Abstract A comprehensive methodology for the imple-mentation of thermal convection into the finite element (FE)analysis of laser direct energy deposition (DED) claddingis developed and validated. Improved convection model-ing will produce improved thermal simulations, which willin turn yield more accurate results from subsequent mod-els seeking to predict microstructural changes, deformation,or residual stresses. Two common convection implementa-tions, considering no convection or free convection only,are compared to three novel forced convection methods:forced convection from heat transfer literature, forced con-vection from lumped capacitance experiments, and forcedconvection from hot-film anemometry measurements.During the cladding process, the exposed surface, the sur-face roughness, and total surface area change due to materialdeposition. The necessity of accounting for the evolutionof the mesh surface in the FE convection model is investi-gated. Quantified error analysis shows that using any of the
M. F. Gouge (�) · J. C. HeigelThe Pennsylvania State University, 17 Reber Building,University Park, PA, 16802 USAe-mail: [email protected]
P. MichalerisThe Pennsylvania State University, 232 Reber Building,University Park, PA, 16802 USAe-mail: [email protected]
T. A. PalmerThe Pennsylvania State University, 4410D AppliedScience Building, University Park, PA, 16802 USAe-mail: [email protected]
three forced convection methodologies improves the accu-racy of the numerical simulations. Using surface-dependenthot-film anemometry measured convection yields the mostaccurate temperature history, with L2 norm errors of 6.25−22.1 ◦C and time-averaged percent errors of 2.80–12.4 %.Using a physically representative convection model appliedto a continually evolving mesh surface is shown to be nec-essary for accuracy in the FE simulation of laser claddingprocesses.
Cladding is the melting of a thin (less than 3 mm) layer ofmetal upon a surface, to act as a protective coating or torepair damaged surfaces. Laser cladding is a sub-applicationof direct energy deposition (DED), which can be used torapidly clad parts [1–4]. The DED process injects metalpowder or wire into a melt pool created by a laser beamfocused on the part surface. A non-reactive gas flow isused to deliver the metal powder for powder-based systems.In both powder and wire systems, an additional shieldinggas flow is used to prevent environmental contaminationof the melt pool. Each of these gas flows creates localizedforced convective cooling on the build surface. In an effortto understand and control the thermal gradients whichdetermine the plastic deformation, residual stresses, andmicrostructure in DED parts, the convective effects areexamined in an ongoing effort to improve DED models[5, 6].
The modeling of DED processes was developed frommulti-pass welding models [7–14]. Prior DED models
which focus on single passes frequently account for naturalconvection alone [15–18] or neglect convective losses alto-gether [19, 20]. Some thin wall models (less than 3 mmthick) which predict microstructure [21] or control melt poolsize [22] also ignore the effect of convection losses but offerno direct validation of thermal history. Han et al. implementnatural convection in a two layer model, yet the com-parison with experiment shows the model is consistentlyunder-cooled [23].
As the number of passes increase, temperatures at the sur-face increase, which result in higher rates of heat transfervia conduction, radiation, and convection. As these temper-atures increase, the convective heat loss due to the shieldingand delivery gases, once negligible, may become signifi-cant. These gas flows are typified as impinging jets in theheat transfer literature. While the recent work on impin-ging jets has focused upon Computational Fluid Dynamics(CFD) models [24–27], the foundational literature reportsexperimental data and algebraic models of convection fromvarious published research sources [28, 29]. These empir-ical functions give the heat transfer coefficient, h(r), asexponential decay functions of radial distance, r , from thenozzle center, which are dependent upon the nozzle-height(H) to nozzle-diameter ratio (D), H/D, and the nozzleReynolds number, ReD . By matching H/D and ReD , thesame value for h(r) should be obtained. In DED processes,however, the presence of the powder and coaxial flows intro-duce factors that are not considered in the existing heattransfer studies. Thus, when applying the effect of forcedconvection to FE models, several different strategies havebeen explored.
Deus and Mazumder are among the first to use forcedconvection in a DED model, but they provide few detailson its implementation [30]. Vasinonta et al. consider cool-ing losses from the shielding gases (0.014 W) in their modelfromwhich they determine the effect of forced convection tobe negligible. However, they offer no experimental valida-tion for the magnitude of these cooling losses, but neverthe-less negate convective effects [31]. Dai and Shaw implementa single heat transfer coefficient (h = 60 W/m2K) thataccounts for cooling by both buoyant effects and the shield-ing gases[32]. Wen and Shin [33, 34] applied unreportedvalues of free and forced convection to their model, whileAggrangsi and Beuth [22] use a free convection value of5 W/m2K and a forced convection value of 10 W/m2K.Neither research group validates their models.
Ghosh and Choi conclude that accurate natural andforced convection bounds are necessary to attain goodmodel results [35, 36]. The authors do not disclose thevalues that they use for free or forced convection, but theirthermal model is experimentally validated, reporting anerror of 13 %. Zekovic, Dwivedi, and Kovacevic performa CFD study to determine the convective bounds for their
FE model, yet they do not detail the convection fieldresulting from their CFD work [37]. However, the inherentcaveat when using cascading models is that the assumptions,approximations, and numeric errors from CFD influencesthe quality of the subsequent thermal FE results.
