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subcooled Yuwen 11 is investigated points solid approximate solution respectively The e ects of porosity Stefan number and subcooling on the surface temperature and solid liquid interface are also investigated The present work provides a strong foundation upon which the investigation of complex selectively fusing a thin layer of the powders with scan imate diameter of the laser beam The particular melting and resolidification induced by a directed laser sity change of the powder bed accompanies the melting process Melting and solidification in 1 D semi infinite body with density change under the boundary condition of the first kind have been investigated by Zckert and Drake 9 Crank 10 Carslaw and Jaeger 11 and Corresponding author Tel 1 573 884 6939 fax 1 573 884 5090 E mail address zhangyu missouri edu Y Zhang Applied Thermal Engineering 26 1359 4311 see front matter C211 2005 Elsevier Ltd All rights reserved ning laser beam After sintering of a layer a new layer of the powder is deposited in the same manner and a 3 D part can be built in a layer by layer process A mixed metal powder bed which contains two types of the metal powders possessing significantly di erent melting points is used extensively in direct SLS of metal powders 2 3 The high melting point powder never melt in the sintering process and plays a significant role as the support structure necessary to avoid boiling phenom enon which is the formation of spheres with the approx beam It is a good starting point to investigate a simpli fied 1 D model to get a better understanding of the melting process in direct SLS before a much more com plicated 3 D model is investigated Fundamentals of melting and solidification have been investigated extensively and detailed reviews are avail able in Refs 7 8 Melting in SLS of the metal powders is significantly di erent from the normal melting since the volume fraction of the gas in the powders decreases significantly after melting Therefore a significant den three dimensional selective laser sintering SLS process can be based C211 2005 Elsevier Ltd All rights reserved Keywords Melting Metal Powder layer 1 Introduction Direct Selective Laser Sintering SLS is an emerging technology of Solid Freeform Fabrication SFF via which 3 D parts are built from the metal based powder bed with CAD data 1 A fabricated layer is created by material properties and methods of material analysis of the metal based powder system for SLS applications are 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 density objects from metal powder direct SLS is realized via Analysis of melting in a metal powder layer with Tiebing Chen Department of Mechanical and Aerospace Engineering University Received 1 February 2005 Available online Abstract Melting of a subcooled two component metal powder layer mixture of two metal powders with significantly di erent melting physical model The temperature distributions in the liquid and doi 10 1016 j applthermaleng 2005 07 034 two component constant heat flux Zhang of Missouri Columbia Columbia MO 65211 United States accepted 18 July 2005 October 2005 analytically The powder bed considered consists of a Shrinkage induced by melting is taken into account in the phases are obtained using an exact solution and an integral 2006 751 765 Nomenclature c p specific heat J kg C01 K C01 h sl latent heat of melting or solidification J kg C01 k thermal conductivity W m C01 K C01 K g dimensionless thermal conductivity of gas K s dimensionless e ective thermal conductivity of unsintered powder q 00 heat flux W m C02 s solid liquid interface location m S dimensionless solid liquid interface location s 0 location of liquid surface m S 0 dimensionless location of liquid surface Sc subcooling parameter Ste Stefan number t time s T temperature K w velocity of liquid phase m s C01 W dimensionless velocity of the liquid phase z coordinate m Z dimensionless coordinate Greek symbols a thermal di usivity m 2 s C01 752 T Chen Y Zhang Applied Thermal Charach and Zarmi 12 It should be noted that melting during SLS occurs under the boundary condition of specified 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 using the heat balance integral method Zhang et al 14 investigated 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 with the boundary condition of the second kind by using a semi exact method Shrinkage formation due to density change during the solidification process in 2 D cavity was investigated numerically by Kim and Ro 16 who concluded 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 constant heating heat flux E ects of the porosity of the solid phase initial subcooling parameter and dimensionless thermal conductivity of the gas were investigated Since SLS 