- Email: [email protected]
Analytical solution for three-dimensional model predicting temperature in the welding cavity of electron beam
ARTICLE IN PRESS
Vacuum 82 (2008) 316–320 www.elsevier.com/locate/vacuum
Analytical solution for three-dimensional model predicting temperature in the welding cavity of electron beam Ching-Yen Hoa,, Mao-Yu Wenb, Yi-Chwen Leec a
Department of Mechanical Engineering, Hwa Hsia Institute of Technology, Taipei 235, Taiwan Graduate Institute of Mechatronics Engineering, Cheng Shiu University, Kaohsiung 833, Taiwan c Department of Architecture and Landscape Design, National Taitung Junior College, Taitung 950, Taiwan b
Received 8 March 2007; received in revised form 16 April 2007; accepted 21 April 2007
Abstract This paper provides an analytical solution for three-dimensional model predicting temperature in the welding cavity of electron beam. It is not easy to measure the temperature on the keyhole of electron-beam welding. Therefore it is essential to develop an analytical model that can accurately predict the temperature in the keyhole. In this study, the keyhole produced by an electron beam is assumed to be a paraboloid of revolution and the intensity of electron beam is supposed to be Gaussian profile. In order to obtain an analytical solution, the parabolic coordinate system is utilized to analyze the temperature in the keyhole and the parameter approximating convection is proposed to account for the effect of convection of molten metal. Considering the momentum balance at the bottom of the keyhole but neglecting the absorption in the plume, an analytical solution is developed for semi-infinite workpieces. As compared with other analytical solutions, the analytical solution obtained by this model provides the temperature distribution more consistent with the experimental data. The effects of various parameters on the temperature distribution in the keyhole are also discussed in this study. r 2007 Published by Elsevier Ltd. Keywords: Electron beam; Temperature; Analytical model; Welding
1. Introduction Temperature fields are required to predict welding processes, microstructure, residual stresses, and distortions of workpieces. Therefore, finding analytical and simple thermal solutions is strongly desired. Incident energy flux had been assumed to be generated by a point source at the surface in order to simplify the model and obtain an analytical solution [1]. Swift-Hook and Gick [2] utilized the analytical and simple line-source model [1] to calculate consistent shapes of electron-beam welds, provided the energy absorbed by the workpiece is known. On the other hand, a line-source model is applicable for high-powerdensity-beam welding [3], which is characterized by the formation of a deep and narrow cavity in the molten pool. Steen et al. [4] combined the point and line source of Corresponding author. Fax: +886 2 29436521.
E-mail addresses: [email protected] (C.-Y. Ho), [email protected] (M.-Y. Wen), [email protected] (Y.-C. Lee). 0042-207X/$ - see front matter r 2007 Published by Elsevier Ltd. doi:10.1016/j.vacuum.2007.04.040
Rosenthal [5] to model more effectively a keyhole weld and estimate the power actually absorbed by the weld. The line source represents absorption down the keyhole and the point source represents the plasma radiation from the plume. It was suggested that the melt width in the lower part of the weld is directly proportional to the strength of the line source. Koleva et al. [6] predicted the regions of weld depth and width for different levels of beam power and welding velocity by a moving linear heat source. Binda et al. [7] proposed a semi-empirical model of the temperature field in laser welding, based on a modification of the Rosenthal solution. The hypothesis of a constant heat rate along the thickness was replaced by a generic unknown function, to be determined from experimental data. While a line-source model can predict the shapes of electron-beam welds [2], the point- and line-source models in their pristine form have certain defects. These are: (1) infinite temperatures occurred near the sources; (2) the distribution of incident flux was not taken into account.