The effect of implementing experimentally derived con-vection models in FE simulations of a DED cladding pro-cess is investigated. A FE model is developed that is capableof applying convection on the continuously evolving sur-face of the deposition material. Five possible convectionboundary conditions are investigated: no convection, naturalconvection alone, forced convection measured by lumpedcapacitance experiments, forced convection extracted fromheat transfer literature, and forced convection from hot-filmanemometry experiments. These three forced convectionmodels are novel to the DED FE modeling. A compar-ison is made between using evolving and non-evolvingsurfaces. Temperature results using the different convectionmodels are compared to experimental measurements. Threemethods for developing convection boundary conditions arerecommended to improve the accuracy of the FE modelingof any DED process.
2 FE model
2.1 Governing equations
2.1.1 Heat conduction equation in the Lagrangianreference frame
For a body with constant density, ρ, and an isotropic specificheat capacity, Cp, the governing equation is:
ρCpdT
dt= −∂(qi(xj , t))
∂xi
+ Q(xj , t) (1)
where temperature is T , time is t, heat flux vector is qi ,position vector is xj , and body heat source is Q. Thenecessary initial condition is T0 = T∞, where T∞ isthe ambient temperature. A two-part Neumann boundarycondition is implemented consisting of the applied heatsource and the surface heat losses due to both convectionand thermal radiation.
The distribution of heat through the part is described byFourier’s conduction equation:
qi = −k(T )∂T
∂xi
(2)
where the isotropic temperature-dependent thermal conduc-tivity is k(T ). Temperature-dependent properties for Cp andk for Inconel® 625 are given in Table 1.
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Table 1 Temperature-dependent thermal properties of Inconel® 625[38]
T (◦C) k(W/m-k) Cp(J/kg)
−18 9.2 402
21 9.8 410
38 10.1 414
93 10.8 427
204 12.5 456
316 14.1 480
427 15.7 496
538 17.5 513
649 19.0 560
760 20.8 590
871 22.8 620
982 25.2 645
2.1.2 Goldak’s heat input model
The ellipsoidal heat source model from Goldak et al. [39] isused in this study, where the heat source is defined as:
Q = 6√3Pη
abcπ√
πe−
[3x2
a2+ 3y2
b2+ 3(z+vwt)2
c2
](3)
where P is the incident laser power, η is the laser absorptionefficiency, a is the transverse axis, b is the depth axis, c isthe longitudinal axis, vw is the laser velocity, and t is thepass duration.
2.2 Convection boundary conditions
Heat loss due to convection qconv is described by Newton’slaw of cooling:
qconv = h(Ts − T∞) (4)
where Ts is surface temperature.Stephan-Boltzmann’s law describes losses due to thermal
radiation, qrad, as:
qrad = εσ(Ts
4 − T∞4)
(5)
where ε is the surface emissivity and σ is the Stephan-Boltzmann constant (5.67× 10−8W/m2K4). A single valueof emissivity is applied as has been common practice inwelding and DED modeling [40–42]. Here, the average oftemperature-dependent emissivities is used, ε = 0.43[43].The above expression neglects secondary radiation modes,shape factor considerations, and re-radiation from surround-ing surfaces. This equation is nonlinear, but it may belinearized into a heat transfer coefficient form:
qrad = hrad(Ts − T∞) (6)
where
hrad = εσ (Ts + T∞)(T 2s + T 2∞
)(7)
Linear superposition of the heat transfer coefficients yields:
h = hfree + hforced + hrad (8)
Thus, the free convection (hfree), forced convection (hforced),and radiation boundary conditions can be integrated into theFE analysis separately, which are then summed to a singletotal heat transfer coefficient. For the present work, hfree andhforced are determined through an experimental explorationof the convective bounds.
3 Experimental cladding procedures
3.1 Cladding process: experimental descriptionand process parameters
DED cladding is performed on an Inconel® 625 plate ofdimensions 152.4 mm × 76.2 mm × 12.7 mm. A YLR-12000 IPG Photonics fiber laser with 1070–1080 nm wave-length is passed through optical fiber (200 μm diameter)to a colimator (200 mm focal length). After the collima-tor, the beam is passed through a focusing optic. Theseexperiments used a 200 mm focus optic. This laser is usedto melt Inconel® 625 powder (44–149 μm or −100/+325sieve size). The powder is injected coaxially at a 19.0 g/minthrough a Precitect® YC50 clad head. The powder is pro-pelled by argon gas flowing at 9 L/min. An additional coax-ial 9 L/min stream of argon gas shields the molten pool fromoxidation while also protecting the laser optics from powderthat might be entrained or reflected from the surface. Theclad head maintains a vertical offset of 10 mm above thebuild, yielding a beam diameter of 4.064 mm on the sub-strate surface. During deposition, the clad head travels at
Fig. 1 Presentation of the clad plate
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Table 2 Experimental process parameters for single layer longitudinalcladding
Laser power (kW) 2.4
Laser velocity (mm/s) 10.6
Laser beam diameter (mm) 4.06
Powder feed rate (g/min) 19.0
Pass length (mm) 109
Stepover (mm) 2.03
Number of passes 36
10.6 mm/s, while it moves at 31.8 mm/s between depositionpasses. Unidirectional cladding is performed in the longitu-dinal deposition direction, as shown in Fig. 1. Experimentalprocess parameters are summarized in Table 2.
