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Development of a theoretical process map for laser cladding using two-dimensional conduction heat transfer model
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Computational Materials Science 41 (2008) 457–466 www.elsevier.com/locate/commatsci
Development of a theoretical process map for laser cladding using two-dimensional conduction heat transfer model Subrata Kumar a, Subhransu Roy b
b,*
a JFWTC, GE India Technology Center, Bangalore 560 066, India Mechanical Engineering Department, Indian Institute of Technology, Kharagpur 721 302, West Bengal, India
Received 18 December 2006; received in revised form 30 April 2007; accepted 5 May 2007 Available online 27 June 2007
Abstract In blown powder laser cladding process, the powder travels across the laser path, gets heated up by absorbing laser energy, and finally melts on the substrate under the intense laser beam; as the substrate moves away this melt pool solidifies to form a continuous built-up layer. In the present study a two-dimensional conduction heat transfer equation has been solved using finite volume method to develop a theoretical process map for laser cladding. The developed process map indicates a range of scanning speed and powder feed rate for the feasibility of the process; the lower limit is dictated by the maximum melt pool temperature, and the higher limit by poor bonding due to lack of melting of the substrate (i.e. low dilution). Parametric regions for thick and thin cladding with low dilution can be decided from the process map. It is found that the process range expands with the increase in total absorbed power as well as power directly absorbed by the powder. Correlations for maximum melt pool temperature and dilution are presented. A process map for identifying the form and scale of the microstructure in the solidified layer is also presented. 2007 Elsevier B.V. All rights reserved. Keywords: Conduction heat transfer; Cladding; Process map; Laser processing
1. Introduction Laser cladding with powder feed is one of the emerging technologies having multiple applications in manufacturing industry; ranging from surface repair and modification with superior coating over a mechanical component [1–4] to direct fabrication of three-dimensional functional metallic components, having simple to complex geometry, with desired compositions [5–9]. The basic working principle of this process is melting a substrate surface by a moving laser beam and adding pneumatically delivered metal/ceramic powders into the melt pool which subsequently melts; as the laser moves away, the melt solidifies and forms the built-up layer over the substrate [10]. Laser cladding with powder feed is a multi-parameter process where the laser parameters (power, beam profile, etc.), powder parameters *
Corresponding author. Tel.: +91 3222 282968. E-mail address: [email protected] (S. Roy).
0927-0256/$ - see front matter 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2007.05.002
(feed rate, size, etc.) and other parameters (scanning speed, material properties, etc.) influence the deposition track characteristics (geometry, dilution, mixing, microstructure, etc.). Most of the studies on laser material deposition with powder feed generally deal with the metallurgical aspects— the relationship between the microstructure and properties, usually hardness and corrosion resistance of the built-up surface layer for a series of added materials [11–19]. It is of great importance to find the range of process parameters such as laser power, scanning speed, powder feed rate, etc., for a low dilution, pore free and continuous deposition. Experimental studies show that the laser power required for good deposition is in between discontinuous track and large dilution power range [20]; built-up height increases with powder feed rate and decreases with scanning speed [21]; deposition width increases with laser power [22]; increase in powder flow rate causes lower dilution [23,24]; global absorption and attenuation of the beam increase
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Nomenclature a aK b cp D F G fl Hc Hs h hc ^i, k ^ k kp m_ p ^n P p Bi Bo Ca Nc q_ 000 h Ste R r r0 T t U U?
fraction of lower mode in the laser beam elements of coefficient matrix deposition width (m) specific heat (J/kg K) dilution irradiation flux vector (W/m2) temperature gradient (K/m) liquid fraction built-up height (m) substrate melt depth (m) specific sensible enthalpy (J/kg) heat transfer coefficient (W/m2 K) unit vectors along x- and z-axis thermal conductivity (W/m K) equilibrium partition coefficient powder feed rate (kg/s) unit surface normal vector laser power (W) dynamic pressure (N/m2) Biot number Bond number capillary number conduction-to-radiation parameter volumetric energy source (W/m3) Stefan number radius of curvature (m) position vector (m) radius of Gaussian beam (m) temperature (K) time (s) scanning speed (m/s) lateral scanning speed (m/s)
with powder feed rate [21]; shorter interaction time produces finer microstructure and higher hardness [25]; some conditions of speed, laser power and material produce tensile stresses during solidification that may lead to microcracks [26]. There also have been some experimental efforts to correlate different process parameters for good quality deposition [22,27,28]. These experimental investigations are generally very expensive and time consuming. To reduce the experimental efforts, development of a process map through theoretical investigations based on appropriate heat transfer model is the aim of this article. In the present study a two-dimensional conduction heat transfer model has been used to develop a theoretical process map for laser cladding process. The governing equation along with boundary conditions are solved using finite volume method in a non-orthogonal grid system with collocated variable arrangements [29]. The developed process map indicates a range of scanning speed and powder feed rate for the feasibility of the process based on the limits of maximum melt pool temperature and low dilution.
velocity of the solidification front (m/s) Vs v velocity vector (m/s) x, y, z Cartesian coordinate system We Weber number Greek symbols a thermal diffusivity (m2/s) Dhsl specific latent heat of fusion (J/kg) DPIZ power absorbed by powder (W) DVm volume of the melt pool (m3) emissivity g fractional absorption of laser power l dynamic viscosity (kg/m s) q density of the medium (kg/m3) r surface tension (N/m) rb Stefan Boltzmann constant (W/m2 K4) x relaxation parameter Subscripts 0 reference 1 ambient condition A, C melt pool end points E eutectic l liquidus m melting point of pure base metal max maximum value p powder Superscripts w dimensionless 0 per unit length
Correlations for maximum melt pool temperature and dilution are presented. Parametric regions for thick and thin cladding are shown in the process map. It is found that the process range expands with the increase in total absorbed power as well as power directly absorbed by the powder. A process map for identifying the form and scale of the microstructure in the solidified layer is also presented. Residual stress state that is responsible for formation of undesirable microcracks has not been addressed while developing the process map.
2. Modeling of laser cladding process A typical set-up for the process is schematically shown in Fig. 1. The laser beam is falling on a substrate where it forms a melt pool. The powder particles travel through the laser–powder interaction zone before reaching the substrate. Most of the powder particles fall into the melt pool on the substrate, gets trapped and melts. As the laser moves
S. Kumar, S. Roy / Computational Materials Science 41 (2008) 457–466
459
Laser Beam Intensity Profile y
z
b Powder + Carrier gas Laser Beam Spot Path
LaserPowder Interaction Zone >
A : Melting ends
n
AB : Freezing Front BC : Melting Front
Build up layer Melting starts : C x
x
Hc : Clad Height Hs : Substrate Melt Depth
B U
MeltingSolidification Zone Substrate
Fig. 1. Set-up for laser deposition of metals with powder feed.
away, the melt solidifies and forms the built-up layer over the substrate. The whole process can be described by its three components—(i) model for the laser beam, (ii) model for the laser–powder interaction zone and (iii) model for the melting–solidification zone. In this study, the first two are incorporated in the model for the melting–solidification zone. The following assumptions are made, (1) The laser beam: The laser beam is assumed to be a mixture of the first two fundamental modes, 40% TEM00 and 60% TEM01*, which gives a near uniformly distributed top-hat type beam profile. (2) The laser–powder interaction zone: Powder particles get heated up by absorption in the laser–powder interaction zone, termed as preheating, and the laser beam attenuates due to absorption and scattering. Here, DP IZ ¼ m_ p ðhp h1 Þ
ð1Þ
is an input parameter where DPIZ is the power absorbed for preheating the powder of mass flow rate m_ p up to the enthalpy level hp. The intensity profile nature of the attenuated beam is assumed to be invariant. (3) Energy source due to mass addition: Preheated powder enters the melt pool and comes in thermal equilibrium with the melt. This has been treated as uniformly distributed volume source in the melt pool to satisfy global energy conservation. (4) Total absorbed laser power: This model is based on total absorbed laser power gP. The efficiency parameter g has been introduced to account for the loss by reflection from the irradiated substrate surface as well as for the energy loss due to scattering by powder particles.