ARTICLE IN PRESS C.-Y. Ho et al. / Vacuum 82 (2008) 316–320
Ho and Wei et al. [8–11] found that the distribution of incident fluxes is an important factor affecting the penetration depth and fusion zone; (3) vertical heat transport was neglected [12]; (4) momentum balance was not taken into account. Hence good agreement between point and line-source models and experimental results can be due to compensating effects between these uncertain factors. In order to avoid weaknesses and obtain additional valuable results, Wei and Shian [13] proposed a threedimensional analytical model around the cavity produced by a moving distributed high-intensity beam, which considered the momentum balance at the cavity base and did not result in infinite temperature. This model successfully predicted the power per unit of penetration as a function of solid Peclet number. Ho [14] also validated the penetration depth and fusion zone induced by a focused electron beam using the Wei and Shian’s model. The above-mentioned models are successful and attractive on the predictions of the penetration depth and fusion zone. However, the errors relative to the experimental data are still large as these analytical models are utilized to predict the temperature in the keyhole of electron-beam welding. Therefore instead of the image method in the Wei and Shian’s model, this study provided a modified model to obtain the more consistent cavity temperatures with experimental results.
2. System model and analysis 2.1. Physical system As sketched in Fig. 1, a workpiece is moved with a constant speed relative to the electron beam. As the electron beam strikes a workpiece, the metal under the beam is vaporized and the electrons are refocused within the metal, resulting in a narrow, deeply penetrating hole surrounded by molten metal. The molten metal flows in a
Workpiece moving velocity U S-L interface
z
hm h
o
m
y x
Fig. 1. Schematic diagram of physical model and coordinate system.
317
thin layer around a vapor-supported cavity to the rear where it cools and solidifies. 2.2. Governing equations and boundary conditions For welding with a moving heat source the geometry becomes slightly asymmetric [11]. The cavity, however, can be roughly observed to be a paraboloid of revolution near the cavity base [15,16], especially for a deep and narrow cavity. Hence, the cavity is idealized by a paraboloid of revolution and a curvilinear orthogonal parabolic coordinate xZf system can be used effectively. Introducing the following transforms: pffiffiffiffiffi x^ 2 xZ cos f x¼ ¼ , (1a) Pe s^ pffiffiffiffiffi y^ 2 xZ sin f , (1b) y¼ ¼ s^ Pe z¼
z^ ðx ZÞ pffiffiffiffi , ¼ s^ Pe S
(1c)
y¼
T^ T^ 1 , T^ T^ m
(1d)
Y ¼ y exp½
pffiffiffiffiffi xZ cos f,
(1e)
where Peclet number and the parameter approximating ^ a and convection, respectively, are defined as Pe U^ s=^ ^ s, ^ a^ , and a^ z correspondingly denote the S a^ =^az . U, dimensional workpeice speed, energy distribution parameter, liquid diffusivity, and enhanced diffusivity in vertical direction. Thermal diffusivity is not isotropic due to convection. Since the flow of liquid enhances energy transport, the diffusivity in the flow direction is increased by a constant multiple of around five [17,18]. Therefore, the parameter approximating convection, S, is proposed. x and Z are the parabolic coordinates. The dimensionless form of a heat conduction equation for quasi-steady state can be expressed in parabolic coordinate system 4 q qY q qY 1 1 1 q2 Y x Z þ þ þ x þ Z qx qx qZ qZ 4 x Z q2 f ¼ Y. ð2Þ The profile on any horizontal cross section of an electron beam is a Gaussian distribution. This was confirmed by measuring distributions of incident fluxes at vertical distances that deviated from the focal spot of an electron beam [19]. Therefore, the energy flux of electron beam can be described by 3 ½3ðr=sÞ2 e , (3) ps2 where 3 is selected to assure 95 percent of incident energy located within the energy-distribution radius. Energy losses due to evaporation can be neglected by comparison with incident beam energy and it is assumed that the incident