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 SLS process In this paper melting of the mixed powder bed with finite thickness subjected to constant heating heat flux will be investigated C22a dimensionless thermal di usivity b parameter to distinguish between two melting cases d thermal penetration depth m D dimensionless thermal penetration depth e volume fraction of gas es porosity for unsintered powder h dimensionless temperature q density kg m C03 s dimensionless time volume fraction of the low melting point powder in the powder mixture Subscripts g gas i initial l liquid phase m melting point p sintered part s unsintered solid mixture of two solid pow ders Engineering 26 2006 751 765 2 Physical model The physical model of the melting problem is shown in Fig 1 A powder bed with finite thickness contains two metal powders with significantly di erent melting points The initial temperature of the powder bed is below the melting point of the low melting point pow der At time t 0 a constant heat flux q 00 is suddenly applied to the top surface of the powder bed and the bottom surface of the powder bed is assumed to be adiabatic Since the initial temperature of the powder bed is below the melting point of the low melting point powder its melting does not start simultaneously with the addition of heat heating Only after a finite period of time of preheating in which the surface tem perature of the powder reaches the melting point of the low melting point powder will the melting start The powder with the high melting point will never melt during the entire process Therefore the problem can be subdivided into two problems one being heat conduction during preheating and the other being melting The physical model is considered as a conduc tion controlled problem The e ect of natural convec tion in the liquid region due to the temperature di erence is not considered since the temperature is highest at the liquid surface and decreases with increas ing z T Chen Y Zhang Applied Thermal Engineering 26 2006 751 765 753 2 1 Duration of preheating During preheating pure conduction heat transfer oc curs in the powder mixture The governing equation and the corresponding initial and boundary conditions for the preheating problem are z Fig 1 Physical q s s 0 H 0 a s o 2 T s oz 2 oT s ot 0 z H 0 t t m 1 T T i 0 z H 0 t 0 2 C0 k s oT s oz q 00 z 0 t t m 3 oT s oz 0 z H 0 t t m 4 2 2 Melting After melting starts the governing equation in the liquid phase is a l o 2 T l oz 2 oT l ot w oT l oz s 0 z t m 5 where w is the velocity of liquid surface induced by the shrinkage Since the liquid is incompressible the shrink age velocity w is w ds 0 dt s 0 z t m 6 Eq 5 is subjected to the following boundary condition C0k l oT l oz q 00 z 0 t t m 7 The governing equation for the solid phase and its cor responding boundary conditions are a s o 2 T s oz 2 oT s ot s t z t m 8 oT s 0 z H 0 t t m 9 Liquid solid interface Low melting point powder High melting point powder model Original surface Liquid surface oz The temperature at the solid liquid interface satisfies T l z t T s z t T m z s t t t m 10 The energy balance at the solid liquid interface is k s oT s oz C0 k l oT l oz 1 C0e s q l h sl ds dt z s t t t m 11 Based on the conservation of mass at the solid liquid interface the shrinkage velocity w and the solid liquid interface velocity ds dt have the following relationship 17 w e s C0e l 1 C0e l ds dt 12 2 3 Non dimensional governing equations By defining the following dimensionless variables h l qc p p T l C0 T m Uq l h sl h s qc p p T s C0 T m Uq l h sl Sc qc p p T m C0 T i Uq l h sl s a p t H 2 Z z H S s H S 0 s 0 H D d H W w C1 H a p K s k s k p 1 C0e s K g k g k p C22a s a s a p Ste q 00 H Uq l h sl a p 13 The non dimensional governing equation and the corre sponding initial and boundary conditions for the pre heating problem become o 2 h s oZ 2 1 C22a s C1 oh s os 0 Z 1 s s m 14 h C0Sc 0 Z 1 s 0 15 oh s oZ C0 Ste K s 1 C0e s Z 0 s s m 16 h s C0Sc Z D s s m 23 h l Z s h s Z s 0 Z S s s s m 24 K s oh s oZ C0 1 C0e l 1 C0e s oh l oZ dS ds Z S s s s m 25 W e s C0e l 1 C0e l dS ds S 0 Z t m 26 3 Approximate solutions When the top surface of the mixed metal powder bed is subjected to constant flux heating the heat flux will penetrate through the top surface and conduct down ward the bottom surface The depth to which the heat flux penetrates at an instant in time is defined as the thermal penetration depth beyond which there is no heat conduction Goodman and Shea 13 introduced a 00 754 T Chen Y Zhang Applied Thermal Engineering 26 2006 751 765 oh s oZ 0 Z D s s m 18 For melting the non dimensional equation and corre sponding boundary conditions are o 2 h l oZ 2 oh l os W oh l oZ S 0 Z s m 19 W dS 0 ds S 0 Z t m 20 oh l oZ C0 Ste 1 C0e l Z S 0 s s m 21 o 