NUMERICAL SIMULATION OF DIRECT METAL LASER SINTERING OF SINGLE- COMPONENT POWDER ON TOP OF SINTERED LAYERS

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1 Proceeding o IMECE6 6 ASME International Mechanical Engineering Congre & Expoition Novemer 5-, 6, Chicago, Illinoi, USA IMECE6-379 NUMERICAL SIMULATION OF DIRECT METAL LASER SINTERING OF SINGLE- COMPONENT POWDER ON TOP OF SINTERED LAERS Bin iao and uwen hang Department o Mechanical and Aeropace Engineering Univerity o Miouri-Columia Columia, MO 65 zhangyu@miouri.edu ABSTRACT A three dimenional model decriing melting and reolidiication o direct metal laer intering o looe powder on top o intered layer with a moving Gauian laer eam i developed. Natural convection in the liquid pool driven y uoyancy and Marangoni eect i taken into account. A temperature tranorming model i employed to model melting and reolidiication in the laer intering proce. The continuity, momentum, and energy equation are olved uing a inite volume method. Eect o dominant proceing parameter including numer o the exiting intered layer underneath, laer canning velocity and initial poroity on the intering proce are invetigated. INTRODUCTION Selective laer intering (SLS i a rapid prototyping/manuacturing technique that allow generating complex 3-D part y uing powdered material layer y layer [-4. During the SLS proce, a thin powder layer i laercanned to ue the two-dimenional lice to an underlying olid piece, which conit o a erie o tacked and ued twodimenional lice. Ater laer canning, a reh powder layer i pread and the canning proce i repeated. SLS i a complex proce that require urther undertanding o the phyical mechanim. Some heat conduction model have een developed to reveal the inluence o proceing condition or the laer induced heating proce in the pat. Cheng and Kar [5 conducted a theoretical tudy or the deniication o ceramic coating proceed with a moving laer eam. A three dimenional quai-teady tate heat conduction model wa developed y applying the Fourier integral tranorm method. An approximate expreion or temperature ditriution wa alo preented and it reult are ound to e in good agreement with the exact olution within a limited region around the laer pot. Li et al. [6 modeled the laer melting prolem o ceramic material. A 3-D quai-teady tate heat conduction equation wa olved y applying the Green unction method, and the ingularitie aociated with the numerical integration are treated uing a linear interpolation method. The predicted reult rom the model uing the volumetric heating ource and urace heating ource are compared with experimental data. ie and Kar [7 analyzed a one-dimenional heat conduction prolem to invetigate the melting rate during laer material proceing. The prolem wa olved approximately to otain a correlation among melt depth, power denity, and laer irradiation time. Baed on thi correlation, the dynamic o melting, a relationhip etween the melt depth and power denity and an average melting velocity are expreed y imple analytic ormula. The time to reach the melting and oiling temperature at the urace o the workpiece are alo otained. The luid low within the melt pool i mainly driven y urace tenion and uoyancy orce, which play important role in the temperature ditriution and the hape o the liquid pool. Numerical tudie o heat traner and luid low in laerinduced melt pool during laer welding and cladding have een carried out in the pat. Chan et al. [8 analyzed a tranient ehavior o heat traner and luid low driven y urace tenion on a tationary laer heating proce. The eect o variou dimenionle parameter on urace temperature, melted pool and cooling rate were tudied. Kou and Wang [9 developed a 3-D convection model or the cae where the workpiece i moving with repect to the laer eam. Fluid low driven y uoyancy orce and urace tenion gradient wa conidered. The model demontrated that the urace teniontemperature coeicient can igniicantly aect oth the convection pattern and the penetration o laer melted pool. Li et al. [ tudied the convection-diuion phae change proce during laer melting o ceramic material. Reult y pure heat conduction model, heat conduction model incorporating latent heat o uion and model involving oth Copyright 6 y ASME

