Analysis of melting in a subcooled two-component metal powder layer with constant heat flux.pdf

Analysis of melting in a subcooled two-component metal powder layer with constant heat fl ux Tiebing Chen, Yuwen Zhang * Department of Mechanical and Aerospace Engineering, University of Missouri-Columbia, Columbia, MO 65211, United States Received 1 February 2005; accepted 18 July 2005 Available online 11 October 2005 Abstract Melting of a subcooled two-component metal powder layer is investigated analytically. The powder bed considered consists of a mixture of two metal powders with signifi cantly diff erent melting points. Shrinkage induced by melting is taken into account in the physical model. The temperature distributions in the liquid and solid phases are obtained using an exact solution and an integral approximate solution, respectively. The eff 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 three-dimensional selective laser sintering (SLS) process can be based. ? 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 selectively fusing a thin layer of the powders with scan- 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 signifi cantly diff 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 signifi cant role as the support structure necessary to avoid boiling phenom- enon, which is the formation of spheres with the approx- imate diameter of the laser beam. The particular 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 melting and resolidifi cation induced by a directed laser beam. It is a good starting point to investigate a simpli- fi ed 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 solidifi cation have been investigated extensively and detailed reviews are avail- able in Refs. 7,8. Melting in SLS of the metal powders is signifi cantly diff erent from the normal melting since the volume fraction of the gas in the powders decreases signifi cantly after melting. Therefore, a signifi cant den- sity change of the powder bed accompanies the melting process. Melting and solidifi cation in 1-D semi-infi nite body with density change under the boundary condition of the fi rst kind have been investigated by Zckert and Drake 9, Crank 10, Carslaw and Jaeger 11 and 1359-4311/$ - 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: +1 573 884 5090. E-mail address: (Y. Zhang). Applied Thermal Engineering 26 (2006) 751765 Charach and Zarmi 12. It should be noted that melting during SLS occurs under the boundary condition of specifi ed heat fl ux instead of specifi ed temperature. Goodman and Shea 13 studied melting and solidifi ca- tion in the fi nite slab under a specifi ed heat fl ux by using the heat balance integral method. Zhang et al. 14 investigated the melting problem in a subcooled semi- infi nite region subjected to constant heat fl ux heating. Zhang et al. 15 solved melting in a fi nite slab with the boundary condition of the second kind by using a semi-exact method. Shrinkage formation due to density change during the solidifi cation 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 solidifi cation process. Zhang and Faghri 17 analytically solved a one- dimensional melting problem in a semi-infi nite two- component metal powder bed subjected to a constant heating heat fl ux. Eff 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 fi nite thickness during the SLS process. In this paper, melting of the mixed powder bed with fi nite thickness subjected to constant heating heat fl ux will be investigated. 2. Physical model The physical model of the melting problem is shown in Fig. 1. A powder bed with fi nite thickness contains two metal powders with signifi cantly diff 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 fl ux, q00, 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 fi nite 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 eff ect of natural convec- tion in the liquid region due to the temperature diff erence is not considered since the temperature is highest at the liquid surface and decreases with increas- ing z. Nomenclature cp specifi c heat (J kg?1K?1) hsllatentheatofmeltingor solidifi cation (J kg?1) kthermal conductivity (W m?1K?1) Kgdimensionless thermal conductivity of gas Ks dimensionless eff ective thermal conductivity of unsintered powder q00 heat fl ux (W m?2) ssolidliquid interface location (m) Sdimensionless solidliquid interface location s0location of liquid surface (m) S0dimensionless location of liquid surface Scsubcooling parameter SteStefan number ttime (s) Ttemperature (K) wvelocity of liquid phase (m s?1) Wdimensionless velocity of the liquid phase zcoordinate (m) Zdimensionless coordinate Greek symbols a thermal diff usivity (m2s?1) ? a dimensionless thermal diff usivity bparameter to distinguish between two melting cases dthermal penetration depth (m) Ddimensionless thermal penetration depth evolumefractionof gas(es) (porosity for unsintered powder) hdimensionless temperature qdensity (kg m?3) sdimensionless time /volume fraction of the low melting point powder in the powder mixture Subscripts ggas iinitial lliquid phase mmelting point psintered part sunsintered solid (mixture of two solid pow- ders) 752T. Chen, Y. Zhang / Applied Thermal Engineering 26 (2006) 751765 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 as o2Ts