2d a Thermal Model of Laser Cladding by Powder Injection

A Thermal Model of Laser Cladding by Powder Injection

A.F.A. HOADLEY and M. RAPPAZ

A two-dimensional (2-D) finite element model is presented for laser cladding by powder injection. The model simulates the quasi-steady temperature field for the longitudinal section of a clad track. It takes into account the melting of the powder in the liquid pool and the liquid/ gas free surface shape and position, which must conform to the thermal field in order to obtain a self-consistent solution. The results for an idealized problem, where there is almost no melting of the substrate material, demonstrate the linear relationship between the laser power, the processing velocity, and the thickness of the deposited layer. The calculated clad heights agreed well with the experimental values for the conditions where a cobalt-based hard-facing alloy is clad onto mild steel with a linearly focused laser source.

I . I N T R O D U C T I O N

THE selection of a material for a given application can be based upon its surface properties when a wear and/or corrosion environment is involved and/or upon its bulk properties when mechanical resistance to trac- tion or creep is required. Modification of the surface layer by laser treatment is therefore an area of much interest to the manufacturing industry, Eu because it makes possible the production of high-quality surfaces while main- taining the mechanical properties of the base material. In the cladding process, a laser beam is used to heat the clad/substrate surface and produce a molten pool (Figure 1). The facing compound is delivered in a powder form into this interaction zone by means of a carrier gas. On impinging on the melt pool surface, the powder is entrained in the melt, exchanging energy, mass, and momentum with the liquid. When melting occurs in both the clad and substrate materials, a chemically homoge- neous surface layer is produced with a metallurgical bond between the clad and the base material.

It is the high degree of control over the laser energy which allows the optimization of the process. I21 An ex- cess of energy which substantially remelts the substrate material leads to the dilution of the clad composition and, thus, the degradation of the clad properties. On the other hand, where there is insufficient energy, a fusion bond may not be formed, leading to poor adhesive properties and the risk of the clad spalling during service. How- ever, it is precisely the localized nature of the laser energy which makes in situ measurements so difficult. Therefore, computer modeling can yield real insight in the laser cladding process, furthering our comprehension of the underlying physics occurring in the laser interaction zone. At the same time, simulation is a tool for quantifying the effects of the operating parameters, thereby aiding the optimization, scaleup, and control of the process.

The injection of the powder into the melt pool distin- guishes cladding from other laser or moving heat source

processes, such as surface hardening or surface remelting. Very few models of cladding have previously been published. Kar and Mazumder t31 considered the disso- lution of the powder and the mixing of the clad in a one- dimensional study. Weerasinghe and Steen 141 developed a finite difference model of laser cladding by powder injection. In their study, they included the effect of pre- heating the powder by the laser beam and also the effect of the powder in shadowing part of the laser energy from the surface. However, in assuming that the powder melted instantaneously on the clad surface, they did not allow for mixing within the melt pool. If mixing is not accounted for, the melted powder must be convected away from the surface by flow in the melt, thus necessitating a full solution of the Navier-Stokes equations, as considered in recent work by Picasso and Hoadley. 151

In this article, a two-dimensional (2-D) model for laser cladding is presented, whereby mixing is assumed to dis- tribute the powder instantaneously in the melt. Although the fluid flow is not solved for rigorously, convective heat transfer is partially accounted for by the uniform redistribution of the latent heat associated with the powder. The advantage of this approach is that the input parameters are thereby greatly simplified in comparison to References 4 and 5, as it is unnecessary to know the powder distribution in the gas stream or its precise temperature at the clad surface. This enables the model to be easily applied to real processing conditions. Whereas in Reference 4 the three-dimensional heat flow was calculated for a single clad track with a very limited number of points, in this work, the longitudinal section is modeled accurately using a deforming finite element method. This situation applies to a linear focusing system, where the laser energy distribution is uniform in the transverse direction and heat loss out of the longitudinal plane may be neglected. Although this is not strictly valid for a circular laser beam, it is still possible to draw from this approach some important conclusions regarding the interplay of the processing parameters.

A.F.A. HOADLEY, formerly Postdoctoral Fellow, Ecole Polytechnique Frdrrale de Lausanne, is Senior Research Engineer, Comalco, Thomastown, Australia. M. RAPPAZ, Professor, is with the Materials Department, Ecole Polytechnique Frdrrale de Lausanne, MX-G, Ecublens, 1015 Lausanne, Switzerland.

Manuscript submitted September 16, 1991.

II. T H E P H Y S I C A L M O D E L

A. Meso-Scale Calculation Domain

Figure 1 shows a profile view of the laser-affected zone during cladding. The laser beam is focused vertically down

METALLURGICAL TRANSACTIONS B VOLUME 23B, OCTOBER 1992--631

Z

l / / / / / . Powder / / /

,~X F1 Xs / " / / / ;./"/.-/ / / / ?+;oo

h c ) / , /4 / - xm

I 3 . . . . . l F3 = = = = = -- = = Substrate 5~e . . . . . . . . . !

I--= Iv2 I- I

I

v4

Fig. i - - P r o c e s s schematic showing the longitudinal section of the laser affected region during cladding by powder injection.

onto the substrate surface. The powder is injected at the melt surface from a nozzle attached to the laser head. The positioning of this nozzle in front of the melt pool means that the powder must at least partially traverse the laser beam before impinging on the surface. A reference frame which moves with the laser beam is defined, such that the positive x direction coincides with the traverse direction of the beam. Hence, the material is moving in the negative sense with a constant velocity, - v o. In this Eulerian reference frame, the field variables are considered to be quasi-stationary; that is to say, over a certain duration, an observer moving with the heat source will not notice significant changes.

