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Some electrical effects in the keyhole plasma in deep penetration cw CO2 laser welding
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science
Applied Surface Science 106 (1996) 240-242
Some electrical effects in the keyhole plasma in deep penetration cw CO 2 laser welding Phiroze Kapadia a,*, John Dowden
b
a Department of Physics, Unitersi O, of Essex, Colchester, Essex C04 3SQ, UK b Department of Mathematics, University of Essex, Colchester, Essex C04 3SQ, UK Received 17 September 1995; accepted 31 December 1995
Abstract The process of laser welding of a material has been extensively studied from an experimental point of view and considerable mathematical modelling has been carried out on the cw aspects of the process as well. For a cw CO 2 laser a partially ionised plasma is likely to exist in the keyhole, emerging from it to produce a turbulent plume in which variations of electrical charge density have been observed. The presence of large temperature gradients near the keyhole walls in conjunction with the Onsager/de Groot relations has the consequence that there must be an electrical potential gradient in the plasma and associated space-charge distribution. These effects are analysed and quantitative relations obtained for the case of a steel work piece.
1. Introduction In the process of deep penetration laser welding the action of the laser beam on the translating specimen oriented perpendicularly to the laser beam results in a hole formed in the material, the hole being referred to as a keyhole. For the configuration, see Fig. 1. The absorption of the laser light in the material is a central quantity of interest in understanding the efficient transfer of energy from the laser beam to the material being welded. This paper analyses the action of some of these mechanisms in the context of laser welding. The real problem is reduced to an idealised form, in which it is possible
* Corresponding author. Tel.: +44-1206-872863; fax: +441206-873598; e-marl: [email protected].
to understand some of the possible electrical effects in the keyhole plasma by first suppressing sources of fluctuation and instability. The analysis makes use of the de Groot irreversible thermodynamic relations applied to the keyhole plasma in an attempt to include the effects of thermal and potential gradients. It considers that the action of the cw CO 2 laser results in an imposed thermal field in the plasma. The combination of temperature gradients, concentration gradients and diffusion suggests the need to examine conditions in the interior of the keyhole more closely using the theory of irreversible thermodynamics, cross-linking processes of heat conduction in the plasma particularly near the walls of the keyhole where temperature gradients of the order of 109 K m - ~ may be expected to be present [1]. This would involve considering the Onsager relations employed in the form generalised by de Groot [2] with a
0169-4332/96/$15.00 Copyright © I996 Elsevier Science B.V. All rights reserved. PH S01 6 9 - 4 3 3 2 ( 9 6 ) 0 0 4 3 6 - 9
P. Kapadia, J. Dowden / Applied SurJDce Science 106 (1996) 240-242 Laser beam ~
: I : . . . .
Direction of translation
~@q-Weldpool ~ Keyhole and plasma
Fig. 1. The configuration of the laser relative to the work piece.
view to analysing these phenomena. These relations imply that spatial variations in the electrostatic potential ~b will occur in the keyhole plasma and that these are likely to propagate up and down the keyhole plasma. Apart from these fluctuations, the steady temperature and concentration gradients are associated with a steady background electrostatic potential and it is this effect that will be considered here. The main purpose of this study is to examine the effects of the cross linking of electrostatic potential gradients with thermal and concentration gradients in the environment of the laser-generated keyhole that is a consequence of the existence of the Onsager coefficients.
241
about the coefficients in the relations, and these ideas can be applied to the keyhole. The flow of material is neglected here for simplicity as are the effects of viscosity with the only forces active being assumed to be electrical, so that F k = -ekVcb. The pressure is assumed to be almost uniform and atmospheric [4] and the effect of gravitational force, which in principle depends on the orientation of the work piece, is small [5]. Euler's equation in the case of mechanical equilibrium reduces to VP=-pV~, where p,~=Y~=lekc k is the total charge per unit mass with Pk = M ~ / V where M k denotes the mass component k and V denotes the volume. Concentrations c i for the ith component are then given by c i = p y p = M J M . De Groot introduces new heat transfer coefficients Q[ and employs forces X , = - V T / T and X~ = F k T V ( t z k / T ) where /z k is the chemical potential of the kth component. For the stationary state the basic equations are
r -(e~-e.)W~}
= O,
k=l,2
.....
n-l,
/
(1)
2. The de Groot relations and their application Theoretical work shows that it is characteristic of the keyhole that there is a very large gradient in the vicinity of the walls and a relatively small gradient in the central region of the keyhole [1]. The temperature distribution itself is approximately uniform across the keyhole and drops steeply to the vaporising temperature T v at the wall. For present purposes the temperature field will be regarded as given. As a result of the high temperatures involved, ionisation takes place and Poisson's equation then can be used to find the potential [3]. Previous analyses of the keyhole, however, leave out the effects of concentration gradients on the potential that arises in the presence of a strong temperature gradient such as occurs near the keyhole wall. In his book, de Groot [2] addresses the problem of thermal diffusion and electrical phenomena in mixtures in a non-uniform temperature field in a stationary state. He generalises Onsager's relations and relates generalised forces linearly to generalised fluxes, obtaining information
together with electrical neutrality ~"k= 1ek c'~ = 0 and the condition ~ _ i ck = 1. These give ekVc k = 0,
(2)
k-I
and 11
E k
= 0.
