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Analysis of melting of alloy powder bed with constant heat fl ux Bin Xiao Yuwen Zhang Department of Mechanical and Aerospace Engineering University of Missouri Columbia Columbia MO 65211 USA Received 13 March 2006 received in revised form 5 November 2006 Available online 11 January 2007 Abstract Melting of an alloy powder bed with constant heat fl ux for application in selective laser sintering SLS is analyzed in this paper Since melting of an alloy occurs in a range of temperatures instead of at a single melting point there will be a mushy zone containing par tially melted powders between the unmelted region and the completely melted region The mushy zone can be further divided into two sub regions 1 a lower part with constant porosity shrinkage takes place and 2 an upper part with constant volume no shrinkage Temperature distributions in diff erent regions and locations of melting interfaces are obtained using an integral approximation method The results show that increasing initial porosity and temperature of the powder bed accelerate the melting process The melting slows down with increasing thermal conductivity of the interstitial gas 2006 Elsevier Ltd All rights reserved 1 Introduction Direct metal laser sintering DMLS is a rapid prototyp ing manufacturing technology which directly fabricates parts via melting and resolidifi cation of metallic powders 1 5 Fundamentals of heat transfer in melting and solidi fi cation have been studied intensively and are well docu mented 6 8 Carslaw and Jaeger 9 discussed melting of a semi infi nite body with constant thermophysical proper ties and obtained an analytical solution for Dirichlet boundary conditions Lax 10 studies the temperature rise under a steady state condition due to a stationary Gaus sian beam in a semi infi nite cylindrical medium Bell 11 developed a one dimensional thermal model for laser annealing over a wide range of laser pulse duration and absorption coeffi cient Yilbas 12 has presented a numeri cal solution for a pulsed CO2laser heating process Ros tamlandRaisi 13 havenumericallysolvedthe temperature distribution and melt pool size in a semi infi nite body due to a moving laser heat source Kim and Sim 14 have studied thermal behavior and fl uid fl ow during laser surface heating of alloys Iwamoto et al 15 have performed numerical analysis of energy transfer and surface modifi cation of a metal surface by pulsed laser heating While the above studies focused on long pulse laser material interactions phase change heat transfer dur ing ultrashort laser processing of metal fi lms has also been investigated 16 17 Thethermalcontactresistance between the mold and substrate during laser assisted imprinting fabrication was numerically investigated by Hsiao et al 18 During the DMLS process the powder bed shrinks as the porosity of the powder bed becomes nearly zero after melting The eff ect of shrinkage during solid liquid phase change processes has been investigated by some research ers Eckert and Drake 19 obtained a similarity solution of a one dimensional solidifi cation problem with the con sideration of the density change in phase change Zhang and Faghri 20 analytically solved a one dimensional melt ing problem in a powder bed containing a powder mixture under a boundary condition of the second kind The results showed that the shrinkage eff ect on the melting of the pow der bed is not negligible Konrad et al 21 obtained an analytical solution of melting and resolidifi cation of a two component metal powder bed subjected to temporal Gaussian heat fl ux Xiao and Zhang 22 analytically solved 0017 9310 see front matter 2006 Elsevier Ltd All rights reserved doi 10 1016 j ijheatmasstransfer 2006 11 012 Corresponding author Tel 1 573 884 6936 fax 1 573 884 5090 E mail address zhangyu missouri edu Y Zhang International Journal of Heat and Mass Transfer 50 2007 2161 2169 rapid melting of a subcooled single component metal pow der bed in the SLS process Under irradiation of a pulse laser beam the surface of the powder particle is molten fi rst and the core of the particle remains solid Melting in SLS of single component metal powder was modeled as melting occurring in a range of temperature with signifi cant density change In contrast to metals with well defi ned melting point alloyed powders exhibits a melting temperature range dur ing the sintering process Therefore mushy zone containing mixture of liquid and solid is formed and the