In Figure 1, the origin is located at the triple point between the clad liquid and the solid substrate surface, denoted Xm. The boundaries are as follows: F~ is the top surface including the solid substrate and solid and liquid clad surface, F2 is the boundary at which the substrate enters on the right side of the calculation domain, F3 is the "downwind" boundary where both the clad and substrate leave the calculation domain, and F4 is the boundary at the base of the calculation domain, which may be the inferior surface of the workpiece or an artificial boundary within the base material. The liquid-free surface is defined between the point Xm, the first point of melting, and Xs, the last point on the surface of the clad to solidify (Figure 1). Hereafter, the properties related to the base material are labeled with the subscript b, the clad with the subscript c, and p is used for the powder. The indices s and l denote solid and liquid, respectively.

B. The Time Scales for Mass and Energy Transfer

Before contemplating a rigorous model of the laser cladding process, it is worthwhile estimating the magnitude of the principle driving forces affecting both heat and mass transfer. In Appendix A, the following time scales are defined:

ta, the time required for the heat to diffuse the distance of the clad height, he; tr, the average residence time for a particle in the melt pool;

tin, the mixing time or time required to disperse fully the powder in the melt; and t I, the fusion time or time required to melt and dis- solve the powder.

Table I summarizes these different time scales for two extreme processing speeds and clad thicknesses, t61 A full discussion of how these times have been calculated is given in Appendix A. From this order of magnitude analysis, it is possible to conclude that at low processing speeds, heat transfer is dominated by thermal diffusion (small Peclet number). Furthermore, the residence time of the powder in the liquid is much longer than the time required to mix and melt the powder, and therefore, the volume fraction of solid particles in the melt should be very low. At the high processing speed, the situation is not so clear. Diffusion and advection are of a similar importance, and it would appear to be more possible for unmelted powder to be incorporated directly into the clad. However, this simple analysis shows quite clearly that the mixing and the melting of the powder occur at about the same rate. Thus, it is difficult to justify the assump- tion of Weerasinghe and Steen t41 that the powder melts instantaneously at the liquid surface. The approach taken in this work is just the opposite and assumes that the powder is completely distributed in the liquid prior to melting. Based upon this hypothesis, it is possible to consider in detail the mass and energy continuity equations which apply to the cladding process.

C. The Clad Mass Balance

Considering the arrival of the powder particles at the surface F~, it is evident that a solid particle which im- pacts with this surface will be absorbed only if it im- pinges on the melt surface, as a solid particle hitting a solid surface will be deflected and lost. A particle may only stick to the solid surface if it arrives as a liquid droplet, having been heated during its passage through the laser beam. However, this situation is easily detected as it leads to a very rough surface and, in fact, is rarely observed, if the powder injection system is well aligned. Accordingly, it is reasonable to assume that only those particles falling onto the liquid surface of the clad are absorbed.

If there is no convection in the liquid region of the clad, then the following mass conservation equation is found for the curved part of the clad surface Fj characterized by the outward pointing normal vector n:

fpppVp, n = -pcvonx [1]

where the powder injection velocity is v e, pp is the powder density (equal to Pc if there is no dilution), and fp is the partial volume fraction of powder in the gas jet. (Note that, by definition, v e ' n is negative.) In using Eq. [1], the powder distribution describes the free surface shape. However, where the deposited mass is quickly redistrib- uted due to convection within the liquid, the surface shape is controlled by a balance of forces (see Section E). Thus, the specific distribution of the powder at the surface (a variable which is extremely difficult to determine with any precision) is of little importance and only the total

632--VOLUME 23B, OCTOBER 1992 METALLURGICAL TRANSACTIONS B

Table I. Laser Cladding Time Scales

~'0 h c

(ram/s) (mm) Pe tr (s) td (s) t,, (s) tf (s)

1.0 2.0 0.3 2.0 0.6 0.002 0.0005 100.0 0.2 3.0 0.002 0.006 0.0002 0.0005

powder deposition rate per unit transverse width of the clad, Mp, is considered:

I - - - LPPVP " n dF1 = pchcvo [2] Ms J F 1

thereby satisfying the global mass balance for the clad.

D. The Thermal Balance

The heat equation for the clad and the substrate in a reference frame attached to the laser beam is

OHc/b OHc/b v 0 - - -- div(Kc/b(T) grad (T)) + Qt [3]

Ot Ox

where K(T) is the temperature-dependent thermal conductivity and H and T are the volumetric enthalpy and temperature fields, respectively. The temperature is con- tinuous at the clad/substrate interface, whereas the enthalpy is not, and thus, the indices c and b have been kept. The term Qz is a heat sink term in the molten region, which is discussed in detail later in this section. Although the stationary melt pool shape and clad free surface is searched, the nonstationary heat-flow equation is used as a means to reach the steady state.

If there is no mixing of the powder within the melt (i.e., no convection), the boundary condition at the surface F1 is

qn = P"/3(T) "p(x) - h(T). (T - Ta)

- (Hcvonx+fp 'Hpvp 'n) onF1 [4]

The first contribution represents the energy coming from the laser beam. In this model, the heat source is uniform in the transverse direction and is described in the travel direction by a function p(x), which is designed to fit a specific laser energy profile. Thus, P is the total power of the laser beam per unit length in the transverse direction, and the integral of p (x ) is normalized to unity. The term/3 is the temperature-dependent absorption which is a material/surface parameter determining the fraction of the laser energy entering the workpiece.