3.2 Measurement methodology
The Inconel® 625 substrate is mounted on the fixture shownin Fig. 2. This fixture was constructed for the measurementof instantaneous deflection by means of a Laser Displace-ment Sensor (LDS) mounted below the plate. The testfixture has a very small contact area (7.26E-4 m2), minimi-zing heat loss by means of conduction, essentially isolatingsurface heat losses to convection and radiation.
Fig. 2 Cladding experimental setup
Omega® 66-k-30 thermocouples are tack welded to theplate in the four locations shown in Fig. 3. These thermo-couples have an uncertainty of 2.2 ◦C (or 0.75 %, whicheveris larger) and a maximum temperature of 1250 ◦C [44].TC 1 is located on the center of the bottom surface. TC2 is located on the bottom surface, at the longitudinalmidpoint, near the edge where the first pass is deposited.TC 1 and TC 2 capture the thermal history of the sub-strate below the cladding region. TC 3 is located on theside nearest the beginning of the deposition passes, at thetransverse midpoint, at the top of the plate. This thermocou-ple, outside of the cladding area, shows the rate at whichheat conducts horizontally through the clad plate. TC 4 islocated on the longitudinal midpoint on the top of the sidenearest the last cladding pass, measuring the temperature asclose to the melt pool as possible without incurring ther-mocouple failure. This array of thermocouples captures theheat transfer through the part along all three cartesian axes.Thus, a simulation which accurately predicts the tempera-tures at the thermocouple locations can be said to accuratelycapture the heat conducting through the substrate in thevertical, longitudinal, and transverse directions, the associ-ated temperatures, internal thermal gradients, and near meltpool behavior.
Fig. 3 Thermocouple location schematic
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4 Numeric implementation
4.1 FE solver
Analysis is performed using CUBIC (Pan Computing LLC),a Newton-Raphson-based FE solver developed specificallyto model multi-pass additive-welding and DED processes.When the laser is on, the time increment is fixed by the laserradius and laser speed:
tj = tj−1 + R
vw
(9)
where tj is the time for the current step, tj−1 is the time ofthe previous step, R is the laser radius, and vw is the laservelocity.
The temperature-dependent properties of Inconel® 625are given in Table 1. The FE solver linearly interpolates fortemperatures between the temperature bounds. For temper-atures beyond the minimum and maximum given values, ituses the closest tabulated value.
4.2 FE mesh
The FE mesh, shown in Fig. 4, is produced in Patran (MSCSoftware). The deposition element thickness corresponds tothat of the deposited material, while the deposition elementlength and width are equal to the laser radius, yieldinga deposited element size of 2.032 mm long, 2.032 mmwide, and 1.168 mm thick. The mesh is coarsened throughthe thickness of the substrate. The model is comprised of16302 Hex8 elements and 19929 nodes. A 3 stage meshconvergence study is performed. The meshes are based on2,3, and 4 elements per laser diameter. The error betweenthe coarsest and finest mesh is 3.30 %, averaged amongstthe four thermocouples locations, which justifies the use ofthe coarsest mesh.
4.3 Element activation strategy
When performing FE analysis of DED processes, it is nec-essary to account for the addition of the deposited materialto the model via the addition or activation of new elements.For the present cladding analysis, the quiet element methodis used [45]. This activation strategy designates elements
Fig. 4 FE mesh
as quiet or active by manipulating their material properties.Elements are multiplied by a coefficient, S, so that:
Cp,quiet = SCpCp (10)
and
kquiet = Skk (11)
Elements are made quiet by setting Scp = 0.01 and Sk =10E-05. These low coefficients minimize the energy thatflows into the quiet elements, effectively removing themfrom the model. The mesh before deposition, with all quietelements, is shown in Fig. 5a. When elements are activatedby the presence of the laser, they are added to the solution bysetting Scp = 1 and Sk = 1, which returns the elements theirtemperature-dependent properties. Both quiet and activeelements are shown in the middle of the cladding processin Fig. 5b. The initial temperature of quiet elements duringactivation is also reset to the ambient temperature in orderto enforce the continuity of the temperature field. A detaileddescription and verification of this method is discussedin [45].
4.4 Evolving free surface
Surface heat losses from thermal convection and radiationmust be applied to the free surfaces of the FE model.While trivial for the thermal modeling of parts with staticsurfaces, in the modeling of additive processes, whichhave continuously evolving surfaces, this adds a furthercomplexity.