2 latent heat o uion and luid were compared. They demontrated that the et prediction accuracy or the melt/olid interace can e achieved y conidering oth the latent heat o uion and luid low in the molten pool. Melting and reolidiication are the mechanim or laer intering o metal intering o metal powder to orm a layer o part and ond dierent layer together to orm unctional part. During metal SLS proce, when the laer eam can and melt a row o powder particle, the melted powder grain tick to each other via urace tenion orce, therey orming a erie o phere with diameter approximately equal to the diameter o the laer eam; thi i reerred to a alling phenomena. In order to overcome the alling phenomenon, everal method including melting o low-melting point powder in a two-component metal powder ytem [-, partial melting o ingle-component metal powder [3-5 are developed. The part produced y partial melting o inglecomponent or two-component metal powder uually are not ully deniied and need time-conuming pot proceing. In order to eliminate the aove limitation, direct metal laer intering (DMLS technique [6-9 ha een developed in the recent year. By applying lower canning velocity and high laer intenity with protection ga to ully melt each ingle track, the entire track o the laer canning can e completely molten without orming pherical tructure [6, 8. The ditinctive eature o DMLS i that it i alway accompanied y the hrinkage due to the igniicant denity change, which in turn conideraly aect the temperature ditriution and the hape o liquid pool. iao and hang [ have etalihed a theoretical model or DMLS in a emiininite powder ed, and the principal character o heat traner and luid low are explored and parametric eect on the intering proce are analyzed. However, in practice, DMLS i a layer-y-layer proce y which the intering proce occur in a reh looe powder layer on top o multiple intered layer. For etter undertanding the inluence o the proceing parameter on the pool dynamic and geometry a well a urace temperature ditriution, the convective heat traner and luid low during DMLS o looe powder on top o exiting intered layer are analyzed numerically in the preent tudy. The eect o hrinkage, laer canning velocity and the numer o exiting intered layer on the intering proce are invetigated. NOMENCLATURE C = dimenionle heat capacity, ρc/( ρc c = peciic heat ( J/kg-K = ource term B = dimenionle ource term g = gravitational acceleration ( m/ h = convective heat traner coeicient ( W/m -K h = latent heat o melting ( J/kg k = thermal conductivity ( W/m-K N = dimenionle moving laer eam intenity, i α q /[ π Rk ( T T a l i N R = radiation numer, 3 εσ e ( T Ti R/ k N t = temperature ratio, T /( T Ti p = preure ( N/m P = dimenionle preure, pr /( ρα q = laer power (W R = radiu o the laer eam ( m = location o the melt/olid interace ( m = location o urace ( m S = dimenionle location o the melt/olid interace, / R S = dimenionle location o urace, / R T = temperature ( K t = time ( u = canning velocity o the laer eam (m/ U = dimenionle canning velocity o laer eam, ur / α v = velocity vector, u i+ v j+ w k (m/ V = dimenionle velocity vector, v R / α V = volume (m 3 w = velocity induced y hrinkage (m/ W = dimenionle velocity induced y hrinkage, wr / α x, y, z = coordinate (m,, = dimenionle coordinate, ( x, yz, / R Greek ymol α = thermal diuivity (m / α a = aorptivity θ = dimenionle temperature δ T = one-hal o phae change temperature range (K δθ = one-hal o dimenionle phae change temperature range, δt /( T Ti ε = initial poroity (volume raction o void, Vg /( Vg + V ε e = emiivity o urace μ = dynamic vicoity (kg/m- ρ = denity (kg/m 3 σ = 8 4 Stean-Boltzman contant ( 57 W/m -K τ = dimenionle time, α t/ R Δ = dimenionle thickne o the looe powder layer Sucript e = eective Copyright 6 y ASME