oz2 oTs ot ;0 z H0; t tm1 T Ti;0 z H0; t 02 ? ks oTs oz q00;z 0; t tm3 oTs oz 0;z H0; t tm4 2.2. Melting After melting starts, the governing equation in the liquid phase is al o2Tl oz2 oTl ot w oTl oz ;s0 z tm5 where w is the velocity of liquid surface induced by the shrinkage. Since the liquid is incompressible, the shrink- age velocity w is w ds0 dt ;s0 tm7 The governing equation for the solid phase and its cor- responding boundary conditions are as o2Ts oz2 oTs ot ;st tm9 The temperature at the solidliquid interface satisfi es Tlz;t Tsz;t Tm;z st; t tm10 The energy balance at the solidliquid interface is ks oTs oz ? kl oTl oz 1 ? es/qlhsl ds dt ;z st; t tm 11 Based on the conservation of mass at the solidliquid interface, the shrinkage velocity, w, and the solidliquid interface velocity, ds/dt, have the following relationship 17: w es? el 1 ? el ds dt 12 2.3. Non-dimensional governing equations By defi ning the following dimensionless variables: hl qcppTl? Tm Uqlhsl hs qcppTs? Tm Uqlhsl Liquid-solid interface q s s0 H0 Low melting point powder High melting point powder z Original surface Liquid surface Fig. 1. Physical model. T. Chen, Y. Zhang / Applied Thermal Engineering 26 (2006) 751765753 Sc qcppTm? Ti Uqlhsl ;s apt H2 ;Z z H S s H ;S0 s0 H ;D d H ;W w ? H ap Ks ks kp1 ? es ;Kg kg kp ; ? a s as ap Ste q00H Uqlhslap 13 The non-dimensional governing equation and the corre- sponding initial and boundary conditions for the pre- heating problem become o2hs oZ2 1 ? a s ? ohs os ;0 Z 1; s sm14 h ?Sc;0 Z 1; s 015 ohs oZ ? Ste Ks1 ? es ;Z 0; s sm16 hs ?Sc;Z D; s sm17 ohs oZ 0;Z D; s sm18 For melting, the non-dimensional equation and corre- sponding boundary conditions are o2hl oZ2 ohl os W ohl oZ ;S0 Z sm19 W dS0 ds ;S0 sm21 o2hs oZ2 1 ? a s ? ohs os ;Ss sm23 hlZ;s hsZ;s 0;Z Ss; s sm24 Ks ohs oZ ? 1 ? el 1 ? es ohl oZ dS ds ;Z Ss; s sm25 W es? el 1 ? el dS ds ;S0 Z tm26 3. Approximate solutions When the top surface of the mixed metal powder bed is subjected to constant fl ux heating, the heat fl ux will penetrate through the top surface and conduct down- ward the bottom surface. The depth to which the heat fl ux penetrates at an instant in time is defi ned as the thermal penetration depth, beyond which there is no heat conduction. Goodman and Shea 13 introduced a parameter, b = q00H/2ks(Tm? Ti), to classify two cases of melting in a fi nite 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- dimensionalparameters defi nedinEq.(13),i.e., b = Ste/2KsSc(1 ? es). It can be seen that the value of b is determined by four basic non-dimensional parame- ters: Stefan number Ste, subcooling parameter Sc, eff ec- tive thermal conductivity of the solid phase Ksand Fig. 2. Validation of analytical solutions. 754T. Chen, Y. Zhang / Applied Thermal Engineering 26 (2006) 751765 volume fraction of gas esin 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. ohs oZ D;s ? ohs oZ 0;s ? 1 ? a s d ds H ScD27 where H RD 0 hsZ;sdZ. hs(Z, s) is assumed to be a second degree polynomial function which satisfi es boundary conditions specifi ed by Eqs. (16)(18). Then hs(Z, s) can be determined hsZ;s ?Sc Q 2KsD1 ? es D ? Z228 The Eqs. (16)(18) and (28) can be substituted into Eq. (27) and then an ordinary diff erential equation for the thermal penetration depth, D, is obtained which can be solved easily. D 6 ? ? as? s1=229 When the thermal penetration depth reaches the bottom surface, i.e., D = 1, the temperature distribution in the powder bed is hsZ;s ?Sc Ste 2Ks1 ? es 1 ? Z2; 0 Z 1; s sD1 sm30 (a) (b) l l l l l l l l Fig. 3. Eff ect of porosity in the liquid phase on surface temperature (Ste = 0.02). T. Chen, Y. Zhang / Applied Thermal Engineering 26 (2006) 751765755 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 fi nite slab. In a manner analogous to that described previously, the temperature of the powder is hsZ;s ?Sc Ste 2Ks1 ? es 1 ? Z2 Ste ? ? as Ks1 ? es ? s ? sD1;0 Z 1; sD1 0; s sm36 hs0;s ?Sc Ste ? ffi ffi ffi ffi ffi ffi ffi ffiffi 6ass p =2 ? Ks? 1 ? es?; Z 0; 0 s sD=1 , the problem becomes melting in a fi nite slab. 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 integral approximate method identical to the case of b 1. 4. Results and discussion Thevalidationoftheanalyticalsolutionwas conducted by comparing the results with the numerical results obtained from Chen and Zhang 20, who inves- tigated the two-dimensional melting and resolidifi cation 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 powder layer subjected to constant heat fl ux, the Gaussian laser beam was replaced by a constant heat- ing heat fl ux 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 liquidsolid 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 (a) (b) l l Fig. 8. Eff ect of subcooling on surface temperature (Ste = 0.15). 760T. Chen, Y. Zhang / Applied Thermal Engineering 26 (2006) 751765 4% longer than that obtained from the numerical solution. The eff 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 solidliquid interface of the powder bed will be investigated. Fig. 3 shows how the surface tempera- ture is infl uenced by the porosity in the liquid phase for Ste = 0.02 and several diff erent subcooling parame- ters. The eff ect of shrinkage is isolated by fi xing 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 ph