qj ¼
ARTICLE IN PRESS C.-Y. Ho et al. / Vacuum 82 (2008) 316–320
318
energy flux is totally absorbed by the cavity. An energy balance between incident flux and conduction on the cavity wall then yields sffiffiffi qY Y x cos f qZ 2 Z Z¼Z0 ! pffiffiffiffi pffiffiffiffiffiffiffi 3Q S 12xZ0 ¼ exp xZ0 cos f , ð4Þ pPe P2e where the dimensionless beam power Q is defined as ^ k^l sð ^ T^ m T^ 1 Þ. The second term on the left-hand side of Q= Eq. (4) is ignored because no error occurs at a location of either an angle of p/2 or the cavity base where the incident flux is the highest. Although errors increase as the workpiece surface is approached, temperatures predicted are still acceptable. The reason for this is that energies impinging on this region have a large incident angle so that the intensity is reduced. It is also assumed to be negligible for heat loss to ambient air on the workpiece surface outside the cavity. Therefore, the adiabatic condition is employed on the workpiece surface. The dimensionless temperature Y is finite at the cavity base (x ¼ 0) and at Z-N. With the boundary conditions as shown in Fig. 2, Eq. (2) can be solved using the separation-of-variables method. Local thermodynamic equilibrium is assumed at the base of the cavity. The momentum balance between the vapor pressure and pressure due to surface tension at the cavity base is utilized to determine the coordinate Z0 of the cavity surface [20]. This yields Pe ½1 þ Y ðyb 1Þ ½ðHðyb yB ÞÞ=ððyB þy1 Þðyb þy1 ÞÞ e , (5) P where yB and yb, respectively, represent the dimensionless temperature at cavity base and boiling. Other dimensionless parameters in Eq. (5) are defined as Z0 ¼
ðT^ m T^ 1 Þd^g=dT^ Y¼ , g^ m
(7) finite at → ∞
∂ = 0 at = e ∂
finite at = 0
∂ − cos ∂ 2
= =
− 3Q S ( e Pe
RðT^ m T^ 1 Þ
.
(8)
3. Results and discussion This study provided a thermal model to predict the cavity temperature during electron-beam welding. The cavity temperature is influenced by some parameters such as the beam power, beam profile, beam diameter, beam focusing characteristics, welding velocity, workpiece material, etc. The effects of beam power, welding velocity, and workpiece material on cavity temperature are analytically investigated in this paper. The cavity temperatures calculated by different models are shown in Fig. 3 and compared with the available experiment data [16]. Evidently, the cavity temperatures predicted by line-source model and the model proposed by this study are more consistent with the measured cavity temperature than these obtained by the point-source model and the model of Wei and Shian. However, the temperature at the cavity base approaches to infinity for the linesource model. The present model cannot only improve the defect of the infinite temperature at cavity base, but also predict the temperature that approximately agrees with the experimental data. The maximal temperature occurs at the cavity base and then decreases toward the cavity opening. 2500
(6)
p^ b s^ , g^ m
h^lg
p^ b is the vapor pressure at boiling and is evaluated by the Clausius–Clapeyron equation. g^ is surface tension and is assumed to be a linearly decreasing function of temperature. When the temperature arrives at the melting point, g^ is equal to g^ m . h^lg is the latent heat of evaporation and R is gas constant.
Infinite
Infinite Experimental data (Schauer and Giedt 1978)
2000 Cavity temperature (°C)
P¼
H¼
This work
1500
Line source Wei and Shian
1000
Point source
500
0 0 cos −12 0 / Pe2 )
0
at = 0 Fig. 2. Boundary conditions.
0
2
4
6
8
10
Distance from cavity base to workpiece surface (mm) Fig. 3. Comparison of the measured and predicted cavity temperatures for different models of welding Al 1100 (S ¼ 0.08, Pe ¼ 0.25, Q ¼ 60).
ARTICLE IN PRESS C.-Y. Ho et al. / Vacuum 82 (2008) 316–320
2500
2500
Cavity temperature (°C)
2000
1500
1000 Pe = 0.25 Pe = 0.125 500
0 2
0
4
6
8
10
Distance from cavity base to workpiece surface (mm) Fig. 5. Temperatures from the cavity base to opening for different Peclet numbers (S ¼ 0.08, Q ¼ 60, Al 1100).