3 g = ga = liquid = uion p = powder material = olid PHSICAL MODELS Prolem tatement Figure how the direct metal laer intering proce o a looe powder layer on top o multiple previouly intered layer under a moving circular Gauian laer eam at a contant canning peed,, along the poitive x-direction. A raction o the laer power i aored y the powder and lead to the ormation o a melt pool. Ater the laer ource move away, the liquid pool cool and reolidiie to orm the ully deniied part. The prolem under conideration i a typical moving heat ource prolem [. Since the ize o heat ource (in the order o mm or le i much maller than the ize o the powder ed, the intering proce appear to e quai-teady tate rom the tandpoint o the oerver located in and traveling with the heat ource [-4. zd y HA z yd x Liquid Pool Moving Laer Beam y z x xd Unintered one Exiting Sintered Layer Fig. Phyical model or metal laer intering on top o intered layer Governing equation A temperature tranorming model uing a ixed grid method [5 i employed to decrie the melting and reolidiication during laer intering. Thi model aume that the melting and olidiication occur over a range o phae change temperature, and it can alo e ucceully applied to phae change prolem at a ixed melting point. Due to axiymmetry with repect to the center plan (y =, the velocity and temperature ield were computed on only hal o the domain (y >. Since the molten pool move with the laer eam, the prolem i more conveniently tudied in a reerence rame move with the laer eam. The governing equation in the moving reerence rame (x,y,z can e expreed a ollowing [6: ρ + ( ρu + ( ρv + ( ρw = ( t x z ( ρu + ρu( u u + ( ρuv + ρu( w+ w t x z [ [ p ( u ( u = + μ + μ + ( μ u ( x x x z z ( ρv + [ ρv( u u + ( ρvv + [ ρv( w+ w t x z (3 p v v v = + ( μ + ( μ + ( μ x x z z ( ρw + [ ρw( u u + ( ρwv + [ ρw( w+ w t x z (4 p w w w = + ( μ + ( μ + ( μ + ρgβ( T T z x x z z ( ρct + [ ρct ( u u + ( ρcvt + [ ρct ( w + w t x z = k + k + k (5 x x z z + t x z ( ρ [ ρ( u u ( ρv [ ρ( w w A poroity equal to zero within the melt pool ater melting, the hrinkage velocity o the melting zone can e determined in the moving coordinate ytem [3: z > w = (6 ε < u z t x By applying temperature tranorming model, the eective heat capacity o the powder ed can e expreed a c T < T δt h c= cm + T δt T T (7 δ T c T > T and in the eq. (5 i deined a T < T δt h = T δt T T h T > T The thermal conductivity in eq. (5 in the muhy zone can e aumed a a linear unction o temperature, i.e., ke T < T δt k ke k = ke + ( T T T δt T T (9 δ T k T > T where k i the eective thermal conductivity o the e unintered powder ed. It can e calculated uing the empirical correction propoed y Hadley [7. (8 3 Copyright 6 y ASME

4 k e where ( p g ( p g ε + k / k ( ε ( α g ε ( + k / k ε ( ( k p / k g ( ε + (+ ε ( k p / kg ( + ε ( k p / k g + ε = k + α ( =.8 +.ε (a 4.898ε ε 87 logα = ( ε ε ( ε ε 8 ( At the olid-liquid interace or in the muhy region, the velocitie are zero. A commonly ued procedure i to precrie a luid vicoity that i equal to the liquid vicoity in the liquid region and increae gradually over the muhy region to a large value in the olid. Thereore the vicoity i expreed a N T < T δt μ N μ = μ + ( T T δ T T δ T T T + δ T ( δ T μ T > T Boundary condition The oundary condition at the top urace ( z = i q x + y 4 4 exp ht ( T ( + + σε e T T = k ( π R R z z= where R i the radiu o the Gauian laer eam. To model the Marangoni convection due to temperature gradient at the top urace, we alance the hear orce and urace tenion at the ree urace v v n γ T μ + = (3 n v v n γ T μ + = (4 n where v and v n in Eq. (3 are the tangential and normal velocity component at the heating urace. The ucript o v, and repreent the two tangential direction. The oundary o preure at the urace o the molten pool i aumed to e the atmopheric preure, and the preure at the liquid-olid interace i determined y Laplace-oung Equation. p = p, z = (5 σ σ p = pv = p, z = (6 re re At the ottom urace ( z = z max, the oundary condition or velocity and temperature are a ollow: u = v = w = (7 k = h( T T (8 z The