temperature in the mushy zone is between solidus and liquidus temper atures Niu and Chang 23 showed that during the DMLS of high speed steel powders a liquid phase forms along the grain boundaries in the particles The liquid fl ows and wets the solid particles or the grain boundaries This results in rapid densifi cation by rearrangement of the solid particles Prabhu and Bourell reported similar mechanism in the DMLS of pre alloyed bronze powders 24 Before DMLS of alloy powders the powder structure is supported by the solid particles Partial melting of the pow der particle during DMLS will result in decrease of the powder bed volume because the void spaces initially occu pied by the gas will be taken up by the liquid phase Con servation of mass principles shows that the porosity of the powder bed defi ned as summation of the volume fraction of the gas and liquid remains constant while the volume of the powder bed shrinks 25 When all gas is driven out the shrinkage process ceases and the volume of the powder will be constant from that moment Therefore the mushy zone between the liquid region and solid region can be further divided into two parts the lower part with constant porosity and the upper part with constant volume In order to understand the mechanisms of DMLS of alloy powders melting of alloy powder bed with shrinkage phenomena will be investigated in this paper The temper ature distribution and locations of melting interfaces will be obtained using an integral approximation method Eff ects of processing parameters such as porosity subcool ing parameters and thermal conductivity of the interstitial gas es on melting process will be investigated 2 Physical model The problem under consideration is shown in Fig 1 A randomly packed metal powder bed which is initially at a Nomenclature c specifi c heat J kg K fmass fraction of solid f0mass fraction of solid on the heating surface hs latent heat of melting J kg kthermal conductivity W m K Kdimensionless thermal conductivity q00 heat fl ux absorbed by the powder bed W m2 s0location of heating surface m smlocation of interface between upper and lower parts of mushy zone m s location of interface between liquid and mushy zones m slocation of interface between solid and mushy zones m S0dimensionless location of heating surface Smdimensionless location of interfacebetween upper and lower parts of mushy zone S dimensionless location of interfacebetween liquid and mushy zones Sdimensionless location of interfacebetween solid and mushy zone Tiinitial temperature on the powder bed Ttemperature K T mliquidus temperature K Tsmsolidus temperature K ttime s Vvolume m3 wshrinkage velocity m s Wdimensionless shrinkage velocity zvertical coordinate m Zdimensionless vertical coordinate Greek symbols a thermal diff usivity m2s a0weighted average in the function for calculating eff ective thermal conductivity of the powder bed dpowder thermal penetration depth m dhdimensionless melting temperature range hdimensionless temperature Ddimensionless thermal penetration depth eporosity volume fraction of void Vg V Vg V Vs qdensity kg m3 sdimensionless time ugvolume fraction of gas Vg Vg V Vs u volume fraction of liquid V Vg V Vs usvolume fraction of solid Vs Vg V Vs Subscripts 0surface eff eff ective ggas ffusion iinitial liquid mmushy zone pmetal powder ssolid 2162B Xiao Y Zhang International Journal of Heat and Mass Transfer 50 2007 2161 2169 uniform temperature Ti below the solidus temperature of the powders Tsm is in a half space z 0 At time t 0 a constant heat fl ux q00 is applied to the surface of the pow der bed Since alloy powders melt in a range of temperature Tsm T m a mushy zone is formed when the surface temperature of the powder bed reaches the solidus temper ature The mushy zone can be divided into two sub regions depending on whether shrinkage is accompanied 1 lower part with constant porosity shrinkage takes place and 2 upper part with constant volume no shrinkage The melt ing of alloy powders is accomplished in three stages Dur ing the fi rst stage the lower part of the mushy zone that contains gas liquid and solid is formed The porosity defi ned as the total volume of void including the volumes of gas and liquid relative to the total volume occupied by the solid matrix and void volume e V Vg V Vg Vs ug u 25 is constant in the lower part of the mushy zone The whole powder bed shrinks as the interstitial gas between solid particles is driven out by the generated liquid phase During the second stage the