2 h s oZ 2 1 C22a s C1 oh s os S s Z s m 22 Fig 2 Validation of analytical parameter b q H 2k s T m C0 T i to classify two cases of melting in a finite slab When b is greater than 1 the top surface temperature reaches the melting point in a shorter time than the thermal penetration depth reaches the bottom surface indicating that a shorter preheating time is needed If b is less than 1 the surface tempera ture is still below the melting point when the thermal penetration depth has reached the bottom surface Pre heating continues until the top surface temperature reaches the melting point of low melting point powder The parameter b can also be expressed using non dimensional parameters defined in Eq 13 i e b Ste 2K s Sc 1 C0 e s It can be seen that the value of b is determined by four basic non dimensional parame ters Stefan number Ste subcooling parameter Sc e ec tive thermal conductivity of the solid phase K s and solutions T Chen Y Zhang Applied Thermal Engineering 26 2006 751 765 755 volume fraction of gas e s in the solid phase Preheating and melting for both b 1 will be discussed 3 1 Preheating 3 1 1 b 1 The heat balance integral method 18 19 is employed here Integrating the heat conduction Eq 14 with re spect to Z from 0 to D the integral equation is obtained oh s oZ D s C0 oh s oZ 0 s C20C21 1 C22a s d ds H ScD 27 where H R D 0 h s Z s dZ h s Z s is assumed to be a second degree polynomial function which satisfies boundary conditions specified by Eqs 16 18 Then h s Z s can be determined b Fig 3 E ect of porosity in the liquid phase l l l l h s Z s C0Sc Q 2K s D 1 C0e s DC0 Z 2 28 The Eqs 16 18 and 28 can be substituted into Eq 27 and then an ordinary di erential equation for the thermal penetration depth D is obtained which can be solved easily D 6 C1 C22a s C1s 1 2 29 When the thermal penetration depth reaches the bottom surface i e D 1 the temperature distribution in the powder bed is h s Z s C0Sc Ste 2K s 1 C0e s 1 C0 Z 2 0 Z 1 s s D 1 s m 30 a l l l l on surface temperature Ste 0 02 756 T Chen Y Zhang Applied Thermal Engineering 26 2006 751 765 which becomes the initial condition of the next stage of preheating After the thermal penetration depth reaches the bottom the problem becomes a conduction problem in a finite slab In a manner analogous to that described previously the temperature of the powder is h s Z s C0Sc Ste 2K s 1 C0e s 1 C0 Z 2 Ste C1 C22a s K s 1 C0e s C2 sC0s D 1 0 Z 1 s D 1 s 1 When b is greater than 1 melting starts before the penetration depth reaches the bottom and therefore the preheating time s m corresponding thermal penetra tion depth D m and temperature distribution at time s m are 17 s m 2 3 1 C0e s 2 K 2 s Sc 2 C1 1 Ste 2 C22a s 34 D m 2 1 C0e s K s Sc C1 1 Ste 35 a l l l l on surface temperature Ste 0 15 T Chen Y Zhang Applied Thermal Engineering 26 2006 751 765 757 h s Z s Sc 1 C0 Z D m C18C19 2 C0 1 Z 0 s s m 36 h s 0 s C0Sc Ste C1 6a s s p 2 C1 K s C1 1 C0e s C138 Z 0 0 s s m 37 where Eq 38 is the surface temperature on the top of the powder bed 3 2 Solution of melting 3 2 1 Temperature distribution in the liquid Melting starts when the surface temperature of the powder bed reaches the melting point of the low melting point powder A liquid layer is formed as the result of melting the temperature distribution of which a b K Fig 5 E ect of porosity in the liquid phase on the location of the l l l l l l l does not depend on the value of b It can be obtained by an exact solution of Eqs 19 21 and 24 17 i e h l Z s 2Ste sC0s m p 1C0e l ierfc ZC0S 0 2 sC0s m p C18C19 C0ierfc SC0S 0 2 sC0s m p C18C19C20C21 38 where S 0 is 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 in semi infinite two component powder bed The solution for melting of an infinite powder bed containing a mixture of two metal powders has been obtained by a l l l l l l l l l liquid surface and the liquid solid interface Ste 0 15 T Chen Y Zhang Applied Thermal Engineering 26 2006 751 765 759 Zhang 17 The temperature distribution in the liquid phase is given by Eq 38 The temperature distribution in the solid region ob tained by 17 h s Z s Sc DC0 Z DC0 S C18C19 2 C0 1 43 The location of solid liquid interface is also obtained by 17 dS ds Ste 1 C0e s erfc 1 C0e s S 2 1 C0e l sC0s m p C18C19 C0 2K s Sc DC0 S 44 The thermal penetration depth satisfies the equation a Fig 7 E ect of subcooling on surface l dD ds 6K s DC0 S 1 2 3 Sc C18C19 C0 2Ste 1 C0e s erfc 1 C0e s S 2 1 C0e l sC0s m p C18C19 45 At the time that the thermal penetration depth reaches the bottom surface i e D 1 the temperature distribu tion in the solid is h s Z s D 1 Sc 1 C0 Z 1 C0 S C18C19 2 C0 1 46 The time that thermal penetration depth reaches to the bottom s D 1 is obtained from D s D 1 1 47 b l temperature Ste 0 02 760 T Chen Y Zhang Applied Thermal