ymmetric condition at the center urace ( y = are valid: v = (9 u w = = = ( At ide urace ( y = y, the oundary condition are max u = v = w = ( k = h( T T ( At x = x (ar ahead o the heat ource: max u = v = w = (3 T = T i (4 At x = x (ar ehind the heat ource: max u = v = w = (5 = (6 x Dimenionle governing equation Introducing the ollowing dimenionle variale ( x, yz, (,, =, ( uvwu,,, R ( UVWU,,, =, S =, R α R τ = α t, T T θ =, δt δθ =, ρc C =, ( ρc C =, R T T T i T i ( ρc l ( ρc B =, k K =, ke Ke =, μ Pr =, T δθ = δ c ( T T i k k ρα T T i c ( T Ti St =,, pr P =, gβ R 3 ( T Ti Ra = (7 h ρα να The governing equation can e rewritten a U V W + + = (8 U [ UU ( U ( UV [ UW ( + W τ (9 P U U U = + Pr + Pr + Pr V [ VU ( U ( VV [ VW ( + W τ (3 P V V V = + Pr + Pr + Pr W [ WU ( U ( WV [ WW ( + W τ (3 P W W W = + Pr + Pr + Pr + Ra Prθ ( Cθ [ Cθ( U U ( CθV [ Cθ( W + W τ θ θ θ = ( K + ( K + ( K (3 B [ ( U U B ( VB [( W + W B τ 4 Copyright 6 y ASME

5 where S = S S (33 W ε < U S τ N θ < δθ ( Pr N (34 Pr = Pr + ( θ δθ δθ θ δθ δθ Pr θ > δθ C θ < δθ (35 C = ( + C + δθ θ δθ Stδθ θ > δθ Ke θ < δθ ( Ke (36 K = Ke + ( θ + δθ δθ θ δθ δθ θ > δθ θ < δθ (37 B = δθ θ δθ St θ > δθ St The oundary condition Eq. ( written in dimenionle orm i a ollow, θ 4 K = N exp i N R ( θ + Nt = S (, (38 4 ( θ + Nt Bi( θ θ The dimenionle orm o other oundary equation can alo e otained uing the dimenionle variale deined in Eq. (6. NUMERICAL SOLUTIONS The arication o unctional part i relying on a layer-ylayer SLS proce. The newly intered layer hould integrate tightly with the exiting layer underneath or trong mechanical propertie. The melting/reolidiication prolem deined y eq. (8 (3 i a teady-tate, three-dimenional, nonlinear prolem. Since the location o the olid-liquid interace and the heating urace are unknown a priori, a ale tranient method i employed to locate variou interace. Steady-tate olution i otained when the temperature ditriution and location o variou interace do not vary with the ale time. Equation (8 (3 with the ale tranient term can e olved y the inite volume method [8. The computational domain i the whole powder ed and a lock-o technique [8 i employed to imulate the exitence o the empty pace created y hrinkage o the powder ed ater melting. The denity and thermal conductivity in the empty pace are et to e zero. The governing equation and oundary condition ormulated in eq. (8 (3 and eq. (38 were dicretized and olved uing the SIMPLEC algorithm [8. The convection and diuion term were dicretized uing the power-law cheme. The oundary condition are merged into energy and momentum equation or the appropriate node uing additional ource term method [7. A non-uniorm 5 5 (in the,, direction, repectively grid numer i ued in computation and the ale time tep i. A ine grid within the melt pool and coare grid in the unintered region were ued. The iterative procedure wa continued until the ollowing convergence criterion wa atiied: old φ φ P < (39 φ P where P Σ denote ummation overall grid point and φ i the variale eing computed, e.g., U, V, W and T. The laer intenity and canning velocity hould enure to otain the ideal intering depth which i eyond the ottom urace o the looe powder layer. To otain the expected intering depth, the optimum comination o dimenionle laer eam intenity and canning velocity i required. Thereore, the dimenionle laer power intenity i increaed in mall increment in order to otain the expected intering depth when the intering depth doe not move down. The computation can e topped when 5% overlap etween HA and exiting intered layer underneath. RESULTS AND DISCUSSIONS The developed computer program i irt ued to imulate laer melting o a non-porou 663 aluminum heet with a dimenion o mm. Figure how the comparion o the uion oundary otained y the preent tudy and experiment conducted y Kou and Wang [9. The nominal eam power indicated on the Spectra Phyic 97 continuou-wave CO laer wa.3 KW. The travel peed o the workpiece wa 43 mm/. The power aored y the workpiece wa meaured calorimetrically. A calorimeter wa made o a mm quare tue o the workpiece material. An 86% heat lo rom the urace area irradiated y the laer eam include thoe y relection, radiation and convection. z ( -4 m y ( -4 m EPERIMENT SIMULATION Fig. Comparion o experimental and calculation reult or laer uion o 663 aluminum heet 5 Copyright 6 y ASME