Figure 6a depicts the cladding model before deposition,while Fig. 6b shows the model mid-clad. If the convection
Fig. 5 Evolution of the FE mesh during cladding simulation
312 Int J Adv Manuf Technol (2015) 79:307–320
bounds were based solely upon the pre-deposition modelthen as elements were activated, the convective surfaceswould be steadily eliminated, inhibiting heat transfer fromthe cladding surface elements. If the convective boundswere based solely upon the fully clad model (shown inFig. 4), the same error would occur, but in the oppositeorder, with no convection being applied over the clad areaat the beginning of the simulation. CUBIC implementsan algorithm which at each time step flags Gauss pointswhich are on the free surface of the mesh then applies theconvection and radiation losses stipulated in the model.
4.5 Heat input model parameters
To ensure that no artificial heat loss occurs during FEquadrature, the heat input is restricted to the layer of thedeposited elements by setting the axes in Eq. 3 to a = R,b = 0.575R, and c = R. The laser absorption efficiency, η,for Inconel® 625 has been shown to be approximately 35 %for wavelengths of 1080 nm using a pulsed Nd:YAG lasersystem [46].
5 Convection models
Five convective models are investigated. These includeno convection, natural convection alone, forced convectionfrom lumped capacitance experiments, forced convectionfrom heat transfer literature, and forced convection mea-sured by hot-film anemometry.
Fig. 6 Evolution of free surfaces during cladding simulation
Using no convection is investigated since it is fre-quently seen in both welding and DED simulation literature[19–22]. Natural convection alone is a common approxi-mation in the DED literature [15–18]. Forced convectionfrom lumped capacitance experiments may offer a low costmethod of measuring heat losses under a DED deposi-tion flow, provided they attain adequate accuracy duringmodel validation. Extracting convection data from literatureavoids the cost of experimentation entirely, but again mustbe tested against experimental DED time-temperature datato ensure their accuracy. Hot-film anemometry attains themost detailed measurements of the DED process itself, butmay be both laborious and expensive to acquire, and like theprevious convection methodologies, the ensuing FE analysismust undergo the same validation process.
5.1 No convection
This method applies neither free nor forced convection tothe part. Therefore, all heat loss is due to thermal radiationalone.
5.2 Natural convection
To determine the rate of natural convection (also knownas free convection), lumped capacitance experiments arecompleted under quiescent conditions for A36 steel, 6061-T6 aluminum, and commercially pure copper plates [47].Table 3 presents the relevant physical parameters, Biotnumbers, and calculated average heat transfer coefficientsfor the three material types. From the three experiments, themean havg = 9 W/m2K.
5.3 Forced convection from lumped capacitanceexperiments
The lumped capacitance method may also be used to mea-sure an area averaged forced convection. Two preheatedtest plates are subjected to the cooling effects of both the
Table 3 Natural convection determination by the lumped capacitancemethod [47]
Property A36 A6061-T6 Copper
Length (mm) 140 144 140
Width (mm) 139 141 141
Thickness (mm) 6.35 3.12 3.20
ρ (kg/m3) 7833 2710 8470
Cp (J/kg K) 465 1256 386
k (W/m K) 54 167 386
Bi 1.0E-3 1.7E-4 7.4E-4
havg (W/m2 K) 9 9 9
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powder-delivery and shielding gases, from which a lumpedcapacitance rate of forced convection is determined [47].The nominal test plate size, 50.8 mm× 50.8 mm, is approxi-mately the average full width at half maximum of the smoothand rough anemometry fits. Table 4 presents the physicalparameters, Biot number, and calculated havg from theseexperiments. A mean havg = 65.5 W/m2K over the area ofthe plate is found by this method.
5.4 Forced convection based on published research
The closest conditions for experimentally determined valuesof h(r) are found in Saad et al.[48]. As stated above,impinging jet heat transfer regimes may be matched usingReD , which is the Reynolds number at the nozzle, andH/D, which is the ratio of the height of the nozzle abovethe surface (H ) over the diameter of the nozzle (D). Thecladding experiments produce a value of ReD = 1594(using ambient temperature properties for pure argon gas)and H/D = 1. Saad et al. present data for an impinging jetwith ReD = 1960 and H/D = 2. The authors plot valuesof h(r) as a function of non-dimensional radial distancer = x/D. These values are extracted, re-dimensionalizedand fitted to a curve of the following form:
h(r) = AeBr + h0 (12)
where A is the peak convective value, B is the exponentialdecay rate, and h0 is the convective offset. From the workof Saad et al.:
hSaad(r) = 92.8e−0.045r + 0.0 W/m2K. (13)
5.5 Hot-film anemometry
Hot-film anemometry experiments are completed to deter-mine local h values [47]. Two sets of anemometer measure-ments are obtained, one for which the sensor is fixed to asmooth plexiglass plate and one for which the sensor is fixed
Table 4 Forced convection determination by the lumped capacitancemethod [47]
Property A6061-T6 Copper
Length (mm) 49.4 50.8
Width (mm) 46.8 50.2
Thickness (mm) 3.12 3.20
ρ (kg/m3) 2710 8470
Cp (J/kg K) 1256 397.5
k (W/m K) 167 386
Bi 1.4E-3 5.3E-4
havg (W/m2 K) 66.2 64.8
to a plexiglass plate that is surrounded with anti-skid tape.The smooth plexiglass simulates the convective losses of avirgin substrate surface, while the anti-skid tape experimentrepresents the surfaced roughed by the deposited material.