(3)
1
Eqs. (1)-(3) form a set of (n + 1) linear equations from which one can obtain Vc~ and Vd~ in terms of VT. Consider e I = - e for electrons, e 2 denoting ions and e 3 = 0 neutrals. Eqs. (1)-(3) can then be written in the form a~Vc 1 + blVc 2 + qlVT + elVflp = O, a2Vc I + beVc z + q2VT + e2V4) = O, elVc I + e2Vc 2 = O, Vc I + Vc 2 + Vc 3 = 0,
242
P. Kapadia, J. Dowden / Applied Surface Science 106 (1996) 240-242
where
q2 = 357 + 0 . 0 3 0 5 T - 0.996 × 10 6T2 - 0.965 × 108(hr.
al = 0c~ ( IdOl- Jz3),
bl = -~c2 ( ] Z l - / z 3 ) ,
ql = Ql* - hi + h3, 0 a2 = 0c----~( / x 2 - " 3 ) ,
0 b2 = ~--~c2( / x 2 - " 3 )
For example, at T = 8 kK, q ~ = 5 2 3 + 0 . 1 7 6 × 1012qSv and q2 = 540 - 0.965 × 10s0v. Thus if any one of ql, q2 and q5r can be determined, so can the other two; these are quantities which are not well known. We may note that it seems likely from these relations that the potential ~ has a very weak dependence on temperature T.
and q2 = Q* - h~ + h 3 Hence
3. Discussion and conclusions
Vr]
~in
e, e:
)
+
e; j
,
(4)
with the subscript zc denoting the stationary state. Note that
ql( a 2 / e l - b J e ? )
- q2( a l / e l - b l / e 2 )
e , e 2 { a , / e ~ - ( a~ + b , ) / e l e 2 + b 2 / e ~ }
"
(5) Since the right-hand sides of Eqs. (4) and (5) at constant pressure are functions of T only, and we are interested only in such conditions, the left-hand sides are, respectively, el(Ocl/OT);, and (Od)/OT)p. For an ideal gas the chemical potential [6] is given by
A consequence of the existence of the O n s a g e r / d e Groot coefficients is the presence of an electric field in those regions of the keyhole where the temperature gradients are high, such as the region near to the walls. The effects estimated are under steady stationary conditions so that for example a measured value of the electrostatic potential gradient could yield information on electron and ion heat transfer coefficients to the keyhole wall. The process of laser welding indicates that keyhole processes of considerable complexity are active but the possibility of detecting electrostatic steady and fluctuating potentials in the keyhole offers an interesting way of trying to track these processes and elucidate their behaviour. Relations between temperature and the rate of change of electric potential with temperature have been found in the steady case.
Acknowledgements so /x e - / x , and /x s -/x~, can be found while c I and c 2 can be expressed in terms of the degree of ionisation, a ( T ) by the relations
C1
C2
me
ms
OL o~(rne+mi)+(1-a)m
. "
The form of c~(T) for iron at 1 bar can be calculated as a function of T so that Eqs. (4) and (5) become a pair of relations between ql and q2 and qSr--= d~o/dT=(V4)/VT)~. They can in principle be solved for q~ and q2; the resulting expressions are very complicated, but over the range of temperatures 6 k K < T < 15 kK, can be approximated accurately by q~ = 352 + 0 . 0 2 8 4 T - 0.926 X 10 6T2
+ 0.176 × 10 lZq~r,
The authors would like to acknowledge the support of the E U R E K A 113 and E U R E K A 194 programmes of the European Community.
References [1] J.M. Dowden, P. Kapadia and N. Postacioglu, J. Phys. D: Appt. Phys. 22 (1989) 741. [2] S.R. de Groot, Thermodynamics of Irreversible Processes (North-Holland, Amsterdam, 1952) ch. 12. [3] R. Finke, P. Kapadia and J.M. Dowden, J. Phys. D: Appl. Phys. 23 (1990) 643. [4] J.M. Dowden and P. Kapadia, J. Phys. D: Appl. Phys. 28 (1995) 2252. [5] J.G. Andrews and D.R. Atthey, J. Phys. D: Appl. Phys. 9 (1976) 2181. [6] L.D. Landau and E.M. Lifshitz, Statistical Physics, Part 1, 3rd ed. (Pergamon, Oxford, 1980).
