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Prediction of pore size in high power density beam welding
International Journal of Heat and Mass Transfer 79 (2014) 223–232
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Prediction of pore size in high power density beam welding P.S. Wei ⇑, T.C. Chao Department of Mechanical and Electro-Mechanical Engineering, National Sun Yat-Sen University, Kaohsiung 80424, Taiwan, ROC
a r t i c l e
i n f o
Article history: Received 12 June 2014 Received in revised form 29 July 2014 Accepted 29 July 2014
Keywords: Laser beam welding Electron beam welding Keyhole welding Pore formation Porosity Two-phase vertical annular flow
a b s t r a c t This study is to investigate parameters responsible for the final pore size during high power density laser and electron beam welding processes. Dimensionless parameters include the surface tension parameter, Mach number and liquid pressure at the keyhole base, friction factor, melting temperature, ratio of specific heats at constant pressure and volume, and loss coefficient. The friction factor can also be treated as a combination of viscous shear stress and thermocapillary force. Although the formation of macroporosity is recognized as an important problem that limits the widespread industrial application of laser and electron beam welding, the physics of the macroporosity formation is not well understood. In order to solve this problem, both the states at the times when the keyhole is about to be enclosed and the pore is completely formed are considered to be governed by the equation of state. The pore shape thus can be determined by the Young–Laplace equation. The gas pressure is determined by calculating a compressible flow of the two-phase, vapor–liquid dispersion in a vertical keyhole of varying cross-section, paying particular attention to transition between the annular and slug flows. It shows that regardless of a supersonic or subsonic flow at the base, the final pore size decreases as the dimensionless surface tension parameter, liquid pressure at the keyhole base, and melting temperature decrease. The pore size also decreases with increasing friction factor subject to a supersonic flow at the base. The effects of the ratio of specific heats at constant pressure and volume, friction factor and loss coefficient on the final pore size are insignificant for a subsonic mixture. This work provides a systematical understanding of the factors affecting the final pore size during keyhole mode welding. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction High power density laser and electron beam welds often contain porosities [1–5]. The resulting degradation of mechanical properties depends on the size distribution and volume fractions of pores. Mechanism of porosity can be simply and interestingly revealed by using the Q-mass spectrograph to analyze gas content in pores via drilling the portion of pores in high vacuum. Pores are found to be mainly composed of metal vapor and some entrained shielding gas and air in the keyhole. Pastor et al. [6] thus explained the mechanism for the formation of porosity to be caused by the collapse of unstable keyhole. When pressure due to surface tension exceeds vapor pressure, projections occur inside the keyhole. The projections increase in size with surface tension until gravitational force dominates projections. The keyhole collapse and pore thus is formed.
⇑ Corresponding author. Tel.: +886 7 5254050. E-mail addresses: [email protected] (P.S. Wei), m993020071@student. nsysu.edu.tw (T.C. Chao). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2014.07.091 0017-9310/Ó 2014 Elsevier Ltd. All rights reserved.
The keyhole oscillates in very complex and unstable manner, as first experimentally observed by Tong and Giedt [7,8] using a high energy pulsed X-ray source. The radiographs not only verified the existence of the keyhole but also showed its oscillations in size and shape. The oscillatory nature of the cavity provided an explanation for welding defects such as spiking, and porosity at the tip of the fusion zone, as shown in Fig. 1 [9]. In this case, serious pores are in lengths of 0.5 to 1 mm in the welding of Al 5083 subject to the focal spot above the surface by 20 mm, and welding speed of 15 mm/s. Al 5083 contains a significant amount of volatile element Mg. Arata [10] and Katayama and Matsunawa [11] also used a microfocused X-ray transmission imaging system to observe pore formation affected by fluid flow, showing that large bubbles were primarily formed at the bottom of the keyhole. A confirmative study of pore formation in keyhole welding requires a systematical and theoretical understanding of fluid flow and heat transfer through the keyhole experiencing its collapse. Lee et al. [12] numerical studied the formation and stability of a stationary laser weld keyhole. The keyhole is formed by displacement of the melt induced by evaporation recoil pressure based on the local equilibrium temperature, while surface tension and
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Nomenclature Ac fim Fr g h H Je K Mc p ~c;a p Q r Rc s Tc T~ c;a Tm U ~c u u‘ V
cross-sectional area of keyhole, Ac ¼ pr 2c ~im =q ~ 2c ~ cu friction factor 2s ~ modified Froude number, Fr Rc T~ cB =g h gravitational acceleration keyhole depth ~ ~cpc T~ cB total energy, H H= local entrainment flux across keyhole qffiffiffiffiffiffiffiffiffiffiffiffi ~ cB Rc T~ cB J e ~J e =q qffiffiffiffiffiffiffiffiffiffiffiffiffi loss coefficient ~ c = jRc T~ c Mach number, Mc u ~=p ~cB pressure p p ~c; max þ p ~cB Þ=2. average pressure ðp ~=~cpc T~ cB absorbed energy = q ~ radial coordinate, r ~r =h specific gas constant arc length along keyhole wall from keyhole tip mixture gas temperature, T c T~ c =T~ cB average temperature ðT~ c; max þ T~ cB Þ=2 melting temperature, T m T~ m =T~ cB qffiffiffiffiffiffiffiffiffiffiffiffi drilling speed, U U= Rc T~ cB qffiffiffiffiffiffiffiffiffiffiffiffiffi ~ c ¼ Mc jRc T~ c mixture velocity in keyhole, u ~ ‘ =U liquid layer velocity, u‘ u welding speed or volume
Wc z
wall,
hydrostatic pressure oppose cavity formation. At laser powers of 500 W and greater, the protrusion occurs on the keyhole wall, which results in keyhole collapse and void formation at the bottom. Initiation of the protrusion is caused mainly by collision of upward and downward flows due to the pressure components in the liquid. Zhou et al. [13] numerically showed that pore formation in pulsed laser welding is caused by two competing factors: one is the solidification rate of the molten metal and the other is the backfilling speed of the molten metal during the keyhole collapse process. Porosity formation was found to be strongly related with the depth-to-width aspect ratio of the keyhole. The larger the ratio, the easier porosity will be formed, and the larger the size of the
axial mass flow rate ~2 ~ c =q ~ cB h ~ cB u Wc W ~ axial coordinate, z ~z=h
through
core
region,
Greek letters ~q ~ cB Rc T~ cB C surface tension parameter, C c=h ~ =q ~ cB q density, q q j specific heat ratio c surface tension / axial velocity component ratio between entrained mixture and keyhole mixture Superscripts dimensional quantity Subscripts a average B keyhole base c core region or keyhole e entrainment ‘ liquid max maximum p pore 1,2 locations at edge of keyhole base and opening of keyhole
voids. Controlling the laser pulse profile is proposed to prevent/ eliminate porosity formation in laser welding. Zhao et al. [14] numerically showed that in continuous laser welding the competition of the dynamic forces and the melt flow results in an unsteady keyhole. Sometimes the keyhole shrinks and collapses suddenly, forms a bubble at the bottom of the molten pool. Porosity will occur if the bubble fails to escape from the molten pool. Pang et al. [15] simulated the self-consistent keyhole shape and weld pool dynamics in deep penetration laser welding. Under certain low heat input welding conditions deep penetration laser welding with a collapsing free keyhole could be obtained and the flow directions near the keyhole wall are upwards and approximately parallel to the keyhole wall. Courtois et al. [16] showed that under high laser power, the keyhole surface undergoes strong instabilities and bubble can appear during the collapse of the keyhole. Depending on the moment of the laser stop, residual porosity may occur. In fact, the initial position of the bubble will determine if the solidification front will have time to capture the bubble. The present work proposes a quasi-steady, axisymmetric, and averaged one-dimensional model, which are identical to the annular two-phase vertical flow [17–19], to simulate the pore formation during keyhole mode welding. This simple, general and flexible method has been extensively and efficiently used to investigate the complicated annular two-phase flows and their transitions from annular to slug flows. Slug flow is a liquid–gas two-phase flow in which the gas phase exists as large bubbles separated by liquid slugs. This work provides a systematical and fundamental step to understand the effects of the compressible mixture gas in the keyhole on the formation and final size of a pore taking place at the base. 2. System model and analysis
Fig. 1. Porosity occurs in the welding of Al 5083 [9].