Engineering 26 2006 751 765 When s s D 1 the problem becomes melting in a finite slab The temperature distribution in the solid h s Z s and the liquid solid interface location S can be ob tained by solving Eqs 22 24 using the integral approximate method identical to the case of b 1 4 Results and discussion The validation of the analytical solution was conducted by comparing the results with the numerical results obtained from Chen and Zhang 20 who inves tigated the two dimensional melting and resolidification of a two component metal powder layer in SLS process subjected to a moving laser beam In order to use the two dimensional code in Ref 20 to solve melting in a b Fig 8 E ect of subcooling on surface l a powder layer subjected to constant heat flux the Gaussian laser beam was replaced by a constant heat ing heat flux on the top of the entire powder bed and the laser scanning velocity was set to zero in numerical solution The parameters used in the present paper were converted into corresponding parameters in Ref 20 for purpose of code validation The comparisons of instantaneous locations of liquid surface and liquid solid interface obtained by analytical and numerical solutions are shown in Fig 2 It can be seen that the preheating time obtained by the analytical and numer ical solutions are almost the same The locations of liquid surface and liquid solid interface obtained by analytical and numerical solutions move at very similar trends The time it takes to completely melt the entire powder layer obtained from analytical solution is about l temperature Ste 0 15 T Chen Y Zhang Applied Thermal Engineering 26 2006 751 765 761 l 4 longer than that obtained from the numerical solution The e ects of porosity subcooling dimensionless thermal conductivity and Stefan number on the surface temperature location of the liquid surface and the loca tion of the solid liquid interface of the powder bed will be investigated Fig 3 shows how the surface tempera ture is influenced by the porosity in the liquid phase for Ste 0 02 and several di erent subcooling parame ters The e ect of shrinkage is isolated by fixing the sub cooling parameter porosity of the solid phase and the dimensionless thermal conductivity It can be seen that the surface temperature increases as porosity in the liquid phase increases This is because the e ective ther mal conductivity decreases with increasing volume frac tion of the gas When Sc 0 1 the preheating time is a b Fig 9 E ect of subcooling on the location of the liquid much shorter compared to when Sc 3 0 The e ect of shrinkage on the surface temperature for Ste 0 15 is 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 preheating time When Sc 3 0 one can observe that the duration of the melting process is shortened significantly when Ste increases from 0 02 to 0 15 Fig 5 shows the loca tions of solid liquid interface and liquid surface corre sponding to the conditions of Fig 3 The solid liquid interface moves faster when more gas is driven out from the liquid It follows that the corresponding location of the liquid surface moves downward significantly due to the shrinkage of the mixed metal powder bed The locations of solid liquid interface and liquid surface corresponding to the conditions of Fig 4 are shown in l surface and the liquid solid interface Ste 0 02 762 T Chen Y Zhang Applied Thermal Engineering 26 2006 751 765 l Fig 6 The decrease of porosity in the liquid phase also expedites the motion of the solid liquid interface and liquid surface downward Fig 7 shows the e ect of the initial subcooling on the surface temperature for Ste 0 02 It can be seen that the preheating time increases when the subcooling parameter Sc is increased from 0 1 to 0 5 The same trend is observed when Sc increases from 1 0 to 3 0 The e ect of the initial subcooling on the surface tem perature for Ste 0 15 is shown in Fig 8 Compared to the case of Ste 0 02 the preheating time for Ste 0 15 is significantly shortened Meanwhile the preheating time for Ste 0 15 increases when Sc is in creased from 1 0 to 3 0 Figs 7 a and 8 a indicate that a lower liquid surface temperature can be obtained if a larger initial subcooling value is used however a b l Fig 10 E ect of subcooling on the location of the liquid these changes are not apparent from Figs 7 b and 8 b Fig 9 shows the location of the solid liquid interface and the liquid surface corresponding to the conditions of Fig 7 It can be seen in Fig 9 a the existence of greater initial subcooling reduces the moving velocity of the solid liquid interface substantially Before the thermal penetration depth reaches the bottom the solid liquid interface moves rather slowly however the liquid solid interface moves much faster after the ther mal penetration depth has reached the bottom