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Analysis of melting in a subcooled two-componentmetal powder layer with constant heat fluxTiebing Chen, Yuwen Zhang*Department of Mechanical and Aerospace Engineering, University of Missouri-Columbia, Columbia, MO 65211, United StatesReceived 1 February 2005; accepted 18 July 2005Available online 11 October 2005AbstractMelting of a subcooled two-component metal powder layer is investigated analytically. The powder bed considered consists of amixture of two metal powders with significantly different melting points. Shrinkage induced by melting is taken into account in thephysical model. The temperature distributions in the liquid and solid phases are obtained using an exact solution and an integralapproximate solution, respectively. The effects of porosity, Stefan number, and subcooling on the surface temperature and solidliquid interface are also investigated. The present work provides a strong foundation upon which the investigation of complexthree-dimensional selective laser sintering (SLS) process can be based.? 2005 Elsevier Ltd. All rights reserved.Keywords: Melting; Metal; Powder layer1. IntroductionDirect Selective Laser Sintering (SLS) is an emergingtechnology of Solid Freeform Fabrication (SFF) viawhich 3-D parts are built from the metal-based powderbed with CAD data 1. A fabricated layer is created byselectively fusing a thin layer of the powders with scan-ning laser beam. After sintering of a layer, a new layer ofthe powder is deposited in the same manner and a 3-Dpart can be built in a layer-by-layer process.A mixed metal powder bed, which contains two typesof the metal powders possessing significantly differentmelting points, is used extensively in direct SLS of metalpowders 2,3. The high melting point powder never meltin the sintering process and plays a significant role as thesupport structure necessary to avoid boiling phenom-enon, which is the formation of spheres with the approx-imate diameter of the laser beam. The particularmaterial properties and methods of material analysisof the metal-based powder system for SLS applicationsare addressed by Storch et al. 4 and Tolochko et al.5. Fundamental issues on direct SLS are thoroughly re-viewed by Lu et al. 6. In fabrication of near full densityobjects from metal powder, direct SLS is realized viamelting and resolidification induced by a directed laserbeam. It is a good starting point to investigate a simpli-fied 1-D model to get a better understanding of themelting process in direct SLS before a much more com-plicated 3-D model is investigated.Fundamentals of melting and solidification have beeninvestigated extensively and detailed reviews are avail-able in Refs. 7,8. Melting in SLS of the metal powdersis significantly different from the normal melting sincethe volume fraction of the gas in the powders decreasessignificantly after melting. Therefore, a significant den-sity change of the powder bed accompanies the meltingprocess. Melting and solidification in 1-D semi-infinitebody with density change under the boundary conditionof the first kind have been investigated by Zckert andDrake 9, Crank 10, Carslaw and Jaeger 11 and1359-4311/$ - see front matter ? 2005 Elsevier Ltd. All rights reserved.doi:10.1016/j.applthermaleng.2005.07.034*Corresponding author. Tel.: +1 573 884 6939; fax: +1 573 8845090.E-mail address: zhangyu (Y. Zhang)./locate/apthermengApplied Thermal Engineering 26 (2006) 751765Charach and Zarmi 12. It should be noted that meltingduring SLS occurs under the boundary condition ofspecified heat flux instead of specified temperature.Goodman and Shea 13 studied melting and solidifica-tion in the finite slab under a specified heat flux by usingthe heat balance integral method. Zhang et al. 14investigated the melting problem in a subcooled semi-infinite region subjected to constant heat flux heating.Zhang et al. 15 solved melting in a finite slab withthe boundary condition