In this study, the co-ordinate system and geometry of the keyhole are illustrated in Fig. 2(a). For convenience, the solid can be
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3. The flow is in quasi-steady state. This is generally adopted in studying two-phase flows [17–19]. 4. The mixture of vapor and liquid particles in the keyhole is neutral, ideal and compressible, as experimentally confirmed by Reznichenko and Verigin [26]. 5. The equations governing the core region filled with vapor and liquid particles are considered homogeneous, and this assumption has been widely used in the modeling of two-phase flows [17–19], and alloy solidification [27]. 6. The friction shear exerted by either a laminar or turbulent liquid layer along the keyhole wall is evaluated by introducing a friction factor [17]. The shear stress induced by thermocapillary force therefore can also be included in the friction factor. In view of complicated phenomena encountered, the friction factor is considered as a constant for a systematical and general analysis. 7. The equations of state is applied at the times when the keyhole is about to be closed and when the temperature drops to melting temperature to form a final pore. 8. Liquid pressure is ignored, in comparison with capillary pressure and gas pressure during the cooling period to facilitate mathematical manipulation. 2.1. Shape of free surface The free surface of the liquid layer or keyhole wall can be determined from the Young–Laplace equation
pc þ
Fig. 2. (a) The physical model and coordinate system, and (b) pore formation from enclosure of the keyhole.
considered to move upward at a constant relative speed U relative to the liquid–solid interface. In view of the incident flux, a molten pool in a thin layer is formed around the keyhole. Mixture flux is ejected at the keyhole base and entrained through the side wall. The core region thus contains a mixture comprised of vapor and droplets [20–24]. The keyhole is often susceptible to collapse and block the incident flux, leading to enclosure of the keyhole and formation of the pore at the base, as illustrated in Fig. 2(b). The pore at the keyhole base is approximately a sphere, characterized by an effective radius rp. Without loss of generality, the major assumptions made are the following: 1. The model is one-dimensional. This is valid for a deep and narrow keyhole produced by a high power density beam or small welding speed. 2. The model is axisymmetric in keyhole welding with a low scanning speed. The reason for this is that the time scale for keyhole collapse is around 103 to 104 s [12,13], which is much less than time scale for welding around 0.1 to 102 s, estimated by the ratio between the fusion zone width and welding speed. Furthermore, the ratio between welding speed and velocity in the liquid layer is usually a small value around 0.01. It is noted that transport processes are quite different in the welding with high and low welding speeds. As presented by Amara et al. [25], different inclination angles between the front and rear walls result in low temperature, pressure and gas velocity at the keyhole base. This is because the incident flux cannot directly impinge on the keyhole base.
1 1 ¼ p‘ þ C þ R1 R2 qc J 2e
ð1Þ
where the terms on the left-hand side are, dimensionless mixture pressure and recoil pressure due to entrainment exerted from the core region, respectively. The terms on the right-hand side are dimensionless liquid and capillary pressures, respectively. The dimensionless first and second principal curvatures are, respectively, defined as
1 1 R1 r
,sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 dr 1þ ; dz
2
1 d r 2 R2 dz
,"
2 #3=2 dr 1þ dz
ð2Þ
Boundary conditions are r = 0, dr/dz ? 1 at z = 0. 2.2. Pore formation The state when the keyhole is about to be closed is determined from equation of state
~c;a V~ ¼ m ~ c Rc T~ c;a p
ð3Þ
The mixture in the final pore shape or pore shape at the melting point is governed by
~p V~ p ¼ m ~ p Rc T~ m p
ð4Þ
Ignoring liquid pressure, Young–Laplace equation (1) applied for a spherical pore is simplified to
~p ¼ p
2c ~rp
ð5Þ
Considering no condensation takes place during the enclosure pro~ p ¼ 4p~r 3 =3, a combination ~c = m ~ p ; and pore volume V cess, m p between Eqs. (3)–(5) leads to
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3pc;a VT m rp ¼ 8pT c;a C
ð6Þ
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Fig. 4. The effects of the surface tension parameter on (a) dimensionless final pore size, maximum length, radius and volume, and (b) dimensionless average temperature and pressure at the time when the keyhole containing a supersonic mixture is closed.
2.3. Vapor flow in keyhole Determination of mixture pressure, temperature and volume in the keyhole in Eq. (6) needs to solve conservation equations of mass, momentum and energy of a compressible flow. Combining conservation equations lead to [28]
Fig. 3. Comparison of axial variations in dimensionless mixture temperature, pressure, Mach number and keyhole radius for (a) different grid systems, and between (b) exact closed-form and numerical results for a supersonic flow, and (c) exact closed-form and numerical results for a subsonic flow subject to different ejected mass fluxes at the base.
which is the dimensionless equation to calculate effective radius or final size of a pore taking place at the base in keyhole welding.
i dpc jM2c dAc h ¼ 1 þ ðj 1ÞM 2c dF c 2 pc 1 M c Ac dz He Hc dW c Q 1 þ ð1 /Þð1 þ ðj 1ÞM2c Þ þ Tc Tc Wc
ð7Þ
( dT c jM2c j 1 dAc 1 jM 2c dz ¼ ðj 1ÞM 2c dF c þ Q 2 Tc j Ac 1 Mc jM2c T c " # ) j1 He Hc 1 jM2c dW c ð8Þ þ ð1 /Þðj 1ÞM 2c j Tc Wc jM2c
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Fig. 5. The effects of the surface tension parameter on (a) dimensionless final pore size, maximum length, radius and volume, and (b) dimensionless average temperature and pressure at the time when the keyhole containing a subsonic mixture is closed.