2500
2000 Cavity temperature (°C)
The measured temperature linearly decreases toward the workpiece surface but abruptly descends at the cavity opening. This possibly attributes to the heat dissipation to the ambient air. The cavity temperature predicted by this work is higher than that from measurement near the cavity opening due to the adiabatic condition on workpiece surface. In this paper the heat dissipation to the ambient air is negligible as compared with the incident flux of electron beam. This assumption should be corrected to establish the better model. On the other hand, the cavity temperature predicted by this work is higher at the cavity base and lower at the middle of cavity than the measured temperature. This reason possibly results from it that the plasma absorption [21] and multiple reflections [8–10] have not been considered in the present model yet. The effect of parameters approximating convection on the cavity temperature is shown in Fig. 4. The large parameter approximating convection means that the effective diffusivity in z direction is smaller than that in other directions. Therefore, the heat diffusion in z direction is small relative to other directions. However, most electron-beam energy concentrates on the neighborhood of the cavity base due to Gaussian profile of electron beam. This is the reason why the temperature is higher at cavity base but lower near the cavity opening for a large parameter approximating convection than small one. The variation of the cavity temperature with Peclet number is illustrated in Fig. 5. The small Peclet number increases the cavity temperature and penetration depth. This is because the welding velocity is slow so that the heat in the cavity has no enough time to diffuse for the small Peclet number. The cavity temperatures and penetration depths for different electron-beam powers are sketched in Fig. 6.
319
1500
1000 Q = 60 Q = 120 500
Cavity temperature (°C)
2000
0 0
1500
5
10
15
20
25
Distance from cavity base to workpiece surface (mm)
1000
Fig. 6. Temperatures from the cavity base to opening for different electron-beam powers (S ¼ 0.08, Pe ¼ 0.25, Al 1100).
S = 0.08 S = 0.2
500
0 0
2
4
6
8
10
Distance from cavity base to workpiece surface (mm) Fig. 4. Temperatures from the cavity base to opening for different parameters approximating convection (Pe ¼ 0.25, Q ¼ 60, Al 1100).
The high power of electron beam leads to high cavity temperature and deep penetration. From this figure it is also found that the temperature from the cavity base to opening descends more quickly for low power than high power of electron beam. Fig. 7 demonstrates the cavity temperatures for welding Al 1100 and SS 304. The cavity temperature is higher and the penetration is deeper for welding SS 304. This attributes to the larger absorptivity of SS 304.
ARTICLE IN PRESS C.-Y. Ho et al. / Vacuum 82 (2008) 316–320
Cavity temperature (°C)
320
3000
direction is increased by a constant multiple of around five [17,18]. The cavity temperature and penetration depth increase with the increasing beam power and workpiece absorptivity and the decreasing Peclet number.
2000
Acknowledgment Support for this work by National Science Council of the Republic of China under Grant no. NSC 94-2212-E-146001- is gratefully acknowledged. Al 1100
1000
References
SS 304
[1] [2] [3] [4]
0 0
2
4
6
8
10
Distance from cavity base to workpiece surface (mm) Fig. 7. Temperatures from the cavity base to opening for welding Al 1100 and SS 304 (S ¼ 0.08, Pe ¼ 0.25, Q ¼ 60).
[5] [6] [7]
4. Conclusions
[8] [9] [10] [11]
An analytical thermal model was proposed to predict the cavity temperature during electron-beam welding. The temperature predicted by this work is more consistent with the experiment data than other analytical models. The effects of beam power, welding velocity, and workpiece material on cavity temperature were analytically investigated in this paper. The parameter approximating convection in this model verifies that the diffusivity in the flow
[12] [13] [14] [15] [16] [17] [18] [19] [20] [21]
Rosenthal D. Welding J 1941;20:220s–34s. Swift-Hook DT, Gick AEF. Welding J 1973;52:492–9. Elmer JW, Giedt WH, Eagar TW. Welding J 1990;69:167s–76s. Steen WM, Dowden J, Davis M, Kapadia P. J Phys D: Appl Phys 1998;21:1255–60. Rosenthal D. Trans ASME 1946;68:849–66. Koleva E, Mladenov G, Vutova K. Vacuum 1999;53:67–70. Binda B, Capello E, Previtali B. J Mater Process Technol 2004;155–156:1235–41. Ho CY, Wei PS. Int J Heat Mass Transfer 1997;40:1895–905. Wei PS, Ho CY. Int J Heat Mass Transfer 1998;41:3299–308. Ho CY. Heat Mass Transfer 2005;42:91–103. Wei PS, Wu TH, Chow YT. Trans ASME J Heat Transfer 1990;112/ 1:163–9. Wei PS, Ho JY. Int J Heat Mass Transfer 1990;33:2207–17. Wei PS, Shian MD. ASME J Heat Transfer 1993;115:848–56. Ho CY. J Mater Process Technol 2005;167:265–72. Schauer DA, Giedt WH. Welding J 1978;57:189s–95s. Schauer DA, Giedt WH, Shintaku SM. Welding J 1978;57:127s–33s. Gau C, Viskanta R. Int J Heat Mass Transfer 1984;27:113–23. Giedt WH, Wei XC, Wei SR. Welding J 1984;63:376s–83s. Hicken GK, Giedt WH, Bentley AE. Welding J 1991;70:695–755. Wei PS, Chow YT. Metall Trans B 1992;23B:81–90. Lacroix D, Jeandel G. J Appl Phys 1998;84:2443–9.
