=F.(12)Here K is a conductive matrix, M is a heat capacity matrix, F is heat supply due to heatconvection on boundary. Under small time step,“oscillation” occurs in numeral solutions. Inorder to overcome this problem, the rational approximate method is adopted.4 Application of Rational Approximate MethodIn the functional (10), though T is fix in the variational operationl6, the various interpolatorscheme of T in time field can be found, such as various finite difference methods. The instabilityof finite difference method leads to“oscillation” in numeral solutions. In rational approximatemethod, we can not find out certain interpolator scheme of T, then treat approximately theanalytic solution of nonlinear heat conduct equation (12). It is convential to write Eq.(12) asfollowsl7l:dT，+KT=P.(13)duUnder initial condition (9), analytic solution of Eq.(13) isT(x,t)=exp(-tM- 1K)(To-K lP)+k-1p.(14)Its incremental form isT(x,ti+△t)= exp(- OtM~ - 1K)(T(x,t)- K-1P)+ k-1p.(15)In order to calculate the exp(- OtM 1K), let C =1 ccrins expansion ofexp(x) is中国煤化工exp(-C)=I-C +::YHCNM HG(16)here I is unit matrix. The exp(-C) can be replaced by its approximate formula. If theapproximate formula is identical with the right side of Eq.(16) in C2, the error range willTemperature and Phase Transformation during Gas Quenching309be in C3. In this paper, Norsette approximation was used to solve Eq.(15). The Norsetteapproximate formula can be taken asexp(-C)≈Cn(C)=(-1)"Z(-1)Ln-j(aC)j(17)Dv(C)j=0();(1+ aC)-1Here a is parameter, Ln-j(1) is the Laguerre polynomial multinomial of 1/a. SubstituteEq.(17) into Eq.(15), T:+1 and T denote T(x,ti+△t) and T(x, ti), respectively. We have(M + aOtK)T+1= ML}-'-(aOtK)j(T- k-1P)+ k-1P. (18)(-1)名()(M+ OtK)j=1Letting W; = a△tK/(M +△tK), Eq.(18) becomes(M + aOtK)T+1= M T:+器()Ln-jw?| + aOtP.(19)Solving Eq.(19) Ti+1 was obtained. In general, n can be taken as 2. As far to the value of a,it is shown that a = 1一√2/2 is better from theoretical analysis and practices7].5 ResultsFor 45 steel, the initial temperature and final temperature of ferrite/ pearlite transformationare 705 °C and 515 °C, respectively; the initial temperature of bainite transformation is 475 °C;Ms= 245 °C, and Mr = -90 °C; the latent heats of bainite and matensite phase transformationsare△HB,100% = 0.314 Ws/mm3 and△HM,100%=0.628 Ws/ mm3, respectively. The thermalphysical properties of 45 steel in various phases are listed in Table 2.Table 2 Nonlinear relationship of thermal physical properties under various phases and tempera-turel.81Temperature(°C)20200 .250300 .350450 .500700900 Phase4.104.304.584.754.90pCw/3.784.231.875.315.74"/P(10-3 W.s. mm-3. K-1)4.2:4.464.87B3.764.22 4.341.501.702.102.302.504.403.803.703.20F/P(10-2 w. mm-1. K-1)4.205.403.90 3.80 3.70 3.60MNote: A一Austenite, F/P- Ferrite/Pearlite, B- Bainite, M- Matensite.Figure 3 denotes the calculated results of temperature. Figure 4 denotes the comparison oftemperature field between the calculated values and experimental values. From the Figs.3 and4, the difference of temperature between the surface and middle of specimen in gas quenchingis smaller. It can be predicted that the thermal stresses and strains will be smaller, and itsdeformation can be controlled easily after gas quenchi;二3 coincide withexperimental values. It is also shown that adopted中国煤化工d the Tationalapproximation can avoid effectual“oscillation” of the:MYHCNMHGrsmallertimestep.Figures 5 and 6 denote the simulated variations of austenite phase ξ and martensite phase5 with quenching time during gas quenching, respectively. Compared with temperature field,310CHENG He-ming, WANG Hong-gang and XIE Jian-bintheir variations agree with actual situation. The percentage of martensite phase is 94%，andcoincides with experimental values.1000900 aExpericentia] vealues--- Caleulated valucx00Fp 710P" -0 mm0 mrn600-5009 um400-300r=10 mu .00[02060100140610uTimc 7iTire isFig.3 Calculated values of tempera-Fig.4 Comparison of temperature be-turetween calculated values and ex-perimental values0.8-g 0.8-台.0.6-0.60.4-0.2-60六100Tine itsIime thFig.5 Variation of austenite phaseFig.6Variation of martensite phasewith quenching times during gaswith quenching times duringquenchinggas quenching6Conclusions(1) The heat conduction during quenching is a nonlinear problem, and the calculation oftemperature is closely related to surface heat-transfer coefficient and phase transformation.For this reason, the coupled effects between nonlinear surface heat-transfer coefficient andphase transformation must be taken into account in simulation of temperature field during gasquenching.(2) In the numerical simulation of temperature during rapid cooling, the rational approxi-mation can avoid effectual“oscillation” of the numerical solutions under smaller time step.References1] Cheng Heming, He Tianchun, Xie Jianbin. Solution of an inverse problem of heat conduction of45 steel with martensite phase transformation in high pressure during gas quenching[J]. Journalof Materials Science and Technology, 2002, 18(4):372- 374.2] Prince R F，Fletcher A J. Determination of surface h中国煤化工ng quenching ofsteel plates[]. Journal of Metals Technology, 1980,3] Cheng Heming, Wang Honggang, Chen Tieli. Solution:THC N M H Gneat conductionof 45 steel during quenching[J]. Acta Metallurgica Sinica, 1997， 33(5):467-472.4] Denis s, Farias D, Simon A. Mathematical model coupling phase transformation and temperatureevolution in steels[J]. ISIJ International, 1992, 32(4):3163- 3172.Temperature and Phase Transformation during Gas Quenching311[5] Ivan Tzitzellkov, Paul H. Eine mathematische methode zur beschreibung des umwandlungsverhal-tens eutektoidischer staehle[J]. Archiv Eisenhuttenwes, 1974, 45(5):5254 5262.6] Wang Honggang. Theory of Thermoelasticity[M]. Tsinghua University Press, Beijing, 1989， 131-138 (in Chinese).7] Cheng Heming, Huang Xieqing, Fan Jiang, Wang Honggang. The application of rational approxi-mation in the calculation of a temperature field with a non-linear surface heat-transfer cofficientsduring quenching for 42CrMo steel cylinder[J]. Metals and Materials, 1999, 5(5):445- 450.8] Bates E. ASM Handbook 4[M]. Heat Treating, ASM International, 1991, 72- 75.9] AdamsJ A, Rogers D F. Computer- Aided Heat Transfer Analysis[M]. McGraw-HillC, New York,1973, 128- -154.中国煤化工MHCNMHG