dM c 2 þ ðj 1ÞM2c ¼ Mc 2ð1 M 2c Þ "
1 q p‘ ¼ p‘B þ q‘ U 2 ð1 u2‘ Þ ‘ z K q‘ U 2 u2‘ 2 Fr ð9Þ
The external force in Eqs. (7)–(9) is comprised of shear stresses at the keyhole wall and gravitational force, given by
dF c
2sim 1 dz þ r c qc Fr jM2c T c
ð10Þ
Interfacial shear stress can also include thermocapillary force. Dimensionless total energy is defined as
Hc ¼ T c þ
j1 2
M2c T c þ
j1 z jFr
ð11Þ
The variation of mass rate through the keyhole is changed due to entrainment. Hence,
dW c 2J e ds pffiffiffiffiffiffiffiffi ¼ Wc r c qc M c jT c
Fig. 6. The effects of friction factor on (a) dimensionless final pore size, maximum length, radius and volume, and (b) dimensionless average temperature and pressure at the time when the keyhole containing a supersonic mixture is closed.
Dimensionless local liquid pressure in Eq. (1) can be determined from the Bernoulli’s equation
(
dAc dz 1 þ jM 2c þ jM 2c dF c þ Q T c 2 þ ðj 1ÞM 2c Ac # ) He Hc 1 þ jM2c dW c þ 1 þ ð1 /ÞjM2c þ Tc 2 þ ðj 1ÞM2c W c
227
ð12Þ
ð13Þ
where the terms on the right-hand side are liquid pressure at the base, difference in dynamic pressures between the base and location considered, and hydrostatic and loss pressures, respectively.
2.4. Numerical procedure Given independent parameters governing properties and geometries of the keyhole and liquid layer, dimensionless pressure, temperature and Mach number of the mixture in the keyhole are found by solving Eqs. (7)–(9) together with Eqs. (10)–(12). With mixture pressure, recoil pressure due to entrainment, and liquid pressure determined from Eq. (13), Young–Laplace equation (1) is used to predict the shape of the keyhole. The maximum length, radius and volume are calculated as the keyhole is about to be closed. Eq. (6) is then used to predict the final size of a pore taking place at the base.
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Fig. 7. The effects of friction factor on (a) dimensionless final pore size, maximum length, radius and volume, and (b) dimensionless average temperature and pressure at the time when the keyhole containing a subsonic mixture is closed.
Fig. 8. The effects of dimensionless liquid pressure at the base on (a) dimensionless final pore size, maximum length, radius and volume, and (b) dimensionless average temperature and pressure at the time when the keyhole containing a supersonic mixture is closed.
3. Results and discussion In this work, the pore size affected by different dimensionless parameters in keyhole welding is theoretically predicted. The dimensionless independent parameters to be investigated are the surface tension parameter (C), Mach number (McB) and liquid pressure (p‘B) at the base, friction factor (fim), ratio of specific heats at constant pressure and volume (j), and melting temperature (Tm), and loss coefficient (K). It is noted that interfacial shear stress exerted by the liquid layer can also include thermocapillary force, leading to a smaller friction factor. Dimensionless entrainment flux on the keyhole wall is specified by
Je ¼
1z ðJ e1 J e2 Þ þ J e2 ð1 þ 1:2 sin 20zÞ 1 z1
ð14Þ
which indicated that entrainment flux is allowed to spatial oscillation. Typical values of dimensionless parameters are chosen to be: Q = 0, McB = 5 or 0.1, C = 0.05, fim = 0, p‘B = 0.05, Tm = 0.7, JeB = 0.01, Je1 = 0.08, Je2 = 0, j = 1.45, He = 0.5, / = 0.15, Fr = 1.84 106, U⁄ = 103, and K = 0, which are estimated from ~JeB = 104 102kg/m2-s [24,29–32], T~ cB ¼ 103 104 K [22,25,33],
q~ c ¼ 103 100 kg=m3 [25], q~ ‘ ¼ 103 104 kg=m3 , U = 0.01 1 m/s ~2 = 103 106 Pa [25]. For a pulsed energy beam, [14–16,24,33], and p supersonic and subsonic flows at the base take place alternatively [34]. The effects of different grid sizes on axial variations of dimensionless keyhole radius, Mach number, temperature and pressure of a supersonic mixture gas in the keyhole are shown in Fig. 3(a). Provided that the total number of grids is greater than 3000, the solutions are converged within a relative deviation of 105. Fig. 3(b) shows that finite-difference predictions agree quite well with exact closed-form solutions for different ejected mass fluxes at the base. Exact closed-form solutions can be readily found from Eqs. (7)–(9) in the absence of entrainment flux, friction force and energy absorption. The predicted temperature and pressure of the supersonic mixture show a decrease and then increase from the lower to the upper region of the keyhole. In view of high mixture pressure, the keyhole is enclosed. Mach number increases and then decreases in the direction from the lower to upper regions. The variations of transport variables are consistent with knowledge of a compressible flow through a divergent and convergent
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Fig. 9. The effects of dimensionless liquid pressure at the base on (a) dimensionless final pore size, maximum length, radius and volume, and (b) dimensionless average temperature and pressure at the time when the keyhole containing a subsonic mixture is closed.
nozzle. Transport variables of a subsonic flow exhibit opposite trends, as shown in Fig. 3(c). The computed results for the subsonic flow also agree well with exact closed-form solutions for different ejected mass fluxes at the keyhole base. The keyhole in either supersonic or subsonic case is enclosed as a pore in an approximately spherical shape. The dimensionless final pore size, and maximum length, radius and volume at the time when the keyhole is closed as functions of the dimensionless surface tension parameter subject to a supersonic mixture in the keyhole are shown in Fig. 4(a). It can be seen that a decrease in the surface tension parameter decreases the final pore size, and the maximum length, radius and volume at the time when the keyhole is enclosed. Eq. (6) indicates that the final pore size is proportional to the square root of the ratio between average pressure and temperature. Both the average pressure and temperature decrease with the surface tension parameter, as shown in Fig. 4(b). The ratio between the average pressure and temperature, however, decreases with the surface tension parameter. In keyhole welding, the penetration depth of the keyhole can be estimated to be
229
Fig. 10. The effects of dimensionless melting temperature on (a) dimensionless final pore size, maximum length, radius and volume, and (b) dimensionless average temperature and pressure at the time when the keyhole containing a supersonic mixture is closed.