Vacuum 82 (2008) 316–320 www.elsevier.com/locate/vacuum
Analytical solution for three-dimensional model predicting temperature in the welding cavity of electron beam Ching-Yen Hoa,, Mao-Yu Wenb, Yi-Chwen Leec a
Department of Mechanical Engineering, Hwa Hsia Institute of Technology, Taipei 235, Taiwan Graduate Institute of Mechatronics Engineering, Cheng Shiu University, Kaohsiung 833, Taiwan c Department of Architecture and Landscape Design, National Taitung Junior College, Taitung 950, Taiwan b
Received 8 March 2007; received in revised form 16 April 2007; accepted 21 April 2007
Abstract This paper provides an analytical solution for three-dimensional model predicting temperature in the welding cavity of electron beam. It is not easy to measure the temperature on the keyhole of electron-beam welding. Therefore it is essential to develop an analytical model that can accurately predict the temperature in the keyhole. In this study, the keyhole produced by an electron beam is assumed to be a paraboloid of revolution and the intensity of electron beam is supposed to be Gaussian profile. In order to obtain an analytical solution, the parabolic coordinate system is utilized to analyze the temperature in the keyhole and the parameter approximating convection is proposed to account for the effect of convection of molten metal. Considering the momentum balance at the bottom of the keyhole but neglecting the absorption in the plume, an analytical solution is developed for semi-infinite workpieces. As compared with other analytical solutions, the analytical solution obtained by this model provides the temperature distribution more consistent with the experimental data. The effects of various parameters on the temperature distribution in the keyhole are also discussed in this study. r 2007 Published by Elsevier Ltd. Keywords: Electron beam; Temperature; Analytical model; Welding
1. Introduction Temperature fields are required to predict welding processes, microstructure, residual stresses, and distortions of workpieces. Therefore, finding analytical and simple thermal solutions is strongly desired. Incident energy flux had been assumed to be generated by a point source at the surface in order to simplify the model and obtain an analytical solution [1]. Swift-Hook and Gick [2] utilized the analytical and simple line-source model [1] to calculate consistent shapes of electron-beam welds, provided the energy absorbed by the workpiece is known. On the other hand, a line-source model is applicable for high-powerdensity-beam welding [3], which is characterized by the formation of a deep and narrow cavity in the molten pool. Steen et al. [4] combined the point and line source of Corresponding author. Fax: +886 2 29436521.
E-mail addresses: [email protected] (C.-Y. Ho), [email protected] (M.-Y. Wen), [email protected] (Y.-C. Lee). 0042-207X/$ - see front matter r 2007 Published by Elsevier Ltd. doi:10.1016/j.vacuum.2007.04.040
Rosenthal [5] to model more effectively a keyhole weld and estimate the power actually absorbed by the weld. The line source represents absorption down the keyhole and the point source represents the plasma radiation from the plume. It was suggested that the melt width in the lower part of the weld is directly proportional to the strength of the line source. Koleva et al. [6] predicted the regions of weld depth and width for different levels of beam power and welding velocity by a moving linear heat source. Binda et al. [7] proposed a semi-empirical model of the temperature field in laser welding, based on a modification of the Rosenthal solution. The hypothesis of a constant heat rate along the thickness was replaced by a generic unknown function, to be determined from experimental data. While a line-source model can predict the shapes of electron-beam welds [2], the point- and line-source models in their pristine form have certain defects. These are: (1) infinite temperatures occurred near the sources; (2) the distribution of incident flux was not taken into account.