(5) Melt pool convection: Convection mode of heat transfer due to fluid flow is present only in a small region, described as the melt pool. It is surrounded by solid material where the heat diffuses by conduction. The melt pool surface is continuously bombarded by the solid powder particles. Moreover, assuming that a large proportion of the melt pool is occupied by solid powder particles undergoing melting, the strength of the melt pool convection can be considered to be very week. Here, the melt pool convection is neglected. 2.1. The governing equation As the laser moves over the substrate along with powder feeding; within few millimeters of travel a steadily moving melt pool forms below the laser beam, which on solidification generates a uniform deposit layer over the substrate [30]. The reference frame is attached to the moving heat source, thus reducing it to a quasi-stationary process in the Eulerian reference frame where the substrate moves with constant velocity U in the x-direction. The laser beam is moving rapidly back-and-forth in the direction perpendicular to the x–z plane (see Fig. 1 inset) with a time scale smaller than the thermal diffusion time scale such that a uniform time-averaged intensity in the y-direction can be assumed. Therefore, a one-dimensional beam intensity profile which varies only in the x-direction is considered and the problem reduces to two dimensions. Under these conditions and based on fixed-grid numerical method [31], the energy equation for the melting–solidification zone is oðqhÞ oðqhÞ k þU ¼$ $h þ q_ 000 ð2Þ h ot ox cp
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S. Kumar, S. Roy / Computational Materials Science 41 (2008) 457–466
The heat source q_ 000 h is zero in the solid region (h < hl) and elsewhere it is defined as oðqfl Þ oðqfl Þ m_ p 000 ^ þ Ui hp ðh þ Dhsl Þ Dhsl q_ h ¼ ot ox bV0m |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Part-II
Part-I
ð3Þ V0m
In Eq. (3), represents two-dimensional melt pool area and m_ p is the powder deposition rate over the width b that gives rise to a clad height Hc at the scanning speed U. Powder feed rate per unit width m_ 0p is defined as m_ 0p ¼
m_ p ¼ qUH c b
pure liquid :
hE < h < hl : fl ¼
Tm T Tm Tl
1k1
p
ð5Þ
h P hl : fl ¼ 1:0
2.2. The boundary conditions
F^ n ¼ ^ n
k $h þ hc ðT T 1 Þ þ rb ðT 4 T 41 Þ cp
ð6Þ
where 1=2 0
8 1 F¼ gP DP 0IZ p ð1 þ aÞr0 2x2 2 2 ^ a þ ð1 aÞ 2 eð2x =r0 Þ k r0 P DP IZ P 0 ¼ ; and DP 0IZ ¼ ¼ m_ 0p ðhp h1 Þ b b Other boundaries :
x ! 1; z ! 1 : h ¼ h1 ;
Hs b ; bH ¼ ; r0 r0 a0 t $ t H ¼ 2 ; $H ¼ r0 r0
HH s ¼
h h1 ; h0 h1
hH p ¼
H H H m_ 0H p ¼ q U Hc ;
hp h1 ; h0 h1
PH ¼
rH ¼
TH ¼
r r0
hH ; cp0
PH H _ 0H ; DP 0H IZ ¼ m p hp bH Dhsl lU 0 Ste ¼ ; Ca ¼ ; h0 h1 r P 0H ¼
q0 gr20 ; r
q0 U 20 r0 r
We ¼
k
ð8Þ
The dimensionless governing equation is
oh x!1: ¼0 ox
ð10Þ
GH ¼ $H T H ^ n
gP q0 a0 r0 ðh0 h1 Þ
Bo ¼
Eq. (7) gives a mixed mode beam profile having fraction a of TEM00 and (1 a) of TEM01* [34]. The one-dimensional laser intensity profile in Eq. (7) is a valid assumption when the time scale for the lateral movement b/U? is smaller than the thermal diffusion time scale r20 =a0 and the lateral scanning width b is much greater than the nominal beam radius r0. The speed U? of lateral motion of the beam is also much greater than the forward scanning speed U. Mathematically, b r0 and U ? U
Hc ; r0 U UH ¼ ; U0 HH c ¼
ð7Þ
ð9Þ
b r2 < 0; U ? a0
The governing equation and boundary conditions are normalized using the properties of the liquid metal at liquidus temperature as reference properties (i.e. q0, a0, k0, cp0, h0, etc.); the radius r0 of the Gaussian component in the laser beam is the length scale, the speed of thermal diffusion (U0 = a0/r0) is the velocity scale and (h0 h1)/cp0 is the temperature scale. The important dimensionless parameters are
hH ¼
Top surface :
r ov þ 2l ^n ¼ p p1 þ qgðH c zÞ R on x 6 xC : z ¼ 0; x P xA : z ¼ H c
xC < x < xA :
3. Dimensionless GE
h 6 hE : fl ¼ 0:0
mushy region :
During material addition for building a surface layer, a curved melt pool free surface is formed that connects the untreated substrate surface to the built-up layer. At steady state the shape of this free surface of the melt pool is derived from the balance of all the normal forces acting on the liquid–gas interface,
ð4Þ
‘Part-I’ in Eq. (3) is due to heating and melting of the solid powder mass injected into the melt pool according to the model assumption and ‘Part-II’ is due to conversion of latent enthalpy to sensible enthalpy at the solidification interface and vice-versa at the melting interface [32]. The equation for the liquid fraction in the mushy region is given by [33] pure solid :
2.3. Top surface shape equation
Nc ¼
rb ðT l T 1 Þ3 r0
oðqH hH Þ þ $H qH vH hH ¼ $H H ot
;
Bi ¼
hc r 0 k
kH H H $ h cH p
! þ q_ 000H h
where q_ 000H h
¼
m_ 0H p
½hH p 0H Vm
H oðqH fl Þ H oðq fl Þ ðh þ SteÞ Ste þU otH oxH H
The dimensionless boundary condition for the top surface is ! kH H H 1 H4 H F ^n ¼ ^n H $ h T þ BiT H þ Nc cp |fflfflffl{zfflfflffl} radiative loss
S. Kumar, S. Roy / Computational Materials Science 41 (2008) 457–466
where rffiffiffi 2 8 1 H H H2 2xH ^ ðP 0H m_ 0H k F ¼ e p hp Þ a þ ð1 aÞ2x p1 þ a The shape of the top surface is given by 1 ovH H H þ 2Ca H ^ n ¼ WeðpH pH 1 Þ þ BoðH c z Þ H on R
ð11Þ
As the melt pool convection is very weak due the presence of large number of particles undergoing melting, the viscous term on the left and the dynamic pressure terms on the right can be neglected in Eq. (11). Additionally, due to low Bond number ðBo ¼ qgr20 =r ¼ 0:03Þ and smaller size of the melt pool, the shape of the free surface is approximated as a circular arc of constant radius Rw connecting the points ‘C’ and ‘A’ in Fig. 1. This approximation is very near to the reported pictures of melt pool by high speed photography and video recording of laser cladding [35,36]. Thus, the simplified top surface shape equation is x 6 xC : zH ¼ 0 xC < x < xA : RH ¼ constant H
x P xA : z ¼
ð12Þ
HH c
4. Solution procedure The dimensionless governing equation with non-linear boundary conditions applied to the irradiated surface are solved using finite-volume method on a boundary fitted non-orthogonal grid system with collocated variable arrangement [29]. The top surface boundary line is generated with some H assumed melt pool end point positions, rH A and rC . Bodyconforming grid lines are generated by an algebraic grid generator. A typical non-orthogonal grid system is shown in Fig. 2. Capital letters (P, E, W, N, S) in this figure are cell centers and the small letters (e, w, n, s) are the face cen-
461
ters. In collocated variable arrangement, all the values of the dependent variables are stored at cell centers, and face center values are evaluated by interpolation. The discretization of the dimensionless governing equation for each control volume gives a set of algebraic equations which can be written for the P’th node of Fig. 2 as X aK hH where K ¼ E; W; N; S ð13Þ aP hH P þ K ¼ qP ; K
The above set of algebraic equation (13) is solved iteratively until the normalized residual falls below 104. Then H the melt pool end points rH A and rC are corrected using relaxation parameters xA and xC, respectively: Hold Hnew Hold rHnew A;C ¼ rA;C þ xA;C ðrA;C rA;C Þ |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl}
ð14Þ
correction term
The whole process is repeated, starting with a new grid generation for the updated free surface, until the correction term in Eq. (14) becomes insignificant and then the surface profile is considered to be converged. To improve solution accuracy, fine grids are taken in the melt pool region where the change of variables is rapid. To check the grid independence of the solution three grid levels, 90 · 40, 135 · 60 and 180 · 80, were tested for few sets of input parameters and finally 180 · 80 grid level was selected for all sets of input parameters. 4.1. Validation of the computer code The computer code is validated with a two-dimensional analytical solution for heating of a semi-infinite body with a moving heat source strip on the surface. The radiative and convective heat losses from the surface are neglected. The analytical solution for steady surface temperature of 0.3
Numerical Analytical
j+1
N j
W
n^
n w
n^
P
0.2
n^
e s n^
j–1
E
0.1
S i–1
j–2 i–2
i 0
i+2 Fig. 2. Non-orthogonal grid system.