~U ~ ~t h
ð15Þ
The welding time can be scaled as [35]
~t r ~ =V
ð16Þ
Substituting Eq. (16) into Eq. (15) gives
~~ ~ Ur h V
ð17Þ
which shows that the penetration depth of a keyhole mode welding increases with penetration speed and beam radius and decreases with welding speed. Provided that welding speed is 0.015 m/s, typical beam radius and drilling speed are 2 104 m and 0.5 m/s, respectively, the penetration depth estimated from Eq. (17) gives 7 103 m, agreeing with an experimental result (see Fig. 1). In this case, the surface tension parameter of Al 5083 can be estimated to be around 0.04 subject to a reasonable mixture pressure of around 3500 Pa [25]. Fig. 4(b) therefore gives the pore size around
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Fig. 11. The effects of dimensionless melting temperature on (a) dimensionless final pore size, maximum length, radius and volume, and (b) dimensionless average temperature and pressure at the time when the keyhole containing a subsonic mixture is closed.
one-tenth of the penetration depth, agreeing with the experimental result (see Fig. 1). Similar results are found in the case of a subsonic mixture as shown in Fig. 5(a) and (b). Referring to Figs. 4(b) and 5(b) indicates that average pressure normalized by pressure at the base is greater than average temperature normalized by temperature at the base when the keyhole containing a supersonic mixture is about to be closed. Dimensionless average pressure and temperature are greater than unity. Dimensionless pressure and temperature of a subsonic flow in the keyhole, however, exhibit opposite results. The dimensionless final pore size, and maximum length, radius and volume at the time when the keyhole is about to be closed increase whereas dimensionless average pressure and temperature decrease as the friction factor subject to a supersonic mixture decreases, as shown in Fig. 6(a) and (b), respectively. Even though the ratio between average pressure and temperature slightly decreases, the increased final pore size is attributed to greater volume at the time when the keyhole is about to be closed. The effects of the friction factor for a subsonic mixture on the dimensionless final pore size, and maximum length, radius and volume at the
Fig. 12. The effects of the ratio of specific heats at constant pressure and volume on (a) dimensionless final pore size, maximum length, radius and volume, and (b) dimensionless average temperature and pressure at the time when the keyhole containing a subsonic mixture is closed.
time when the keyhole is about to be closed are insignificant, as shown in Fig. 7(a). A decrease in the friction factor, however, reduces averaged temperature and pressure at the time when the keyhole is closed, as shown in Fig. 7(b). In the case of a supersonic mixture in the keyhole, a decrease in dimensionless liquid pressure at the base reduces the final pore size, and maximum length, radius, volume, and averaged pressure and temperature at the time when the keyhole is about to be closed, as shown in Fig. 8(a) and (b), respectively. Since decrease in average pressure is more than that of temperature in the case of a subsonic mixture, the final pore size slightly decreases with liquid pressure at the base, as shown in Fig. 9(a) and (b). The dimensionless melting temperature is independent of the maximum length, radius, volume and average temperature and pressure at the time when the keyhole containing a supersonic mixture is closed, as can be seen from Fig. 10(a) and (b), respectively. Evidently, deformation of the keyhole during drilling and welding periods is not related to melting temperature (see Eqs. (7)–(14)). A decrease in dimensionless melting temperature reduces the final pore size, as can be seen from Eq. (6). In the case of a subsonic mixture in the keyhole, the dimensionless melting temperature is also independent of the maximum length, radius,
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distinguished by supersonic and subsonic flows at the base. The conclusions drawn are as follows: 1. In the case of the keyhole containing a supersonic mixture, the final pore size decreases for a lower surface tension parameter, liquid pressure at the base and melting temperature, and higher friction factor. The friction factor becomes smaller if thermocapillary force is included. 2. For a subsonic mixture in the keyhole, the final pore size decreases for a lower surface tension parameter, liquid pressure at the base, and melting temperature. 3. The effects of the ratio of specific heats at constant pressure and volume, friction factor and loss coefficient subject to a subsonic flow in the keyhole on the final pore size are insignificant. 4. The final pore size for a keyhole containing a supersonic mixture is bigger than that of a keyhole containing a subsonic mixture. This is because the ratio between dimensionless average pressure and temperature when the keyhole containing a supersonic mixture is about to be closed is greater than that for the keyhole containing a subsonic mixture. 5. Average pressure normalized by pressure at the base is greater than average temperature normalized by temperature at the base when the keyhole containing a supersonic mixture is about to be closed. The keyhole containing a subsonic mixture exhibits opposite results. 6. Average pressure and temperature of a supersonic mixture when the keyhole is about to be closed are greater than pressure and temperature at the base, respectively. However, average pressure and temperature when the keyhole is about closed are, respectively, less than pressure and temperature at the base.
Acknowledgments This work was supported by NSC 96-2221-E-110-068-MY3, ROC. Superior and sincere directions from Professor T. DebRoy, Materials Science and Engineering, the Penn. State University, are also deeply appreciated. Fig. 13. The effects of loss coefficient on (a) dimensionless final pore size, maximum length, radius and volume, and (b) dimensionless average temperature and pressure at the time when the keyhole containing a subsonic mixture is closed.
volume, and average temperature and pressure at the moment when the keyhole is closed, as shown in Fig. 11(a) and (b), respectively. The final pore size also decreases with dimensionless melting temperature. Fig. 12(a) shows that the effects of the ratio of specific heats at constant pressure and volume of a subsonic mixture on the dimensionless final pore size, and maximum length, radius and volume at the time when the keyhole is about to be closed are insignificant. The average temperature and pressure, however, increase with decreasing the specific heat ratio, as shown in Fig. 12(b). The effects of the loss coefficient on the final pore size, and maximum length, radius and volume at the time when the keyhole containing a subsonic mixture is closed are insignificant, as shown in Fig. 13(a). An increase in the loss coefficient increases the average temperature and pressure, as shown in Fig. 13(b).
4. Conclusions Supersonic and subsonic flows at the keyhole base alternatively occur for a pulsed high intensity beam. Thermal and flow phenomena as well as the pore formation in the keyhole are therefore
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[15] S. Pang, L. Chen, J. Zhou, Y. Yin, T. Chen, A three-dimensional sharp interface model for self-consistent keyhole and weld pool dynamics in deep penetration laser welding, J. Phys. D: Appl. Phys. 44 (2011) 025301. [16] M. Courtois, M. Carin, P. Le Masson, S. Gaied, A two-dimensional axiallysymmetric model of keyhole and melt pool dynamics during spot laser welding, Rev. Métall. 110 (2013) 165–173. [17] G.B. Wallis, One-Dimensional Two-Phase Flow, McGraw-Hill, New York, 1969. [18] G.F. Hewitt, Chapter 2. Liquid–gas systems, in: G. Hetsroni (Ed.), Handbook of Multiphase Systems, Hemisphere Pub., New York, 1982. [19] M. Ishii, Chapter 2.4. Wave phenomena and two-phase flow instabilities, in: G. Hetsroni (Ed.), Handbook of Multiphase Systems, Hemisphere Pub, New York, 1982. [20] B.S. Yilbas, Study of liquid and vapor ejection processes during laser drilling of metals, J. Laser Appl. 7 (1995) 147–152. [21] D.L. DeBastiani, M.F. Modest, V.S. Stubican, Mechanism of material removal from silicon carbide by carbon dioxide laser heating, J. Am. Ceram. Soc. 73 (1990) 1947–1952. [22] X. Zeng, X. Mao, S.S. Mao, J.H. Yoo, R. Greif, Laser–plasma interactions in fused silica cavities, J. Appl. Phys. 95 (2004) 816–822. [23] S. Basu, T. DebRoy, Liquid metal expulsion during laser irradiation, J. Appl. Phys. 72 (1992) 3317–3322. [24] C.L. Chan, J. Mazumder, One-dimensional steady-state model for damage by vaporization and liquid expulsion due to laser-material interaction, J. Appl. Phys. 62 (1987) 4579–4586. [25] E.H. Amara, R. Fabbro, A. Bendib, Modeling of the compressible vapor flow induced in a keyhole during laser welding, J. Appl. Phys. 93 (2003) 4289–4296.