ARTICLE IN PRESS C.-Y. Ho et al. / Vacuum 82 (2008) 316–320
Ho and Wei et al. [8–11] found that the distribution of incident fluxes is an important factor affecting the penetration depth and fusion zone; (3) vertical heat transport was neglected [12]; (4) momentum balance was not taken into account. Hence good agreement between point and line-source models and experimental results can be due to compensating effects between these uncertain factors. In order to avoid weaknesses and obtain additional valuable results, Wei and Shian [13] proposed a threedimensional analytical model around the cavity produced by a moving distributed high-intensity beam, which considered the momentum balance at the cavity base and did not result in infinite temperature. This model successfully predicted the power per unit of penetration as a function of solid Peclet number. Ho [14] also validated the penetration depth and fusion zone induced by a focused electron beam using the Wei and Shian’s model. The above-mentioned models are successful and attractive on the predictions of the penetration depth and fusion zone. However, the errors relative to the experimental data are still large as these analytical models are utilized to predict the temperature in the keyhole of electron-beam welding. Therefore instead of the image method in the Wei and Shian’s model, this study provided a modified model to obtain the more consistent cavity temperatures with experimental results.
2. System model and analysis 2.1. Physical system As sketched in Fig. 1, a workpiece is moved with a constant speed relative to the electron beam. As the electron beam strikes a workpiece, the metal under the beam is vaporized and the electrons are refocused within the metal, resulting in a narrow, deeply penetrating hole surrounded by molten metal. The molten metal flows in a
Workpiece moving velocity U S-L interface
z
hm h
o
m
y x
Fig. 1. Schematic diagram of physical model and coordinate system.
317
thin layer around a vapor-supported cavity to the rear where it cools and solidifies. 2.2. Governing equations and boundary conditions For welding with a moving heat source the geometry becomes slightly asymmetric [11]. The cavity, however, can be roughly observed to be a paraboloid of revolution near the cavity base [15,16], especially for a deep and narrow cavity. Hence, the cavity is idealized by a paraboloid of revolution and a curvilinear orthogonal parabolic coordinate xZf system can be used effectively. Introducing the following transforms: pffiffiffiffiffi x^ 2 xZ cos f x¼ ¼ , (1a) Pe s^ pffiffiffiffiffi y^ 2 xZ sin f , (1b) y¼ ¼ s^ Pe z¼
z^ ðx ZÞ pffiffiffiffi , ¼ s^ Pe S
(1c)
y¼
T^ T^ 1 , T^ T^ m
(1d)
Y ¼ y exp½
pffiffiffiffiffi xZ cos f,
(1e)
where Peclet number and the parameter approximating ^ a and convection, respectively, are defined as Pe U^ s=^ ^ s, ^ a^ , and a^ z correspondingly denote the S a^ =^az . U, dimensional workpeice speed, energy distribution parameter, liquid diffusivity, and enhanced diffusivity in vertical direction. Thermal diffusivity is not isotropic due to convection. Since the flow of liquid enhances energy transport, the diffusivity in the flow direction is increased by a constant multiple of around five [17,18]. Therefore, the parameter approximating convection, S, is proposed. x and Z are the parabolic coordinates. The dimensionless form of a heat conduction equation for quasi-steady state can be expressed in parabolic coordinate system 4 q qY q qY 1 1 1 q2 Y x Z þ þ þ x þ Z qx qx qZ qZ 4 x Z q2 f ¼ Y. ð2Þ The profile on any horizontal cross section of an electron beam is a Gaussian distribution. This was confirmed by measuring distributions of incident fluxes at vertical distances that deviated from the focal spot of an electron beam [19]. Therefore, the energy flux of electron beam can be described by 3 ½3ðr=sÞ2 e , (3) ps2 where 3 is selected to assure 95 percent of incident energy located within the energy-distribution radius. Energy losses due to evaporation can be neglected by comparison with incident beam energy and it is assumed that the incident