The second term in Eq. [4] corresponds to the heat exchange, with the atmosphere due to natural convection and radiation. The term h is the surface heat transfer coefficient, and Ta is the ambient temperature. The third term represents the enthalpy jump at the clad free surface between the volumetric enthalpy of the incoming powder, lip, and that of the clad, Hc. This last contribution assumes that the transfer of energy to the powder occurs instantaneously at the flee surface. The magnitude of this term may cause the clad surface to resolidify locally, if the mass of powder arriving at any one point is not bal- anced by the heat input from the laser beam. It is for

this reason that the model of Weerasinghe and Steen [4] predicted melt lengths significantly shorter than the diameter of their laser beam.

In the current model, where the powder is assumed to be mixed rapidly in the liquid, the transfer of energy occurs within the melt volume. Analogous with the mass balance, the distribution of the powder does not need to be ascertained, as it is only the total deficiency in the powder energy, as integrated over the whole clad surface, which is important. Hence, the total energy transfer in order to heat and melt the powder is now accounted for by a uniform heat sink, Qt, for the whole liquid region, ~t , defined such that

- ( (Hcvonx + fp "HpVp" n) dF1 = Qzftl [5] J F l

and, therefore, the third term in Eq. [4] is eliminated. At the boundaries within the workpiece, it is necessary

to prescribe either a temperature or heat flux condition. For the right-hand boundary, F2 (Figure 1), a Dirichlet condition was chosen:

T = T~ o n F 2 [6]

where T~ is the temperature or the temperature distribution of the fresh material arriving in the calculation domain. Conversely, the downwind boundary, F3, is chosen sufficiently far from the laser beam, so that in comparison with the advected energy, diffusion can be neglected, leading to an adiabatic condition

OT q , = - - K - - = 0 onF3 [7]

Ox

If the substrate is not too thick with respect to the depth of the heat-affected zone, the through thickness can be modeled, leading to a convective condition on the bot- tom surface, F4,

q, = - h ( T ) . ( T - Ta) on F 4 [81

Otherwise, the temperature distribution obtained from a 2-D point heat source analytical solution can be imposed at a given depth within the metal, tTj

E. The Free Surface Shape

The clad liquid surface shape is governed by the balance of the forces acting normal to it

( 0v, ) . . . . n + yK [9] Pt Pp + ptgz 2txl On

where g is the gravitational acceleration assumed to be acting in the z direction,/x is the viscosity, y is the surface tension coefficient, and K is the curvature of the


clad. The terms on the left represent the pressure difference between the clad liquid and the powder/gas mixture, while those on the right represent the normal stress exerted by the fluid flow, vt, and the surface tension. The effect of the fluid momentum is assumed to be negligible as the capillary number is small (Ca = ] v t l # J y < 0.01). I81 If the contribution of gravity is also ignored, Eq. [9] then yields a surface curvature which is related only to the pressure difference (P~ - Pp). Assuming that this difference is constant, the free surface is an arc of a circle, def'med between the points Xm and Xs (Figure 1).

Figure 2 is a single frame taken from a high-speed film (6000 frames/s) during cladding of a cobalt-based alloy. The camera was positioned at an angle of ap- proximately 45 deg to the horizontal so that the melt is viewed in profile. The free surface, seen as the bright area in the photograph, conforms closely to that of a circular arc of 3 .0-mm radius.

F. The Temperature of the Powder

For a self-consistent model of the cladding process, it is also necessary to take into account the temperature of the incoming powder at the free surface, Fx. It has been observed that the volume fraction of powder in the incoming jet, fp, can significantly influence the energy coupling of the process. 191 In the model of Weerasinghe and Steen, 141 the powder had a shadowing effect which resulted in only a slight distortion of the energy distribution and no substantial increase in the total absorption. A similar approach is taken here, only applied to the average conditions for a single particle. For simplicity, convective and radiative losses are not considered, as these are insignificant in comparison with the laser heat flux. Assuming the laser intensity is uniform over the beam width, o'x, then the temperature of the particle, Tp, having moved through a distance, lp, of the laser beam is

3~319Pip Tp = T, + [101

4o~,R?cpVp

where Rp is the radius of the particle, Cp is its volumetric specific heat, and vp is its mean velocity. The term 19 is

Fig. 2 - - A single frame from a high-speed film of laser cladding showing the injection nozzle and the melt surface (for a cobalt-based alloy onto a steel substrate). Processing conditions: v0 = 13.3 m m / s , o'r = 1 . 0 m m , a n d P - - 1 . 5 k W .

a parameter which takes into account the proportion of the spherical surface irradiated. If only the energy direct from the laser is incident on the particle, 19 is unity, and the energy coupling is not increased due to the powder. If energy reflected from the melt surface or from other particles is included instead, 19 is greater than 1. The maximum value of 19 is that of a black body, when the multiple reflection is so important that all of the energy of the laser is absorbed by the powder particles, in which case, @ = 4//3.

The calculation of the powder temperature is illustrated by the following example. Consider a cobalt-based powder particle (Table III) with a diameter of 80 /zm, injected with an angle of 45 deg and a velocity of 5 m/s . The laser power is 1500 W, and the beam area is 0.9 x 7.5 mm; the flux absorbed by the particle due to the laser, assuming no multiple reflection, 19 = 1, is 1.7 x 10 7 W / m 2. If the path length, lp, is ~x/cos 45, then the change in the powder temperature, Tp - Ta, is only 80 ~ However, it is important to note that a lower particle velocity, Vp, or a greater laser intensity, P/~r,, could lead to a significantly higher powder temperature.