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Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Prediction of pore size in high power density beam welding P.S. Wei ⇑, T.C. Chao Department of Mechanical and Electro-Mechanical Engineering, National Sun Yat-Sen University, Kaohsiung 80424, Taiwan, ROC
a r t i c l e
i n f o
Article history: Received 12 June 2014 Received in revised form 29 July 2014 Accepted 29 July 2014
Keywords: Laser beam welding Electron beam welding Keyhole welding Pore formation Porosity Two-phase vertical annular flow
a b s t r a c t This study is to investigate parameters responsible for the final pore size during high power density laser and electron beam welding processes. Dimensionless parameters include the surface tension parameter, Mach number and liquid pressure at the keyhole base, friction factor, melting temperature, ratio of specific heats at constant pressure and volume, and loss coefficient. The friction factor can also be treated as a combination of viscous shear stress and thermocapillary force. Although the formation of macroporosity is recognized as an important problem that limits the widespread industrial application of laser and electron beam welding, the physics of the macroporosity formation is not well understood. In order to solve this problem, both the states at the times when the keyhole is about to be enclosed and the pore is completely formed are considered to be governed by the equation of state. The pore shape thus can be determined by the Young–Laplace equation. The gas pressure is determined by calculating a compressible flow of the two-phase, vapor–liquid dispersion in a vertical keyhole of varying cross-section, paying particular attention to transition between the annular and slug flows. It shows that regardless of a supersonic or subsonic flow at the base, the final pore size decreases as the dimensionless surface tension parameter, liquid pressure at the keyhole base, and melting temperature decrease. The pore size also decreases with increasing friction factor subject to a supersonic flow at the base. The effects of the ratio of specific heats at constant pressure and volume, friction factor and loss coefficient on the final pore size are insignificant for a subsonic mixture. This work provides a systematical understanding of the factors affecting the final pore size during keyhole mode welding. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction High power density laser and electron beam welds often contain porosities [1–5]. The resulting degradation of mechanical properties depends on the size distribution and volume fractions of pores. Mechanism of porosity can be simply and interestingly revealed by using the Q-mass spectrograph to analyze gas content in pores via drilling the portion of pores in high vacuum. Pores are found to be mainly composed of metal vapor and some entrained shielding gas and air in the keyhole. Pastor et al. [6] thus explained the mechanism for the formation of porosity to be caused by the collapse of unstable keyhole. When pressure due to surface tension exceeds vapor pressure, projections occur inside the keyhole. The projections increase in size with surface tension until gravitational force dominates projections. The keyhole collapse and pore thus is formed.
⇑ Corresponding author. Tel.: +886 7 5254050. E-mail addresses: [email protected] (P.S. Wei), m993020071@student. nsysu.edu.tw (T.C. Chao). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2014.07.091 0017-9310/Ó 2014 Elsevier Ltd. All rights reserved.
The keyhole oscillates in very complex and unstable manner, as first experimentally observed by Tong and Giedt [7,8] using a high energy pulsed X-ray source. The radiographs not only verified the existence of the keyhole but also showed its oscillations in size and shape. The oscillatory nature of the cavity provided an explanation for welding defects such as spiking, and porosity at the tip of the fusion zone, as shown in Fig. 1 [9]. In this case, serious pores are in lengths of 0.5 to 1 mm in the welding of Al 5083 subject to the focal spot above the surface by 20 mm, and welding speed of 15 mm/s. Al 5083 contains a significant amount of volatile element Mg. Arata [10] and Katayama and Matsunawa [11] also used a microfocused X-ray transmission imaging system to observe pore formation affected by fluid flow, showing that large bubbles were primarily formed at the bottom of the keyhole. A confirmative study of pore formation in keyhole welding requires a systematical and theoretical understanding of fluid flow and heat transfer through the keyhole experiencing its collapse. Lee et al. [12] numerical studied the formation and stability of a stationary laser weld keyhole. The keyhole is formed by displacement of the melt induced by evaporation recoil pressure based on the local equilibrium temperature, while surface tension and
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Nomenclature Ac fim Fr g h H Je K Mc p ~c;a p Q r Rc s Tc T~ c;a Tm U ~c u u‘ V
cross-sectional area of keyhole, Ac ¼ pr 2c ~im =q ~ 2c ~ cu friction factor 2s ~ modified Froude number, Fr Rc T~ cB =g h gravitational acceleration keyhole depth ~ ~cpc T~ cB total energy, H H= local entrainment flux across keyhole qffiffiffiffiffiffiffiffiffiffiffiffi ~ cB Rc T~ cB J e ~J e =q qffiffiffiffiffiffiffiffiffiffiffiffiffi loss coefficient ~ c = jRc T~ c Mach number, Mc u ~=p ~cB pressure p p ~c; max þ p ~cB Þ=2. average pressure ðp ~=~cpc T~ cB absorbed energy = q ~ radial coordinate, r ~r =h specific gas constant arc length along keyhole wall from keyhole tip mixture gas temperature, T c T~ c =T~ cB average temperature ðT~ c; max þ T~ cB Þ=2 melting temperature, T m T~ m =T~ cB qffiffiffiffiffiffiffiffiffiffiffiffi drilling speed, U U= Rc T~ cB qffiffiffiffiffiffiffiffiffiffiffiffiffi ~ c ¼ Mc jRc T~ c mixture velocity in keyhole, u ~ ‘ =U liquid layer velocity, u‘ u welding speed or volume
Wc z
wall,
hydrostatic pressure oppose cavity formation. At laser powers of 500 W and greater, the protrusion occurs on the keyhole wall, which results in keyhole collapse and void formation at the bottom. Initiation of the protrusion is caused mainly by collision of upward and downward flows due to the pressure components in the liquid. Zhou et al. [13] numerically showed that pore formation in pulsed laser welding is caused by two competing factors: one is the solidification rate of the molten metal and the other is the backfilling speed of the molten metal during the keyhole collapse process. Porosity formation was found to be strongly related with the depth-to-width aspect ratio of the keyhole. The larger the ratio, the easier porosity will be formed, and the larger the size of the
axial mass flow rate ~2 ~ c =q ~ cB h ~ cB u Wc W ~ axial coordinate, z ~z=h
through
core
region,