qj ¼
ARTICLE IN PRESS C.-Y. Ho et al. / Vacuum 82 (2008) 316–320
318
energy flux is totally absorbed by the cavity. An energy balance between incident flux and conduction on the cavity wall then yields sffiffiffi qY Y x cos f qZ 2 Z Z¼Z0 ! pffiffiffiffi pffiffiffiffiffiffiffi 3Q S 12xZ0 ¼ exp xZ0 cos f , ð4Þ pPe P2e where the dimensionless beam power Q is defined as ^ k^l sð ^ T^ m T^ 1 Þ. The second term on the left-hand side of Q= Eq. (4) is ignored because no error occurs at a location of either an angle of p/2 or the cavity base where the incident flux is the highest. Although errors increase as the workpiece surface is approached, temperatures predicted are still acceptable. The reason for this is that energies impinging on this region have a large incident angle so that the intensity is reduced. It is also assumed to be negligible for heat loss to ambient air on the workpiece surface outside the cavity. Therefore, the adiabatic condition is employed on the workpiece surface. The dimensionless temperature Y is finite at the cavity base (x ¼ 0) and at Z-N. With the boundary conditions as shown in Fig. 2, Eq. (2) can be solved using the separation-of-variables method. Local thermodynamic equilibrium is assumed at the base of the cavity. The momentum balance between the vapor pressure and pressure due to surface tension at the cavity base is utilized to determine the coordinate Z0 of the cavity surface [20]. This yields Pe ½1 þ Y ðyb 1Þ ½ðHðyb yB ÞÞ=ððyB þy1 Þðyb þy1 ÞÞ e , (5) P where yB and yb, respectively, represent the dimensionless temperature at cavity base and boiling. Other dimensionless parameters in Eq. (5) are defined as Z0 ¼
ðT^ m T^ 1 Þd^g=dT^ Y¼ , g^ m
(7) finite at → ∞
∂ = 0 at = e ∂
finite at = 0
∂ − cos ∂ 2
= =
− 3Q S ( e Pe
RðT^ m T^ 1 Þ
.
(8)
3. Results and discussion This study provided a thermal model to predict the cavity temperature during electron-beam welding. The cavity temperature is influenced by some parameters such as the beam power, beam profile, beam diameter, beam focusing characteristics, welding velocity, workpiece material, etc. The effects of beam power, welding velocity, and workpiece material on cavity temperature are analytically investigated in this paper. The cavity temperatures calculated by different models are shown in Fig. 3 and compared with the available experiment data [16]. Evidently, the cavity temperatures predicted by line-source model and the model proposed by this study are more consistent with the measured cavity temperature than these obtained by the point-source model and the model of Wei and Shian. However, the temperature at the cavity base approaches to infinity for the linesource model. The present model cannot only improve the defect of the infinite temperature at cavity base, but also predict the temperature that approximately agrees with the experimental data. The maximal temperature occurs at the cavity base and then decreases toward the cavity opening. 2500
(6)
p^ b s^ , g^ m
h^lg
p^ b is the vapor pressure at boiling and is evaluated by the Clausius–Clapeyron equation. g^ is surface tension and is assumed to be a linearly decreasing function of temperature. When the temperature arrives at the melting point, g^ is equal to g^ m . h^lg is the latent heat of evaporation and R is gas constant.
Infinite
Infinite Experimental data (Schauer and Giedt 1978)
2000 Cavity temperature (°C)
P¼
H¼
This work
1500
Line source Wei and Shian
1000
Point source
500
0 0 cos −12 0 / Pe2 )
0
at = 0 Fig. 2. Boundary conditions.
0
2
4
6
8
10
Distance from cavity base to workpiece surface (mm) Fig. 3. Comparison of the measured and predicted cavity temperatures for different models of welding Al 1100 (S ¼ 0.08, Pe ¼ 0.25, Q ¼ 60).