The roughness of the clad and unclad surfaces along withtheir hot-film anemometry experimental representations arequantified through optical profliometry as in reference [49].Measurement values are tabulated, along with their standarddeviations (STD) in Table 5. PV is the maximum peak-to-valley distance, rms is the root-mean-squared roughnessvalues, and Ra is the average deviation from the mean.Representative optical profliometry measurement samplesfor each of the four surfaces are given in Fig. 7. Note that theanti-skid tape was very difficult to measure, as is evidencedby the small area of measurement shown in Fig. 7c. Theplexiglass and unclad Inconel® 625 surfaces both havelow values of PV (4.122 and 28.99 μm, respectively), rms(0.0071 and 1.110 μm), and Ra (0.018 and 0.967 μm).This similarity in surface roughness implies a similarityin boundary layer growth, and thus justifies the use ofplexiglass for the convection measurements. The clad andanti-skid tape show agreement as well, particularly in therms (33.10 and 32.85μm, respectively) and Ra (25.62 and26.91 μm) measurements. Again, this shows the validity ofusing the anti-skid tape to represent the clad surface in thehot-film anenometry convection measurements.
The measured h values are fitted to axisymmetric expo-nential decay curves of the form in Eq. 12. The equation forthe smooth plate is:
hsmooth(r) = 75.3e−0.0471r + 33.7 W/m2K (14)
while that for the rough surface is:
hrough(r) = 69.0e−0.0697r + 21.7 W/m2K. (15)
The axisymmetric experimental and curve fit values forboth the smooth and rough surfaces are plotted in Fig. 8.To fit the above curves, the errors are taken, for boththe integrated cooling values (Q) and at a point-by-pointaverage, along the observed plane. These errors are:
where Qfit is the area under the curve of the exponentialfit function and Qtest is the area under the curve for theanemometry measurements, in W/m K, n is the number ofmeasured increments for each data set, hfit(r) is the localfit function convection value, and hmeasured(r) is convectionfrom the two anemometry experiments.
As the exponential decay functions cannot capture theasymmetry of the anemometry data (due to the asymmetryin the gas flows), an integrated convection comparison isused to ensure an equivalent rate of cooling is applied in thesimulations to the measured convection. The point-by-pointerror gives the average percent of the fit function deviationfrom the convection measurements. The integrated valuesand percent errors are presented in Table 6.
For the three forced convection models, natural convec-tion is applied to those regions unaffected by the gas flows.
Table 6 Anemometer curve fit error
Anemometer Qtest Qfit Integrated Point-by-point
(W/m K) (W/m K) % error % error
Rough 0.0061661 0.0061664 0.005 10.1
Smooth 0.0096670 0.0096677 0.007 5.17
Int J Adv Manuf Technol (2015) 79:307–320 315
The bottom and sides of the substrate. The three convectionregions for the hot-film method, natural, smooth surface,and rough surface, are depicted in Fig. 9. On the sides andthe bottom, hfree = 9 W/m2K, is applied. For regionson the surface of the mesh, either in the areas with nodeposition or before the activation of deposition elements,the smooth surface convection is applied. Activated depo-sition element free surfaces (shown in Fig. 6) are giventhe rough surface convection values. This methodologyattempts to capture the evolving nature of the clad surfaceand its impact upon the rate of cooling on the part duringcladding. The five convective bounds considered are shownin Fig. 10.
5.6 Analysis cases
There are six simulation cases considered, which are:
Case 1: No convectionCase 2: Natural convection aloneCase 3: Forced convection from lumped capacitance experimentsCase 4: Forced convection from heat transfer literatureCase 5: Forced convection measured by hot-film anemometryCase 6: Forced convection with a non-evolving surface
The six cases are described below.
5.7 Case 1: No convection
The Case 1 simulation has no convection losses duringthe entirety of the deposition process, as explained inSection 5.1
Fig. 9 Hot-film anemometry convection diagram: natural convectionalone is applied to the substrate side and bottom surfaces. Smoothsurface convection is applied to the top substrate surface. Roughsurface convection is applied to the activated deposition elementsshown on the back half of the plate
5.8 Case 2: Natural convection alone
Case 2 applies hfree = 9W/m2K to all free surfaces at eachiteration, as developed in Section 5.2
5.9 Case 3: Forced convection from lumped capacitanceexperiments
For the Case 3 simulation, the average forced convec-tion measured by the lumped capacitance method, havg =65.5 W/m2K, is applied to the 50.8 × 50.8 mm squareregion, centered at the laser position, on the top surfaceof the FE model, as developed in Section 5.3 All otherfree surfaces are given the measured natural convection ofhfree = 9 W/m2K.