Greek letters ~q ~ cB Rc T~ cB C surface tension parameter, C c=h ~ =q ~ cB q density, q q j specific heat ratio c surface tension / axial velocity component ratio between entrained mixture and keyhole mixture Superscripts dimensional quantity Subscripts a average B keyhole base c core region or keyhole e entrainment ‘ liquid max maximum p pore 1,2 locations at edge of keyhole base and opening of keyhole
voids. Controlling the laser pulse profile is proposed to prevent/ eliminate porosity formation in laser welding. Zhao et al. [14] numerically showed that in continuous laser welding the competition of the dynamic forces and the melt flow results in an unsteady keyhole. Sometimes the keyhole shrinks and collapses suddenly, forms a bubble at the bottom of the molten pool. Porosity will occur if the bubble fails to escape from the molten pool. Pang et al. [15] simulated the self-consistent keyhole shape and weld pool dynamics in deep penetration laser welding. Under certain low heat input welding conditions deep penetration laser welding with a collapsing free keyhole could be obtained and the flow directions near the keyhole wall are upwards and approximately parallel to the keyhole wall. Courtois et al. [16] showed that under high laser power, the keyhole surface undergoes strong instabilities and bubble can appear during the collapse of the keyhole. Depending on the moment of the laser stop, residual porosity may occur. In fact, the initial position of the bubble will determine if the solidification front will have time to capture the bubble. The present work proposes a quasi-steady, axisymmetric, and averaged one-dimensional model, which are identical to the annular two-phase vertical flow [17–19], to simulate the pore formation during keyhole mode welding. This simple, general and flexible method has been extensively and efficiently used to investigate the complicated annular two-phase flows and their transitions from annular to slug flows. Slug flow is a liquid–gas two-phase flow in which the gas phase exists as large bubbles separated by liquid slugs. This work provides a systematical and fundamental step to understand the effects of the compressible mixture gas in the keyhole on the formation and final size of a pore taking place at the base. 2. System model and analysis
Fig. 1. Porosity occurs in the welding of Al 5083 [9].
In this study, the co-ordinate system and geometry of the keyhole are illustrated in Fig. 2(a). For convenience, the solid can be
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3. The flow is in quasi-steady state. This is generally adopted in studying two-phase flows [17–19]. 4. The mixture of vapor and liquid particles in the keyhole is neutral, ideal and compressible, as experimentally confirmed by Reznichenko and Verigin [26]. 5. The equations governing the core region filled with vapor and liquid particles are considered homogeneous, and this assumption has been widely used in the modeling of two-phase flows [17–19], and alloy solidification [27]. 6. The friction shear exerted by either a laminar or turbulent liquid layer along the keyhole wall is evaluated by introducing a friction factor [17]. The shear stress induced by thermocapillary force therefore can also be included in the friction factor. In view of complicated phenomena encountered, the friction factor is considered as a constant for a systematical and general analysis. 7. The equations of state is applied at the times when the keyhole is about to be closed and when the temperature drops to melting temperature to form a final pore. 8. Liquid pressure is ignored, in comparison with capillary pressure and gas pressure during the cooling period to facilitate mathematical manipulation. 2.1. Shape of free surface The free surface of the liquid layer or keyhole wall can be determined from the Young–Laplace equation
pc þ
Fig. 2. (a) The physical model and coordinate system, and (b) pore formation from enclosure of the keyhole.
considered to move upward at a constant relative speed U relative to the liquid–solid interface. In view of the incident flux, a molten pool in a thin layer is formed around the keyhole. Mixture flux is ejected at the keyhole base and entrained through the side wall. The core region thus contains a mixture comprised of vapor and droplets [20–24]. The keyhole is often susceptible to collapse and block the incident flux, leading to enclosure of the keyhole and formation of the pore at the base, as illustrated in Fig. 2(b). The pore at the keyhole base is approximately a sphere, characterized by an effective radius rp. Without loss of generality, the major assumptions made are the following: 1. The model is one-dimensional. This is valid for a deep and narrow keyhole produced by a high power density beam or small welding speed. 2. The model is axisymmetric in keyhole welding with a low scanning speed. The reason for this is that the time scale for keyhole collapse is around 103 to 104 s [12,13], which is much less than time scale for welding around 0.1 to 102 s, estimated by the ratio between the fusion zone width and welding speed. Furthermore, the ratio between welding speed and velocity in the liquid layer is usually a small value around 0.01. It is noted that transport processes are quite different in the welding with high and low welding speeds. As presented by Amara et al. [25], different inclination angles between the front and rear walls result in low temperature, pressure and gas velocity at the keyhole base. This is because the incident flux cannot directly impinge on the keyhole base.
1 1 ¼ p‘ þ C þ R1 R2 qc J 2e
ð1Þ
where the terms on the left-hand side are, dimensionless mixture pressure and recoil pressure due to entrainment exerted from the core region, respectively. The terms on the right-hand side are dimensionless liquid and capillary pressures, respectively. The dimensionless first and second principal curvatures are, respectively, defined as
1 1 R1 r
,sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 dr 1þ ; dz
2
1 d r 2 R2 dz
,"
2 #3=2 dr 1þ dz
ð2Þ
Boundary conditions are r = 0, dr/dz ? 1 at z = 0. 2.2. Pore formation The state when the keyhole is about to be closed is determined from equation of state
~c;a V~ ¼ m ~ c Rc T~ c;a p
ð3Þ
The mixture in the final pore shape or pore shape at the melting point is governed by
~p V~ p ¼ m ~ p Rc T~ m p
ð4Þ
Ignoring liquid pressure, Young–Laplace equation (1) applied for a spherical pore is simplified to
~p ¼ p
2c ~rp
ð5Þ
Considering no condensation takes place during the enclosure pro~ p ¼ 4p~r 3 =3, a combination ~c = m ~ p ; and pore volume V cess, m p between Eqs. (3)–(5) leads to
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3pc;a VT m rp ¼ 8pT c;a C
ð6Þ
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Fig. 4. The effects of the surface tension parameter on (a) dimensionless final pore size, maximum length, radius and volume, and (b) dimensionless average temperature and pressure at the time when the keyhole containing a supersonic mixture is closed.