6 The eam diameter, mm, wa meaured uing plit anode method. Detail o the experimental procedure and treatment o the data can e ound in Re. [9. A Shown in Fig., ome dicrepancie were oerved etween the imulated and experimental proile or the uion oundarie. The main reaon may e due to deviation in the modeled Gauian model rom the actual laer energy ditriution. Otherwie, the imulated and meaured uion oundarie are in good agreement with each other. Numerical calculation i perormed or intering o AISI 34 tainle teel powder in preent tudy. Tale lit the material phyical propertie or AISI 34 [9-3. Tale Thermophyical propertie o AISI 34 Nomenclature Symol Unit Value Denity ρ p kg/m 3 7 Thermal conductivity k p W/m-K 4.9 Speciic heat c p J/kg-K 46 Melting point T K 67 Latent heat o uion h kj/kg 47 Vicoity μ kg/-m 5 Surace tenion at melting point γ N/m.943 Dependence o urace tenion on γ temperature N/m K Figure 3 illutrate the urace temperature ditriution o the powder ed at the quai-teady tate ( N = 5. The peak temperature at the powder ed urace i near the trailing edge o the laer eam rather than at the center o the laer eam due to motion o the laer eam. Becaue the thermal conductivity in the molt pool i much larger than that in the unintered zone, the temperature change moothly in the molten pool ut harply in the unintered zone near the molt pool. Figure 4 how the three-dimenional hape o the powder ed urace, molt pool and HA at the ame condition o Fig. 3 when N = and N = 5. The depth o urace and HA decreae with the increaing in the laer canning direction. Surace Temperature Fig. 3 The temperature ditriution at the urace o the powder layer ( Δ =, U =, ε =, N = (a ( Fig. 4 Three-dimenional hape o the HA ( Δ =, U =, ε =, (a N = ; ( N = 5 Figure 5(a-(c how the velocity vector in the liquid pool plotted in three dierent view ( ε =, N = 5. Since the urace tenion i a decreaing unction o temperature, i.e., γ / T <, the higher urace tenion o the cooler liquid metal near the edge o the liquid pool tend to pull the liquid metal away rom the center o the liquid pool, where the liquid metal i hotter and the urace tenion i lower. Thereore, luid low on the urace o liquid pool i radially outward a can e een in Fig. 5(a. The liquid metal low i alo driven y uoyancy orce a illutrated in Fig. 5( and 5(c. The hotter liquid metal near the central region o the molten pool low up to the urace, while the cooler liquid metal near the pool oundary ink along the melt/olid interace to the ottom o the pool. Thi circulation o luid low induced y the urace tenion gradient and uoyancy orce i conitent with the typical natural convection pattern ound in the literature. Figure 6(a-(c preent the temperature contour in the powder ed plotted in three dierent view ( N = 5. The Marangoni convection (radially outward at the top urace and the hrinkage phenomenon reulted in a igniicant amount o heat low rom the hotter region to the colder region epecially in the z direction, which in turn reult in a wider and deeper melt pool. Another oervation i that the iotherm near the 6 Copyright 6 y ASME

7 melting ront are more cloely paced compared with thoe ar away rom the melt/olid interace. The eect o hrinkage comined with the canning velocity o the moving laer eam and the numer o the exiting intered layer underneath on the ormation o the Heat Aected one (HA are invetigated. The overlap etween the liquid pool and exiting intered layer elow are conidered in order to ond the newly depoited layer with the exiting intered layer tightly. Figure 7 how the eect o the dimenionle velocity on the intering proce with only one exiting intered layer. It can e een rom Figure 7(a that the overlapped region o the liquid pool ha reached the ottom