upper part of the mushy zone that contains liquid and solid only is formed on top of the lower part of the mushy zone when the volume fraction of gas ug on the heating surface becomes zero While shrinkage takes place in the lower part where the porosity remains constant there is no shrinkage in the upper part if the densities of the solid and liquid phases of the powder material are the same When the surface temperature of the powder bed reaches the liquidus temperature the liquid region appears on top of the upper part of mushy zone and the process enters the third stage During this stage the powder bed is divided into four regions from the bottom to the heating surface 1 unsintered region 2 lower part of mushy zone with constant porosity shrinkage takes place 3 upper part of mushy zone with constant volume no shrinkage and 4 liquid region see Fig 1 The third stage of the melting represents the most complex situation and its gov erning equations and solutions are presented below 2 1 Governing equations in diff erent regions and interfaces 2 1 1 The unsintered region The energy equation of the unsintered region under neath the mushy zone is qc p 1 e oTs ot keff o2Ts oz2 s t z 4 2 1 2 The lower part of the mushy zone The energy and continuity equations are o qcTm ot o qcwTm oz o oz k oTm oz qhs of ot sm t z s t 5 oq ot o qw oz 0 sm t z s t 6 Neglecting the contribution of gas to the density and heat capacity the thermal properties of the lower part of the mushy zone can be expressed as Solid Zone Mushy Zone Original Surface Liquid Zone S0 S1 Sm S q Gas Liquid Solid Z Z Fig 1 Physical model of direct metal laser sintering with constant heat fl ux B Xiao Y Zhang International Journal of Heat and Mass Transfer 50 2007 2161 21692163 qc m 1 ug qc p 7 qm 1 ug qp 8 km 1 ug kp 9 where the subscript p represents the properties of powder material The thermal conductivity in the partially molten region km is much higher than that before melting because the contact area between the liquid and solid is signifi cantly increased The degree of melting in the mushy zone can be mea sured by the solid fraction f a value of 1 indicates full solid 0 indicates full liquid and the value between 1 and 0 indicates mixture of solid and liquid The solid fraction is related to the local temperature by f 0T P T m T m T T m Tsm Tsm T 10 The solid mass fraction f is related to the volume fractions of solid and liquid by f qpsus qpsus qp u 11 where qpsand qp are densities of the solid and liquid phases of the powder material which are assumed to be the same in the present study qps qp At the lower part of the mushy zone porosity in the mushy zone remains constant so that always equals to 1 e Eq 11 can there fore be simplifi ed as f 1 e 1 ug 12 Substituting Eqs 7 9 and Eq 12 into Eqs 5 and 6 the energy and continuity equations become o ot Tm f o oz w Tm f ap o oz 1 f oTm oz hs cp 1 f of ot sm t z s t 13 o ot 1 f o oz w 1 f 0 sm t z s t 14 2 1 3 The upper part of the mushy zone The energy equation is oTm ot w0 oTm oz ap o2Tm oz2 hs cp of ot s t z sm t 15 Substituting Eq 10 into Eq 15 the energy equation in the upper part becomes 1 hs 2cpDT oTm ot w0 oTm oz ap o2Tm oz2 s t z sm t 16 where DT T m Tsmand w0 ds0 dt 2 1 4 The liquid region The energy equation is oT ot w0 oT oz ap o2T oz2 s0 t z s t 17 which is subject to the following boundary condition kp oT oz q00 z s0 t 18 2 1 5 Interfacial boundary conditions The temperature at the interface between the mushy zone and unsintered regions is equal to the solidus temper ature i e Ts Tm Tsm z s t 19 and the energy balance at the interface between mushy zone and unsintered regions is kp 1 e oTm oz keff oTs oz z s t 20 At the interface between the upper and lower parts of the mushy zone the volume fraction of the gas ug is zero and the solid fraction f is equal to 1 e according to Eq 12 Therefore the temperature at the interface be tween the upper and lower parts can be obtained from Eq 10 i e Tm Tsm e T m Tsm z sm t 21 The energy balance at the interface of the upper and lower parts of mushy zone is oTm oz z s m oTm oz z s m z sm t 22 The temperature at the interface between mushy zone and liquid region is equal to the liquidus temperature T Tm T m z s t 23 The energy balance at the interface between mushy and liquid zones is oT oz oTm oz z s t 24 3 Non dimensional governing equations Introducing the following dimensionless variables s apt apqphs q00 2 h cp T T m hs Z z apqphs q00 S s apqphs q00 W wapqphs q00 ap W0 w0apqphs q00 