of the second kind by using asemi-exact method. Shrinkage formation due to densitychange during the solidification process in 2-D cavitywas investigated numerically by Kim and Ro 16, whoconcluded that the density change played a more impor-tant role than convection in the solidification process.Zhang and Faghri 17 analytically solved a one-dimensional melting problem in a semi-infinite two-component metal powder bed subjected to a constantheating heat flux. Effects of the porosity of the solidphase, initial subcooling parameter and dimensionlessthermal conductivity of the gas were investigated. SinceSLS of the metal powder is actually a layer-by-layer pro-cess, it is necessary to investigate melting in a mixed me-tal powder bed with the finite thickness during the SLSprocess. In this paper, melting of the mixed powder bedwith finite thickness subjected to constant heating heatflux will be investigated.2. Physical modelThe physical model of the melting problem is shownin Fig. 1. A powder bed with finite thickness containstwo metal powders with significantly different meltingpoints. The initial temperature of the powder bed isbelow the melting point of the low melting point pow-der. At time t = 0, a constant heat flux, q00, is suddenlyapplied to the top surface of the powder bed, and thebottom surface of the powder bed is assumed to beadiabatic. Since the initial temperature of the powderbed is below the melting point of the low meltingpoint powder, its melting does not start simultaneouslywith the addition of heat heating. Only after a finiteperiod of time of preheating, in which the surface tem-perature of the powder reaches the melting point ofthe low melting point powder, will the melting start.The powder with the high melting point will nevermelt during the entire process. Therefore, the problemcan be subdivided into two problems: one being heatconduction during preheating and the other beingmelting. The physical model is considered as a conduc-tion-controlled problem. The effect of natural convec-tion in the liquid region due to the temperaturedifference is not considered since the temperature ishighest at the liquid surface and decreases with increas-ing z.Nomenclaturecpspecific heat (J kg?1K?1)hsllatentheatofmeltingorsolidification(J kg?1)kthermal conductivity (W m?1K?1)Kgdimensionless thermal conductivity of gasKsdimensionless effective thermal conductivityof unsintered powderq00heat flux (W m?2)ssolidliquid interface location (m)Sdimensionless solidliquid interface locations0location of liquid surface (m)S0dimensionless location of liquid surfaceScsubcooling parameterSteStefan numberttime (s)Ttemperature (K)wvelocity of liquid phase (m s?1)Wdimensionless velocity of the liquid phasezcoordinate (m)Zdimensionless coordinateGreek symbolsathermal diffusivity (m2s?1)? adimensionless thermal diffusivitybparameter to distinguish between two meltingcasesdthermal penetration depth (m)Ddimensionless thermal penetration depthevolumefractionof gas(es) (porosity forunsintered powder)hdimensionless temperatureqdensity (kg m?3)sdimensionless time/volume fraction of the low melting pointpowder in the powder mixtureSubscriptsggasiinitiallliquid phasemmelting pointpsintered partsunsintered solid (mixture of two solid pow-ders)752T. Chen, Y. Zhang / Applied Thermal Engineering 26 (2006) 7517652.1. Duration of preheatingDuring preheating, pure conduction heat transfer oc-curs in the powder mixture. The governing equation andthe corresponding initial and boundary conditions forthe preheating problem areaso2Tsoz2oTsot;0 z H0; t tm1T Ti;0 z H0; t 02? ksoTsoz q00;z 0; t tm3oTsoz 0;z H0; t tm42.2. MeltingAfter melting starts, the governing equation in theliquid phase isalo2Tloz2oTlot woTloz;s0 z tm5where w is the velocity of liquid surface induced by theshrinkage. Since the liquid is incompressible, the shrink-age velocity w isw ds0dt;s0 z tm6Eq. (5) is subjected to the following boundary condition:?kloTloz q00;z 