5.10 Case 4: Forced convection from heat transfer literature
Case 4 uses a radially decaying function for convectionof the type in Eq. 12, extracted from the heat transferstudy by Saad et al., as described in Section 5.4 Thisforced convection is applied to the free substrate top anddeposited material surfaces of the part at each iteration.Natural convection, hfree = 9W/m2K, is applied to all otherfree surfaces.
5.11 Case 5: Forced convection measured by hot-filmanemometry
The hot-film anemometry measurement functions, pre-sented in Section 5.5, are used in the Case 5 simulation. Asstated before, the smooth fit is applied to the undepositedfree surfaces on the top of the substrate, the rough fit isapplied to the free deposition element surfaces once acti-vated, and natural convection is applied to the sides andbottom of the substrate, as done in Cases 3 and 4 (lumpedcapacitance forced convection and forced convection fromheat transfer literature, respectively).
−150 −100 −50 0 50 100 1500
20
40
60
80
100
120
Distance from laser center (mm)Forc
ed c
onve
ctio
n (W
/m2 K
)
No convectionNatural convectionLumped capicitance forced convectionConvection from Saad et al.Anenometry measurements, smooth surfaceAnenometry measurements, rough surface
Fig. 10 Convection boundary conditions
316 Int J Adv Manuf Technol (2015) 79:307–320
5.12 Case 6: Forced convection with a non-evolving surface
Case 6 is implemented to show the necessity of applyingconvection to the evolving free surface at each increment.This case applies convection to a non-evolving surface,as is the case for many commercial FE codes. Case 6uses the same hot-film anemometry model given in Case5. The smooth surface data fit is applied to the uncladregion of the substrate while the rough surface convectionis applied to the external surfaces of the completely cladmesh. Therefore, neither free nor forced convective lossesare accounted for in the cladding region until the elementsat that location have been activated.
The six FE analysis cases presented in this paperare summarized in Table 7 according to the form ofEq. 12.
Area integrated values of cooling are used to com-pare all of the convection models. Each convection modelis integrated over the smallest applied cooling area, the50.8 × 50.8 mm square from Case 3. The method of shellsis used to integrate the exponential decay-based coolingrates over an equivalent circular area. This method inte-grates the h(r) values over annular rings. These rings aretaken from the midpoints between specified locations. Theexponential fit functions are integrated over the regions1/10th of the anemometry steps (r ± 0.3 mm) for improvedaccuracy of the numeric integration. The applied areas andintegrated rates of cooling of the Cases 1–6 are in given inTable 8.
5.13 Error analysis
Two error metrics are employed, L2 error norm and percenterror. The L2 error norm is used to measure the total devi-ation from measured values, as unlike mean-error methods,
Table 7 Analysis cases
Case Description A B h0 Evolving
(W/m2K) (W/m2K) surface
1 No convection N/A N/A 0.0 Yes
2 Free convection N/A N/A 9.0 Yes
3 Lumped N/A N/A 65.5 Yes
capacitance
4 Saad et al. 92.8 −0.045 0.0 Yes
5 Anemometry
Smooth 75.3 −0.0471 33.7 Yes
Rough 69.7 −0.0697 21.7 Yes
6 Anemometry
Smooth 75.3 −0.0471 33.7 No
Rough 69.7 −0.0697 21.7 No
peaks in errors are not lost in the averaging process. The L2
value is calculated as: where the temperatures are taken ateach iteration of the deposition process, in ◦C.
6 Results and discussion
6.1 The effect of convection boundary conditions
Figure 11 illustrates the outcome of the six analysis casespresented above compared with each of the thermocouplelocations. Quantified errors for all six cases at all fourthermocouples are presented in Table 9.
6.1.1 Case 1: No convection
Case 1, in which no convection is applied, is the least accu-rate of all cases considered, with simulated temperatures atall four thermocouples exceeding the in situ measurementsby 100 ◦C at least once during each build. The percenterrors for Case 1 span 16.4–27.5 %, while the L2 errorsare all in excess of 50 ◦C. The rapid divergence of thetemperature histories of case 1 from the measured tem-peratures in Fig. 11 dramatically illustrates the inadequacyof completely ignoring convective losses in DED claddingsimulations.