0
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15
Fig. 3. Comparison between numerical and analytical solution for the test problem.
462
S. Kumar, S. Roy / Computational Materials Science 41 (2008) 457–466
the semi-infinite body for high value of Peclet number (Pe) is given by [37] 8 pffiffiffiffiffiffiffiffihpffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffii H ðT surface T 1 ÞpkU < pPe xH xH 1 for x P 1 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi : 2aF for xH 6 1 pPexH
12
ð15Þ
7
0.2
6
0.1
10
0.5 0.4
8
0.3
5 4 3 2 1 0
0
1
2
3
4
5
6
7
8
9
10 11 12 13 14
Fig. 4. Iso-enthalpy contours for P 0 w = 3.0, Uw = 2.0, m_ 0H p ¼ 0:4 and hH p ¼ 0:5.
5. Results The material used for this simulation is steel, an alloy of iron and carbon, containing small % of carbon. This material melts over a small temperature range depending on the % of carbon as per the temperature–composition phase diagram [38]. The thermo-physical properties of liquid iron at melting temperature, listed in Table 1, are used for this simulation. kp was taken as 0.3 which is typical of steel. A top-hat beam intensity profile is selected by choosing parameter a = 0.4. The powder preheat level hH p varies between 0 and 1; hH p ¼ 0 (no preheat) is an ideal case and hH p ¼ 1 means that the powder is heated to near meltingpoint temperature. Constant thermo-physical properties of liquid iron at melting point are used. The energy equation with the appropriate boundary conditions is solved and the enthalpy field is calculated for three dimensionless power levels P 0 w = 2.4, 3.0 and 4.0 for a range of m_ 0H p and 0H Uw varied from a starting value of m_ 0H and p ¼ 0:1 P Uw = 0.25P 0 w, respectively, until they reach a limit, i.e. the substrate does not melt and there is no fusion bonding between the deposit and the substrate. 5.1. Enthalpy field and the selection of parameters Fig. 4 shows typical contours of specific sensible enthalpy for a set of input parameters. The shaded part in this figure is the melt pool, having hw P 1. This figure reveals that part of the laser beam is falling ahead of the melt pool and rapidly heating up the substrate in the Table 1 Properties of liquid iron at melting point and other used values [42,43]
cp Tm hm h1
0.8 0.7 .6 0
9
Here F is the uniform heat flux from x = 0 to x = l, which is moving with speed U; Pe (=Ul/a) is the Peclet number; and xw = x/l. For the numerical solution, the present computer code is used in a domain of size 15 · 5. Other inputs are Fw = 1 for w 3 6 xw 6 4, Ste = 0, m_ 0H p ¼ 0 and U = 20. The finest grid level, i.e. 180 · 80, is chosen for the solution. It is clear from Fig. 3 that the numerical solution for the test problem is matching with the analytical solution.
q
Laser Beam
11
7.03 · 103 kg/m3 824 J/kg K 1538 C 1050 · 103 J/kg
Dhsl
247 · 103 J/kg
r
k a l
36 W/m K 6.2 · 106 m2/s 5.6 · 103 kg/ms
hc rb
1800 · 103 N/m 0.19 250 W/m2 K 5.67 · 108 W/m2 K4
upstream region. Due to the presence of the convective term in the energy equation as a result of the relative motion between the laser and the workpiece, the temperature gradient is large in the upstream region and there is a long hot zone trailing behind the laser beam. From the geometry of the melt pool, the depth of the remelted substrate layer and the height of the deposited layer can be find out. These two data are important to calculate dilution D. Dilution is the fraction of substrate material in the deposited layer and for two-dimensional temperature field it is defined as D¼
Hs HH ¼ H s H Hc þ Hs Hc þ Hs
ð16Þ
where Hs is the substrate remelted depth and Hc is the built-up height as shown in Fig. 1. D is an important parameter to decide the combination of the values of the process parameters P 0 w, Uw and m_ 0H p as low dilution means better quality of deposition. Similarly, the enthalpy field also gives the maximum melt pool sensible enthalpy hH max which is an important parameter to decide the feasibility of the process; it must be above the melting point (hw = 1) but below the boiling point (hw 2). These two parameters (D and hH max ) are selected to develop the process map for laser cladding. w Fig. 5a shows the variation of D and hH for max with U H 0w 0H P = 4.0 and m_ p ¼ 0:8 at different hp and Fig. 5b shows 0w _ 0H the variation of D and hH = 4.0 and max with m p for P w H U = 2.0 at different hp . The left y-axis shows D between 0 and 1 and the right y-axis shows hH max between 0 and 2. The bottom curves are for D and the top curves are for w hH or m_ 0H max . In general, D decreases with increasing U p , the linear term is stronger than the non-linear term for w H the variation of D with m_ 0H p , and hmax also reduces as U H 0H or m_ p increases. These figures also reveal that hp has significant effect on these, specially D. A detailed study of these can be found in reference [39].
S. Kumar, S. Roy / Computational Materials Science 41 (2008) 457–466
a
463
0.7
2 =0 1.8 = 0.25 = 0.5 1.6 =1 1.4
0.6
1.2
0.6
1.2
0.5
1
0.5
1
0.4
0.8
0.4
0.8
0.3
0.6
0.3
1 0.9 0.8
b
1 0.9
1.8
0.8
1.6
0.7
1.4
0.2
0.4
0.2
0.1
0.2
0.1
0
1
1.5
2
2.5
3
3.5
4
4.5
5
0 5.5
2
0 0.4
0.6 =0 = 0.25 = 0.5 =1 0.5
0.6
0.7
0.4 0.2 0.8
0.9
1
1.1
0 1.2
w 0w _ 0H _ 0H Fig. 5. Variation of D (lower curves) and hH = 4.0 at different hH max (upper curves) for P p . (a) Variation with U for m p ¼ 0:8 and (b) variation with m p w for U = 2.0.