2.3. Vapor flow in keyhole Determination of mixture pressure, temperature and volume in the keyhole in Eq. (6) needs to solve conservation equations of mass, momentum and energy of a compressible flow. Combining conservation equations lead to [28]
Fig. 3. Comparison of axial variations in dimensionless mixture temperature, pressure, Mach number and keyhole radius for (a) different grid systems, and between (b) exact closed-form and numerical results for a supersonic flow, and (c) exact closed-form and numerical results for a subsonic flow subject to different ejected mass fluxes at the base.
which is the dimensionless equation to calculate effective radius or final size of a pore taking place at the base in keyhole welding.
i dpc jM2c dAc h ¼ 1 þ ðj 1ÞM 2c dF c 2 pc 1 M c Ac dz He Hc dW c Q 1 þ ð1 /Þð1 þ ðj 1ÞM2c Þ þ Tc Tc Wc
ð7Þ
( dT c jM2c j 1 dAc 1 jM 2c dz ¼ ðj 1ÞM 2c dF c þ Q 2 Tc j Ac 1 Mc jM2c T c " # ) j1 He Hc 1 jM2c dW c ð8Þ þ ð1 /Þðj 1ÞM 2c j Tc Wc jM2c
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Fig. 5. The effects of the surface tension parameter on (a) dimensionless final pore size, maximum length, radius and volume, and (b) dimensionless average temperature and pressure at the time when the keyhole containing a subsonic mixture is closed.
dM c 2 þ ðj 1ÞM2c ¼ Mc 2ð1 M 2c Þ "
1 q p‘ ¼ p‘B þ q‘ U 2 ð1 u2‘ Þ ‘ z K q‘ U 2 u2‘ 2 Fr ð9Þ
The external force in Eqs. (7)–(9) is comprised of shear stresses at the keyhole wall and gravitational force, given by
dF c
2sim 1 dz þ r c qc Fr jM2c T c
ð10Þ
Interfacial shear stress can also include thermocapillary force. Dimensionless total energy is defined as
Hc ¼ T c þ
j1 2
M2c T c þ
j1 z jFr
ð11Þ
The variation of mass rate through the keyhole is changed due to entrainment. Hence,
dW c 2J e ds pffiffiffiffiffiffiffiffi ¼ Wc r c qc M c jT c
Fig. 6. The effects of friction factor on (a) dimensionless final pore size, maximum length, radius and volume, and (b) dimensionless average temperature and pressure at the time when the keyhole containing a supersonic mixture is closed.
Dimensionless local liquid pressure in Eq. (1) can be determined from the Bernoulli’s equation
(
dAc dz 1 þ jM 2c þ jM 2c dF c þ Q T c 2 þ ðj 1ÞM 2c Ac # ) He Hc 1 þ jM2c dW c þ 1 þ ð1 /ÞjM2c þ Tc 2 þ ðj 1ÞM2c W c
227
ð12Þ
ð13Þ
where the terms on the right-hand side are liquid pressure at the base, difference in dynamic pressures between the base and location considered, and hydrostatic and loss pressures, respectively.
2.4. Numerical procedure Given independent parameters governing properties and geometries of the keyhole and liquid layer, dimensionless pressure, temperature and Mach number of the mixture in the keyhole are found by solving Eqs. (7)–(9) together with Eqs. (10)–(12). With mixture pressure, recoil pressure due to entrainment, and liquid pressure determined from Eq. (13), Young–Laplace equation (1) is used to predict the shape of the keyhole. The maximum length, radius and volume are calculated as the keyhole is about to be closed. Eq. (6) is then used to predict the final size of a pore taking place at the base.
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Fig. 7. The effects of friction factor on (a) dimensionless final pore size, maximum length, radius and volume, and (b) dimensionless average temperature and pressure at the time when the keyhole containing a subsonic mixture is closed.
Fig. 8. The effects of dimensionless liquid pressure at the base on (a) dimensionless final pore size, maximum length, radius and volume, and (b) dimensionless average temperature and pressure at the time when the keyhole containing a supersonic mixture is closed.
3. Results and discussion In this work, the pore size affected by different dimensionless parameters in keyhole welding is theoretically predicted. The dimensionless independent parameters to be investigated are the surface tension parameter (C), Mach number (McB) and liquid pressure (p‘B) at the base, friction factor (fim), ratio of specific heats at constant pressure and volume (j), and melting temperature (Tm), and loss coefficient (K). It is noted that interfacial shear stress exerted by the liquid layer can also include thermocapillary force, leading to a smaller friction factor. Dimensionless entrainment flux on the keyhole wall is specified by
Je ¼
1z ðJ e1 J e2 Þ þ J e2 ð1 þ 1:2 sin 20zÞ 1 z1
ð14Þ
which indicated that entrainment flux is allowed to spatial oscillation. Typical values of dimensionless parameters are chosen to be: Q = 0, McB = 5 or 0.1, C = 0.05, fim = 0, p‘B = 0.05, Tm = 0.7, JeB = 0.01, Je1 = 0.08, Je2 = 0, j = 1.45, He = 0.5, / = 0.15, Fr = 1.84 106, U⁄ = 103, and K = 0, which are estimated from ~JeB = 104 102kg/m2-s [24,29–32], T~ cB ¼ 103 104 K [22,25,33],
q~ c ¼ 103 100 kg=m3 [25], q~ ‘ ¼ 103 104 kg=m3 , U = 0.01 1 m/s ~2 = 103 106 Pa [25]. For a pulsed energy beam, [14–16,24,33], and p supersonic and subsonic flows at the base take place alternatively [34]. The effects of different grid sizes on axial variations of dimensionless keyhole radius, Mach number, temperature and pressure of a supersonic mixture gas in the keyhole are shown in Fig. 3(a). Provided that the total number of grids is greater than 3000, the solutions are converged within a relative deviation of 105. Fig. 3(b) shows that finite-difference predictions agree quite well with exact closed-form solutions for different ejected mass fluxes at the base. Exact closed-form solutions can be readily found from Eqs. (7)–(9) in the absence of entrainment flux, friction force and energy absorption. The predicted temperature and pressure of the supersonic mixture show a decrease and then increase from the lower to the upper region of the keyhole. In view of high mixture pressure, the keyhole is enclosed. Mach number increases and then decreases in the direction from the lower to upper regions. The variations of transport variables are consistent with knowledge of a compressible flow through a divergent and convergent
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Fig. 9. The effects of dimensionless liquid pressure at the base on (a) dimensionless final pore size, maximum length, radius and volume, and (b) dimensionless average temperature and pressure at the time when the keyhole containing a subsonic mixture is closed.
nozzle. Transport variables of a subsonic flow exhibit opposite trends, as shown in Fig. 3(c). The computed results for the subsonic flow also agree well with exact closed-form solutions for different ejected mass fluxes at the keyhole base. The keyhole in either supersonic or subsonic case is enclosed as a pore in an approximately spherical shape. The dimensionless final pore size, and maximum length, radius and volume at the time when the keyhole is closed as functions of the dimensionless surface tension parameter subject to a supersonic mixture in the keyhole are shown in Fig. 4(a). It can be seen that a decrease in the surface tension parameter decreases the final pore size, and the maximum length, radius and volume at the time when the keyhole is enclosed. Eq. (6) indicates that the final pore size is proportional to the square root of the ratio between average pressure and temperature. Both the average pressure and temperature decrease with the surface tension parameter, as shown in Fig. 4(b). The ratio between the average pressure and temperature, however, decreases with the surface tension parameter. In keyhole welding, the penetration depth of the keyhole can be estimated to be
229
Fig. 10. The effects of dimensionless melting temperature on (a) dimensionless final pore size, maximum length, radius and volume, and (b) dimensionless average temperature and pressure at the time when the keyhole containing a supersonic mixture is closed.