6.1.2 Case 2: Natural convection
Natural convection alone, case 2, is the least accurate ofthe models which account for convection. Temperaturesare consistently over-predicted, showing the failure of thefree convection model to cool the simulated process in amanner consistent with the physical process. As the heataddition continues, errors at the beginning propagate over
Table 8 Cooling rate comparison
Case Description Applied Integrated cooling Evolving
area (m2) rate (W/K) surface
1 No convection N/A 0.000 Yes
2 Natural convection 0.00258 0.023 Yes
3 Forced lumped 0.00258 0.169 Yes
capacitance
4 Saad et al. 0.00257 0.106 Yes
5 Anemometry fit
Smooth 0.00257 0.109 Yes
Rough 0.00257 0.170 Yes
6 Anemometry fit
Smooth 0.00257 0.109 No
Rough 0.00257 0.170 No
Int J Adv Manuf Technol (2015) 79:307–320 317
0 100 200 300 400 500 600 700 800 900 10000
50
100
150
200
250
300
350
400
450
Time (s)
Tem
pera
ture
( ° C
)
Experimental DataCase 1: No convectionCase 2: Natural convectionCase 3: Lumped capacitanceCase 4: From Saad et al.Case 5: Hot−film anemometryCase 6: Non−evolving surface
(a) TC1
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450
Time (s)
Tem
pera
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( ° C
)
Experimental DataCase 1: No convectionCase 2: Natural convectionCase 3: Lumped capacitanceCase 4: From Saad et al.Case 5: Hot−film anemometryCase 6: Non−evolving surface
(b) TC2
0 100 200 300 400 500 600 700 800 900 10000
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Time (s)
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Experimental DataCase 1: No convectionCase 2: Natural convectionCase 3: Lumped capacitanceCase 4: From Saad et al.Case 5: Hot−film anemometryCase 6: Non−evolving surface
(c) TC3
0 100 200 300 400 500 600 700 800 900 10000
100
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Time (s)
Tem
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)
Experimental DataCase 1: No convectionCase 2: Natural convectionCase 3: Lumped capacitanceCase 4: From Saad et al.Case 5: Hot−film anemometryCase 6: Non−evolving surface
(d) TC4
Fig. 11 Comparison of simulated versus experimental temperatures atTC1-4
the remainder of the simulation. Each thermocouple has aL2 error ranging from 22.3 to 48.3 ◦C and percent errors of9.01–16.6 %.
6.1.3 Case 3: Forced convection from lumped capacitanceexperiment
The forced convection from lumped capacitance simulation,Case 3 attains accurate results despite being the simplestdescription of forced convection. This convection modelattains percent errors below 10 % for all four thermocou-ples. The L2 error spans 7.89–29.2 ◦C while the percenterror is 4.22–9.56 %. Like the natural convection onlymethod (Case 2), Case 3 consistently overpredicts tempera-tures, especially at the end of the simulation.
The results from Case 3 underlines the importance ofthe location at which convection values are applied, notmerely the rate of cooling. In Table 8, it was shown thatCase 3 has an integrated cooling rate nearly identical tothe smooth surface anemometry fit of Case 5. However, theforced convection is limited to a 50.8 × 50.8 mm square,with a convection value lower than both of the hot-filmanemometry data sets, so that Case 3 applies a slightlylower cooling rate at a much smaller location than hot-filmanemometry-based simulation (Case 5).
6.1.4 Case 4: Forced convection from heat transferliterature
The plot of Case 4 shows the first radially decaying convec-tion simulation considered, where the value of h diminisheswith planar distance from the point of impingement (fromSaad et al.). This results in the third most accurate sim-ulation (with percent errors ranging from 4.60 to 7.50 %and L2 error spans 7.32–26.3◦C), and like the natural con-vection and forced convection from lumped capacitanceresults (Cases 2 and 3, respectively), this analysis consis-tently over-predicts temperatures. This is the reverse of whatwould be expected from studying both the theoretical andexperimental convective research. Impinging jet researchsuggests that h increases with increasing ReD and withH/D (to some maximum value near H/D = 7) [29]. Thisresearch informs the a priori assumption that the coolingcurve from Saad et al. used here should instead over-coolthe part during this FE analysis. However, the pure expo-nential decay, with no convective offset, limits the forcedconvective effects to a smaller region than for Case 5 (basedon the hot-film anemometry measurements) which leads tolower overall cooling. The conclusion may be drawn thatthe DED process convection behavior, comprised of twocoaxial, shearing flows and a highly localized heating zone,cannot be perfectly modeled merely by matching dimen-sionless parameters (ReD and H/D). However, in lieu of
318 Int J Adv Manuf Technol (2015) 79:307–320
Table 9 Simulation error for the six analysis cases
Thermocouple Error Case 1: Case 2: Case 3: Case 4: Case 5: Case 6:
metric No Natural Lumped From Saad Hot-film Hot-film
convection convection capacitance et al. anemometry anemometry
Evolving Yes Yes Yes Yes Yes No
surface
TC1 L2 (◦C) 60.3 33.3 13.4 9.07 6.25 21.1
% error 16.4 9.00 4.22 5.27 2.80 5.94
TC2 L2 (◦C) 71.3 41.1 23.1 18.2 13.0 27.0
% error 27.5 16.6 9.56 7.50 4.74 11.2
TC3 L2 (◦C) 50.9 22.3 7.89 7.32 18.6 7.29
% error 19.9 10.2 6.49 6.33 12.4 6.24
TC4 L2 (◦C) 72.9 48.2 29.2 26.3 22.1 41.9
% error 19.3 11.2 5.54 4.60 6.03 10.2
the experimental work, approximate boundary conditionsmay be developed by matching non-dimensional parame-ters from existent heat transfer literature, which will equalor exceed the accuracy of previous thermal DED modelingwork.