5.2. Development of parametric correlations and theoretical process map 1.9
Gathering all the simulated data for D and hH max at various Uw and m_ 0H for a particular combination of P 0 w and p H hp , an equation in the form of
1.7 1.6
1.5 1.3
H _ 0H D; hH max ¼ f ðU ; m p Þ
1.4
1.1
H2 ¼ C 1 þ C 2 U H þ C 3 m_ 0H þ C 5 m_ 0H2 p þ C4 U
0.9
ð17Þ
0.7
1.2 0.6
can be generated, where C1,2,. . . are the coefficients for different terms of the above equation. By least square method these coefficients are derived and listed in Table 2 for four different combinations of P 0 w and hH p. The correlations for D and hH max , expressed by Eq. (17), with the respective coefficients C1,2,. . . from Table 2 give surface profiles above the U H –m_ 0H p plane. Such surfaces for P 0 w = 3.0 and hH p ¼ 1 are shown in Fig. 6. The z-axis is for both D and hH max on the same scale from 0 to 2. The upper shaded surface is for hH max and the lower shaded surface for D. Few contour lines along with their values are also shown on the respective surfaces in Fig. 6. For each combination of P 0 w and hH p this type of surface plot can be drawn. In two-dimensional plot on the U H –m_ 0H p plane, these surface plots are represented by their respective contour lines only. Fig. 7 shows the contours of D and hH max for H 0H _ P 0 w = 4.0 and hH ¼ 1 on the U – m plane. The operating p p
0.5 0.3
0.5
0.1 0.4
0.75 1
0.3
1.25 1.5 1.75 2 2.25
0.4 0.6 0.8
2.5 1 1.2 3
2.75
H 0w _ 0H Fig. 6. D and hH = 3.0 and max surfaces above the U –m p plane for P ¼ 1. hH p
range of Uwand m_ 0H p for feasibility of the process can be found from this figure—the upper limits of Uw and m_ 0H p are decided from the low dilution limit (say D = 0.2),
Table 2 Coefficients of the correlation for D and hH max P0w
hH p
Coefficients for D C1
C2
C3
C4
C5
C1
C2
C3
C4
C5
2.4 3.0 4.0 4.0
1 1 1 0
0.946 1.012 1.053 1.402
0.222 0.116 0.005 0.121
1.502 1.101 0.678 1.215
0.211 0.087 0.022 0.011
0.394 0.134 0.082 0.315
2.142 2.432 2.993 3.602
0.899 0.799 0.827 1.322
0.004 0.024 0.162 0.081
0.204 0.130 0.115 0.200
0.485 0.362 0.134 0.075
Coefficients for hH max
464
S. Kumar, S. Roy / Computational Materials Science 41 (2008) 457–466 1.6
1.4
1.4
1.2
1.2
1. 8
1.6
0.3
1. 5 1.6 1. 7
it thi ck de po s
0.4 0.2 0
0
0.5
0. 2
0.8 0.2
0. 5
1.8 1.9
0.6
1
0.2
0.4
2
0.8
2
0.4
1
low dilution limit 0.2
0. 3
0.6
0. 2
0.6
2
0.4
thin
os dep
1
it 0.2
1.5
2
2.5
3
3.5
4
4.5
5
5.5
0
high melt pool temperature limit 0
H 0w _ 0H Fig. 7. D and hH = 4.0 and p plane for P max contour lines on the U –m ¼ 1. hH p
beyond this there is lack of good fusion bond between the build-up layer and the substrate, and the lower limits of Uw and m_ 0H p are based on the requirement that the maximum specific sensible enthalpy must be below the boiling point (say hH max ¼ 2). The parametric regions of thick and thin H deposition are also shown in Fig. 7 as m_ 0H represents p =U deposit thickness. Fig. 8 shows the effect of total absorbed laser power P 0 w on the operating range for scanning speed Uw and powder deposition rate m_ 0H p . This figure reveals that the window of operating range for Uw and m_ 0H p expands as the power level is increased. The operating range for Uwand m_ 0H p also increases with the increase in powder preheat hH p as shown in Fig. 9. This figure shows that the process zone for P 0 w = 4.0 defined by the upper boundary at D = 0.2 as well as the lower boundH ary at hH max ¼ 2:0 expands as hp increases from 0 to 1. So, with powder preheating, thinner as well as thicker deposition with low dilution is easier to achieve. The correlation for D given by Eq. (17) with the coefficients in Table 2 is in good agreement with the results found from solving the governing equation as shown in Fig. 10. In this figure DCorrelation on the x-axis is found
1.6 1.4
it
low dilution limit low 0.2 dil uti on
thi ck de po s
1.2 1 0.8
s epo nd thi
0.2
0.6
it
0.2
0.4 0.2 0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
Fig. 8. Effect of P 0 w on the upper limit of the operating range of m_ 0H p and Uw at hH p ¼ 1.
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
0w Fig. 9. Effect of hH = 4.0. p on the process map for P
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Fig. 10. Error graphs of the fitted curve for dilution.
using Eq. (17) and DNumerical on the y-axis is found from the solution of the governing equation. The parameters that determine the form and scale of the microstructure during directional solidification of alloys are mainly the temperature gradient G and growth velocity Vs at the solidification interface [40]. G is the temperature gradient ð¼ $T ^nÞ in the direction normal to the solidification interface and Vs is the velocity of the solidification front. G can be calculated from the solution field and Vs = U cos h [41] can be calculated by knowing the inclination of the solidification interface as shown in Fig. 11. Vs is very small at the bottom of the solidified layer and reaches its peak at the top. G/Vs determines the form of the microstructure; as G/Vs increases, the solidification morphology changes from planar front to cellular to columnar dendrite to equiaxed dendrites [40]. The scale of the microstructure depends on the cooling rate T_ ¼ GV s at the solid–liquid interface. Higher value of T_ gives rise to finer grains. The _H non-dimensional parameters Gw, V H s and T over the depth
S. Kumar, S. Roy / Computational Materials Science 41 (2008) 457–466
465
regions of thick and thin deposition with low dilution are also identified in the process map. A graph showing _ H at the solidification interthe variation of Gw, V H s and T face over the depth of the built-up layer is presented. These parameters are important in predicting the form and the scale of the microstructure over the thickness of the resolidified layer. References
Fig. 11. Velocity of the solidification front AB.
_H Fig. 12. Variation of Gw, V H s and T .
of the built-up layer is shown in Fig. 12 for four different combinations of P 0 w and V H s . It is observed that these graphs are most sensitive to the variation in P 0 w/Uw values; two bands at values of P 0 w/Uw equal to 1 and 2 are clearly discernible. 6. Conclusion Based on the analysis of results from a simplified twodimensional conduction model of material deposition, parametric correlations for D and hH max and a theoretical process map for laser cladding process is developed. The lowest value of Uw or m_ 0H p in the process map is guided by the boiling temperature in the melt pool and the highest value of Uw or m_ 0H p is limited by the low dilution limit. It is observed that the window of operating range for Uw 0w and m_ 0H is increased. This operating p expands as the P range also expands with the increase of hH p . Parametric
[1] L. Shepeleva, B. Medres, W.D. Kaplan, M. Bamberger, A. Weisheit, Surface and Coatings Technology 125 (2000) 45–48. [2] Y.P. Kathuria, Surface and Coatings Technology 132 (2000) 262– 269. [3] L. Sexton, S. Lavin, G. Byme, A. Kennedy, Journal of Materials Processing Technology 122 (2002) 63–68. [4] N.B. Dahotre, Advanced Materials and Processes (2002) 35–39. [5] E. Breinan, H. Kear, in: Proceedings of Conference on Rapid Solidification Processing, 1978, p. 87. [6] E.W. Kreutz, G. Backes, A. Gasser, K. Wissenbach, Applied Surface Science 86 (1995) 310–316. [7] B. Kieback, A. Neubrand, H. Riedel, Materials Science and Engineering A 362 (2003) 81–105. [8] W.U.H. Syed, A.J. Pinkerton, L. Lin, Applied Surface Science 252 (13) (2006) 4803–4808. [9] A. Yakovlev, E. Trunova, D. Grevey, M. Pilloz, I. Smurov, Surface and Coatings Technology 190 (2005) 15–24. [10] W.M. Steen, Laser Material Processing, second ed., Springer-Verlag, 1991. [11] J. Th. M. De Hosson, L. Mol. van. De Otterloo, Surface Engineering 13 (6) (1997) 471–481. [12] G.Q. Song, F. Hlawka, A. Cornet, T. Marchoine, Surface Engineering 14 (1) (1998) 67–72. [13] J.M. Pelletier, F. Oucherif, P. Sallamand, A.B. Vannes, Materials Science and Engineering A 202 (1995) 142–147. [14] F. Audebert, R. Colaco, R. Vilar, H. Sirkin, Scripta Materialia 40 (5) (1999) 551–557. [15] Y.T. Pei, J.H. Ouyang, T.C. Lei, Surface and Coatings Technology 81 (1996) 131–135. [16] Y. Wang, G. Li, C. Wang, Y. Xia, B. Sandip, C. Dong, Surface and Coatings Technology 176 (2003) 284–289. [17] X. Wu, B. Xu, Y. Hong, Materials Letters 56 (2002) 838–841. [18] T.M. Yue, A.H. Wang, H.C. Man, Scripta Materialia 40 (3) (1999) 303–311. [19] A.H. Wang, T.M. Yue, Composites Science and Technology 61 (2001) 1549–1554. [20] Y. Liu, J. Mazumder, K. Shibata, Metallurgical and Materials Transactions B 25 (1994) 749–759. [21] A. Frenk, M. Vandyoussefi, J.D. Wagniere, A. Zryd, W. Kurz, Metallurgical and Materials Transactions B 28 (1997) 501–508. [22] U. de Oliveira, V. Ocelik, J. Th. M. De Hosson, Surface and Coatings Technology 197 (2005) 127–136. [23] M. Qian, L.C. Lim, Z.D. Chen, W.L. Chen, Journal of Materials Processing Technology 63 (1997) 590–593. [24] S. Sun, Y. Durandet, M. Brandt, Surface and Coatings Technology 194 (2005) 225–231. [25] Y.P. Kathuria, Materials Science and Technology 17 (2001) 1451– 1454. [26] U. De Oliveira, V. Ocelı´k, J. Th. M. De Hosson, Surface and Coatings Technology 201 (2007) 6363–6371. [27] F.J. Kahlen, A. Kar, ASME Journal of Manufacturing Science and Engineering 123 (2001) 38–44. [28] V. Ocelı´k, U. de Oliveira, M. de Boer, J. Th. M. de Hosson, Surface and Coatings Technology 201 (2007) 5875–5883. [29] J.H. Ferziger, M. Peric, Computational Methods for Fluid Dynamics, third ed., Springer, 2002.