0; t tm7The governing equation for the solid phase and its cor-responding boundary conditions areaso2Tsoz2oTsot;st z tm8oTsoz 0;z H0; t tm9The temperature at the solidliquid interface satisfiesTlz;t Tsz;t Tm;z st; t tm10The energy balance at the solidliquid interface isksoTsoz? kloTloz 1 ? es/qlhsldsdt;z st; t tm11Based on the conservation of mass at the solidliquidinterface, the shrinkage velocity, w, and the solidliquidinterface velocity, ds/dt, have the following relationship17:w es? el1 ? eldsdt122.3. Non-dimensional governing equationsBy defining the following dimensionless variables:hlqcppTl? TmUqlhslhsqcppTs? TmUqlhslLiquid-solid interface q ss0H0Low melting point powder High melting point powder zOriginal surface Liquid surface Fig. 1. Physical model.T. Chen, Y. Zhang / Applied Thermal Engineering 26 (2006) 751765753Sc qcppTm? TiUqlhsl;s aptH2;Z zHS sH;S0s0H;D dH;W w ? HapKskskp1 ? es;Kgkgkp;? asasapSte q00HUqlhslap13The non-dimensional governing equation and the corre-sponding initial and boundary conditions for the pre-heating problem becomeo2hsoZ21? as?ohsos;0 Z 1; s sm14h ?Sc;0 Z 1; s 015ohsoZ ?SteKs1 ? es;Z 0; s sm16hs ?Sc;Z D; s sm17ohsoZ 0;Z D; s sm18For melting, the non-dimensional equation and corre-sponding boundary conditions areo2hloZ2ohlos WohloZ;S0 Z sm19W dS0ds;S0 Z tm20ohloZ ?Ste1 ? el;Z S0; s sm21o2hsoZ21? as?ohsos;Ss Z sm22ohsoZ 0;Z 1; s sm23hlZ;s hsZ;s 0;Z Ss; s sm24KsohsoZ?1 ? el1 ? esohloZdSds;Z Ss; s sm25W es? el1 ? eldSds;S0 Z tm263. Approximate solutionsWhen the top surface of the mixed metal powder bedis subjected to constant flux heating, the heat flux willpenetrate through the top surface and conduct down-ward the bottom surface. The depth to which the heatflux penetrates at an instant in time is defined as thethermal penetration depth, beyond which there is noheat conduction. Goodman and Shea 13 introduced aparameter, b = q00H/2ks(Tm? Ti), to classify two casesof melting in a finite slab. When b is greater than 1, thetop surface temperature reaches the melting point in ashorter time than the thermal penetration depth reachesthe bottom surface, indicating that a shorter preheatingtime is needed. If b is less than 1, the surface tempera-ture is still below the melting point when the thermalpenetration depth has reached the bottom surface. Pre-heating continues until the top surface temperaturereaches the melting point of low melting point powder.The parameter b can also be expressed using non-dimensionalparametersdefinedinEq.(13),i.e.,b = Ste/2KsSc(1 ? es). It can be seen that the value ofb is determined by four basic non-dimensional parame-ters: Stefan number Ste, subcooling parameter Sc, effec-tive thermal conductivity of the solid phase KsandFig. 2. Validation of analytical solutions.754T. Chen, Y. Zhang / Applied Thermal Engineering 26 (2006) 751765volume fraction of gas esin the solid phase. Preheatingand melting for both b 1 will be discussed.3.1. Preheating3.1.1. b 1The heat-balance integral method 18,19 is employedhere. Integrating the heat-conduction Eq. (14) with re-spect to Z from 0 to D, the integral equation is obtained.ohsoZD;s ?ohsoZ0;s?1? asddsH ScD27where H RD0hsZ;sdZ.hs(Z, s) is assumed to be a second degree polynomialfunction which satisfies boundary conditions specifiedby Eqs. (16)(18). Then hs(Z, s) can be determinedhsZ;s ?Sc Q2KsD1 ? esD ? Z228The Eqs. (16)(18) and (28) can be substituted intoEq. (27) and then an ordinary differential equation forthe thermal penetration depth, D, is obtained whichcan be solved easily.D 6 ? ? as? s1=229When the thermal penetration depth reaches the bottomsurface, i.e., D = 1, the temperature distribution in thepowder bed ishsZ;s ?Sc Ste2Ks1 ? es1 ? Z2;0 Z 1; s sD1 sm30(a)(b)llllllllFig. 