ARTICLE IN PRESS C.-Y. Ho et al. / Vacuum 82 (2008) 316–320
2500
2500
Cavity temperature (°C)
2000
1500
1000 Pe = 0.25 Pe = 0.125 500
0 2
0
4
6
8
10
Distance from cavity base to workpiece surface (mm) Fig. 5. Temperatures from the cavity base to opening for different Peclet numbers (S ¼ 0.08, Q ¼ 60, Al 1100).
2500
2000 Cavity temperature (°C)
The measured temperature linearly decreases toward the workpiece surface but abruptly descends at the cavity opening. This possibly attributes to the heat dissipation to the ambient air. The cavity temperature predicted by this work is higher than that from measurement near the cavity opening due to the adiabatic condition on workpiece surface. In this paper the heat dissipation to the ambient air is negligible as compared with the incident flux of electron beam. This assumption should be corrected to establish the better model. On the other hand, the cavity temperature predicted by this work is higher at the cavity base and lower at the middle of cavity than the measured temperature. This reason possibly results from it that the plasma absorption [21] and multiple reflections [8–10] have not been considered in the present model yet. The effect of parameters approximating convection on the cavity temperature is shown in Fig. 4. The large parameter approximating convection means that the effective diffusivity in z direction is smaller than that in other directions. Therefore, the heat diffusion in z direction is small relative to other directions. However, most electron-beam energy concentrates on the neighborhood of the cavity base due to Gaussian profile of electron beam. This is the reason why the temperature is higher at cavity base but lower near the cavity opening for a large parameter approximating convection than small one. The variation of the cavity temperature with Peclet number is illustrated in Fig. 5. The small Peclet number increases the cavity temperature and penetration depth. This is because the welding velocity is slow so that the heat in the cavity has no enough time to diffuse for the small Peclet number. The cavity temperatures and penetration depths for different electron-beam powers are sketched in Fig. 6.
319
1500
1000 Q = 60 Q = 120 500
Cavity temperature (°C)
2000
0 0
1500
5
10
15
20
25
Distance from cavity base to workpiece surface (mm)
1000
Fig. 6. Temperatures from the cavity base to opening for different electron-beam powers (S ¼ 0.08, Pe ¼ 0.25, Al 1100).
S = 0.08 S = 0.2
500
0 0
2
4
6
8
10
Distance from cavity base to workpiece surface (mm) Fig. 4. Temperatures from the cavity base to opening for different parameters approximating convection (Pe ¼ 0.25, Q ¼ 60, Al 1100).
The high power of electron beam leads to high cavity temperature and deep penetration. From this figure it is also found that the temperature from the cavity base to opening descends more quickly for low power than high power of electron beam. Fig. 7 demonstrates the cavity temperatures for welding Al 1100 and SS 304. The cavity temperature is higher and the penetration is deeper for welding SS 304. This attributes to the larger absorptivity of SS 304.
ARTICLE IN PRESS C.-Y. Ho et al. / Vacuum 82 (2008) 316–320
Cavity temperature (°C)
320
3000
direction is increased by a constant multiple of around five [17,18]. The cavity temperature and penetration depth increase with the increasing beam power and workpiece absorptivity and the decreasing Peclet number.
2000
Acknowledgment Support for this work by National Science Council of the Republic of China under Grant no. NSC 94-2212-E-146001- is gratefully acknowledged. Al 1100
1000
References
SS 304
[1] [2] [3] [4]
0 0
2
4
6
8
10
Distance from cavity base to workpiece surface (mm) Fig. 7. Temperatures from the cavity base to opening for welding Al 1100 and SS 304 (S ¼ 0.08, Pe ¼ 0.25, Q ¼ 60).
[5] [6] [7]
4. Conclusions
[8] [9] [10] [11]
An analytical thermal model was proposed to predict the cavity temperature during electron-beam welding. The temperature predicted by this work is more consistent with the experiment data than other analytical models. The effects of beam power, welding velocity, and workpiece material on cavity temperature were analytically investigated in this paper. The parameter approximating convection in this model verifies that the diffusivity in the flow
[12] [13] [14] [15] [16] [17] [18] [19] [20] [21]
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