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[30] J. Choi, L. Han, Y. Hua, ASME Journal of Heat Transfer 127 (2005) 978–986. [31] V.R. Voller, C. Prakash, International Journal of Heat and Mass Transfer 30 (8) (1987) 1709–1719. [32] C. Prakash, M. Simonds, A.K. Singal, International Journal of Heat and Mass Transfer 30 (12) (1987) 2690–2694. [33] M.C. Flemings, Solidification Processing, McGraw Hill, New York, 1974. [34] R.L. Philips, C.L. Andrews, Applied Optics 22 (5) (1983) 643–644. [35] A.F.A. Hoadley, M. Rappaz, Metallurgical Transactions B 23B (1992) 631–642. [36] H. Gedda, Laser Surface Cladding: A Literature Survey, Division of Materials Processing, Lulea Tekniska Universitet, 2000..
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Computational Materials Science 41 (2008) 457–466 www.elsevier.com/locate/commatsci
Development of a theoretical process map for laser cladding using two-dimensional conduction heat transfer model Subrata Kumar a, Subhransu Roy b
b,*
a JFWTC, GE India Technology Center, Bangalore 560 066, India Mechanical Engineering Department, Indian Institute of Technology, Kharagpur 721 302, West Bengal, India
Received 18 December 2006; received in revised form 30 April 2007; accepted 5 May 2007 Available online 27 June 2007
Abstract In blown powder laser cladding process, the powder travels across the laser path, gets heated up by absorbing laser energy, and finally melts on the substrate under the intense laser beam; as the substrate moves away this melt pool solidifies to form a continuous built-up layer. In the present study a two-dimensional conduction heat transfer equation has been solved using finite volume method to develop a theoretical process map for laser cladding. The developed process map indicates a range of scanning speed and powder feed rate for the feasibility of the process; the lower limit is dictated by the maximum melt pool temperature, and the higher limit by poor bonding due to lack of melting of the substrate (i.e. low dilution). Parametric regions for thick and thin cladding with low dilution can be decided from the process map. It is found that the process range expands with the increase in total absorbed power as well as power directly absorbed by the powder. Correlations for maximum melt pool temperature and dilution are presented. A process map for identifying the form and scale of the microstructure in the solidified layer is also presented. 2007 Elsevier B.V. All rights reserved. Keywords: Conduction heat transfer; Cladding; Process map; Laser processing
1. Introduction Laser cladding with powder feed is one of the emerging technologies having multiple applications in manufacturing industry; ranging from surface repair and modification with superior coating over a mechanical component [1–4] to direct fabrication of three-dimensional functional metallic components, having simple to complex geometry, with desired compositions [5–9]. The basic working principle of this process is melting a substrate surface by a moving laser beam and adding pneumatically delivered metal/ceramic powders into the melt pool which subsequently melts; as the laser moves away, the melt solidifies and forms the built-up layer over the substrate [10]. Laser cladding with powder feed is a multi-parameter process where the laser parameters (power, beam profile, etc.), powder parameters *
Corresponding author. Tel.: +91 3222 282968. E-mail address: [email protected] (S. Roy).
0927-0256/$ - see front matter 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2007.05.002
(feed rate, size, etc.) and other parameters (scanning speed, material properties, etc.) influence the deposition track characteristics (geometry, dilution, mixing, microstructure, etc.). Most of the studies on laser material deposition with powder feed generally deal with the metallurgical aspects— the relationship between the microstructure and properties, usually hardness and corrosion resistance of the built-up surface layer for a series of added materials [11–19]. It is of great importance to find the range of process parameters such as laser power, scanning speed, powder feed rate, etc., for a low dilution, pore free and continuous deposition. Experimental studies show that the laser power required for good deposition is in between discontinuous track and large dilution power range [20]; built-up height increases with powder feed rate and decreases with scanning speed [21]; deposition width increases with laser power [22]; increase in powder flow rate causes lower dilution [23,24]; global absorption and attenuation of the beam increase
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Nomenclature a aK b cp D F G fl Hc Hs h hc ^i, k ^ k kp m_ p ^n P p Bi Bo Ca Nc q_ 000 h Ste R r r0 T t U U?
fraction of lower mode in the laser beam elements of coefficient matrix deposition width (m) specific heat (J/kg K) dilution irradiation flux vector (W/m2) temperature gradient (K/m) liquid fraction built-up height (m) substrate melt depth (m) specific sensible enthalpy (J/kg) heat transfer coefficient (W/m2 K) unit vectors along x- and z-axis thermal conductivity (W/m K) equilibrium partition coefficient powder feed rate (kg/s) unit surface normal vector laser power (W) dynamic pressure (N/m2) Biot number Bond number capillary number conduction-to-radiation parameter volumetric energy source (W/m3) Stefan number radius of curvature (m) position vector (m) radius of Gaussian beam (m) temperature (K) time (s) scanning speed (m/s) lateral scanning speed (m/s)
with powder feed rate [21]; shorter interaction time produces finer microstructure and higher hardness [25]; some conditions of speed, laser power and material produce tensile stresses during solidification that may lead to microcracks [26]. There also have been some experimental efforts to correlate different process parameters for good quality deposition [22,27,28]. These experimental investigations are generally very expensive and time consuming. To reduce the experimental efforts, development of a process map through theoretical investigations based on appropriate heat transfer model is the aim of this article. In the present study a two-dimensional conduction heat transfer model has been used to develop a theoretical process map for laser cladding process. The governing equation along with boundary conditions are solved using finite volume method in a non-orthogonal grid system with collocated variable arrangements [29]. The developed process map indicates a range of scanning speed and powder feed rate for the feasibility of the process based on the limits of maximum melt pool temperature and low dilution.
velocity of the solidification front (m/s) Vs v velocity vector (m/s) x, y, z Cartesian coordinate system We Weber number Greek symbols a thermal diffusivity (m2/s) Dhsl specific latent heat of fusion (J/kg) DPIZ power absorbed by powder (W) DVm volume of the melt pool (m3) emissivity g fractional absorption of laser power l dynamic viscosity (kg/m s) q density of the medium (kg/m3) r surface tension (N/m) rb Stefan Boltzmann constant (W/m2 K4) x relaxation parameter Subscripts 0 reference 1 ambient condition A, C melt pool end points E eutectic l liquidus m melting point of pure base metal max maximum value p powder Superscripts w dimensionless 0 per unit length
Correlations for maximum melt pool temperature and dilution are presented. Parametric regions for thick and thin cladding are shown in the process map. It is found that the process range expands with the increase in total absorbed power as well as power directly absorbed by the powder. A process map for identifying the form and scale of the microstructure in the solidified layer is also presented. Residual stress state that is responsible for formation of undesirable microcracks has not been addressed while developing the process map.