3. Effect of porosity in the liquid phase on surface temperature (Ste = 0.02).T. Chen, Y. Zhang / Applied Thermal Engineering 26 (2006) 751765755which becomes the initial condition of the next stage ofpreheating. After the thermal penetration depth reachesthe bottom, the problem becomes a conduction problemin a finite slab. In a manner analogous to that describedpreviously, the temperature of the powder ishsZ;s ?Sc Ste2Ks1 ? es1 ? Z2Ste ? ? asKs1 ? es? s ? sD1;0 Z 1; sD1 s 1When b is greater than 1, melting starts before thepenetration depth reaches the bottom and therefore,the preheating time, sm, corresponding thermal penetra-tion depth, Dm, and temperature distribution at time smare 17sm231 ? es2K2sSc2?1Ste2? as34Dm 21 ? esKsSc ?1Ste35(a)(b)llllllllFig. 4. Effect of porosity in the liquid phase on surface temperature (Ste = 0.15).756T. Chen, Y. Zhang / Applied Thermal Engineering 26 (2006) 751765hsZ;s Sc1 ?ZDm?2? 1#;Z 0; s sm36hs0;s ?Sc Ste ?ffiffiffiffiffiffiffiffiffi6assp=2 ? Ks? 1 ? es?;Z 0; 0 s sm37where Eq. (38) is the surface temperature on the top ofthe powder bed.3.2. Solution of melting3.2.1. Temperature distribution in the liquidMelting starts when the surface temperature of thepowder bed reaches the melting point of the low meltingpoint powder. A liquid layer is formed as the resultof melting, the temperature distribution of whichdoes not depend on the valueof b. It can beobtained by an exact solution of Eqs. (19)(21) and(24) 17, i.e.,hlZ;s2Steffiffiffiffiffiffiffiffiffiffiffiffiffis?smp1?elierfcZ?S02ffiffiffiffiffiffiffiffiffiffiffiffiffis?smp?ierfcS?S02ffiffiffiffiffiffiffiffiffiffiffiffiffis?smp?38where S0is dimensionless location of liquid surface.3.2.2. Temperature distribution in the solid (b 1)Melting begins before the heat flux reaches the bot-tom of the powder bed, thus the problem is melting insemi-infinite two-component powder bed. The solutionfor melting of an infinite powder bed containing amixture of two metal powders has been obtained by(a)(b)llllllllllllllllFig. 6. Effect of porosity in the liquid phase on the location of the liquid surface and the liquidsolid interface (Ste = 0.15).758T. Chen, Y. Zhang / Applied Thermal Engineering 26 (2006) 751765Zhang 17. The temperature distribution in the liquidphase is given by Eq. (38).The temperature distribution in the solid region ob-tained by 17,hsZ;s ScD ? ZD ? S?2? 1#43The location of solidliquid interface is also obtainedby 17,dSdsSte1 ? eserfc1 ? esS21 ? elffiffiffiffiffiffiffiffiffiffiffiffiffis ? smp?2KsScD ? S44The thermal penetration depth satisfies the equationdDds6KsD ? S1 23Sc?2Ste1 ? eserfc1 ? esS21 ? elffiffiffiffiffiffiffiffiffiffiffiffiffis ? smp?45At the time that the thermal penetration depth reachesthe bottom surface, i.e., D = 1, the temperature distribu-tion in the solid ishsZ;sD1 Sc1 ? Z1 ? S?2? 1#46The time that thermal penetration depth reaches to thebottom, sD=1, is obtained fromDsD1 147(a)(b)llFig. 7. Effect of subcooling on surface temperature (Ste = 0.02).T. Chen, Y. Zhang / Applied Thermal Engineering 26 (2006) 751765759When s sD=1, the problem becomes melting in a finiteslab. The temperature distribution in the solid, hs(Z, s),and the liquidsolid interface location, S can be ob-tained by solving Eqs. (22)(24) using the integralapproximate method identical to the case of b 1.4. Results and discussionThevalidationoftheanalyticalsolutionwasconducted by comparing the results with the numericalresults obtained from Chen and Zhang 20, who inves-tigated the two-dimensional melting and resolidificationof a two-component metal powder layer in SLS processsubjected to a moving laser beam. In order to use thetwo-dimensional code in Ref. 20 to solve melting ina powder layer subjected to constant heat flux, theGaussian laser beam was replaced by a constant heat-ing heat flux on the top of the entire powder bed andthe laser scanning velocity was set to zero in numericalsolution. The parameters used in the present paper wereconverted into corresponding parameters in Ref. 20for purpose of code validation. The comparisons ofinstantaneous locations of liquid surface and liquidsolid interface obtained by analytical and numericalsolutions are shown in Fig. 2. It can be seen that thepreheating time obtained by the analytical and numer-ical solutions are almost the same. The locations ofliquid surface and liquidsolid interface obtained byanalytical and numerical solutions move at very similartrends. The time it takes to completely melt the entirepowder layer obtained from analytical solution is about(a)(b)llFig. 