6.1.5 Case 5: Forced convection from hot-film anemometry
Case 5, based upon the hot-film anemometry experiments,is the overall most accurate simulation, with an L2 rangeof 6.25–22.1 ◦C and a percent error ranging from 2.80 to12.4 %. Case 5’s temperature histories in Fig. 11 consis-tently match both the trend and magnitude of the thermo-couple measurements. While the lumped capacitance-basedsimulation (Case 3) has similarly small errors, it is unableto match the experimental temperature trends. Again, thistestifies to the importance of having a FE convection imple-mentation which captures both the total rate of cooling andthe location at which the convection is applied during theDED process.
In this vein, there are several improvements that canbe made upon the Case 5 simulation. Some of the depar-ture from experiment may be attributable to differences inroughness between the actual part and the representativeanti-skid tape surface in the anemometry measurements,which could be accounted for in future anemometry exper-iments. Comparing Case 5 with Cases 3 and 4 (lumpedcapacitance forced convection and from Saad’s heat trans-fer study, respectively) at TC 3 indicates that the rate ofconvection decays somewhat beyond the 100 mm radialdistance measured in the anemometry experiments, a dis-tance which again further anemometry experiments or CFDcould determine. Other improvements could include mod-ified boundary conditions on the bottom of the plate. Thebottom substrate surface will experience re-radiation by
the test fixture (Fig. 2). Furthermore, the substrate bottom,being a downward facing surface with an elevated temper-ature in a semi-enclosure, will experience a different rateof convection due to buoyant forces and the entrainment ofheated gases. Further experimental and modeling work isnecessary to pursue these hypotheses.
6.1.6 Case 6: Forced convection from hot-film anemometrywith a non-evolving surface
Case 6 implements a physically realistic convection modelin a non-physically realistic manner. The Case 6 tempera-ture curves presented in Fig. 11 show that this convectionmethodology leads to a consistently over-heated claddingsimulation, due to the insulating effect of applying con-vection only upon surfaces that are external in the finalmesh. Excluding TC3 (which is significantly overcooled byCase 5) both the percent error and L2 errors nearly doublefrom Case 5 to Case 6, ranging from 5.94 to 11.2 % and6.24 to 41.9 ◦C, respectively. This shows that to accuratelycapture the thermal behavior of DED cladding in FE simu-lations, convection must be not only fully representative ofthe actual cooling rates, but must be applied to an evolvingsurface just as occurs during the cladding process.
7 Conclusions
Applying experimentally measured convection to a con-tinually evolving mesh surface improves the accuracy ofthermal simulations in the FE modeling of DED processes.A methodology for implementing physically representa-tive convection rates in a physically realistic manner forthe FE simulation of DED processes has been presented.A total of six analysis cases are evaluated: no convection,
Int J Adv Manuf Technol (2015) 79:307–320 319
natural convection alone, forced convection based uponlumped capacitance experiments, forced convection basedupon a non-dimensionally similar published heat transferstudy, forced convection from hot-film anemometry mea-surements, and hot-film measurement-based forced convec-tion using a non-evolving surface. The accuracy of the sim-ulations is evaluated by comparing simulated temperaturesto those of four thermocouples placed at four locations uponthe substrate. A summary of the conclusions is presentedbelow:
1. Ignoring all convective losses has shown to yield inac-curate simulation temperatures.
2. Using natural convection alone is a better approx-imation than ignoring all convective losses, butis inferior to the models which include forcedconvection.
3. Forced convection extracted from literature and convec-tion from lumped capacitance experiments can improvethe accuracy of thermal DED simulations over thecommon practice of neglecting forced convection.
4. Using hot-film anemometry measured convectionyields the most accurate simulation. This convectionmodel applies an axisymmetric exponentially decayingconvection function upon an evolving surface, apply-ing measured smooth surface convection upon uncladfree surface elements and measured rough surface con-vection upon the activated clad elements. This methodproduces errors of 2.80–12.4 %, and L2 between 6.25and 22.1◦C.
5. It has been shown that the convection model must beapplied to an evolving surface to capture the change ofpart geometry due to the addition of material during thecladding process.
Acknowledgments The material is based upon work supportedby the Office of Naval Research through the Naval Sea SystemsCommand under Contract No. N00024-02-D-6604, Delivery orderNo. 0611. Jarred C. Heigel is supported by the National ScienceFoundation under Grant No. DGE1255832. Any opinions, findings,and conclusions or recommendations expressed in this material arethose of the authors and do not necessarily reflect the views of theNational Science Foundation. The authors would like to thank PanComputing LLC for the generous use of their computing resources andthe CUBIC FE code. The authors would also like to thank Jeff Irwinfor his assistance producing Fig. 9 and Guy Showers and Douglas E.Wolfe for providing the surface roughness measurements.
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