I I I . N U M E R I C A L F O R M U L A T I O N

A 2-D finite element method is used to solve Eq. [3] on a single domain, containing both the solid and the melt phases of the clad and the base material. The specific details of the numerical algorithm are contained in Appendix B. Please note that the phase change is taken into account using a special integration technique, first developed by Crivelli and Idelsohn. I1~

Although the solution is calculated over a mesh having a constant topology, it is necessary to deform this mesh in order to follow the points Xm and Xs. The point Xm is the first point to have melted on the substrate and therefore corresponds to the liquidus isotherm for the base material. The point X, is the last point to start to resolidify on the clad surface pertaining to the liquidus of the clad. For a constant clad height, the latter point must also coincide with the horizontal tangent on the surface. Where the clad and substrate are different materials, it is unlikely that both points will occur on the same isotherm. (This often leads to real problems when cladding materials with very dissimilar melting points.) The following strategy is used to ensure that the free surface conforms with the thermal solution at steady state: (1) A starting mesh is generated by giving a first esti- mate of the curvature of the surface, e.g., K = (2hc) -1. The beginning of the free surface is fixed at the origin (x = 0), and the laser beam is initially positioned close to the origin at XL. An initial mesh of 800 nodes (one quarter of the number normally used) is depicted in Figure 3(a). Note that in order to apply the integration technique described in Appendix B, the mesh contains triangular elements in the regions where a phase change is possible. (2) When the melting isotherm (Xm) has moved away from the front of the free surface, indicating fusion in the base material, the laser beam is permitted to move from its initial position. The new position of the center of the laser beam is chosen as

XnL = XnL -1 -- O'Xm [111


Fig. 3 - - D e m o n s t r a t i n g the ad jus tment to the laser b e a m posi t ion and the deformat ion of the finite e l ement mesh dur ing the c ladding calcula t ion. Parameters : hc = 0 .6 mm, P = 400 W / m m , Vo = 12 m m / s , ~r, = 0 .5 m m , 1", = 20 ~ T, = 20 ~ and h = 100 W / m 2 / ~ (a) The ini t ial mesh wi th the laser centered at - 0 . 1 5 mm. (b) In te rmedia te mesh , as yet no deformat ion , Xs = - 1 . 0 m m , XL = - 0 . 2 5 mm. (c) Steady state, X~ = - 1 . 5 4 m m and XL = - 0 . 3 7 mm.

where 0 is a relaxation parameter which is normally set to 0.5 to ensure a smooth convergence for the point Xm at the origin. Thus, depending on whether X,, is positive or negative, the laser beam is moved behind or ahead, respectively, of its previous position. This intermediate stage in the calculation is depicted in Figure 3(b). (3) When the horizontal length of the melt is greater than the clad height, (the center of curvature is in the substrate), the mesh begins to deform, in order to obtain agreement at the point Xs. The new curvature, K n, of the free surface may be calculated directly from the points Xm and Xs found at the previous time step, n - 1,

2hc K n = [12]

h~ + (XZ' -XT-') 2

although it is usual to introduce some relaxation also, to prevent too great a change to the calculation domain in any one time step. This new radius of curvature being found, all the points of the enmeshment are moved accordingly, but in order not to alter the finite element matrix structure, the topology of the enmeshment is not changed.

The steady state is attained when convergence is achieved in the first iteration and the changes in the laser position and the mesh are negligible. The final mesh is shown in Figure 3(c). Note that the liquidus isotherm is in agreement with the two extremities of the free surface, Xm and Xs, the liquid pool having grown substantially in comparison to Figure 3(b).

IV. EXPERIMENTAL

In this study, both numerical and physical experiments have been performed. With the former, an idealized situation has been designed specifically to demonstrate the effects of the different process parameters. So as not to obscure the results of the parametric analysis, the physical properties did not vary with temperature, with the exception of the liquid fraction, which decreased linearly from the liquidus to the solidus (Table II). The laser beam was given by a single Gaussian distribution of standard deviation, ~rx. This and the other process conditions are given in the legend of Figure 4.

In addition to the numerical experiments, results from real cladding of a cobalt-based alloy onto mild steel are reported. In this case, the model has been used to predict the laser power required to deposit a given clad thickness. The actual experimental setup for cladding is described in detail elsewhereY 1"~2J The maximum power available from the CO2 laser was 1.75 kW. The mirror had an effective width of 7.5 ram, over which the energy distribution could be assumed to be uniform. As the laser beam length in the travel direction was only 0.9 mm (measured by burn prints), the heat flow along the center of the track was essentially contained within the longitudinal plane, thereby allowing the comparison with the 2-D simulation.

Using Eq. [10], the average powder temperature arriving at the surface was estimated to be 100 ~ This temperature allows only for direct irradiation inflight,

METALLURGICAL T R A N S A C T I O N S B VOLUME 23B, OCTOBER 1992--635

XL= -0.35ram

(a)

XL= -0.47ram

(b) XL= -0.61ram

(c)

Fig. 4 - - I d e a l i z e d problem: the influence of the laser power per unit width on the shape of the melt pool and the degree of dilution due to remelting of the base material: (a) P = 460 W / m m , (b) P = 510 W / r a m , and (c) P = 610 W / r a m . Other parameters: hc = 1.0 mm. vo = 12 m m / s , o'~ = 0.5 ram, T~ = 20 ~ T~ = 20 ~ and h = 100 W/m2/~

Table II. Physical Properties for Case A

L i q u i d u s t e m p e r a t u r e So l idus t e m p e r a t u r e T h e r m a l c o n d u c t i v i t y V o l u m e t r i c spec i f ic hea t L a t e n t hea t o f f u s i o n A b s o r p t i o n

1300 [~ 1265 [~

14.7 [W/m/~ 3.53 X 106 [ j / m 3 F C ]

2.45 x 109 IJ /m 3] 20 [pct]

O = 1. It is necessary to reduce accordingly the laser energy incident directly on the surface to account for the shadowing effect of the powder. Table III summarizes the physical properties of both the clad material and the mild steel. I~3.t41 The absorption for the clad liquid was measured for the processing conditions using an in situ calorimetric technique. 1~SI All the experiments were made on 10-mm-thick mild steel plate, which was sandblasted to give a uniform and relatively high absorption of the 10.6-/xm radiation. The absorption of the steel substrate was also obtained calorimetrically, under conditions where the laser power was insufficient to melt the surface.