8. Effect of subcooling on surface temperature (Ste = 0.15).760T. Chen, Y. Zhang / Applied Thermal Engineering 26 (2006) 7517654% longer than that obtained from the numericalsolution.The effects of porosity, subcooling, dimensionlessthermal conductivity and Stefan number on the surfacetemperature, location of the liquid surface, and the loca-tion of the solidliquid interface of the powder bed willbe investigated. Fig. 3 shows how the surface tempera-ture is influenced by the porosity in the liquid phasefor Ste = 0.02 and several different subcooling parame-ters. The effect of shrinkage is isolated by fixing the sub-cooling parameter, porosity of the solid phase, and thedimensionless thermal conductivity. It can be seen thatthe surface temperature increases as porosity in theliquid phase increases. This is because the effective ther-mal conductivity decreases with increasing volume frac-tion of the gas. When Sc = 0.1, the preheating time ismuch shorter compared to when Sc = 3.0. The effectof shrinkage on the surface temperature for Ste = 0.15is shown in Fig. 4. As we can see, the increase of poros-ity in the liquid phase results in higher surface tempera-tures and that a higher Sc requires a longer preheatingtime. When Sc = 3.0, one can observe that the durationof the melting process is shortened significantly whenSte increases from 0.02 to 0.15. Fig. 5 shows the loca-tions of solidliquid interface and liquid surface corre-sponding to the conditions of Fig. 3. The solidliquidinterface moves faster when more gas is driven out fromthe liquid. It follows that the corresponding locationof the liquid surface moves downward significantlydue to the shrinkage of the mixed metal powder bed.The locations of solidliquid interface and liquid surfacecorresponding to the conditions of Fig. 4 are shown in(a)(b)llFig. 9. Effect of subcooling on the location of the liquid surface and the liquidsolid interface (Ste = 0.02).T. Chen, Y. Zhang / Applied Thermal Engineering 26 (2006) 751765761Fig. 6. The decrease of porosity in the liquid phase alsoexpedites the motion of the solidliquid interface andliquid surface downward.Fig. 7 shows the effect of the initial subcooling onthe surface temperature for Ste = 0.02. It can be seenthat the preheating time increases when the subcoolingparameter, Sc, is increased from 0.1 to 0.5. The sametrend is observed when Sc increases from 1.0 to 3.0.The effect of the initial subcooling on the surface tem-perature for Ste = 0.15 is shown in Fig. 8. Comparedto the case of Ste = 0.02, the preheating time forSte = 0.15 is significantly shortened. Meanwhile, thepreheating time for Ste = 0.15 increases when Sc is in-creased from 1.0 to 3.0. Figs. 7(a) and 8(a) indicatethat a lower liquid surface temperature can be obtainedif a larger initial subcooling value is used; however,these changes are not apparent from Figs. 7(b) and8(b).Fig. 9 shows the location of the solidliquid interfaceand the liquid surface corresponding to the conditionsof Fig. 7. It can be seen in Fig. 9(a), the existence ofgreater initial subcooling reduces the moving velocityof the solidliquid interface substantially. Before thethermal penetration depth reaches the bottom, thesolidliquid interface moves rather slowly, however theliquidsolid interface moves much faster after the ther-mal penetration depth has reached the bottom. At ahigher subcooling parameter, melting occurs only afterthe thermal penetration depth has reached the bottomas shown in Fig. 9(b). The reason for these phenomenais that the preheating brings the average temperature ofthe entire powder bed very close to the melting point of(a)(b)llFig. 