science
Applied Surface Science 106 (1996) 240-242
Some electrical effects in the keyhole plasma in deep penetration cw CO 2 laser welding Phiroze Kapadia a,*, John Dowden
b
a Department of Physics, Unitersi O, of Essex, Colchester, Essex C04 3SQ, UK b Department of Mathematics, University of Essex, Colchester, Essex C04 3SQ, UK Received 17 September 1995; accepted 31 December 1995
Abstract The process of laser welding of a material has been extensively studied from an experimental point of view and considerable mathematical modelling has been carried out on the cw aspects of the process as well. For a cw CO 2 laser a partially ionised plasma is likely to exist in the keyhole, emerging from it to produce a turbulent plume in which variations of electrical charge density have been observed. The presence of large temperature gradients near the keyhole walls in conjunction with the Onsager/de Groot relations has the consequence that there must be an electrical potential gradient in the plasma and associated space-charge distribution. These effects are analysed and quantitative relations obtained for the case of a steel work piece.
1. Introduction In the process of deep penetration laser welding the action of the laser beam on the translating specimen oriented perpendicularly to the laser beam results in a hole formed in the material, the hole being referred to as a keyhole. For the configuration, see Fig. 1. The absorption of the laser light in the material is a central quantity of interest in understanding the efficient transfer of energy from the laser beam to the material being welded. This paper analyses the action of some of these mechanisms in the context of laser welding. The real problem is reduced to an idealised form, in which it is possible
* Corresponding author. Tel.: +44-1206-872863; fax: +441206-873598; e-marl: [email protected].
to understand some of the possible electrical effects in the keyhole plasma by first suppressing sources of fluctuation and instability. The analysis makes use of the de Groot irreversible thermodynamic relations applied to the keyhole plasma in an attempt to include the effects of thermal and potential gradients. It considers that the action of the cw CO 2 laser results in an imposed thermal field in the plasma. The combination of temperature gradients, concentration gradients and diffusion suggests the need to examine conditions in the interior of the keyhole more closely using the theory of irreversible thermodynamics, cross-linking processes of heat conduction in the plasma particularly near the walls of the keyhole where temperature gradients of the order of 109 K m - ~ may be expected to be present [1]. This would involve considering the Onsager relations employed in the form generalised by de Groot [2] with a
0169-4332/96/$15.00 Copyright © I996 Elsevier Science B.V. All rights reserved. PH S01 6 9 - 4 3 3 2 ( 9 6 ) 0 0 4 3 6 - 9
P. Kapadia, J. Dowden / Applied SurJDce Science 106 (1996) 240-242 Laser beam ~
: I : . . . .
Direction of translation
~@q-Weldpool ~ Keyhole and plasma
Fig. 1. The configuration of the laser relative to the work piece.
view to analysing these phenomena. These relations imply that spatial variations in the electrostatic potential ~b will occur in the keyhole plasma and that these are likely to propagate up and down the keyhole plasma. Apart from these fluctuations, the steady temperature and concentration gradients are associated with a steady background electrostatic potential and it is this effect that will be considered here. The main purpose of this study is to examine the effects of the cross linking of electrostatic potential gradients with thermal and concentration gradients in the environment of the laser-generated keyhole that is a consequence of the existence of the Onsager coefficients.
241
about the coefficients in the relations, and these ideas can be applied to the keyhole. The flow of material is neglected here for simplicity as are the effects of viscosity with the only forces active being assumed to be electrical, so that F k = -ekVcb. The pressure is assumed to be almost uniform and atmospheric [4] and the effect of gravitational force, which in principle depends on the orientation of the work piece, is small [5]. Euler's equation in the case of mechanical equilibrium reduces to VP=-pV~, where p,~=Y~=lekc k is the total charge per unit mass with Pk = M ~ / V where M k denotes the mass component k and V denotes the volume. Concentrations c i for the ith component are then given by c i = p y p = M J M . De Groot introduces new heat transfer coefficients Q[ and employs forces X , = - V T / T and X~ = F k T V ( t z k / T ) where /z k is the chemical potential of the kth component. For the stationary state the basic equations are
r -(e~-e.)W~}
= O,
k=l,2
.....
n-l,
/
(1)
2. The de Groot relations and their application Theoretical work shows that it is characteristic of the keyhole that there is a very large gradient in the vicinity of the walls and a relatively small gradient in the central region of the keyhole [1]. The temperature distribution itself is approximately uniform across the keyhole and drops steeply to the vaporising temperature T v at the wall. For present purposes the temperature field will be regarded as given. As a result of the high temperatures involved, ionisation takes place and Poisson's equation then can be used to find the potential [3]. Previous analyses of the keyhole, however, leave out the effects of concentration gradients on the potential that arises in the presence of a strong temperature gradient such as occurs near the keyhole wall. In his book, de Groot [2] addresses the problem of thermal diffusion and electrical phenomena in mixtures in a non-uniform temperature field in a stationary state. He generalises Onsager's relations and relates generalised forces linearly to generalised fluxes, obtaining information
together with electrical neutrality ~"k= 1ek c'~ = 0 and the condition ~ _ i ck = 1. These give ekVc k = 0,
(2)
k-I
and 11
E k
= 0.