V. RESULTS AND DISCUSSION

Figure 4 shows a closeup of the laser interaction zone for three laser powers of increasing intensity: 460, 510, and 610 W /m m . The clear area represents the melt pool defined by the liquidus isotherm, 1300 ~ In Figure 4(a), the liquidus just touches the substrate at the point Xm, the power being insufficient to melt the base material. In this case, a fusion bond would not be formed between the clad and the substrate. By contrast in Figure 4(c), the laser melted not only the clad but also a significant depth into the base material. Assuming per- fect mixing within the melt pool, the dilution factor, d, is defined by the ratio between the transverse section in the base material to the total surface of the laser trace. In a 2-D problem, d is simply given by

d = x 100 pct [13]

where hb and h~ are the melted depth in the base material and the clad height, respectively. Figure 4(b) shows the optimum situation. Here, the liquidus isotherm has a near horizontal tangent at the front of the melt pool, Xm, giving rise to only superficial melting of the substrate, while still forming a fusion bond between the two materials.

Given that a low dilution (d < 1.0 pet), is a requirement for the operating point of the laser cladding process, Figure 5 presents the interaction between the three primary processing parameters for cladding: the laser power, P, the clad height, he, and the processing speed, v0. From a practical viewpoint, a horizontal line in Figure 5 corresponding to a fixed laser power will inter- sect each of the lines plotted in this figure at a point indicating the optimum processing speed for producing a required clad thickness [i.e., a lower value of the velocity increases the dilution, whereas bonding is lost at larger values (Figure 4)].

It is clearly seen in Figure 5 that the laser power is a linear function of the clad height. The intercept p0 at zero clad height may be interpreted as the power required to heat the substrate from T~ to its fusion temperature. For a single track, this intercept point is a function of the processing speed and of the substrate properties. In the case of overlapping passes, part of the previous track is reheated, so that the thermal properties of the clad will also be important. The slope of each line in Figure 5 is again a function of the processing speed


Composition Liquidus temperature Eutectic temperature Solid conductivity Liquid conductivity Latent heat of fusion Absorption

Cobalt Alloy Mild Steel

1000

28 Cr, 4 W, 1 C, bal. Co 1354 1265

13.39 + 0.028T 48.8

2.60 • 109

31.0 (-+1.o)

C 0.08, Mn, Si, bal. Fe [wt pct] 1530 [~ 1493 [~

6 5 . 0 - 5 • 10-2T + 2 x 10 5TE [W/m/~ 35.0 [W/m/~

1.93 X 10 9 [ J /m 3] 16.0 + 8.17 • 10 3T ( • [pct]

Volumetric specific heat (Co/Cr) 3.525 • 106 + 2.064 X 103T + 0.69T 2 [ j / m 3 / ~ Volumetric specific heat (Fe/C) 3.485 x 10 6 + 3.334 x 103T + 1.25T 2 [ j / m 3 / ~

900 mv=18'mm/s ' ' ' S , " ~ / ~ _ " o v=15 mm/s / " ~ �9 + v=12 mm/s t ~ / - / a v= 9 mm/s / / ~ / . o v = 6 m m / s /j~r" ~ - " / -

800

~600

500

4oo

.~ 300 ,..1

200

100

0.0

T a b l e III . P h y s i c a l P r o p e r t i e s f or C a s e B

' i ' L 0.4 0 8 1.2 1.6 2.0 Clad height [mm]

Fig. 5 - - I d e a l i z e d problem: the effect of the clad height, h, , and the laser velocity, v0, on the laser power for opt imum conditions, i .e., dilution <1 .0 pcI. Parameters: ~ = 0.5 mm, Tp = 20 ~ T= = 20 ~ and h = I00 W/m2/~

and also a function of the clad properties. For this simplified case, the cladding parameters may be represented by the equation

P = A + Bvo + hc(C + Dvo) [14]

where the constants A, B, C, and D obtained from a linear regression analysis are A = 91.5 W / m m , B = 7.15 J / m m 2, C = 22.0 W / m m 2, and D = 25.14 J / m m 3. It should be noted that the value at zero clad height, p0 = A + Bvo, increases with processing speed, v0 (B > 0). However, due to the significant loss by diffusion (coefficient A), the ratio P~ which represents the energy dissipated into the substrate per unit length of clad, decreases on increasing the cladding velocity; i.e., at 6 ram/s , P~ o = 22.5 J / m m 2, whereas at 18 m m / s , P ~ o = 13.0 J / m m 2. From an energy point of view, it is clear that the highest cladding speed is desirable, as less energy is lost by diffusion. This value is, however, limited by the power of the laser being used and by other factors, such as residual stresses.

The parameters (C + Dvo) are the part of the laser power used to form the clad. In particular, the parameter D may be considered as the specific energy required for a given volume of clad. It is interesting that the product of D and the absorption,/3, is ( /3-D) = 5.0 x 10 9 J / m 3,

while the enthalpy change between the temperature of the incoming powder and the liquidus temperature is

7.0 x 109 J / m 3. The fact that the clad specific energy is less indicates that some energy is recovered, probably in heating the substrate, which is always at a lower temperature than the liquid clad.

In Figure 6, the standard deviation of the laser distribution, ~rx, is varied through a factor of 3, affecting both the intercept position and the slope of the operating line. Clearly, with a more focused beam, it is easier to heat the substrate to its fusion temperature, thus reducing p0. However, reducing the beam width in the x direction also decreases the pool length. This, in turn, leads to an increase in the free surface curvature, thereby increasing the angle which the clad surface makes with the substrate at X,,. Thus, the lines in Figure 6 are converging, with increasing clad height, as it becomes more difficult to melt at the triple point Xm.