ap Keff keff kp 1 e dh cp T m Tsm hs 25 2164B Xiao Y Zhang International Journal of Heat and Mass Transfer 50 2007 2161 2169 the governing equations and boundary conditions can be nondimensionalized as follows Keff o2hs oZ2 ohs os S s Z 1 26 o os hm f o oZ W hm f o oZ 1 f ohm oZ 1 f of os Sm s Z S s 27 o os 1 f o oZ W f 0 Sm s Z S s 28 1 1 dh ohm os W0 ohm oZ o2hm oZ2 S s Z Sm s 29 o2h oZ2 oh os W0 oh oZ S0 s Z S s 30 oh oZ 1 1 e Z S0 s 31 hs hm dh Z S s 32 Keff ohs oZ ohm oZ Z S s 33 hm e 1 dh Z Sm s 34 ohm oZ Z S m ohm oZ Z S m Z Sm s 35 h hm 0 Z S s 36 oh oZ ohm oZ Z S s 37 4 Integral approximate solutions The integral approximate method for unsintered region requires introduction of the dimensionless thermal penetra tion depth D beyond which the temperature of the powder bed is not aff ected by the surface heating i e hs D s 0 38 ohs D s oZ 0 39 The integral equation of unsintered region is obtained by integrating Eq 26 in the interval of S D with respect to Z Keff oh D s oZ oh S s oZ d ds Z D S hdZ hiD dhS 40 Assuming a second degree polynomial temperature distri bution in the unsintered region and using Eqs 32 38 and 39 to determine the constants the temperature distri bution in the unsintered region is obtained hs Z s hi hi Dhm D Z D S 2 41 Substituting Eq 41 into Eq 40 one obtains the follow ing diff erential equation for thermal penetration depth and the interface between the unsintered region and mushy zone 6Keff D S dD ds 2 dS ds 42 The integral equations in the lower part of the mushy zone are obtained by integrating Eqs 27 and 28 from Smto S with respect to Z d ds Z S Sm hm f dZ Z S Sm lnf dZ hm f Z S S lnfjZ SmSm 1 f ohm oZ S Sm 43 d ds Z S Sm 1 f dz 1 f Z S S 1 f Z Sm Sm W f S Sm 0 44 The expression of solid fraction f as a single value of tem perature in the lower part of the mushy zone defi ned in Eq 10 is diffi cult to be used in Eqs 43 and 44 because of the complexity of the integral equations The eff ect of the distribution of the solid fraction in the mushy zone using the integral method was investigated by Tien and Geiger 27 who concluded that the result deviations are only four to eight percent when diff erent forms of solid fraction dis tributions were used For the ease of integration the in verse of solid fraction is assumed to be a linear function of in the mushy zone i e 1 f 1 e 1 e S Z S Sm 45 It is also assumed that temperature distribution in the lower part of the mushy zone is a second degree polynomial function hm f B0 B1 S Z S Sm B2 S Z S Sm 2 Sm s Z s 46 where the constants B0 B1and B2can be obtained by using the boundary conditions specifi ed by Eqs 32 34 B0 dh 47 B1 e 1 e dh 2Keff hi dh S Sm D S 48 B2 e 1 e dh 2Keff hi dh S Sm D S 49 Substituting Eqs 45 49 into Eqs 43 and 44 one obtains Sm 2 e S0 S 50 d ds Keff hi dh 3 D S S Sm 2 edh 6 1 e 1 ln 1 e e S Sm dhdS0 ds ln 1 e dSm ds 4Keff hi dh D S e 2 e dh 1 e S Sm 51 Integrating Eq 29 in the interval of S Sm with respect to Z the integral equation for the upper part of the mushy zone becomes B Xiao Y Zhang International Journal of Heat and Mass Transfer 50 2007 2161 21692165 1 1 dh d ds Z Sm S hmdZ e 1 dh dSm ds W0 e 1 dh ohm oZ Z Sm ohm oZ Z S 52 Assuming the temperature distribution in the upper part of the mushy zone as a second degree polynomial function and applying the boundary conditions of Eqs 34 36 yields hm e 1 dh Dm Sm Z 1 e dh Dm Sm S Sm Z Sm S 2 53 where Dm e 2 e dh S Sm 2Keff hi dh 1 e D S 54 Substituting Eq 53 into Eq 52 yields 1 1 dh d ds 1 6 Dm Sm S 2 1 e dh 3 4Sm S 1 e dh dS0 ds 2 1 e dh Sm S 2Dm 55 Since the entire liquid layer moves with the velocity W0 the layer is stationary from the coordinate Z0that moves with velocity W0 The relationship between the two coordinate systems is Z0 Z Z s s W0ds Z S0 56 The energy equation and the corresponding boundary con ditions can be rewritten as o2h oZ02 oh os 0 Z0 s 57 oh oZ0 1 1 e Z0 0 s s 58 h Z0 s 0 Z0 S S0 s s 59 The exact solution of Eqs 57 59 is 28 h0 Z0 s 2 ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi s s p 1 e ierfc Z0 2 ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi s s p ierfc S S0 2 ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi s s p 60 Changing back to the Z coordinate system Eq 60 can be rewritten as h Z s 2 ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi s s p 1 e ierfc Z S0 2 ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi s s p ierfc S S0 2 ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi s s p 61 Substituting Eqs 53 and 61 into Eq 37 th