Numerical Simulation of Fz Single Crystal Growth Process with Radio-Frequency Induction Heating
- PDF / 380,290 Bytes
- 6 Pages / 414.72 x 648 pts Page_size
- 60 Downloads / 217 Views
_
157
Mat. Res. Soc. Symp. Proc. Vol. 398 ©01996 Materials Research Society
could not strongly affect the temperature field in the melt. MATHEMATICAL MODEL AND NUMERICAL METHODS Any quantitative analysis of electromagnetic phenomena must commence with Maxwell's equations. The followings are assumed in the analysis: (1) The system is axially symmetric. (2) The media are linear, isotropic and stationary. (3) The displacement current is unimportant in this situation. (4) There is no net charge in the system. Under these assumptions, Maxwell's equation can be transformed into the following non-dimensional equation in cylindrical coordinates.
)
a (r dA a2Ai 1Srr r ( Týr iý+ z2
Ai_ r2
I i= s,m,f,c or a
(1)
where the single crystal, the melt, the feed rod, the RF coil and free space are denoted by the subscripts 's', 'i', 'f', 'c', and 'a', respectively. Ji and A, are the non-dimensional angular components of the current density and the vector potential, respectively, and are normalized with the single crystal radius and the forced current density in the RF coil. In the conduction-dominated model of heat transfer in the FZ system, we assume that the system is axisymmetric and quasi-steady. Under these assumptions, the non-dimensional energy equations are given as follows, where the single crystal radius and the melting-point temperature are used as the characteristic values. In the single crystal and feed rod: -Pei 0
1 0a
aTi
rW
r
=rarr(Kirar
az
In the melt:
a (0Ti)I
+ a-Kia1z-z+
iNQ(Ai'A*)
i=s,f
(2)
3 (Am'A*) 0Tn + •-1-mNQ a' Kmr~r aTm\ + az ýKm--~m a T 0mm SD m (3) N( r aKr ar) + az ~, az The last terms in the above equations are the heat generation rate by the Joulean heating, where A* is the complex conjugate of A. N0 is a non-dimensional number and Pe is Peclet number. The boundary conditions for temperature fields are expressed as follows. At the melt/single-crystal and melt/feed-rod interfaces: (4) Pei St (e z.n ) •-n - --W - + Ki aTi -Km maTm
n Tm
=- T
ni=s,f (5)
= 1
At the melt free surface and crystal surface: -a K,--n -=RTi a(Ti -TT4) Rai(
(6)
i = m,sr sorf
At the ends of the feed-rod and the single crystal: (7)
=0 , i = s or f
an In equation (4), the sign of the right hand side is negative for the single crystal 's', and positive for the feed-rod 'f. The length of the feed-rod and the single crystal should be long enough that the boundary condition, equation(7), is satisfied. St is the Stefan number and Ra is the Radiation number. All interface coordinates in the system are also treated as part of the solution. The coordinates of the melt/single crystal and melt/feed-rod interfaces are determined so that equation (4) is satisfied, i.e. the interfaces coincide with the melting isotherm. The shape of the melt free surface can be determined by solving the equation for the normal force balance at the surface, i.e. the Young-Laplace equation taking the electromagnetic force into account;
158
2H
-
Bo z + I Bo, Bs2 2
(8)
+
where B, is the non-dimensional tangential magnetic fi
Data Loading...