A very significant problem associated with many thermal surfacing processes is the formation of residual stresses caused by the difference in thermal expansion and thermal history of the coating and the base material. In the extreme situation, these stresses can cause crack- ing in the clad and/or in the substrate, leading to early failure of the component. Surface cracks are particularly detrimental in corrosive environments. To reduce the stress level, the substrate is often preheated. Figure 7 shows the optimum operating curves, with respect to dilution, for different substrate temperatures, T~. As would be ex- pected, p0 is reduced when increasing T~, because a lower

1000

_ 800 E E

6oo O e ~

8 400

0

._~

.2 200

i I i I

- - - - Focus 0.25 mm - - Focus 0.50 mm . . . . Focus 0.75 mm _ , ~

I

00 014 018 112 16 20 Clad height [ram]

Fig. 6 - - I d e a l i z e d problem: the effect of the energy dispersion, o'x, on the laser power, for opt imum conditions, i.e., dilution < 1 .0 pet, vo = 12 m m / s .


1000 i i - i i

- - Substrate 20~ . . . . Substrate 400~ 800

E Substrate 800~

~ 400 ~ " / . / "

_ , , , , , , , , 0.0 0.4 0.8 1.2 1.6 2.0

Clad height [ ram]

Fig. 7 - - I d e a l i z e d problem: the effect of the substrate temperature, T~, on the laser power, for opt imum conditions, i.e., dilution <1 .0 pct, v0 = 12 m m / s .

energy is now required to heat the substrate to its melting point. Perhaps more surprising is that the slope is hardly affected by such an extreme increase in the substrate temperature. This is a further proof that the slope (P - P~ depends primarily on the clad properties and the laser geometry.

Figure 7 also shows the importance of controlling the substrate temperature. Elevated substrate temperatures may be experienced when large areas are covered by overlapping clads, due to the diffusion of heat ahead of the current track. I f during cladding the substrate temperature rises substantially, either the clad thickness or the dilution or both may increase, depending on the powder feeding. Consider, in the case of Figure 7, the processing conditions are chosen in order to obtain a clad thickness of 1.0 ram, with the substrate at ambient temperature. Now if the temperature in the base material rises to 400 ~ a clad of up to 1.25 mm would be produced with the same laser power. However, by reducing the laser power by 13 pct, the clad thickness could be maintained at 1.0 mm.

Turning now to the simulation of the actual experiments performed in the laboratory, Figure 8 depicts the steady-state temperature when cladding at 1.67 mm/s . The first two isotherms correspond to the liquidus of the steel, 1530 ~ and the liquidus of the cobalt alloy, 1354 ~ Note that the former passes through the point Xm, which controls the melting of the substrate, while the latter isotherm passes through the point X~ on the surface of the clad. In this situation where there is low dilution, the clad liquidus isotherm defines the extent of the melt pool. In addition, this interface can be used to calculate the solidification velocity, t~61 The temperature gradient is also easily obtained from Figure 8, and using these two parameters, the scale of the microstructure can be estimated, t6,17l Finally, the temperature gradient also gives some insight into the magnitude of the stresses produced during cladding.

Now comparing the model with the experiments performed with the line focus mirror, Figure 9 shows the agreement for two laser speeds, 1.67 and 3.33 m m / s . With the experimental data, only the clads having less

Xs=-0"89mmNN~ ~ = - 0 . 1 9 r a m

8,0 mm ~1

Fig. 8 - - T h e calculated quasi-steady temperature field near the laser beam during cladding of a cobalt-based alloy onto mild steel. The max imum isotherm corresponds to the substrate liquidus temperature, 1530 ~ Parameters: hc = 0.5 mm, v0 = 1.67 m m / s , P = 171 W / m m , ere = 0.9 mm, Tp = 100 ~ T, = 20 ~ and h = 100 W/m2l~

300 i i i 1 i i , ~ - ' . /

[] v=1.67 mm/s . / 280 o v=3.33 mm/s . / I /

~260 v=5.00 mm/s t / E f / ~ 240 t / - / " Max. power

220 - / / f f

~200 /

~ 180 a

,40 z - - - ' - - /

120 Min. clad height

0 0 0 1 0 2 013 01s 0 '7 ~

Clad height [mm]

Fig. 9 - - C o m p a r i s o n between the experimental data and simulation results for laser cladding a cobalt-based alloy onto mild steel. The experimental points for vo = 1.67 and 3.33 m m / s correspond to conditions where the dilution was less than 5 pct. The dashed line corresponds to the calculated power for the processing speed v0 = 5.0 m m / s .

than 5 pct dilution are plotted in this figure. The con- tinuous curves have been calculated with the model as for the idealized case. The horizontal line corresponds to the maximum power achievable with the laser (1.75 kW). The vertical line corresponds to the minimum clad height (240 /xm) which could be produced with a low dilution. This minimum clad thickness is probably related to the powder size, which, having an average diameter of about 80 /zm, would be a consid- erable fraction of the melt volume. The superheat of the liquid is also important, as it must melt the particle. If the dilution is low, the melt is also relatively cold, thereby increasing the difficulty for melting and dissolving the powder. The fact that the simulation power per width at 5 m m / s for 240 /~m is near to the maximum laser power explains why no clad was achieved at this speed. However, thinner coatings can be prepared if one relaxes the requirement of low dilution, as now both the melt temperature and volume are increased.