(3)
1
Eqs. (1)-(3) form a set of (n + 1) linear equations from which one can obtain Vc~ and Vd~ in terms of VT. Consider e I = - e for electrons, e 2 denoting ions and e 3 = 0 neutrals. Eqs. (1)-(3) can then be written in the form a~Vc 1 + blVc 2 + qlVT + elVflp = O, a2Vc I + beVc z + q2VT + e2V4) = O, elVc I + e2Vc 2 = O, Vc I + Vc 2 + Vc 3 = 0,
242
P. Kapadia, J. Dowden / Applied Surface Science 106 (1996) 240-242
where
q2 = 357 + 0 . 0 3 0 5 T - 0.996 × 10 6T2 - 0.965 × 108(hr.
al = 0c~ ( IdOl- Jz3),
bl = -~c2 ( ] Z l - / z 3 ) ,
ql = Ql* - hi + h3, 0 a2 = 0c----~( / x 2 - " 3 ) ,
0 b2 = ~--~c2( / x 2 - " 3 )
For example, at T = 8 kK, q ~ = 5 2 3 + 0 . 1 7 6 × 1012qSv and q2 = 540 - 0.965 × 10s0v. Thus if any one of ql, q2 and q5r can be determined, so can the other two; these are quantities which are not well known. We may note that it seems likely from these relations that the potential ~ has a very weak dependence on temperature T.
and q2 = Q* - h~ + h 3 Hence
3. Discussion and conclusions
Vr]
~in
e, e:
)
+
e; j
,
(4)
with the subscript zc denoting the stationary state. Note that
ql( a 2 / e l - b J e ? )
- q2( a l / e l - b l / e 2 )
e , e 2 { a , / e ~ - ( a~ + b , ) / e l e 2 + b 2 / e ~ }
"
(5) Since the right-hand sides of Eqs. (4) and (5) at constant pressure are functions of T only, and we are interested only in such conditions, the left-hand sides are, respectively, el(Ocl/OT);, and (Od)/OT)p. For an ideal gas the chemical potential [6] is given by
A consequence of the existence of the O n s a g e r / d e Groot coefficients is the presence of an electric field in those regions of the keyhole where the temperature gradients are high, such as the region near to the walls. The effects estimated are under steady stationary conditions so that for example a measured value of the electrostatic potential gradient could yield information on electron and ion heat transfer coefficients to the keyhole wall. The process of laser welding indicates that keyhole processes of considerable complexity are active but the possibility of detecting electrostatic steady and fluctuating potentials in the keyhole offers an interesting way of trying to track these processes and elucidate their behaviour. Relations between temperature and the rate of change of electric potential with temperature have been found in the steady case.
Acknowledgements so /x e - / x , and /x s -/x~, can be found while c I and c 2 can be expressed in terms of the degree of ionisation, a ( T ) by the relations
C1
C2
me
ms
OL o~(rne+mi)+(1-a)m
. "
The form of c~(T) for iron at 1 bar can be calculated as a function of T so that Eqs. (4) and (5) become a pair of relations between ql and q2 and qSr--= d~o/dT=(V4)/VT)~. They can in principle be solved for q~ and q2; the resulting expressions are very complicated, but over the range of temperatures 6 k K < T < 15 kK, can be approximated accurately by q~ = 352 + 0 . 0 2 8 4 T - 0.926 X 10 6T2
+ 0.176 × 10 lZq~r,
The authors would like to acknowledge the support of the E U R E K A 113 and E U R E K A 194 programmes of the European Community.
References [1] J.M. Dowden, P. Kapadia and N. Postacioglu, J. Phys. D: Appt. Phys. 22 (1989) 741. [2] S.R. de Groot, Thermodynamics of Irreversible Processes (North-Holland, Amsterdam, 1952) ch. 12. [3] R. Finke, P. Kapadia and J.M. Dowden, J. Phys. D: Appl. Phys. 23 (1990) 643. [4] J.M. Dowden and P. Kapadia, J. Phys. D: Appl. Phys. 28 (1995) 2252. [5] J.G. Andrews and D.R. Atthey, J. Phys. D: Appl. Phys. 9 (1976) 2181. [6] L.D. Landau and E.M. Lifshitz, Statistical Physics, Part 1, 3rd ed. (Pergamon, Oxford, 1980).