2. Modeling of laser cladding process A typical set-up for the process is schematically shown in Fig. 1. The laser beam is falling on a substrate where it forms a melt pool. The powder particles travel through the laser–powder interaction zone before reaching the substrate. Most of the powder particles fall into the melt pool on the substrate, gets trapped and melts. As the laser moves
S. Kumar, S. Roy / Computational Materials Science 41 (2008) 457–466
459
Laser Beam Intensity Profile y
z
b Powder + Carrier gas Laser Beam Spot Path
LaserPowder Interaction Zone >
A : Melting ends
n
AB : Freezing Front BC : Melting Front
Build up layer Melting starts : C x
x
Hc : Clad Height Hs : Substrate Melt Depth
B U
MeltingSolidification Zone Substrate
Fig. 1. Set-up for laser deposition of metals with powder feed.
away, the melt solidifies and forms the built-up layer over the substrate. The whole process can be described by its three components—(i) model for the laser beam, (ii) model for the laser–powder interaction zone and (iii) model for the melting–solidification zone. In this study, the first two are incorporated in the model for the melting–solidification zone. The following assumptions are made, (1) The laser beam: The laser beam is assumed to be a mixture of the first two fundamental modes, 40% TEM00 and 60% TEM01*, which gives a near uniformly distributed top-hat type beam profile. (2) The laser–powder interaction zone: Powder particles get heated up by absorption in the laser–powder interaction zone, termed as preheating, and the laser beam attenuates due to absorption and scattering. Here, DP IZ ¼ m_ p ðhp h1 Þ
ð1Þ
is an input parameter where DPIZ is the power absorbed for preheating the powder of mass flow rate m_ p up to the enthalpy level hp. The intensity profile nature of the attenuated beam is assumed to be invariant. (3) Energy source due to mass addition: Preheated powder enters the melt pool and comes in thermal equilibrium with the melt. This has been treated as uniformly distributed volume source in the melt pool to satisfy global energy conservation. (4) Total absorbed laser power: This model is based on total absorbed laser power gP. The efficiency parameter g has been introduced to account for the loss by reflection from the irradiated substrate surface as well as for the energy loss due to scattering by powder particles.
(5) Melt pool convection: Convection mode of heat transfer due to fluid flow is present only in a small region, described as the melt pool. It is surrounded by solid material where the heat diffuses by conduction. The melt pool surface is continuously bombarded by the solid powder particles. Moreover, assuming that a large proportion of the melt pool is occupied by solid powder particles undergoing melting, the strength of the melt pool convection can be considered to be very week. Here, the melt pool convection is neglected. 2.1. The governing equation As the laser moves over the substrate along with powder feeding; within few millimeters of travel a steadily moving melt pool forms below the laser beam, which on solidification generates a uniform deposit layer over the substrate [30]. The reference frame is attached to the moving heat source, thus reducing it to a quasi-stationary process in the Eulerian reference frame where the substrate moves with constant velocity U in the x-direction. The laser beam is moving rapidly back-and-forth in the direction perpendicular to the x–z plane (see Fig. 1 inset) with a time scale smaller than the thermal diffusion time scale such that a uniform time-averaged intensity in the y-direction can be assumed. Therefore, a one-dimensional beam intensity profile which varies only in the x-direction is considered and the problem reduces to two dimensions. Under these conditions and based on fixed-grid numerical method [31], the energy equation for the melting–solidification zone is oðqhÞ oðqhÞ k þU ¼$ $h þ q_ 000 ð2Þ h ot ox cp
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S. Kumar, S. Roy / Computational Materials Science 41 (2008) 457–466
The heat source q_ 000 h is zero in the solid region (h < hl) and elsewhere it is defined as oðqfl Þ oðqfl Þ m_ p 000 ^ þ Ui hp ðh þ Dhsl Þ Dhsl q_ h ¼ ot ox bV0m |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Part-II
Part-I
ð3Þ V0m
In Eq. (3), represents two-dimensional melt pool area and m_ p is the powder deposition rate over the width b that gives rise to a clad height Hc at the scanning speed U. Powder feed rate per unit width m_ 0p is defined as m_ 0p ¼
m_ p ¼ qUH c b
pure liquid :
hE < h < hl : fl ¼
Tm T Tm Tl
1k1
p
ð5Þ
h P hl : fl ¼ 1:0
2.2. The boundary conditions
F^ n ¼ ^ n
k $h þ hc ðT T 1 Þ þ rb ðT 4 T 41 Þ cp
ð6Þ
where 1=2 0
8 1 F¼ gP DP 0IZ p ð1 þ aÞr0 2x2 2 2 ^ a þ ð1 aÞ 2 eð2x =r0 Þ k r0 P DP IZ P 0 ¼ ; and DP 0IZ ¼ ¼ m_ 0p ðhp h1 Þ b b Other boundaries :
x ! 1; z ! 1 : h ¼ h1 ;
Hs b ; bH ¼ ; r0 r0 a0 t $ t H ¼ 2 ; $H ¼ r0 r0
HH s ¼
h h1 ; h0 h1
hH p ¼
H H H m_ 0H p ¼ q U Hc ;
hp h1 ; h0 h1
PH ¼
rH ¼
TH ¼
r r0
hH ; cp0
PH H _ 0H ; DP 0H IZ ¼ m p hp bH Dhsl lU 0 Ste ¼ ; Ca ¼ ; h0 h1 r P 0H ¼
q0 gr20 ; r
q0 U 20 r0 r
We ¼
k
ð8Þ
The dimensionless governing equation is
oh x!1: ¼0 ox
ð10Þ
GH ¼ $H T H ^ n
gP q0 a0 r0 ðh0 h1 Þ
Bo ¼
Eq. (7) gives a mixed mode beam profile having fraction a of TEM00 and (1 a) of TEM01* [34]. The one-dimensional laser intensity profile in Eq. (7) is a valid assumption when the time scale for the lateral movement b/U? is smaller than the thermal diffusion time scale r20 =a0 and the lateral scanning width b is much greater than the nominal beam radius r0. The speed U? of lateral motion of the beam is also much greater than the forward scanning speed U. Mathematically, b r0 and U ? U
Hc ; r0 U UH ¼ ; U0 HH c ¼
ð7Þ
ð9Þ
b r2 < 0; U ? a0
The governing equation and boundary conditions are normalized using the properties of the liquid metal at liquidus temperature as reference properties (i.e. q0, a0, k0, cp0, h0, etc.); the radius r0 of the Gaussian component in the laser beam is the length scale, the speed of thermal diffusion (U0 = a0/r0) is the velocity scale and (h0 h1)/cp0 is the temperature scale. The important dimensionless parameters are
hH ¼
Top surface :
r ov þ 2l ^n ¼ p p1 þ qgðH c zÞ R on x 6 xC : z ¼ 0; x P xA : z ¼ H c
xC < x < xA :
3. Dimensionless GE
h 6 hE : fl ¼ 0:0
mushy region :
During material addition for building a surface layer, a curved melt pool free surface is formed that connects the untreated substrate surface to the built-up layer. At steady state the shape of this free surface of the melt pool is derived from the balance of all the normal forces acting on the liquid–gas interface,
ð4Þ
‘Part-I’ in Eq. (3) is due to heating and melting of the solid powder mass injected into the melt pool according to the model assumption and ‘Part-II’ is due to conversion of latent enthalpy to sensible enthalpy at the solidification interface and vice-versa at the melting interface [32]. The equation for the liquid fraction in the mushy region is given by [33] pure solid :
2.3. Top surface shape equation
Nc ¼
rb ðT l T 1 Þ3 r0
oðqH hH Þ þ $H qH vH hH ¼ $H H ot
;
Bi ¼
hc r 0 k
kH H H $ h cH p
! þ q_ 000H h
where q_ 000H h
¼
m_ 0H p
½hH p 0H Vm
H oðqH fl Þ H oðq fl Þ ðh þ SteÞ Ste þU otH oxH H
The dimensionless boundary condition for the top surface is ! kH H H 1 H4 H F ^n ¼ ^n H $ h T þ BiT H þ Nc cp |fflfflffl{zfflfflffl} radiative loss
S. Kumar, S. Roy / Computational Materials Science 41 (2008) 457–466
where rffiffiffi 2 8 1 H H H2 2xH ^ ðP 0H m_ 0H k F ¼ e p hp Þ a þ ð1 aÞ2x p1 þ a The shape of the top surface is given by 1 ovH H H þ 2Ca H ^ n ¼ WeðpH pH 1 Þ þ BoðH c z Þ H on R
ð11Þ