Considering that there were no adjustable parameters in the model, the calculated curves in Figure 9 fit rather


well the experimental data. This is particularly the case at the higher clad speed (3.33 mm/s) , where heat loss in the transverse direction is much less significant. Also, at the lower speed, more time is required to achieve steady-state conditions, as heat diffuses a longer distance in front of the beam. For example, Figure 8 shows that 4 mm in front of the melt pool, the temperature is about 400 ~ (the actual specimen was only 60-mm long). Fi- nally, it is necessary to consider how convection within the melt could alter the heat flow. During cladding, im- pact from the powder and shear from the carrier gas have the effect of pushing the hot liquid on the surface toward the back of the pool.t51 Although this enhances the heat transfer, the result would be a longer, but also shallower, melt pool, and thus, more energy would be required to achieve a given clad thickness.

In Figure 8, the laser energy distribution is depicted above the surface. The position of the center of this distribution relative to the front of the melt pool is a result of the simulation. Figure 10 shows the calculated distance between the center of the melt surface, 0.5 �9 (Xm + Xs), relative to the laser beam center, XL, which was obtained from the calculations used to construct Figure 9. This information may be used for positioning the powder injection nozzle in order to maximize the amount of powder absorbed into the clad. It is important to remember that these results have been obtained based on the criterion of low dilution, and hence, the power is not constant. As the shape of the melt pool for a given clad height is not significantly affected by the processing speed, the relative position of the molten surface is not altered markedly either. In fact, the clad height has a much stronger effect. Between a clad height of 0.3 and 0.6 mm, the center of the melt is shifted 75/zm or 8 pet of the laser beam width in the travel direction. These results indicate that mass and energy cannot be decou- pied completely, as done here, by specifying the clad height.

VI. CONCLUSIONS

A comprehensive model of laser cladding by powder injection has been presented which considers the mixing of the powder in the melt pool as an integral part of the process. In addition to being a physically sound as- sumption, it has made the cladding process more ame- nable to simulation, as the powder mass and energy distribution need not be ascertained. The results of the model give the steady-state temperature field, the shape of the melt pool, and the position of the melt surface relative to the laser beam. This information, which is not readily available by direct measurement, provides the starting data for calculations of the microstructure and the residual stress field.

Repeated calculations using the model have given an insight into the influence of the processing parameters. In the idealized case, where the conditions were pur- posely chosen to give minimal dilution, the required laser power varied linearly with the clad thickness and the processing speed. The significance of this result is that a complete map of the processing conditions can be made from a very limited number of data points. The model

was compared with experimental results for cladding a cobalt-based alloy onto mild steel over a reasonable range of conditions. The predicted laser power for a given clad thickness agreed well with the actual laser power required. However, to further validate this work, different clad/substrate couples should be investigated.

Extensions to the cladding model could also be en- visaged. The powder feeding system could be modeled, allowing the prediction of the clad height and the powder temperature within a single self-consistent model. In addition, the effect of convection on the heat transfer in the melt pool should be taken into account in future work.

APPENDIX A

An order of magnitude analysis is used to identify the time scales associated with mass and energy transfer occurring in the melt pool. Considering first the energy transfer, there are two primary competing mechanisms: the advection of the piece with a velocity relative to the laser beam of -v0 and the diffusion of energy in the solid, which may be related to the thermal diffusivity, ab. Thus, the diffusion time, td, for heat conducted through the solid is defined as

ha td ---- _ _

Od b

[All

where the clad height, he, is used for the length scale for energy diffusion. The interaction time of the liquid beneath the laser beam is related to the length of the melt surface in the travel direction. Similarly, the average residence time, tr, of a particle in the melt pool is given by

~, h c t~ = - [A2I

V 0 V0

where [~ is the average horizontal length of the cross- hatched region in Figure 1, which is of similar order to

0.35 E

~ 0.30

0.25 o

'~ 0.20

-,.~

~0.15

,.A

i i i

o v= 1.67 mm/s [] o v= 3.33 mm/s

O a v= 5.00 mm/s a

0.0 01.2 01.4 016 0.8

Clad height [mm]

Fig. 1 0 - - T h e calculated position of the laser beam center, XL, relative to the melt pool center, obtained from the simulation results of Fig. 9.


that of h~. The ratio of these time scales leads to the well-known Peclet number,

td vohc Pe = - - -

t~ a b

Considering now that the powder arriving at the liquid surface, again two competing processes have to be considered: the entrainment of the powder in the melt flow and the melting of the powder particles in the superheated melt pool. The mixing time, tin, is characterizing the entrainment of a powder particle over the clad height, h~, and is inversely proportional to the velocity field in the liquid, vt:

h~ t m = - - [A3]

V1

On the other hand, the melting of a small spherical particle of radius, Rp, in a superheated melt can be calculated analytically making two assumptions: the solid particle has an initial temperature equal to the melting point and only heat diffusion occurs. The fusion time, tf, is found to be

tf - - - [A4I 2KtATo

where Af is the enthalpy of fusion of the particle per unit volume, Kj is the liquid conductivity, and A To is the superheat of the melt. If the temperature of the incoming powder is not equal to the melting temperature, Eq. [A4] can be modified such that

R~(H~ - Hp) tf - [A5]

2KtAT0

where Hc and H e are the volumetric enthalpies of the liquid clad and incoming powder, respectively.

The values in Table I correspond to cladding a cobalt alloy using an 80-/xm powder (see Table III for the thermal properties). The mixing time was calculated assuming an average velocity for the fluid flow of 1 m/s near to the surface. This is consistent with recent measurements using tracer particles and high-speed video pho- tography I181 and with numerical calculations of the Marangoni effect, t5,191 The melting time was calculated assuming an average melt superheat of 200 ~ estimated from surface temperatures measured by pyrometry during actual cladding.