o the phyical domain when the quai-teady tate i achieved. The higher moving laer eam intenity i needed or the intering proce with higher canning velocity in order to otain the ame intering depth. Figure 7(a alo illutrate that, when the canning velocity increae, the whole melt pool hit toward the oppoite direction o the laer canning due to the enhanced advection low caued y the moving o the laer eam relative to the powder ed. Beide, the hape o HA with dierent velocitie are imilar ut the liquid pool i a litter narrow when the canning velocity increae a hown in Fig. 7(. That i due to the act that the time interaction etween the moving laer eam and the powder layer ecome horter when canning velocity increae (a ( (c Fig. 6 Dimenionle temperature contour ( Δ =, U =, ε =, N = 5, (a Top view; ( Longitudinal view at y =; (c Cro-ectional view at x = U =, N i =.4 U =, N i =.44 - ε= N= (a - - (a U =, N i =.4 U =, N i =.44 ε= ϕ g = N= ( (c Fig. 5 Dimenionle velocity vector ( Δ =, U =, ε =, N = 5 (a Top view;( Longitudinal view at y =; (c Cro-ectional view at x = - - ( Fig. 7 Eect o laer intenity and canning velocity on the intering proce with only one exiting intered layer ( ε =, N =, (a Longitudinal view at y =; ( Cro-ectional view at x = 7 Copyright 6 y ASME

8 U =, N i =3 U =, N i =48 ε= N=3 component metal powder. The optimized comination o the laer intenity and the canning velocity or the cae with dierent numer o exiting intered layer to achieve the required intering depth wa analyzed. It howed that, when the numer o exiting intered layer underneath i increaed, higher intenity i needed to achieve required overlap etween newly intered layer and exiting layer. A the increae o canning velocity will horten the powder/laer interaction time, higher intenity i needed to achieve the required overlap when canning velocity increae. - - (a U =, N i =38 U =, N i =46 U =, N i =.8 U =, N i =.88 ε= N=5 ε= N=3 - - ( Fig. 8 Eect o laer intenity and canning velocity on the intering proce with only one exiting intered layer ( ε =, N = 3, (a Longitudinal view at y =; ( Cro-ectional view at x = Figure 8-9 how the eect o the dimenionle velocity on the intering proce with three exiting intered layer ( N = 3 and ive exiting intered layer ( N = 5. Compared with Figure 7, it can e een that the laer intenity increae igniicantly when the numer o exited intered layer underneath the looe powder layer increae. In order to get the deired intering depth to comine the newly intered layer with the exiting intered lay, higher laer intenity i need or the cae with higher canning velocity under the ame other condition. The liquid pool move lightly toward the poitive direction o the -direction ecaue o the advection heat low caued y the moving laer eam. The ottom o the overlapped region o HA i not lat ince the required overlap etween newly intered layer and exiting lay ha een achieved eore it reache the ottom urace o the powder ed. CONCLUSIONS Three dimenional numerical modeling or direct metal laer intering on top o intered layer ha een perormed. The computer code wa validated y comparing the predicted cro-ection or the melt/olid interace during laer melting o a non-porou 663 aluminum heet with experimental reult. The luid low greatly inluence the temperature ditriution and the hape o liquid pool during laer intering o ingle- - - (a U =, N i =.8 U =, N i = ε= N=5 ( Fig. 9 Eect o laer intenity and canning velocity on the intering proce with only one exiting intered layer ( ε =, N = 5, (a Longitudinal view at y =; ( Cro-ectional view at x = ACKNOWLEDGEMENT Support or thi work y the Oice o Naval Reearch under grant numer N i grateully acknowledged. REFERENCES [ Kruth, J. P., 99, Material Ingre Manuacturing y Rapid Prototyping Technique, Manuacturing Technology CIRP Annal, Vol. 4, No., pp [ Hauer, C., Child, T. H. C., Dalgarno, K. W. and Eane, R. B., 999, Atmopheric Control during Direct Selective Laer Sintering o Stainle Steel 34S Powder, 8 Copyright 6 y ASME

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