As the melt pool convection is very weak due the presence of large number of particles undergoing melting, the viscous term on the left and the dynamic pressure terms on the right can be neglected in Eq. (11). Additionally, due to low Bond number ðBo ¼ qgr20 =r ¼ 0:03Þ and smaller size of the melt pool, the shape of the free surface is approximated as a circular arc of constant radius Rw connecting the points ‘C’ and ‘A’ in Fig. 1. This approximation is very near to the reported pictures of melt pool by high speed photography and video recording of laser cladding [35,36]. Thus, the simplified top surface shape equation is x 6 xC : zH ¼ 0 xC < x < xA : RH ¼ constant H
x P xA : z ¼
ð12Þ
HH c
4. Solution procedure The dimensionless governing equation with non-linear boundary conditions applied to the irradiated surface are solved using finite-volume method on a boundary fitted non-orthogonal grid system with collocated variable arrangement [29]. The top surface boundary line is generated with some H assumed melt pool end point positions, rH A and rC . Bodyconforming grid lines are generated by an algebraic grid generator. A typical non-orthogonal grid system is shown in Fig. 2. Capital letters (P, E, W, N, S) in this figure are cell centers and the small letters (e, w, n, s) are the face cen-
461
ters. In collocated variable arrangement, all the values of the dependent variables are stored at cell centers, and face center values are evaluated by interpolation. The discretization of the dimensionless governing equation for each control volume gives a set of algebraic equations which can be written for the P’th node of Fig. 2 as X aK hH where K ¼ E; W; N; S ð13Þ aP hH P þ K ¼ qP ; K
The above set of algebraic equation (13) is solved iteratively until the normalized residual falls below 104. Then H the melt pool end points rH A and rC are corrected using relaxation parameters xA and xC, respectively: Hold Hnew Hold rHnew A;C ¼ rA;C þ xA;C ðrA;C rA;C Þ |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl}
ð14Þ
correction term
The whole process is repeated, starting with a new grid generation for the updated free surface, until the correction term in Eq. (14) becomes insignificant and then the surface profile is considered to be converged. To improve solution accuracy, fine grids are taken in the melt pool region where the change of variables is rapid. To check the grid independence of the solution three grid levels, 90 · 40, 135 · 60 and 180 · 80, were tested for few sets of input parameters and finally 180 · 80 grid level was selected for all sets of input parameters. 4.1. Validation of the computer code The computer code is validated with a two-dimensional analytical solution for heating of a semi-infinite body with a moving heat source strip on the surface. The radiative and convective heat losses from the surface are neglected. The analytical solution for steady surface temperature of 0.3
Numerical Analytical
j+1
N j
W
n^
n w
n^
P
0.2
n^
e s n^
j–1
E
0.1
S i–1
j–2 i–2
i 0
i+2 Fig. 2. Non-orthogonal grid system.
0
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15
Fig. 3. Comparison between numerical and analytical solution for the test problem.
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S. Kumar, S. Roy / Computational Materials Science 41 (2008) 457–466
the semi-infinite body for high value of Peclet number (Pe) is given by [37] 8 pffiffiffiffiffiffiffiffihpffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffii H ðT surface T 1 ÞpkU < pPe xH xH 1 for x P 1 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi : 2aF for xH 6 1 pPexH
12
ð15Þ
7
0.2
6
0.1
10
0.5 0.4
8
0.3
5 4 3 2 1 0
0
1
2
3
4
5
6
7
8
9
10 11 12 13 14
Fig. 4. Iso-enthalpy contours for P 0 w = 3.0, Uw = 2.0, m_ 0H p ¼ 0:4 and hH p ¼ 0:5.
5. Results The material used for this simulation is steel, an alloy of iron and carbon, containing small % of carbon. This material melts over a small temperature range depending on the % of carbon as per the temperature–composition phase diagram [38]. The thermo-physical properties of liquid iron at melting temperature, listed in Table 1, are used for this simulation. kp was taken as 0.3 which is typical of steel. A top-hat beam intensity profile is selected by choosing parameter a = 0.4. The powder preheat level hH p varies between 0 and 1; hH p ¼ 0 (no preheat) is an ideal case and hH p ¼ 1 means that the powder is heated to near meltingpoint temperature. Constant thermo-physical properties of liquid iron at melting point are used. The energy equation with the appropriate boundary conditions is solved and the enthalpy field is calculated for three dimensionless power levels P 0 w = 2.4, 3.0 and 4.0 for a range of m_ 0H p and 0H Uw varied from a starting value of m_ 0H and p ¼ 0:1 P Uw = 0.25P 0 w, respectively, until they reach a limit, i.e. the substrate does not melt and there is no fusion bonding between the deposit and the substrate. 5.1. Enthalpy field and the selection of parameters Fig. 4 shows typical contours of specific sensible enthalpy for a set of input parameters. The shaded part in this figure is the melt pool, having hw P 1. This figure reveals that part of the laser beam is falling ahead of the melt pool and rapidly heating up the substrate in the Table 1 Properties of liquid iron at melting point and other used values [42,43]
cp Tm hm h1
0.8 0.7 .6 0
9
Here F is the uniform heat flux from x = 0 to x = l, which is moving with speed U; Pe (=Ul/a) is the Peclet number; and xw = x/l. For the numerical solution, the present computer code is used in a domain of size 15 · 5. Other inputs are Fw = 1 for w 3 6 xw 6 4, Ste = 0, m_ 0H p ¼ 0 and U = 20. The finest grid level, i.e. 180 · 80, is chosen for the solution. It is clear from Fig. 3 that the numerical solution for the test problem is matching with the analytical solution.
q
Laser Beam
11
7.03 · 103 kg/m3 824 J/kg K 1538 C 1050 · 103 J/kg
Dhsl
247 · 103 J/kg
r
k a l
36 W/m K 6.2 · 106 m2/s 5.6 · 103 kg/ms
hc rb
1800 · 103 N/m 0.19 250 W/m2 K 5.67 · 108 W/m2 K4
upstream region. Due to the presence of the convective term in the energy equation as a result of the relative motion between the laser and the workpiece, the temperature gradient is large in the upstream region and there is a long hot zone trailing behind the laser beam. From the geometry of the melt pool, the depth of the remelted substrate layer and the height of the deposited layer can be find out. These two data are important to calculate dilution D. Dilution is the fraction of substrate material in the deposited layer and for two-dimensional temperature field it is defined as D¼
Hs HH ¼ H s H Hc þ Hs Hc þ Hs
ð16Þ
where Hs is the substrate remelted depth and Hc is the built-up height as shown in Fig. 1. D is an important parameter to decide the combination of the values of the process parameters P 0 w, Uw and m_ 0H p as low dilution means better quality of deposition. Similarly, the enthalpy field also gives the maximum melt pool sensible enthalpy hH max which is an important parameter to decide the feasibility of the process; it must be above the melting point (hw = 1) but below the boiling point (hw 2). These two parameters (D and hH max ) are selected to develop the process map for laser cladding. w Fig. 5a shows the variation of D and hH for max with U H 0w 0H P = 4.0 and m_ p ¼ 0:8 at different hp and Fig. 5b shows 0w _ 0H the variation of D and hH = 4.0 and max with m p for P w H U = 2.0 at different hp . The left y-axis shows D between 0 and 1 and the right y-axis shows hH max between 0 and 2. The bottom curves are for D and the top curves are for w hH or m_ 0H max . In general, D decreases with increasing U p , the linear term is stronger than the non-linear term for w H the variation of D with m_ 0H p , and hmax also reduces as U H 0H or m_ p increases. These figures also reveal that hp has significant effect on these, specially D. A detailed study of these can be found in reference [39].