APPENDIX B

The approach taken here follows closely the work of Crivelli and Idelsohn; I~~ however, it has been especially adapted to the problem of laser cladding, where in addition, there is advection of the workpiece and fusion of the powder in the melt pool. After formulating Eq. [3] with the finite element method, the different contribu- tions can be rearranged in terms of a residual vector, r. This yields the following nonlinear equation:

(H-+1 _ H n) r - + [K]T "+'

At

+ Vn+l _ Qn+l _ Fn+l = 0 [B1]

where for the implicit formulation, the vectors, including temperature T, are calculated for the end of the time step, i.e., n + 1. Applying the Galerkin approximation, each of the terms in the above equation are as follows:

Kij = f VNiK(T)VNj dUt (conductivity matrix) J f l e

H i = ff~, NiH(T) dlq (enthalpy vector)

Vi = -fa~ Niv- grad H(T) d~

Q~ = Qz fa ' N~ df~ (heat sink vector)

F~ = fr~ N~q, dF (flux boundary condition vector)

[B21

(advection vector)

where N is the linear parametric shape function. The term v is the nodal velocity calculated at each node point, which for a nondeforming mesh and constant laser position, is simply -v0.

The element matrices in Eq. [B2] are generated depending on whether or not an interface between two phases cuts across the element. This is necessary, in particular, for the heat sink vector, Q, which is calculated for the liquid and is set to zero in the solid. In the case where the element is fully solid or fully liquid, the usual element integration procedure is employed, whereby the influence of temperature is accounted for by evaluating the physical properties at a number of internal "Gauss" points. For elements containing more than one phase, i.e., when the liquidus, solidus, and/or eutectic isotherm intersects with the element, a special integration technique is employed.

First, the position, s c*, of a particular isotherm, T*, is obtained directly by linear interpolation for the element edge.

T* = No(~*)To + N,(~*)T~ [B3]

where No and N~ are the one-dimensional linear parametric shape functions for the edge and To and T, are the corresponding temperatures at the edge extremities. The element is then subdivided by assuming that the interface is a straight line joining the two interface points. This situation is depicted for a triangular element in Figure B 1 (a), which divides the area into a quadrilateral region of phase/3 and a subtriangular region of phase a. So long as the difference in properties between the two regions is an additive function of the nodal variable (in this case, temperature), the integration procedure can account for this property difference in the following way.

�9 e - �9 r First, the whole or master element, f~ , is integrated fo the properties of phase/3. The subelement, fU/~, is then integrated for the difference in properties between the two phases, a and/3, and these values are added with the correct weighting to the master element. This procedure is illustrated for the heat sink vector, Q,

6 4 0 - - V O L U M E 2 3 B , O C T O B E R 1 9 9 2 M E T A L L U R G I C A L T R A N S A C T I O N S B

Q i = Q l [ f n N ~ d ~ e + f n o / ( 0 - 1 ) N T / ~ d ~ / ~ ]

[B41

where, in this instance, it has been assumed that the master element, fU, is liquid and the subelement, ~ / ~ , is solid. As the sink in the solid is zero, the last term in Eq. [B4] is negative.

Figure B l(b) shows how this technique can be ex- tended to alloys, where there may be a mushy region and, thus, two interfaces across the same element. An example of this is the enthalpy, which is characterized by the liquid fraction, f~. For a binary or a pseudo-binary mixture, the variation of the liquid fraction with temperature may be obtained from a solute diffusion model, t17] The integration of the triangular element in Figure B 1 (b) involves first the integration of the master element (12 e) for the enthalpy as a function o f f l (phase/3) and then the integration of the two subelements (~"/~, ~r/#),

Af[(Neft(T) d~ ~ H~= LJn e

+ ( NT/~(O -f~(T)) d~ ~/~ Jn

+ fn,/ NYt~(1- fl(T)) d~T/~] [B5]

A where Aj is the latent heat of fusion and H~ is the latent heat part of the enthalpy.

Equation [B 1] is solved using a Newton iteration scheme with line minimization to ensure convergence.[ 10] Each time step is said to have converged when the normalized sum of the residuals given by

r(T) < [ K ] . T - e r [B6]

where the tolerance, er, in this work was preset to 0.001.

Fig. B1--Schematic diagram illustrating the special integration procedure for elements containing more than one phase: (a) The element is cut by a single interface, corresl~nding to a solid/liquid phase change. The master element is liquid (phase/3); the subelement represents the difference between the phases, a//3. (b) The element is cut by two interfaces, corresponding to an alloy having a finite solidification in- terval. The master element is the mushy phase/3; i.e., 0 < ft < 1. The subelements reflect the differences between the solid and the liquid and the mushy phase, a//3 and 3'//3, respectively.

The time step, At, is adjusted so that if only three iterations are required, the length of the next time step is increased, while if four or more iterations are used, the time step is reduced.

A C K N O W L E D G M E N T S

The authors are indebted to Mr. J.-D. Wagni6re for his help in obtaining the experimental results. They would also like to thank both Mr. A. Frenk and Dr. C. Marsden in this regard and also for their constructive comments on both the model and this manuscript. In addition, they would like to acknowledge the close collaboration of the Mathematics Department of the EPFL and, in particular, Mr. M. Picasso. The authors are most grateful for the financial support of the Swiss government, Commission pour l'Encouragement de la Recherche Scientifique, Beme, and the company, Sulzer Innotec Ltd., Winterthur.

REFERENCES

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mit CO2-Laserstrahlung, Diplomarbeit, Lehrstuhl ftir Lasertechnik, RWTH, Aachen, Germany, 1989, pp. 76-82.

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2d a Thermal Model of Laser Cladding by Powder Injection

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