~U ~ ~t h
ð15Þ
The welding time can be scaled as [35]
~t r ~ =V
ð16Þ
Substituting Eq. (16) into Eq. (15) gives
~~ ~ Ur h V
ð17Þ
which shows that the penetration depth of a keyhole mode welding increases with penetration speed and beam radius and decreases with welding speed. Provided that welding speed is 0.015 m/s, typical beam radius and drilling speed are 2 104 m and 0.5 m/s, respectively, the penetration depth estimated from Eq. (17) gives 7 103 m, agreeing with an experimental result (see Fig. 1). In this case, the surface tension parameter of Al 5083 can be estimated to be around 0.04 subject to a reasonable mixture pressure of around 3500 Pa [25]. Fig. 4(b) therefore gives the pore size around
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Fig. 11. The effects of dimensionless melting temperature on (a) dimensionless final pore size, maximum length, radius and volume, and (b) dimensionless average temperature and pressure at the time when the keyhole containing a subsonic mixture is closed.
one-tenth of the penetration depth, agreeing with the experimental result (see Fig. 1). Similar results are found in the case of a subsonic mixture as shown in Fig. 5(a) and (b). Referring to Figs. 4(b) and 5(b) indicates that average pressure normalized by pressure at the base is greater than average temperature normalized by temperature at the base when the keyhole containing a supersonic mixture is about to be closed. Dimensionless average pressure and temperature are greater than unity. Dimensionless pressure and temperature of a subsonic flow in the keyhole, however, exhibit opposite results. The dimensionless final pore size, and maximum length, radius and volume at the time when the keyhole is about to be closed increase whereas dimensionless average pressure and temperature decrease as the friction factor subject to a supersonic mixture decreases, as shown in Fig. 6(a) and (b), respectively. Even though the ratio between average pressure and temperature slightly decreases, the increased final pore size is attributed to greater volume at the time when the keyhole is about to be closed. The effects of the friction factor for a subsonic mixture on the dimensionless final pore size, and maximum length, radius and volume at the
Fig. 12. The effects of the ratio of specific heats at constant pressure and volume on (a) dimensionless final pore size, maximum length, radius and volume, and (b) dimensionless average temperature and pressure at the time when the keyhole containing a subsonic mixture is closed.
time when the keyhole is about to be closed are insignificant, as shown in Fig. 7(a). A decrease in the friction factor, however, reduces averaged temperature and pressure at the time when the keyhole is closed, as shown in Fig. 7(b). In the case of a supersonic mixture in the keyhole, a decrease in dimensionless liquid pressure at the base reduces the final pore size, and maximum length, radius, volume, and averaged pressure and temperature at the time when the keyhole is about to be closed, as shown in Fig. 8(a) and (b), respectively. Since decrease in average pressure is more than that of temperature in the case of a subsonic mixture, the final pore size slightly decreases with liquid pressure at the base, as shown in Fig. 9(a) and (b). The dimensionless melting temperature is independent of the maximum length, radius, volume and average temperature and pressure at the time when the keyhole containing a supersonic mixture is closed, as can be seen from Fig. 10(a) and (b), respectively. Evidently, deformation of the keyhole during drilling and welding periods is not related to melting temperature (see Eqs. (7)–(14)). A decrease in dimensionless melting temperature reduces the final pore size, as can be seen from Eq. (6). In the case of a subsonic mixture in the keyhole, the dimensionless melting temperature is also independent of the maximum length, radius,
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distinguished by supersonic and subsonic flows at the base. The conclusions drawn are as follows: 1. In the case of the keyhole containing a supersonic mixture, the final pore size decreases for a lower surface tension parameter, liquid pressure at the base and melting temperature, and higher friction factor. The friction factor becomes smaller if thermocapillary force is included. 2. For a subsonic mixture in the keyhole, the final pore size decreases for a lower surface tension parameter, liquid pressure at the base, and melting temperature. 3. The effects of the ratio of specific heats at constant pressure and volume, friction factor and loss coefficient subject to a subsonic flow in the keyhole on the final pore size are insignificant. 4. The final pore size for a keyhole containing a supersonic mixture is bigger than that of a keyhole containing a subsonic mixture. This is because the ratio between dimensionless average pressure and temperature when the keyhole containing a supersonic mixture is about to be closed is greater than that for the keyhole containing a subsonic mixture. 5. Average pressure normalized by pressure at the base is greater than average temperature normalized by temperature at the base when the keyhole containing a supersonic mixture is about to be closed. The keyhole containing a subsonic mixture exhibits opposite results. 6. Average pressure and temperature of a supersonic mixture when the keyhole is about to be closed are greater than pressure and temperature at the base, respectively. However, average pressure and temperature when the keyhole is about closed are, respectively, less than pressure and temperature at the base.
Acknowledgments This work was supported by NSC 96-2221-E-110-068-MY3, ROC. Superior and sincere directions from Professor T. DebRoy, Materials Science and Engineering, the Penn. State University, are also deeply appreciated. Fig. 13. The effects of loss coefficient on (a) dimensionless final pore size, maximum length, radius and volume, and (b) dimensionless average temperature and pressure at the time when the keyhole containing a subsonic mixture is closed.
volume, and average temperature and pressure at the moment when the keyhole is closed, as shown in Fig. 11(a) and (b), respectively. The final pore size also decreases with dimensionless melting temperature. Fig. 12(a) shows that the effects of the ratio of specific heats at constant pressure and volume of a subsonic mixture on the dimensionless final pore size, and maximum length, radius and volume at the time when the keyhole is about to be closed are insignificant. The average temperature and pressure, however, increase with decreasing the specific heat ratio, as shown in Fig. 12(b). The effects of the loss coefficient on the final pore size, and maximum length, radius and volume at the time when the keyhole containing a subsonic mixture is closed are insignificant, as shown in Fig. 13(a). An increase in the loss coefficient increases the average temperature and pressure, as shown in Fig. 13(b).
4. Conclusions Supersonic and subsonic flows at the keyhole base alternatively occur for a pulsed high intensity beam. Thermal and flow phenomena as well as the pore formation in the keyhole are therefore
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