10. Effect of subcooling on the location of the liquid surface and the liquidsolid interface (Ste = 0.15).762T. Chen, Y. Zhang / Applied Thermal Engineering 26 (2006) 751765the low melting powder so that the melting process canproceed very quickly. The relationship between thesolidliquid interface and the liquid surface, however,is the same for all subcooling parameters since it onlydepends on the volume fractions of the gas in the solidand liquid phase (see Eq. (26). The location of thesolidliquid interface and the liquid surface correspond-ing to the conditions of Fig. 8 are also plotted in Fig. 10.A similar trend can also be observed in Fig. 9.In order to prevent the sintered part from oxidizationby the air, Su et al. 21 used argon as protective gas inthe powder bed. Compared with air that has a dimen-sionless thermal conductivity of Kg= 3.7 10?4, theargon has a much lower dimensionless thermal conduc-tivity of Kg= 2.5 10?4. The surface temperature forvarying dimensionless thermal conductivity at differentSte is shown in Fig. 11. It can be seen that the preheat-ing time decreases as the thermal conductivity of the gasis decreased for both Ste = 0.02 and 0.15. However, thedifference of the surface temperatures for different gasesis insignificant.The effect of the dimensionless thermal conductivityof the gas on the locations of the solidliquid interfaceand the liquid surface corresponding to the conditionsof Fig. 11 are shown in Fig. 12. When Ste = 0.02, thevelocity of the solidliquid interface increases whenargon is employed as protective gas. When Ste = 0.15,the velocity of the solidliquid interface is faster for(a)(b)llFig. 11. Effect of dimensionless thermal conductivity of gas on surface temperature.T. Chen, Y. Zhang / Applied Thermal Engineering 26 (2006) 751765763Kg= 2.5 10?4before the thermal penetration depthreach to the bottom of the powder layer. Compared withthe case of Kg= 3.7 10?4, the time it takes for the ther-mal penetration depth reach the bottom of the powderlayer is longer for Kg= 2.5 10?4. Once the thermalpenetration depth reaches the bottom of the powderlayer, the melting processes for both Kg= 3.7 10?4and Kg= 2.5 10?4proceed rapidly since more heatcan be used to supply the latent of melting.5. ConclusionMeltingofasubcooledtwo-componentmetalpowder layer subjected to a constant heat flux heatingwas investigated analytically. The shrinkage inducedby melting was also taken into account. Increases ofthe Stefan number and decreases of subcooling num-ber accelerate the melting process significantly. Meltingof the powder layer filled with argon with lowerdimensionless thermal conductivity is also investigatedin order to avoid oxidization. Melting in a finite pow-der bed is different from melting in a semi-infinite slabsince the solidliquid interface during melting in afinite powder bed moves faster than that in a semi-infiniteslab.Thephysicalmodelandresultsofthis investigation provides a strong foundation uponwhich further investigation of the complex three-dimensionalselectivelasersintering(SLS)processcan be based.(a)(b)llFig. 12. Effect of dimensionless thermal conductivity of gas on the location of the liquid surface and the liquidsolid interface.764T. Chen, Y. Zhang / Applied Thermal Engineering 26 (2006) 751765AcknowledgementsSupport for this work by the Office of Naval Re-search (ONR) under grant number N00014-04-1-0303is greatly acknowledged.References1 J.G. Conley, H.L. Marcus, Rapid prototyping and solid freeformfabrication, Journal of Manufacturing Science and Engineering119 (1997) 811816.2 T. Manzur, T. DeMaria, W. Chen, C. Roychoudhuri, PotentialRole of High Powder Laser Diode in Manufacturing, SPIEPhotonics West Conference, San Jose, CA, 1996.3 D.E. Bunnell, Fundamentals of Selective Laser Sintering ofMetals, Ph.D. Dissertation, University of Texas at Austin, 1995.4 S. Storch, D. Nellessen, G. Schaefer, R.