S. Kumar, S. Roy / Computational Materials Science 41 (2008) 457–466
a
463
0.7
2 =0 1.8 = 0.25 = 0.5 1.6 =1 1.4
0.6
1.2
0.6
1.2
0.5
1
0.5
1
0.4
0.8
0.4
0.8
0.3
0.6
0.3
1 0.9 0.8
b
1 0.9
1.8
0.8
1.6
0.7
1.4
0.2
0.4
0.2
0.1
0.2
0.1
0
1
1.5
2
2.5
3
3.5
4
4.5
5
0 5.5
2
0 0.4
0.6 =0 = 0.25 = 0.5 =1 0.5
0.6
0.7
0.4 0.2 0.8
0.9
1
1.1
0 1.2
w 0w _ 0H _ 0H Fig. 5. Variation of D (lower curves) and hH = 4.0 at different hH max (upper curves) for P p . (a) Variation with U for m p ¼ 0:8 and (b) variation with m p w for U = 2.0.
5.2. Development of parametric correlations and theoretical process map 1.9
Gathering all the simulated data for D and hH max at various Uw and m_ 0H for a particular combination of P 0 w and p H hp , an equation in the form of
1.7 1.6
1.5 1.3
H _ 0H D; hH max ¼ f ðU ; m p Þ
1.4
1.1
H2 ¼ C 1 þ C 2 U H þ C 3 m_ 0H þ C 5 m_ 0H2 p þ C4 U
0.9
ð17Þ
0.7
1.2 0.6
can be generated, where C1,2,. . . are the coefficients for different terms of the above equation. By least square method these coefficients are derived and listed in Table 2 for four different combinations of P 0 w and hH p. The correlations for D and hH max , expressed by Eq. (17), with the respective coefficients C1,2,. . . from Table 2 give surface profiles above the U H –m_ 0H p plane. Such surfaces for P 0 w = 3.0 and hH p ¼ 1 are shown in Fig. 6. The z-axis is for both D and hH max on the same scale from 0 to 2. The upper shaded surface is for hH max and the lower shaded surface for D. Few contour lines along with their values are also shown on the respective surfaces in Fig. 6. For each combination of P 0 w and hH p this type of surface plot can be drawn. In two-dimensional plot on the U H –m_ 0H p plane, these surface plots are represented by their respective contour lines only. Fig. 7 shows the contours of D and hH max for H 0H _ P 0 w = 4.0 and hH ¼ 1 on the U – m plane. The operating p p
0.5 0.3
0.5
0.1 0.4
0.75 1
0.3
1.25 1.5 1.75 2 2.25
0.4 0.6 0.8
2.5 1 1.2 3
2.75
H 0w _ 0H Fig. 6. D and hH = 3.0 and max surfaces above the U –m p plane for P ¼ 1. hH p
range of Uwand m_ 0H p for feasibility of the process can be found from this figure—the upper limits of Uw and m_ 0H p are decided from the low dilution limit (say D = 0.2),
Table 2 Coefficients of the correlation for D and hH max P0w
hH p
Coefficients for D C1
C2
C3
C4
C5
C1
C2
C3
C4
C5
2.4 3.0 4.0 4.0
1 1 1 0
0.946 1.012 1.053 1.402
0.222 0.116 0.005 0.121
1.502 1.101 0.678 1.215
0.211 0.087 0.022 0.011
0.394 0.134 0.082 0.315
2.142 2.432 2.993 3.602
0.899 0.799 0.827 1.322
0.004 0.024 0.162 0.081
0.204 0.130 0.115 0.200
0.485 0.362 0.134 0.075
Coefficients for hH max
464
S. Kumar, S. Roy / Computational Materials Science 41 (2008) 457–466 1.6
1.4
1.4
1.2
1.2
1. 8
1.6
0.3
1. 5 1.6 1. 7
it thi ck de po s
0.4 0.2 0
0
0.5
0. 2
0.8 0.2
0. 5
1.8 1.9
0.6
1
0.2
0.4
2
0.8
2
0.4
1
low dilution limit 0.2
0. 3
0.6
0. 2
0.6
2
0.4
thin
os dep
1
it 0.2
1.5
2
2.5
3
3.5
4
4.5
5
5.5
0
high melt pool temperature limit 0
H 0w _ 0H Fig. 7. D and hH = 4.0 and p plane for P max contour lines on the U –m ¼ 1. hH p
beyond this there is lack of good fusion bond between the build-up layer and the substrate, and the lower limits of Uw and m_ 0H p are based on the requirement that the maximum specific sensible enthalpy must be below the boiling point (say hH max ¼ 2). The parametric regions of thick and thin H deposition are also shown in Fig. 7 as m_ 0H represents p =U deposit thickness. Fig. 8 shows the effect of total absorbed laser power P 0 w on the operating range for scanning speed Uw and powder deposition rate m_ 0H p . This figure reveals that the window of operating range for Uw and m_ 0H p expands as the power level is increased. The operating range for Uwand m_ 0H p also increases with the increase in powder preheat hH p as shown in Fig. 9. This figure shows that the process zone for P 0 w = 4.0 defined by the upper boundary at D = 0.2 as well as the lower boundH ary at hH max ¼ 2:0 expands as hp increases from 0 to 1. So, with powder preheating, thinner as well as thicker deposition with low dilution is easier to achieve. The correlation for D given by Eq. (17) with the coefficients in Table 2 is in good agreement with the results found from solving the governing equation as shown in Fig. 10. In this figure DCorrelation on the x-axis is found
1.6 1.4
it
low dilution limit low 0.2 dil uti on
thi ck de po s
1.2 1 0.8
s epo nd thi
0.2
0.6
it
0.2
0.4 0.2 0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
Fig. 8. Effect of P 0 w on the upper limit of the operating range of m_ 0H p and Uw at hH p ¼ 1.
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
0w Fig. 9. Effect of hH = 4.0. p on the process map for P
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Fig. 10. Error graphs of the fitted curve for dilution.
using Eq. (17) and DNumerical on the y-axis is found from the solution of the governing equation. The parameters that determine the form and scale of the microstructure during directional solidification of alloys are mainly the temperature gradient G and growth velocity Vs at the solidification interface [40]. G is the temperature gradient ð¼ $T ^nÞ in the direction normal to the solidification interface and Vs is the velocity of the solidification front. G can be calculated from the solution field and Vs = U cos h [41] can be calculated by knowing the inclination of the solidification interface as shown in Fig. 11. Vs is very small at the bottom of the solidified layer and reaches its peak at the top. G/Vs determines the form of the microstructure; as G/Vs increases, the solidification morphology changes from planar front to cellular to columnar dendrite to equiaxed dendrites [40]. The scale of the microstructure depends on the cooling rate T_ ¼ GV s at the solid–liquid interface. Higher value of T_ gives rise to finer grains. The _H non-dimensional parameters Gw, V H s and T over the depth
S. Kumar, S. Roy / Computational Materials Science 41 (2008) 457–466
465
regions of thick and thin deposition with low dilution are also identified in the process map. A graph showing _ H at the solidification interthe variation of Gw, V H s and T face over the depth of the built-up layer is presented. These parameters are important in predicting the form and the scale of the microstructure over the thickness of the resolidified layer. References
Fig. 11. Velocity of the solidification front AB.
_H Fig. 12. Variation of Gw, V H s and T .
of the built-up layer is shown in Fig. 12 for four different combinations of P 0 w and V H s . It is observed that these graphs are most sensitive to the variation in P 0 w/Uw values; two bands at values of P 0 w/Uw equal to 1 and 2 are clearly discernible. 6. Conclusion Based on the analysis of results from a simplified twodimensional conduction model of material deposition, parametric correlations for D and hH max and a theoretical process map for laser cladding process is developed. The lowest value of Uw or m_ 0H p in the process map is guided by the boiling temperature in the melt pool and the highest value of Uw or m_ 0H p is limited by the low dilution limit. It is observed that the window of operating range for Uw 0w and m_ 0H is increased. This operating p expands as the P range also expands with the increase of hH p . Parametric
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